==== TABLE OF CONTENTS ====
Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//01-introduction.tex
Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//02-inductive-reduction.tex
Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03-programs.tex
Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03a-program-foundations.tex
Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03b-projective-good-model.tex
Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03c-base-presentations.tex
Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03c-restricted-induction.tex
Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//04-boundary.tex
Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//05-projective-program-input.tex
Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//05-rank-one.tex
Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//06-simple-nonvanishing.tex
Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//07-signed-rigidity.tex
Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//main.tex
Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//preamble.tex
Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//two-diagonals.tex
Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//references.bib
===========================
==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//01-introduction.tex ====
\section{Introduction}\label{sec:introduction}
The abundance problem asks when the numerical positivity of a log canonical
bundle produces a holomorphic map. More precisely, a nef adjoint should have
a positive multiple generated by global sections. On a compact Kähler space,
nefness is an analytic condition on a cohomology class, whereas generation
is a statement about an actual holomorphic line bundle. The distinction is
essential in the nonprojective setting.
We prove log abundance for compact Kähler spaces under logarithmic Iitaka
subadditivity for smooth projective varieties with reduced boundary. The
argument passes through a stronger birational statement: every
pseudo-effective smooth log adjoint has a semiample positive part and an
effective rational fixed divisor equal to its analytic divisorial negative
part. This form both supports the induction and retains the line-bundle
information needed for generation on the original space.
\subsection{The assumption and the theorem}
A compact Kähler space is a compact complex analytic space with a Kähler
form in the sense of local strictly plurisubharmonic potentials on local
embeddings. For a rational line bundle \(L\), analytic nefness means that
\(c_1(L)\) belongs to the closure of the Kähler cone. Equivalently, fix a
Kähler form \(\omega\) and an integer \(r>0\) for which \(rL\) is a line
bundle. For every \(\varepsilon>0\), that line has a smooth Hermitian metric
\(h_\varepsilon\) satisfying
\[
\frac{\sqrt{-1}}{2\pi r}\Theta_{h_\varepsilon}
\geq-\varepsilon\omega.
\]
Smooth metrics and forms on a singular space are understood through local
embeddings. We call \(L\) semiample if an actual positive integral multiple
is a holomorphic line bundle generated by its global sections.
\begin{assumption}[Logarithmic Iitaka subadditivity]
\label{asm:log-iitaka}
Let \(f:X\to Y\) be a surjective morphism with connected fibers between
smooth connected projective complex varieties. Let \(D_X\) and \(D_Y\) be
reduced effective simple normal crossing divisors, allowing zero divisors,
such that
\[
\Supp(f^*D_Y)\subseteq\Supp(D_X).
\]
For a very general smooth fiber \(F\), put \(D_F=D_X|_F\). Then
\[
\kappa(X,K_X+D_X)\geq
\kappa(F,K_F+D_F)+\kappa(Y,K_Y+D_Y).
\]
Here \(D_F\) is reduced SNC and
\(K_F+D_F\sim (K_X+D_X)|_F\). The Iitaka dimension is \(-\infty\) when
all positive integral systems are empty, with
\((-\infty)+b=-\infty\) also for \(b=-\infty\); the zero divisor on a
point has Iitaka dimension zero.
\end{assumption}
\begin{theorem}[Log abundance for compact Kähler spaces]\label{thm:main}
Assume Assumption~\ref{asm:log-iitaka}. Let \(X\) be a normal irreducible
compact Kähler complex analytic space, and let \(\Delta\geq0\) be a rational
divisor such that \((X,\Delta)\) is log canonical and \(K_X+\Delta\) is
rational Cartier. If \(K_X+\Delta\) is analytically nef, then it is
semiample. Explicitly, there is an integer \(m>0\) such that
\(m(K_X+\Delta)\) is Cartier and
\[
H^0\bigl(X,\OO_X(m(K_X+\Delta))\bigr)\otimes_{\C}\OO_X
\longrightarrow \OO_X(m(K_X+\Delta))
\]
is surjective at every point of \(X\).
\end{theorem}
The theorem includes every finite dimension and the zero-boundary case.
Assumption~\ref{asm:log-iitaka} is used through the projective good-model
theorem stated in Proposition~\ref{proj:good-models}. The cited proof of
that projective theorem uses only its zero-boundary case, on a resolved
projective Albanese fibration.
\subsection{Context and relation to earlier work}
In the minimal model program, abundance complements the construction of
minimal models: the nef adjoint on a minimal model should determine a
canonical fibration. For minimal projective threefolds, Miyaoka proved the
case of numerical dimension one and Kawamata completed canonical abundance
\cite{Miyaoka1988,Kawamata1992}. Keel, Matsuki, and McKernan established
log abundance for threefolds
\cite{KMM1994,KMM2004}. In arbitrary dimension, Birkar, Cascini, Hacon, and
McKernan established log terminal models for projective klt pairs with big
boundary and pseudo-effective adjoint, and finite generation for big
adjoints \cite{BCHM2010}.
The compact Kähler history already points to the importance of simple
spaces. Peternell's threefold theorem isolated the possible case of a simple
space not bimeromorphic to a finite quotient of a torus
\cite{Peternell2001}; Demailly and Peternell subsequently proved canonical
nonvanishing for nef terminal Kähler threefolds, including that case
\cite{DemaillyPeternell2003}. Höring and Peternell constructed minimal models
for normal \(\Q\)-factorial terminal compact Kähler threefolds with pseudo-effective canonical class
\cite{HoeringPeternell2016}. Campana, Höring, and Peternell treated canonical abundance
\cite{CHP2016,CHP2023}, and Das and Ou
established log abundance for compact Kähler log canonical threefolds
\cite[Corollary~1.3]{DasOu2026}.
With \(D_X=D_Y=0\), Assumption~\ref{asm:log-iitaka} is Iitaka's classical
\(C_{n,m}\) inequality
\(\kappa(X)\geq\kappa(F)+\kappa(Y)\) for algebraic fiber spaces
\cite[Equation~(1.0.1)]{CaoPaun2017}. Cao and Păun proved the analogous
logarithmic inequality over an abelian base when the pair on the total space
has an effective rational boundary and is klt
\cite[Theorem~1.1]{CaoPaun2017}. Fujino derives
logarithmic subadditivity from the conjectured equality
\(\kappa_\sigma(X,K_X+D)=\kappa(X,K_X+D)\) for smooth projective
\(X\) with reduced SNC \(D\)
\cite{Fujino2017Subadditivity,Fujino2020Corrigendum}. Hashizume proves the
reduced-SNC subadditivity statement when the log canonical divisor of the
very general fiber is abundant \cite[Theorem~1.2]{Hashizume2020}. Here the
implication is used in the other direction: the projective companion
\cite{LI} derives good models for pseudo-effective projective lc adjoints
from Assumption~\ref{asm:log-iitaka}, and those good models anchor the
Kähler induction.
Hacon and Xie's cone theorem, adjunction, relative positivity, and
modified-big preparation \cite{HX} provide analytic inputs for our
program constructions. Section~\ref{sec:programs} constructs effective
generalized base data on a higher carrier, with an exact adjoint
pullback identity and a multiplier-ideal inclusion. Fibrewise
$L^2$ positivity and regularization give this presentation; its
Stein-factor consequence supplies the non-klt induction.
The section then proves the restricted semiampleness, proper relative
good-model, and finite-model statements needed here, following the
dimension-induction strategy of \cite{HX}.
The nef data remain globally nef on a fixed carrier, and the total
generalized boundary is globally modified big. Actual rational lines
make the selected ray contractions projective; ordinary projective
analytic results \cite{FujinoAnalyticBCHM} then construct their flips
and the relative models. This produces a Kähler pullback description of
the relevant Bott--Chern class. Generation of the prescribed holomorphic
line requires the additional divisorial decomposition and boundary
arguments. The fibration comparison and period positivity come from
\cite{CI}; the boundary and lifting constructions adapt inputs in
\cite{CB,BL}. The precise imported results are stated at their points
of use.
\subsection{The proof and its main constructions}
For a pseudo-effective \((1,1)\)-class \(\alpha\), \(N(\alpha)\) records
the least generic vanishing forced along prime divisors. Given a smooth
compact Kähler manifold \(X\) and a rational SNC boundary \(B\) with
\(J=K_X+B\) pseudo-effective, the induction constructs a modification
\(\mu:Y\to X\) with \(Y\) smooth and
\[
\mu^*J\sim_{\Q}P+R,\qquad P\text{ semiample},\qquad
R=N(c_1(\mu^*J)),
\]
as an identity of rational holomorphic line bundles, with \(R\) an effective
rational divisor. Section~\ref{sec:inductive-reduction} uses transfer rules,
algebraic reduction, and generically finite covers to reduce the induction
to projective manifolds, nontrivial fibrations, and simple spaces of
algebraic dimension zero. Here simple means that no positive-dimensional
proper compact subvariety passes through a very general point.
Section~\ref{sec:programs} supplies two program results. For the fibration
case it contracts a known negative part while preserving the prescribed nef
line. For the simple case it proves special termination for a chosen scaling
and reaches a nef model when the pseudo-effective adjoint is rationally
equivalent to a divisor supported on the reduced boundary, allowing negative
coefficients. The termination argument uses lower-dimensional induction to
construct one model on which an interval of perturbed adjoints is nef. On a
common resolution their negative multiplicities are affine, whereas every
nontrivial wall of the restricted program changes a slope. Hence only
finitely many such walls occur.
Section~\ref{sec:boundary} turns generation on separate strata into generation
on the entire reduced dlt boundary, adapting Fujino's method of admissible
sections \cite[Section~4]{Fujino2000}. When a comparison is needed, a Mori
contraction realizes Kollár's residue comparison at the two coefficient-one
points of a general \(\PP^1\) fiber
\cite[Definition~13 and Proposition~14]{Kollar2013Sources}. On a common
resolution, the pulled-back restrictions of the ambient Kähler class differ
by a real linear combination of Chern classes of line bundles. On a stratum
whose boundary dominates the image of the map
defined by its semiample adjoint, restriction determines sections. For the
remaining strata, we prove finite image for self-comparisons on sufficiently
divisible pluricanonical systems using an invariant integral, period and
lattice arguments, and a uniform cohomological bound for cyclic covers.
Products over these finite images give compatible generating sections, used
in both closing arguments.
Section~\ref{sec:rank-one} first reduces to fibrations whose very general
fiber has log Kodaira dimension zero. On a prepared fibration \(g:X\to W\),
a relative generating form gives
\[
K_X+B\sim_{\Q}g^*H+A_*,\qquad H=K_W+T+M,
\]
where \(T\) is a rational SNC boundary and \(M\) is pulled back from a nef
rational line on a projective quotient of \(W\). Fiber induction allows us
to subtract the part of \(A_*\) dominating \(W\) from a positive current for
\(K_X+B\). The resulting metric descends along connected smooth fibers; a
zero among the nonnegative coefficients at primes dominating each base prime
allows extension, proving \(H\) pseudo-effective. If \(a(W)=0\), the
projective quotient is a point, so \(M\sim_{\Q}0\) and lower-dimensional
induction makes \(H\) rationally equivalent to its negative divisor. For
projective \(W\), a reduction via a chosen generalized program derived from
Assumption~\ref{asm:log-iitaka} either lowers the positive base dimension,
handled by secondary induction, or reaches a nef line that is big or torsion.
At a stopping case, an intersection argument identifies the full negative
divisor upstairs. A torsion positive part completes the decomposition
directly. In the big nef case, Section~\ref{sec:programs} contracts that
divisor while preserving the nef line; boundary generation, extension, and
a new log canonical place at a hypothetical base locus prove semiampleness.
Section~\ref{sec:nonvanishing} proves, independently of
Assumption~\ref{asm:log-iitaka}, that a positive canonical power has a nonzero
meromorphic section on every smooth connected simple compact Kähler manifold
of algebraic dimension zero. Ou's uniruledness and foliation results supply
the cotangent slope control \cite{Ou2025}. A point-threshold bound for big
classes of volume one is contradicted using an auxiliary projective bundle
over \(X\times X\). The diagonal in its square produces a subsheaf occupying
a fixed positive fraction of a symmetric cotangent power of that bundle.
A second construction over the diagonal of \(X\times X\) forces determinant
vanishing which, after restriction to a blowup of \(X\), violates the point
bound.
Finally, Section~\ref{sec:signed} uses the meromorphic section to obtain,
after resolving and enlarging to a reduced SNC boundary, a pseudo-effective
adjoint with a signed representative supported there.
Section~\ref{sec:programs} gives a nef dlt model, and
Section~\ref{sec:boundary} gives generation on its reduced boundary.
Pseudo-effectivity of the canonical pullback and an intersection argument
isolate positive-dimensional boundary fibers away from components whose
signed coefficients are nonpositive. A local root construction adapted from
\cite[Proposition~3.3]{LI} separates the positive and negative divisor
supports. An SNC Hodge-module calculation extends the lifting method of
\cite[Sections~10--12]{BL} to residual poles and kills every finite-order
obstruction to lifting such a fiber. Douady space and Artin approximation
give compact deformations leaving the boundary; compactness of the cycle-space
components and relative compactness of bounded-volume cycles allow a Baire
argument to produce a covering family. Simplicity
forces the actual nef line to be torsion. The program comparison and
Lemma~\ref{bir:unique-current} then permit subtraction of the added boundary,
yielding the inductive decomposition.
==== END Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//01-introduction.tex ====
==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//02-inductive-reduction.tex ====
\section{Birational decompositions and the geometric reduction}
\label{sec:inductive-reduction}
The proof will produce a semiample line bundle on a smooth model, together
with its entire fixed divisorial part. This section makes that statement
precise and reduces the induction to two cases: spaces admitting a
nontrivial fibration, and simple spaces of algebraic dimension zero.
\subsection{The inductive decomposition and its negative part}
We use additive notation for rational holomorphic line bundles. Thus
\(L\sim_{\Q}L'\) means that \(mL\) and \(mL'\) are holomorphically
isomorphic for some positive integer \(m\). If a rational divisor occurs in such
an identity, it denotes its associated rational line bundle. In contrast,
\(c_1(L)\) and \(\{D\}=c_1(\OO_X(D))\) denote real Bott--Chern classes.
Canonical comparisons are made with the usual local meromorphic canonical
identifications. In particular, an identity of lines below contains more
information than equality of their Chern classes.
Let \(X\) be a smooth compact Kähler manifold and let \(\alpha\) be a
pseudo-effective real \((1,1)\)-class. Fix a Kähler form \(\omega\). For a
prime divisor \(D\), its minimal multiplicity is
\[
\nu_D(\alpha)=\lim_{\varepsilon\downarrow0}
\inf\bigl\{\nu_D(T):T\in\alpha,\ T\geq-\varepsilon\omega\bigr\}.
\]
Here \(T\) ranges over closed currents, and \(\nu_D(T)\) is its generic
Lelong number; one may equivalently use currents with analytic singularities.
The limit is independent of \(\omega\). Boucksom's divisorial decomposition
is
\[
N(\alpha)=\sum_D\nu_D(\alpha)D,
\qquad Z(\alpha)=\alpha-\{N(\alpha)\}.
\]
The sum is a finite effective real divisor, and \(Z(\alpha)\) is modified
nef: its minimal multiplicity at every prime is zero. We write \(N(L)\) and
\(\nu_D(L)\) for \(N(c_1(L))\) and \(\nu_D(c_1(L))\). These are the
analytic notions, with small Kähler
perturbations, throughout the paper. We use the foundational results in
\cite[Sections~2--3 and~5]{Boucksom2004}.
The induction asks for a decomposition that retains both the actual line
bundle and this analytic negative divisor.
\begin{definition}\label{def:good-decomposition}
For \(n\geq0\), let \(\Ggood_n\) be the following assertion. If \(X\)
is a connected smooth compact Kähler manifold of dimension \(n\), \(B\)
is a rational simple normal crossing boundary with coefficients in
\([0,1]\), and \(J=K_X+B\) is pseudo-effective, then there is a smooth
compact Kähler modification \(\mu:Y\to X\) and an actual rational line
identity
\begin{equation}\label{bir:good-decomposition}
\mu^*J\sim_{\Q}P+R,
\qquad P\text{ semiample},\qquad R=N(\mu^*J)\geq0,
\end{equation}
where \(R\) is a rational divisor.
\end{definition}
The next lemmas make this decomposition stable under further resolutions
and allow exceptional resolution errors to be removed. They also provide
the fixed-section statement needed to descend the case in which the
positive part is torsion.
\begin{lemma}[Negative-part calculus]\label{bir:calculus}
Let \(\alpha\) be pseudo-effective on a smooth compact Kähler manifold.
\begin{enumerate}
\item Every positive current in \(\alpha\) contains the divisorial current
\([N(\alpha)]\). If \(0\leq F\leq N(\alpha)\), then
\[
\alpha-\{F\}\text{ is pseudo-effective},\qquad
N(\alpha-\{F\})=N(\alpha)-F.
\]
\item Minimal multiplicities are homogeneous and subadditive. In
particular, if \(\beta\) is nef, then
\(N(\alpha+\beta)\leq N(\alpha)\).
\item If \(\mu:Y\to X\) is a smooth modification and \(D'\) is the strict
transform of a prime \(D\), then
\[
\nu_{D'}(\mu^*\alpha)=\nu_D(\alpha).
\]
\item If \(\gamma\) is modified nef and \(j:\widetilde D\to X\) is a
resolution of a prime divisor, then \(j^*\gamma\) is pseudo-effective.
\end{enumerate}
\end{lemma}
\begin{proof}
The first assertion about currents follows from the definition of minimal
multiplicity and Siu decomposition. Homogeneity and subadditivity follow by
scaling and adding testing currents. The subtraction identity is the
corresponding property in the big cone, followed by a small Kähler
perturbation. More explicitly, for \(\alpha_\varepsilon=\alpha+
\varepsilon\{\omega\}\), subtract
\(F_\varepsilon=\min\{F,N(\alpha_\varepsilon)\}\) coefficientwise.
In the big cone all positive currents contain this divisor, so subtracting
it translates each minimal multiplicity by its coefficient. As
\(\varepsilon\downarrow0\), \(F_\varepsilon\to F\). The class of
\(F-F_\varepsilon\) can be absorbed in a Kähler perturbation tending to
zero (there are only finitely many components). This gives the upper bound
for the asserted identity; subadditivity applied after adding \(F\) gives
the reverse bound. Weak compactness of positive currents gives
pseudo-effectivity of the limit. This is also the subtraction property of
the divisorial decomposition in \cite[Section~3]{Boucksom2004}.
For the strict-transform formula, pull almost-positive testing currents
to \(Y\). Near the generic point of \(D'\), the map is an isomorphism,
so the generic order is unchanged. This proves one inequality. Conversely,
push a testing current on \(Y\) to \(X\). To control its negative error,
write the error as a small multiple of the fixed positive current
\(\mu_*\omega_Y\), and add a fixed smooth Kähler representative of
\(C\omega_X-\{\mu_*\omega_Y\}\) for \(C\) sufficiently large. The
result is a positive test in a Kähler perturbation tending to zero. The
pushforward of \(\omega_Y\) has no divisorial order at the generic point
of \(D\), where \(\mu\) is an isomorphism. Thus this test has the same
generic order as the original one at \(D'\). Regularization, if needed,
returns to tests with analytic singularities. This proves the other
inequality and the formula. It is the analytic form of the strict-transform
comparison in \cite[Lemma~5.8]{Boucksom2004}.
Finally, choose analytic-singularity tests for a modified-nef class whose
generic order at the chosen prime tends to zero. Subtract that generic
divisorial order before restricting to a resolution of the prime. The
remaining singular set does not contain its generic point, so restriction
gives an almost-positive current there. The subtracted multiple and the
Kähler error tend to zero. Taking the limit proves the last assertion; see
also \cite[Propositions~2.4 and~3.8]{Boucksom2004}.
\end{proof}
For a rational line \(J\) on a normal compact Kähler space \(X\) whose pullback
to a smooth resolution is pseudo-effective, we also use minimal
multiplicities for divisorial places. If \(Q\) appears on a smooth
resolution \(p:Y\to X\), set
\[
\nu_Q(J):=\nu_Q(p^*J).
\]
The strict-transform formula on a common smooth refinement makes this
independent of the chosen resolution. When a real class is already on a
fixed smooth model, \(\nu_Q\) continues to denote its coefficient in the
negative part on that model.
\begin{lemma}[Exceptional translation]\label{bir:exceptional}
Let \(p:Y\to X\) be a proper bimeromorphic morphism from a smooth compact
Kähler manifold to a normal compact Kähler space. Let \(M\) be a real
\((1,1)\)-class on \(X\) represented by smooth local potentials, and let
\(E\geq0\) be a real \(p\)-exceptional divisor. If
\(\alpha=p^*M+\{E\}\) is pseudo-effective, then \(p^*M\) is
pseudo-effective and
\[
N(\alpha)=N(p^*M)+E.
\]
\end{lemma}
\begin{proof}
It suffices by Lemma~\ref{bir:calculus} to prove \(E\leq N(\alpha)\).
Write
\[
E-N(\alpha)=U-V,
\qquad U,V\geq0,
\]
with no common component, and suppose \(U\neq0\). Its components are
\(p\)-exceptional. Put \(n=\dim Y\) and
\(\ell=\max\{\dim p(U_i):U_i\text{ a component of }U\}\). Normality
gives \(\ell\leq n-2\). Let \(\eta\) be the pullback of a Kähler form
on \(X\), let \(\omega\) be Kähler on \(Y\), and set
\[
q(\gamma,\delta)=
\int_Y\gamma\,\delta\,\eta^{\ell}\omega^{n-\ell-2}.
\]
The restriction property in Lemma~\ref{bir:calculus} gives
\(q(Z(\alpha),U)\geq0\). The term \(q(p^*M,U)\) is zero: on each
component of \(U\), it contains \(\ell+1\) factors pulled back from an
image of dimension at most \(\ell\). Distinct effective divisors have
nonnegative intersection against the semipositive and Kähler factors in
\(q\). Hence
\[
0\leq q(Z(\alpha),U)=q(U,U)-q(V,U)\leq q(U,U).
\]
On the other hand, \(q(U,\eta)=0\) by the same dimension count, whereas
\(q(\eta,\eta)>0\). The mixed Hodge index theorem, applied first with
Kähler factors and then by a semipositive limit, says that \(q\) is
negative semidefinite on \(\eta^\perp\). It follows that \(q(U,U)=0\)
and that \(U\) is in the radical of \(q\): the radical assertion follows
from negative semidefiniteness on \(\eta^\perp\), and then from
\(q(U,\eta)=0\). But
\[
q(U,\omega)=\sum_i u_i\int_{U_i}
\eta^{\ell}\omega^{n-\ell-1}>0.
\]
Indeed, a component with image dimension \(\ell\) has strictly positive
integrand on a dense open. This is a contradiction. The mixed Hodge index
statement used here is the mixed Hodge--Riemann relation
\cite[Theorems~A and~C]{DinhNguyen}; the limit preserves the assertion that
there is at most one positive direction.
\end{proof}
\begin{lemma}[Pulling up a known decomposition]\label{bir:pullup}
Let \(\alpha\) be a pseudo-effective real \((1,1)\)-class on a smooth compact
Kähler manifold, and suppose
\[
\alpha=\beta+\{R\},\qquad \beta\text{ nef},\qquad R=N(\alpha)\geq0.
\]
For every smooth modification \(\mu:Y\to X\),
\[
N(\mu^*\alpha)=\mu^*R.
\]
In particular, this applies to the Chern classes of an actual rational line
decomposition with a nef rational positive part.
\end{lemma}
\begin{proof}
Subadditivity and nefness give \(N(\mu^*\alpha)\leq\mu^*R\). By the
strict-transform formula, the difference
\(U=\mu^*R-N(\mu^*\alpha)\) is effective and exceptional. Moreover
\(Z(\mu^*\alpha)=\mu^*\beta+\{U\}\) is modified nef. Apply
Lemma~\ref{bir:exceptional} to this class. It gives
\(U\leq N(Z(\mu^*\alpha))=0\), proving the assertion.
\end{proof}
\begin{lemma}[Fixed sections]\label{bir:sections}
Let \(L\) be a pseudo-effective rational line on a smooth compact Kähler
manifold. Every section \(s\) of an integral multiple \(mL\) satisfies
\[
\ord_D(s)\geq m\nu_D(L)
\]
at every prime \(D\). If \(L\sim_{\Q}P+R\), with \(P\) semiample and
\(R=N(L)\) rational, then in every sufficiently divisible degree
multiplication by the canonical section of \(mR\) identifies
\(H^0(X,mP)\) with \(H^0(X,mL)\).
\end{lemma}
\begin{proof}
The divisor current \([\operatorname{div}(s)]/m\) is a positive current
in \(c_1(L)\), so Lemma~\ref{bir:calculus} gives the order bound. In the
stated decomposition, every section therefore divides holomorphically by
the canonical section of \(mR\). Conversely, multiplication gives a
section of \(mL\). The actual rational line identity makes these operations
inverse after clearing denominators.
\end{proof}
\begin{lemma}[Uniqueness when the positive part is torsion]
\label{bir:unique-current}
Suppose \(L\sim_{\Q}R\), where \(R=N(L)\) is an effective rational
divisor on a smooth compact Kähler manifold. Then the only positive current
in \(c_1(L)\) is \([R]\). If \(F\geq0\) is a rational divisor and
\(L-F\) is pseudo-effective, then
\[
F\leq R,\qquad L-F\sim_{\Q}R-F=N(L-F).
\]
The same conclusions hold if \(L\sim_{\Q}P+R\) with \(P\) torsion.
\end{lemma}
\begin{proof}
Write \(X\) for the manifold. The assertion is immediate if \(\dim X=0\),
so assume \(\dim X>0\).
Every positive current \(T\) in \(c_1(L)\) contains \([R]\). The
residual \(T-[R]\) is positive with zero cohomology class, so its mass
against a Kähler form to the power \(\dim X-1\) is zero. Thus it vanishes.
For the second assertion, add \([F]\) to any positive current in
\(c_1(L-F)\). Uniqueness gives \(F\leq R\), and subtraction in
Lemma~\ref{bir:calculus} gives the asserted negative part. A torsion line
has zero Chern class and becomes trivial after taking a positive multiple,
so the last statement is the same argument.
\end{proof}
\subsection{Birational transfer of the decomposition}
All modifications resolving spaces, maps, or ideals can be chosen
projective. A projective modification of a compact Kähler space is Kähler;
graphs between compact Kähler models can likewise be resolved by smooth
compact Kähler manifolds. We use these standard analytic resolution and
flattening results without further mention.
For log discrepancies our convention is
\[
a(E;X,\Delta)=1+\ord_E\bigl(K_Y-p^*(K_X+\Delta)\bigr)
\]
on a smooth model \(p:Y\to X\). In particular, on a log resolution of an
lc pair, giving each new exceptional divisor coefficient one produces an
effective exceptional error. The next proposition explains why this
convention is harmless.
\begin{proposition}[Resolution and descent]\label{bir:resolution-transfer}
Let \((X,\Delta)\) be a normal compact Kähler lc pair with rational
boundary and rational Cartier adjoint \(J\). Let \(p:Y\to X\) be a log
resolution, and put
\[
B_Y=p_*^{-1}\Delta+\sum_{E\text{ exceptional}}E.
\]
If \(p^*J\) is pseudo-effective and the conclusion of
Definition~\ref{def:good-decomposition} holds for \((Y,B_Y)\), then there
is a smooth compact Kähler modification \(r:V\to X\) with an actual
rational line identity
\[
r^*J\sim_{\Q}P+R,\qquad P\text{ semiample},\qquad
R=N(r^*J)\geq0,
\]
where \(R\) is a rational divisor.
If \(J\) is analytically nef, it is semiample on \(X\).
\end{proposition}
\begin{proof}
With compatible canonical choices there is an actual rational divisor
identity
\[
K_Y+B_Y=p^*J+E_p,\qquad
E_p=\sum_{E\text{ exceptional}}a(E;X,\Delta)E\geq0.
\]
Suppose \(q:V\to Y\) realizes the decomposition for the left side.
Lemma~\ref{bir:exceptional}, applied over \(X\), gives
\[
N\bigl(q^*(K_Y+B_Y)\bigr)
=N(q^*p^*J)+q^*E_p.
\]
Cancelling \(q^*E_p\) in the actual rational line identities proves the
first assertion. This argument also shows that further resolutions using
the reduced-exceptional convention do not change the assertion
\(\Ggood_n\).
If \(J\) is nef, its pullback is nef, so its negative part is zero. The
semiample line in the decomposition is therefore \(q^*p^*J\) itself, up
to rational line isomorphism. Choose an actual Cartier multiple generated
on \(V\). Normality gives \((p q)_*\OO_V=\OO_X\), and projection formula
identifies its sections with the sections downstairs. A base point
downstairs would make every pullback section vanish on its nonempty fiber.
There is no such point, so that multiple of \(J\) is globally generated.
\end{proof}
The second transfer concerns a torsion positive part. Its proof uses the
following local construction for finite maps, which will also be used for
meromorphic sections and for covers of spaces of algebraic dimension zero.
\begin{lemma}[Finite analytic norms and characteristic polynomials]
\label{bir:finite-norm}
Let \(\nu:Z\to X\) be a finite surjective morphism of normal irreducible
complex spaces, of degree \(d>0\), and let \(L\) be a holomorphic line
bundle on \(X\). A nonzero meromorphic section \(s\) of \(\nu^*L\)
has a nonzero meromorphic norm \(\operatorname{Nm}_\nu(s)\) in
\(L^{\otimes d}\), with
\[
\operatorname{div}(\operatorname{Nm}_\nu(s))
=\nu_*\operatorname{div}(s).
\]
If \(s\) is holomorphic, its norm is holomorphic. If, in addition,
\(s\) is nonzero at every point of \(\nu^{-1}(x)\), then its norm is
nonzero at \(x\).
For a global meromorphic function \(f\) on \(Z\), there is a monic
polynomial \(\chi_f(T)\) of degree \(d\) with global meromorphic
coefficient functions on \(X\) such that \(\chi_f(f)=0\) after pulling
the coefficients to \(Z\).
\end{lemma}
\begin{proof}
At \(x\in X\), choose a holomorphic frame of \(L\) and put
\[
R=\OO_{X,x},\qquad K=\operatorname{Frac}(R),\qquad
A=(\nu_*\OO_Z)_x.
\]
The generic algebra \(A\otimes_RK\) is a product of finite field
extensions of \(K\), of total dimension \(d\). The local meromorphic
coefficient \(a\) of \(s\) belongs to this algebra. Since \(Z\) is
irreducible and \(s\) is not identically zero, its restriction is nonzero
on every generic local component. Thus \(a\) is nonzero in every field
factor, and the determinant of multiplication by \(a\) is nonzero.
Under a change of frame by a unit \(g\), this determinant changes by
\(g^{-d}\). The determinants therefore glue to the asserted
meromorphic section. The valuation formula for a norm over a discrete
valuation ring gives the displayed divisor identity at every prime of
the normal space \(X\).
If \(s\) is holomorphic, then \(a\in A\) is integral over \(R\).
Its norm is integral over \(R\) and belongs to \(K\), hence belongs to
\(R\) by normality. This does not require \(A\) to be locally free.
If \(s\) is nonzero at every point over \(x\), then \(a\) is a unit
in the finite semilocal algebra \(A\); applying the same argument to
\(a^{-1}\) shows that its norm is a unit in \(R\).
For a meromorphic function \(f\), use instead the determinant of
\(T\) minus multiplication by its local coefficient. These monic
polynomials agree on overlaps, and Cayley--Hamilton shows that they
annihilate \(f\). All these constructions use fractions of local analytic
germs, so they remain available when the global meromorphic functions on
\(X\) are constant.
\end{proof}
We now descend a decomposition whose positive part is torsion, the form
needed for a covering family on a space of algebraic dimension zero.
\begin{proposition}[Descent of a purely negative decomposition]
\label{bir:finite-descent}
Let \(e:Y\to X\) be a proper generically finite surjective morphism
between connected smooth compact Kähler manifolds. Let \(L\) be a
pseudo-effective rational line on \(X\), and let \(F\geq0\) be a
rational divisor on \(Y\). Suppose that on a smooth model over \(Y\),
the line \(e^*L+F\) is rationally linearly equivalent to its rational
negative part. Then there is an effective rational divisor \(D\) on
\(X\) such that
\[
L\sim_{\Q}D=N(L).
\]
\end{proposition}
\begin{proof}
Replace \(Y\) by the smooth model in the hypothesis and pull \(F\) back.
Write \(e^*L+F\sim_{\Q}R=N(e^*L+F)\). Pull back a positive current in
\(c_1(L)\) and add \([F]\). Lemma~\ref{bir:unique-current} gives
\(F\leq R\). Put \(D_Y=R-F\). Subtraction in
Lemma~\ref{bir:calculus} gives
\[
e^*L\sim_{\Q}D_Y=N(e^*L)\geq0.
\]
In particular \([D_Y]\) is the unique positive current in \(c_1(e^*L)\).
Choose \(m\) so that \(mL\) and \(mD_Y\) are integral and the latter is
the divisor of a holomorphic section \(s\) of \(e^*(mL)\). Factor \(e\)
through its normal Stein space:
\[
Y\xrightarrow{h}Z\xrightarrow{\nu}X,
\qquad \nu\text{ finite of degree }d=\deg e.
\]
The connected birational map \(h\) descends \(s\) to a holomorphic
section \(s_Z\) of \(\nu^*(mL)\), by normality and projection formula.
Lemma~\ref{bir:finite-norm} gives a holomorphic section of \(mdL\) whose
divisor is
\[
\operatorname{div}(\operatorname{Nm}_\nu(s_Z))
=\nu_*\operatorname{div}(s_Z)=m e_*D_Y.
\]
Thus \(D=d^{-1}e_*D_Y\) is effective and satisfies \(L\sim_{\Q}D\).
Both \(e^*D\) and \(D_Y\) are positive divisor currents in
\(c_1(e^*L)\), so uniqueness gives \(e^*D=D_Y\). We already know
\(N(L)\leq D\). If this inequality were strict at a prime \(A\subset X\),
choose analytic-singularity tests for \(L\) whose generic orders at \(A\)
tend to \(\nu_A(L)\), and pull them to \(Y\). For any prime \(A'\)
dominating \(A\), the generic orders are multiplied by
\(\operatorname{mult}_{A'}e^*A\). The pulled-back Kähler errors can be
enlarged to Kähler errors upstairs tending to zero. Consequently
\[
\nu_{A'}(e^*L)
\leq \operatorname{mult}_{A'}(e^*A)\,\nu_A(L)
<\operatorname{mult}_{A'}(e^*A)\,\operatorname{coeff}_A(D),
\]
contrary to \(N(e^*L)=D_Y=e^*D\). This proves \(D=N(L)\).
\end{proof}
We will also use Proposition~\ref{bir:finite-descent} when \(e\) is a
modification and \(F=0\). Once the positive part of a decomposition is
known to be torsion, it gives \(L\sim_{\Q}N(L)\) on the original smooth
space; Lemma~\ref{bir:pullup} then identifies the negative divisor upstairs
with the pullback of \(N(L)\).
\subsection{The projective anchor and the two geometric cases}
\begin{proposition}[Projective good models]\label{proj:good-models}
Under Assumption~\ref{asm:log-iitaka}, every projective lc pair over
\(\C\) with rational boundary and pseudo-effective rational Cartier
adjoint has a good log minimal model. In particular, \(\Ggood_n\) holds
for projective manifolds in every dimension, and every nef rational lc
adjoint on a projective variety is semiample as an actual rational line.
\end{proposition}
\begin{proof}
The good-model assertion is the rational-boundary consequence of the full
argument in \cite[Proposition~2.5, Theorem~9.6, and Section~10]{LI}. That
argument establishes the additional inductive and nonvanishing premises of
its Proposition~2.5 before applying it, and proves a stronger real-boundary
induction. Its only use of Assumption~\ref{asm:log-iitaka} is in its
Lemma~6.1, on a resolved projective Albanese fibration with both boundaries
zero. Thus its premise is precisely available here.
For clarity, its good-model conclusion gives on a common smooth projective
resolution
\[
u^*(K_X+B)=v^*(K_{X_m}+B_m)+F,
\qquad F\geq0\text{ and }v\text{-exceptional},
\]
as an actual rational divisor identity; the line on \(X_m\) is semiample.
This is the comparison in \cite[Lemma~2.2]{LI}. Algebraic nefness of a
rational line on a projective variety implies analytic nefness, by adding
arbitrarily small ample rational classes. Lemma~\ref{bir:exceptional}
therefore identifies \(F\) with the analytic negative part of the left
side. On a smooth projective variety, analytic pseudo-effectivity of a
divisor class agrees with algebraic pseudo-effectivity
\cite[Proposition~1.2]{BDPP}; hence the good-model assertion applies to
every projective instance of \(\Ggood_n\). This proves the stated form of
\(\Ggood_n\). If the original adjoint
is nef, the same comparison and normal descent give its semiampleness.
\end{proof}
The remainder of the paper proves the following two propositions. They
are stated here so that the global induction can be completed before its
technical components are developed. A compact space is \emph{simple} if
no positive-dimensional proper compact analytic subvariety passes through
a very general point. We write \(a(X)\) for its algebraic dimension.
\begin{proposition}[The fibration case]\label{rank:fibration-case}
Assume Assumption~\ref{asm:log-iitaka}. Fix \(n>0\) and assume
\(\Ggood_j\) for \(j<n\). Let \(X\) be a connected
smooth compact Kähler manifold of dimension \(n\), let \(B\) be a rational
SNC boundary with coefficients in \([0,1]\), and suppose \(K_X+B\) is pseudo-effective. If \(X\) admits
a dominant meromorphic map to an irreducible compact space \(Y\) in Fujiki
class \(\mathcal C\), with \(0<\dim Y<n\), then the conclusion of
\(\Ggood_n\) holds for \((X,B)\).
\end{proposition}
\begin{proposition}[The simple case]\label{simple:case}
Fix \(n>0\) and assume \(\Ggood_j\) for \(j<n\). Let \(X\) be a connected
smooth simple compact Kähler manifold of dimension \(n\) with \(a(X)=0\).
For a rational SNC boundary \(B\) with coefficients in \([0,1]\) and
\(J=K_X+B\) pseudo-effective,
there is a smooth compact Kähler modification \(\mu:Y\to X\) such that
\[
\mu^*J\sim_{\Q}N(\mu^*J),
\]
and this negative part is a rational divisor. In particular, the conclusion
of \(\Ggood_n\) holds with torsion positive part.
\end{proposition}
Proposition~\ref{rank:fibration-case} is proved in
Section~\ref{sec:rank-one}; it reduces the fibration to relative log Kodaira
dimension zero and then descends the adjoint to its base.
Proposition~\ref{simple:case} is proved in Section~\ref{sec:signed}, using
the meromorphic nonvanishing theorem of Section~\ref{sec:nonvanishing}.
Both propositions use \(\Ggood\) only in dimensions below \(n\).
\begin{theorem}[Good divisorial decomposition]
\label{thm:good-decomposition}
Under Assumption~\ref{asm:log-iitaka}, the assertion \(\Ggood_n\) holds
for every finite \(n\).
\end{theorem}
\begin{proof}
We use Propositions~\ref{rank:fibration-case} and~\ref{simple:case}, whose
proofs occupy the rest of the paper. The assertion is immediate in dimension
zero. Assume it in smaller dimensions and let \((X,B)\) be as in
Definition~\ref{def:good-decomposition} in dimension \(n>0\).
If \(a(X)=n\), the Kähler Moishezon theorem makes \(X\) projective, and
Proposition~\ref{proj:good-models} applies. If \(0<a(X)<n\), the algebraic
reduction of \(X\) is a dominant meromorphic map to a projective model of
dimension \(a(X)\) \cite[Theorem and Definition~3.1]{CampanaPeternell1999}.
Proposition~\ref{rank:fibration-case} applies.
It remains to consider \(a(X)=0\). If \(X\) is simple, use
Proposition~\ref{simple:case}. Otherwise Campana's maximal covering-family
theorem supplies a generically finite evaluation map from an incidence
space which itself has a nontrivial fibration; see
\cite[Lemma~17 and Corollary~18]{Campana2026}. After resolving the incidence,
the graph, and the base, we obtain a smooth compact Kähler source \(X'\) and maps
\[
e:X'\longrightarrow X,\qquad f:X'\longrightarrow Y',
\qquad 0<\dim Y'<n,
\]
where \(e\) is proper generically finite and \(Y'\) is in class
\(\mathcal C\) \cite[(1.1.1), (1.5)(3)--(4), (2.2)(1),(3), and (2.3)]{Fujiki1982}.
The algebraic dimension of \(X'\) is zero. Indeed, a
meromorphic function on \(X'\) descends through the birational part of the
normal Stein factorization of \(e\). Lemma~\ref{bir:finite-norm} gives its
characteristic polynomial over the finite part, with meromorphic
coefficients on \(X\). These coefficients are constant because \(a(X)=0\),
so the function is constant on the irreducible space \(X'\). Bimeromorphic
modifications do not change this conclusion.
Choose a reduced SNC divisor \(B'\) containing the inverse image of
\(\Supp B\), after a further resolution. Pullback of logarithmic
differentials gives the actual rational identity
\[
K_{X'}+B'=e^*(K_X+B)+F,\qquad F\geq0.
\]
For example, this follows locally by pulling back logarithmic top forms
for the reduced support of \(B\); decreasing its coefficients to those of
\(B\) only increases the error. Thus the adjoint upstairs is
pseudo-effective. Proposition~\ref{rank:fibration-case} applies to
\((X',B')\). Its semiample positive part is torsion, since a nonconstant
semiample map would give a nonconstant meromorphic function on \(X'\).
Proposition~\ref{bir:finite-descent} now gives the required decomposition
for \(K_X+B\). This exhausts the cases and proves the induction.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
Apply Theorem~\ref{thm:good-decomposition} on a log resolution of the
given lc pair, with the reduced-exceptional boundary. The adjoint is
pseudo-effective because the original one is nef.
Proposition~\ref{bir:resolution-transfer} then gives generation of an
actual Cartier multiple at every point of the original normal space.
\end{proof}
==== END Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//02-inductive-reduction.tex ====
==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03-programs.tex ====
\section{Programs with a fixed nef part and special termination}
\label{sec:programs}
\label{prog:section}
Fix a dimension \(n\), and assume \(\Ggood_d\) for every \(d<n\).
This section supplies two program constructions used in the induction.
An ordinary dlt adjoint that is already the sum of a nef rational line and
its divisorial negative part has a nef model, reached by contracting that
negative part while preserving the nef line. A suitably chosen scaling of
a pseudo-effective ordinary dlt adjoint starting on a smooth space has
special termination. Its proof uses \(\Ggood_d\) only on proper log
canonical strata and compares two descriptions of negative multiplicities
as the scaling parameter tends to zero. Along the way, cohomology transport
supplies both the generic scaling directions and the exceptional splitting
carried by the terminal models used in Section~\ref{sec:boundary}.
The restricted generalized model input is stated first for use in these
ordinary constructions. Its proof occupies the last three subsections:
polarization and descent, an already-projective relative construction,
and the dimension induction. That proof is independent of the assumptions
\(\Ggood_d\) used for ordinary special termination.
\subsection{Program inputs and descent of actual lines}
A normal compact Kähler space is \emph{globally \(\Q\)-factorial} if
every global Weil divisor has a positive Cartier multiple. It is
\emph{globally strongly \(\Q\)-factorial} if every global coherent
rank-one reflexive sheaf has an invertible reflexive power. The strong
property implies the first one by applying it to the sheaf associated to a
Weil divisor. Both properties concern global objects; the strong property
makes no assertion about reflexive sheaves defined only on arbitrary
analytic open subsets. For a rational line \(J\) we write
\(\{J\}=c_1(J)\), and use the same braces for the class of a rational
Cartier divisor. A trace of a rational line under a bimeromorphic map
means its reflexive transform. Bott--Chern transforms will be used only
along the detected birational steps described below; on first Chern classes they
agree with reflexive transforms whenever the step is ordinary.
All resolutions in the compact arguments below are smooth compact
Kähler spaces, and their indicated maps to the spaces being resolved
are projective. A common resolution is projective over each
indicated model.
\begin{definition}[A resolution adapted to log canonical strata]
\label{def:lc-strata-resolution}
An effective rational dlt pair \((T,B)\) on a normal compact Kähler space
satisfies the \emph{lc-strata resolution convention} if it admits a
projective log resolution \(r:W\to T\) with smooth compact Kähler source
such that the strict boundary and the exceptional support together have
distinct smooth components forming a simple normal crossing divisor.
Writing
\[
K_W+B_W^{\mathrm{cr}}=r^*(K_T+B)
\]
with compatible canonical choices, every \(r\)-exceptional coefficient
of \(B_W^{\mathrm{cr}}\) is strictly less than one, and \(r\) is an
isomorphism at the general point of every log canonical center of
\((T,B)\).
\end{definition}
This convention asserts the existence of one such resolution. It imposes
no condition on later resolutions chosen for other purposes. The ordinary
dlt models supplied below for the applications in
Sections~\ref{sec:boundary}, \ref{sec:rank-one}, and~\ref{sec:signed} admit
one by choosing a defining dlt resolution that preserves the simple normal
crossing open set meeting the general points of all log canonical centers,
and resolving any remaining data away from that open set.
An \emph{ordinary negative step} for \(J=K_T+B\) is a projective
bimeromorphic divisorial contraction, or a diagram
\[
T\xrightarrow{f} Z\xleftarrow{f^+}T^+
\]
of projective small bimeromorphic morphisms, with connected fibers and
normal base, such that \(-J\) is \(f\)-ample and, in the small case,
the trace \(J^+\) is \(f^+\)-ample. The contracted curves on \(T\)
span one nonzero ray for degrees of global rational lines. We require
this ray condition on the contracting side only. All spaces occurring
in these steps are compact Kähler, and the working spaces are globally
strongly \(\Q\)-factorial. The boundary is pushed forward in a
divisorial step and strictly transformed in a small step.
On a common smooth compact Kähler resolution, projective over \(T\)
and the next space with maps \(p\) and \(q\), the natural meromorphic comparisons of
canonical bundles give the actual rational-line comparison
\begin{equation}
p^*J\sim_{\Q}q^*J^+ +F,\qquad F\geq0,\qquad q_*F=0.
\label{prog:comparison}
\end{equation}
Here and below these adjoint identities use the local meromorphic
canonical identifications. In particular they do not choose a global
meromorphic frame for an arbitrary line bundle. Negativity proves
Equation~\eqref{prog:comparison}; moreover, its support contains the full
inverse image of the non-isomorphism locus in the contraction base.
Thus discrepancies increase strictly for a place whose center maps
into that locus, and do not decrease elsewhere. This preserves klt
and dlt singularities. These assertions, including strictness, are
proved in \cite[Lemma~7.3]{CB}. Their proof is local over the
contraction base and has no dimension restriction.
For use on strata, we record why the full support assertion holds
for a detected small step. Choose a sufficiently divisible integer
\(r>0\) and the evaluation ideal
\[
\operatorname{im}\bigl(f^*f_*\OO_T(rJ)\longrightarrow\OO_T(rJ)\bigr)
=\mathcal I\otimes\OO_T(rJ).
\]
Outside the exceptional locus of \(f\), evaluation is an isomorphism.
On a positive-dimensional projective fiber, every section vanishes:
\(rJ\) has negative degree on every fiber curve, and curves cover
every positive-dimensional fiber component. Connectedness gives the
same vanishing on the whole nontrivial fiber. Hence the zero set of
\(\mathcal I\) is exactly the exceptional locus. On a common
resolution principalizing \(\mathcal I\), relative generation on
the positive side identifies \(rF\) in
Equation~\eqref{prog:comparison} with the divisor of
\(\mathcal I\OO_W\). Thus \(\Supp F\) is the full inverse image
of that locus. On a common resolution of a longer negative program,
discrepancy monotonicity makes the cumulative comparison dominate
this first-step divisor. These are support statements for the actual
comparison, stronger than effectiveness and exceptionality alone.
We next state the restricted model inputs used in this section. A nef
b-class is a Bott--Chern class that is globally nef on one fixed smooth
compact Kähler carrier, with its linear traces on the other models. The
generalized adjoint on \(T\) is
\(\alpha=\{K_T+B\}+\beta_T\). Generalized klt, abbreviated gklt,
means that all generalized log discrepancies, computed on that carrier,
are positive. The modified-bigness hypothesis below concerns the whole
boundary-plus-nef trace \(\{B\}+\beta_T\). We use the convention of
\cite[Definition~2.8]{HX}: a modified-big trace is the pushforward of
a big Bott--Chern class on a modification. The trace itself may be
a current; the generalized adjoint is required to define a
Bott--Chern class.
For a proper map $f:T\to S$, let $q_f$ be the quotient map to
$H^{1,1}_{\BC}(T,\R)/f^*H^{1,1}_{\BC}(S,\R)$. Relative
pseudo-effectivity means
$q_f(\alpha)\in\overline{q_f(\operatorname{Psef}(T))}$, where
$\operatorname{Psef}(T)$ is the absolute pseudo-effective cone.
Equivalently, for Kähler classes $\omega_T,\omega_S$, the class
$\alpha$ is pseudo-effective over $S$ if for every $\epsilon>0$
some $c_\epsilon\geq0$ makes
$\alpha+\epsilon\omega_T+c_\epsilon f^*\omega_S$ big.
This imposes no projectivity hypothesis on $f$. For the actual
relative line classes of an already projective map, this is the
usual relative pseudo-effective cone.
\begin{proposition}[Restricted Kähler model inputs]\label{prog:mori}
Let \((T,B+\boldsymbol\beta)\) be an effective compact Kähler gklt
pair of dimension \(d\), with \(T\) globally strongly
\(\Q\)-factorial. Suppose that its nef b-data are globally nef on a
fixed smooth carrier and that \(\{B\}+\beta_T\) is globally
modified big. Write \(\alpha=\{K_T+B\}+\beta_T\).
\begin{enumerate}
\item If \(\alpha\) is nef, there is a proper morphism with connected
fibers \(g:T\to Z\) to a normal compact Kähler space and a Kähler
class \(\omega_Z\) such that \(\alpha=g^*\omega_Z\).
\item For a proper morphism \(\pi:T\to V\) to a normal compact
Kähler space, if \(\alpha\) is pseudo-effective over \(V\), there
is a chosen good log terminal model over \(V\). It is nonextracting,
compact Kähler and globally strongly \(\Q\)-factorial. Its adjoint
is nef over \(V\), and on a common projective resolution its comparison
with \(\alpha\) is effective and exceptional over the new model,
with strictly positive coefficient at every prime contracted from \(T\).
The construction retains compatible forward Bott--Chern traces of the
specified data on a common smooth carrier.
\item A compact polytope of these data on one fixed carrier has
finitely many marked relative canonical models and relative weak log
canonical models with normal compact Kähler targets.
\end{enumerate}
The relative assertion applies to proper morphisms, without assuming
that \(\pi\) is projective. It is an assertion about a chosen model,
not termination of every generalized flip sequence.
\end{proposition}
\begin{proof}
These are the semiampleness, proper relative model and finite-model
assertions of Theorem~\ref{prog:restricted-package}, proved below
by dimension induction independently of \(\Ggood\).
Finite marked weak-model geography is recorded in
Lemma~\ref{prog:finite-geography}. Projective analytic model
constructions used in that induction are supplied by
Proposition~\ref{prog:projective-good-model}. None of these statements
promotes an undetected generalized-ray contraction to a projective map.
\end{proof}
We use separately the analytic cone theorem
\cite[Theorem~1.3]{HX}: an \(\alpha\)-negative analytic extremal ray
of an effective gklt adjoint has a rational curve generator \(C\) with
\(0<-\alpha\cdot C\leq2d\). Existence of the supporting contraction
in the applications below follows from the nef assertion above.
When an ordinary rational adjoint is negative on that ray,
Lemma~\ref{prog:detected-projectivity} and
Proposition~\ref{prog:detected-flip} give its projective contraction and ordinary
flip. We call these \emph{detected ordinary steps}.
The Bott--Chern transform through such a birational step has a concrete
description which does not require the source to be smooth. Write the
step as \(T\xrightarrow{f}Z\xleftarrow{f^+}T^+\), with
\(f^+=\id\) for a divisorial contraction, and let \(J\) be its
negative rational adjoint. For \(\gamma\in H^{1,1}_{\BC}(T,\R)\),
choose the unique real number \(c\) for which
\(\gamma-c\{J\}\) annihilates the contracted analytic ray.
Every contracted curve is on that ray. Lemma~\ref{prog:bc-descent}
therefore gives a unique class \(\delta\) on \(Z\) with
\(\gamma-c\{J\}=f^*\delta\), and we put
\[
\gamma^+=(f^+)^*\delta+c\{J^+\}.
\]
The rational-singularity and vanishing hypotheses of that lemma hold:
slightly lowering the rational Cartier floor gives a klt adjoint still
antiample over \(Z\), and relative vanishing gives rational
singularities on the base. On a common resolution, the pullback
difference is \(c\) times the adjoint comparison in
Equation~\eqref{prog:comparison}; it is exceptional over the new
model, and over both models in a flip. This defines a linear transform.
For the Chern class of a rational line it agrees with its reflexive trace,
by the coherent line comparison \cite[Lemma~7.2]{CB} and exceptional
negativity. The already-projective relative program below only needs a
ray for degrees of global rational lines; this Bott--Chern construction
is used for its subsequent detected analytic-ray programs.
When such a class difference lies in an exceptional divisor span, we write
it as \(\{F\}\) for the representing real exceptional divisor \(F\).
This divisor is unique: an exceptional divisor with zero class has zero
degree on every contracted curve, and exceptional negativity applied to
both signs makes it zero.
We also use local projective analytic results. For a projective
analytic morphism over a Stein neighborhood of a compactum with
Fujino's property (P), relative klt base point freeness says that, if
\(L\) is a relatively nef Cartier line and
\(aL-(K+\Gamma)\) is relatively ample for some positive integer
\(a\), then all sufficiently high powers of \(L\) are relatively
generated after shrinking
\cite[Theorems~6.2 and 6.5]{FujinoAnalyticBCHM}.
We use the finite negative-ray truncation in the relative cone theorem
in the same setting. Finally, the multigraded adjoint finite-generation
theorem applies to a projective morphism from a smooth space, for simultaneous
effective SNC klt boundaries with a common relatively ample rational
summand \cite[Theorem~3.1]{DHP}. Finitely many smaller Stein
neighborhoods suffice over a compact base.
\begin{lemma}[Descent without changing a Cartier exponent]
\label{prog:integral-descent}
Let \(f:T\to Z\) be a projective bimeromorphic contraction with normal
target. Suppose an ordinary rational klt adjoint is \(f\)-antiample.
If a Cartier line \(L\) has degree zero on every contracted curve,
then \(f_*L\) is an invertible sheaf and evaluation is an isomorphism
\[
f^*(f_*L)\simeq L.
\]
The same conclusion holds for an ordinary dlt adjoint whose floor is
rational Cartier and which is \(f\)-antiample. In a flip, a line with
zero contracted degree therefore has the same Cartier exponent on
both sides, and the two lines are pullbacks of a line on the common
base.
\end{lemma}
\begin{proof}
For dlt input, decrease the floor by a sufficiently small positive
rational multiple. The resulting pair is klt and remains
\(f\)-antiample. For this klt pair the base point free hypothesis
holds for \(L\), because \(L\) has zero fiber degrees and the negative
adjoint is relatively ample. Over a smaller base neighborhood every
sufficiently high power of \(L\) is therefore generated. The
associated morphism is constant on each connected fiber: its
tautological line has degree zero on every fiber curve, and every
positive-dimensional projective image contains a curve. Its graph
then factors through \(Z\); the graph projection is finite
bimeromorphic and \(Z\) is normal. Descend two consecutive generated
powers \(L^k,L^{k+1}\) to lines \(M_k,M_{k+1}\). The line
\(M_{k+1}\otimes M_k^{-1}\) pulls back to \(L\). Projection formula
identifies this quotient with
\(f_*L\), so these local descents and their evaluation maps agree on
overlaps. Pull the descended line to the positive side of a flip.
This is the argument of \cite[Lemma~7.4]{CB}.
\end{proof}
\subsection{Relative klt programs and small models}
The local finite-generation theorem also supplies an ordinary klt program
over a compact base. We use it to obtain globally strongly
\(\Q\)-factorial small models of strata. The construction keeps track of
degrees of global rational lines, which are the degrees needed for exact
Cartier descent.
\begin{proposition}[Relative klt programs over a compact base]
\label{prog:relative-klt}
Let \(\pi:T\to V\) be projective bimeromorphic, where \(V\) is a
normal compact Kähler space and \(T\) is normal, compact Kähler, and
globally strongly \(\Q\)-factorial. Let \((T,B)\) be an effective
rational klt pair with rational-line adjoint \(J=K_T+B\).
There is a finite sequence of ordinary \(J\)-negative steps over
\(V\), all projective over \(V\), whose last adjoint is curve-nef
over \(V\), meaning that it has nonnegative degree on every curve
contracted over \(V\). Every working space is compact Kähler and globally
strongly \(\Q\)-factorial. On a common resolution the comparison
from the first adjoint to the last is effective and exceptional over
the last space.
\end{proposition}
\begin{proof}
We give the reduction to the local finite-generation input, including why
it yields a single finite program over the compact base. Let
\(\mathcal N_{\R}\) be the finite-dimensional space of degrees of global
rational lines on curves over \(V\), and put \(r=\dim\mathcal N_{\R}\).
If \(r=0\), then \(J\) is already curve-nef. Assume \(r>0\). Choose
rational lines \(H_1,\ldots,H_r\) whose degree classes form a basis and
such that both \(H_i\) and \(J+H_i\) are relatively ample. Such a basis
exists because the relatively ample classes whose sum with \(J\) is also
relatively ample form a nonempty open set containing sufficiently positive
classes. Define the rational simplex of adjoints
\[
\Pi=\operatorname{conv}\{J,J+H_1,\ldots,J+H_r\}.
\]
We will choose a general direction \(H=\sum_i\theta_iH_i\) with
\(\theta_i>0\) and \(\sum_i\theta_i=1\). Its entire scaling segment lies
in this simplex:
\[
J+tH=(1-t)J+\sum_i t\theta_i(J+H_i)\in\Pi
\qquad(0\leq t\leq1),
\]
and it lies in the relative interior for \(0<t<1\).
We produce the klt representatives for these exact vertices on each
sufficiently small Stein neighborhood separately. A relative Cartier line
and its inverse have local effective representatives there: their proper
direct images are coherent of generic rank one, so Cartan generation
supplies nonzero local sections. This uses that \(\pi\) is bimeromorphic.
Represent each \(H_i\) by a divided general free effective divisor
\(G_i\sim_{\Q}H_i\), using a sufficiently divisible relatively generated
multiple. These finitely many general divisors preserve klt with \(B\).
Choose a relatively ample integral line \(A\), and local effective
representatives \(A^+\sim A\), \(A^-\sim -A\) by the preceding argument.
For one sufficiently small positive rational \(\epsilon\), all boundaries
\[
\Delta_0=B+\epsilon(A^++A^-),\qquad
\Delta_i=B+G_i+\epsilon(A^++A^-)
\]
are effective and klt. Their adjoints represent \(J\) and \(J+H_i\),
respectively, and \(\epsilon A^+\) is a common effective relatively ample
summand. In particular, every rational adjoint in \(\Pi\) is relatively big.
On this Stein member, choose one simultaneous projective log resolution
of the finite family. Give each exceptional prime a coefficient in
\((\max\{0,c_0,\ldots,c_r\},1)\), where the \(c_j\) are its crepant
coefficients for the vertex pairs. If
\(E\geq0\) is resolution-exceptional with \(-E\) resolution-ample, then
subtracting a sufficiently small multiple of \(E\) from the pullback of
\(\epsilon A^+\) leaves a relatively ample rational divisor. A sufficiently
small positive rational multiple of this divisor can be removed from every
boundary while leaving all coefficients effective and below one; the strict
exceptional margins ensure this along the exceptional primes. A divided
general free representative of this common summand preserves the simultaneous SNC
condition. The smooth multigraded theorem now applies to these vertices.
Exceptional corrections leave the local adjoint rings unchanged by
projection to a normal space; clearing the finitely many line
identifications simultaneously makes them multiplicative. This is the
reduction in \cite[Section~7.2]{CB}, using \cite[Theorem~3.1]{DHP}.
Here are the two consequences of finite generation that we need.
The normalized main relative Proj of any rational adjoint \(\Theta\in\Pi\)
extracts no prime divisor, and its trace is a relatively
ample rational line. Indeed, resolve the degree-one base ideal after
a common Veronese and write
\[
p^*(m\Theta)\sim q^*H_\Theta+G,\qquad G\geq0,
\]
with \(H_\Theta\) tautological and relatively ample. Generation says that
all sections in degree \(k\) vanish at least along \(kG\).
If a component of \(G\) were not \(q\)-exceptional, relative
generation of \(q_*\OO(G)\otimes H_\Theta^k\) would supply a section with
smaller vanishing there. If a \(p\)-exceptional prime \(Q\) were not
\(q\)-exceptional, the same argument with \(q_*\OO(Q)\otimes H_\Theta^k\)
would contradict \(p_*\OO(Q)=\OO_T\). This proves nonextraction.
There are also only finitely many \emph{marked} normalized main Proj
models, where the marking records their common bimeromorphic open
over \(V\). To see this, choose finitely many homogeneous generators
locally. Proj charts of each diagonal ring use homogeneous monomials
as denominators; the supports of those monomials in the finite set of
generators determine the degree-zero localizations and their gluing.
Only finitely many support patterns occur. Normalization and taking
the main component preserve this finiteness. A finite cover of the
compact base gives finitely many global marked models, since marked
local identifications agree on the common dense open and glue
uniquely. These are \cite[Lemmas~7.5 and 7.6]{CB}.
For completeness, we construct the scaling program controlled by this
finite list. On a working model, let \(A_0\) be a relatively ample line.
The closed curve cone in the dual of its global degree space has compact
slice \(A_0\cdot z=1\): for every line \(D\), both \(kA_0+D\) and \(kA_0-D\)
are relatively ample for \(k\) large, which bounds every coordinate.
Local cone truncations on the finite Stein cover express the negative
part of this slice, after any positive ample truncation, using
finitely many actual curve classes. A negative extremal ray can
therefore be separated by a rational nef support annihilating just
that ray. A positive multiple of the support minus the klt adjoint
is relatively ample. Relative base point freeness contracts exactly
the ray. If the contraction is divisorial, its single exceptional
prime and Lemma~\ref{prog:integral-descent} prove the global strong
property downstairs. If it is small, the relative Proj of the
adjoint is its small positive model. For any global line \(D\) on
the negative model, killing its ray degree and applying the same
lemma gives an actual rational identity
\[
D\sim_{\Q}cJ+f^*D_Z,\qquad c\in\Q.
\]
It transforms to the positive side, proving the global strong
property there. This is the continuation construction of
\cite[Proposition~7.7]{CB}; it also preserves projectivity over \(V\)
and compact Kählerness.
The transforms of \(H_1,\ldots,H_r\) continue to span the degree spaces:
use pullback in a divisorial step and the preceding decomposition of
each line in a small step. A line of zero global relative degrees
stays so by Lemma~\ref{prog:integral-descent}; curves in the
contraction bases can be lifted through projective morphisms.
There are only countably many possible finite sequences of curve rays,
and each determines its contractions and flips uniquely. We may therefore
choose \(H\) in the relative interior of
\(\operatorname{conv}\{H_1,\ldots,H_r\}\) so that it avoids all ties
between independent ray degrees in advance.
At a positive nef wall, write \(w=J+tH\) for the traces on the current
model, and choose a preceding parameter \(s>t\) for which \(J+sH\) is
relatively ample. On the zero face of the normalized compact curve slice,
\[
J=-\frac{t}{s-t}(J+sH)
\]
in degrees. Thus \(J\) is uniformly negative on that face. Choose the
positive ample truncation so that its compact remainder \(K\) misses the
face. The finite rational polyhedral test and the general choice of \(H\)
make the face a single ray \(R\). The wall class has a positive minimum on
\(K\) and is positive on every other truncated curve generator. Hence a
sufficiently small open neighborhood of \(w\) inside the annihilator
\(R^\perp\) consists of nef classes with zero face exactly \(R\).
Suppose first that \(R^\perp\) has positive dimension. It is a rational
hyperplane because \(R\) is represented by an integral curve. Choose a
rational simplex of support classes in that neighborhood containing the
degree class of \(w\) in its relative interior, and represent each vertex
by a rational line \(N\). For each such \(N\), compactness gives an
integer \(a>0\) for which \(aN-J\) is relatively ample. Relative base
point freeness contracts exactly \(R\) and descends \(N\) to a relatively
ample rational line on the contraction base. These bases agree because
the contracted curves agree. The difference between \(w\) and the
corresponding convex combination of these rational lines has zero
relative degree. In the finite rational span of the lines involved, the
degree map has rational kernel, since rational lines have rational degrees
on integral curves. The difference is therefore a real combination of
rational lines of zero relative degree. Lemma~\ref{prog:integral-descent}
descends these lines, and lifting curves from the base shows that their
descents still have zero degree over \(V\). They do not affect relative
ampleness. Convexity now shows that \(w\) descends to a relatively ample
real line class over \(V\). If \(R^\perp=0\), the same zero-degree descent
applies to \(w\); the contraction base has no curves over \(V\), hence is
finite over \(V\), and relative ampleness is automatic.
After a flip, let \(w^+\) be the pullback of this wall class to the positive
side. For sufficiently small \(\delta>0\),
\[
J^++(t-\delta)H^+
=\left(1-\frac{\delta}{t}\right)w^+
+\frac{\delta}{t}J^+
\]
is relatively ample over \(V\): the wall is pulled back from a relatively
ample class on the contraction base, and \(J^+\) is relatively ample over
that base. After a divisorial contraction, openness of the ample cone on
the base gives the same nonempty interval of relative ampleness.
Each working model is consequently one of the marked ample models
already counted. Indeed, for a fixed finite prefix and an interior
parameter \(0<t<1\), replace the direction and parameter by nearby
rational ones. They remain in \(\Pi\), and all strict signs in that prefix
and the final relative ampleness persist; the associated relative section
ring has this working model as its Proj. A repeated marked working model
would have the same trace of \(B\), because all steps are nonextracting,
and hence the same discrepancies. This contradicts the strict increase in
Equation~\eqref{prog:comparison} at an intervening nontrivial step. The
finite list therefore forces termination.
The last adjoint is curve-nef, since the continuation construction
applies whenever it is not. Composing
Equation~\eqref{prog:comparison} proves the final comparison.
\end{proof}
\begin{corollary}[Small strong models]\label{prog:small-model}
Let \((S,\Delta)\) be an effective rational dlt pair on a normal
compact Kähler space, with rational-line adjoint. Suppose that an effective
rational Cartier divisor \(D\) satisfies
\(\Supp D=\Supp\lfloor\Delta\rfloor\). There is a projective small morphism
\(h:Y\to S\), with \(Y\) globally strongly \(\Q\)-factorial and
compact Kähler, which is crepant for the full adjoint and is an
isomorphism over the smooth locus of \(S\). The transformed full
pair is dlt. The same assertion for an ordinary klt pair allows
\(D=0\).
\end{corollary}
\begin{proof}
Choose a small rational \(\epsilon>0\) for which
\((S,\Delta-\epsilon D)\) is effective and klt. On a projective log
resolution \(r:W\to S\), use the strict lowered boundary and give
every exceptional prime a coefficient strictly between its crepant
coefficient and \(1\), and at least zero. The resulting klt adjoint
has the actual form
\[
K_W+\Gamma\sim_{\Q}r^*(K_S+\Delta-\epsilon D)+F,
\qquad F\geq0,
\]
where every \(r\)-exceptional prime has positive coefficient in
\(F\). Apply Proposition~\ref{prog:relative-klt} over \(S\).
On its endpoint the trace \(F_Y\) is effective, exceptional over
\(S\), and relatively nef. Negativity makes \(F_Y=0\).
Since all initially exceptional primes had positive coefficient,
\(Y\to S\) is small and the lowered adjoint is crepant.
Restoring \(\epsilon D\) gives the full crepant identity.
A projective small morphism to a smooth germ is an isomorphism:
transport a relatively ample line to the smooth germ, where it is
Cartier; smallness makes the original line its pullback, contradicting
relative ampleness on a nontrivial fiber. Thus \(h\) is an isomorphism
over an SNC open meeting all lc centers of the dlt pair. The full
crepant comparison then proves dlt on \(Y\).
\end{proof}
\subsection{Scaling toward zero and keeping a nef rational line}
The next construction runs every positive truncation of an ordinary
dlt scaling. It also explains why a sufficiently large multiple of
a prescribed nef rational line forces every step to be trivial for
that line. The integer that clears the line will be the same at
every step.
For the detected ordinary scalings constructed below, let
\(\alpha_0(\lambda)=\{H\}+\lambda\omega\) on the first space
\(T_0\), with initial parameter \(1\), and let \(\alpha_i(\lambda)\)
be its linear transform on a working model \(T_i\). Put
\(\lambda_{-1}=1\). If the next step occurs at threshold
\(\lambda_i\), its ray is annihilated by \(\alpha_i(\lambda_i)\)
and is negative for \(\alpha_i(0)\). The trace \(\alpha_i(\lambda)\)
is nef on \([\lambda_i,\lambda_{i-1}]\), which we call its
\emph{working interval}. If \(\alpha_i(0)\) is nef, we set
\(\lambda_i=0\) and stop. Working intervals are allowed to be
degenerate; strict decrease of positive thresholds will be arranged only
in the proof of special termination.
\begin{lemma}[Positive truncations and a fixed nef line]
\label{prog:nef-trivial}
Let \((T,B)\) be an effective rational dlt pair on a globally strongly
\(\Q\)-factorial compact Kähler \(d\)-fold, and put \(J=K_T+B\).
\begin{enumerate}
\item For a sufficiently large Kähler class \(\omega\), there is an
ordinary \(J\)-program with scaling of \(\omega\), starting with
\(\{J\}+\omega\) Kähler. Every positive truncation is finite.
If \(J\) is pseudo-effective, the program either reaches a nef adjoint
or has positive thresholds tending to zero. If \(J\) is not
pseudo-effective, the program is finite and ends with a Mori fiber
contraction.
\item Let \(P\) be a nef rational line with \(mP\) Cartier, and choose
\(b\in\Q\) with \(b>4dm\). The same alternatives hold for
\(H=J+bP\), according to pseudo-effectivity of \(H\), with
\(\omega\) enlarged if necessary. Every birational step is an ordinary
\(J\)-negative step on which \(P\) is trivial, and the final Mori
ray, if present, is also \(P\)-trivial. The same line \(mP\) descends
through every birational contraction and pulls back to its positive
side. Its traces remain analytically nef; if \(P\) is semiample, its
same generated multiple and associated morphism are preserved, also
through the final Mori contraction.
\end{enumerate}
\end{lemma}
\begin{proof}
Write \(H=J\) in the first case and \(H=J+bP\) in the second.
Fix \(\omega\) with \(\{H\}+\omega\) Kähler. For each
\(0<\eta<1\), choose a positive rational \(\epsilon\) so small that,
with \(F=\lfloor B\rfloor\),
\[
(T,B-\epsilon F)\ \text{is klt},\qquad
\eta\omega+\epsilon\{F\}\ \text{is Kähler}.
\]
For \(\lambda\in[\eta,1]\) use the presentations
\[
\{H\}+\lambda\omega
=\{K_T+B-\epsilon F\}
+\bigl(b\{P\}+\lambda\omega+\epsilon\{F\}\bigr),
\]
omitting \(b\{P\}\) in the first case. Their b-data are globally
nef on the fixed initial carrier, and their boundary-plus-nef traces
are globally modified big. These are effective gklt data.
We first construct a step whenever the unscaled working trace is not
nef. Choose a positive shift strictly below the current wall and use the
gklt presentation of its current trace \(A\). The ray at the wall is
\(A\)-negative. On a compact slice of the analytic curve cone, the
cone theorem makes the \(A\)-negative part locally polyhedral after
a sufficiently small positive Kähler truncation. Separating the
chosen extremal ray from the other finitely many rays in that
truncation gives a Kähler class \(\omega_R\) such that
\(A+\omega_R\) is nef and its zero face is precisely this ray;
see the supporting construction in Proposition~\ref{prog:ordinary-step}.
Add the pullback of this new Kähler class to the fixed nef carrier.
The resulting adjoint still has globally nef data and modified-big
boundary-plus-nef trace, so the nef assertion of
Proposition~\ref{prog:mori} supplies its supporting contraction.
This uses a new supporting direction on the current space; it does
not require the trace of the original scaling direction to be Kähler.
We verify a rational ordinary detector before asserting projectivity
or constructing a flip.
In the second case nefness of \(P\) makes \(J\) negative on the ray;
the first case has this property directly. Lower the floor by a small
rational amount, retaining negativity of \(J_\epsilon\), so that
\(-J\cdot R\leq2(-J_\epsilon\cdot R)\). The ordinary klt cone
length bound gives a rational curve generator \(C\) with
\[
0<-J\cdot C\leq4d.
\]
In the second case, if \(P\cdot C>0\), integrality of \(mP\) gives
\(P\cdot C\geq1/m\), contradicting
\[
H\cdot C\geq-4d+b/m>0.
\]
Thus in that case the ray is \(P\)-trivial. The lowered ordinary adjoint is a
global rational detector. The detected projectivity and flip
constructions in Lemma~\ref{prog:detected-projectivity}
and Proposition~\ref{prog:detected-flip} supply the projective ordinary step.
The difference of \(J\) and a positive rational multiple of the
lowered adjoint has zero contracted degree; exact descent in
Lemma~\ref{prog:integral-descent} shows that restoring the floor
preserves the positive-side sign. This is the ordinary replacement
of Proposition~\ref{prog:ordinary-replacement}, and the full pair
remains dlt by Equation~\eqref{prog:comparison}.
In the second case, Lemma~\ref{prog:integral-descent} descends the actual Cartier line
\(mP\) without changing \(m\). Analytic nefness descends and pulls
back under these projective contractions \cite[Lemma~2.44]{HX}.
A generated multiple descends as a generated line by projection
formula and pulls back as one. Thus the same length estimate and the
same \(b\) apply at the next step. The corresponding assertion on
a final Mori ray follows from exactly the same estimate. For a
semiample \(P\), the map of its fixed generated multiple is constant
on each connected projective contracted fiber: otherwise its image
contains a curve of positive degree. The map therefore factors through
the normal contraction base, including in the fiber-type case.
The traces used in this construction are linear by the degree-zero
descent description preceding Lemma~\ref{prog:integral-descent}.
At a wall the scaled class is pulled back from its base. The comparison
for an earlier parameter is a nonnegative multiple of the ordinary
adjoint comparison. Consequently the gklt presentations on any fixed
positive segment remain gklt along its program, and every working
model is a weak model for a member of that segment. The finite marked
weak-model assertion in Proposition~\ref{prog:mori} applies.
A marked model cannot recur: its boundary trace would be the same,
whereas a nontrivial intervening ordinary step strictly increases a
discrepancy. There are therefore only finitely many steps with
threshold at least \(\eta\).
For decreasing cutoffs, any already-constructed finite prefix remains
valid. The floor perturbations cancel in the displayed classes and
continue to cancel under their linear transforms; their nonpositive
comparisons preserve the new gklt data. Starting at its last nef wall
therefore extends the prefix to the smaller cutoff. This gives one
compatible scaling program. A positive limiting threshold would lie
in a finite positive truncation, so an infinite program has limit zero.
Pseudo-effectivity excludes a negative Mori endpoint. If \(H\) is
not pseudo-effective, its pseudo-effective threshold in the initial
Kähler direction is positive. Every nef working trace, together with
its effective exceptional comparison, makes the initial scaled class
pseudo-effective, so the scaling thresholds are bounded below by
that positive number. The program is therefore finite. Its endpoint
cannot be nef, since that comparison would make \(H\)
pseudo-effective; it is the stated Mori fiber contraction.
\end{proof}
We call these programs \emph{detected ordinary scalings}. Their steps
are ordinary negative steps of one analytic extremal ray, with the
linear Bott--Chern transform described above.
The next lemma connects their nef working traces
to the analytic negative part, including its limit at parameter zero.
\begin{lemma}[Negative parts along a detected ordinary scaling]
\label{prog:negative-along-scaling}
In a scaling from Lemma~\ref{prog:nef-trivial}, write \(H=J\) in its first
case and \(H=J+bP\) in its second, and use the notation
\(\alpha_0(\lambda)=\{H\}+\lambda\omega\) above. For a parameter
\(\lambda\) in the working interval of \(T_i\), a common smooth
resolution \(p:W\to T_0\), \(q:W\to T_i\) gives
\begin{equation}
p^*\alpha_0(\lambda)=q^*\alpha_i(\lambda)+\{E_i(\lambda)\},
\qquad E_i(\lambda)\geq0,\qquad q_*E_i(\lambda)=0.
\label{prog:working-negative}
\end{equation}
Here \(E_i(\lambda)\) is an actual real exceptional divisor, and
\[
N(p^*\alpha_0(\lambda))=E_i(\lambda).
\]
For the limit statement, let \(H\) be any rational line on a normal compact
Kähler space \(T\) whose pullback to a smooth resolution is
pseudo-effective, and let \(A\) be a Kähler class on \(T\). On any fixed
smooth resolution \(r:U\to T\) and for every prime \(Q\subset U\),
\[
\lim_{\lambda\downarrow0}\nu_Q(r^*(\{H\}+\lambda A))
=\nu_Q(r^*\{H\}).
\]
Thus the same convergence holds for every divisorial place, after choosing
a resolution on which it appears.
\end{lemma}
\begin{proof}
For \(\lambda\) in the working interval of \(T_i\), every earlier
threshold is at least \(\lambda\). Each earlier step is therefore negative
or trivial for this parameter. Composing its comparisons gives
Equation~\eqref{prog:working-negative}. Since \(\alpha_i(\lambda)\) is
nef, Lemma~\ref{bir:exceptional} identifies the exceptional divisor with
the negative part.
On the fixed resolution \(U\), choose a Kähler class \(\Omega\) and
\(c>0\) such that \(c\Omega-r^*A\) is Kähler. Monotonicity of each
minimal multiplicity under addition of a nef class gives
\[
\nu_Q(r^*\{H\}+\lambda c\Omega)
\leq \nu_Q(r^*(\{H\}+\lambda A))
\leq \nu_Q(r^*\{H\}).
\]
The left side tends to the right side by the definition using small Kähler
perturbations. This proves the convergence on \(U\), and the same argument
applies on a resolution representing any chosen divisorial place.
\end{proof}
\begin{proposition}[Contracting a known negative part]
\label{prog:contract-negative}
Let \((T,B)\) be an effective rational dlt pair on a globally
strongly \(\Q\)-factorial compact Kähler space, and put \(J=K_T+B\).
Suppose there is an actual rational-line identity
\[
J\sim_{\Q}P+E,
\]
where \(P\) is analytically nef, \(E\) is an effective rational
Cartier divisor, and, on one smooth compact Kähler resolution
\(\pi:W\to T\) projective over \(T\),
\[
N(\pi^*\{J\})=\pi^*E.
\]
There is a finite ordinary \(J\)-negative program to a nef model
\(T'\) which contracts precisely the prime components of \(E\)
among the primes of \(T\). Every step is \(P\)-trivial, with a
fixed Cartier exponent as in Lemma~\ref{prog:nef-trivial}, and
\[
J_{T'}\sim_{\Q}P_{T'}.
\]
If \(P\) is semiample, then so is this actual last adjoint.
\end{proposition}
\begin{proof}
Lemma~\ref{bir:pullup} makes the stated equality valid on any
higher resolution. Choose \(b\) as in
Lemma~\ref{prog:nef-trivial}. Adding \(bP\) leaves the same
negative part. Indeed, on a smooth resolution write
\(\alpha=\pi^*\{J\}\), \(p=\pi^*\{P\}\), and \(e=\pi^*E=N(\alpha)\).
Subadditivity and homogeneity of negative multiplicities give
\begin{align*}
N(\alpha+bp)&\leq e,\\
(1+b)e=N((1+b)\alpha)
&\leq N(\alpha+bp)+bN(\{e\})
\leq N(\alpha+bp)+be.
\end{align*}
Thus \(N(\alpha+bp)=e\).
Run the scaling for \(H=J+bP\).
Let \(Q\) be the strict transform on a fixed resolution of a prime
component of \(E\). Its multiplicity in \(N(\pi^*\{H\})\) is
positive. The limit in Lemma~\ref{prog:negative-along-scaling} makes its multiplicity
positive for every sufficiently small positive parameter. If
that prime still survived on the corresponding working model,
Equation~\eqref{prog:working-negative} would make its multiplicity zero,
because the comparison there is exceptional over that model.
It must therefore be contracted after finitely many steps. The
same conclusion holds if a nef endpoint is reached before taking
the limit. There are only finitely many components of \(E\), so
after a finite prefix all have disappeared.
Throughout the program \(P\) descends and remains nef. The
actual identity \(J_i\sim_{\Q}P_i+E_i\), with \(E_i\) the
codimension-one pushforward of \(E\), follows from the reflexive
trace convention. Once \(E_i=0\), we have \(J_i\sim_{\Q}P_i\); since
\(P_i\) is nef, the program stops. Conversely, a divisorial negative step can
contract only a prime of \(E_i\): its ray is \(P_i\)-trivial,
so the effective divisor \(E_i\) has negative degree on every
contracted curve, and those curves are contained in its support.
This proves the exact assertion about contracted primes.
\end{proof}
The next lemma records the cohomology carried by a detected ordinary scaling
that begins on a smooth space. Its surjectivity will let us choose one
generic initial scaling direction for all later models. Its exceptional
splitting will also carry the span of line classes to the terminal models
used in Section~\ref{sec:boundary}.
\begin{lemma}[Cohomology through ordinary steps]
\label{prog:cohomology-transport}
Let a finite prefix of one of the detected ordinary scalings constructed
in Lemma~\ref{prog:nef-trivial} start on a smooth compact Kähler
space, and let \(V\) be any working space.
For every smooth compact Kähler resolution \(p:U\to V\)
projective over \(V\),
\begin{align}
H^2(U,\R)
&=p^*H^2(V,\R)\ \oplus\
\bigoplus_{Q\ {\rm exceptional\ for}\ p}\R\{Q\}, \label{prog:H2}\\
H^{1,1}(U,\R)
&=p^*H^{1,1}_{\BC}(V,\R)\ \oplus\
\bigoplus_{Q\ {\rm exceptional\ for}\ p}\R\{Q\}.
\label{prog:H11}
\end{align}
The linear transform from the first space is surjective onto
\(H^2(V,\R)\) and \(H^{1,1}_{\BC}(V,\R)\), respectively. For a small correspondence between
working spaces, the difference of the pulled-back classes and
their transforms on a common resolution lies in the span of the
primes exceptional over both spaces.
Under the displayed decompositions, the Chern class of a global
holomorphic line on \(U\) is the sum of the pullback of a rational
holomorphic line class on \(V\) and an exceptional rational divisor
class. In particular, passage to the exceptional quotient preserves
the real span of line Chern classes.
\end{lemma}
\begin{proof}
For a modification of a smooth compact Kähler manifold, the
degree-two modification theorem gives these decompositions, and
the exceptional classes have type \((1,1)\). Suppose they hold
at one step. Its contraction base and both working spaces have
rational singularities: slight lowering of the rational Cartier
floor makes the ordinary pair klt, and the detected contraction has
a base with rational singularities. These contractions are
bimeromorphic, so a common resolution and Leray give
\(R^if_*\OO=0\) for \(i>0\); this verifies the additional
hypothesis in Lemma~\ref{prog:bc-descent}.
The ray of this detected contraction is one ray for Bott--Chern degrees.
This also suffices for degree-two classes. Indeed, pull a class
\(\gamma\in H^2(V,\R)\) to a smooth resolution \(p:U\to V\)
and take its Hodge decomposition. By the decomposition already proved for
this working model in Equation~\eqref{prog:H11}, its \((1,1)\)-part is \(p^*\beta+\{E\}\) for a
Bott--Chern class \(\beta\) and an exceptional real divisor
\(E\). The other Hodge components have zero degree on curves.
On every \(p\)-contracted curve the degree of \(E\) is therefore
zero, so both signs of negativity make \(E=0\). Lifting any
curve of \(V\) through the projective resolution now shows that
\(\gamma\) and \(\beta\) have the same curve degrees. Thus
subtracting a multiple of \(\{J\}\) from either a degree-two or
a Bott--Chern class can annihilate every curve of the contracted
fiber. By
Lemma~\ref{prog:bc-descent}, a degree-two class, or a Bott--Chern class,
with these zero degrees is pulled back from the base.
Subtract a multiple of \(\{J\}\) to make any given class
annihilate that ray, descend the remainder, and use the trace
\(\{J^+\}\) to define its transform. On a common resolution the
difference is a multiple of the exceptional adjoint comparison in
Equation~\eqref{prog:comparison}. For a flip the two exceptional prime
sets on that resolution are the same. For a divisorial
contraction the target resolution has just the one additional
exceptional prime of the step. The old decomposition therefore
spans the new one and proves surjectivity of the transform.
The sums are direct. If the pullback of a class from \(V\) is
an exceptional divisor class, that divisor has zero degree on
all curves over \(V\). Applying exceptional negativity to both
signs makes the divisor zero; injectivity of pullback gives that
the class is zero. The pullback injectivity in degree two and in
Bott--Chern cohomology is also
Lemma~\ref{prog:bc-descent}, applied to the resolution. Passing to
higher smooth resolutions proves the assertion for every \(p\).
For the line assertion, let \(\mathscr H\) be a holomorphic line on
\(U\), and put \(\mathscr Q=(p_*\mathscr H)^{**}\). Global strong
\(\Q\)-factoriality gives an invertible sheaf
\(\mathscr M=\mathscr Q^{[m]}\) for some \(m>0\). The coherent
evaluation comparison of \cite[Lemma~7.2]{CB} gives
\[
\mathscr H^{\otimes m}\simeq p^*\mathscr M\otimes\OO_U(E),
\qquad E\ \text{an integral \(p\)-exceptional divisor}.
\]
Indeed \((p_*\mathscr H^{\otimes m})^{**}=\mathscr M\), since the
two sheaves agree at the general points of all divisors of \(V\).
Local generators of \(p_*\mathscr H^{\otimes m}\) in a frame
of \(\mathscr M\) compare their evaluations with the pulled-back
coefficients by meromorphic quotients. Those quotients agree on the
isomorphism locus and hence glue; their zeros and poles are exceptional.
Taking Chern classes proves the assertion without choosing a meromorphic
section of \(\mathscr H\).
\end{proof}
Before returning to special termination, we record the consequence used in
the boundary argument: a lower-dimensional manifold of Kodaira dimension
zero has a terminal torsion model carrying the preceding exceptional
splitting.
\begin{corollary}[Terminal models in Kodaira dimension zero]
\label{prog:terminal-kappa-zero}
Let \(X\) be a smooth compact Kähler \(d\)-fold with \(d<n\)
and \(\kappa(X,K_X)=0\). A finite ordinary \(K_X\)-negative
program reaches a globally strongly \(\Q\)-factorial compact
Kähler terminal space \(X_{\min}\) whose actual canonical
rational line is torsion. For every smooth compact Kähler
resolution projective over \(X_{\min}\), the decompositions in
Equations~\eqref{prog:H2} and~\eqref{prog:H11} hold.
\end{corollary}
\begin{proof}
A pluricanonical section makes \(K_X\) pseudo-effective.
By \(\Ggood_d\), on a smooth modification
\(\mu:\widetilde X\to X\) there is an actual identity
\[
\mu^*K_X\sim_{\Q}P+E,\qquad P\ \text{semiample},\quad
E=N(\mu^*\{K_X\}).
\]
Lemma~\ref{bir:sections} identifies the divisible section spaces
with those of \(P\). Hence \(\kappa(P)=0\); a semiample line of
Iitaka dimension zero has a trivial positive multiple.
Proposition~\ref{bir:finite-descent}, applied with \(e=\mu\) and \(F=0\),
gives an effective rational divisor \(E_X=N(K_X)\) with
\(K_X\sim_{\Q}E_X\).
Lemma~\ref{bir:pullup} then gives
\(N(\mu^*\{K_X\})=\mu^*E_X\).
Apply Proposition~\ref{prog:contract-negative} with \(P=0\).
It reaches a model with torsion canonical line. It is terminal:
exceptional places over the initial smooth space have log
discrepancy greater than \(1\), and a divisor contracted from
that space starts with log discrepancy \(1\) and acquires a
strict increase in Equation~\eqref{prog:comparison}. Finally apply
Lemma~\ref{prog:cohomology-transport}.
\end{proof}
\subsection{A comparison model for the restricted scaling}
We now prepare the lower-dimensional argument that excludes an
infinite sequence of small transformations on a log canonical
stratum. We first construct a semiample small model when the
negative multiplicities have centers away from the floor. We then
make one model nef on an entire interval of perturbations. This
interval is the precise lower-dimensional conclusion needed below.
\begin{lemma}[A semiample small comparison model]
\label{prog:stratum-model}
Let \((S,\Delta)\) be an effective rational dlt pair on a normal
compact Kähler space of dimension \(d<n\). Suppose
\(J=K_S+\Delta\) is a pseudo-effective rational line, and that an
effective rational Cartier divisor \(D\) satisfies
\(\Supp D=\Supp\lfloor\Delta\rfloor\). Assume for every divisorial
place \(Q\) over \(S\) that
\begin{equation}
\nu_Q(J)=0
\quad\text{if the center of \(Q\) is a prime of \(S\), or meets
\(\Supp\lfloor\Delta\rfloor\).}
\label{prog:away-floor}
\end{equation}
There is a globally strongly \(\Q\)-factorial compact Kähler dlt
pair \((M,\Delta_M)\), small bimeromorphic with \((S,\Delta)\),
whose adjoint \(J_M\) is an actual semiample rational line.
Over a neighborhood \(U\) of the floor in \(S\), the comparison is
a projective small crepant morphism \(M_U\to U\).
For every \(\gamma\in H^{1,1}_{\BC}(S,\R)\) there is a transformed
class \(\gamma_M\in H^{1,1}_{\BC}(M,\R)\) which is its pullback over
\(U\). On a common smooth resolution \(p:W\to S,\ q:W\to M\),
\[
q^*\gamma_M-p^*\gamma=\{E_\gamma\},
\]
where \(E_\gamma\) is a real divisor supported on primes exceptional over
both spaces, and \(\Supp E_\gamma\) is disjoint from \(p^{-1}(U)\).
\end{lemma}
\begin{proof}
Resolve the dlt pair, giving new exceptional primes coefficient
one. The resolved adjoint is the pullback of \(J\) plus an
effective exceptional divisor. Apply \(\Ggood_d\) to this
smooth pair. Lemma~\ref{bir:exceptional} subtracts the
exceptional resolution error from its negative part. On a
smooth higher model \(r:\widetilde S\to S\) we obtain the actual
identity
\begin{equation}
r^*J\sim_{\Q}P+E,\qquad P\ \text{semiample},\qquad
E=N(r^*\{J\}).
\label{prog:stratum-decomposition}
\end{equation}
Every component of \(E\) is exceptional over \(S\) and has
center disjoint from the floor, by Equation~\eqref{prog:away-floor}.
Choose \(m\) divisible enough that \(mJ,mP,mE\) are integral and
\(mP\) is generated. Lemma~\ref{bir:sections} shows that
multiplication by the section of \(mE\) identifies the complete
section spaces. Over a neighborhood of the floor, \(E\) is
absent upstairs. The line \(mr^*J\) is generated there, and
these sections descend to \(S\) by normality. Thus \(mJ\)
is generated on a neighborhood \(U\) of the entire floor.
Let \(\rho_\Gamma:\Gamma\to S\) be the normalized graph of the map defined by
the complete system \(|mJ|\), and write
\(\varphi:\Gamma\to\PP(H^0(S,mJ)^*)\) for its morphism to projective
space. The map \(\rho_\Gamma\) is projective and is an isomorphism over \(U\).
The rational line
\[
P_\Gamma=\frac1m\varphi^*\OO(1)
\]
is semiample. The generated moving system \(|mP|\) induces a morphism
\(\psi:\widetilde S\to\Gamma\) whose pullback of \(P_\Gamma\) is \(P\).
Choose a small positive rational \(\epsilon\)
for which \((S,\Delta-\epsilon D)\) is effective and klt.
On a projective log resolution \(W_0\to\Gamma\to S\), give
every exceptional prime a nonnegative coefficient strictly
above its crepant coefficient for the lowered pair and strictly
below \(1\). Its klt adjoint equals the pullback of
\(J-\epsilon D\) plus an effective error positive on every
prime exceptional over \(S\).
Run Proposition~\ref{prog:relative-klt} over \(\Gamma\), and write
\(g:Y\to\Gamma\) for the endpoint morphism and \(h=\rho_\Gamma g:Y\to S\).
The surviving
error \(F_Y\) is effective and has positive coefficient on
every prime of \(Y\) exceptional over \(S\). Over \(U\), where
\(\Gamma=S\), it is also exceptional and relatively nef, since
the lowered adjoint is relatively nef and its other term is
pulled back from \(S\). Local exceptional negativity makes it
zero there. Hence \(h\) is small over \(U\), and it is an
isomorphism over the smooth part of \(U\) by the argument in
Corollary~\ref{prog:small-model}. Restore the floor. The full
pair on \(Y\) is dlt near it, by the crepant small comparison
and the SNC open, and is klt elsewhere. No exceptional prime
has center meeting the floor, so \(h^*D\) is its strict
transform and
\[
J_Y\sim_{\Q}h^*J+F_Y.
\]
Put \(P_Y=g^*P_\Gamma\). On a common smooth model
\(a:\widehat S\to\widetilde S\), \(b:\widehat S\to Y\) over \(S\),
the maps \(\psi a\) and \(g b\) agree on the common dense open and hence
everywhere by separatedness of \(\Gamma\). Thus \(a^*P\sim_{\Q}b^*P_Y\).
Define \(E_Y^0=b_*a^*E\). This is an effective rational divisor,
exceptional over \(S\) and supported away from the floor. Pushing
Equation~\eqref{prog:stratum-decomposition} to \(Y\) gives
\[
h^*J\sim_{\Q}P_Y+E_Y^0.
\]
The global strong property on \(Y\) makes \(E_Y^0\) rational Cartier.
The divisor \(a^*E-b^*E_Y^0\) is \(b\)-exceptional and
rationally linearly trivial. Exceptional negativity applied to both signs therefore
gives \(a^*E=b^*E_Y^0\).
Define \(E_Y=E_Y^0+F_Y\). Adding the restored-boundary error gives
\begin{equation}
J_Y\sim_{\Q}P_Y+E_Y,\qquad E_Y\geq0,
\label{prog:graph-adjoint}
\end{equation}
where \(E_Y\) is exceptional over \(S\), is supported away from the floor,
and contains every prime exceptional for \(h\). It is rational Cartier by
the global strong property. On \(\widehat S\), Lemma~\ref{bir:pullup}
gives
\[
N(a^*r^*\{J\})=a^*E=b^*E_Y^0.
\]
Exceptional translation for \(b^*F_Y\) over \(S\) then gives
\(N(b^*\{J_Y\})=b^*E_Y\). The hypotheses of
Proposition~\ref{prog:contract-negative} are now satisfied.
Each contracted curve lies in \(\Supp E_Y\), since its
\(P_Y\)-degree is zero and its \(E_Y\)-degree is negative.
Nontrivial connected projective fibers are covered by curves,
so this contraction and its positive replacement are unchanged
over a neighborhood of the floor. The same assertion persists
with the pushed-forward effective divisor at each step.
All primes exceptional over \(S\) are contracted and no prime
coming from \(S\) is contracted. The endpoint \(M\) is
therefore small with \(S\), has the asserted morphism over
\(U\), and has \(J_M\sim_{\Q}P_M\) semiample.
Pull \(\gamma\) from \(S\) to \(Y\). Although \(Y\) may be singular,
its subsequent program consists of the detected ordinary analytic-ray
steps in Proposition~\ref{prog:contract-negative}. At step \(i\),
choose \(c_i\in\R\) so that
\(\gamma_i-c_i\{J_i\}\) has degree zero on its contracted ray.
All curves in a fiber have class on that same analytic ray. The
working spaces have rational singularities after slightly lowering
the Cartier floor, and relative vanishing gives rational singularities
on the contraction base. Lemma~\ref{prog:bc-descent} therefore
descends this class uniquely to the base. Pull it to the positive side
and add \(c_i\{J_{i+1}\}\), defining \(\gamma_{i+1}\).
The pullback difference is \(c_i\) times the ordinary adjoint
comparison, hence is an actual real exceptional divisor class.
No smooth-start cohomology splitting is needed for this construction.
Telescope these comparisons on a common resolution. The original
map \(Y\to S\) was a morphism, and \(\gamma\) was pulled back
through it. Since \(S\) and \(M\) match in codimension one, the
resulting divisor is exceptional over both. Each step is an
isomorphism over the chosen neighborhood \(U\) of the floor.
There the class remains the pullback from \(S\); the corresponding
comparison divisor has zero class and is exceptional. Negativity
applied to both signs makes it zero there. Thus its support is
disjoint from the inverse image of \(U\), as claimed.
\end{proof}
\begin{lemma}[One nef model for an interval]\label{prog:nef-interval}
Under the hypotheses of Lemma~\ref{prog:stratum-model}, let \(A\) be any Kähler
class on \(S\). There are a normal globally strongly
\(\Q\)-factorial compact Kähler space \(V\), small bimeromorphic
with \(S\), a number \(\delta>0\), and transformed classes
\(\{J_V\},A_V\) such that
\begin{equation}
\{J_V\}+tA_V\ \text{is nef for every }0\leq t\leq\delta.
\label{prog:whole-nef-interval}
\end{equation}
On a common smooth resolution of \(S\) and \(V\), the
differences of pullbacks of \(J,A\) and these transforms lie
in the common exceptional span.
\end{lemma}
\begin{proof}
Let \(P_M=J_M\) denote the semiample rational line from
Lemma~\ref{prog:stratum-model}, and use its transform \(A_M\).
On a common resolution \(p:W\to S,\ q:W\to M\), let \(F_A\) be the
real divisor in the common exceptional span defined by
\[
q^*A_M-p^*A=\{F_A\}.
\]
It is effective: \(-F_A\) is \(q\)-nef, because \(p^*A\) is
nef and the other term is pulled back from \(M\); apply
exceptional negativity. Its support is disjoint from \(p^{-1}(U)\).
In particular \(A_M\) is pseudo-effective, and its negative
multiplicity at every prime of \(M\) is zero. Indeed,
\(q^*A_M=p^*A+\{F_A\}\), and
Lemma~\ref{bir:exceptional} applied over \(S\) gives
\(N(q^*A_M)=F_A\).
Let \(D_M\) be the rational Cartier trace of \(D\).
Choose \(\tau>0\) so small that \(A+\tau\{D\}\) is Kähler.
For a fixed sufficiently small \(t>0\), consider on \(M\) the
boundary \(\Delta_M-t\tau D_M\) and the nef b-class represented on
\(W\) by \(t p^*(A+\tau\{D\})\). Its trace on \(M\) is
\(t(A_M+\tau\{D_M\})\), so its generalized adjoint is
\begin{equation}
\{P_M\}+tA_M.
\label{prog:interval-gadjoint}
\end{equation}
These are effective gklt data. Over \(U\) the b-class
descends to \(M\), so its generalized discrepancy
contribution is zero there, and subtracting \(t\tau D_M\)
raises every zero log discrepancy of the dlt pair.
Away from the floor the ordinary pair is klt; on one fixed
log resolution its finitely many subunit coefficients stay
subunit for small \(t\). This proves gklt. The generalized
boundary is big. On a common resolution the pullback of the
trace of \(A+\tau\{D\}\) minus \(p^*(A+\tau\{D\})\) is effective exceptional
by the same negativity argument as for \(F_A\). The latter
pullback is nef and big, so the trace is big; the remaining
boundary is effective.
Choose an integer \(m>0\) such that \(mP_M\) is generated, and
let \(g:M\to Z\) be the Stein factor of its morphism. Then \(Z\)
is normal projective and \(mP_M=g^*H\) for an ample generated Cartier line
\(H\) on \(Z\). Set \(L=\{P_M\}+tA_M\). The preceding
preparation makes \(L\) an effective gklt adjoint with globally nef
b-data and globally modified-big boundary-plus-nef trace. It is
pseudo-effective because \(P_M\) is nef and \(A_M\) is modified
nef. Apply the proper relative assertion of
Proposition~\ref{prog:mori} over \(Z\). This produces a chosen
nonextracting good log terminal model \(\phi:M\dashrightarrow V\)
and a morphism \(g_V:V\to Z\), with \(L_V\) nef over \(Z\).
The hypothesis here is properness: the semiample morphism \(M\to Z\)
need not be projective. Put
\[
mP_V=g_V^*H,\qquad A_V=(L_V-\{P_V\})/t.
\]
The relative construction preserves the actual line pulled back from
\(Z\), and its forward Bott--Chern transport preserves the displayed
linear relation; these are the transforms of \(P_M,A_M\).
The map \(\phi\) is small. On a common projective resolution
\(a:W'\to M\), \(c:W'\to V\), its comparison is
\[
a^*L=c^*L_V+\{E\},\qquad E\geq0,
\]
where \(E\) is exceptional over \(V\) and has positive coefficient
at the strict transform of each prime contracted from \(M\).
The forward trace \(L_V\) is pseudo-effective, so exceptional
translation gives
\[
N(a^*L)=N(c^*L_V)+E\geq E.
\]
But \(L\) is modified nef on \(M\); the coefficient of
\(N(a^*L)\) at a strict prime of \(M\) is zero. Thus no prime of
\(M\) is contracted. Nonextraction then proves smallness.
Fix \(b>2dm\). We claim that \(L_V+b\{P_V\}\) is absolutely
nef. Otherwise the cone theorem for the effective gklt adjoint
\(L_V\) gives an \(L_V\)-negative rational extremal generator
\(C\) such that
\[
(L_V+b\{P_V\})\cdot C<0,\qquad 0<-L_V\cdot C\leq2d.
\]
Indeed the cone summand on which \(L_V\) is nonnegative also has
nonnegative \(P_V\)-degree, so a negative ray summand must account
for any failure of nefness. If \(P_V\cdot C=0\), ampleness of
\(H\) makes \(C\) vertical over \(Z\), contradicting relative
nefness of \(L_V\). Otherwise Cartier integrality gives
\(P_V\cdot C\geq1/m\), contradicting
\[
(L_V+b\{P_V\})\cdot C\geq-2d+b/m>0.
\]
This proves the claim. Both \(\{P_V\}\) and
\((1+b)\{P_V\}+tA_V\) are nef. By convexity,
\[
\{P_V\}+sA_V\ \text{is nef for }0\leq s\leq\frac{t}{1+b}.
\]
Thus \(\delta=t/(1+b)\) works. The actual line \(P_V\) has the
same generated multiple as \(P_M\), since both are pulled back from
\(H/m\). Its class is the transform of \(J\).
The relative comparison and linear transport give exceptional-divisor
comparisons for \(P_M,A_M\). Since \(S,M,V\) agree in codimension
one, their final comparison divisors are exceptional over both \(S\)
and \(V\), as required.
\end{proof}
\subsection{Special termination}
\begin{theorem}[Special termination for a chosen dlt scaling]
\label{prog:special-termination}
Assume \(\Ggood_d\) for every \(d<n\). Let \((X,B)\) be an
effective rational dlt pair on a smooth compact Kähler
\(n\)-fold, and suppose \(L=K_X+B\) is pseudo-effective.
There is a Kähler scaling direction \(\omega\), with
\(\{L\}+\omega\) Kähler, and a scaling as in
Lemma~\ref{prog:nef-trivial}, such that after finitely many
steps the exceptional, flipping, and flipped loci are disjoint
from the reduced floors of the working pairs.
\end{theorem}
\begin{proof}
We first choose a scaling with strict positive walls and Kähler interior
classes. We then reduce the transformations on each log canonical stratum
to small diagrams away from its smaller strata. Finally, the single nef
interval supplied above rules out infinitely many such diagrams.
\smallskip\noindent\emph{Choice of scaling.}
For every possible finite prefix of the detected ordinary truncation scalings from \(X\),
Lemma~\ref{prog:cohomology-transport} makes the transform of
\(H^{1,1}(X,\R)\) onto the working Bott--Chern space
surjective. There are countably many such sequences: curve
rays belong to countably many integral homology classes, and
a ray determines its contraction and its flip uniquely.
On each working space, two distinct curve rays \(C,C'\) with nonzero
\(L\)-degrees give a proper hyperplane of initial directions defined by
\[
(L\cdot C)(\omega\cdot C')-(L\cdot C')(\omega\cdot C)=0,
\]
where the degrees use the traces on that working space. Whenever both rays
define finite walls, this equation says that their wall parameters agree.
Avoid all these
hyperplanes, and also avoid \(\omega\cdot C=0\) whenever
\(L\cdot C=0\). A general Kähler direction, enlarged to
make the initial scaled class Kähler, has these properties.
At a wall the scaled class is pulled back from the contraction
base. Immediately after a flip, curves in an opposite-side
fiber have positive \(L\)-degree. Another negative ray at
the same wall would satisfy the excluded wall equation with that
opposite-side ray. Immediately after a
divisorial contraction, lift any putative new wall curve
through the projective morphism. Its lifted ray is distinct
from the contracted ray and satisfies the same wall equation; if its
\(L\)-degree were zero, its \(\omega\)-degree would also be
zero. Both possibilities were excluded. Hence consecutive
positive thresholds strictly decrease, and each working
interval has nonempty interior. Its interior nef class is
big: the class on \(X\) is a positive Kähler perturbation of
the pseudo-effective class \(L\), and bigness is preserved
under the birational comparison. No rational curve can have
zero interior degree. An \(L\)-negative or \(L\)-positive
such curve would violate nefness at one of the two nearby
parameters; the zero-\(L\) case was excluded. The gklt
presentation of a positive truncation and
\cite[Theorem~4.3]{HLX} therefore make the interior class
Kähler. By Lemma~\ref{prog:nef-trivial}, infinitely many
thresholds would tend to zero.
\smallskip\noindent\emph{Reduction to small transformations on a stratum.}
Suppose the sequence is infinite. The homological divisor
count \cite[Lemma~7.8]{CB} discards all divisorial ambient
steps after a finite prefix. We next reduce the remaining
flips to small transformations on a stratum.
For a working ambient pair \((X_i,B_i)\), ordinary dlt
adjunction on the normalization \(S\) of an lc center gives
an effective dlt pair \((S,\Delta_S)\) and the
actual rational-line residue identity
\[
K_S+\Delta_S\sim_{\Q}(K_{X_i}+B_i)|_S.
\]
Its lc centers are the proper nested lc centers. The
coefficients of these differents form a DCC set depending
only on the original coefficients and the chain length. At
one restriction they have the form
\[
1-\frac1r+\frac{\sum_j k_jb_j}{r},
\qquad 0\leq\sum_j k_jb_j\leq1,
\]
which preserves DCC under iteration. These assertions,
including the actual residue comparisons, are
\cite[Lemma~7.16]{CB}. Restricting
Equation~\eqref{prog:comparison} along a strict adjunction chain
gives an effective comparison on strata, strictly positive
for a place whose center maps into the ambient exceptional
locus \cite[Lemma~7.17]{CB}. Both lemmas are
dimension-free.
Identify a surviving lc center with its strict transform on the common
isomorphism open. There are finitely many lc centers at each stage, and
their surviving list stabilizes. A new zero-discrepancy place was
already a zero-discrepancy place, and strictness keeps the
general point of its center outside the surgery. Starting with the smallest
centers, induct on their dimensions. Fix a surviving center and suppose the
loci already avoid all its proper subcenters. We will prove that the loci
eventually avoid this center as well. The induced diagrams on its normalized
strata are then isomorphisms near their non-klt loci. On
each stratum, let \(D_S\) be the sum of the restrictions of the ambient
floor components not used in its generic adjunction chain. Strong
\(\Q\)-factoriality of the ambient space makes \(D_S\) rational Cartier,
and the nested-center description gives
\(\Supp D_S=\Supp\lfloor\Delta_S\rfloor\). Slightly lowering this
effective divisor makes the stratum pair klt.
After a further tail the induced transformations on this
stratum are small and their boundary transforms agree.
Indeed, a prime extracted in a restricted transformation
has discrepancy strictly less than \(1\) before extraction:
its new boundary is effective and strict comparison
increases its discrepancy. Its center is away from the
unchanged non-klt locus. At the start of this tail there are
only finitely many such places. On a fixed compact log
resolution, the positive SNC weights \(1-b_j\) have a
positive minimum, and the Jacobian inequality bounds every
place below the strict cutoff \(1\); include also the
finitely many nonexceptional positive-boundary primes.
This is \cite[Lemma~7.9]{CB}, with its lc assertion away
from the non-klt locus. Discrepancies do not decrease, so
all later extractions belong to this same finite list.
Repeated extraction of one place would give a strictly
decreasing sequence of its boundary coefficients, contrary
to the adjunction DCC property.
Once extractions stop, \cite[Lemma~7.8]{CB} makes
divisorial contractions finite. Its dimension-free count
is the dimension of the span of prime divisor cycles in
\(H_{2d-2}\) of the compact \(d\)-dimensional stratum.
Restriction to the common isomorphism open uses
Borel--Moore homology: removing codimension at least two
from the target has no effect in this degree, while a lost
prime has nonzero class by its positive Kähler volume.
The count strictly drops. The coefficients on the
finitely many matched boundary primes then stabilize by
monotonicity and DCC. Neither morphism of a restricted
diagram can still contract a divisor to the common
intermediate space, since strict comparison would change
the discrepancy of that matched divisor. The diagrams are
therefore small, with identical boundary transforms.
We justify the passage from an isomorphic restricted diagram to
absence of ambient surgery near the stratum, also in higher
codimension. Its identified boundaries give the same adjunction data,
not only isomorphic underlying spaces. Choose a common projective log
resolution preserving the general points of the surviving lc centers,
and restrict its adjoint comparisons along the strict adjunction
chain described above. The generic point of each surviving center
lies outside the surgery: a zero-discrepancy place remains such a
place, whereas a center contained in the surgery would acquire
strictly larger discrepancy. Thus the strict transform \(S_W\) of
the normalized stratum is not contained in the comparison support.
For a zero-dimensional stratum this already excludes any intersection
with the surgery, so suppose its dimension is positive.
Its restriction is consequently a well-defined effective divisor.
Strict adjunction identifies its class with the difference of the
two pulled-back stratum adjoints. When the restricted diagram is an
isomorphism with the same boundary, that difference is zero.
An effective nonzero divisor on the compact Kähler resolution of
\(S_W\) has positive mass against a Kähler power, so the restricted
divisor is zero.
If an ambient step nevertheless met the stratum, take the first such
step. The full-support assertion following
Equation~\eqref{prog:comparison}, and surjectivity of \(S_W\) onto
the stratum, force its comparison support to meet \(S_W\).
Noncontainment makes this a nonzero effective restriction, and the
cumulative comparison dominates it. This contradicts the vanishing
just proved. Earlier steps are isomorphisms on a neighborhood of the
stratum by the choice of this first step. The ambient diagram is
therefore an isomorphism near the whole stratum. This verifies the
support conclusion used in the induction on proper subcenters;
for a higher-codimension stratum it is noncontainment in the support,
not merely being different from a divisor component, that is needed.
\smallskip\noindent\emph{Excluding the remaining walls.}
It remains to show that only finitely many of these small
diagrams are nontrivial. Write \(S\) for the first stratum
in this last tail and \(J=K_S+\Delta\) for its adjoint.
Let its restricted original scaling be
\(\{J\}+\lambda H\). Choose an interior parameter
\(\lambda_0>0\) on this working space, above the remaining
walls, and set
\[
A=\{J\}+\lambda_0H.
\]
It is Kähler, as the restriction of the interior ambient
Kähler class. The exact change of parameter is
\begin{equation}
\{J\}+tA=(1+t)(\{J\}+\lambda H),\qquad
\lambda=\frac{\lambda_0t}{1+t},\qquad
t=\frac{\lambda}{\lambda_0-\lambda}.
\label{prog:reparameterization}
\end{equation}
It applies for \(0\leq\lambda<\lambda_0\). The new walls
\(t_i\) strictly decrease toward zero, and the traces
\(\{J_i\}+tA_i\) are nef on their successive restricted
working intervals.
On a common resolution of a nontrivial small stratum step, let \(D_i\)
be the real exceptional divisor defined by
\[
p_i^*\{J_i\}-q_i^*\{J_{i+1}\}=\{D_i\}.
\]
The strict adjoint comparison makes \(D_i\) an effective
nonzero divisor exceptional over both strata. Equality of
the scaled pullbacks at its wall gives the exact
Bott--Chern equality
\begin{equation}
p_i^*(\{J_i\}+tA_i)-q_i^*(\{J_{i+1}\}+tA_{i+1})
=\left(1-\frac{t}{t_i}\right)\{D_i\}.
\label{prog:restricted-comparison}
\end{equation}
For \(t\) in a later working interval all preceding factors
on the right are nonnegative. Telescoping on a common
resolution expresses the pullback of \(\{J\}+tA\) as the
pullback of its nef working trace plus an effective divisor
exceptional over that working stratum.
Lemma~\ref{bir:exceptional} identifies the latter divisor
with the negative part.
Push the resulting positive currents to one fixed smooth
resolution of \(S\) and let \(t\downarrow0\). Closedness of
the pseudo-effective cone proves that \(J\) is
pseudo-effective. The negative multiplicities converge
valuation by valuation by the limit statement in
Lemma~\ref{prog:negative-along-scaling}. The comparison divisors
have no strict prime of \(S\), since the maps are small.
Their coefficients also vanish at every place whose center
meets the floor: each finite composition is unchanged on a
neighborhood of the floor by the induction on smaller
centers. Passing to the limit proves
Equation~\eqref{prog:away-floor}.
The stratum has dimension less than \(n\), and the divisor \(D_S\)
constructed above is rational Cartier with support equal to its floor.
Lemmas~\ref{prog:stratum-model} and
\ref{prog:nef-interval} give one small model \(V\) on
which all transformed classes of \(\{J\}+tA\) are nef
for \(0\leq t\leq\delta\). On a common smooth
resolution \(p:W\to S,\ q:W\to V\), define the real divisor \(F(t)\)
in the common exceptional span by
\begin{equation}
p^*(\{J\}+tA)-q^*(\{J_V\}+tA_V)=\{F(t)\}.
\label{prog:affine-comparison}
\end{equation}
Uniqueness of the representing exceptional divisor makes \(F(t)\)
coefficientwise affine in \(t\). It is effective on
\([0,\delta]\): \(-F(t)\) is \(p\)-nef because the
class pulled back from \(V\) is nef, so exceptional
negativity applies. It is exceptional over \(V\) as
well. Lemma~\ref{bir:exceptional} gives
\[
N\bigl(p^*(\{J\}+tA)\bigr)=F(t)
\qquad(0\leq t\leq\delta).
\]
Every divisorial negative multiplicity is thus affine on
this whole interval. This remains true on higher
resolutions by Lemma~\ref{bir:pullup}.
Choose a nontrivial stratum wall \(t_i\in(0,\delta)\).
On a common resolution of \(S,V\) and the adjacent
strata, their nef working models compute the same
negative part on their respective open intervals. Their
two affine expressions agree at \(t_i\). But
Equation~\eqref{prog:restricted-comparison} says that, at a prime
with coefficient \(d_i>0\) in \(D_i\), their difference
is
\[
\left(1-\frac{t}{t_i}\right)d_i.
\]
Their slopes differ by \(-d_i/t_i\). A single affine
function on \([0,\delta]\) cannot agree with both on
two open intervals. This contradiction excludes every
such wall. Infinitely many walls would tend to zero, so
the nontrivial diagrams on this stratum are finite.
There are finitely many surviving strata. Induction on
their dimensions makes all sufficiently late ambient
flipping and flipped loci disjoint from the floor. This
proves the theorem.
\end{proof}
\begin{corollary}[A signed adjoint supported on the floor]
\label{prog:signed-nef}
Assume \(\Ggood_d\) for every \(d<n\). In the situation of
Theorem~\ref{prog:special-termination}, suppose moreover
that \(L\) is rationally linearly equivalent, as an actual
rational line, to a signed rational divisor supported on
\(\lfloor B\rfloor\). The scaling can be chosen to reach
a nef ordinary dlt model after finitely many steps.
\end{corollary}
\begin{proof}
The signed representation persists under codimension-one
pushforward. A curve disjoint from the transformed floor
has degree zero for each component in this representation,
and hence for the transformed adjoint. Every negative
operation must therefore meet the floor. After the finite
prefix supplied by Theorem~\ref{prog:special-termination}
no further negative operation is possible. The
continuation in Lemma~\ref{prog:nef-trivial} makes the
last adjoint nef.
\end{proof}
\input{sections/03a-program-foundations}
\input{sections/03b-projective-good-model}
\input{sections/03c-restricted-induction}
==== END Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03-programs.tex ====
==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03a-program-foundations.tex ====
\subsection{Polarizations, ordinary replacements, and descent}
\label{prog:foundations}
The program constructions below use the analytic cone
\(\overline{\mathrm{NA}}(T)\), dual to the nef cone in
\(H^{1,1}_{\BC}(T,\R)\) for spaces with rational singularities
\cite[Proposition~2.4]{DHP}. Degrees of global line bundles form a
possibly smaller numerical space. We pass from the analytic cone to
projective geometry by producing an actual relatively ample line.
The ensuing descent statements apply on singular working models as
well as on smooth initial spaces.
\begin{lemma}[Degree-zero Bott--Chern descent]
\label{prog:bc-descent}
Let \(f:T\to Z\) be a proper surjective morphism with connected fibers
between normal compact complex spaces with rational singularities.
Suppose either
\begin{enumerate}
\item \(f\) is bimeromorphic and both spaces belong to Fujiki's class
\(\mathcal C\); or
\item \(f\) is projective and an effective rational boundary
\(\Delta\) makes \((T,\Delta)\) klt with
\(-(K_T+\Delta)\) relatively nef and big.
\end{enumerate}
Pullback is injective on both \(H^2(-,\R)\) and
\(H^{1,1}_{\BC}(-,\R)\). In either group its image consists exactly
of the classes with degree zero on every curve contracted by \(f\).
\end{lemma}
\begin{proof}
These are, respectively, the two cases of
\cite[Lemma~2.6(1),(2)]{DH2024}. The first case is birational descent
between spaces with rational singularities and does not require a
projectivity criterion. The second case uses relative vanishing for
the ordinary rational klt pair and the projective relative curve
space. In particular, it can be applied to a projective log-Fano
contraction before constructing generalized data on its base.
\end{proof}
In applications to an ordinary birational negative contraction,
the rationality hypotheses are automatic. The source is locally klt.
For a dlt pair, first decrease its rational Cartier floor slightly;
relative antiampleness persists. Relative vanishing gives
\(R^if_*\OO_T=0\) for \(i>0\)
\cite[Theorem~5.2]{FujinoAnalyticBCHM}. Composing a resolution of \(T\)
with \(f\), the Leray spectral sequence proves that \(Z\) has rational
singularities. The same reasoning works when the klt boundary is
chosen separately on finitely many Stein base neighborhoods. For
fiber-type contractions in case~(ii), rationality of the base follows
from \cite[Lemma~8.8(i)]{DHP}; thus that application also precedes
the construction of generalized data on the base.
\begin{lemma}[Projectivity from a rational line detecting the ray]
\label{prog:detected-projectivity}
Let $f:T\to Z$ be a proper contraction between normal compact
K\"ahler spaces, with $T$ having rational singularities. Suppose
$\gamma$ is K\"ahler on $Z$ and
\[
\alpha=f^*\gamma,\qquad
\overline{\mathrm{NA}}(T)\cap\alpha^\perp=R
\]
is a nonzero ray. If a global rational line bundle $D$ has
$D\cdot R<0$, then $-D$ is relatively ample and $f$ is projective.
\end{lemma}
\begin{proof}
Fix a K\"ahler class $\omega$ on $T$ and normalize $\overline{\mathrm{NA}}(T)$ by
$\omega\cdot z=1$. The resulting slice $S$ is compact. Its unique
point on $R$ has negative $D$-degree, so $d=c_1(D)$ is negative on
a neighborhood of that point in $S$. On the complementary compact
set, $\alpha$ has a strictly positive minimum, while $d$ is bounded.
For all sufficiently small $s>0$, the class $\alpha-sd$ is therefore
strictly positive on $S$ and is K\"ahler by cone duality.
Choose an integer $m>0$ making $mD$ integral. Then
\[
c_1(-mD)+f^*((m/s)\gamma)=(m/s)(\alpha-sd)
\]
is K\"ahler. On a neighborhood in the base with a potential for
$\gamma$, absorb that potential in a smooth metric on the same global
line bundle $-mD$. Its curvature is positive on the fibres. The analytic
positive-line-bundle criterion and the fibrewise relative-ampleness
criterion make $-mD$ relatively ample. The line bundle was global
throughout, so no gluing of unrelated local polarizations is required.
\end{proof}
An analytic contraction supplied by a supporting class therefore becomes
projective as soon as one rational line has negative degree on its ray.
For a small contraction, that same line will define the flip globally.
The local ordinary adjoints used to prove finite generation need not
glue; their section algebras will be Veronese subalgebras of one algebra.
\begin{lemma}[Divisor representatives over a Stein base]
\label{prog:stein-representatives}
Let $\pi:Y\to U$ be a projective surjective morphism with connected fibres
between normal irreducible complex spaces, where $U$ is Stein, and let
$\mathcal L$ be a holomorphic line bundle on $Y$.
If $\pi$ is bimeromorphic, $\mathcal L$ has a holomorphic section
which is not identically zero, and hence an effective Cartier divisor
representative. More generally, after shrinking $U$ around any specified
point, $\mathcal L$ has a nonzero meromorphic section.
\end{lemma}
\begin{proof}
Suppose first that $\pi$ is bimeromorphic. Grauert's proper direct-image
theorem makes $\mathcal F=\pi_*\mathcal L$ coherent. On the nonempty
open subset $U^\circ$ where $\pi$ is an isomorphism, it is a line bundle.
Choose $u_*\in U^\circ$. Cartan's theorem~A says that global sections
generate $\mathcal F_{u_*}$, so one global section has nonzero image in
$\mathcal F_{u_*}/\mathfrak m_{u_*}\mathcal F_{u_*}$.
Under
\[
H^0(U,\pi_*\mathcal L)=H^0(Y,\mathcal L)
\]
it gives a section $s$ nonzero at the unique point above $u_*$.
Its zero divisor is effective Cartier and represents $\mathcal L$.
The section may vanish elsewhere; only its meromorphic inverse is
being used to obtain a meromorphic trivialization.
For the general case, fix $u\in U$ and $y\in\pi^{-1}(u)$, and choose
a $\pi$-ample line bundle $\mathcal H$. Relative Serre generation,
after shrinking to a Stein neighbourhood of $u$, gives an integer
$n$ for which both $\mathcal L\otimes\mathcal H^n$ and
$\mathcal H^n$ are relatively generated. Their proper direct images
are coherent. Cartan's theorem~A and the evaluation maps supply
sections $s$ and $t$ of these two bundles which are both nonzero at
$y$. The quotient $s/t$ is the required nonzero meromorphic section
of $\mathcal L$. At no point is $Y$ assumed Stein.
\end{proof}
\begin{proposition}[A global algebra for a detected flip]
\label{prog:detected-flip}
Let $(T,B+\mathbf M)$ be a gklt pair on a normal compact K\"ahler
space which is globally strongly $\Q$-factorial, and let $A\in H^{1,1}_{\BC}(T,\R)$
be its adjoint class. Let $f:T\to Z$ be a projective small contraction
onto a normal compact K\"ahler space. Suppose that the classes of all
$f$-vertical curves lie on one ray $R\subset\overline{\mathrm{NA}}(T)$, that $A\cdot R<0$,
and that a global line bundle $L$ satisfies $L\cdot R<0$.
Then $\bigoplus_{m\geq0}f_*L^m$ is locally finitely generated. The
relative Proj of a sufficiently divisible Veronese is a projective
small contraction $f^+:T^+\to Z$, where $T^+$ is compact K\"ahler and
globally strongly $\Q$-factorial. The reflexive transform $L^+$ is a
rational line bundle, with a positive power equal to the relatively
ample tautological bundle. The transformed boundary and the same
b-nef datum define a gklt pair on $T^+$, with relatively K\"ahler
adjoint $A^+$. This is the generalized flip, with the usual
strict discrepancy increase at centres in either exceptional locus.
Moreover, if $t=(A\cdot C)/(L\cdot C)>0$ for an $f$-vertical curve $C$,
then there is a unique $\eta\in H^{1,1}_{\BC}(Z,\R)$ such that
\[
A=t\,c_1(L)+f^*\eta,\qquad
A^+=t\,c_1(L^+)+(f^+)^*\eta.
\]
Every class on $T$ has the corresponding forward transport with an
actual exceptional-divisor correction. No assertion of surjectivity
on Bott--Chern groups is required.
\end{proposition}
\begin{proof}
Since $f$ has a global relatively ample bundle and all its vertical
curves lie on $R$, numerical invariance of ampleness on the projective
fibres shows that $-L$ is $f$-ample.
\smallskip\noindent
\emph{A local rational ordinary adjoint.}
Fix $z\in Z$ and a relatively compact Stein neighbourhood $U$.
Take a projective log resolution $p:V\to T_U$ carrying $\mathbf M$,
and put $\rho=f p$. Write its actual structure boundary as $B_V$, so
\[
[K_V+B_V]+[\mathbf M_V]=p^*A.
\]
Every coefficient of $B_V$ is less than one. Apply
Lemma~\ref{prog:stein-representatives} to obtain a Cartier
representative $D_L$ of $L$ on $T_U$. The same birational direct-image
argument applies to the coherent rank-one reflexive canonical sheaf:
it supplies a canonical form nonzero at a chosen point of the common
smooth isomorphism locus. Use this meromorphic form on $T_U$ and its
pullback to $V$ to choose compatible canonical representatives, and set
\[
N=t p^*D_L-K_V-B_V.
\]
For every $\rho$-vertical curve, its degree equals that of $\mathbf M_V$.
Thus $N$ is $\rho$-nef.
The relative bigness of $N$ uses the additional fact that
$\rho=f\circ p$ is \emph{bimeromorphic}: both the small contraction
$f$ and the resolution $p$ are bimeromorphic. It does not follow from
relative nefness alone. In fact every real Cartier divisor is relatively
big for this projective bimeromorphic map over the Stein neighbourhood.
Here is the required ample-plus-effective decomposition explicitly.
Write $N=\sum_j r_jN_j$, where $r_j>0$, $\sum_jr_j=1$, and the $N_j$
are rational Cartier divisors in its finite Cartier span. Choose a
$\rho$-ample line bundle and a Cartier representative
$P$ for it using Lemma~\ref{prog:stein-representatives}.
If $q_jN_j$ is integral, apply the birational case of that lemma to
$\mathcal O_V(q_jN_j-P)$. Its nonzero holomorphic section has an
effective Cartier zero divisor $E_j$, giving the actual relation
$q_jN_j\sim P+E_j$. This use depends on $\rho$ being bimeromorphic;
the section is allowed to vanish along exceptional fibres. Therefore
\[
N\sim_{\R}
\underbrace{\left(\sum_j\frac{r_j}{q_j}\right)P}_{H}
+\underbrace{\sum_j\frac{r_j}{q_j}E_j}_{E}.
\]
Here $H$ is $\rho$-ample and $E\geq0$, proving $\rho$-bigness.
For a map with positive-dimensional general fibres the direct image
used above can be zero; no such bigness claim is made for that setting.
Make the finitely many section choices on a slightly larger Stein
neighbourhood, and now shrink $U$ so its closure is compact in that
neighbourhood. Properness makes the inverse image of this closure
compact. A log resolution of the finitely many resulting divisors
therefore gives a uniform positive bound for the following choice.
Choose $\epsilon>0$ small enough that $(V,B_V+\epsilon E)$ is sub-klt.
The divisor $(1-\epsilon)N+\epsilon H$ is relatively ample. Express it
as a positive combination of rational ample divisors and take general
members of sufficiently high multiples. Relative Bertini
\cite[Theorem~2.20]{DHP}, applied on a log resolution of $B_V+E$,
gives an effective $\Theta\sim_{\R}N$ such that $(V,B_V+\Theta)$ is
sub-klt. The high multiples make the added coefficients arbitrarily
small. All choices are made near the compact fibre; shrink $U$ once
to retain them.
Put $\Delta=p_*(B_V+\Theta)\geq0$. Write the actual linear equivalence
as $\Theta=N+\sum_j a_j\operatorname{div}_V(g_j)$.
The bimeromorphic map $p$ identifies meromorphic function fields, so
$g_j=p^*h_j$. Pushing down gives
$K_{T_U}+\Delta=tD_L+\sum_j a_j\operatorname{div}_{T_U}(h_j)$;
in particular this divisor is real Cartier. Pulling back this same
identity gives
\[
K_{T_U}+\Delta\sim_{\R}tD_L,
\qquad p^*(K_{T_U}+\Delta)=K_V+B_V+\Theta.
\]
Thus $(T_U,\Delta)$ is klt. Any finitely many base line bundles in a
relative linear equivalence can first be trivialized by shrinking $U$.
Record the resulting equality as
\begin{equation}\label{prog:detector-rationalization}
K_{T_U}+\sum_i d_iD_i=tD_L+\sum_j b_j\operatorname{div}(g_j),
\qquad d_i>0,
\end{equation}
using the positive support of $\Delta$. After a further relatively
compact shrinking, the displayed divisors have finitely many prime
components: their locally finite supports meet a compact inverse image.
Equality of coefficients in \eqref{prog:detector-rationalization} is a
finite rational affine system in $(d_i,t,b_j)$. The klt condition is
open within this system, as seen on one log resolution; the pullbacks
of its adjoints vary linearly by the displayed identity. A nearby
rational solution therefore gives a rational effective klt boundary
$\Delta_q$ and $t_q\in\Q_{>0}$ with
\begin{equation}\label{prog:detector-local-adjoint}
K_{T_U}+\Delta_q\sim_{\Q}t_qD_L.
\end{equation}
The individual local primes $D_i$ need not be $\Q$-Cartier: the identity
ensures that the total adjoint is $\Q$-Cartier.
\smallskip\noindent
\emph{One global flip algebra.}
The adjoint in \eqref{prog:detector-local-adjoint} is $f_U$-antiample.
Its ordinary flip and local finite generation follow from
\cite[Theorems~1.14 and~1.18]{FujinoAnalyticBCHM}. Clearing the displayed
linear equivalence identifies a Veronese of its canonical algebra
with a Veronese of $\mathcal R:=\bigoplus_{m\geq0}f_*L^m$ over $U$.
Hence $\mathcal R$ is locally finitely generated by
\cite[Lemma~2.26]{FujinoAnalyticBCHM}. The graded pieces are coherent. Over a connected normal base open
the inverse image is irreducible; products of nonzero sections of line
bundles are nonzero there. Thus its graded section algebra is an
integral domain, without choosing any global meromorphic frame of $L$.
Compactness supplies one sufficiently divisible Veronese
$\mathcal R^{(r)}$ generated in degree one. Its relative Proj $T^+$
restricts to the ordinary flip over every such $U$. It is therefore
normal, locally klt, and small over $Z$, and has the global relatively
ample bundle
$\mathcal O_{T^+}(1)$. On the common big open this bundle is $L^r$.
Reflexivity gives
\[
(L^+)^{[r]}\simeq\mathcal O_{T^+}(1).
\]
This proves that $L^+$ is a rational line bundle before any assertion
of factoriality. Relative positivity over the compact K\"ahler base
makes $T^+$ compact K\"ahler.
For any global rank-one reflexive sheaf $\mathcal F^+$ on $T^+$,
take its coherent reflexive transform $\mathcal F$ on $T$ using a
common resolution. Some $M=\mathcal F^{[m]}$ is a line bundle.
Choose $c\in\Q$ so that $(M+cL)\cdot C=0$, and clear its denominator
to obtain an integral bundle $J=n(M+cL)$ numerically trivial over $f$.
The local rational klt pairs \eqref{prog:detector-local-adjoint} have
antiample adjoint, so Lemma~\ref{prog:integral-descent} yields
one global line bundle $J_Z$ with $J=f^*J_Z$.
On $T^+$, pull back $J_Z$, undo the rational twist $ncL^+$, and compare
on the common big open. After clearing the already established index
of $L^+$, reflexivity gives an invertible positive reflexive power of
$\mathcal F^+$. This proves global strong factoriality.
\smallskip\noindent
\emph{The original generalized pair.}
The local log-Fano pairs above give $R^if_*\mathcal O_T=0$ for $i>0$
by \cite[Theorem~5.2]{FujinoAnalyticBCHM}. They make $T$ locally klt, hence rational.
Leray for a projective resolution then shows that $Z$ is rational.
Lemma~\ref{prog:bc-descent}(i) therefore gives
$f^*H^{1,1}_{\BC}(Z,\R)=R^\perp$, with injective pullback. Thus $A=t c_1(L)+f^*\eta$ for a unique $\eta$.
Define $A^+=t c_1(L^+)+(f^+)^*\eta$; it is relatively K\"ahler.
To construct its actual discrepancy divisor, let $\mathcal I$ be the
ideal defined by
\[
\operatorname{im}(f^*f_*L^r\longrightarrow L^r)
=\mathcal I\otimes L^r.
\]
Principalize $\mathcal I$ on a common resolution $p:W\to T$,
$q:W\to T^+$ carrying $\mathbf M$. If
$\mathcal I\mathcal O_W=\mathcal O_W(-F_L)$, the tautological quotient
gives the actual bundle identity
\[
q^*\mathcal O_{T^+}(1)=p^*L^r\otimes\mathcal O_W(-F_L).
\]
Hence $E_L=F_L/r$ is effective and exceptional over both sides, and
$[E_L]=p^*c_1(L)-q^*c_1(L^+)$. If $B_W$ is the original structure
boundary, put $B_W^+=B_W-tE_L$. Then
\[
[K_W+B_W^+]+[\mathbf M_W]=q^*A^+.
\]
Its pushforward is $B^+$, and its discrepancies do not decrease.
The local ordinary comparison in \eqref{prog:detector-local-adjoint}
has discrepancy divisor $t_qE_L$, so its strictness proves the stated
strictness for $tE_L$ as well. Thus the original pair remains gklt.
Finally, write any $\theta\in H^{1,1}_{\BC}(T,\R)$ uniquely as
$\theta=f^*\eta_\theta+s c_1(L)$ and set
$\theta^+=(f^+)^*\eta_\theta+s c_1(L^+)$. Its correction is the actual
divisor $sE_L$. For a rational line $M$, the number $s$ is rational. Applying
Lemma~\ref{prog:integral-descent} to a Cartier multiple of $M-sL$
gives $M=sL+f^*M_Z$ as actual rational lines. Reflexive extension
on the common big open gives $M^+=sL^++(f^+)^*M_Z$.
Consequently the cohomological transport agrees with the reflexive
transform on Chern classes.
\end{proof}
\subsubsection{Real boundaries and their linear transport}
The rational ordinary step constructed below also determines the step
for every sufficiently close real boundary. The relevant comparison is
an identity of actual rational lines over the contraction base.
\begin{proposition}[Ordinary real boundaries]
\label{prog:ordinary-replacement}
Let $T$ be normal compact K\"ahler and globally Weil $\Q$-factorial,
with canonical sheaf a rational line bundle. Let $B\geq0$ be a real
boundary with $(T,B)$ klt. Assume the rational ordinary-step result
of Proposition~\ref{prog:ordinary-step} for $T$ and its subsequent models. Then every
$(K_T+B)$-negative extremal analytic ray has an ordinary step with
projective contractions, compact K\"ahler models, and the expected
opposite relative ample signs. Global Weil $\Q$-factoriality is preserved;
global strong $\Q$-factoriality is preserved when imposed initially.
\end{proposition}
\begin{proof}
Put $D=K_T+B$ and fix a $D$-negative extremal ray $R$. On the finite
positive support of $B$, effectiveness, the klt condition and negativity
on $R$ persist under small coefficient changes. Klt openness is checked
on one log resolution of this support. Choose nearby effective rational
klt boundaries $B_0,\ldots,B_m$ and positive real numbers $u_j$ with
\[
B=\sum_j u_jB_j,\qquad \sum_j u_j=1,
\qquad D_j\cdot R<0,\quad D_j=K_T+B_j.
\]
For $B=0$ take just $B_0=0$. Every $D_j$ is a global rational line
bundle. The rational-step theorem for $(T,B_0)$ provides a projective
contraction $f:T\to Z$ onto a compact K\"ahler base, contracting
exactly $R$, with $-D_0$ relatively ample. The base is rational by
\cite[Lemma~8.8]{DHP}, so Lemma~\ref{prog:bc-descent} gives
$f^*H^{1,1}_{\BC}(Z,\R)=R^\perp$; use case~(ii) when $f$ is of
fiber type.
Let $C$ be a rational curve spanning $R$, and set
$a_j=(D_j\cdot C)/(D_0\cdot C)\in\Q_{>0}$. On every projective
fibre, $-D_j$ is numerically equivalent to the positive multiple
$-a_jD_0$. Numerical invariance of ampleness on that fibre, followed
by the relative-ampleness criterion, makes $-D_j$ relatively ample.
The positive combination $-D$ is relatively ample as a real line
bundle. This also handles a fibre-type contraction.
Suppose henceforth that $f$ is birational. A Cartier multiple
$M_j=n_j(D_j-a_jD_0)$ is numerically trivial over $Z$ and
$M_j-D_0$ is relatively ample. Lemma~\ref{prog:integral-descent},
applied to the rational klt pair $(T,B_0)$ gives a rational line bundle
$A_j$ on $Z$ with the actual identity
\begin{equation}\label{prog:real-line-descent}
D_j=a_jD_0+f^*A_j.
\end{equation}
Here and below an identity of rational line bundles means an isomorphism
after a common integral multiple. The descent lemma is used only for
birational $f$.
For a small $f$, let $f^+:T^+\to Z$ be its rational $D_0$-flip.
The rational theorem supplies a global relatively ample adjoint
$D_0^+$ and preserves the stated factoriality and canonical-sheaf
conditions. The other transformed adjoints $D_j^+$ are therefore
rational line bundles. Transform \eqref{prog:real-line-descent} on the common
big open and extend by reflexivity to obtain
\[
D_j^+=a_jD_0^++(f^+)^*A_j.
\]
All $D_j^+$ are relatively ample, as is
$D^+=\sum_j u_jD_j^+$. Thus this same small map is the required
ordinary real-boundary flip. The real ordinary discrepancy comparison
proves that $(T^+,B^+)$ is klt and gives the strict increases at
exceptional centres.
For a divisorial contraction, write
$D_0=f^*D_0'+e_0E$ with $e_0>0$. Transforming
\eqref{prog:real-line-descent} on the target and summing yields
\[
D-f^*D'=\left(\sum_j u_ja_j\right)e_0E.
\]
This coefficient is positive. The same discrepancy comparison proves
the required assertions. The ambient category was already preserved
by the rational step in both cases.
\end{proof}
For later use, let \(D=K_T+B\) drive one of the birational steps just
constructed, and write \(f':T'\to Z\) for its positive morphism
(or the identity for a divisorial contraction). Lemma~\ref{prog:bc-descent}
gives the unique decomposition and forward transform
\begin{equation}
\theta=f^*\eta+s\{D\},\qquad
\theta'=(f')^*\eta+s\{D'\}.
\label{prog:ordinary-forward-transport}
\end{equation}
On a common resolution \(p:W\to T\), \(q:W\to T'\), the ordinary
comparison \(p^*D-q^*D'=F\) is an actual effective real divisor,
exceptional over \(T'\) and over both sides for a flip. Hence
\(p^*\theta-q^*\theta'=s\{F\}\). On Chern classes this is the
reflexive transform, by the line identities in the proof above and
Lemma~\ref{prog:integral-descent}. This construction gives forward
transport without assuming surjectivity onto the next Bott--Chern group.
\subsubsection{Actual real lines and relative numerical spaces}
A second polarization argument will be used for maps whose fibers are
Moishezon. It applies to a real combination of global line bundles,
provided that its degrees on the fibers agree with a relatively Kähler
class.
\begin{lemma}[Polarization by an actual real line bundle]
\label{prog:actual-line-polarization}
Let $f:T\to Z$ be a proper Moishezon map, where $T$ is a compact
K\"ahler space. Suppose that $L\in\operatorname{Pic}(T)\otimes\R$
and a relatively K\"ahler class $\omega$ satisfy
\[
L\cdot C=\omega\cdot C
\quad\text{for every compact curve $C$ contracted by $f$.}
\]
Then $L$ is relatively ample and $f$ is projective.
\end{lemma}
\begin{proof}
Let $V$ be an irreducible positive-dimensional subspace of a reduced
fibre. It is Moishezon. Normalize $V$ and take a smooth projective
modification $\pi:\widetilde V\to V^{\mathrm n}$, chosen so that
an effective exceptional divisor $E$ has $-E$ relatively ample.
The pullback of the K\"ahler class from $V$ to its normalization is
K\"ahler. Consequently
$\pi^*\omega-\delta[E]$ is K\"ahler for sufficiently small
$\delta>0$. On the projective manifold $\widetilde V$ the actual
real line bundle $\pi^*L-\delta E$ has the same curve degrees as
that K\"ahler class, and is therefore ample. For example, subtract
a sufficiently small positive multiple of a fixed ample class from
the K\"ahler class; the corresponding real line bundle is nef by
the projective numerical criterion. Also $\pi^*L$ is nef, and the
decomposition
\[
\pi^*L=(\pi^*L-\delta E)+\delta E
\]
shows that it is big. Thus
\[
L^{\dim V}\cdot V=(\pi^*L)^{\dim V}>0.
\]
The real Nakai--Moishezon criterion for proper algebraic spaces
\cite[Theorem~1.6]{FujinoMiyamoto2020}, applied to the algebraizations
of the Moishezon fibres, proves that $L$ is ample on every reduced
fibre. The criterion is insensitive to nilpotents and reducible
components. In the coefficient space of the finitely many global
line bundles occurring in $L$, choose, for each fibre, a rational
simplex containing $L$ in its interior whose vertices restrict to
ample rational lines on that fibre. Openness of fibrewise ampleness
for each of these finitely many rational lines gives a base
neighbourhood on which the entire simplex is relatively ample.
Choose finitely many such neighborhoods covering the base and
intersect their coefficient neighborhoods of $L$. Every rational
point of the resulting neighborhood is relatively ample globally.
A rational simplex in this intersection also expresses $L$ as a
positive combination of relatively ample rational lines. Clearing
the denominator of one rational point proves projectivity.
\end{proof}
\begin{remark}
The proof uses only positivity of the top self-intersections of
$L$. Equality of curve degrees with $\omega$ does not assert
equality of their higher intersection numbers or of their
Bott--Chern classes.
\end{remark}
The projective morphism needed for the relative program can be chosen
on a smooth model of the rational quotient. Here the Hodge-theoretic
polarization has an elementary construction, independent of any
birational projectivity-descent assertion.
\begin{lemma}[Smooth Hodge-theoretic polarization]
\label{prog:smooth-polarization}
Let \(g:W\to S\) be a surjective morphism with connected fibers
between smooth compact Kähler manifolds. If
\(g^*:H^0(S,\Omega_S^2)\to H^0(W,\Omega_W^2)\) is an isomorphism,
then \(g\) is projective. This applies when its general fiber is
rationally connected.
\end{lemma}
\begin{proof}
We use the smooth argument of \cite[Theorem~3.1, Step~1]{CH}.
Let \(d=\dim W-\dim S\). Choose a rational class \(u\in H^2(W,\Q)\)
close to a Kähler class. Write its Hodge decomposition as
\(u=b+g^*v\), where \(b\) is Kähler and \(v\) has types \((2,0)\)
and \((0,2)\). Hodge type and the projection formula give
\[
c=g_*(u^d)=g_*(b^d)>0,\qquad
g_*(u^{d+1})=g_*(b^{d+1})+(d+1)c\,v.
\]
Since pushforward is rational, the class
\[
\ell=u-g^*\frac{g_*(u^{d+1})}{(d+1)c}
=b-g^*\frac{g_*(b^{d+1})}{(d+1)c}
\]
is rational and of type \((1,1)\). Lefschetz's theorem makes it the
Chern class of a rational line. The displayed equality supplies a
metric with positive curvature locally over \(S\), by adding a
potential for the base class. Thus this line is relatively ample.
For rationally connected general fibers, holomorphic two-forms are
pulled back from the smooth base, so the hypothesis holds.
\end{proof}
\begin{lemma}[A rational Hodge correction over a smooth base]
\label{prog:relative-hodge}
Let $f:T\to S$ be an already projective surjective morphism of normal
compact K\"ahler spaces. Assume that $S$ is smooth, $T$ has globally
strongly $\Q$-factorial klt singularities, and, for a projective resolution
$r:W\to T$, the morphism $g=f\circ r$ satisfies
\[
g^*:H^0(S,\Omega_S^2)\xrightarrow{\ \simeq\ }
H^0(W,\Omega_W^2).
\]
Then every $\alpha\in H^{1,1}_{\BC}(T,\R)$ can be written
\[
\alpha=c_1(L)+f^*\gamma,
\]
where $L\in\operatorname{Pic}(T)\otimes_{\mathbb Z}\mathbb R$ is a
finite real combination of global line bundles and
$\gamma\in H^{1,1}(S,\mathbb R)$.
\end{lemma}
\begin{proof}
The composition $g$ is projective, since all the spaces are compact.
Choose a $g$-ample line bundle on $W$, with integral Chern class $H$.
Set $d=\dim W-\dim S$ and
\[
c=g_*(H^d)>0.
\]
Thus $c$ is the positive degree of $H^d$ on a general fibre. The
cohomological pushforward here is the usual pushforward between the
smooth compact manifolds $W$ and $S$, defined over $\mathbb Q$ and of
Hodge bidegree $(-d,-d)$.
Define a rational linear operator on degree-two cohomology by
\[
\Pi(\eta)=\eta-g^*\!\left(\frac{g_*(\eta\smile H^d)}{c}\right).
\]
Every class of type $(2,0)$ on $W$ is pulled back from $S$, and the
same holds for type $(0,2)$. The projection formula therefore shows
that $\Pi$ kills both types. It preserves type $(1,1)$. Consequently
\[
\Pi(H^2(W,\mathbb Q))\subset
H^2(W,\mathbb Q)\cap H^{1,1}(W).
\]
By the Lefschetz $(1,1)$ theorem the image consists of rational Chern
classes of global line bundles. Applying $\Pi$ to $r^*\alpha$ gives
\[
r^*\alpha=c_1(L_W)+g^*\gamma,
\qquad L_W\in\operatorname{Pic}(W)\otimes\mathbb R,
\qquad
\gamma=\frac{g_*(r^*\alpha\smile H^d)}{c}\in H^{1,1}(S,\mathbb R).
\]
For example, express $r^*\alpha$ in a rational basis of $H^2(W,\mathbb Q)$
and apply $\Pi$ to each basis vector. This gives the required finite
real combination $L_W$.
Write $L_W=\sum_j a_jL_j$ with actual line bundles $L_j$ on $W$.
For each $j$, let
\[
\mathcal F_j=(r_*L_j)^{**}.
\]
It is a rank-one reflexive coherent sheaf. Strong
$\mathbb Q$-factoriality gives an integer $m_j>0$ for which
$M_j=\mathcal F_j^{[m_j]}$ is a line bundle on $T$.
The canonical identification over the isomorphism locus of $r$ gives
\[
L_j^{\otimes m_j}\simeq r^*M_j\otimes\mathcal O_W(E_j)
\]
for an integral $r$-exceptional divisor $E_j$.
One may obtain this comparison directly from the evaluation map for
$r_*L_j$: the two coherent rank-one sheaves agree away from the
exceptional locus, so their invertible transforms differ by an
exceptional divisor. No global meromorphic frame of $L_j$ or of the
canonical sheaf is required.
Set $L=\sum_j(a_j/m_j)M_j$ and
$E=\sum_j(a_j/m_j)E_j$. The preceding identities yield
\[
r^*(\alpha-c_1(L)-f^*\gamma)=[E].
\]
The divisor $E$ is $r$-exceptional and is numerically trivial on every
$r$-contracted curve. The exceptional negativity lemma applied in both
signs gives $E=0$. Injectivity of pullback by a resolution, or pushforward
of the equality of currents, now gives
$\alpha=c_1(L)+f^*\gamma$.
\end{proof}
The hypothesis of Lemma~\ref{prog:relative-hodge} persists on each
projective globally strongly \(\Q\)-factorial klt model over this fixed
smooth base: on a common smooth resolution, holomorphic two-forms are
invariant under modifications. Thus the lemma applies separately on
every working model. It supplies real combinations of global lines;
Lemma~\ref{prog:stein-representatives} supplies their Cartier-divisor
representatives on sufficiently small Stein base neighborhoods.
\begin{lemma}[The projective relative cone in Bott--Chern cohomology]
\label{prog:projective-cone}
Let \(f:T\to S\) be projective between normal compact Kähler spaces
with rational singularities, and assume
\[
H^{1,1}_{\BC}(T,\R)
=c_1\bigl(\Pic(T)\otimes\R\bigr)
+f^*H^{1,1}_{\BC}(S,\R).
\]
Let \(N_1^{\mathrm{line}}(T/S)\) denote the space of real combinations
of \(f\)-contracted curves modulo degrees of global line bundles,
and let \(\overline{\mathrm{NE}}^{\mathrm{line}}(T/S)\) be their
closed effective cone. Their analytic classes give an isomorphism
onto the span of the face
\[
\mathcal F_f=
\overline{\mathrm{NA}}(T)\cap(f^*\omega_S)^\perp,
\]
where \(\omega_S\) is any Kähler class on \(S\); this isomorphism
identifies \(\overline{\mathrm{NE}}^{\mathrm{line}}(T/S)\) with
\(\mathcal F_f\). In particular, relative extremal rays in the line
space are extremal rays of the full analytic cone.
\end{lemma}
\begin{proof}
The assumed decomposition makes analytic equivalence and line-degree
equivalence identical on combinations of vertical curves. Their map
into the analytic numerical space is therefore well-defined and
injective. We must show that its closed effective cone is the entire
face, including its classes represented by positive currents.
Fix \(z\in\mathcal F_f\). For every \(\gamma\in
H^{1,1}_{\BC}(S,\R)\), both \(a\omega_S+\gamma\) and
\(a\omega_S-\gamma\) are Kähler for large \(a\). Positivity then
shows that \(z\) annihilates \(f^*\gamma\). Fix a global relatively
ample line \(H\). If a real line \(D\) has nonnegative degree on
every vertical curve, the projective relative numerical criterion
makes \(D+\epsilon H\) relatively ample for every \(\epsilon>0\).
Its Chern class becomes Kähler after adding a sufficiently large
multiple of \(f^*\omega_S\). Therefore
\[
(c_1(D)+\epsilon c_1(H))\cdot z\geq0,
\qquad c_1(D)\cdot z\geq0.
\]
Applying this to both signs of a relatively numerically trivial line
shows that pairing with \(z\) factors through the finite-dimensional
space of relative line degrees. It is nonnegative on its nef cone.
By duality for this projective relative space, it is represented by
an element of \(\overline{\mathrm{NE}}^{\mathrm{line}}(T/S)\).
The decomposition in the statement makes its analytic image equal
to \(z\). Conversely every vertical effective curve lies in
\(\mathcal F_f\), and finite-dimensional injectivity preserves
closedness. This proves the cone equality and the extremality claim.
\end{proof}
==== END Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03a-program-foundations.tex ====
==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03b-projective-good-model.tex ====
\subsection{Good models for already projective relative adjoints}
\begin{proposition}[Projective relative completion]
\label{prog:projective-good-model}
Let \(f:X\to S\) be an already projective surjective morphism of
normal compact Kähler spaces, with \(X\) smooth. Let
\((X,B+\mathbf M)\) be an effective generalized klt pair with adjoint
\(A\), carried by a projective resolution \(p:W\to X\) with smooth
\(W\), so that
\[
p^*A=c_1(K_W)+[B_W]+\mathbf M_W.
\]
Suppose that there are an actual real line bundle
\(J\in\Pic(X)\otimes\R\) and
\(\beta\in H^{1,1}_{\BC}(S)\) such that
\[
A=c_1(J)+f^*\beta,
\qquad
N:=p^*J-K_W-B_W
\text{ is nef and big over }fp.
\]
Assume that \(J\) is pseudoeffective over \(S\).
Then there is a finite chosen \(A\)-negative projective program over
\(S\), with globally strongly \(\Q\)-factorial compact Kähler
working spaces, and a nonextracting endpoint \(X_m\) such that
\[
J_m\sim_{\R}g^*H,
\qquad
A_m=g^*\bigl(c_1(H)+\rho^*\beta\bigr).
\]
Here \(J_m\) is the actual real-line transform of \(J\),
\(g:X_m\to Z\) is a projective connected-fibre morphism,
\(\rho:Z\to S\) is projective, and \(H\) is an actual real line
bundle ample over \(S\). Thus the class on \(Z\) in this formula is
relatively Kähler over \(S\). The generalized pair on \(X_m\) is
gklt. On a common resolution the comparison with \(A\) is effective
and exceptional over \(X_m\), with strictly positive coefficient
at the strict transform of every prime contracted from \(X\).
\end{proposition}
\begin{proof}
We reduce to fixed ordinary pairs, and construct one global program.
Choose finitely many Stein open sets \(U_\ell\subset S\) and
semianalytic Stein compact subsets \(W_\ell\Subset U_\ell\) whose
interiors cover \(S\). Such compacts are obtained by intersections
with closed polydiscs in local embeddings of \(S\). In particular
they satisfy condition~(P) of \cite{FujinoAnalyticBCHM}: the source
is normal, the base is Stein, the compact is Stein, and its
intersection with any analytic subset defined near it has finitely
many connected components. This remains true on every normal
projective model. Shrinking \(U_\ell\) around \(W_\ell\) preserves
the finite cover.
\smallskip\noindent
\emph{Actual ordinary replacements.}
On each chart represent the finitely many line bundles in the data
by Cartier divisors. For a projective map to a Stein space this can
be done by twisting a line bundle and its prospective meromorphic
frame by sufficiently positive relative lines and using nonzero
sections. The generic-rank-one direct-image argument is an
alternative only when the map is bimeromorphic.
Relative bigness gives
\(N\sim_{\R}P+E\), with \(P\) relatively ample and \(E\geq0\).
For sufficiently small \(\epsilon>0\),
\((W,B_W+\epsilon E)\) is sub-klt near the compact inverse image.
The line \((1-\epsilon)N+\epsilon P\) is relatively ample.
General divided relative sections give an effective representative
\(T\) such that
\[
\Theta:=\epsilon E+T\sim_{\R}N,
\qquad (W,B_W+\Theta)\text{ is sub-klt}.
\]
All supports and coefficient margins are fixed near that compact.
With compatible canonical representatives, push the finite
principal-divisor expression for \(\Theta-N\) down and pull it
back again. For \(\Delta_\ell=p_*(B_W+\Theta)\) this gives
\[
\Delta_\ell\geq0,
\qquad K_X+\Delta_\ell\sim_{\R}J|_{X_{U_\ell}},
\qquad
p^*(K_X+\Delta_\ell)=K_W+B_W+\Theta.
\]
In the last equality the chosen principal correction is included
in the representative of \(K_X+\Delta_\ell\).
Thus \((X_{U_\ell},\Delta_\ell)\) is ordinary klt and its boundary
is relatively big. Lemma~11.15 of \cite{FujinoAnalyticBCHM}, with
its big part equal to \(\Delta_\ell\) and its remaining part zero,
replaces it, in its actual real linear equivalence class, by a klt
boundary with an effective general relatively ample rational
summand. Its condition on non-klt centres is vacuous.
Choose the reserved ample summand to be \(\Q\)-linearly equivalent
to a small rational multiple of a fixed global \(f\)-ample line.
Its later strict transform is consequently \(\Q\)-Cartier by the
transport of that global line, without a local factoriality claim.
We make these choices for a finite rational polytope of actual
global line data. Start with a finite rational span containing
\(J\), and add relatively ample rational lines spanning the global
relative degree space. The above construction is valid in a
neighbourhood of \(J\): retain a small part of the ample summand
and use openness of ampleness and the strict klt inequalities.
It is also valid along a segment from \(J\) to a sufficiently
positive adjoint, by adding divided general ample divisors.
Use finitely many such representatives and one log resolution on
each chart. More explicitly, represent a finite simplex inside the
open ample neighbourhood used above; varying its positive
coefficients represents all nearby global line parameters on the
same finite support. Form the joint finite affine system consisting
of the divisor identities on all charts, with common global line
coordinates and separate boundary and principal-divisor
coordinates. This system is rational. Allow all boundary
coefficients to vary, including those originally contributed by
the real boundary \(B\); fix only the reserved rational ample part
and the zero-support constraints. Its projection to the global
line coordinates contains the just constructed neighbourhood, so
it has a rational affine section. Choose that section sufficiently
close to the constructed real affine lift, by rational approximation
in its affine space of sections, so that the strict klt and
effectiveness margins are retained along the compact segment.
In the joint solution space the
strict klt and effectiveness margins permit a rational polytope
containing the scaling segment. This supplies on every chart
a rational polytope of ordinary klt boundaries with a fixed
effective relatively ample rational summand, representing the
restrictions of the same global line parameters. These boundaries,
including the scaling representatives, are fixed before the
program starts. Their divisors need not agree on different charts.
\smallskip\noindent
\emph{One global scaling construction.}
Apply the global degree-space construction in the proof of
Proposition~\ref{prog:relative-klt} to \(J\) and a general ample
scaling direction in the chosen span. Here are the modifications
that allow real line data and a nonbirational map to \(S\).
The global relatively ample line still gives a compact normalized
slice of the curve cone. The fixed ordinary chart pairs give finite
negative-ray truncations. A general direction avoids equal wall
parameters for independent ray-degree functionals, exactly as in
that proof. A positive wall therefore selects a single ray in the
global degree space. A rational nef support is represented by a
global rational line; a large multiple minus the chartwise ordinary
adjoint is relatively ample. Ordinary relative base point freeness
contracts the ray using the evaluation of this one global line.
If the contraction is small, choose a global rational line having
negative ray degree and construct its flip by its one global
graded algebra. The local ordinary proof of
Proposition~\ref{prog:detected-flip} applies to this algebra:
all curves of the contraction have proportional degrees for global
lines, so the detector and the driving line are positively
proportional over the contraction; ordinary rational replacement
gives local finite generation. This part of that proof needs only
the already projective contraction and these line degrees.
It does not require a prior assertion about the full Bott--Chern
cone. The relative Proj is the next global model. In a divisorial
step the exceptional prime and
Lemma~\ref{prog:integral-descent} give the same continuation as in
Proposition~\ref{prog:relative-klt}. Killing the degree of any global
line by a rational multiple of the detector, then applying that
lemma, preserves global strong \(\Q\)-factoriality in both cases.
The global lines continue to span the relative degree spaces.
The rational support-simplex argument in that proposition makes
the wall a descended relatively ample real line on its contraction
base and gives a nonempty interval of ampleness immediately after
each step. This argument uses relative line data and projectivity,
not birationality of the map to \(S\).
We check explicitly the persistence of the ordinary pairs. Expand
the line parameters in finitely many global rational lines. Their
reflexive transforms are actual rational lines by the preceding
strong property. Pushing forward the fixed principal-divisor
identities gives
\[
K_{X_i}+\Delta_{\ell,i}\sim_{\R}J_i|_{(X_i)_{U_\ell}}.
\]
Hence these ordinary adjoints are \(\R\)-Cartier; no local
factoriality of individual prime divisors is being assumed.
On a common resolution their comparison minus the appropriate
positive multiple of the detector comparison is exceptional and
relatively numerically trivial. Negativity in both signs makes
this difference zero. Every chosen step is consequently negative
for the fixed ordinary pairs. They remain effective and klt, and
their boundaries remain relatively big. The same equality of
comparison divisors, after adding the fixed base pullback,
preserves the generalized pair and its discrepancy inequalities.
For termination, every already constructed working model is a weak
log canonical model on each chart of a boundary in the fixed
polytope: choose a parameter in its nonempty ample scaling interval.
Indeed, actual descent at a preceding wall \(\lambda_j\) makes
the comparison for \(J+tH\) equal to
\((1-t/\lambda_j)\) times its \(J\)-comparison. For a parameter
in the current interval all preceding \(\lambda_j\) are at least
\(t\), so these comparisons are effective.
Theorem~E of \cite{FujinoAnalyticBCHM} gives finitely many marked
models on each chart. That theorem does not require locally
\(\Q\)-factorial targets. There are therefore finitely many possible
global marked working models. Indeed, isomorphisms between the
restrictions of two existing marked models agree on their common
dense open, and hence on overlaps. A repeated marked working model
would have the same boundary transform and discrepancies, contrary
to strict discrepancy increase at an intervening nontrivial step.
The global program is finite. Relative pseudoeffectivity excludes
a Mori fibre endpoint, so \(J_m\) is nef over \(S\).
This is a finite-cover argument for a single constructed program;
it does not glue independent minimal models.
\smallskip\noindent
\emph{Semiampleness and the global polarization.}
On the endpoint the fixed ordinary pair is klt, its effective
boundary is relatively big, and its adjoint is nef over
\(W_\ell\). Lemma~11.16 of \cite{FujinoAnalyticBCHM} applies to
this already existing weak model. More explicitly, Lemma~11.15
replaces the endpoint boundary by \(A_\ell+B_\ell\), where
\(A_\ell\geq0\) is relatively ample, \(B_\ell\geq0\), and the
pair is klt. Theorem~8.3 then gives semiampleness near \(W_\ell\):
its log canonical, klt-existence, real Cartier, ample-summand and
nef-over-compact hypotheses have all been checked.
The actual equivalences identify these local semiample adjoints
with restrictions of \(J_m\). Their ample models agree on overlaps
by \cite[Lemma~11.3]{FujinoAnalyticBCHM}. To retain a global
polarization, shrink the initial parameter polytope around \(J\)
after the finite program: all its finitely many strict step signs
persist, so the transformed ordinary boundaries remain klt and big.
On each chart the pushforward of its initial common ample rational
summand is effective and relatively big. Apply
\cite[Lemma~11.13]{FujinoAnalyticBCHM} to this common big rational
part and the small endpoint boundary polytope. All pairs are klt,
so the condition on non-klt centres is vacuous. The lemma gives a
common general relatively ample rational summand by one
\(\Q\)-linearly trivial translation. In particular the rational
parametrization is preserved; separate applications of
Lemma~11.15 to individual parameters would not suffice here.
In this finite rational span Theorem~11.17 of \cite{FujinoAnalyticBCHM}
makes the endpoint nef region a rational polytope. Intersect the
finitely many inverse images of these regions and express \(J_m\)
as a positive combination of rational points in its minimal face.
The corresponding global rational lines are semiample on every
chart by the same ordinary base point free argument. After clearing
denominators and taking common sufficiently divisible powers over
the finite cover, each is relatively generated globally: generation on
the charts is generation of its global coherent relative
evaluation map. The product of these finitely many relative
morphisms, followed by Stein factorization, gives a projective
\(g:X_m\to Z\), a projective \(\rho:Z\to S\), and descended
global lines whose positive combination \(H\) is relatively ample.
It realizes the glued ample model and gives
\(J_m\sim_{\R}g^*H\). Adding back \(f_m^*\beta\) proves the
asserted Bott--Chern identity. Composing the effective step
comparisons proves the final discrepancy statement.
\end{proof}
\begin{corollary}[Semiampleness on an existing projective model]
\label{prog:projective-semiample}
Let \(f:T\to S\) be a projective morphism of normal compact Kähler
spaces with \(T\) globally strongly \(\Q\)-factorial, and let
\(J\in\Pic(T)\otimes\R\) be nef over \(S\). Suppose that on a
finite Stein-compact cover satisfying condition~(P), there are
effective ordinary klt boundaries \(\Delta_\ell\), big over the
base, such that
\(K_T+\Delta_\ell\sim_{\R}J|_{T_{U_\ell}}\).
Then \(J\) has the projective relative semiample contraction and
actual descended ample real line of
Proposition~\ref{prog:projective-good-model}.
In particular this applies to any already existing weak model of
the fixed ordinary chart pairs in that proof; one need not run a
new program for the adjoint in question.
\end{corollary}
\begin{proof}
Lemma~11.15 and Theorem~8.3 of \cite{FujinoAnalyticBCHM} give
local semiampleness. For an existing weak model this is also
Lemma~11.16. To globalize the polarization, take a finite rational
span of the global lines defining \(J\). On each chart the ample
part of the normalized boundary allows small perturbations in
this span. The joint rational affine construction in the preceding
proof gives a rational family with a common rational ample part;
all boundary coefficients are allowed to vary. Theorem~11.17 then
gives a rational nef region on each chart. The finite-intersection,
rational-vertex and global-evaluation argument in that proof gives
one projective contraction and its actual descended ample real
line. Only its local ample models are identified by uniqueness.
\end{proof}
\begin{remark}[The two uses of projective completion]
\label{prog:projective-good-model-uses}
For a selective extraction over an already constructed weak model,
the carrier map is projective bimeromorphic. The inherited
exceptional splitting expresses the nef datum, modulo this base,
by actual real lines. Relative nefness follows from the carrier
datum, while relative bigness follows from birationality, using the
ample-plus-effective construction in the ordinary replacement.
The prepared adjoint is a base pullback plus an effective
exceptional divisor supported on exactly the unwanted primes.
Proposition~\ref{prog:projective-good-model} applies. On its endpoint
the remaining error is effective, exceptional over the fixed base,
and relatively nef; negativity makes it zero. Strict discrepancy
improvement prevents contraction of a requested zero-error prime.
The chosen extraction boundary must be effective and klt; in
particular its prescribed primes have the usual admissible
discrepancy range.
For the graph construction, suppose the already projective graph
resolution \(h:V\to Y_m\) has smooth source and
\[
A_V=h^*A_{Y_m}+[G],\qquad G\geq0.
\]
Then the global relative real line is \(J=\OO_V(G)\), the base
class is \(A_{Y_m}\), and its carrier line is
\(N=G-K_V-B_V\). Its relative degrees equal those of the prepared
nef datum. That datum is nef and big on the prepared carrier, for
example a pullback of its prepared Kähler class; thus \(N\) is nef
and big over \(Y_m\). This bigness is a separate hypothesis check,
since \(h\) can have positive-dimensional general fibres.
Effectivity of \(G\) gives relative pseudoeffectivity. Apply the
proposition over \(Y_m\). The subsequent negativity argument must
still remove \(G_m\); only then is the endpoint adjoint the pullback
of \(A_{Y_m}\) and good over the original base.
For transport of all classes, verify separately that all
Bott--Chern classes on the graph carrier are real-line classes
modulo \(Y_m\), using the smooth rationally connected fibration
and the inherited exceptional splitting on the lower-dimensional
base model. Lemma~\ref{prog:projective-cone} then identifies
the relative degree rays with full Bott--Chern rays, so the
detected-step transport applies. The driving identity involving
\(G\) alone does not give this additional property.
\end{remark}
\begin{corollary}[Ordinary projective Mori output]
\label{prog:projective-mori}
Let \(f:X\to S\) be an already projective surjective morphism of
compact Kähler spaces, with \(X\) smooth. Let \((X,B)\) be an
effective ordinary real klt pair, and let \(D=K_X+B\). If \(D\)
is not pseudoeffective over \(S\), then its projective program over
\(S\), with scaling of a relatively ample actual real line \(N\),
terminates with a Mori fibre contraction. The initial scaling
parameter is chosen so that \(D+TN\) is nef over \(S\).
The working spaces are globally strongly \(\Q\)-factorial and
compact Kähler, and the actual line comparisons and integral
descent of Proposition~\ref{prog:projective-good-model} apply.
\end{corollary}
\begin{proof}
The line \(N\) is fixed; it need not be a general scaling direction.
In particular several extremal rays may have the same wall
parameter. Choose \(\epsilon>0\) such that \(D+2\epsilon N\) is
still not pseudoeffective over \(S\). On the finite Stein-compact
cover choose divided general effective representatives of \(N\)
so that all the resulting ordinary pairs representing \(D+tN\),
\(0\leq t\leq T\), are klt. For
\(\epsilon\leq t\leq T\) reserve a common effective relatively
ample rational summand. The joint rational-family construction in
Proposition~\ref{prog:projective-good-model} puts these fixed pairs
in the polytopes required by Theorem~E of
\cite{FujinoAnalyticBCHM}.
Suppose a finite prefix has reached \(X_i\), and put
\(\mu_i=\lambda_{i-1}\), with \(\mu_0=T\). Its transformed
\(D_i+\mu_iN_i\) is nef. Define
\[
\lambda_i=\min\{t\in[0,\mu_i]:D_i+tN_i
\text{ is nef over }S\}.
\]
The feasible set is a nonempty closed interval. Inductively
\(\lambda_i>2\epsilon\): all previous steps have nonpositive
comparison for \(D+2\epsilon N\), so its transform is still not
pseudoeffective. A first threshold at most \(2\epsilon\) would
make this transform nef by convexity between the current and
preceding thresholds, a contradiction.
There is a \(D_i\)-negative extremal ray \(R_i\) with
\((D_i+\lambda_iN_i)\cdot R_i=0\). To see attainment without a
generality assumption, let \(t<\lambda_i\) tend to \(\lambda_i\).
A ray negative for \(D_i+tN_i\) has positive \(N_i\)-degree,
because \(D_i+\mu_iN_i\) is nef. When \(t>\lambda_i/2\), it is
also negative for \(D_i+(\lambda_i/2)N_i\). The latter has fixed
ordinary klt big-boundary representatives on the charts, since
\(\lambda_i/2>\epsilon\). Normalize each of these boundaries by
\cite[Lemma~11.15]{FujinoAnalyticBCHM} as \(A_\ell+B'_\ell\),
with \(A_\ell\geq0\) relatively ample and the new pair klt.
The \(A_\ell\)-truncation of the cone theorem for
\(K+B'_\ell\) gives finitely many rays negative for the entire
adjoint in question. The finite cover therefore gives finitely
many global negative rays. One of these rays attains the wall,
as required.
Choose one such ray, even if the wall vanishes on other rays.
The rational-support argument for an individual negative ray in
Proposition~\ref{prog:relative-klt} gives a global rational nef
support annihilating exactly \(R_i\), and ordinary relative base
point freeness contracts it. For the real ordinary boundary one
may first choose a nearby rational klt boundary that is still
negative on \(R_i\). The contraction therefore has a rational
ordinary antiample adjoint. Its flip, when small, is constructed
from one global detector algebra as above.
If the contraction is of fibre type, stop. Otherwise the wall
\(w_i=D_i+\lambda_iN_i\) has degree zero on \(R_i\).
In the finite rational span of its actual line factors, this
zero-degree subspace is rational: its defining functional has
rational values on rational lines. Thus \(w_i\) is a real
combination of rational lines of zero ray degree.
Lemma~\ref{prog:integral-descent} descends those lines to the
contraction base. Their combination is nef over \(S\), by lifting
curves through the projective contraction. Pulling it to the
positive side shows that the transformed wall stays nef. The next
threshold therefore satisfies \(\lambda_{i+1}\leq\lambda_i\);
equality is permitted. No interval of ampleness between successive
thresholds is required.
If \(E_i\geq0\) is the ordinary \(D_i\)-comparison on a common
resolution, this actual wall descent gives, for every real \(t\),
\[
p_i^*(D_i+tN_i)
\sim_{\R}q_i^*(D_{i+1}+tN_{i+1})
+(1-t/\lambda_i)E_i.
\]
For \(t\leq\lambda_i\) the error is effective. This proves all
the nonpositivity and fixed-pair persistence assertions used in
the induction, including the lower bound on thresholds.
At every working model use the \emph{wall parameter} \(t=\lambda_i\),
rather than an interior ample parameter. Since
\(\lambda_i\leq\lambda_j\) for every earlier step, the displayed
comparisons show that \(X_i\) is a weak log canonical model of
the initial fixed chart pairs representing \(D+\lambda_iN\).
Its trace is nef by definition. Theorem~E counts all these marked
weak log canonical models, including the ones at repeated wall
parameters. The finite chart cover gives a finite global marked
list. Repetition would contradict the strict ordinary
\(D\)-discrepancy increase at an intervening step. Thus this
program with the specified \(N\) is finite. Non-pseudoeffectivity
excludes a nef endpoint for \(D\), so the last contraction is a
Mori fibre contraction. The comparison at \(t=1=\lambda_i\) is
crepant: such a step supplies no strict discrepancy improvement
for \(D+N\), while it is still strictly negative for \(D\).
\end{proof}
==== END Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03b-projective-good-model.tex ====
==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03c-base-presentations.tex ====
\subsubsection{Base presentations and multiplier ideals}
The following constructions take place in the category of generalized
data specified above. In particular, the whole adjoint is a
Bott--Chern class, whereas the boundary and nef traces on a singular
space are interpreted as currents. On a smooth carrier the structure
boundary is a real divisor; its negative components may be exceptional
over the space of the pair. The nef part is specified on that carrier
and pulled back to higher models.
\begin{lemma}[Adjoint-preserving preparation]
\label{prog:base-presentation-preparation}
Let $(X,B+\mathbf M)$ be a generalized pair on a normal compact
K\"ahler space, with globally modified-big total boundary
$B+\mathbf M_X$. Write
\[
A=[K_X+B+\mathbf M_X],\qquad
\mathcal J_o=\mathcal J(X,B+\mathbf M).
\]
There are generalized data on $X$ with the same adjoint class, a
projective carrier $p:U\to X$ with $U$ smooth and compact K\"ahler, an SNC real
divisor $C$ whose negative components are $p$-exceptional, and a
K\"ahler class $\beta$ such that
\begin{equation}\label{prog:base-presentation-prepared}
p^*A=[K_U+C]+\beta,\qquad
\mathcal J_1=p_*\mathcal O_U(-\lfloor C\rfloor)
\supseteq\mathcal J_o.
\end{equation}
The carrier may be required to dominate any prescribed projective
modification of $X$, with the structure and exceptional divisors
having simultaneous SNC support. The prepared total boundary is
globally modified big. In particular, the preparation preserves the
gklt condition.
\end{lemma}
\begin{proof}
Choose a common projective log resolution $p_0:U_0\to X$ of the
original nef carrier and the prescribed modification, and write
\[
p_0^*A=[K_{U_0}+C^0]+\mathbf M_{U_0}.
\]
The class $\mathbf M_{U_0}$ is nef, and the negative part $(C^0)^-$
is exceptional over $X$. The modified-big preparation
\cite[Lemmas~2.10 and~2.23]{HX} can be made with the prescribed
resolution. We give its divisor and ideal details. Choose an effective divisor $E_0$ with sufficiently large coefficients
and full $p_0$-exceptional support so that the class
\[
[(C^0)^+]+\mathbf M_{U_0}+[E_0]
\]
is big. Resolve a Kodaira decomposition of this class. On a further
projective modification $a:U\to U_0$ with $U$ smooth one obtains
\[
a^*\bigl([(C^0)^+]+\mathbf M_{U_0}+[E_0]\bigr)
=\kappa+[D],\qquad D\geq0,
\]
where $\kappa$ is K\"ahler. The support can be made SNC together
with all the other divisors. On any additional blowup, subtract a
sufficiently small effective exceptional class from the pulled-back
K\"ahler class and add it to the divisor; an exceptional divisor
whose negative is relatively ample makes the resulting class
K\"ahler again.
Put $p=p_0a$ and
\[
C^{\mathrm{alt}}=D-a^*(C^0)^--a^*E_0-K_{U/U_0}.
\]
Then
\[
p^*A=[K_U+C^{\mathrm{alt}}]+\kappa,
\]
and every negative component of $C^{\mathrm{alt}}$ is
$p$-exceptional. Let $C^{\mathrm{old}}$ be the crepant structure
boundary of the original data on this same model. For $0<t\ll1$,
set
\[
C=(1-t)C^{\mathrm{old}}+tC^{\mathrm{alt}},\qquad
\beta=(1-t)a^*\mathbf M_{U_0}+t\kappa.
\]
The class $\beta$ is K\"ahler, the adjoint identity is unchanged,
and negative coefficients still occur only on exceptional divisors.
There are only finitely many coefficients. Thus $t$ can be chosen so
that
\[
\lfloor C\rfloor\leq\lfloor C^{\mathrm{old}}\rfloor.
\]
Pushing forward the corresponding inclusion of invertible sheaves
gives~\eqref{prog:base-presentation-prepared}. The K\"ahler class
$\beta$ plus the strict transform of the effective trace boundary is
big; its pushforward is the prepared total boundary. If the original pair
is gklt, its multiplier ideal is the unit ideal, and hence so is
$\mathcal J_1$.
\end{proof}
\begin{proposition}[Base presentations for projective contractions]
\label{prog:base-presentation}
Let $f:X\to Z$ be a projective contraction between normal irreducible
compact K\"ahler spaces. Suppose that $(X,B+\mathbf M)$ has globally
modified-big total boundary and that
\[
A=[K_X+B+\mathbf M_X]=f^*\gamma,
\qquad \gamma\in H^{1,1}_{\mathrm{BC}}(Z).
\]
There is an adjoint-preserving preparation as in
Lemma~\ref{prog:base-presentation-preparation}, with multiplier ideal
$\mathcal J_1\supseteq\mathcal J_o$, for which
\begin{equation}\label{prog:base-presentation-Stein}
R^if_*\mathcal J_1=0\quad(i>0),\qquad
f_*\mathcal O_{N_1}=\mathcal O_Z/(f_*\mathcal J_1),
\qquad N_1=V(\mathcal J_1).
\end{equation}
If $N_1$ dominates $Z$, then $f_*\mathcal J_1=0$.
Otherwise there are generalized data $(Z,B_Z+\mathbf N)$, a
projective carrier $\tau:T\to Z$ with $T$ smooth and compact K\"ahler, a real
SNC structure divisor $C_T$, and a K\"ahler class $\beta_T$ such that
\begin{equation}\label{prog:base-presentation-identities}
\begin{gathered}
[K_Z+B_Z+\mathbf N_Z]=\gamma,\qquad
\tau^*\gamma=[K_T+C_T]+\beta_T,\qquad
\mathbf N_T=\beta_T,\\
B_Z=\tau_*C_T\geq0,\qquad
B_Z+\mathbf N_Z\ \text{is globally modified big}.
\end{gathered}
\end{equation}
The negative components of $C_T$ are $\tau$-exceptional, and
\begin{equation}\label{prog:base-presentation-ideal}
\mathcal J_Z:=\mathcal J(Z,B_Z+\mathbf N)
=\tau_*\mathcal O_T(-\lfloor C_T\rfloor)
\supseteq f_*\mathcal J_1.
\end{equation}
In particular the base data are gklt wherever $f_*\mathcal J_1$ is
the unit ideal. If the source data are gklt, the base data are gklt.
All adjoint identities are preserved by pullback to higher models.
\end{proposition}
\begin{proof}
Choose a projective resolution $\mu:W\to Z$, with $W$ smooth
compact K\"ahler and reduced exceptional divisor $F$ having SNC
support. Apply Lemma~\ref{prog:base-presentation-preparation} on a
carrier dominating the main component of the base change. We obtain
projective maps $p:U\to X$ and $q:U\to W$ with $fp=\mu q$ and
\begin{equation}\label{prog:base-presentation-source}
[K_U+C]+\beta=p^*A=q^*\mu^*\gamma,
\qquad \mathcal J_1=p_*\mathcal O_U(-\lfloor C\rfloor).
\end{equation}
Here $U$ is smooth compact K\"ahler, $\beta$ is K\"ahler, and all
negative components of $C$ are $p$-exceptional. The maps are
projective by composition and factorization over the separated base
$W$. The map $q$ has connected fibres: this holds over the open set
where $\mu$ is an isomorphism, and its Stein factorization is finite
and bimeromorphic over the normal space $W$.
The integral divisor $-\lfloor C\rfloor$ has class
\[
[-\lfloor C\rfloor]
=[K_U+\{C\}]+\beta-(fp)^*\gamma.
\]
The pair $(U,\{C\})$ is klt with SNC boundary. Relative
Kawamata--Viehweg vanishing, first for $p$ and then for $fp$, applies
because the positive class modulo pullbacks is K\"ahler
\cite[Theorem~2.2]{HX}. Equivalently, the real line bundle
$-K_U-C$ is relatively ample,
so the vanishing follows from \cite[Theorem~1.2]{FujinoVanishing2025}
with the identity as its first morphism. Leray gives $R^if_*\mathcal J_1=0$ for
$i>0$. Applying $f_*$ to the sequence defining $N_1$, and using
$f_*\mathcal O_X=\mathcal O_Z$, proves
\eqref{prog:base-presentation-Stein}. If $N_1$ dominates $Z$, the
ideal $f_*\mathcal J_1\subseteq\mathcal O_Z$ vanishes generically
and therefore vanishes. This proves that alternative.
Suppose from now on that $N_1$ does not dominate $Z$. Every
$q$-horizontal coefficient of $C$ is then less than one. Indeed,
a component with coefficient at least one forces the multiplier
ideal to be proper along its image in $X$, and its image would make
$N_1$ dominate $Z$. If $Z$ is a point, the asserted base presentation
is the dimension-zero presentation. We therefore assume
$b=\dim Z>0$.
\paragraph{The fibre metric.}
Choose a smooth representative $\rho$ of
$\mu^*\gamma-c_1(K_W)$, a K\"ahler form $\omega_W$, and a
K\"ahler form representing $\beta$, still denoted by $\beta$.
For some $a>0$,
\[
\beta\geq a q^*\omega_W.
\]
Consider the holomorphic line bundle
\[
L=\mathcal O_U(-K_{U/W}-\lfloor C\rfloor).
\]
Equation~\eqref{prog:base-presentation-source} gives a singular
Hermitian metric $h$ on $L$ whose curvature current is
\begin{equation}\label{prog:base-presentation-curvature}
\Theta_h(L)=\beta+[\{C\}]-q^*\rho.
\end{equation}
Its weights are smooth functions plus the divisorial logarithms
with coefficients in $\{C\}$. We normalize $dd^c$ so that
$dd^c\log|z|^2=[z=0]$ and write squared-norm weights as $e^{-\varphi}$.
Let $e$ be the canonical meromorphic section of
\[
K_{U/W}\otimes L=\mathcal O_U(-\lfloor C\rfloor),
\]
so that $\operatorname{div}(e)=-\lfloor C\rfloor$.
Choose an analytic Zariski open set $W^\circ\subseteq W$ on which
$q$ is a submersion, the horizontal boundary is relatively SNC,
there are no vertical boundary components, $\mu$ is an isomorphism
onto an open subset of $Z$, and coherent base change holds in degree
zero. On $q^{-1}(W^\circ)$, the divisor $-\lfloor C\rfloor$ is
effective and exceptional over $X$. Normality and the contraction
property therefore identify
\[
q_*(K_{U/W}\otimes L)|_{W^\circ}
=\mathcal O_{W^\circ}e.
\]
Here $K_{U/W}$ is viewed as the relative canonical line bundle. The
fibre norm
\[
I(w)=\int_{U_w}|e_w|_h^2
\]
is positive and finite. The fractional SNC weights are integrable;
any zeros of $e_w$ only improve integrability. Relative SNC
coordinates and a partition of unity show that $I$ is smooth:
parameter derivatives act on smooth factors multiplying fixed
integrable monomial weights.
Set $\psi=-\log I$ on $W^\circ$. Then
\begin{equation}\label{prog:base-presentation-open-positivity}
\rho+dd^c\psi\geq a\omega_W.
\end{equation}
Here are the details for the singular weights. Locally choose
$\ell$ with $dd^c\ell=\rho-a\omega_W$, and put
$h'=he^{-q^*\ell}$. Its weights are psh, and its fibre norm satisfies
$I'=e^{-\ell}I$. Restrict the family to a small coordinate line disk
$\Delta_R$, with centre $0$. The sharp $L^2$ extension theorem gives
a twisted top form extending $e_0$ and satisfying
\begin{equation}\label{prog:base-presentation-sharp-extension}
\int_{q^{-1}(\Delta_R)}|E|_{h'}^2
\leq\pi R^2 I'(0).
\end{equation}
Top-form norms here are normalized to use Euclidean area in the
base coordinate. We use the Stein hypersurface form of the optimal
extension theorem \cite[Theorem~2.2 and \S3.5, (3.1)]{GuanZhouOptimal2015},
with defining function the disk coordinate divided by $R$.
To check applicability, embed the restricted projective family into
projective space over the disk and remove a relative hyperplane
section that does not contain the central fibre. The complement is
Stein. A holomorphic section of $L$ on this Stein space can be chosen
nonzero at a point of the central fibre; deleting its zero divisor
trivializes $L$ and leaves a Stein space. The central fibre is smooth
and connected, hence irreducible, so both deletions leave a dense
open subset of it. For a zero-dimensional fibre choose the deletions
to avoid its point. In the resulting trivialization the weight is
psh, and the sharp extension theorem applies to canonical forms.
The extension extends across the deleted hypersurfaces in reverse
order by local $L^2$ removability: in every original frame the metric
is bounded below by a smooth positive metric. The extension retains
its norm bound and its prescribed restriction to the central fibre.
After dividing $E$ by the base differential, the rank-one direct
image identifies it with $b(t)e$, where $b$ is holomorphic and
$b(0)=1$. Thus the disk average of $|b(t)|^2I'(t)$ is at most $I'(0)$.
Jensen's inequality and the submean inequality for $\log|b|^2$ give
\[
\log I'(0)
\geq\operatorname{avg}_{\Delta_R}
\bigl(\log|b(t)|^2+\log I'(t)\bigr)
\geq\operatorname{avg}_{\Delta_R}\log I'(t).
\]
The same argument applies to every sufficiently small disk at every
centre and in every complex line direction. Hence $-\log I'=\psi+\ell$ is psh, which proves
\eqref{prog:base-presentation-open-positivity}.
\paragraph{Extension over the base.}
Let $D$ be a prime divisor in $W\setminus W^\circ$. Choose a component
$Q$ of $q^*D$ dominating $D$. When $D$ is not $\mu$-exceptional, choose
$Q$ not $p$-exceptional: take a component over the corresponding prime
on $Z$ already on $X$ and then its strict transform. Put
\[
u=\operatorname{ord}_Q(q^*D),\qquad c=\operatorname{coeff}_Q(C).
\]
At a general point of $Q$, avoiding all other relevant divisors,
there are local coordinates with
\[
t_1=z_1^u,\qquad t_j=z_j\quad(2\leq j\leq b).
\]
Indeed the map $Q\to D$ is submersive at a general smooth point,
and a local unit in the first equation is absorbed by a holomorphic
$u$th root. Such points can be chosen above every point of a dense
analytic open subset of $D$ by generic smoothness and properness.
Integration over a fixed small polydisk in the remaining fibre
coordinates gives
\begin{equation}\label{prog:base-presentation-divisor-bound}
I(t)\geq c_*|t_1|^{-2(c+u-1)/u}
\end{equation}
for some $c_*>0$ near each such base point. The section and metric
contribute the exponent $c$, while division by the base differential
in the relative canonical form contributes $u-1$.
For a nonexceptional $D$ the chosen $Q$ has $c\geq0$, so
\eqref{prog:base-presentation-divisor-bound} bounds $\psi$ from above
near its general points. For the finitely many exceptional divisors,
the same estimate supplies a nonnegative integer $m$ such that, if
$k$ is a smooth metric on $\mathcal O_W(F)$ and $s_F$ its divisor
section,
\[
\varphi=\psi+m\log|s_F|_k^2
\]
is locally bounded above near general points of every divisor.
Adding local potentials of $\rho+m c_1(k)-a\omega_W$ to
$\varphi$ gives psh weights by
\eqref{prog:base-presentation-open-positivity}. These weights
extend across the divisors and then across the remaining analytic
set of codimension at least two, giving
\begin{equation}\label{prog:base-presentation-positive-current}
S=\rho+m c_1(k)+dd^c\varphi\geq a\omega_W
\end{equation}
on all of $W$.
\paragraph{A K\"ahler carrier.}
Regularize the current in
\eqref{prog:base-presentation-positive-current} in its class, retaining
a lower bound for the potentials. More precisely, choose an
approximation with analytic singularities such that
\begin{equation}\label{prog:base-presentation-regularization}
S'=\rho+m c_1(k)+dd^c\varphi',\qquad
S'\geq\frac a2\omega_W,\qquad
\varphi'\geq\varphi-O(1).
\end{equation}
These simultaneous properties are the Bergman regularization
properties of \cite{DemaillyRegularization1992}; see also
\cite[pp.~59--60]{PopoviciRegularization2004}.
Resolve the analytic singularities and the support of $F$ by a
sequence of blowups with smooth centres $r:T\to W$, with simultaneous
SNC support, and write
\[
r^*S'=[H]+\zeta,\qquad H\geq0.
\]
The residual current $\zeta$ has locally bounded potentials. Removing
the divisorial part from the positive current
$r^*S'-(a/2)r^*\omega_W$ shows that
\[
\zeta\geq\frac a2r^*\omega_W.
\]
The difference has vanishing Lelong numbers, so its class is nef by
regularization. Choose an effective $r$-exceptional divisor $G$ with
$-G$ relatively ample. For $0<\eta\ll1$, the class
$(a/2)r^*\omega_W-\eta[G]$ is K\"ahler. Therefore
\begin{equation}\label{prog:base-presentation-carrier-data}
\begin{split}
\tau&=\mu r,\qquad
\beta_T=[\zeta]-\eta[G],\\
C_T&=H+\eta G-mr^*F-K_{T/W}
\end{split}
\end{equation}
has K\"ahler $\beta_T$. If $r$ is an isomorphism, omit $G$ and its
correction. The choice of $\eta$ will be decreased once more below.
The exact class calculation is
\[
\begin{split}
[K_T+C_T]+\beta_T
&=r^*c_1(K_W)+[H]+[\zeta]-m[r^*F]\\
&=r^*(c_1(K_W)+[\rho])=\tau^*\gamma.
\end{split}
\]
All possibly negative terms in $C_T$ are $\tau$-exceptional.
Consequently $B_Z=\tau_*C_T$ is effective. Specify the nef b-data by
$\mathbf N_T=\beta_T$; the displayed identity defines generalized
data on $Z$ with adjoint class $\gamma$. At the level of currents,
pushing the identity gives the required trace identity on $Z$:
exceptional divisor currents push to zero, and a $dd^c$ difference
pushes to a $dd^c$ difference. Canonical traces can be computed using
local meromorphic canonical frames; a change of frame changes the
corresponding Chern potentials by the usual logarithmic transition
term. Thus only the whole adjoint, rather
than its individual traces, needs to be a Bott--Chern class. The
class $\beta_T$ plus the strict transform of $B_Z$ is big on $T$,
so its pushforward $B_Z+\mathbf N_Z$ is globally modified big.
These are~\eqref{prog:base-presentation-identities}.
\paragraph{The multiplier ideal.}
Take a local holomorphic function $v\in f_*\mathcal J_1$. By the
pushforward description of $\mathcal J_1$,
\begin{equation}\label{prog:base-presentation-source-orders}
\operatorname{div}(q^*\mu^*v)\geq\lfloor C\rfloor.
\end{equation}
Thus $q^*\mu^*v\,e$, tensored with a local base canonical frame,
is a holomorphic $L$-valued top form on $U$. Its squared norm is
locally integrable by the fractional SNC weights in
\eqref{prog:base-presentation-curvature}. Over the submersion locus,
fibre integration followed by base integration is precisely this
top-form integral. Properness and a finite coordinate covering
therefore give
\begin{equation}\label{prog:base-presentation-base-integral}
\int |\mu^*v|^2e^{-\psi}<\infty
\end{equation}
locally on $W$; the integral is taken where the original fibre
metric is defined.
The lower bound on $\varphi'$ in
\eqref{prog:base-presentation-regularization} preserves this
integrability after replacing $\psi$ by
$\varphi'-m\log|s_F|_k^2$. Change variables through $r$. Along a
prime divisor of $T$, let $h,j,k_D,d$ be respectively the coefficients
of $H$, $\operatorname{div}(\tau^*v)$, $K_{T/W}$, and $r^*F$.
The local integrand has exponent
$2(j-h+md+k_D)$, up to a bounded positive factor. Its integrability
implies $j-h+md+k_D>-1$, or, since $j$ is integral,
\[
j\geq\lfloor h-md-k_D\rfloor.
\]
Therefore
\[
\operatorname{div}(\tau^*v)
\geq\lfloor H-mr^*F-K_{T/W}\rfloor.
\]
Choose $\eta>0$ sufficiently small that adding $\eta G$ does not
change any of these finitely many round-downs. The resulting
inequality is exactly
$\operatorname{div}(\tau^*v)\geq\lfloor C_T\rfloor$, proving
\eqref{prog:base-presentation-ideal} as an inclusion of ideal sheaves.
In particular, if $f_*\mathcal J_1$ is the unit ideal on an open set,
then so is $\mathcal J_Z$, and the base data are gklt there.
Finally, the nef data descend to $T$, and the structure boundaries
on higher models are defined by crepant pullback of
\eqref{prog:base-presentation-identities}. On every common refinement
of the source and base carriers, both adjoints are the pullback of
$\gamma$. Thus the identities remain exact on higher models.
\end{proof}
\begin{lemma}[Contractions of closed subspaces]
\label{prog:base-emc}
Let $D$ be a compact complex space, possibly nonreduced, and let
$\delta\in H^2(D,\R)$ have nonnegative degree on compact curves.
Use the definitions of a Moishezon morphism and of a contraction
defined by a class in \cite[Definitions~2.32 and~2.34]{HX}.
Then the following hold.
\begin{enumerate}
\item If $\delta$ is endowed with a Moishezon contraction, so is
its restriction to every closed analytic subspace of $D$.
\item Let $g:D\to V$ be a projective contraction of compact
complex spaces, so $g_*\OO_D=\OO_V$, and suppose
$\delta=g^*\epsilon$ for a class $\epsilon\in H^2(V,\R)$
having nonnegative degree on compact curves. Then $\delta$ is
endowed with a Moishezon contraction if and only if $\epsilon$
is endowed with one.
\end{enumerate}
\end{lemma}
\begin{proof}
Write $h:D\to S$ for a Moishezon contraction defined by
$\delta$. For a closed subspace $D_0\subset D$, take the Stein
factorization
\[
D_0\longrightarrow S_0\longrightarrow S
\]
of its restriction. The first map is a contraction, and the second
is finite. Thus a curve in $D_0$ is contracted by the first map
exactly when its $\delta$-degree is zero. Choose a projective
surjection $P\to D$ witnessing that $h$ is Moishezon, with
$P\to S$ projective. The closed subspace
$P_0=P\times_D D_0$ is projective over $S$ and projective
and surjective over $D_0$. It is also projective over $S_0$:
a relatively ample line for $P_0\to S$ remains ample on the
fibres of $P_0\to S_0$. This proves the first assertion,
including its scheme structure.
For the descent direction of the second assertion, every curve
in a fibre of $g$ has $\delta$-degree zero. A connected
projective complex space is connected by chains of compact curves
after passing to its reduction. Therefore $h$ is constant on
the underlying set of each fibre of $g$. It factors as
$h=k\circ g$ for a holomorphic map $k:V\to S$.
To see that this factorization holds also for nonreduced spaces,
choose a coordinate neighbourhood in $S$ of the image of a fibre.
Properness permits restriction to a neighbourhood of its point
in $V$ whose inverse image maps into that coordinate
neighbourhood. The coordinate functions of $h$ then descend
uniquely through $g_*\OO_D=\OO_V$; their analytic relations
descend through the same equality. These local factorizations
glue. Furthermore,
\[
k_*\OO_V=h_*\OO_D=\OO_S.
\]
Every irreducible compact curve $C\subset V$ is dominated by
a compact curve $C'\subset D$. Indeed, after base change to
the normalization of $C$, take a component dominating that curve
and a projective multisection. The projection formula gives
\[
\delta\cdot C'
=\deg(C'\to C)\,\epsilon\cdot C.
\]
Consequently $k$ contracts exactly the curves of
$\epsilon$-degree zero. The composite $P\to D\to V$
is a projective surjection, and $P\to S$ is projective, so
it witnesses that $k$ is Moishezon.
Conversely, suppose $k:V\to S$ is a Moishezon contraction
defined by $\epsilon$. Then $k\circ g$ is a contraction,
and the projection formula verifies its curve criterion, treating
separately curves contracted by $g$. If $Q\to V$ is a
projective surjection witnessing that $k$ is Moishezon, then
$D\times_V Q\to D$ is projective and surjective, while
$D\times_V Q\to Q\to S$ is projective. This supplies the
required witness.
\end{proof}
\begin{lemma}[Crepant transport of the multiplier closed subspace]
\label{prog:multiplier-transport}
Let $g:X\to Y$ be a projective bimeromorphic morphism of
normal compact K\"ahler spaces. Suppose effective generalized
pair data on $X$ and $Y$ have the same globally nef b-part
and are crepant over $Y$. In particular their adjoint classes
$A_X$ and $A_Y$ satisfy $A_X=g^*A_Y$ in Bott--Chern
cohomology. Write $J_X,J_Y$ for their multiplier ideals and
$N_X=V(J_X)$, $N_Y=V(J_Y)$ for their non-klt closed
subspaces. Then
\[
g_*J_X=J_Y,\qquad R^ig_*J_X=0\quad(i>0),
\qquad g_*\OO_{N_X}=\OO_{N_Y}.
\]
The restriction $N_X\to N_Y$ is a projective contraction.
Hence, whenever $A_Y$ is nef, the restriction of $A_X$ to
$N_X$ is endowed with a Moishezon contraction if and only if
the restriction of $A_Y$ to $N_Y$ is endowed with one.
\end{lemma}
\begin{proof}
Take a projective common log resolution $r:U\to X$ carrying
the nef datum, and put $s=g\circ r$. By crepancy there is
one real SNC divisor $C$ such that
\[
K_U+[C]+\beta=s^*A_Y=r^*A_X,
\]
where $\beta$ is globally nef. The multiplier ideals are
\[
J_X=r_*\OO_U(-\lfloor C\rfloor),\qquad
J_Y=s_*\OO_U(-\lfloor C\rfloor).
\]
Their direct-image equality follows at once. For vanishing, set
$L=\OO_U(-\lfloor C\rfloor)$ and
$\Delta=C-\lfloor C\rfloor$. The pair $(U,\Delta)$
is klt, and the actual real line
\[
L-(K_U+\Delta)=-K_U-C
\]
has class $\beta-s^*A_Y$. It is relatively nef over $Y$
and relatively big because $s$ is bimeromorphic. The same
statements hold over $X$. Relative Kawamata--Viehweg vanishing,
in the form of \cite[Theorem~1.2]{FujinoVanishing2025} with
the first morphism the identity and the real line
$\mathcal H=-K_U-C$, therefore gives
\[
R^is_*L=R^ir_*L=0\qquad(i>0).
\]
The real-line identity in that theorem is the displayed definition
of $\mathcal H$; its relative positivity is checked by the class
identity above. The klt condition means that there are no proper log
canonical strata to test for relative log bigness. Leray gives
$R^ig_*J_X=0$ for $i>0$. Pushing forward
$0\to J_X\to\OO_X\to\OO_{N_X}\to0$ and using
$g_*\OO_X=\OO_Y$ now proves the asserted structure-sheaf
equality. It also identifies the scheme-theoretic image with
$N_Y$, giving the projective contraction between these closed
subspaces. Apply Lemma~\ref{prog:base-emc} to conclude.
\end{proof}
\begin{remark}
The preceding lemma also transports the contraction hypothesis
through a crepant projective flip: apply it to the two contractions
to their common base, equipped with the descended adjoint and
the common nef b-data. On that base the boundary is the effective
pushforward of the source boundary, and the nef data remain on
the same carrier; the descended Bott--Chern adjoint and its
identity on a common resolution define the crepant base data
in the whole-adjoint and current-trace category used here.
All assertions concern the fixed adjoint
and its multiplier ideal. No conclusion about the multiplier
ideal of a different driving adjoint is used.
\end{remark}
\begin{corollary}[The contraction step over a lower-dimensional base]
\label{prog:base-emc-induction}
Use the hypotheses and notation of
Proposition~\ref{prog:base-presentation}. Suppose that
$A=f^*\gamma$ is nef and that its restriction to the original
non-klt closed subspace $N_o=V(J_o)$ is endowed with a
Moishezon contraction. Put
\[
I=f_*J_1,\qquad N_*=V(I)\subset Z.
\]
Then $\gamma|_{N_*}$ is endowed with a Moishezon contraction.
If the prepared non-klt closed subspace $N_1=V(J_1)$ dominates
$Z$, then $\gamma$ and $A$ are endowed with Moishezon
contractions. Otherwise the generalized pair supplied on $Z$
has a non-klt closed subspace endowed with the contraction
defined by the restricted class of $\gamma$. In this latter
case, if $\gamma$ is nef and NQC and the assertion
$\mathsf C_{\dim Z}$ is available, then $\gamma$ and $A$
are endowed with Moishezon contractions.
\end{corollary}
\begin{proof}
The inclusion $J_o\subset J_1$ gives $N_1\subset N_o$
as closed analytic subspaces. By
Lemma~\ref{prog:base-emc}, the given contraction restricts to
$N_1$. Proposition~\ref{prog:base-presentation} identifies
\[
f_*\OO_{N_1}=\OO_Z/I=\OO_{N_*}.
\]
Thus the restriction $N_1\to N_*$ is a projective
contraction. The restricted adjoint is the pullback of
$\gamma|_{N_*}$, so Lemma~\ref{prog:base-emc} descends
the contraction to $N_*$. Nonnegative degrees on curves of
$N_*$ follow from projective multisections and nefness of
$A$.
If $N_1$ dominates the normal irreducible space $Z$, then
$I=0$: a local function in $I$ pulls back to zero on the
dominating closed subspace and hence vanishes generically on
$Z$. Normality, and in particular reducedness, gives its
vanishing. Therefore $N_*=Z$, which proves the assertion in
this case.
If $N_1$ does not dominate $Z$, the constructed base ideal
satisfies $I\subset J_Z$. Hence
\[
V(J_Z)\subset V(I)=N_*
\]
as closed analytic subspaces. Restriction by
Lemma~\ref{prog:base-emc} provides the required contraction
on $V(J_Z)$, with its complete scheme structure. The
generalized pair on $Z$ has globally nef carrier data and
globally modified-big total boundary by
Proposition~\ref{prog:base-presentation}. When the stated
nef and NQC conditions hold, these facts are precisely the
hypotheses of $\mathsf C_{\dim Z}$, which gives the
contraction on $Z$. Finally Lemma~\ref{prog:base-emc}
pulls it back through the projective contraction $f$ to $X$.
The case of an empty non-klt closed subspace has the same
interpretation, with its contraction condition vacuous.
\end{proof}
==== END Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03c-base-presentations.tex ====
==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03c-restricted-induction.tex ====
\subsection{A restricted dimension induction}
\label{prog:dimension-induction}
All generalized nef data in this subsection are globally nef on a fixed
compact K\"ahler carrier. For the degree arguments we use weak NQC:
a positive combination of rational degree-two classes having
nonnegative degree on compact curves. Strong NQC implies this
condition, and the contraction assertion admits this weak form
\cite[Definition~2.43]{HX}. The nef adjoint itself remains nef in
Bott--Chern cohomology throughout. A modified-big boundary means the total trace
$B+\mathbf M_X$ is globally modified big. Relative assertions retain
these absolute hypotheses. A marked model records its bimeromorphic map
from the specified source; two markings agree only under an isomorphism
commuting with those maps. Weak models below may be arbitrary normal compact K\"ahler weak log
canonical models. Their generalized nef traces are understood as
closed currents pushed forward from the fixed nef carrier; only the
whole adjoint is required to be a Bott--Chern class. In particular no
factoriality, and no separate Bott--Chern class for either $K_Y+B_Y$
or the nef trace, is required of an auxiliary weak target. This
category contains both the working models and the adjunction strata
used in special termination. Chosen program outputs remain globally
strongly $\mathbb Q$-factorial.
For dimension at most $d$, write $\mathsf B_d$ for semiampleness of a
nef gklt adjoint with modified-big boundary, including a Moishezon
connected contraction onto a normal compact K\"ahler target. It makes
no assertion that every such contraction is projective.
Write $\mathsf C_d$ for the NQC non-klt contraction assertion of
\cite[Theorem~1.6]{HX}, and $\mathsf C_{d,\mathrm{big}}$ for its
big-adjoint case. The relative assertions are:
\begin{itemize}
\item $\mathsf M_d$: a relatively pseudo-effective gklt adjoint over
any proper map to a normal compact K\"ahler base $S$, with the globally
nef and modified-big data just specified, has a chosen nonextracting
good log terminal model over $S$. Its working space is globally
strongly $\mathbb Q$-factorial and compact K\"ahler; its connected
relative canonical morphism has normal compact K\"ahler target and
is Moishezon. From a smooth common carrier,
all Bott--Chern classes have forward traces, and the chosen model has
the rational degree-two and Bott--Chern exceptional splittings proved
below.
\item $\mathsf F_d$: a compact polytope of these data on one fixed
carrier has a polyhedral relative effective locus, finitely many
relative canonical chambers with chosen good terminal models, and
finitely many marked weak models, accounting for every weak model
in the category specified above.
\end{itemize}
Passing to the proper image and then to its normal Stein factor allows
all base maps to be taken surjective with connected fibres. These
operations preserve the compact K\"ahler base category.
The independent inputs are the analytic cone theorem, the projective
results of Proposition~\ref{prog:projective-good-model}, relative
vanishing, adjunction, relative-canonical positivity, and the
nef-and-big K\"ahler criterion. The base presentations and
multiplier-ideal descent used in the induction are proved below.
The projective constructions use the actual-line lemmas of the preceding
subsection. No assertion of termination of an arbitrary sequence of
flips is used.
\input{sections/03c-base-presentations}
\subsubsection{A fixed integral grid and cohomology transport}
\begin{lemma}[Integral degree-two descent for a locally log-Fano map]
\label{prog:integral-h2}
Let $f:X\to Z$ be a projective surjective morphism with connected fibres
between normal complex analytic spaces. Assume that $f_*\mathcal O_X=
\mathcal O_Z$, that $R^if_*\mathcal O_X=0$ for $i>0$, and that every point
of $Z$ has a neighbourhood $U$ on which there is a klt pair
$(X_U,\Delta_U)$ with $-(K_{X_U}+\Delta_U)$ ample over $U$.
Then $f^*:H^2(Z,\mathbb Z)\to H^2(X,\mathbb Z)$ is injective, and its
image consists exactly of the integral classes having degree zero on
every compact curve contracted by $f$.
\end{lemma}
\begin{proof}
Properness, normality and connected fibres identify both
$f_*\mathcal O_X=\mathcal O_Z$ and
$f_*\mathcal O_X^*=\mathcal O_Z^*$. Apply $Rf_*$ to the analytic
exponential sequence. The local surjectivity of
$\mathcal O_Z\to\mathcal O_Z^*$ and the stated coherent vanishing give
\[
R^1f_*\mathbb Z_X=0,
\qquad
R^1f_*\mathcal O_X^*\simeq R^2f_*\mathbb Z_X.
\]
Let $\xi\in H^2(X,\mathbb Z)$ have zero degrees on contracted curves.
A germ of its edge image in $R^2f_*\mathbb Z_X$ is, under this
isomorphism, represented after shrinking $U$ by an actual line bundle
$L$ on $X_U$. Equality of these germs means that, after further
shrinking, $c_1(L)$ and $\xi|_{X_U}$ have the same class. In particular
$L$ has degree zero on every curve in every fibre above this smaller
neighbourhood.
On a sufficiently small Stein base neighbourhood, twisting by a
relatively ample line and taking a quotient of two nonzero relative
sections gives a meromorphic Cartier representative for $L$. Thus
the Cartier-divisor form of base point freeness applies.
The line bundle $L$ is relatively nef, and
$aL-(K_{X_U}+\Delta_U)$ is relatively ample for every real $a$.
The projective analytic base-point-free theorem applies to the Cartier
line $L$. Its conclusion is generation for \emph{every} sufficiently
large integer power, not only for a divisible subsequence
\cite[Theorems~6.2 and~6.5]{FujinoAnalyticBCHM}. Thus, over a further relatively compact
neighbourhood, both $L^m$ and $L^{m+1}$ are relatively generated.
A generated line of degree zero on every curve of a projective connected
fibre defines a constant projective map on that fibre: the restriction
of the generated line is the pullback of $\mathcal O(1)$, and a
positive-dimensional projective image contains a curve of positive
degree. The corresponding relative image is therefore finite and
bimeromorphic over the normal base, and is the base itself. Consequently
\[
L^m\simeq f^*M_m,\qquad L^{m+1}\simeq f^*M_{m+1}
\]
for line bundles on the neighbourhood in $Z$. Taking their quotient
proves $L\simeq f^*(M_{m+1}\otimes M_m^{-1})$. Its germ in
$R^1f_*\mathcal O_X^*$ is zero, so the edge image of $\xi$ is zero.
The integral Leray spectral sequence, using
$R^1f_*\mathbb Z_X=0$, gives
\[
0\longrightarrow H^2(Z,\mathbb Z)
\xrightarrow{f^*}H^2(X,\mathbb Z)
\longrightarrow H^0(Z,R^2f_*\mathbb Z_X).
\]
Exactness at the middle term proves the assertion. The argument is
integral throughout and does not discard torsion. The converse follows
by restriction of a pulled-back class to a fibre.
\end{proof}
\begin{lemma}[A fixed integral grid along existing crepant steps]
\label{prog:fixed-integral-grid}
Let $X_0$ be a compact K\"ahler space, and suppose that
\[
\alpha_0=\sum_{j=1}^r a_j\gamma_{0j},\qquad a_j>0,
\qquad \gamma_{0j}\in H^2(X_0,\mathbb Q),
\]
where each $\gamma_{0j}$ has nonnegative degree on compact curves.
Fix integers $m_j>0$ such that $m_j\gamma_{0j}$ is the image of an
integral cohomology class. Consider an existing sequence of projective
birational steps
\[
X_i\xrightarrow{f_i}Z_i\xleftarrow{f_i^+}X_{i+1},
\]
where $f_i^+$ is the identity for a divisorial step, $f_i$ satisfies
Lemma~\ref{prog:integral-h2}, and $\alpha_i$ is trivial on its
contracted ray. Assume every $f_i$-contracted curve has degree
proportional to that ray for the classes in the display. Then the
classes $\gamma_{ij}$ descend through $f_i$ and pull back through
$f_i^+$, the same integers $m_j$ remain integral multiples at every
stage, and the transported classes remain nonnegative on curves.
In particular, for every integral compact curve $C\subset X_i$,
\[
\alpha_i\cdot C>0\quad\Longrightarrow\quad
\alpha_i\cdot C\geq\delta,
\qquad \delta:=\min_j\frac{a_j}{m_j}>0.
\]
\end{lemma}
\begin{proof}
Positivity of the coefficients and nonnegativity of the summands imply
that every $\gamma_{ij}$ annihilates the contracted ray. Apply the
integral descent lemma to the chosen integral lift of
$m_j\gamma_{ij}$ and pull its descended class back to $X_{i+1}$.
This keeps $m_j$ fixed.
To verify nonnegativity, take a curve on $X_{i+1}$. If it is contracted
by $f_i^+$, every descended summand has degree zero. Otherwise its
image is a curve $\Gamma\subset Z_i$. The projective map $f_i$ has a
compact curve mapping onto $\Gamma$ with positive degree: take a
component of its inverse image dominating $\Gamma$ and intersect with
sufficiently many relatively ample hyperplanes. Nonnegativity of the
old summand on this curve implies nonnegativity on $\Gamma$, hence on
the curve on $X_{i+1}$. This argument uses a positive-degree lift, not
a degree-one section. Finally $m_j\gamma_{ij}\cdot C$ is a
nonnegative integer, giving the stated lower bound.
\end{proof}
\begin{corollary}[Uniform crepancy parameter; no termination assertion]
\label{prog:uniform-canonical-parameter}
Suppose additionally that the working spaces are $d$-dimensional klt
spaces, that $\alpha_i$ are nef, and that the ordinary canonical
negative-ray length bound $0<-K_{X_i}\cdot C\leq2d$ is available.
For a fixed real number $t>2d/\delta$, every negative extremal ray
chosen in an existing $(K_{X_i}+t\alpha_i)$-program is
$\alpha_i$-trivial. If the steps are projective ordinary canonical
steps, Lemma~\ref{prog:fixed-integral-grid} therefore applies
inductively with this same value of $t$.
If $\alpha_0$ is not big and $\alpha_0-c_1(K_{X_0})$ is big, then
$K_{X_i}+t\alpha_i$ remains non-pseudo-effective along these crepant
steps. Consequently a terminating such program ends with a Mori
fibre contraction. This corollary does not assert that the requisite
contractions exist or that the program terminates.
\end{corollary}
\begin{proof}
A negative ray is $K_{X_i}$-negative because $\alpha_i$ is nef. Choose
its rational curve generator with the ordinary length bound. If its
$\alpha_i$-degree were positive, then
\[
(K_{X_i}+t\alpha_i)\cdot C\geq-2d+t\delta>0,
\]
a contradiction. The step is thus an ordinary canonical step with
$\alpha_i$ pulled back from its base, and the preceding lemma applies.
On a common resolution the pullback of
$\alpha_{i+1}-c_1(K_{X_{i+1}})$ is the pullback of
$\alpha_i-c_1(K_{X_i})$ plus the effective canonical comparison.
Thus it stays big. Equality of the crepant pullbacks of $\alpha_i$
preserves its nonbigness. The
identity
\[
(t+1)\alpha_i=(c_1(K_{X_i})+t\alpha_i)
+(\alpha_i-c_1(K_{X_i}))
\]
excludes pseudo-effectivity of its first summand. A nef endpoint is
therefore impossible.
\end{proof}
\begin{lemma}[Cohomology splitting along detected steps]
\label{prog:detected-cohomology}
Let $X_0$ be a smooth compact K\"ahler manifold. Consider a finite
sequence of divisorial contractions and small flips
\[
X_0\dashrightarrow X_1\dashrightarrow\cdots
\dashrightarrow X_r
\]
between normal globally strongly $\mathbb Q$-factorial compact
K\"ahler spaces with rational singularities.
Assume that every negative contraction $f_i:X_i\to Z_i$ is a
projective contraction of one ray of the analytic cone, that $Z_i$
has rational singularities, and that each small flip is the detected
flip of a global rational line $L_i$ with $L_i\cdot R_i<0$.
For a divisorial contraction assume the usual single exceptional
prime. Then, for every projective smooth resolution $p:U\to X_i$,
\begin{align*}
H^2(U,\mathbb R)
&=p^*H^2(X_i,\mathbb R)
\oplus\bigoplus_{E\text{ exceptional for }p}\mathbb R[E],\\
H^{1,1}(U,\mathbb R)
&=p^*H^{1,1}_{\rm BC}(X_i,\mathbb R)
\oplus\bigoplus_{E\text{ exceptional for }p}\mathbb R[E].
\end{align*}
The first equality also holds over $\mathbb Q$. There are compatible
forward linear transforms on $H^2(-,\mathbb R)$ and on Bott--Chern
cohomology. They are surjective, are isomorphisms for a small flip,
and preserve rational degree-two classes. On a common resolution the
pullback difference is an actual exceptional divisor class; for a
flip it is exceptional over both spaces.
The exceptional quotient also takes the Chern class of a holomorphic
line bundle on $U$ to a rational holomorphic line class on $X_i$.
\end{lemma}
\begin{proof}
For a modification of a smooth compact K\"ahler manifold the two
splittings are the degree-two modification formula. Suppose that they
hold for $X=X_i$. We use only the following birational descent fact:
for a proper bimeromorphic map between normal compact spaces in
Fujiki's class with rational singularities, pullback is injective on
$H^2(-,\mathbb R)$ and on Bott--Chern cohomology, and its image in
either group consists of the classes having degree zero on every
contracted curve. This is the bimeromorphic case of
\cite[Lemma~2.6(1)]{DH2024}.
First note that a degree-two class $\xi\in H^2(X,\mathbb R)$ has
the same degrees on curves as a Bott--Chern class. Indeed, pull it to
a smooth projective resolution $a:W\to X$ and take the $(1,1)$-part
of its Hodge decomposition. The induction hypothesis writes that
part as $a^*\beta+[F]$, with $F$ exceptional over $X$. On every
$a$-contracted curve, $a^*\xi$ and its $(2,0)$ and $(0,2)$ parts
have degree zero. Hence $F$ has degree zero on every such curve.
Exceptional negativity applied to both signs gives $F=0$.
Projective multisections lifting a curve of $X$ then show that
$\xi$ and $\beta$ have the same degrees on every curve of $X$.
Suppose first that $X\dashrightarrow X^+$ is a small detected flip.
Write $f:X\to Z$ and $f^+:X^+\to Z$, and fix an integral curve
$C$ on its negative ray. For $\xi\in H^2(X,\mathbb R)$ set
\[
s=\frac{\xi\cdot C}{c_1(L)\cdot C}.
\]
By the preceding paragraph, $\xi-s c_1(L)$ vanishes on every
$f$-contracted curve. Thus it is $f^*\eta$ for a unique
$\eta\in H^2(Z,\mathbb R)$. Define
\[
T\xi=(f^+)^*\eta+s c_1(L^+).
\]
For a Bott--Chern class, the same descent takes place in
Bott--Chern cohomology, so the two transforms agree. The detected
flip construction gives an effective rational divisor $E_L$ on a
common projective resolution $a:W\to X$, $b:W\to X^+$ such that
\[
a^*c_1(L)-b^*c_1(L^+)=[E_L].
\]
Consequently
\[
a^*\xi=b^*T\xi+s[E_L].
\]
If $\xi$ is rational, then $s$ is rational. Since the image of
$f^*$ on real degree-two cohomology is the realification of a
rational subspace, its unique preimage $\eta$ is rational as well.
This proves rationality of $T\xi$.
The sets of prime divisors exceptional for $a$ and $b$ are identical:
the two working spaces are small bimeromorphic. The old splitting
and the displayed pullback identity therefore span
\[
H^2(W,\mathbb R)
=b^*H^2(X^+,\mathbb R)
+\operatorname{span}_{\mathbb R}\{[E]:E\text{ is }b
\text{-exceptional}\},
\]
and similarly in bidegree $(1,1)$. The sum is direct. If a pullback
from $X^+$ equals an exceptional divisor class, that divisor has
degree zero on all $b$-contracted curves, so both signs of negativity
make it zero; pullback injectivity then makes the original class
zero. In particular the dimensions on the two sides agree, $T$ is
injective by the old direct sum, and $T$ is surjective. The rational
version follows either from the same argument or from realification.
For a divisorial contraction $f:X\to X^+$, let $D$ be its
exceptional prime. Its degree on the contracted ray is negative:
otherwise $D$ would be $f$-nef, contrary to exceptional negativity.
For any $\xi$, subtract
\[
\frac{\xi\cdot C}{D\cdot C}[D]
\]
and descend the result by the same birational cohomology lemma.
This defines $T\xi$. On a common resolution its pullback difference
is a multiple of the total transform of $D$. The target's exceptional
prime set is the old one together with the strict transform of $D$.
The same spanning and directness argument gives the target splitting
and surjectivity; its kernel is $\mathbb R[D]$. Rationality is again
immediate from the quotient of rational intersection numbers.
These arguments initially prove the splitting on a chosen common
resolution. To obtain it on any projective smooth resolution
$U\to X^+$, dominate that resolution and the chosen one by a smooth
common projective resolution. Apply the smooth degree-two
modification formula upstairs and push down to $U$. An exceptional
divisor pushes either to zero or to a divisor exceptional over $X^+$.
This proves spanning on $U$; negativity gives directness there.
Finally let $\mathcal H$ be a holomorphic line on $U$ and put
$\mathcal Q=(p_*\mathcal H)^{**}$. Global strong
$\mathbb Q$-factoriality gives an $m>0$ for which
$\mathcal M=\mathcal Q^{[m]}$ is invertible. The coherent evaluation
comparison gives
\[
\mathcal H^{\otimes m}
\simeq p^*\mathcal M\otimes\mathcal O_U(F)
\]
for an integral $p$-exceptional divisor $F$. This comparison is
obtained from the coherent direct image and evaluation map; it does
not assume that $\mathcal H$ has a global meromorphic section.
Dividing first Chern classes by $m$ proves the line assertion.
\end{proof}
\begin{corollary}[Stability under projective extraction]
\label{prog:extraction-cohomology}
Let $r:Y\to X$ be a projective bimeromorphic morphism of normal compact
K\"ahler spaces with rational singularities. Suppose that $Y$ is globally
strongly $\mathbb Q$-factorial and that $X$ has the splitting property in
the preceding lemma. Then $Y$ has that property as well. More precisely,
if $E_1,\ldots,E_s$ are the prime divisors exceptional for $r$, then
\[
H^2(Y,\mathbb R)
=r^*H^2(X,\mathbb R)\oplus
\bigoplus_{j=1}^{s}\mathbb R[E_j],
\]
and the analogous statements hold over $\mathbb Q$ and in
Bott--Chern cohomology.
\end{corollary}
\begin{proof}
Choose any projective smooth resolution $q:W\to Y$. The composite
$p=rq$ is a projective resolution of $X$. Its exceptional primes are
exactly the $q$-exceptional primes together with the strict transforms
$\widetilde E_j$ of the $E_j$. Since every $E_j$ is $\mathbb Q$-Cartier,
\[
[\widetilde E_j]=q^*[E_j]+[F_j]
\]
for a rational $q$-exceptional divisor $F_j$. Substitute these identities
in the splitting for $p$. The result spans $H^2(W,\mathbb R)$ by
$q^*H^2(Y,\mathbb R)$ and the $q$-exceptional divisor classes. Their sum
is direct: an exceptional divisor whose class is a pullback has degree
zero on every $q$-contracted curve, and applying exceptional negativity
to both signs makes the divisor zero. Pullback injectivity then applies.
The same argument in bidegree $(1,1)$ and over $\mathbb Q$ proves the
splitting property for $Y$.
Now expand $q^*\xi$, for $\xi\in H^2(Y,\mathbb R)$, using the original
$p$-splitting and replace the $[\widetilde E_j]$ as above. The difference
between $q^*\xi$ and a class in
$q^*(r^*H^2(X,\mathbb R)+\sum_j\mathbb R[E_j])$ is $q$-exceptional,
and is therefore zero by the direct sum just proved. Thus the displayed
formula for $H^2(Y,\mathbb R)$ spans. Its directness follows from
exceptional negativity for $r$ and pullback injectivity. The rational
and Bott--Chern statements follow by the identical argument. The line
bundle assertion of the preceding lemma uses only global strong
$\mathbb Q$-factoriality and hence applies to $Y$ as well.
\end{proof}
\paragraph{Support in the special-termination comparison.}
\label{prog:special-support}
The following support comparison applies to the detected steps in
the dimension induction. Let $\psi:X\dashrightarrow X^+$ be a detected
small negative step with maps $f:X\to Z$ and $f^+:X^+\to Z$,
and let $L$ be its negative detector. For a sufficiently
divisible $r>0$, put
\[
\operatorname{im}(f^*f_*L^r\longrightarrow L^r)
=\mathcal I\otimes L^r.
\]
The zero set of $\mathcal I$ is exactly the exceptional locus of $f$.
Outside that locus $f$ is an isomorphism, so the evaluation map is
surjective. On a positive-dimensional projective fibre, every section
of $L^r$ vanishes: its restriction to each integral curve has negative
degree, and those curves cover every positive-dimensional fibre
component. Connectedness supplies the whole nontrivial fibre.
On a resolution principalizing $\mathcal I$, the detector comparison
is a positive multiple of the effective divisor defined by
$\mathcal I\mathcal O_W$. Consequently its support is the full
inverse image of the exceptional locus. The driving adjoint differs
from a positive real multiple of $c_1(L)$ by a class pulled back from
the contraction base. Its comparison divisor is therefore the same
positive multiple of the detector comparison: their class difference
is exceptional and zero, so both-sign negativity makes it zero as a
divisor. The comparison for a longer
negative program is at least this first-step divisor, by discrepancy
monotonicity on a common resolution.
Suppose the beginning and end restrictions to a surviving normalized
lc stratum $S$ are isomorphic and their strict adjunction data agree.
The general point of this stratum avoids every intervening surgery:
a zero-discrepancy place remains such a place, whereas strict
comparison would increase its discrepancy if its center were contained
in the exceptional locus. Thus its strict transform $S_W$ on a common
resolution is not contained in the comparison support. Iterated strict
adjunction identifies the restriction of the effective comparison with
the difference of the identified adjoints, so its class is zero.
A nonzero effective divisor on a positive-dimensional compact K\"ahler
stratum has positive K\"ahler mass. The restricted divisor is therefore
zero. If the first step meeting the stratum were nontrivial there,
the full inverse-image support and surjectivity $S_W\to S$ would make
this restriction nonzero, a contradiction. For a zero-dimensional
stratum, generic-point avoidance already excludes an intersection.
The whole program is consequently an isomorphism near the stratum.
This is the same strict-adjunction support argument used in the proof
of Theorem~\ref{prog:special-termination}; it does not use its
lower-dimensional abundance assumption.
\begin{proposition}[A uniform relative NQC bound]
\label{prog:relative-nqc-grid}
Let $f_0:U\to Z$ be an already projective surjective morphism with
connected fibres between smooth compact
K\"ahler manifolds, and suppose that
\[
f_0^*:H^0(Z,\Omega_Z^2)\longrightarrow H^0(U,\Omega_U^2)
\]
is an isomorphism. Let $\alpha_0$ be a nef class with an expression
\[
\alpha_0=\sum_{j=1}^k r_j\eta_{j,0},\qquad
r_j>0,\quad \eta_{j,0}\in H^2(U,\mathbb Q),
\]
where $\eta_{j,0}$ has nonnegative degree on every curve vertical
over $Z$. Let $D_0=K_U+\Delta_U+\mathbf M_U$ be a generalized klt
adjoint. Consider its projective relative program for
$D_0+a\alpha_0$, under the projective local ordinary reduction described in the
proof of Lemma~\ref{prog:base-point-free-induction}.
If $\alpha_0=0$, triviality is immediate. Otherwise there are
positive integers $m_j$, chosen once on $U$, and a number
\[
\delta=\min_j\frac{r_j}{m_j}>0
\]
such that, for $a\delta>2\dim U$, the entire relative program is
$\alpha_0$-trivial. The same number $a$ works at every step.
\end{proposition}
\begin{proof}
\emph{Rational classes modulo the fixed base.}
Choose an $f_0$-ample line bundle with integral Chern class $H$, and
put $e=\dim U-\dim Z$ and $c=f_{0*}(H^e)>0$. The rational operator
\[
\Pi(\eta)=\eta-
f_0^*\!\left(\frac{f_{0*}(\eta\smile H^e)}{c}\right)
\]
preserves type $(1,1)$ and kills types $(2,0)$ and $(0,2)$.
Indeed these latter classes are pulled back from $Z$, and the
projection formula applies. Thus
$\Pi(H^2(U,\mathbb Q))\subset H^{1,1}(U)\cap H^2(U,\mathbb Q)$.
By the Lefschetz $(1,1)$ theorem, choose an actual line bundle
$L_{j,0}$ and an integer $m_j>0$ with
\[
\eta_{j,0}=\frac1{m_j}c_1(L_{j,0})+f_0^*\gamma_j,
\qquad \gamma_j\in H^2(Z,\mathbb Q).
\]
The original summands $\eta_{j,0}$ need not have type $(1,1)$.
Their base components account for this. Their weighted sum gives
\begin{equation}\label{prog:eq:relative-nqc}
\alpha_0=\sum_j\frac{r_j}{m_j}c_1(L_{j,0})+f_0^*\Gamma,
\qquad \Gamma=\sum_jr_j\gamma_j\in H^{1,1}(Z,\mathbb R).
\end{equation}
The final assertion follows because $f_0^*$ is an injective Hodge
map and the other terms have type $(1,1)$. The identity is therefore
also one of Bott--Chern classes.
Each $L_{j,0}$ has nonnegative integral degree on every
$Z$-vertical curve.
\smallskip\noindent
\emph{Induction across an actual contraction.}
Suppose at a finite stage $f_i:U_i\to Z$ we have actual line bundles
$L_{j,i}$, nef over $Z$, and
\[
\alpha_i=\sum_j\frac{r_j}{m_j}c_1(L_{j,i})+f_i^*\Gamma.
\]
Assume also that $\alpha_i$ is nef and that the transformed
$D_i$ is generalized klt. These assertions hold initially.
Let $R$ be a $(D_i+a\alpha_i)$-negative extremal ray over $Z$.
It is $D_i$-negative because $\alpha_i$ is nef. Lemma~\ref{prog:relative-hodge} and Lemma~\ref{prog:projective-cone}
identify the relative curve cone with the corresponding face of the
full analytic cone, at the initial model and after every step. The
analytic cone theorem therefore supplies a rational generator $C$ with
\[
0<-D_i\cdot C\leq2\dim U.
\]
All $L_{j,i}\cdot C$ are nonnegative integers. If
$\alpha_i\cdot C>0$, one of them is at least one, and hence
$\alpha_i\cdot C\geq\delta$. Therefore
\[
(D_i+a\alpha_i)\cdot C
\geq-2\dim U+a\delta>0,
\]
a contradiction. We have $\alpha_i\cdot C=0$ and
$L_{j,i}\cdot C=0$ for every $j$. Every curve contracted by the
projective extremal contraction $h_i:U_i\to Y_i$ has class on
$R$ when tested by global line bundles, so the same vanishing
holds for all of them.
On each fixed Stein chart of $Z$, choose the effective real
ordinary representative of the initial relatively ample nef
datum once, before running the program. Its actual real linear
equivalence to the fixed real line-bundle representative persists
under pushforward. The Bott--Chern comparison with the generalized
adjoint modulo the fixed base class persists separately by
exceptional negativity, as detailed below.
Thus every $h_i$ is, locally on $Y_i$, a contraction with an
ordinary real klt log-Fano boundary. On a relatively compact base
chart, rational approximation in the finite affine system of Cartier
adjoint identities gives a rational effective klt boundary with the
same antiample sign, as in Proposition~\ref{prog:detected-flip}.
Lemma~\ref{prog:integral-descent} therefore applies.
Consequently
\[
M_{j,i}:=(h_i)_*L_{j,i}\ \text{is a line bundle},\qquad
h_i^*M_{j,i}=L_{j,i}.
\]
For a divisorial contraction set $L_{j,i+1}=M_{j,i}$.
For a flip $h_i^+:U_{i+1}\to Y_i$, set
$L_{j,i+1}=(h_i^+)^*M_{j,i}$.
These are actual line bundles with the \emph{same} integers $m_j$.
The descended Bott--Chern class
\[
\alpha_{Y_i}
=\sum_j\frac{r_j}{m_j}c_1(M_{j,i})+f_{Y_i}^*\Gamma
\]
satisfies $h_i^*\alpha_{Y_i}=\alpha_i$.
It is nef by descent of nefness through the projective surjection
$h_i$; its pullback to the next model is $\alpha_{i+1}$.
This proves that the step is crepant for $\alpha_i$. Hence it is
also a $D_i$-negative step, and the original generalized pair
remains gklt.
Finally each $M_{j,i}$ is nef over $Z$. For a $Z$-vertical curve
$B\subset Y_i$, projectivity supplies a curve $B'\subset U_i$
mapping onto it with some positive degree $d_B$. Then
\[
d_B(M_{j,i}\cdot B)=L_{j,i}\cdot B'\geq0.
\]
No assertion $d_B=1$ is required. Pullback preserves this relative
nefness, so the induction applies on $U_{i+1}$. Its curve degrees
are again integers because the transported objects are line
bundles, not because any curves were lifted with degree one.
This proves the fixed bound through the whole program.
\end{proof}
\begin{remark}[Scope of the degree grid]
The transported rational classes
\[
\eta_{j,i}=\frac1{m_j}c_1(L_{j,i})+f_i^*\gamma_j
\]
have degrees in $m_j^{-1}\mathbb Z_{\geq0}$ on curves vertical over
the fixed smooth base $Z$. This relative assertion is exactly what
the initial projective program requires. It does not assert
nonnegativity of these individual classes on all curves of every
birational model, and uses neither rational connectedness of
resolution fibres nor Graber--Harris--Starr.
\end{remark}
\subsubsection{The negative-part comparison for a nef adjoint}
We write $N_\sigma(\xi)$ for the divisorial negative part of a
pseudo-effective class. We use its homogeneity, subadditivity,
continuity after adding a vanishing K\"ahler perturbation, and the
identities
\[
N_\sigma(r^*\xi+[E])=N_\sigma(r^*\xi)+E,
\qquad r_*N_\sigma(r^*\xi)=N_\sigma(\xi)
\]
for an effective $r$-exceptional divisor $E$. A nef class has zero
negative part. These are the divisorial Zariski-decomposition
properties of \cite[Section~3]{Boucksom2004} and
\cite[Appendix~A]{DHY}. No identity
$N_\sigma(r^*\xi)=r^*N_\sigma(\xi)$ for arbitrary $\xi$ is assumed.
\begin{lemma}[The absolute negative part after a relative program]
\label{prog:negative-part-program}
Let $\mu:U\to X$ be a projective resolution, let $\alpha$ be nef on $X$,
let $c>0$, and let $F_U\geq0$ be $\mu$-exceptional. Put
\[
D_U=c\mu^*\alpha+[F_U].
\]
Suppose that $\phi:U\dasharrow V$ is a finite $D_U$-negative program,
possibly relative to another base, and denote its transformed class and
divisor by $D_V$ and $F_V=\phi_*F_U$. Then
\[
N_\sigma(D_V)=F_V.
\]
\end{lemma}
\begin{proof}
Take a common projective resolution $p:W\to U$, $q:W\to V$.
The negativity comparison for the finite program is
\[
p^*D_U=q^*D_V+[E],
\qquad E\geq0,\quad E\text{ is }q\text{-exceptional}.
\]
This comparison is valid for a relative negative program: its proof
uses the negativity of the contracted rays and the positive adjoint on
each flipped side, not absolute nefness of the final adjoint.
Since $p^*F_U$ is effective and exceptional over $X$, the exceptional
identity and nefness of $(\mu p)^*\alpha$ give
\[
N_\sigma(p^*D_U)=p^*F_U
=N_\sigma(q^*D_V)+E.
\]
Pushing forward by $q$ proves the assertion. In particular, the
negative part in this statement is absolute, even when the program
that produced $V$ was relative.
\end{proof}
\subsubsection{Descent of the effective vertical divisor}
\begin{lemma}[Vertical numerical triviality]
\label{prog:vertical-divisor}
Let $g:V\to S$ be a projective surjective morphism with connected fibres.
Assume that $S$ is globally Weil $\mathbb{Q}$-factorial. Let $F\geq0$ be
an effective real Cartier divisor on $V$, vertical over $S$, such that
\[
F\cdot C=0
\quad\text{for every curve }C\text{ contracted by }g.
\]
Then there is an effective real Cartier divisor $F_S$ on $S$ such that,
as actual divisors,
\[
F=g^*F_S.
\]
\end{lemma}
\begin{proof}
For a prime divisor $P\subset S$, write $m_Q$ for the multiplicity of
$Q$ in $g^*P$, where $Q$ ranges over the prime divisors dominating $P$.
These multiplicities may be computed over the smooth generic locus of
$P$, where $P$ is Cartier. Put
\[
b_P=\min_{g(Q)=P}\frac{\operatorname{coeff}_Q F}{m_Q},
\qquad F_S=\sum_P b_PP.
\]
Only finitely many $b_P$ are nonzero, since any such $P$ is the image of
a component of $F$. Thus $F_S$ is a genuine effective Weil real divisor.
Global Weil $\mathbb{Q}$-factoriality makes this finite divisor real
Cartier.
We first check equality over codimension one in $S$. Work near a
general point of $P$ and restrict to a transverse disk. Resolving the
source, and cutting by general relative ample hypersurfaces if the
relative dimension exceeds one, reduces to a projective surface over
that disk with connected fibres. These operations can be performed
over a relatively compact Stein neighbourhood; they do not require
global sections on $S$. The intersection matrix of the components of
the special fibre is negative semidefinite, with kernel generated by
the full fibre with its multiplicities. The restriction of $F$ has
zero intersection with every fibre component. Its coefficients are
therefore proportional to those multiplicities. Equivalently, all the
ratios in the definition of $b_P$ are equal. When $g$ is birational this
assertion over the generic point of $P$ is immediate.
Consequently
\[
G=F-g^*F_S
\]
is a signed real Cartier divisor supported over a subset of codimension
at least two in $S$. Moreover, $G$ is numerically trivial over $S$.
We check that such a signed exceptional divisor is zero. The assertion
is local on $S$. Over a relatively compact Stein neighbourhood, take
$d=\dim V-\dim S$ general relative ample hypersurfaces and resolve their
intersection. They give a projective generically finite morphism
$H\to S$. The cuts may be chosen to meet any prescribed component of
$G$ in a nonzero divisorial trace. After normalization and Stein
factorization, $H\to S$ factors as a projective birational morphism
$H\to S'$ followed by a finite morphism $S'\to S$. The restricted
divisor $G|_H$ is exceptional for $H\to S'$ and numerically trivial
over $S'$. Applying the ordinary exceptional negativity lemma to both
$G|_H$ and $-G|_H$ gives $G|_H=0$. The choice of the cuts therefore
excludes every nonzero component of $G$. Hence $G=0$.
The surface argument above is also the usual proof of the
degenerate-divisor negativity statement: after subtracting the minimum
multiple of the full fibre, an effective residual fibre divisor omits
a component and cannot be numerically trivial. Notice that projectivity
of $g$ was a hypothesis; it was not deduced from factoriality.
\end{proof}
\subsubsection{The negative part on the lower-dimensional base}
\begin{lemma}[Detecting the negative part by pullback currents]
\label{prog:negative-part-base}
Let $g:V\to S$ be a surjective morphism. Suppose that $\alpha_S$ is nef,
$F_S\geq0$ is real Cartier, $c>0$, and
\[
D_S=c\alpha_S+[F_S],\qquad D_V=g^*D_S,\qquad
N_\sigma(D_V)=g^*F_S.
\]
Then $N_\sigma(D_S)=F_S$.
\end{lemma}
\begin{proof}
Nefness gives $N_\sigma(D_S)\leq F_S$. Fix a prime divisor $P\subset S$
and a prime divisor $Q\subset V$ dominating it. Write
$m=\operatorname{mult}_Q(g^*P)>0$ and $b=\operatorname{coeff}_P F_S$.
At their generic smooth points, pullback of positive currents satisfies
\[
\nu(g^*T,Q)=m\nu(T,P).
\]
Indeed, the divisorial term $\nu(T,P)[P]$ pulls back with multiplicity
$m$, and the residual current has zero generic Lelong number there.
Choose K\"ahler forms $\omega_S,\omega_V$ and a constant $C>0$ for which
$C\omega_V-g^*\omega_S$ is K\"ahler. For $\varepsilon>0$, let $T_\varepsilon$
be any positive current in the big class $D_S+\varepsilon[\omega_S]$.
Minimality of the multiplicity and monotonicity under adding a nef
class give
\begin{align*}
m\nu(T_\varepsilon,P)
&=\nu(g^*T_\varepsilon,Q)\\
&\geq\nu(D_V+\varepsilon g^*[\omega_S],Q)\\
&\geq\nu(D_V+C\varepsilon[\omega_V],Q).
\end{align*}
Taking the infimum over $T_\varepsilon$, and then letting
$\varepsilon\downarrow0$, gives
\[
m\nu(D_S,P)\geq\nu(D_V,Q)
=\operatorname{coeff}_Q(g^*F_S)=mb.
\]
This proves the reverse inequality for every $P$.
All multiplicities are unchanged by resolving away from the generic
points in question, so the same argument applies to the normal spaces
under consideration.
The $\varepsilon$ perturbation is necessary: at a pseudoeffective
boundary class, the infimum over its exact positive currents need not
equal its minimal multiplicity. The argument uses that equality only
for the big perturbed classes.
\end{proof}
\begin{lemma}[Pullback when the positive part is nef]
\label{prog:negative-part-pullback}
Suppose
\[
\xi=P+[F],\qquad P\text{ nef},\quad F\geq0\text{ real Cartier},
\qquad N_\sigma(\xi)=F.
\]
For every projective resolution $p:W\to S$,
\[
N_\sigma(p^*\xi)=p^*F.
\]
\end{lemma}
\begin{proof}
Write $N'=N_\sigma(p^*\xi)$. Since $p^*P$ is nef, $N'\leq p^*F$.
Birational invariance gives $p_*N'=F$, so
$E=p^*F-N'\geq0$ is $p$-exceptional. The positive part of $p^*\xi$ is
\[
p^*P+[E],
\]
and is modified nef. If $E\neq0$, exceptional negativity in its
covering-curve form provides a component of $E$ covered by curves $C_t$
contracted by $p$, with $E\cdot C_t<0$
\cite[Appendix~A, Lemma~A.3]{DHY}. A modified nef class has nonnegative
intersection with a general member of a family of curves covering a
prime divisor: use currents with arbitrarily small negative part and
zero generic divisorial Lelong number, restrict to a general member,
and let the negative bound tend to zero. But here
\[
(p^*P+E)\cdot C_t=E\cdot C_t<0,
\]
a contradiction. Thus $E=0$.
\end{proof}
\begin{proposition}[Any lower-dimensional good model suffices]
\label{prog:nef-good-comparison}
Suppose the initial relative program has the data in
Lemma~\ref{prog:negative-part-program}. Assume in addition that
its class $\alpha_V$ is nef and agrees with $\mu^*\alpha$ on a common
resolution, and that there is an already projective connected-fibre
morphism $g:V\to S$ with
\[
\alpha_V=g^*\alpha_S,\qquad D_V=g^*D_S.
\]
Suppose there is a projective small modification
$s:S^{\mathrm q}\to S$ such that $S^{\mathrm q}$ is globally Weil
$\mathbb Q$-factorial; the identity is allowed. Assume that the
lower-dimensional adjoint $s^*D_S$ has a good model:
there are a normal compact K\"ahler space $S_m$, a common projective
resolution $p:W\to S^{\mathrm q}$, $q:W\to S_m$, and a nef semiample
class $D_m$ such that
\[
p^*s^*D_S=q^*D_m+[E],
\qquad E\geq0,\quad E\text{ is }q\text{-exceptional}.
\]
Then $\alpha$ is semiample on $X$. If the contraction defining
semiampleness of $D_m$ is Moishezon, the resulting contraction of
$\alpha$ is Moishezon as well. In particular, it is unnecessary to
lift a program on $S$ to $V$, or to require that a chosen program
producing $S_m$ preserve $\alpha_S$ at every intermediate step.
\end{proposition}
\begin{proof}
Nefness descends under the surjective morphism $g$, so $\alpha_S$ is nef.
The class of $F_V$ is pulled back from $S$, so $F_V$ is numerically
trivial over $S$. It is therefore vertical: an effective horizontal
component would restrict to a nonzero effective divisor on a general
projective fibre, with positive intersection against a suitable power
of an ample class, contradicting numerical triviality. For a
birational $g$, verticality is automatic by dimension.
First take a projective resolution $\pi:\widetilde V\to V$ of the main
component of $V\times_S S^{\mathrm q}$. The induced map
$\widetilde g:\widetilde V\to S^{\mathrm q}$ is projective and has
connected fibres. Pull back $\alpha_V,D_V,F_V$ along $\pi$, and pull
back $\alpha_S,D_S$ along $s$. All displayed class identities are
preserved, and $\pi^*F_V$ remains an effective vertical divisor.
Lemma~\ref{prog:negative-part-program} gives
$N_\sigma(D_V)=F_V$ before this modification. Because
$D_V=c\alpha_V+[F_V]$ with $\alpha_V$ nef,
Lemma~\ref{prog:negative-part-pullback} then gives
\[
N_\sigma(\pi^*D_V)=\pi^*F_V.
\]
We may therefore replace $(V,S,g)$ by
$(\widetilde V,S^{\mathrm q},\widetilde g)$ and suppress the new
superscripts in the rest of the proof. The good-model comparison now
reads $p^*D_S=q^*D_m+[E]$. No assertion that the crepant subboundary on
$\widetilde V$ is effective is needed: this space is used only for
classes, currents and the effective divisor $\pi^*F_V$. The
lower-dimensional generalized pair is the crepant pullback to the
small model $S^{\mathrm q}$.
The class identity $[F_V]=g^*(D_S-c\alpha_S)$ shows that $F_V$ is
numerically trivial over $S$. Lemma~\ref{prog:vertical-divisor}
gives an actual effective real Cartier divisor $F_S$ with
$F_V=g^*F_S$. Injectivity of pullback yields
\[
D_S=c\alpha_S+[F_S].
\]
The established identity $N_\sigma(D_V)=F_V$ and
Lemma~\ref{prog:negative-part-base} therefore give
$N_\sigma(D_S)=F_S$, and
Lemma~\ref{prog:negative-part-pullback} gives
\[
N_\sigma(p^*D_S)=p^*F_S.
\]
On the other hand $q^*D_m$ is nef, so the exceptional-divisor identity
applied to the good-model comparison gives
\[
N_\sigma(p^*D_S)=E.
\]
Thus $E=p^*F_S$ as actual divisors. Subtracting them from the comparison
proves the exact class equality
\[
q^*D_m=c\,p^*\alpha_S.
\]
This proves the needed preservation of $\alpha_S$ from the final good
model, without an assumption about its intermediate models.
Choose a contraction $h:S_m\to T$ to a normal compact K\"ahler space
and a K\"ahler class $\kappa$ on $T$ with $D_m=h^*\kappa$. Taking a
resolution of the main component of $V\times_S W$, and then a common
projective resolution with $U$, produces a projective modification
$r:R\to X$ and a holomorphic map $\ell:R\to T$ satisfying
\[
r^*\alpha=\frac1c\ell^*\kappa.
\]
For every curve $C$ in a fibre of $r$ this equality implies
$\ell(C)$ is a point, since a K\"ahler class has positive degree on
every nonconstant image curve. The fibres of the projective modification
$r$ are connected projective complex spaces, hence are connected by
chains of curves. Therefore $\ell$ is
constant on each fibre of $r$. The factorization theorem for a proper
map onto a normal space gives a holomorphic map $f:X\to T$ with
$\ell=f\circ r$. Pushing forward the class equality by $r$ gives
\[
\alpha=f^*(\kappa/c).
\]
The maps from the main fibre-product component to $S_m$ have connected
general fibres; their Stein factorizations are finite birational over
the normal space $S_m$, hence have connected fibres everywhere.
Together with the connected fibres of $h$, this shows that $\ell$,
and therefore $f$, has connected fibres. This is the required
semiampleness of $\alpha$.
Finally, $R\to S_m$ is projective: it is obtained from the projective
map $g$ by base change, followed by projective resolutions and the
projective map $q$. If $h$ is Moishezon, choose a projective surjection
$H\to S_m$ such that $H\to T$ is projective. A component of
$R\times_{S_m}H$ dominating $R$ is projective and surjective over $X$,
and its map to $T$ is projective. This is a Moishezon witness for $f$.
\end{proof}
\begin{lemma}[The contraction step in the dimension induction]
\label{prog:base-point-free-induction}
Assume $\mathsf C_{d-1}$, $\mathsf B_{d-1}$ and $\mathsf M_{d-1}$.
Then $\mathsf C_{d,\mathrm{big}}$ and $\mathsf B_d$ hold.
\end{lemma}
\begin{proof}
\smallskip\noindent
\emph{The big non-klt contraction.}
Apply the construction of \cite[Section~3]{HX}. On its smooth
modification $\nu:U\to X$ it writes
\[
\nu^*\alpha=[D_U]+\eta_U,
\qquad D_U\geq0,\quad \eta_U\text{ K\"ahler}.
\]
The induction over the jumping coefficients of the multiplier
ideal reduces the new reduced stratum to dimension at most $d-1$.
The given contraction on the old non-klt subspace and
$\mathsf C_{d-1}$ provide the Moishezon maps of those strata.
Every map that is promoted to a projective map in that construction
contracts exactly the curves on which $\nu^*\alpha$ vanishes.
On each such curve the actual real line bundle $-D_U$ has degree
$\eta_U\cdot C$. On a further projective common resolution
$q:U''\to U$, choose an effective exceptional divisor $E$ with
$-E$ relatively ample and choose $\varepsilon>0$ sufficiently small.
Use the decomposition
\[
q^*\nu^*\alpha=[D'']+\eta'',\qquad
D''=q^*D_U+\varepsilon E,\qquad
\eta''=q^*\eta_U-\varepsilon[E].
\]
Here $\eta''$ is K\"ahler, and the actual real line bundle $-D''$
has degree $\eta''\cdot C$ on every contracted curve. Restrict this
decomposition to the smooth reduced components in the gluing
construction. Lemma~\ref{prog:actual-line-polarization} then supplies
precisely the relative ampleness needed in place of
\cite[Lemma~2.42]{HX}, including the common-resolution step in its
Claim~3.5. The exceptional term $-\varepsilon[E]$ makes the
positive part $\eta''$ K\"ahler on the common resolution.
The exact sequences of multiplier ideals, relative vanishing,
finite pushouts, and extension from sufficiently thickened
$\operatorname{Supp}D_U$ in that proof then apply unchanged.
They give the birational Moishezon contraction in
$\mathsf C_{d,\mathrm{big}}$. In particular the gluing step uses
neither generalized termination nor a projectivity criterion for
an undetected ray.
\smallskip\noindent
\emph{Preparation for both cases of $\mathsf B_d$.}
The modified-big perturbation
\cite[Lemma~2.23]{HX} supplies a nef datum that dominates a
K\"ahler class on its carrier. Subtracting a sufficiently small
multiple of the pullback of a K\"ahler class $\omega_X$ still leaves
a nef datum on that carrier. Thus
$\alpha$ is a gklt adjoint plus $\varepsilon\omega_X$, and the cone
argument of \cite[Lemma~2.45]{HX} makes $\alpha$ NQC.
This preparation applies whether or not $\alpha$ is big.
\smallskip\noindent
\emph{The nef-and-big case of $\mathsf B_d$.}
For a gklt pair the multiplier ideal is the unit ideal, so the
preceding big non-klt assertion now applies and gives a birational
contraction $h:X\to Y$.
On a projective resolution $p:W\to X$ chosen projective over $Y$,
relative Kawamata--Viehweg vanishing for the effective exceptional
divisor $-\lfloor B_W\rfloor$ gives
$R^ih_*\mathcal O_X=0$ for $i>0$. Thus $Y$ has rational
singularities. Proper birational Bott--Chern descent
\cite[Lemma~8.7]{DHP} gives $\alpha=h^*\gamma$.
The descended adjoint is gklt, nef and big, and has no trivial
curve. The criterion \cite[Theorem~4.3]{HLX} makes $\gamma$
K\"ahler.
For completeness, the projectivity input in that criterion is
restricted to a map $S\to Z'$ between smooth compact K\"ahler
manifolds with rationally connected general fibre: $S$ is the
chosen smooth divisor on a resolution and $Z'$ is the resolution
of the null-locus component. The smooth-source, smooth-base
argument in \cite[Theorem~3.1, Step~1]{CH} applies. Its subsequent
relative program starts with this projective map. Thus this application of the criterion uses only the
smooth-source, smooth-base projectivity argument and an
already-projective relative program.
\smallskip\noindent
\emph{The non-big case of $\mathsf B_d$.}
Use a projective small factorialization and the modified-big
perturbation of the pair. The non-pseudo-effectivity of $K_X$
gives an MRC fibration by \cite{Ou2025}. Choose a smooth K\"ahler
MRC base $Z$ and a smooth modification $\mu:U\to X$ such that
$U\to Z$ is projective. Only the smooth case of
\cite[Theorem~3.1, Step~1]{CH} is needed: pullback identifies
the holomorphic two-forms since the general fibre is rationally
connected. Lemma~\ref{prog:relative-hodge} identifies any $(1,1)$-class on $U$,
modulo a class pulled back from $Z$, with an actual real line bundle.
Lemma~\ref{prog:projective-cone} identifies its relative rays
with rays of the analytic cone; both statements persist along the
projective relative construction.
Write, as in \cite[Theorem~4.1, Claim~4.1]{HX},
\[
D_U(a)=K_U+\Delta_U+\mathbf M_U+a\alpha_U
=(a+1)\alpha_U+F_U,
\quad \alpha_U=\mu^*\alpha,\quad
\Delta_U,F_U\geq0,
\]
where $\Delta_U$ is klt and $F_U$ is $\mu$-exceptional.
The modified-big replacement is chosen, as in the cited Claim~4.1,
so that the nef datum $\mathbf M_U$ on this smooth carrier is
K\"ahler. If a further projective resolution is needed, subtract a
sufficiently small effective exceptional divisor from its nef part
and add that divisor to the subboundary; this preserves the
adjoint and the klt inequalities.
Choose $a_0$ using the uniform bound of
Proposition~\ref{prog:relative-nqc-grid}. It makes the relative
$D_U(a_0)$-program $\alpha_U$-trivial and preserves that choice
through every step. This is a
\emph{projective} relative program from its first step: the nef
data are represented by real line bundles modulo $Z$.
Lemma~\ref{prog:relative-hodge} applies on every
intermediate model over this fixed smooth base.
More precisely, put $r:U\to Z$ and choose one actual global real
line bundle $L$ with
\[
\mathbf M_U+a_0\alpha_U=c_1(L)+r^*\gamma.
\]
The bundle $L$ is relatively ample, because $\mathbf M_U$ is
K\"ahler and $\alpha_U$ is nef. On each of a fixed finite collection
of relatively compact Stein charts of $Z$, choose an effective
real divisor $\Theta$ representing $L$, with $(U,\Delta_U+\Theta)$
klt. The representatives can retain a positive relatively ample
part in their boundaries. Here the equality
$K_U+\Delta_U+\Theta\sim_{\R}K_U+\Delta_U+L$ is an actual
real linear equivalence of line bundles; the equality with
$D_U(a_0)$ modulo $r^*\gamma$ is separately an equality of
Bott--Chern classes.
Fix these ordinary pairs on the initial charts. Their actual
linear equivalences and their Bott--Chern comparisons with $D_U(a_0)$
modulo the fixed base class persist separately throughout a global
relative program. Proposition~\ref{prog:projective-good-model} applies:
the source is smooth, its adjoint is represented by an actual global
real line modulo $Z$, its carrier nef part is relatively ample, and
$D_U(a_0)=(a_0+1)\mu^*\alpha+[F_U]$ is pseudo-effective.
It gives a finite program $U\dasharrow V$ and a good relative endpoint.
This is one global program controlled by fixed ordinary chart pairs.
Its construction uses global line algebras, not a choice of unrelated
local minimal models. Proposition~\ref{prog:relative-nqc-grid} makes
every step $\alpha_U$-trivial with the same value $a_0$.
Here is an explicit ample-model comparison that gives the required
descent of $\alpha_V$. Both $D_V(a_0)$ and $\alpha_V$ are nef over
$Z$. Put $a_1=a_0+1$ and $a_2=a_0+2$. Before the program starts,
choose fixed ordinary representatives on the finite chart cover for
all three relatively ample nef parts $\mathbf M_U+a_j\alpha_U$,
$j=0,1,2$, on one simultaneous log resolution. The
$\alpha_U$-crepancy of the constructed program makes its endpoint
$V$ a weak model for each of these three fixed ordinary adjoints.
Their traces $D_V(a_j)=D_V(a_0)+(a_j-a_0)\alpha_V$ are nef.
Corollary~\ref{prog:projective-semiample} therefore makes
\[
D_V(a_j)=D_V(a_0)+(a_j-a_0)\alpha_V
\]
semiample over $Z$. Their zero curves are exactly the common zero
curves of $D_V(a_0)$ and $\alpha_V$. Their projective ample-model
contractions therefore have the same fibres: these fibres are
connected by curves, and each contraction is constant on the
fibres of the other. Identify the two normal targets, writing the
common contraction as $g:V\to S$. If
$D_V(a_j)=g^*\lambda_j$, with $\lambda_j$ relatively K\"ahler over
$Z$, subtraction gives
\[
\alpha_V=g^*(\lambda_2-\lambda_1)=g^*\alpha_S.
\]
This is an equality of Bott--Chern classes, not merely of curve
degrees. Since the initial program is $\alpha_U$-trivial, it is
also $D_U(a_1)$-negative. From now on put $a=a_1$ and suppress
the argument in $D_U(a)$ and $D_V(a)$.
The maps $V\to Z$ and $S\to Z$ are projective. Moreover $g$ is
projective: a relatively ample bundle for $V\to Z$ restricts to
an ample bundle on every fibre of $g$, and hence is $g$-ample.
Each $D_U(a_j)$ is not big globally: pushing a big such class to
$X$ would make $(a_j+1)\alpha$ big. If it were big over the smooth
MRC base $Z$, relative-canonical positivity
\cite[Theorem~2.48]{HX}, together with the pseudo-effectivity of
$K_Z$, would make it big globally. It is therefore not big over
$Z$. This property persists to its relative semiample model, so
its relative canonical morphism has $\dim S<d$.
Apply Proposition~\ref{prog:base-presentation} to $g$ and the
whole adjoint $D_V$. Its generalized data are effective and gklt.
The nef datum is represented on the initial carrier by
$\mathbf M_U+a\alpha_U$, a K\"ahler class; on a common carrier over
$V$ its pullback is nef and big. Adding the strict transform of the
effective boundary proves that the total boundary on $V$ is globally
modified big. The proposition therefore gives
\[
D_S=K_S+B_S+\mathbf N_S,\qquad D_V=g^*D_S,
\]
with effective gklt data on $S$, a globally nef carrier, and globally
modified-big total boundary. Independently, $\alpha_S$ is nef by
descent of nefness through the surjective projective map $g$.
The lower-dimensional model assertion will be applied to this whole
adjoint $D_S$.
Take a projective small factorialization $s:S^{\mathrm q}\to S$
and resolve the main component of $V\times_S S^{\mathrm q}$.
The resulting maps $\pi:\widetilde V\to V$ and
$\widetilde g:\widetilde V\to S^{\mathrm q}$ are projective,
and $\widetilde V$ is compact K\"ahler. Pull back $D_V$,
$\alpha_V$ and $F_V$ to $\widetilde V$, and $D_S$ and
$\alpha_S$ to $S^{\mathrm q}$. The pair on $S^{\mathrm q}$ is
gklt and its boundary-plus-nef class remains modified big.
Lemma~\ref{prog:negative-part-program} first gives
$N_\sigma(D_V)=F_V$. Lemma~\ref{prog:negative-part-pullback} then gives
$N_\sigma(\pi^*D_V)=\pi^*F_V$. The new total space is used only as a
carrier for this equality and for pullbacks of positive currents;
no effective boundary on it is needed. Rename these spaces $V$
and $S$. No additional relative program is needed for this modification.
The vertical-divisor descent and negative-part lemmas above now
give actual effective divisors $F_S,F_V$ with
\[
F_V=g^*F_S,\qquad
D_S=(a+1)\alpha_S+[F_S],\qquad N_\sigma(D_S)=F_S.
\]
Apply $\mathsf M_{d-1}$ to the generalized pair with adjoint $D_S$.
On a common resolution $p:W\to S$, $q:W\to S_m$ of the resulting
good model, write
\[
p^*D_S=q^*D_m+[E],\qquad E\geq0
\text{ exceptional over }S_m.
\]
The negative-part comparison proved above gives
$E=p^*F_S$, and hence
\[
q^*D_m=(a+1)p^*\alpha_S.
\]
Since $D_m$ is semiample, $\alpha_S$ is semiample after descent
through $p$. Pulling back by $g$ and using the
$\alpha$-trivial initial program gives semiampleness of
$\alpha$ on $X$. The target is compact K\"ahler, and the
resulting map is Moishezon, as follows by taking common projective
modifications of the maps just constructed. The comparison uses only the chosen lower good model; no
lower-dimensional program is lifted to $V$.
\end{proof}
\begin{proposition}[A negative ray with an actual detector]
\label{prog:ordinary-step}
Assume $\mathsf B_d$. Let $A$ be a gklt adjoint in dimension at most
$d$ on a globally strongly $\mathbb Q$-factorial compact K\"ahler
space, with globally nef carrier data. Let $R$ be an $A$-negative
extremal ray of the full analytic cone. If an actual global rational
line $L$ has $L\cdot R<0$, the ray has a projective contraction;
a small contraction has its detected flip. The working models retain
the strong factoriality and compact K\"ahler properties.
In particular this applies to an ordinary rational adjoint
$A=c_1(K_X+B)$ with detector $K_X+B$, and to ordinary real boundaries
by Proposition~\ref{prog:ordinary-replacement}.
\end{proposition}
\begin{proof}
The local polyhedrality of the negative cone makes $R$ an exposed ray.
Choose a nef support $\lambda$ whose null analytic face is $R$.
Normalize the analytic cone by a K\"ahler class. On a neighbourhood
of the point representing $R$, the class $-A$ is positive. On the
remaining compact part of this slice, $\lambda$ has a positive
minimum. Consequently, for a sufficiently large $b$, the class
$\omega_R=b\lambda-A$ is K\"ahler. Thus
$A+\omega_R=b\lambda$ is nef with precisely the null face $R$.
The gklt presentation with nef carrier datum increased by the pullback
of $\omega_R$ has modified-big total boundary. Apply $\mathsf B_d$
to construct its contraction. Lemma~\ref{prog:detected-projectivity}
makes the same global line $-L$ relatively ample, and
Proposition~\ref{prog:detected-flip} constructs the flip when needed.
For a divisorial contraction the exceptional-prime negativity and
integral line descent give strong factoriality on the target: pull
back a rank-one reflexive sheaf, remove its degree by a rational
multiple of the exceptional divisor, descend the resulting line,
and compare on the common big open. Local ordinary log-Fano
replacement gives rational singularities by relative vanishing.
The supporting target and projective flip are compact K\"ahler.
\end{proof}
\subsubsection{Relative good models and finite geography}
\begin{lemma}[Removing the preparation error]
\label{prog:remove-preparation}
Let $\mu:W\to X$ be a projective smooth preparation over $S$ with
$A_W=\mu^*A_X+[F]$, where $F\geq0$ is $\mu$-exceptional and has
positive coefficient on every added prime. If $W\dashrightarrow Y$
is a chosen good log terminal model of $A_W$ over $S$, then it induces
a good log terminal model of $A_X$ over $S$.
\end{lemma}
\begin{proof}
On a common projective resolution $p:V\to W$, $q:V\to Y$, write
$p^*A_W=q^*A_Y+[E]$ with $E\geq0$ exceptional over $Y$.
Put $r=\mu p$ and $D=E-p^*F$. Then
\[
[D]=r^*A_X-q^*A_Y,\qquad r_*D=r_*E\geq0.
\]
The divisor $-D$ is $r$-nef because $A_Y$ is nef over $S$.
Negativity gives $D\geq0$, hence $E\geq p^*F$. Every added prime
is therefore exceptional over $Y$. This proves nonextraction from
$X$ and gives its effective exceptional comparison with $A_Y$.
Strictness at any contracted prime of $X$ follows from strictness
for the chosen log terminal model of $W$, since $F$ has coefficient
zero there. The same semiample endpoint makes the model good.
\end{proof}
\begin{lemma}[Recovering the uncontracted original primes]
\label{prog:weak-to-terminal}
Let a gklt pair on $X$ have a nonextracting weak good model
$X\dashrightarrow Y$ over $S$. Assume $Y$ is compact K\"ahler,
globally strongly $\mathbb Q$-factorial and has the exceptional
splitting of Lemma~\ref{prog:detected-cohomology}. Then a projective
crepant extraction $Y'\to Y$ gives a good log terminal model of the
original pair. The exceptional splitting persists on $Y'$.
\end{lemma}
\begin{proof}
On a common resolution write the actual comparison
$p^*A_X=q^*A_Y+[E]$, where $E\geq0$ is $q$-exceptional.
There are finitely many primes of $X$ contracted by its marking.
Let $\mathcal P$ consist of those with coefficient zero in $E$.
Take a projective log resolution $r:W\to Y$ which contains all
these valuations and dominates the fixed nef carrier. In the crepant
structure boundary $B_W$, the coefficient of each member of
$\mathcal P$ is its original effective boundary coefficient, hence
lies in $[0,1)$. Keep these coefficients unchanged. Increase each
other $r$-exceptional coefficient, if necessary from a negative
value, to a number in $(\max\{0,b_Q\},1)$. Keep all nonexceptional
coefficients unchanged. The resulting effective klt boundary
$\Gamma_W$ satisfies
\[
[K_W+\Gamma_W]+\mathbf M_W=r^*A_Y+[F],
\]
where $F\geq0$ has precisely the unwanted exceptional primes as its
support. The inherited splitting gives actual real-line representatives
for every class modulo $Y$. Since $r$ is birational, the nef carrier
datum is relatively big; the adjoint is relatively represented by
$F$ and is pseudo-effective. Apply
Proposition~\ref{prog:projective-good-model} over $Y$.
Its projective endpoint $r':Y'\to Y$ has an effective relatively nef
exceptional trace $F'$, so negativity gives $F'=0$. A prime outside
$\operatorname{Supp}F$ cannot be contracted by an $F$-negative
birational program: on a general curve through its generic point the
effective divisor has nonnegative degree. Thus all members of
$\mathcal P$ survive, and all unwanted exceptional primes disappear.
No new prime is extracted by the program. It follows that $r'$ is
crepant and that the marking from $X$ contracts exactly primes with
strictly positive original comparison coefficient. It is nonextracting
and is therefore a log terminal model. The pullback of the semiample
adjoint on $Y$ makes it good. Finally apply
Corollary~\ref{prog:extraction-cohomology}.
\end{proof}
\begin{lemma}[A detector throughout the big-adjoint construction]
\label{prog:big-good-model}
Assume $\mathsf B_d$ and $\mathsf F_{d-1}$. The big-adjoint
construction of \cite[Section~5.1]{HX} gives a chosen good model in
dimension $d$ using only detected steps. It also gives the case of
$\mathsf M_d$ in which $A+c f^*\omega_S$ is big for some $c>0$,
with the map to $S$ retained throughout.
\end{lemma}
\begin{proof}
After the smooth preparation in \cite[Proposition~5.12]{HX}, write the
adjoint as
\[
A=D+\omega,\qquad D\geq0,\quad \omega\text{ K\"ahler},
\qquad \operatorname{Supp}D=N_+(A).
\]
Here $N_+(A)$ denotes the reduced divisor with support
$\bigcap_{0<\epsilon\ll1}\operatorname{Supp}N_\sigma(A-\epsilon\omega)$;
this support is constant for sufficiently small positive $\epsilon$
and is independent of the chosen Kähler class $\omega$
\cite[Lemma~5.3]{HX}. At a nonterminal scaling step,
\[
A_i\cdot R_i<0,\qquad
(A_i+t_i\omega_i)\cdot R_i=0,\qquad t_i>0.
\]
Consequently $\omega_i\cdot R_i>0$ and $D_i\cdot R_i<0$.
A component of the actual effective divisor $D_i$ supplies a global
rational Cartier detector. The supporting contraction follows from
$\mathsf B_d$, and its projectivity and flip follow from
Proposition~\ref{prog:detected-flip}. This argument does not require
$\omega_i$ to remain nef.
To apply $\mathsf B_d$ to a gdlt presentation, lower its finitely many
floor coefficients on the initial smooth carrier and add the same small
class to the K\"ahler nef datum. The adjoint does not change, and the
resulting gklt presentation persists through the same negative steps.
The original gdlt presentation is retained for adjunction to the floor.
Thus the special-termination proof of \cite[Lemma~5.8]{HX} applies with
$\mathsf F_{d-1}$ on its smooth lower-dimensional carrier. Its other
inputs are adjunction, discrepancy monotonicity, and the local
Cartier-index calculation. The last neighbourhood-isomorphism step
uses the evaluation-ideal support argument in
the \hyperref[prog:special-support]{support comparison above}: the comparison of
the first detected flip has support equal to the full inverse image
of its exceptional locus, and later comparisons dominate it.
One does not infer disjointness of images merely from a vanishing
restriction of an arbitrary exceptional divisor. The theta induction of
\cite[Proposition~5.11]{HX} now gives a weak model. Applying
$\mathsf B_d$ to its nef adjoint makes it good, and
Lemma~\ref{prog:weak-to-terminal} gives the log terminal model.
All its steps are detected or belong to the already-projective
relative extraction in that lemma.
For completeness, the negative-part identity needed in this argument
is only
\[
N(p^*\xi+E)=N(p^*\xi)+E\quad(E\geq0\text{ $p$-exceptional}),
\]
not the stronger formula $N(p^*\xi)=p^*N(\xi)$ for an arbitrary
pseudo-effective class. The comparison with a nef endpoint identifies
its exceptional error with the negative part. The formula for
$N_+$ follows by writing a K\"ahler form upstairs as
$p^*\omega-F$, with $F$ positive on every exceptional prime, and applying
the displayed identity to
$p^*(\xi-\epsilon\omega)+E+\epsilon F$. Pushforward identifies the
nonexceptional coefficients; $\epsilon F$ supplies all exceptional
primes. These are exactly the uses in the theta argument.
Now put $P=f^*\omega_S$ and suppose $A+cP$ is big for some $c$.
Fix $\delta>0$ smaller than the $\omega_S$-degree of every compact
integral curve on $S$ and enlarge $c$ so $c>4d/\delta$.
Use the following direct preparation, which keeps the original
unshifted globally nef datum. On a smooth projective carrier
$\nu:W\to X$ write
\[
\nu^*(A+cP)=[G]+\omega,\qquad G\geq0,\quad\omega\text{ K\"ahler},
\qquad [K_W+B_W]+M_W=\nu^*A,
\]
with the non-K\"ahler support property used in the big-class
preparation. Here $\operatorname{Ex}(\nu)$ below denotes the reduced
divisor formed by all $\nu$-exceptional primes. Write
$B_W=B_W^+-B_W^-$. For small $\epsilon>0$ put
\[
\begin{split}
B'&=B_W^++\epsilon B_W^-+
\epsilon(G+\operatorname{Ex}(\nu)),\\
M'_W&=M_W+c\nu^*P+\epsilon\omega,\\
b&=B_W^++\frac{\epsilon}{1+\epsilon}\operatorname{Ex}(\nu).
\end{split}
\]
The new boundary is klt and $M'_W$ is K\"ahler. Direct addition gives
\[
A'=[K_W+B']+M'_W
=(1+\epsilon)\bigl([K_W+b]+M_W+c\nu^*P\bigr).
\]
Moreover $B'\geq b$. In every theta branch the cumulative added
boundary satisfies $0\leq C\leq1-B'$, so
\[
b+\frac{C}{1+\epsilon}
\leq b+\frac{1-b}{1+\epsilon}<1.
\]
Thus the unshifted presentation
$[K_W+b+C/(1+\epsilon)]+M_W$ is genuinely gklt with the
\emph{original} globally nef datum. This is the presentation used
for the horizontal length bound. No base form is subtracted from
an arbitrary newly prepared nef datum.
A horizontal negative ray would have $P$-degree at least $\delta$;
the bound $0<-A_{\rm unshifted}\cdot C\leq4d$ would contradict
$c\delta>4d$. Every constructed step is therefore over $S$.
Its projective connected fibres are contracted by $f$, so $f$ factors
through its base and through the flip. The final finite extraction is
also over $S$.
The shifted nef endpoint is semiample by $\mathsf B_d$. Its zero face
contains no horizontal curve: decompose such a curve by the unshifted
adjoint's cone theorem; nefness of the shifted class forces all
summands into the zero face, and a summand of positive $P$-degree again
contradicts the length bound. The shifted class is big, so the
connected semiampleness map is bimeromorphic. Its connected fibres are Moishezon and are connected
by chains of compact curves. The preceding zero-face argument makes
the map to $S$ constant on each such curve, hence on every fibre.
The analytic factorization lemma therefore factors the map to $S$
through the semiampleness map. Subtracting $c\omega_S$ from the
normalized class on its target gives the required relative K\"ahler class for the unshifted adjoint. The effective
resolution error from the preparation is removed by the usual
negativity comparison, giving a model of the original data.
The forward traces and exceptional splittings persist by
Lemma~\ref{prog:detected-cohomology}. In the last extraction,
the newly extracted prime classes move from the old exceptional span
into the pullback span; they are actual rational Cartier classes.
Both-sign negativity proves directness as before.
\end{proof}
\begin{lemma}[Changing the base of a relatively trivial adjoint]
\label{prog:graph-good-model}
Suppose $g:X\to Y$ is projective, its general fibre is rationally
connected, and a gklt adjoint satisfies $A_X=g^*A_Y$. Assume the
prepared nef datum is K\"ahler on a smooth carrier. Let
$Y\dashrightarrow Y_m$ be the chosen nonextracting good model
supplied by $\mathsf M_{d-1}$ over $S$, with its exceptional
cohomology splitting, and let $\dim Y<d$. Then a single projective relative construction gives a weak good model of $(X,A_X)$ over $S$ with that splitting.
Lemma~\ref{prog:weak-to-terminal} makes it log terminal. No step
of the base program is lifted.
\end{lemma}
\begin{proof}
Take a common projective smooth resolution $T\to Y$, $T\to Y_m$
and resolve the main component of $X\times_Y T$, further dominating
the prepared nef carrier; denote the resulting smooth space by $V$.
Its morphism $h:V\to Y_m$ is projective. Before constructing a
program, verify algebraicity for all classes. The projective RC map
$V\to T$ and Lemma~\ref{prog:relative-hodge} express every class
modulo $T$ by actual real lines. The splitting of the chosen $Y_m$
expresses classes from $T$, modulo $Y_m$, by exceptional divisor
classes. Hence
\[
H^{1,1}_{\rm BC}(V)=c_1(\operatorname{Pic}(V)\otimes\mathbb R)
+h^*H^{1,1}_{\rm BC}(Y_m).
\]
Lemma~\ref{prog:projective-cone} identifies the relative cone
with the full analytic face. These properties and the exceptional
splitting persist along the projective relative steps by
Lemma~\ref{prog:detected-cohomology}.
Choose the effective gklt preparation on this final carrier. Its
adjoint has the identity
\[
A_V=h^*A_{Y_m}+[G],\qquad G=G_{\rm base}+F\geq0,
\]
where $G_{\rm base}$ is pulled back from the actual effective base
comparison and $F$ is exceptional over $X$, with positive coefficient
on every added prime. Its nef datum is nef and big on the prepared
carrier. Modulo $Y_m$ it is the actual real-line class
$[G]-c_1(K_V)-[B_V]$. Thus every hypothesis of
Proposition~\ref{prog:projective-good-model} holds, with the effective
relative adjoint $G$. Denote its global relative endpoint by $V_m$
and the strict transform of $G$ by $G_m$.
On a common resolution $p:W\to V$, $q:W\to V_m$, the actual adjoint
comparison has effective $q$-exceptional error $E$. The difference
$p^*G-q^*G_m-E$ has zero Bott--Chern class and is $q$-exceptional,
because $q_*p^*G=G_m$. Negativity in both signs makes it zero:
\[
p^*G=q^*G_m+E. \tag{*}
\]
Choose a big common-isomorphism open $U\subset Y_m$ for
$Y\dashrightarrow Y_m$. Such an open exists because that map is
nonextracting. On $U$ we have $G_{\rm base}=0$. Let
$r:W_U\to X_U$ be the induced projective birational map. The divisor
$q^*G_m$ is $r$-nef, since the endpoint adjoint is nef over $U$ and
the base class has zero degree on $r$-fibres. Equation $(*)$ gives
\[
r_*q^*G_m=-r_*E\leq0,
\]
since $F$ is exceptional over $X$. Negativity yields $q^*G_m\leq0$.
Effectivity yields $G_m|_U=0$. This step removes the possible
\emph{horizontal} components of $F$; they were not assumed very
exceptional at the start.
Now $G_m$ is supported over codimension at least two in $Y_m$.
It is nef over $Y_m$, so very-exceptional negativity gives $G_m=0$.
The endpoint adjoint is the pullback of $A_{Y_m}$, hence is nef and
good over $S$. Its relative canonical morphism is Moishezon: it is
the connected factor of the composite of the projective morphism to
$Y_m$ and the lower-dimensional Moishezon canonical morphism.
Lemma~\ref{prog:remove-preparation} removes the added
source-exceptional errors and proves nonextraction. All these operations are over $S$.
The all-class invariant was established before the relative program;
its preservation follows at each step from the same cone identification
and forward cohomology splitting. If the construction has contracted
any original prime crepantly, apply
Lemma~\ref{prog:weak-to-terminal} to retain it.
\end{proof}
\begin{lemma}[The global rational quotient over a fixed base]
\label{prog:relative-model}
Assume $\mathsf M_{d-1}$ and the case of $\mathsf M_d$ in which a
base shift makes the adjoint big. Then $\mathsf M_d$ holds when no
class $A+c f^*\omega_S$, $c\geq0$, is big.
\end{lemma}
\begin{proof}
First take the usual smooth carrier with effective gklt boundary
and adjoint $\mu^*A+[F_0]$, where $F_0$ is effective exceptional
with full exceptional support. Its boundary-plus-nef trace is big and its adjoint
is not big, so its canonical class is not pseudo-effective. Take
its global MRC and resolve the MRC graph and the fixed carrier,
obtaining a projective map to a smooth compact K\"ahler $Z$ with
$K_Z$ pseudo-effective.
Perform the K\"ahler-nef-part preparation \emph{after} these graph
resolutions, preserving the maps to $Z$ and $S$. This order ensures
that the final scaling datum is K\"ahler, rather than merely the nef
pullback of an earlier K\"ahler datum. Write the resulting adjoint as
\[
A_0=K_U+B_U+M_U=\mu^*A+[F],
\qquad F\geq0\text{ exceptional},\qquad M_U\text{ K\"ahler}.
\]
The global modified-big hypothesis makes a preliminary base shift
unnecessary. Relative pseudo-effectivity is preserved.
No base shift of $A_0$ is big, since pushing a big such class down
would make a base shift of $A$ big. The fixed morphism $h:U\to Z$
is projective and has RC general fibres. All these preparations
precede the next, fresh base shift.
Put $P=f_U^*\omega_S$ and choose $c>2d/\delta$, where every
integral curve on $S$ has $\omega_S$-degree greater than
$\delta>0$. Set
\[
A_c=A_0+cP=D+N,\qquad D=K_U+B_U,\quad N=M_U+cP.
\]
Since $A_c$ is not big globally, relative-canonical positivity
\cite[Theorem~2.48(2)]{HX}, together with pseudo-effectivity of $K_Z$,
shows that $A_c$ is not big over $Z$, whether or not it is globally
pseudo-effective. The class $N$ is big over $Z$. Consequently $D$
cannot be pseudo-effective over $Z$: otherwise $D+N=A_c$ would be
big over $Z$. The smooth projective RC Hodge projection represents
every class modulo $Z$ by an actual real line. The ordinary global klt adjoint $D$ is therefore relatively
non-pseudo-effective. The projective ordinary Mori construction of
Corollary~\ref{prog:projective-mori}, with scaling of the actual
relatively ample real line represented by $N$, terminates in a Mori
fibre space. Lemma~\ref{prog:relative-hodge} and
Lemma~\ref{prog:projective-cone} identify every relative ray
with a full analytic ray. Its nef b-data retain the fixed globally
nef carrier $U$.
First suppose $A_c$ is not pseudo-effective globally. The retained
relative-canonical positivity theorem on $h$, together with
pseudo-effectivity of $K_Z$, then says that $A_c$ is not
pseudo-effective over $Z$. The terminating relative program ends
in a Mori contraction at a threshold $\tau>1$. At every step its
scaled zero ray has $A_c$-degree negative. Inductively the map to
$S$ is retained: if such a ray were horizontal, its $P$-degree
would exceed $\delta$, while the gklt cone theorem for $A_0$ gives
a rational generator of $-A_0$-degree at most $2d$, contradicting
$A_c\cdot R\leq0$. Once $P\cdot R=0$, the step is negative or
crepant for $A_0$, so its gklt presentation persists. The same
argument applies to the final Mori ray. This would be an
$A_c$-negative Mori contraction over $S$, contradicting the
preserved pseudo-effectivity over $S$. Hence $A_c$ is
pseudo-effective globally.
It is not big globally. Relative-canonical positivity now implies
that it is not big over $Z$. Its pseudo-effective threshold for
$D+tN$ over $Z$ is exactly one: for $t>1$ the class is big, while
pseudo-effectivity for $t<1$ would make $A_c$ big because $N$ is
big over $Z$. The same finite relative program therefore ends in
a Mori contraction $g:U'\to Y$ at threshold one.
All birational steps, including any steps at threshold one, and
the final zero ray are over $S$ by the preceding length argument
with $A_c\cdot R\leq0$. For a zero ray with positive $P$-degree,
$A_0$ would again have degree less than $-c\delta$.
Thus $Y$ carries a map to $S$ and $\dim Y<d$.
The class $A_c$ has zero degree on every $g$-contracted curve.
This is an ordinary projective log-Fano contraction: rationalize its
effective real ordinary boundary, preserving klt and relative
antiampleness. Relative vanishing gives rational singularities on
$Y$, which is compact K\"ahler because it is projective over $Z$.
Lemma~\ref{prog:bc-descent} therefore gives an actual equality
$A_c=g^*A_Y$ in Bott--Chern cohomology. This descent precedes the
base presentation. Apply Proposition~\ref{prog:base-presentation}
to $g$ and the whole adjoint $A_c$. The negative or crepant steps
preserve its effective gklt data. Its nef part is the trace of
$M_U+cP$ from the fixed carrier; on a common carrier it is nef and
big, so its total boundary is globally modified big. The proposition
gives effective gklt data for $A_Y$ with globally nef carrier and
globally modified-big total boundary. Relative pseudo-effectivity
descends through the projective surjection $g$, so $\mathsf M_{d-1}$
applies over $S$, with the base shift retained. For the source data in
Lemma~\ref{prog:graph-good-model}, use
Lemma~\ref{prog:base-presentation-preparation} to choose a K\"ahler
nef datum on a smooth carrier while keeping $A_c$ fixed. Apply that
graph lemma to the chosen lower-dimensional output.
Adjoint-preserving preparation is compatible with the original pair
in this comparison. On a common resolution of a nonextracting good
output, keep the original nef datum and subtract the effective model
comparison divisor from the original structure divisor. The resulting
structure divisor has the same adjoint as the output; its trace is
effective by nonextraction, and its discrepancies have not decreased.
Strictness on contracted original primes is preserved. Thus the
prepared presentation supplies a comparison for the original data as
well.
Steps at threshold one can be $A_c$-crepant divisorial contractions.
Consequently the composite constructed so far need only be a weak
good model of the original prepared adjoint. Apply
Lemma~\ref{prog:weak-to-terminal} to extract the finitely many original
primes with zero comparison coefficient. This gives the strict log
terminal model and preserves the cohomology invariant. The shift
changes neither relative comparisons nor relative nefness.
Lemma~\ref{prog:remove-preparation} removes the full-support
preparation error and proves the claimed $\mathsf M_d$.
\end{proof}
\subsubsection{Finite geography and closing the induction}
\begin{lemma}[Canonical models first, weak models second]
\label{prog:finite-geography}
Assume $\mathsf B_d$ and $\mathsf M_d$. Then $\mathsf F_d$ holds for compact polytopes
of globally nef b-data on a fixed carrier and globally modified-big
boundary, over the fixed base.
\end{lemma}
\begin{proof}
First prove, by induction on the parameter-polytope dimension,
polyhedrality of the effective locus, finite canonical-model chambers,
and finitely many \emph{chosen} terminal models on those chambers,
following \cite[Theorem~5.15]{HX}.
The zero-dimensional case uses $\mathsf M_d$ and uniqueness of the
canonical model; it does not yet assert finiteness of all weak models.
At a central point choose $X\dashrightarrow X_m\to X_c$ over
$S$. The transport invariant in $\mathsf M_d$ supplies the traces
of the whole nearby polytope on $X_m$. Shrink so the finitely many
strict exceptional inequalities remain strict. Add a common pullback
from $S$ so that the central canonical class on $X_c$ is K\"ahler
absolutely. On the boundary of the parameter polytope, apply the
smaller-parameter induction \emph{over $X_c$}; its existence input
is $\mathsf M_d(X_c)$, just proved independently. No projectivity of
$X_m\to X_c$ or Hodge projection over the possibly singular $X_c$
is used.
The central adjoint is zero over $X_c$, so the relative effective
locus is the radial cone on its boundary effective locus. Apply the
cone-length estimate of \cite[Claim~5.1]{HX} on the finitely chosen
\emph{gklt terminal models} $h_i:Y_i\to X_c$, all of dimension $d$.
Write $A_{u,i}$ for the boundary-parameter adjoint, nef over $X_c$,
and $\psi_i:Y_i\to Z_i$ for its relative canonical morphism.
The cone theorem is not applied to the possibly lower-dimensional
canonical targets $Z_i$.
Let $H_i=h_i^*\omega_c$ and choose $\delta>0$ below the
$\omega_c$-degree of every compact integral curve in $X_c$. Fix $\bar\lambda<\delta/(\delta+2d)$ and for
$0<\lambda\leq\bar\lambda$ put
\[
A_{\lambda,i}=(1-\lambda)H_i+\lambda A_{u,i}.
\]
The same cone estimate shows more than nefness: for a sufficiently
small $\epsilon>0$, uniform on the finite list and the chosen radial
interval, $A_{\lambda,i}-\epsilon H_i$ is nef. Indeed on a
horizontal $A_{u,i}$-negative rational ray its degree is greater than
$(1-\lambda-\epsilon)\delta-2d\lambda>0$; on the nonnegative
part of the cone and on the vertical cone it is nonnegative.
It follows for every class $\xi\in\overline{\mathrm{NA}}(Y_i)$ that
\[
A_{\lambda,i}\cdot\xi=0
\quad\Longleftrightarrow\quad
H_i\cdot\xi=A_{u,i}\cdot\xi=0.
\]
In particular its null curves are exactly those contracted by
$\psi_i$. The central data are crepant through these maps over $X_c$;
thus the convex data defining $A_{\lambda,i}$ remain gklt with
globally modified-big boundary. Apply $\mathsf B_d$ to $A_{\lambda,i}$. Its connected
semiampleness map and $\psi_i$ both have Moishezon fibres, and each
connected such fibre is connected by chains of compact curves.
Equality of their contracted curves makes each map constant on the
fibres of the other. The analytic factorization lemma identifies
the two maps. Thus the selected relative canonical model is also
the canonical model over $S$ on this radial interval.
This proves local polyhedrality and the finite local list; compactness
gives the global finite list of canonical models and chosen terminal
models.
\paragraph{Uniform preparation preserving every weak model.}
\label{prog:uniform-preparation}
Write $A_u=[K_X+B_u+\mathbf M_{u,X}]$ for the adjoints in the
parameter polytope. We first record why a weak model of these gklt data
extracts no divisor. On a common resolution $p:U\to X$, $q:U\to Y$
put $D=p^*A_u-q^*A_{u,Y}$, using the actual structure-boundary
comparison. The weak-model discrepancy inequalities give $p_*D\geq0$,
and $-D$ is $p$-nef because $A_{u,Y}$ is nef over the fixed base.
Negativity therefore gives $D\geq0$. An extracted prime on $Y$ has
coefficient one in the weak-model boundary, whereas its coefficient in
the structure boundary of the gklt source is less than one; it would
give a negative coefficient of $D$. Hence there is no such prime.
Consequently $D$ is $q$-exceptional and the target is itself gklt.
Fix a parameter $u_0$. The boundary plus nef part is globally
modified big by hypothesis. The modified-big preparation
\cite[Lemma~2.23]{HX} gives, on a fixed projective smooth carrier
$a:W\to X$, an effective klt boundary $\Gamma$, a K\"ahler class
$\gamma_0$, and an effective $a$-exceptional real divisor $F$ such that
\begin{equation}\label{prog:eq:uniform-preparation}
[K_W+\Gamma]+\gamma_0=a^*A_{u_0}+[F].
\end{equation}
Here is the effectivity detail in this use of the preparation. Its new
nef datum dominates a K\"ahler form on a carrier. After resolving the
new structure boundary, choose an effective exceptional divisor whose
negative is relatively ample and subtract a sufficiently small multiple
from the pulled-back datum to make it K\"ahler. Add the same multiple
to the structure boundary,
then replace its negative coefficients by zero and, if needed, add
small positive coefficients on the remaining exceptional primes.
All coefficients stay below one. Only exceptional coefficients have
been increased; their increase is the divisor $F\geq0$ in
\eqref{prog:eq:uniform-preparation}.
On a sufficiently small closed polyhedral neighbourhood of $u_0$ put
\[
\gamma_u=\gamma_0+a^*(A_u-A_{u_0}).
\]
These classes remain K\"ahler by openness, and the same $\Gamma$ and
the same $F$ satisfy
\[
[K_W+\Gamma]+\gamma_u=a^*A_u+[F].
\]
Thus this is one affine family of gklt pairs on one smooth carrier,
with a fixed effective boundary and K\"ahler nef data.
Let $\psi:X\dashrightarrow Y$ be \emph{any} weak model at a parameter
$u$ in this neighbourhood, and resolve the induced map $W\dashrightarrow
Y$ by $r:T\to W$, $q:T\to Y$. Its original comparison is
\[
(ar)^*A_u=q^*A_{u,Y}+E_u,
\qquad E_u\geq0\text{ is }q\text{-exceptional}.
\]
Since $\psi$ is nonextracting, every $a$-exceptional prime is exceptional
over $Y$. Hence $r^*F$ is also $q$-exceptional, and
\begin{equation}\label{prog:eq:prepared-weak}
r^*\bigl([K_W+\Gamma]+\gamma_u\bigr)
=q^*A_{u,Y}+E_u+r^*F.
\end{equation}
This proves that $W\dashrightarrow Y$ is a weak model of the prepared
pair. In detail, subtract the effective exceptional error on the right
from its structure boundary on $T$ and push down to $Y$. The resulting
boundary is precisely the transform of $\Gamma$, it is effective, the
adjoint is the already existing Bott--Chern class $A_{u,Y}$, and all
discrepancies are at least those of the prepared gklt pair. This uses generalized data at the level of currents. On $T$ keep
the nef datum $r^*\gamma_u$, and subtract the displayed effective
$q$-exceptional error from the prepared structure boundary. Pushing
forward by $q$ gives the boundary $\Gamma_Y$ and the trace of this
fixed nef b-class. Their sum with the canonical current represents
the already existing locally exact adjoint $A_{u,Y}$; the resolution
identity defines its structure boundary. The generalized discrepancies
have not decreased. This argument neither requires $K_Y+\Gamma_Y$ to
be real Cartier nor asserts that the nef trace is separately a
Bott--Chern class. It therefore applies to every K\"ahler weak target,
including the adjunction strata used in special termination.
\paragraph{Every weak model occurs in the finite canonical list.}
Choose a small box in $H^{1,1}(W,\mathbb R)$ which spans that vector
space and whose addition to the compact family $\{\gamma_u\}$ stays
inside the K\"ahler cone. For any $Y$ above choose a K\"ahler class
$\eta_Y$ and define $\eta_W=r_*q^*\eta_Y$. Smooth-source cohomology
puts $\eta_W$ in $H^{1,1}(W,\mathbb R)$ and gives an actual
$r$-exceptional real divisor $D_\eta$ with
\[
r^*\eta_W-q^*\eta_Y=[D_\eta].
\]
Its negative is $r$-nef, so $D_\eta\geq0$ by exceptional negativity.
As $W\dashrightarrow Y$ extracts no divisors, $D_\eta$ is also
$q$-exceptional. Scale $\eta_Y$ by a sufficiently small positive number
so that $\eta_W$ lies in the fixed box. Adding this identity to
\eqref{prog:eq:prepared-weak} yields an effective
$q$-exceptional comparison with target class $A_{u,Y}+\eta_Y$, which
is K\"ahler over the original base. Thus $W\dashrightarrow Y$ is the
canonical model of a member of this one enlarged prepared polytope.
Its canonical-model list is finite by the first part of the geography
proof. The induced list of maps from $X$ is therefore finite as well.
A finite cover of the original compact parameter polytope completes
the proof for all weak models.
\end{proof}
\begin{theorem}[The restricted package in every dimension]
\label{prog:restricted-package}
The assertions $\mathsf B_d$, $\mathsf C_d$, $\mathsf M_d$ and
$\mathsf F_d$ hold in every finite dimension, for the globally nef
carrier data and globally modified-big boundary traces specified above.
All birational steps used to prove them are detected steps or belong
to an already-projective relative construction with the stated
relative-line algebraicity. A general semiample morphism is asserted
to be Moishezon; projectivity is asserted only when the construction
provides an actual relatively ample line.
\end{theorem}
\begin{proof}
The assertions in dimensions zero and one follow from degrees and
factorization of maps of compact curves. Suppose the package holds
in dimensions less than $d$. Lemma~\ref{prog:base-point-free-induction}
first proves $\mathsf C_{d,\mathrm{big}}$ and then $\mathsf B_d$.
The former uses $\mathsf C_{d-1}$; the nonbig part of the latter
uses only a chosen $\mathsf M_{d-1}$ model and the explicit
negative-part comparison. In particular, no lower program is lifted.
With $\mathsf B_d$ available, Proposition~\ref{prog:ordinary-step}
constructs detected negative steps in dimension $d$.
Lemma~\ref{prog:big-good-model}, using the already known
$\mathsf F_{d-1}$, gives the big-shift case of $\mathsf M_d$.
Lemma~\ref{prog:relative-model} gives its remaining case, using
$\mathsf M_{d-1}$ and the projective graph construction. This proves
$\mathsf M_d$ over every proper compact K\"ahler base, with its
chosen-model invariant. Lemma~\ref{prog:finite-geography}, whose
hypotheses are \emph{both} $\mathsf B_d$ and $\mathsf M_d$, now
proves $\mathsf F_d$.
It remains to prove $\mathsf C_d$ when its nef NQC adjoint $\alpha$
is not big. First use
Lemma~\ref{prog:base-presentation-preparation}, preserving the
adjoint $\alpha$ and making the nef datum K\"ahler on a smooth
carrier. Its multiplier-ideal inclusion allows the given contraction
to restrict to the prepared non-klt closed subspace by
Lemma~\ref{prog:base-emc}. Next take the projective dlt preparation
in \cite[Section~6]{HX}, keeping its crepant structure data. The
working space is klt and globally strongly $\mathbb Q$-factorial.
The retained crepant structure boundary need not be lc; it is
effective, including the extracted primes of coefficient at least one. On a common carrier the nef datum is
nef and big. Consequently the total boundary trace is modified big,
and $\alpha-c_1(K_X)$ is a big Bott--Chern class. The contraction on
the non-klt closed subspace transports through this preparation by
Lemma~\ref{prog:multiplier-transport}. Choose an NQC expression with
positive coefficients and integral multiples of its rational
degree-two summands. Corollary~\ref{prog:uniform-canonical-parameter}
provides a single number $t$ such that each negative ray of
$K_X+t\alpha$ is $\alpha$-trivial. Choose a K\"ahler scaling class
$\omega$ with $K_X+t\alpha+\omega$ nef. Every selected negative
ray is ordinary $K_X$-negative; the actual rational canonical line
is its detector. Proposition~\ref{prog:ordinary-step} supplies its
projective contraction or flip, and the fixed integral grid persists.
The descended $\alpha$ remains nef by projective descent of nefness.
The driving class is non-pseudo-effective throughout, by the
corollary. Its initial pseudo-effective scaling threshold is positive;
choose a cutoff $\varepsilon>0$ below that threshold. On the fixed
initial carrier the compact segment of globally nef data
\[
t\alpha+[\varepsilon,1]\omega
\]
has modified-big total boundary. At its scaling parameter each working
model is a marked weak model of a member of this segment. Each selected
step is negative for the fixed driving adjoint, so its discrepancies
do not decrease and increase at a valuation affected by that step.
An infinite sequence would repeat a marked model by $\mathsf F_d$;
the cumulative nonzero effective comparison for the fixed adjoint
would then contradict equality of that marking. Hence this program
terminates. Non-pseudo-effectivity excludes a nef endpoint, leaving
an $\alpha$-trivial ordinary Mori contraction $f:X'\to Z$.
The structure data of the prepared contraction problem remain
crepant throughout this $\alpha$-trivial program. Their effective
traces and globally nef and big carrier datum therefore persist.
Lemma~\ref{prog:multiplier-transport} transports the contraction on
the non-klt closed subspace through each birational step.
Lemma~\ref{prog:bc-descent} gives
\[
\alpha_{X'}=f^*\gamma
\]
in Bott--Chern cohomology. The class $\gamma$ is nef by projective
descent; Lemma~\ref{prog:integral-h2} descends the fixed rational
NQC summands, and projective multisections preserve their
nonnegative degrees.
Apply Corollary~\ref{prog:base-emc-induction} to this projective
Mori contraction. In its notation the prepared non-klt subspace
$N_1$ has Stein space $V(f_*J_1)$ over $Z$. If $N_1$ dominates
$Z$, this Stein space is $Z$ itself and its inherited contraction
already supplies the contraction of $\gamma$. Otherwise the base
presentation has multiplier ideal $J_Z\supset f_*J_1$, so the
contraction restricts to $V(J_Z)$ as a closed subspace of
$V(f_*J_1)$. Since $\dim Z<d$, the assertion $\mathsf C_{d-1}$
then gives the contraction of $\gamma$. Pullback supplies the
contraction on $X'$. Its pullback to projective common resolutions
factors through the original space by Lemma~\ref{prog:base-emc},
because the adjoint classes are crepant. This gives the required
Moishezon contraction on the original space.
This proves $\mathsf C_d$ and completes the induction.
\end{proof}
\begin{corollary}[Finite positive parts of detected scalings]
\label{prog:detected-positive-truncation}
Let a gklt adjoint with globally nef carrier data be run with scaling
of the trace of a K\"ahler class on a fixed smooth carrier. Suppose
that every chosen negative analytic ray has an actual global rational
line detector and that the total boundary trace on each positive
scaling segment is globally modified big. Every segment whose scaling
parameters lie in a fixed interval $[\varepsilon,T]$, with
$\varepsilon>0$, is finite. If the driving adjoint is not
pseudo-effective, the scaling terminates with a Mori fibre space.
\end{corollary}
\begin{proof}
Proposition~\ref{prog:ordinary-step} constructs each step and
Lemma~\ref{prog:detected-cohomology} supplies the forward traces from
the fixed carrier. Each reached model at a scaling parameter is a
marked weak model of that parameter's adjoint. The compact parameter
segment satisfies $\mathsf F_d$, so there are finitely many such
markings. Strict increase for the fixed driving adjoint rules out
repetition. If that adjoint is not pseudo-effective, the positive
pseudo-effective scaling threshold gives a fixed positive cutoff
containing every parameter of a continuing program. The finite
program cannot end nef, hence ends with the claimed Mori map.
\end{proof}
==== END Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//03c-restricted-induction.tex ====
==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//04-boundary.tex ====
\section{Generation along the dlt boundary}\label{sec:boundary}
We prove that the restrictions of a nef adjoint to the separate log
canonical strata fit together to generate its restriction to the whole
reduced boundary. Semiampleness on the normal components does not by itself
supply compatible sections on their union. The proof has three
parts. A Mori contraction reduces the obstruction to restriction from a
stratum to one comparison of residues. Crepant transport respects all
lower residues. Finally a finiteness argument permits us to impose every
comparison simultaneously by taking products of sections.
Throughout this section, \(n\geq1\) and \(\Ggood_j\) is assumed for every
\(j<n\). We use ordinary dlt pairs with effective rational boundary.
A \emph{stratum} is an irreducible log canonical center; we also allow the
ambient space itself when describing adjunction. A \emph{boundary stratum}
is a stratum contained in the reduced floor. An \emph{incidence} is an
inclusion of strata, and it is \emph{immediate} when the smaller stratum has
codimension one in the larger. On a crepant SNC model, a \emph{unit
component} is a boundary component of coefficient one, and a \emph{unit
stratum} is an irreducible component of an intersection of unit components.
All residue comparisons are in a common divisible even adjoint degree.
\begin{theorem}[Generation on the reduced boundary]\label{floor:generation}
Assume \(\Ggood_j\) for every \(0\leq j<n\).
Let \((V,B)\) be a normal irreducible compact K\"ahler dlt \(n\)-fold with
effective rational boundary \(B\). Suppose that \(V\) is globally
\(\Q\)-factorial, that \((V,B)\) satisfies the lc-strata resolution
convention of Definition~\ref{def:lc-strata-resolution}, and that the actual
\(\Q\)-Cartier adjoint \(J=K_V+B\) is analytically nef.
Then the restriction of an actual positive Cartier multiple of \(J\) to
the whole reduced space \(S=\lfloor B\rfloor\) is globally generated.
\end{theorem}
The assertion is vacuous if \(S=\varnothing\). The remainder of the
section proves it when the floor is nonempty. The construction of
compatible sections follows the admissible-section method of
Fujino~\cite[Section~4]{Fujino2000}; the main work here is to establish its
restriction and finiteness inputs for strata of arbitrary dimension in the
compact K\"ahler setting.
\subsection{Adjunction, the semiample systems, and their comparisons}
We first fix precisely which sections have to agree. The following local
facts retain the actual line, including the residue identifications at
every intersection.
\begin{lemma}[Strata and descent of residues]\label{floor:adjunction}
For each stratum \(Z\subseteq V\) there is an effective rational boundary
\(B_Z\) on the normal compact K\"ahler space \(Z\) such that \((Z,B_Z)\)
is dlt and, in every sufficiently divisible even degree \(q\),
\begin{equation}\label{floor:residue-line}
\mathcal O_Z\bigl(q(K_Z+B_Z)\bigr)
\simeq \mathcal O_V(qJ)|_Z.
\end{equation}
This is the actual meromorphic iterated-residue identification. The prime
components of \(C_Z:=\lfloor B_Z\rfloor\) are precisely the strata
\(D\subset Z\) in an immediate incidence. Sections of \(\mathcal O_V(qJ)|_D\) on
these components \(D\) which have equal residues on every common lower
stratum descend uniquely to a section of the restricted line on the entire
reduced \(C_Z\). The same statement applies to the components of
\(S\subseteq V\).
\end{lemma}
\begin{proof}
Choose one resolution \(p:\widehat V\to V\) witnessing
Definition~\ref{def:lc-strata-resolution}, and write
\(K_{\widehat V}+\widehat B=p^*(K_V+B)\) using the actual meromorphic
identification. Its unit components are the strict transforms
\(\widehat S_i\) of the finitely many components \(S_i\) of \(S\).
Every component \(\widehat Z\) of an intersection of distinct
\(\widehat S_i\)'s maps birationally onto a stratum \(Z\), because the
map is an isomorphism at its general point. Conversely every stratum is
obtained this way. The general SNC locus makes both the component and its
index set unique. Compactness makes the collection finite.
The empty intersection is \(\widehat V\) itself. Write
\(p_Z=p|_{\widehat Z}:\widehat Z\to Z\), so \(p_V=p\).
Here are the adjunction and normality details, including the facts used at
deeper intersections. For a chosen set of indices, keep their coefficients
one and allow each unused coefficient to be either one or \(1/2\). Write
\(B^{\mathbf e}=B-\sum_i(1-e_i)S_i\). Global \(\Q\)-factoriality makes
this an actual rational-line operation, and
\[
\widehat B^{\mathbf e}=\widehat B-
p^*\sum_i(1-e_i)S_i.
\]
The subtracted divisor is effective. All exceptional coefficients remain
below one, and the remaining unit components are exactly the retained
strict transforms. Successive SNC adjunction on \(\widehat Z\) gives
\[
(K_{\widehat V}+\widehat B^{\mathbf e})|_{\widehat Z}
=K_{\widehat Z}+\widehat B_{\widehat Z}^{\mathbf e}.
\]
Here and below such a restriction equality means equality of the
meromorphic residue maps in degree \(q\), and hence equality of the
actual lines. If two orders of residue differ by an ordering sign, its
\(q\)-th power is one.
We induct on the number of chosen indices. At each stratum already
constructed, we retain normality, effectivity of every weighted different,
and the crepant SNC residue comparison on \(\widehat Z\). For this chosen
comparison the exceptional coefficients are below one, the unit components are the retained strict
intersections, and the full-weight pair is dlt. These assertions hold for
\(Z=V\). Lowering unused unit components preserves dlt; lowering all of
them makes the effective pair at the current stratum klt. It has rational
singularities and is Cohen--Macaulay by the dimension-free
floor-connectedness lemma \cite[Lemma~2.1]{BL}. If just one unused
coefficient is retained, the unit divisor of the induced SNC boundary is a
disjoint union \(R\) of smooth divisors on \(\widehat Z\). The same lemma gives
\[
(p_Z)_*\mathcal O_R=\mathcal O_{p_Z(R),\mathrm{red}}.
\]
The fibers are connected, so the images of different connected components
of \(R\) cannot meet. Each component maps birationally to its image; the
displayed equality, factored through finite normalization, makes that
image normal. It is compact K\"ahler by restriction of the local K\"ahler
potentials from \(V\).
For every allowed weight choice, push forward the SNC different to define
the rational different on this new stratum. In codimension one, the residue of a local frame of the
ambient invertible line identifies its divisorial adjoint with the
restricted line. Reflexive extension on the normal stratum gives
Equation~\eqref{floor:residue-line} everywhere. Pulling it back agrees with the
original SNC residue on a dense open and therefore everywhere, proving
crepancy as an actual meromorphic identification.
The different is effective. At a general point of any of its primes,
the normal surface-slice calculation of \cite[Lemma~6.2]{BL}, which is
stated in every dimension, preserves exactly the residue order and reduces
it to adjunction on a normal surface germ with effective boundary and a
smooth marked curve. For completeness, on a minimal resolution of that
surface, write the crepant boundary as the effective strict transform plus
an exceptional divisor \(E\). For each exceptional curve \(D\),
\[
E\cdot D=-(K_{\widetilde T}+B^{\rm str})\cdot D\leq0.
\]
Indeed \(B^{\rm str}\cdot D\geq0\), and adjunction gives
\(K_{\widetilde T}\cdot D=2p_a(D)-2-D^2\geq0\); minimality excludes a
smooth rational exceptional \((-1)\)-curve. The negative-definite
exceptional intersection matrix gives \(E\geq0\): if \(E=P-N\) with
disjoint nonnegative parts and \(N\ne0\), then
\(E\cdot N=P\cdot N-N^2>0\), contrary to the preceding inequalities.
The strict transform is finite birational over the smooth marked curve,
and hence isomorphic to it. Adjunction to that smooth strict transform
thus has nonnegative coefficient. This is the original coefficient by the
surface-slice residue calculation.
The full-weight pair is dlt and has the asserted floor. On an SNC model,
at the general point of the center of a divisorial valuation \(v\), take
normal parameters \(x_1,\ldots,x_a\), giving an unmarked parameter boundary
weight zero. The SNC discrepancy inequality is
\[
a(v;\widehat Z,\widehat B_{\widehat Z})
\geq \sum_{i=1}^a(1-b_i)v(x_i).
\]
Every \(v(x_i)>0\). Thus a zero-discrepancy center is an intersection
of unit components. Those intersections meet the isomorphism locus,
while every exceptional component has coefficient below one. To produce
a dlt resolution for the restricted map, principalize its nonisomorphism
locus and resolve together with the ordered SNC boundary. That locus
contains no entire log canonical center, so the displayed discrepancy
inequality makes every new exceptional coefficient strictly below one.
This is the dlt resolution criterion. It also identifies the unit primes
downstairs with the next incident strata.
Finally let \(R_{\rm all}\) be the whole unit union on \(\widehat Z\). Sections
on its smooth components which agree on all their SNC intersections glue
by the elementary equalizer for the coordinate ideals of an SNC union.
Even iterated residues make the equalities independent of the chosen chain.
Apply the floor-connectedness lemma to the full-weight pair and this full
unit union. It gives
\((p_Z)_*\mathcal O_{R_{\rm all}}=\mathcal O_{C_Z}\).
Projection formula therefore descends the glued section uniquely to the
restricted invertible line on the reduced \(C_Z\). The proof with \(Z=V\)
gives the assertion for \(S\).
\end{proof}
Each boundary stratum has dimension less than \(n\). The class of
\(J_Z:=K_Z+B_Z\) is nef, since it is the restriction of the nef adjoint
class. Proposition~\ref{bir:resolution-transfer}, applied using
\(\Ggood_{\dim Z}\) on a log resolution, makes the actual rational line
\(J_Z\) semiample. Increase \(q\) once
for all so that
\(L_Z:=\mathcal O_Z(qJ_Z)=\mathcal O_V(qJ)|_Z\) is generated for every
boundary stratum. Its map and Stein factorization give
\begin{equation}\label{floor:semiample-map}
f_Z:Z\longrightarrow Y_Z,\qquad L_Z=f_Z^*N_Z,
\end{equation}
where \(Y_Z\) is normal projective, \(f_Z\) has connected fibers, and
\(N_Z\) is ample. These are actual line identities. We call \(C_Z\)
\emph{vertical} if \(f_Z(C_Z)\ne Y_Z\); the empty floor is vertical.
If \(C_Z\) dominates \(Y_Z\), then for every \(k\geq1\) the restriction
\[
H^0(Z,L_Z^k)\longrightarrow H^0(C_Z,L_Z^k|_{C_Z})
\]
is injective. Indeed, every section comes from \(Y_Z\); if its pullback
vanishes on \(C_Z\), surjectivity makes it vanish at every point of \(Y_Z\).
Fix a K\"ahler class \(c\) on \(V\), and let \(c_Z\) be its restriction
to each stratum. For boundary strata \(Z,Z'\) of the same dimension, an
\emph{arrow} is a proper bimeromorphic comparison with a projective
resolution whose source is smooth and compact K\"ahler,
\[
\begin{tikzcd}[column sep=large]
& T\arrow[dl,"p"']\arrow[dr,"p'"]&\\
Z&&Z'
\end{tikzcd}
\]
such that the two pulled-back adjoint lines are equal as invertible
subsheaves of meromorphic \(q\)-pluricanonical forms on \(T\), and
\begin{equation}\label{floor:class-comparison}
p^*c_Z-(p')^*c_{Z'}\in\NS(T)_{\R}.
\end{equation}
Here \(\NS(T)_{\R}\) is the real span of first Chern classes of
holomorphic lines, with torsion removed. The degree-two formula for a
smooth modification says that its new classes are exceptional divisor
classes. It follows that Equation~\eqref{floor:class-comparison} is independent
of further resolution and is preserved by composition. These arrows
therefore form a groupoid on the finite set of strata of each dimension.
An arrow \(\gamma:Z\dashrightarrow Z'\) transports sections by
\[
(p')^*(\gamma_*s)=p^*s.
\]
Normality gives existence and uniqueness. Transport commutes with
multiplication and identifies the complete systems and their Stein targets
in Equation~\eqref{floor:semiample-map}.
The equality of meromorphic lines defines this transport of adjoint sections; the class
congruence will control self-arrows when the floor is vertical.
\subsection{One residue comparison suffices for restriction}
The only obstruction to extending a section from \(C_Z\) is its possible
variation among the connected components of a general floor fiber. We show
that all these components meet one general Mori fiber, whose floor is
connected or consists of two points. Thus at most one residue comparison
is needed.
\begin{lemma}[Restriction and a Mori link]\label{floor:restriction}
For each positive-dimensional boundary stratum \(Z\), there is either no
comparison or one arrow between two components of \(C_Z\), allowing a
component to be compared with itself, with the following property. In all
sufficiently large divisible degrees \(k\), a section of \(L_Z^k|_{C_Z}\)
extends to \(Z\) if its two residues agree under that arrow. With no
arrow, every such section extends. The degree bound is uniform over
sections. When present, the arrow compares the two coefficient-one points
on general \(\PP^1\) fibers of a projective Mori contraction on a
bimeromorphic compact K\"ahler model of \(Z\).
\end{lemma}
\begin{proof}
We use the dimension-free torsion-free restriction lemma
\cite[Lemma~7.1]{BL}. In its notation, for an effective compact K\"ahler
lc pair \((T,G)\), klt away from its reduced floor \(C\), and a connected
map \(f:T\to P\) to a normal compact base from which an actual positive
adjoint multiple is pulled back, it asserts that
\(R^1f_*\mathcal O_T(-C)\) is torsion-free. The ideal sequence gives
\begin{equation}\label{floor:restriction-obstruction}
\mathscr Q:=\coker(\mathcal O_P\longrightarrow f_*\mathcal O_C)
\lhook\joinrel\longrightarrow R^1f_*\mathcal O_T(-C).
\end{equation}
For \(T=Z\) and \(P=Y_Z\), if \(C_Z\) is vertical, \(\mathscr Q\) has
proper support and is zero. Serre vanishing for the kernel of
\(\mathcal O_P\to f_*\mathcal O_C\), tensored with \(N_Z^k\), extends
every restriction in a uniform large-degree tail. If \(C_Z\) dominates,
the same display says that a section extends as soon as its values are
constant on the reduced floor fiber over a dense open of \(Y_Z\): its
class in the torsion-free \(\mathscr Q\otimes N_Z^k\) then vanishes, and
the local lifts glue. They are unique because \(C_Z\) dominates \(Y_Z\),
so \(\mathcal O_{Y_Z}\to(f_Z)_*\mathcal O_{C_Z}\) is injective.
This uses neither reduced special scheme
fibers nor base change at their points.
Suppose now that \(C_Z\) dominates \(Y_Z\). There is an effective rational
Cartier divisor \(D\) with \(\Supp D=\Supp C_Z\): restrict to \(Z\) the
globally \(\Q\)-Cartier floor primes of \(V\) not containing \(Z\).
For small rational
\(\epsilon>0\), \((Z,B_Z-\epsilon D)\) is effective klt. Apply
Corollary~\ref{prog:small-model} to obtain a projective small strong
\(\Q\)-factorialization \(\rho:Z_0\to Z\), crepant also for the full pair.
Write \(\Delta_0\) for its full boundary and \(D_0=\rho^*D\).
The full adjoint
\[
P=K_{Z_0}+\Delta_0
\]
is the actual semiample rational line pulled back from \(Y_Z\), and the
lowered klt adjoint is \(J_{\rm low}=P-\epsilon D_0\). Put
\(d=\dim Z\), choose \(m>0\) with \(mP\) Cartier and generated, and take
\(b>4md\). Consider
\[
H=J_{\rm low}+bP=(1+b)P-\epsilon D_0.
\]
On a general fiber of the map to \(Y_Z\), its class is
\(-\epsilon\{D_0\}\). This is the negative of a nonzero effective
divisor because the floor dominates \(Y_Z\), so its pairing with a
K\"ahler power is negative. If \(H\) were pseudo-effective, a positive
current representing it would restrict to a positive current on almost
every resolved smooth fiber, contradicting this pairing. Thus \(H\)
is not pseudo-effective.
Apply Lemma~\ref{prog:nef-trivial} to the lowered klt pair and the nef
rational line \(P\). Its \(H\)-program with K\"ahler scaling has ordinary
\(J_{\rm low}\)-negative steps, is finite, and ends in a projective Mori
contraction. Every step, including the final Mori ray, is \(P\)-trivial.
The same actual Cartier line \(mP\) descends at each birational step.
Consequently its semiample map to the fixed base \(Y_Z\) persists on
every working model. That map is constant on each connected final Mori
fiber and hence factors through the Mori contraction. We obtain
\[
T\xrightarrow{u}W\xrightarrow{g}Y_Z.
\]
Both maps have connected fibers. The full adjoint on each model is the
trace of \(P\). On a common resolution of a birational step, its natural
meromorphic canonical comparison is an exceptional divisor with zero
class, since the two actual lines descend from the same line on the
contraction base. Exceptional negativity applied to both signs makes
that divisor zero. Thus the full pair remains crepant, including its
meromorphic adjoint identifications. The full transformed pair
\((T,G)\) is effective lc and is klt away from its full floor \(C^+\):
there it agrees with the transformed lowered klt pair. The program is
nonextracting, so the transformed divisor \(D^+\) has
\(\Supp D^+=\Supp C^+\). Since the full adjoint is pulled back from
\(Y_Z\), the lowered adjoint on a \(u\)-fiber is
\(-\epsilon D^+\). Hence \(D^+\) is relatively ample over \(W\).
Compare the connected components of a general floor fiber on the original
small model and on \(T\). On a common projective log resolution the
full crepant subboundary is the same. Its unit union maps to both floors
with connected fibers by \cite[Lemma~2.1]{BL}. Proper closed surjections
with connected fibers preserve connected components, also after restricting
over a point of \(Y_Z\). Thus the connected components of the two
underlying floor fibers correspond.
Every connected component of the floor fiber on \(T\) meets one common
general Mori fiber. Indeed \(C^+\to W\) is surjective because \(C^+\)
is the support of the relatively ample \(D^+\). Apply
Equation~\eqref{floor:restriction-obstruction} to \(u\). Surjectivity of
the floor makes \(\mathcal O_W\to u_*\mathcal O_{C^+}\) injective; the
displayed torsion-free cokernel then makes \(u_*\mathcal O_{C^+}\)
torsion-free. Let
\[
C^+\longrightarrow\overline C\longrightarrow W
\]
be the Stein factorization. The finite space \(\overline C\) is reduced,
and every irreducible component of
\(\overline C\) dominates \(W\). To see the
last implication algebraically, a vertical minimal prime of a finite
reduced algebra over a domain would give, by prime avoidance, a nonzero
element killed by a nonzero element of the domain. After shrinking \(Y_Z\)
to a dense open, the fiber-dimension theorem gives every component of
\(\overline C_y\) dimension \(\dim W-\dim Y_Z\). The fiber \(W_y\) is irreducible of that
dimension: a resolution of \(W\) still has connected fibers over \(Y_Z\),
and a general one is smooth and connected and surjects onto \(W_y\).
Finiteness now makes each component of \(\overline C_y\) dominate \(W_y\).
Consequently each connected component of \(C^+_y\) meets \(u^{-1}(w)\)
for the same general \(w\in W_y\).
The reduced general fiber \(F\) of \(u\) is irreducible by the same
resolution argument. If \(\dim F\geq2\), the support of the effective
ample Cartier multiple of \(D^+|_F\), whose support is \(C^+\cap F\),
is connected. One elementary
proof, valid even if \(F_{\rm red}\) is not normal, cuts by general
hyperplanes to an integral projective surface. If the ample divisor
split into disjoint nonzero Cartier parts \(D_1,D_2\), their pullbacks
to a resolution would be orthogonal, while the projection formula gives
\(v^*(D_1+D_2)\cdot v^*D_i>0\) for both \(i\). Thus both
\((v^*D_i)^2>0\), contradicting the surface Hodge index theorem.
If \(\dim F=1\), choose the fiber also off the singular, boundary
intersection, and ramification loci. It is a smooth connected curve and
\[
0=\deg(K_F+G|_F)=2g(F)-2+\deg(G|_F).
\]
There is a unit point, so \(F\simeq\PP^1\). Its floor has one or two
points; if there are two they exhaust the horizontal boundary. In every
connected case the preceding common-fiber observation makes the general
floor fiber connected, so no comparison is needed. In the remaining case
the two unit points represent all its connected components.
Normalize the horizontal floor primes and take their Stein factorizations
over \(W\). If there are two primes, each finite Stein map has degree one
and is an isomorphism over normal \(W\); their main fiber product gives
a proper bimeromorphic comparison. If there is one prime, its normal Stein
space is a double cover of \(W\). The exchange on its general fiber
extends over the branch locus: the reduced horizontal non-diagonal
component of the double fiber product has finite birational projections
to the normal cover and hence both projections are isomorphisms. Its graph
lifts to the normalized prime. Composing with the program graphs transports
the comparison to two original normal floor primes, possibly the same one.
All graph projections are projective, and the primes survive birationally
because the program extracts no divisors.
This comparison is crepant as a meromorphic residue identity. On a smooth
ruled open, order the markings after an \'etale local cover, choose a base
volume \(\xi\), and put the markings at \(z=0,\infty\). A pulled-back
\(q\)-pluriadjoint frame has the form
\[
a(w)(dz/z)^{\otimes q}\otimes\xi^{\otimes q}.
\]
Its two residues are \(a(w)\xi^{\otimes q}\) and
\((-1)^q a(w)\xi^{\otimes q}\), which agree. This is the two-marking
Poincar\'e-residue comparison of
\cite[Section~3, Definition~13 and Proposition~14]{Kollar2013Sources}.
On a resolved graph both restricted actual lines are pulled back from
\(Y_Z\); their meromorphic embeddings agree on this dense open and hence
everywhere, including along exceptional primes. Equal residues therefore
give a constant value on the entire general floor fiber. The horizontal
consequence of Equation~\eqref{floor:restriction-obstruction} extends the section.
Normality descends it from the small model. Its restriction is the
prescribed section on the original reduced floor, since every original
floor prime is covered birationally and the two restrictions agree after
that pullback.
It remains to check Equation~\eqref{floor:class-comparison}. Let \(\widehat T\)
be a common smooth resolution of the parent and Mori models and let
\(a,b:H\to\widehat T\) be the two maps from a resolved branch graph.
The resolution of the Mori model is an isomorphism near a general entire
\(\PP^1\) fiber: its centers have codimension at least two, hence cannot
dominate the ruling base. Every holomorphic two-form on \(\widehat T\)
comes from the base on that ruled open. In fact its vertical one-form
terms vanish on \(\PP^1\), and its remaining coefficients are constant
on that compact fiber. Thus \(a^*-b^*\) kills \(H^{2,0}\), and also
\(H^{0,2}\) by conjugation. This rational Hodge map on degree two has
image of type \((1,1)\). The Lefschetz \((1,1)\) theorem puts its real
image in \(\NS(H)_{\R}\). Apply this to the pulled-back parent class
\(c_Z\). The link is therefore an arrow as defined above.
\end{proof}
\subsection{Transport respects every lower residue}
An arrow can contract a divisor to a higher-codimension stratum. The next
local observation explains how to compare residues even in that case.
All valuations in its proof are local monomial valuations in an SNC chart.
\begin{lemma}[Unit strata above an SNC stratum]\label{floor:unit-strata}
Let \(p:T\to Z\) be a projective log resolution of a dlt pair, with
crepant SNC subboundary on the smooth \(T\). Let \(R\subseteq T\) be a
unit stratum and let \(D=p(R)\) be its center. Then
\(D\) is a dlt stratum. There is a unit stratum \(R'\subseteq R\) such
that \(p(R')=D\) and \(R'\to D\) is birational. In the iterated residue
comparison along \(R'\), the normal logarithmic Jacobian is \(\pm1\).
\end{lemma}
\begin{proof}
The center is a dlt stratum by the SNC discrepancy calculation in the
proof of Lemma~\ref{floor:adjunction}. Work over the generic SNC part of \(D\),
of codimension \(a\), with normal coordinates \(x_1,\ldots,x_a\) for
the unit divisors. The zero-discrepancy divisorial valuations centered
there are exactly the primitive rational monomial rays in
\(\R_{>0}^a\). Here is a local verification that also fixes their
normalization. The discrepancy inequality in the proof of
Lemma~\ref{floor:adjunction} says on every log-smooth model that the center
of a zero-discrepancy valuation is exactly a unit stratum: an additional
unmarked normal parameter, or a parameter with coefficient below one, would
give positive discrepancy. Blow up that stratum. If the positive normal
orders of the valuation are \(w_1,\ldots,w_b\), the chart indexed by a
smallest order \(w_k\) replaces these by \(w_k\) and the positive members
of \(w_i-w_k\). The exceptional divisor is again unit, and the center
is exactly the new unit stratum. While there is more than one positive
order their sum decreases. Eventually the center is the generic point of
one unit divisor, where a normalized divisorial valuation has order one.
Reversing these ordinary toric blowups proves monomiality and primitive
normalization. Conversely this subtraction algorithm for any primitive
positive integer vector terminates at order one, and reversing it extracts
that monomial valuation. Comparison on a common graph gives uniqueness
of the valuation with those weights and of its center on any model.
For a unit component \(E\) meeting the inverse image of the generic SNC
part of \(D\), put
\[
v_E=(\ord_E x_1,\ldots,\ord_E x_a)\in\Z_{\geq0}^a.
\]
If the unit components through a unit stratum \(Q\) meeting that inverse image are
\(E_1,\ldots,E_b\), define
\[
\phi_Q:\R_{\geq0}^b\longrightarrow\R_{\geq0}^a,\qquad
(t_1,\ldots,t_b)\longmapsto\sum_{j=1}^b t_jv_{E_j},
\qquad \sigma_Q=\phi_Q(\R_{\geq0}^b).
\]
The source carries its standard lattice \(\Z^b\). For the fixed
resolution \(T\), these cones subdivide the orthant over its interior.
On rational rays, \(\phi_Q\) sends a
monomial valuation on \(T\) to the same normalized valuation on \(Z\),
so it is injective on each cone. Interiors of different cones are
disjoint, because the center and the weights of a monomial valuation on
\(T\) are unique. Every rational ray in the orthant occurs: view its
valuation on \(T\) and use the same SNC description there. There are
only finitely many cones over the generic part of \(D\); density of
rational rays therefore gives coverage. Taking subsets of unit components
gives the faces, and uniqueness of valuations also on faces identifies
intersections as common faces. Consequently \(\sigma_R\) is a face
of an \(a\)-dimensional cone \(\sigma_{R'}\) with the same downstairs
center, for a unit stratum \(R'\subseteq R\).
Let \(E_1,\ldots,E_a\) be the unit components through \(R'\).
The integer matrix \(A=(\ord_{E_j}x_i)\) of this full cone is unimodular.
Indeed every positive
primitive integral vector in the source cone is the normalized vector of
a divisorial valuation, so its image is primitive. If a prime divided
\(\det A\), a nonzero kernel vector modulo that prime, lifted to a
positive primitive integral vector, would have nonprimitive image. Hence
\(\det A=\pm1\). The strata \(R'\) and \(D\) have the same dimension.
We next prove that their generically finite map has degree one. In local coordinates at a
general point of \(R'\),
\[
x_i=u_i(y,z)\prod_{j=1}^a y_j^{A_{ij}},
\]
where the \(u_i\) are units, the \(y_j\) are normal coordinates, and
\(z\) are coordinates along the stratum. The tangential map is
generically \'etale in characteristic zero. Since \(A^{-1}\) is integral,
replace \(y_j\) by
\(\widetilde y_j=y_j\prod_i u_i^{(A^{-1})_{ji}}\). Then
\(x_i=\prod_j\widetilde y_j^{A_{ij}}\), and the map consisting of
\(\widetilde y\) and the downstairs tangential coordinates is locally
biholomorphic at a general \'etale point of \(R'\to D\). Two distinct
points over a general point of \(D\) would therefore give two lifts of
nearby target points whose normals approach with any fixed positive rate
vector in the interior of this cone. Varying the nonzero leading
coefficients fills an open set of nearby torus points, so such a target can
be chosen in the locus where \(p\) is an isomorphism. This contradiction
proves that \(R'\to D\) is birational. Finally, after taking residue
along \(R'\), the normal coefficient of
\(\bigwedge_i dx_i/x_i\) is \(\det A\) times that of
\(\bigwedge_jdy_j/y_j\); terms differentiating the units vanish in that
residue. Since \(\det A=\pm1\), the even pluriresidue removes the sign.
\end{proof}
\begin{lemma}[Preservation of lower restrictions]\label{floor:transport}
Fix \(0\leq d<n\) and \(k\geq1\). For each boundary stratum \(Z\) of
dimension at most \(d\), let \(s_Z\in H^0(Z,L_Z^k)\). Suppose these
sections agree under the residue restriction for every incidence, and that
every arrow in dimensions below \(d\) carries its source section to its
target section. For any \(d\)-arrow \(\gamma:Z\dashrightarrow Z'\) and
every component \(D'\) of \(C_{Z'}\),
\[
(\gamma_*s_Z)|_{D'}=s_{D'}.
\]
Moreover verticality of \(C_Z\) is invariant under \(d\)-arrows.
\end{lemma}
\begin{proof}
The assertions are immediate for \(d=0\), when every floor is empty.
For \(d>0\), choose a common log resolution
\(p:T\to Z\), \(p':T\to Z'\) of the arrow. For a target floor prime
\(D'\), start with its strict transform \(R\). It is a unit stratum and
maps birationally to \(D'\). Its source center \(p(R)\) is a stratum by
Lemma~\ref{floor:unit-strata}. If \(R\to p(R)\) is not birational, that
lemma replaces \(R\) by a unit stratum inside it which maps birationally
onto the same source center. The target center may then shrink. If the
map to that center is not birational, apply the lemma on the target side,
and continue alternately. Each necessary replacement strictly decreases
dimension. The process therefore ends with one unit stratum mapping
birationally to lower strata \(D_0,D_0'\) on the two sides.
The image in the common system base remains unchanged throughout. Indeed
the two maps from the resolved graph to the Stein targets are identified
by the arrow, and a replacement keeps its entire image on the side then
being treated. Initially the image was \(f_{Z'}(D')\). The terminal
comparison of \(D_0\) and \(D_0'\) is an arrow: restricting the actual
meromorphic identification to iterated residues gives crepancy, with the
logarithmic determinant sign removed by Lemma~\ref{floor:unit-strata};
restricting the line Chern classes in Equation~\eqref{floor:class-comparison}
gives the same condition modulo \(\NS_{\R}\). Lower invariance thus
identifies the transported and assigned residues on \(D_0'\).
This equality determines the entire section on \(D'\). The Stein map
of \(L_{D'}=L_{Z'}|_{D'}\) is the Stein factor of
\(f_{Z'}|_{D'}\), because the line induced on that Stein space is ample;
its target is finite over \(f_{Z'}(D')\). The image of \(D_0'\) in this
target is a closed irreducible subset of full dimension, since it still
maps onto \(f_{Z'}(D')\), and therefore is the whole target. Both
sections on \(D'\) come from it. Equality after restriction to
\(D_0'\) proves equality on \(D'\).
The same construction shows that the image of every target floor prime
in the system base is the image of a lower source stratum, and the reverse
statement follows from the inverse arrow. A floor dominates the system
base on one side exactly when it does on the other. This proves the last
assertion.
\end{proof}
\subsection{Finite actions on vertical strata}
Once lower-dimensional sections are compatible and invariant,
Lemma~\ref{floor:transport} preserves their restrictions under every arrow.
If \(C_Z\) dominates \(Y_Z\), injectivity of restriction forces any such
transported extension to equal the assigned extension. Thus only vertical
strata require the following argument for finite image.
\begin{proposition}[Finite image of self-arrows]\label{floor:finite-action}
Let \(Z\) be a boundary stratum with \(C_Z\) vertical. For all sufficiently
large divisible \(k\), the self-arrows of \(Z\) have finite image on
\(H^0(Z,L_Z^k)\).
\end{proposition}
For \(m=qk\), a pluricanonical eigenform \(s\in H^0(Z,L_Z^k)\) with
\(\int_Z|s|^{2/m}<\infty\) becomes the \(m\)-th power of a holomorphic top
form on a smooth resolution of a cyclic root cover, so the auxiliary proposition below makes its
eigenvalue a root of unity. A uniform bound on the middle Betti numbers of
selected resolutions of these covers in the fixed degree will then bound
the possible orders; boundedness of the representation will give finite
image. The auxiliary proposition uses the polarization congruence in
Equation~\eqref{floor:class-comparison}.
\subsubsection{Top-form scalars}
\begin{proposition}[A polarized top-form scalar]\label{floor:top-form-scalar}
Let \(U\) be a smooth connected compact K\"ahler manifold of dimension
\(d<n\), and let \(g:U\dashrightarrow U\) be bimeromorphic. Suppose
that on a common smooth resolution the two pullbacks of some K\"ahler
class on \(U\) differ by an element of \(\NS_\R\). If
\(0\ne\alpha\in H^0(U,K_U)\) and \(g^*\alpha=\mu\alpha\), then
\(\mu\) is a root of unity.
\end{proposition}
We first control the canonical Iitaka base; this step uses \(\Ggood_d\)
and the top form, with no polarization congruence. To specify that base,
let \(U\) be a smooth connected compact K\"ahler \(d\)-fold with \(d<n\)
and \(\kappa(U,K_U)\geq0\). A pluricanonical section makes \(K_U\)
pseudo-effective, so \(\Ggood_d\) gives a smooth
modification \(\pi:U'\to U\) and
\[
\pi^*K_U\sim_{\Q}P+E_U,\qquad
P\ \text{semiample},\qquad E_U=N(\pi^*K_U).
\]
Choose \(\ell>0\) sufficiently divisible that the connected semiample map
and its Stein target satisfy
\[
f:U'\longrightarrow Y,\qquad
\mathcal O_{U'}(\ell P)=f^*N,
\]
where \(Y\) is normal projective and \(N\) is ample, and that
Lemma~\ref{bir:sections} identifies every graded piece in
\[
R_\ell(U):=\bigoplus_{j\geq0}H^0(U,j\ell K_U)
\simeq \bigoplus_{j\geq0}H^0(Y,N^j).
\]
This is the full section ring of the ample line \(N\), so
\(Y\simeq\Proj R_\ell(U)\). Passing to a further divisible Veronese gives
the same Proj. We call \(Y\) the \emph{canonical Iitaka base}. A
bimeromorphic self-map of \(U\) acts on this graded ring and therefore
induces an automorphism of \(Y\).
\begin{lemma}[Finite action on the canonical Iitaka base]\label{floor:finite-base}
Let \(U\) be a smooth connected compact K\"ahler manifold of dimension
\(d<n\), let \(0\ne\alpha\in H^0(U,K_U)\), and let \(g:U\dashrightarrow U\)
be bimeromorphic with \(g^*\alpha=\mu\alpha\). The induced automorphism
\(h\) of the canonical Iitaka base \(Y=\Proj R_\ell(U)\), for a sufficiently
divisible \(\ell\) as above, has finite order.
\end{lemma}
\begin{proof}
The top form gives \(\kappa(U,K_U)\geq0\), so the preceding construction
applies. The assertion is
immediate for a point base. Otherwise use
\(f:U'\to Y\) above, and write \(\alpha\) also for its pullback to \(U'\).
For each positive integer \(k\) the
canonical integral \(s\mapsto\int_U|s|^{2/k}\) on \(H^0(U,kK_U)\) is
positive, continuous, and invariant under bimeromorphic change of variables.
Choose \(j\) such that \(N^j\) is very ample. The powers of \(g\) and
their inverses are bounded on the corresponding degree \(j\ell\), which
defines \(Y\). This linear action is semisimple with eigenvalues of
absolute value one. Also \(|\mu|=1\), by integration of
\(\alpha\overline\alpha\). Hence the finite measure
\[
\nu=f_*(i^{d^2}\alpha\wedge\overline\alpha)
\]
is invariant under \(h\); exceptional sets have measure zero here.
Suppose that \(h\) has infinite order. In the projective linear group of
this degree, the algebraic closure of a power of \(h\) is
then a positive-dimensional torus \(T\), acting faithfully on \(Y\).
Indeed a bounded linear operator is diagonalizable, and the connected
algebraic closure of a cyclic diagonal group is a torus. We will obtain a
contradiction from any nontrivial one-parameter subgroup of \(T\).
Let \(Y^\circ\) be a dense Zariski open in the smooth locus where \(f\)
is smooth and the relative top form obtained from \(\alpha\) is nonzero.
Put \(r=\dim U'-\dim Y\). A general fiber \(F\) of an Iitaka map has
\(\kappa(F)=0\). It has a nonzero holomorphic top form here, so
\(h^0(F,K_F)=1\): two such forms would have a nonconstant ratio and give
positive Iitaka dimension. If \(\xi\) is a local holomorphic volume
frame on \(Y^\circ\) and \(\sigma=\alpha/\xi\) is the relative top
form, fiber integration writes the measure as
\begin{equation}\label{floor:hodge-density}
\nu=\|\sigma\|_{\rm Hdg}^{2}\,i^{(\dim Y)^2}\xi\wedge\overline\xi.
\end{equation}
The coefficient is smooth and strictly positive after this shrink. We
include \(r=0\), when the top Hodge line is the one-dimensional degree-zero
line.
Set
\[
Y^*=\bigcup_{j\in\Z}h^j(Y^\circ).
\]
Invariance of \(\nu\) under \(h\) makes the smooth positive densities
agree on overlaps, extending the density in
Equation~\eqref{floor:hodge-density} to \(Y^*\).
This open is invariant under \(T\). Its complement is the
intersection of the closed algebraic sets \(h^j(Y\setminus Y^\circ)\),
which is closed algebraic by Noetherianity. Its algebraic stabilizer
contains the cyclic group generated by \(h\), and therefore contains its
algebraic closure. Choose a one-parameter subgroup \(\C^*\subset T\)
which acts nontrivially, and a general whole orbit in \(Y^*\).
The \(T\)-invariance of the domain supplies this whole orbit. We now
construct a horizontal period metric on \(Y^*\) and prove that its
restriction to the orbit vanishes.
For a smooth K\"ahler family a total K\"ahler class polarizes the primitive
real variation in middle cohomology. On a simply connected open of
\(Y^\circ\), take the smallest real subvariation of the full middle
cohomology containing the line \(F^r\) of top forms and its conjugate.
It is the intersection of the flat real Hodge subspaces with this property;
finite dimensionality reduces the intersection to a finite one. This
construction commutes with restriction and continuation: in a flat
trivialization the condition of being a Hodge subspace, and of containing
the indicated line, is a real analytic equality for the Hodge projections,
so equality on one open continues on the simply connected cover. The
primitive variation is one such subvariation, so the smallest one lies in
primitive middle cohomology. Its polarization is the middle cup pairing
with the usual sign, independent of the total K\"ahler class, because no
power of that class occurs in the middle primitive pairing. The horizontal
period metric of this polarized real variation is therefore intrinsic.
This possibly degenerate metric extends consistently to \(Y^*\).
To compare the metrics on an overlap of two translates of \(Y^\circ\),
shrink the overlap further so that a resolution of the fiberwise birational
graph is a smooth family over it. The two pullbacks on middle cohomology are
flat Hodge embeddings into the cohomology of this common resolution and
carry the top lines to the same line. The smallest subvariation there is
the pullback of the smallest subvariation on either side: it is contained
in both images, and applying the inverse embedding gives the reverse
inclusion. Projection formula identifies their middle cup polarizations.
Their horizontal metrics therefore agree on this smaller open and, by
continuity, on the overlap.
We claim that the period metric restricts to zero on every whole orbit
of this one-parameter subgroup in \(Y^*\); constant orbits are immediate.
For a nonconstant orbit, pull it back under
\(\exp:\C\to\C^*\). The negative upper bound for holomorphic sectional
curvature in horizontal directions of a period domain, together with the
curvature inequality for a pulled-back curve metric, applies to its local
period representations; it holds for real polarizations as well
\cite[Theorem~9.1]{GriffithsSchmid1969}. On a disk of radius \(R\)
centered at an arbitrary point, translate the center to \(z=0\) and
write the pulled-back metric as \(\chi(z)|dz|^2\), continuous and smooth
where positive, with curvature
\(-2\chi^{-1}\partial_z\partial_{\bar z}\log\chi\le-\kappa<0\).
Compare it with
\[
\chi_R(z)=\frac{4R^2}{\kappa(R^2-|z|^2)^2}.
\]
Direct calculation gives
\(\partial_z\partial_{\bar z}\log\chi_R=\kappa\chi_R/2\).
If the maximum of \(\chi/\chi_R\) exceeded one, it would occur where
\(\chi>0\) in the interior, since \(\chi_R\) diverges at the boundary
and \(\chi\) is bounded on the closed disk. At that point,
\[
0\ge\partial_z\partial_{\bar z}\log(\chi/\chi_R)
\ge\frac{\kappa}{2}(\chi-\chi_R)>0,
\]
a contradiction. Thus the Ahlfors--Schwarz comparison gives
\(\chi(0)\le4/(\kappa R^2)\) \cite{Ahlfors1938}.
Letting \(R\to\infty\) forces the metric to vanish
on \(\C\). The argument is local and allows zeros of the
induced metric, so the local representations just constructed suffice.
Consequently their period maps are locally constant along each such orbit.
In any holomorphic volume frame the logarithm of the coefficient in
Equation~\eqref{floor:hodge-density} is harmonic along each orbit: the top Hodge line is
flat there, and \(\sigma\) is a nonzero holomorphic multiple of a flat
generator.
This is incompatible with finite mass. Diagonalize the one-parameter
action in projective coordinates, and choose two coordinates nonzero at a
general point with different weights, of difference \(a\ne0\). A small
smooth transverse slice \(Q\) on which their ratio is one gives a
holomorphic local biholomorphism onto its image
\[
\Phi:\C^*\times Q\longrightarrow Y^*
\]
with fibers of size at most \(|a|\): the ratio at \(\Phi(z,t)\) is
\(z^a\). The entire \(\C^*\) factor is allowed because \(Y^*\) is
invariant. In the product frame \((dz/z)\wedge dt_1\wedge\cdots\wedge
dt_{\dim Q}\), write \(\Phi^*\nu\) with coefficient \(\rho(z,t)>0\).
For fixed \(t\), \(\log\rho(z,t)\) is harmonic on \(\C^*\). Its
circular mean at \(|z|=e^u\) is \(A(t)u+B(t)\). Jensen's inequality
therefore gives, with an irrelevant positive normalization,
\[
\int_{\C^*}\rho(z,t)\frac{i\,dz\wedge d\overline z}{|z|^2}
\ \geq\ c\int_{\R}e^{A(t)u+B(t)}\,du=\infty.
\]
Tonelli's theorem contradicts the finite mass of \(\Phi^*\nu\), which
is at most \(|a|\nu(Y)\) by the bounded covering degree. This proves
that \(h\) has finite order.
\end{proof}
After a power, the scalar problem can thus be restricted to a fiber with
Kodaira dimension zero. The polarization congruence then passes to that
fiber; the next two lattice arguments control its nonprojective factors.
In the next lemma, the hypothesis
on the origin of the model is what provides the degree-two exceptional
splitting; rational singularities alone are not used for that splitting.
\begin{lemma}[The scalar on a symplectic factor]\label{floor:symplectic-scalar}
Let \(Y\) be a terminal compact K\"ahler primitive symplectic space of
dimension \(2e<n\), obtained from a smooth model by the program of
Corollary~\ref{prog:terminal-kappa-zero}. Let \(g:Y\dashrightarrow Y\) be
bimeromorphic and let \(0\ne\eta\in H^0(Y,\Omega_Y^{[2]})\). Suppose
there is a class \(c\in H^{1,1}(Y,\R)\) whose two pullbacks on a common
resolved graph of \(g\) differ by real line Chern classes, and such that
on some projective resolution \(p:R\to Y\) with smooth compact K\"ahler source
\[
p^*c=[\beta]+\{E\},
\]
where \(E\) is an exceptional real divisor and \(\beta\) is a smooth
closed semipositive \((1,1)\)-form, strictly positive on a nonempty open.
Then the scalar in \(g^*\eta=\lambda\eta\) is a root of unity.
\end{lemma}
\begin{proof}
We use the pure Hodge structure on \(H^2(Y,\Q)\) and its rational
Beauville--Bogomolov form \(q_Y\), up to positive rational scale. This
form is positive on the real symplectic plane and Lorentzian on
\(H^{1,1}(Y,\R)\); see \cite[Lemma~2.1, Definition~5.2, and
Lemmas~5.3 and~5.7]{BakkerLehn}. The pullback and \((1,1)\)
identifications for these rational singularities, extension of the
two-form \cite[Corollary~1.8]{KebekusSchnell2021}, and
Lemma~\ref{prog:cohomology-transport} give
\begin{equation}\label{floor:symplectic-splitting}
H^2(R,\Q)=p^*H^2(Y,\Q)\oplus
\langle[p\text{-exceptional primes}]\rangle_{\Q}.
\end{equation}
Indeed that program lemma gives the real \((1,1)\) equality with a direct
exceptional sum; the \((2,0)\) and \((0,2)\) summands compare by form
extension, and all the summands in the resulting equality are rational.
Let \(\mathcal E_R\) be the rational span of the \(p\)-exceptional prime
classes. Identify \(H^2(R,\Q)/\mathcal E_R\) with \(H^2(Y,\Q)\) by
Equation~\eqref{floor:symplectic-splitting}, and define the lattice
\[
\Lambda_Y=
\im\bigl(H^2(R,\Z)/\mathrm{tors}\longrightarrow
H^2(R,\Q)/\mathcal E_R\bigr)
\subset H^2(Y,\Q).
\]
This amounts to quotienting the integral lattice by its saturated
exceptional sublattice. The lattice is independent of a refinement
\(v:R'\to R\): integral Gysin satisfies \(v_*v^*=1\), and the smooth
degree-two modification formula says that
\(x'-v^*v_*x'\) is exceptional and that the new exceptional span is
\(v^*\) of the old one plus the \(v\)-exceptional span. A common
smooth refinement therefore identifies the quotient lattices from any two
resolutions over \(Y\).
The form \(\eta^e\) trivializes \(K_Y\). On a resolution \(R_g\) of
the graph of \(g\), with projections \(p_1,p_2\), the divisors of the
two pulled-back volume forms agree. Terminality says that their positive
components are exactly the exceptional primes. Thus the exceptional lists
for the two projections are identical; in particular \(g\) is small in
both directions. Let \(\mathcal E\) be their common exceptional rational
span, and let
\[
\overline p_i^*:H^2(Y,\Q)\xrightarrow{\sim}H^2(R_g,\Q)/\mathcal E.
\]
Both maps carry \(\Lambda_Y\) onto the image of integral
graph cohomology in this quotient. Hence
\(G=(\overline p_1^*)^{-1}\overline p_2^*\) is a rational Hodge
automorphism of \(H^2(Y,\Q)\) preserving \(\Lambda_Y\).
It also preserves \(q_Y\). The two symplectic forms pull back up to the scalar
\(\lambda\), and integration of their top powers gives \(|\lambda|=1\).
Use the usual positive normalization of
\(W=(p_1^*\eta\,p_1^*\overline\eta)^{e-1}
=(p_2^*\eta\,p_2^*\overline\eta)^{e-1}\). For \((1,1)\) classes
\(x,y\), the differences between \(p_1^*Gx\) and \(p_2^*x\) are
exceptional for both projections. Expand
\(p_1^*Gx\,p_1^*Gy-p_2^*x\,p_2^*y\) into two terms, each with one such
exceptional factor and one factor pulled back by the appropriate projection.
Projection formula makes both pairings with \(W\) zero. This is precisely
the \((1,1)\) part of the Beauville--Bogomolov formula. The symplectic
plane is preserved isometrically because \(|\lambda|=1\), and the Hodge
decomposition is orthogonal. Thus \(G\) is a \(q_Y\)-isometry.
The class \(c\) in the statement has \(q_Y(c)>0\). Since \(\beta\)
is nef over \(Y\), exceptional negativity applied to the divisor \(E\),
with \(\{E\}=p^*c-[\beta]\), gives \(E=\sum_i e_iE_i\geq0\). Pair with the positive
normalization \(W_R=(p^*\eta\,p^*\overline\eta)^{e-1}\). Exceptional
classes pair to zero with \(p^*c\) by projection formula, and hence for
some constant \(b_e>0\)
\begin{equation}\label{floor:positive-bb}
q_Y(c)=b_e\left(\int_R\beta^2 W_R+\sum_i e_i\int_{E_i}\beta W_R\right)>0.
\end{equation}
Every term is nonnegative. The first is strictly positive on the common
open where \(\beta\) is positive and the symplectic form is nondegenerate.
Set \(H_\Q=H^2(Y,\Q)\) and
\(D_\Q=H_\Q\cap H^{1,1}(Y,\C)\). Lefschetz \((1,1)\) on \(R\), together
with the line-trace assertion of Lemma~\ref{prog:cohomology-transport},
identifies \(D_\Q\) with the rational span of line Chern classes.
It is \(G\)-invariant because \(G\) is a rational Hodge automorphism.
The same line-trace assertion applied to the graph congruence gives
\(Gc-c\in D_\R\). The line \(\C\eta\) survives in \(H_\C/D_\C\);
our immediate aim is to prove that all eigenvalues on this quotient have
absolute value one. Write \(\Sigma=(H^{2,0}\oplus H^{0,2})_\R\), the
positive real two-plane. There are three possibilities for the restriction
of the Lorentz form to \(D_\R\).
If \(D_\R\) is negative definite, its orthogonal projection \(c_0\) of
\(c\) to \(D_\R^\perp\) is fixed by \(G\) and has positive square by
Equation~\eqref{floor:positive-bb}. On \(D_\R^\perp\), the invariant space
\(\Sigma\oplus\R c_0\) is positive definite and its complement is
negative definite. All eigenvalues there have absolute value one. If
\(D_\R\) contains a positive vector, it is nondegenerate with one
positive direction. Its orthogonal complement is the positive plane
\(\Sigma\) plus a negative-definite space, giving the same conclusion.
In either case \(H_\R/D_\R\simeq D_\R^\perp\).
In the remaining case \(D_\R\) is negative semidefinite with radical a
rational isotropic line \(\ell\). The lattice action on \(\ell\) is
\(\pm1\). On \(\ell^\perp/\ell\) the form is the positive plane
\(\Sigma\) plus a negative-definite \((1,1)\) space, and the quotient
\(H_\R/\ell^\perp\) is dual to \(\ell\). The invariant flag
\(0\subset\ell\subset\ell^\perp\subset H_\R\) therefore shows that
all eigenvalues on \(H_\R\) have absolute value one; this argument also
allows unipotent parts. In all cases the eigenvalues on
\(H_\C/D_\C\) have absolute value one. The rational quotient
\(H_\Q/D_\Q\) carries the preserved lattice
\[
\overline\Lambda_Y=\Lambda_Y/(\Lambda_Y\cap D_\Q),
\]
where the intersection is saturated. Thus
\(\lambda\) is an algebraic integer and all its algebraic conjugates have
absolute value one. Kronecker's theorem makes it a root of unity.
\end{proof}
\begin{lemma}[The scalar on a torus]\label{floor:torus-scalar}
Let \(A\) be the integral Hodge automorphism of \(H^1(T,\Z)\) induced
by an automorphism of a compact complex torus \(T\). Suppose a positive
Hermitian class \(c\in H^{1,1}(T,\R)\) satisfies
\((\bigwedge^2 A)c-c\in\NS(T)_\R\). Then
\(\det(A|H^{1,0}(T))\) is a root of unity.
\end{lemma}
\begin{proof}
Put \(V=H^1(T,\Q)\) and \(D_\Q=\NS(T)_\Q\). For each eigenvalue
\(\theta\) of \(A_\C\), let \(V_\theta\) be its generalized eigenspace and set
\(d_\theta=\dim V_\theta\), \(p_\theta=\dim(V_\theta\cap H^{1,0}(T))\).
Complex conjugation gives
\(p_\theta+p_{\overline\theta}=d_\theta=d_{\overline\theta}\).
For \(|\theta|\ne1\), project \(c\) to the block
\(V_\theta\wedge V_{\overline\theta}\), with \(\bigwedge^2V_\theta\) understood for
real \(\theta\). This projection preserves \(D_{\overline\Q}\). Indeed
the Chinese-remainder projectors onto the generalized spaces are
polynomials in \(A\) with algebraic coefficients. Apply them to the
two tensor slots separately and then alternate. The resulting maps are
algebraic linear combinations of rational Hodge maps on \(\bigwedge^2V\);
these preserve rational \((1,1)\) classes, hence \(D_\Q\) by Lefschetz
\((1,1)\). Separate slot projections avoid any collision of products of
eigenvalues in degree two.
On this block \(A_2=\bigwedge^2 A\) has the single generalized
eigenvalue \(\theta\overline\theta=|\theta|^2\ne1\). The inverse of \(A_2-1\)
there is a finite polynomial in its nilpotent part. Project the congruence
for \(c\) to the block and apply this inverse. It follows that its block
\(c_{\theta,\overline\theta}\) belongs to \(D_\C\). Positivity says that this
block is a perfect tensor between \(V_\theta\) and \(V_{\overline\theta}\),
pairing their opposite Hodge types: in a basis of \(H^{1,0}\) the relevant
matrices are positive-definite principal Hermitian blocks. For real
\(\theta\) the corresponding alternating tensor is nondegenerate.
The intersection of this block with \(D_\C\) is defined over
\(\overline\Q\). Nondegeneracy is the nonvanishing of a determinant,
so it contains a nondegenerate tensor with algebraic coefficients. For
any Galois automorphism \(\gamma\), the conjugate tensor remains in
\(D_{\overline\Q}\), hence is still of type \((1,1)\) for the given
Hodge decomposition, and is perfect between \(V_{\gamma(\theta)}\) and
\(V_{\gamma(\overline\theta)}\). Opposite-type perfection yields
\begin{equation}\label{floor:torus-hodge-count}
p_{\gamma(\theta)}+p_{\gamma(\overline\theta)}
=d_\theta\qquad(|\theta|\ne1).
\end{equation}
This does not replace \(\gamma(\overline\theta)\) by
\(\overline{\gamma(\theta)}\), and assumes no Galois invariance of the Hodge
decomposition.
The top-form scalar is the algebraic integer
\(z=\prod_\theta \theta^{p_\theta}=\det(A|H^{1,0})\), also when generalized eigenspaces
are nontrivial. For any Galois automorphism \(\delta\), reindex by the
image roots and pair complex conjugates:
\[
\begin{aligned}
\log|\delta z|
&=\frac12\sum_\theta
\bigl(p_{\delta^{-1}(\theta)}
+p_{\delta^{-1}(\overline\theta)}\bigr)\log|\theta|\\
&=\frac12\sum_\theta d_\theta\log|\theta|
=\tfrac12\log|\det A|=0.
\end{aligned}
\]
For \(|\theta|\ne1\) the second equality uses
Equation~\eqref{floor:torus-hodge-count} with \(\gamma=\delta^{-1}\); the other
terms vanish. The last equality is unimodularity of the integral action.
Kronecker's theorem now proves the assertion.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{floor:top-form-scalar}]
Apply Lemma~\ref{floor:finite-base}. After a power, \(g\) acts trivially
on the canonical Iitaka base. Use the modification \(\pi:U'\to U\) and
the map \(f:U'\to Y\) in its construction, and restrict over a general
point where the graph and the fibration are smooth and the fiber meets the
isomorphism locus of \(\pi\). Dividing \(\alpha\) by a local base volume
gives a nonzero top form on the smooth compact fiber \(F\), which has
\(\kappa(F)=0\); its returning map scales the form by the corresponding
power of \(\mu\). A point fiber gives scalar one. For a positive-dimensional
fiber, let \(\omega_U\) be a K\"ahler form representing the class in the
proposition and put \(\beta_F=(\pi|_F)^*\omega_U\). This is a smooth
semipositive form, strictly positive on the dense open where \(\pi\) is
an isomorphism. Pulling the original graph congruence to \(U'\) and
restricting to the fiber shows that the two pullbacks of \([\beta_F]\) on
a common resolved graph of the returning map of \(F\) differ by real line classes.
This weaker positivity suffices below.
Since \(\dim F\leq d<n\), Corollary~\ref{prog:terminal-kappa-zero}, using
\(\Ggood_{\dim F}\), supplies a terminal compact K\"ahler minimal model
\(M\) of \(F\) with torsion canonical line. The descended
nonzero reflexive top form trivializes \(K_M\) itself. Indeed a power of
that form, in a trivialization of a multiple of \(K_M\), is a nonzero
holomorphic function on the compact connected \(M\), hence a nonzero
constant. On a graph resolution of the returning map the two divisors of
this volume form agree; terminality makes their positive components exactly
the exceptional primes. The returning map on \(M\) is therefore small
in both directions.
The K\"ahler Beauville--Bogomolov decomposition
\cite[Theorem~A]{BGL} supplies a finite cover, \'etale in codimension one,
of \(M\) of the form
\[
P=T\times\prod_i H_i\times C,
\]
where \(T\) is a torus, the \(H_i\) are primitive symplectic factors,
and \(C\) is the product of the strict Calabi--Yau factors of dimension
at least three. One-dimensional factors are elliptic curves and are
absorbed into \(T\); two-dimensional factors are placed among the
symplectic ones. An absent factor is a point and contributes scalar one.
Reflexive forms on these klt spaces can be computed on resolutions by
\cite[Corollary~1.8]{KebekusSchnell2021}. A power of the
small returning map lifts bimeromorphically to \(P\). To justify this,
purity makes the restriction of the cover over \(M_{\rm reg}\) finite
\'etale \cite[Section~3.2, before Lemma~3.7]{BGL}. This restriction is
connected and, after choosing base points, determines a finite-index subgroup
of \(\pi_1(M_{\rm reg})\), well-defined up to conjugacy.
Remove the codimension-two exceptional sets from the smooth loci. Removing
an analytic subset of complex codimension at least two from a manifold does
not change its fundamental group. The compact normal connected \(M\) is
reduced and paracompact, with bounded local tangent dimension. Its analytic
pair \((M,M_{\rm sing})\) therefore has a real analytic embedding
\cite[Theorem~1]{AcquistapaceBrogliaTognoli1979}
and a compatible locally finite triangulation
\cite[Section~3, Theorem~1]{Lojasiewicz1964}. Compactness makes the
triangulation finite; barycentric subdivision gives the smooth locus finite
CW homotopy type, so its fundamental group is finitely generated.
There are only finitely many subgroups of
the fixed index of the cover. A power preserves the conjugacy class of its covering subgroup,
so it lifts on this open, and normalization over the proper graph extends
the lift bimeromorphically. We may take further powers throughout.
We explain why the factors can be considered separately. All holomorphic
one-forms on \(P\) come from \(T\). Their pullbacks show on the smooth
open that the torus coordinate of the map depends only on the torus
coordinate. Modulo the ideal generated by positive-degree torus forms,
the two-forms have basis the symplectic forms \(\eta_i\). Write
\(\dim H_i=2e_i\). The highest nonzero power of \(\sum a_i\eta_i\)
has order \(\sum_{a_i\ne0}e_i\); thus the locus where its full power
vanishes is the union of the coordinate hyperplanes. The map permutes
these hyperplanes and hence the lines \(\C\eta_i\), matching their
nilpotence orders \(e_i+1\). No torus two-form can be added to an image
of \(\eta_i\), since wedging that term with the \(e_i\)-th power of the
corresponding target symplectic form would
contradict that nilpotence order. The kernels of the forms now show that
each symplectic coordinate depends only on the corresponding symplectic
factor. Take a power to remove the permutations.
If \(C\) is positive-dimensional, it has no holomorphic form of degree
\(\dim C-1\): the
form algebra of a strict factor has only degrees zero and its top degree,
and no factor has dimension one. In the pullback of the volume form of
\(C\), the component with one differential outside \(C\) and all the
others in \(C\) therefore vanishes. The derivative in the \(C\)
directions has full rank, because the whole map is bimeromorphic and the
other coordinates have just been separated. Its cofactors then force the
derivative of the \(C\) coordinate in outside directions to vanish.
Thus that coordinate also depends only on \(C\). These conclusions on a
dense smooth open give factor bimeromorphic maps by taking their proper
graphs. Their degrees are one because their product has degree one.
The scalar for \(C\) is a root of unity by projective pluricanonical
finiteness. In detail, \(H^{2,0}\) of a smooth compact K\"ahler
resolution of \(C\) vanishes by the strict-factor form algebra and
reflexive extension. Rational approximation of a K\"ahler class and
Kodaira's theorem make this resolution projective. Thus \(C\) is
Moishezon; it is projective by the K\"ahler projectivity criterion for
rational singularities \cite[Corollary~6']{Namikawa2002}. Its canonical
line is trivial and its returning comparison is crepant, as seen from the
volume form. Chow's theorem algebraizes its proper analytic graph. The
projective log pluricanonical representation theorem
\cite[Theorem~1.1]{FujinoGongyo2014} gives finite scalar image.
We spell out how the polarizing class reaches the other factors without
asserting a cohomology splitting for the singular product \(P\). Choose
projective resolutions of its factors with smooth compact K\"ahler sources,
and form the smooth compact
K\"ahler product
\[
X_P=T\times\prod_i\widetilde H_i\times\widetilde C.
\]
Resolve the map \(P\dashrightarrow F\) together with \(X_P\to P\). This
gives a smooth compact K\"ahler space \(R\), a modification \(v:R\to X_P\),
and a morphism \(r:R\to F\). Put \(\beta=r^*\beta_F\). It is smooth
semipositive and strictly positive on a dense open, since \(r\) is
generically finite. Put \(a=v_*[\beta]\). The smooth
modification formula gives \([\beta]=v^*a+\{E_v\}\) for an exceptional real
divisor \(E_v\). The graph congruence for \([\beta_F]\) pulls to the cover and,
by Gysin pushforward, gives the graph congruence for \(a\) modulo real
line classes; the exceptional differences are themselves line classes.
Pass to a common refinement and use the product of resolutions of the
separated factor graphs to compute this congruence; refinements change it
only by exceptional line classes. The restriction \(a_i\) of \(a\) to one factor slice
is independent of the other coordinates, by K\"unneth on the smooth
product \(X_P\). Restricting the product graph therefore compares
\(a_i\) to itself modulo line classes on the factor graph.
Choose the factor slice generally so that it meets the open where \(\beta\)
is strictly positive and meets each \(v\)-exceptional center in codimension at least two,
or avoids that center. This is possible by the fiber dimension theorem
applied to each center and the projection to the other factors. On the
resolved strict transform of this slice, the restriction \(\beta_i\)
is still semipositive and strictly positive on an open, and the restriction
of \(E_v\) is exceptional over the factor. Thus the pullback of \(a_i\)
differs from \([\beta_i]\) only by the class of an exceptional real divisor. This
gives both the required positivity and the congruence on each smooth
factor model.
For a symplectic factor, its symplectic volume trivializes its canonical
line. Canonical singularities preserve the canonical ring on resolution,
so a smooth resolution has Kodaira dimension zero. Its dimension is
\(\dim H_i\leq\dim F<n\). Corollary~\ref{prog:terminal-kappa-zero}, using
\(\Ggood_{\dim H_i}\), runs from that resolution to a terminal model. It is still primitive
symplectic: the unique two-form extends, and its top power trivializes the
canonical line and is nondegenerate on the smooth locus; irregularity and
the form algebra are unchanged. Take a common projective resolution of
this terminal model carrying the pullback of \(\beta_i\).
Lemma~\ref{prog:cohomology-transport} decomposes its class as the pullback
of a class \(c\) on the terminal model plus the class of an exceptional real divisor.
The line-trace assertion of that lemma sends the line Chern classes in
the factor graph congruence to rational line classes on the terminal model.
Pushing through the exceptional quotient therefore shows that the returning
graph compares \(c\) to itself modulo line classes. These are the
hypotheses of Lemma~\ref{floor:symplectic-scalar}.
That lemma makes the scalar on its two-form, and therefore on its top form,
a root of unity.
For the torus, push the semipositive form to the torus as a positive current
and average it over translations. The average is a smooth invariant
positive Hermitian representative: for every nonzero tangent direction its
average is strictly positive because the original form is strictly positive
on an open. A bimeromorphic map of a torus is an affine holomorphic
automorphism; its linear part preserves the integral \(H^1\) lattice.
The comparison modulo \(\NS_\R\) and Lemma~\ref{floor:torus-scalar}
make its top-form scalar a root of unity. The product of the factor
scalars is the appropriate power of \(\mu\). It is a root of unity,
and hence so is \(\mu\).
\end{proof}
\subsubsection{Finite image on vertical strata}
\begin{proof}[Proof of Proposition~\ref{floor:finite-action}]
Put \(d=\dim Z\) and \(m=qk\), so sections of \(L_Z^k\) are meromorphic
\(m\)-pluricanonical forms. Let
\[
E_m=\left\{s\in H^0(Z,mJ_Z):\int_Z|s|^{2/m}<\infty\right\}.
\]
The integral is computed on the regular locus, or on any resolution by
change of variables. This is a vector space: for \(2/m\leq1\) the
elementary power inequality preserves integrability under sums. On a log
resolution an adjoint section is locally
\[
h(z)\frac{(dz_1\wedge\cdots\wedge dz_d)^{\otimes m}}
{\prod_i z_i^{mb_i}},\qquad b_i\leq1.
\]
It is integrable if \(h\) vanishes on every unit component; smaller
coefficients cause no obstruction. Every unit valuation has center in
\(C_Z\). Thus \(E_m\) contains pullbacks of base sections vanishing
along \(f_Z(C_Z)\). For large \(k\) these define \(Y_Z\) birationally:
multiply one nonzero section of a high ample power vanishing on that proper
subset by a complete very ample system. If \(Y_Z\) is a point,
verticality forces \(C_Z=\varnothing\) and a nonzero section is
integrable. In particular \(E_m\ne0\).
The function \(s\mapsto\int|s|^{2/m}\) is continuous on \(E_m\) and
positive away from zero. In an integrable basis, the densities of bounded
linear combinations are dominated by a constant times the sum of the basis
densities, proving continuity by dominated convergence. Its invariance
under arrows makes the action and its inverse uniformly bounded in any
norm on \(E_m\). Moreover this representation detects the representation
on all of \(H^0(Z,mJ_Z)\). If an arrow acts identically on \(E_m\), it
acts identically on \(Y_Z\) because that system is birational. For
\(0\ne s\in E_m\) and any other section \(t\), the ratio \(t/s\) comes
from the rational function field of \(Y_Z\). The arrow fixes both this
ratio and \(s\), and hence fixes \(t\).
Take an eigenform \(s\in E_m\setminus\{0\}\) for the transport
\(\gamma_*\) of one self-arrow, with eigenvalue \(\lambda\). In the
orientation fixed above, \(\gamma_*=(\gamma^{-1})^*\); put
\(g=\gamma^{-1}\), so \(g^*s=\lambda s\). Form the full normalized analytic cyclic root
cover of this meromorphic pluriform and take a projective resolution of
all its components. Locally, if \(s=a\eta^{\otimes m}\) in a
meromorphic canonical frame, the cover is the normalization of
\(t^m=a\), and its tautological top form is \(\alpha=t\eta\).
These descriptions glue when the frame changes. The finite analytic
normalization and this full-root construction are also described in the
proof of \cite[Lemma~7.1]{BL}. On the smooth compact K\"ahler resolution
\(U_s\) of the full cover, let \(\pi_s:U_s\to Z\) be the composite map.
Change of variables gives
\[
\int_{U_s}|\alpha|^2=(\deg\pi_s)\int_Z|s|^{2/m}<\infty.
\]
A meromorphic top form with finite local \(L^2\) norm has no pole, by
the one-variable integral transverse to a putative pole. Thus \(\alpha\)
is holomorphic. Choosing \(\mu^m=\lambda\) lifts \(g\)
bimeromorphically to the full cover, possibly permuting components, with
\(\widetilde g^*\alpha=\mu\alpha\).
The lift preserves a K\"ahler class modulo \(\NS_\R\). A sufficiently
large multiple of the pullback of \(c_Z\), plus a relatively ample line
class for the finite cover and its projective resolution, is K\"ahler on
\(U_s\). Pulling up Equation~\eqref{floor:class-comparison} compares the first
summand, and the changes of the relatively ample classes are line classes.
After a power fixes one connected component, that component has dimension
\(d=\dim Z<n\), so Proposition~\ref{floor:top-form-scalar} applies there.
It follows that
\(\mu\), and hence \(\lambda\), is a root of unity.
The possible orders are bounded in this fixed degree \(m\). We give the
parameter argument because pointwise torsion alone would not make a bounded
group finite. Fix a smooth resolution \(X\) of \(Z\) and an effective
divisor \(D\) clearing the poles of a basis of \(E_m\). Put
\(Q_m=\PP(E_m^\vee)\), the projective parameter space of lines of forms
under our quotient convention. The universal section on \(X\times Q_m\)
belongs to
\[
(K_X(D))^{\otimes m}\boxtimes\mathcal O_{Q_m}(1).
\]
Pull the parameter space back by the finite coordinate-power map
\([z_0:\cdots:z_b]\mapsto[z_0^m:\cdots:z_b^m]\). Its pullback of
\(\mathcal O(1)\) is \(\mathcal O(m)\), so the displayed line now has
an \(m\)-th root \(\mathscr L\). The cyclic algebra
\(\bigoplus_{i=0}^{m-1}\mathscr L^{-i}\), with multiplication defined
by the universal section, defines a finite locally free family
\(\mathcal C\) of rank \(m\).
Every parameter is a nonzero form. Each fiber is therefore reduced:
its algebra is torsion-free over the smooth \(X\) and is generically
squarefree. Every fiber component meets the inverse image
\(W\subset\mathcal C\) of the nonvanishing locus of the universal
section; there the cover is finite \'etale. Normalize \(\mathcal C\)
and resolve the normalization projectively. The composite
\(\rho:\widetilde{\mathcal C}\to\mathcal C\) can be chosen to be an isomorphism
over \(W\). Put \(E_\rho=\rho^{-1}(\mathcal C\setminus W)\).
The locus
\(\{t:\dim(E_\rho)_t\ge d\}\) is proper analytic by the proper
fiber-dimension theorem: every component of the total family meets
\(W\), and its general fiber has dimension \(d\). Remove this locus
and the critical values of the resolved map. Over the remaining dense
open the family is smooth and proper, and each fiber is the disjoint
union of smooth resolutions of the corresponding root-cover components,
because every component meets the locus where \(\rho\) is an isomorphism.
Proper smooth transport makes the sum of their middle Betti numbers
constant there. Resolve
the finitely many irreducible components of the proper analytic complement.
On each such resolution pull back the original finite cyclic family, take
its reduction, and normalize and resolve anew; restricting the previous
normalization is not required. The parameter dimension decreases,
so finitely many such steps cover all parameters. There is consequently a
uniform bound \(B_m\) for the middle Betti number of the entire disjoint
union of selected smooth resolutions of every full cover. This does not assert a bound for
arbitrary further resolutions.
On the disjoint union of the selected resolutions, let \(p_1,p_2\) be the
projections of a smooth resolved graph of the lift. The integral Gysin
endomorphism \((p_1)_!p_2^*\) on middle cohomology has \(\mu\) as an eigenvalue:
the nonzero holomorphic top form has a nonzero cohomology class,
\(p_2^*[\alpha]=\mu p_1^*[\alpha]\), and \((p_1)_!p_1^*=\id\).
If the order of \(\mu\) is \(a\), its
cyclotomic minimal polynomial therefore has degree \(\varphi(a)\leq B_m\).
There are only finitely many such \(a\). Thus the eigenvalues
\(\lambda=\mu^m\) on \(E_m\) lie in a fixed finite set.
The bounded group on \(E_m\) has compact closure in \(\mathrm{GL}(E_m)\).
Every element of that closure still has its eigenvalues in this finite set,
by continuity of characteristic polynomials. On the identity component
the characteristic polynomial must be that of the identity. A compact
linear group is unitarizable, so an element all of whose eigenvalues are
one is the identity. The identity component is trivial; a compact Lie
group with this property is finite. The image on \(E_m\), and therefore
on the full section space, is finite.
\end{proof}
\subsection{Compatible sections on the whole boundary}
We now use restriction, transport, and finite self-arrow image to construct
sections simultaneously on all boundary strata.
For \(d\geq0\) and \(k\geq1\), a system of tuples in degree \(k\) through dimension
\(d\) means a vector subspace
\[
\mathcal V_d(k)\subseteq
\bigoplus_{\substack{Z\subset S\text{ a stratum}\\\dim Z\leq d}}
H^0(Z,L_Z^k).
\]
It is \emph{compatible} if every tuple agrees under the residue restrictions
for every incidence, and \emph{invariant} if every arrow carries its source
component to its target component. It \emph{generates} if for every point
of every stratum some tuple has nonzero value there. Compatibility and
invariance are linear conditions. If such a system generates, the span
of componentwise \(a\)-th powers of its tuples gives a compatible
invariant generating system in degree \(ak\). Thus we may always pass to
a sufficiently large divisible later degree.
\begin{proposition}[Generating tuples]\label{floor:tuples}
There is a degree \(k>0\) and a compatible invariant generating system
of tuples through dimension \(n-1\).
\end{proposition}
\begin{proof}
At the point strata use the same scalar in the canonical zero-form
generator \(1\). The even residue convention identifies these generators
along every path and under every point arrow. This gives the required
system in dimension zero; if there are no points the empty system is
understood.
Suppose the system has been constructed through dimension \(d-1\).
Replace it by powers and their span in a common degree \(k\) chosen so that
Lemma~\ref{floor:restriction} applies on
every \(d\)-stratum, Proposition~\ref{floor:finite-action} applies on every
vertical \(d\)-stratum, and
\(\mathcal I_{f_Z(C_Z)}\otimes N_Z^k\) is generated for every
\(d\)-stratum \(Z\). These conditions hold in a common divisible tail:
there are finitely many strata, and the last condition is Serre's theorem.
For a \(d\)-stratum \(Z\), Lemma~\ref{floor:adjunction} glues
the lower tuple to a section on its whole \(C_Z\). Its residues satisfy
the one link in Lemma~\ref{floor:restriction}, if that link is present,
because the lower tuple is invariant. That lemma extends the section to
\(Z\).
Let \(\mathcal P_d(k)\) be the vector space of all tuples through
dimension \(d\) whose lower part is in \(\mathcal V_{d-1}(k)\) and whose
new components restrict to that lower part. Call its elements
\emph{pre-tuples}. Its projection to
\(\mathcal V_{d-1}(k)\) is surjective, since the finitely many extensions
can be chosen independently. This system already generates. To check a
point \(z\in Z\), put \(y=f_Z(z)\). If \(y\in f_Z(C_Z)\), choose a
floor point over \(y\) and a lower tuple nonzero there. Every extension is
the pullback of a section of \(N_Z^k\), so it is nonzero at every point
over \(y\), including \(z\). If \(y\notin f_Z(C_Z)\), the chosen generation
of \(\mathcal I_{f_Z(C_Z)}\otimes N_Z^k\) supplies a base section nonzero
at \(y\). It pulls back to a section
vanishing on \(C_Z\); combine it with the zero lower tuple and zero
components on the other new strata. This also treats an empty floor.
Surjectivity to the lower system preserves generation at lower points.
We impose invariance in dimension \(d\). For a dominating floor,
injectivity of restriction and Lemma~\ref{floor:transport} already make
every pre-tuple invariant on that stratum.
Verticality is invariant under arrows, so the remaining orbits consist
entirely of vertical strata.
For each such orbit choose a representative \(Z\), and let \(G_Z\) be
the finite isotropy image on \(H^0(Z,L_Z^k)\) given by
Proposition~\ref{floor:finite-action}. For a pre-tuple with component
\(s_Z\), form the norm
\[
P_Z(s_Z)=\prod_{g\in G_Z}g(s_Z)
\in H^0(Z,L_Z^{k|G_Z|}).
\]
Choose a common positive integer \(a\) divisible by all \(|G_Z|\), use
\(P_Z(s_Z)^{a/|G_Z|}\), and transport it to each member of that orbit.
This is independent of the chosen arrow: changing that arrow by an
isotropy arrow merely permutes the factors in degree \(k\). No finiteness
assertion in degree \(ak\) is needed. On strata with dominating floor
use \(s_Z^a\), and on all lower strata use the componentwise \(a\)-th
power of the assigned tuple.
The resulting tuple is compatible. By Lemma~\ref{floor:transport},
every transported factor in a norm has exactly the prescribed lower
restriction. The norm and its indicated power therefore restrict to
the \(a\)-th power of that restriction, the same value used on all lower
strata and in all other orbits. It is invariant in dimension \(d\) by
construction, and remains invariant below it.
These tuples still generate. Fix a point \(z\) in an orbit member.
For each of the finitely many transported factors, choose a point over
\(z\) on a resolution of its comparison graph. Equality of the
pulled-back invertible adjoint lines identifies the fiber values. The
factor is nonzero at \(z\) exactly when the representative component
\(s_Z\) is nonzero at the corresponding point of \(Z\). Evaluation at each
such point is a nonzero linear functional on the generating space
\(\mathcal P_d(k)\).
A finite union of their proper kernels cannot cover a complex vector
space. One pre-tuple makes all factors nonzero and gives a norm nonzero
at \(z\). At a lower point a nonzero assigned value stays nonzero after
its \(a\)-th power. Taking the linear span of all the constructed tuples
therefore gives a compatible invariant generating system in degree \(ak\).
This completes the induction on \(d\).
\end{proof}
\begin{proof}[Proof of Theorem~\ref{floor:generation}]
Take the system in Proposition~\ref{floor:tuples}. Its components on the
prime components of \(S\) agree on every common lower stratum, so
Lemma~\ref{floor:adjunction} descends each tuple uniquely to a section of
the actual line \(\mathcal O_V(qkJ)|_S\). At any \(x\in S\), choose a
prime component through \(x\) and a tuple nonzero there. It is the pullback
of the descended value in the same invertible line fiber at \(x\), so
that value is nonzero. The finite-dimensional span of these descended
sections generates at every point of the reduced \(S\). This is the
claimed global generation of an actual Cartier multiple.
\end{proof}
==== END Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//04-boundary.tex ====
==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//05-projective-program-input.tex ====
\subsection{Projective programs under the assumption}
\label{rank:projective-program-input}
The argument for a projective base uses a terminating program to reach a nef
adjoint and, after decreasing the nef data, a program ending in a Mori fiber
space. The conditional input is termination in the pseudo-effective case.
We first state the exact program result and then derive that case from the
projective good models in Proposition~\ref{proj:good-models}.
\begin{proposition}[A conditional generalized program]
\label{proj:generalized-mmp}
Assume Assumption~\ref{asm:log-iitaka}. Let
\((V,B+\boldsymbol M)\) be a projective generalized dlt rational pair,
where \(V\) is \(\Q\)-factorial, \(B\geq0\), and the nef b-divisor
\(\boldsymbol M\) is determined by a nef rational Cartier divisor on a
projective birational model. Put \(D=K_V+B+M_V\).
There is a choice of \(D\)-MMP with ample scaling which terminates. If
\(D\) is pseudo-effective, its endpoint \(V_m\) has nef adjoint \(D_m\).
If \(D\) is not pseudo-effective, its endpoint has a Mori fiber contraction
to a projective variety of smaller dimension. The birational steps and their
endpoints stay \(\Q\)-factorial generalized dlt, and the forward birational
map extracts no divisors.
In the pseudo-effective case, on a common resolution one has an actual
rational divisor identity
\[
u^*D=v^*D_m+F,\qquad F\geq0\text{ and }v\text{-exceptional}.
\]
\end{proposition}
For the pseudo-effective alternative, we first derive the relative weak
Zariski decompositions used by the minimal-model existence theorem.
For a projective morphism, first replace its image by its normal Stein
factor. We use the relative convention of \cite[Section~2]{TsakanikasXie}:
a divisor is pseudo-effective over the base when its restriction to a very
general fiber of this surjective morphism is pseudo-effective. An NQC weak Zariski decomposition of \(D\) over the
base is a birational equality \(h^*D\equiv_Z P+N\), where \(P\) is a
nonnegative real combination of relatively nef rational Cartier divisors
and \(N\geq0\). The decomposition below has rational Cartier parts.
\begin{lemma}[Relative canonical decomposition]\label{proj:relative-wzd}
Assume Assumption~\ref{asm:log-iitaka}. Let \(V\) be a smooth
quasi-projective complex variety and let \(f:V\to Z\) be projective, with
\(Z\) normal and quasi-projective. If \(K_V\) is pseudo-effective over
\(Z\), then on a normal variety \(Y\) with a projective birational morphism \(h:Y\to V\) there
are rational Cartier divisors \(P,N\) such that
\[
h^*K_V\equiv_Z P+N,\qquad P\text{ nef over }Z,\qquad N\geq0.
\]
\end{lemma}
\begin{proof}
We use the construction in the proof of
\cite[Proposition~3.3]{External037}. Its absolute input is a nef-plus-
effective rational decomposition for the canonical divisor of a smooth
projective variety with pseudo-effective canonical class. This is supplied
here by Proposition~\ref{proj:good-models} and its common-resolution
comparison.
Replace the image of \(f\) by its normal Stein factor. This does not change
relative numerical classes or contracted curves, and now \(f\) is
surjective with connected fibers. Its very general fiber is smooth
projective with pseudo-effective canonical divisor, hence is non-uniruled
by \cite[Theorem~0.2 and Corollary~0.3]{BDPP}. Compactify the projective
morphism and resolve away from \(V\). We obtain
\(\bar f:\bar V\to\bar Z\), with \(\bar V\) smooth projective and
\(\bar Z\) normal projective, unchanged over \(Z\). If \(\dim Z=0\),
\(V\) is already projective and the absolute input proves the lemma.
Suppose \(\dim Z>0\), and let \(r:\widetilde Z\to\bar Z\) be a smooth
projective resolution. Choose a sufficiently positive very ample divisor
\(A_Z\) on \(\bar Z\) and a general \(B\in|2A_Z|\). The pulled-back systems
on both \(\widetilde Z\) and \(\bar V\) are base point free. Bertini makes
\(r^*B\) and \(\bar f^*B\) smooth nonempty reduced divisors. Their double
covers
\[
\pi_Z:Z^\sharp\to\widetilde Z,
\qquad \pi:V^\sharp\to\bar V
\]
are smooth integral projective varieties. The canonical formulas are
\[
K_{Z^\sharp}\sim_{\Q}\pi_Z^*(K_{\widetilde Z}+r^*A_Z),
\qquad
K_{V^\sharp}\sim_{\Q}\pi^*(K_{\bar V}+\bar f^*A_Z).
\]
The first line is big when \(A_Z\) is sufficiently positive. The rational
map \(V^\sharp\dashrightarrow Z^\sharp\) has the same very general fibers
as \(f\). A covering family of rational curves on \(V^\sharp\) would
either dominate a covering family of rational curves on \(Z^\sharp\), or
cover its very general fibers by vertical rational curves. Both are
impossible. Thus \(V^\sharp\) is non-uniruled and \(K_{V^\sharp}\) is
pseudo-effective by BDPP.
Apply the absolute decomposition to \(V^\sharp\). Moving the rational
principal difference in the canonical formula into the nef part gives an
actual rational divisor decomposition for a pullback of
\(\pi^*(K_{\bar V}+\bar f^*A_Z)\). Take a common equivariant resolution of
that model and its conjugate under the covering involution. On this smooth
projective resolution \(g:T\to V^\sharp\), average the two decompositions.
We obtain
\[
g^*\pi^*(K_{\bar V}+\bar f^*A_Z)=\bar P+\bar N,
\qquad \bar P\text{ nef},\quad \bar N\geq0,
\]
where \(\bar P\) and \(\bar N\) are invariant rational Cartier divisors.
Let \(q:T\to\bar Y\) be the finite quotient by the involution. The invariant
composite \(\pi g\) induces a projective birational morphism
\(\bar h:\bar Y\to\bar V\), with \(\bar Y\) normal and projective. An invariant
rational divisor descends by dividing its coefficient at an upstairs prime
by the ramification index. The descended divisor is rational Cartier when
the original divisor is: after clearing denominators, choose one Cartier
equation on a semilocal neighborhood of the finite orbit above a point
(a Cartier divisor is principal on a semilocal ring), and multiply all its
translates. The product is invariant and has divisor
equal to the group order times the original divisor. It descends to a local
equation for a multiple of the downstairs divisor. This argument includes
stabilizers and ramification primes.
Accordingly \(\bar P=q^*P\) and \(\bar N=q^*N\) for rational Cartier
divisors on \(\bar Y\). Lifting curves through the finite map shows that
\(P\) is nef, and coefficientwise descent shows that \(N\) is effective.
Finite pullback is injective on rational divisors, so the displayed identity
descends. Restrict it to \(Y=\bar h^{-1}(V)\). The term pulled
back from \(A_Z\) is numerically trivial over \(Z\), giving
\(h^*K_V\equiv_ZP+N\), as required.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{proj:generalized-mmp}]
Lemma~\ref{proj:relative-wzd} and
\cite[Theorem~5.2]{TsakanikasXie}, applied to the generalized pair
\((U/Z,0+0)\) with \(U\) smooth, prove relative minimal-model existence for smooth varieties
by dimension induction. The hypothesis of that theorem is relative smooth
minimal-model existence one dimension lower; the lemma supplies its other
hypothesis, an NQC weak Zariski decomposition. The induction begins in
dimension zero. Then \cite[Theorem~5.4]{TsakanikasXie} supplies minimal
models for every pseudo-effective NQC generalized lc pair in the relevant
dimension. Their Theorem~2.7 gives existence in the Birkar--Shokurov sense.
The nef data in the statement are NQC, because a nef rational Cartier
divisor is a positive multiple of a nef Cartier divisor. In the
pseudo-effective case the preceding paragraph therefore gives the
Birkar--Shokurov model required by
\cite[Theorem~4.2]{TsakanikasXie}. In the other case, the non-pseudo-effective
alternative of that theorem applies directly. To meet its scaling
hypothesis, take an effective sufficiently positive ample rational divisor
\(A\) whose general components have sufficiently small coefficients. On
a fixed log resolution carrying \(\boldsymbol M\), these components are
transverse to the boundary and preserve generalized log canonicity, while
their ample class can be chosen large enough that \(D+A\) is nef. The
theorem supplies a terminating MMP starting on \(V\).
We verify the step types inductively. Suppose \(V_i\) is
\(\Q\)-factorial generalized dlt. By
\cite[Remark~2.15]{TsakanikasXie}, each birational step has the form
\[
V_i\xrightarrow{g_i}Z_i\xleftarrow{h_i}V_{i+1},
\]
where \(g_i\) contracts a negative extremal ray and \(h_i\) is small and projective.
Suppose \(g_i\) is divisorial, and write \(R\) for its ray. An exceptional
prime \(E\) has \(E\cdot R<0\): otherwise it would be effective,
exceptional, and \(g_i\)-nef, contrary to negativity. There is only one
exceptional prime. Indeed, a combination of two distinct exceptional
primes can be chosen to have degree zero on \(R\); negativity applied to
both signs of that combination would make it zero.
We show directly that \(Z_i\) is \(\Q\)-factorial. For a prime divisor
\(T\) on \(Z_i\), let \(\widetilde T\) be its strict transform, choose a
curve \(C\) spanning \(R\), and put
\[
a=-\frac{\widetilde T\cdot C}{E\cdot C}\in\Q.
\]
Choose a positive integer \(m\) for which \(m(\widetilde T+aE)\) is
Cartier. Its line has degree zero on \(R\), so
\cite[Theorem~1.5]{Xie2022} gives an actual Cartier line \(\mathcal L_T\)
on \(Z_i\) and an isomorphism
\[
\OO_{V_i}\bigl(m(\widetilde T+aE)\bigr)\simeq g_i^*\mathcal L_T.
\]
On an open set trivializing \(\mathcal L_T\), the Cartier divisor
\(m(\widetilde T+aE)\) is principal upstairs. Pushing this principal
divisor identity down through the birational \(g_i\) gives a principal
divisor \(mT\) on that open set.
Thus \(T\) is rational Cartier. It follows that \(h_i\) is an isomorphism:
the pushforward of an \(h_i\)-ample Cartier divisor is rational Cartier
on \(Z_i\), and its pullback is the original divisor because \(h_i\)
is small. It has degree zero on every contracted curve, so relative
ampleness rules out positive-dimensional fibers; a finite birational
map to the normal \(Z_i\) is an isomorphism. If \(g_i\) is small,
the relatively ample canonical model \(h_i\) is the flip of
\cite[Definition~3.2]{HanLi}. Thus the chosen birational steps are
divisorial contractions and flips. The underlying variety of a
generalized dlt pair is klt, and these steps preserve the
\(\Q\)-factorial generalized dlt category
\cite[Remark~2.3 and Lemma~3.7]{HanLi}. They introduce no prime
divisors on the new models. For completeness, on a common resolution
of one step \(V_i\dashrightarrow V_{i+1}\) carrying \(\boldsymbol M\),
taking the difference of the two generalized discrepancy formulas cancels
their common nef divisor. The resulting difference of the adjoint pullbacks is exceptional over the new
model and anti-nef over it. The negativity lemma makes this difference
effective. Composing these actual rational divisor comparisons gives
\(u^*D=v^*D_m+F\) with \(F\) effective and exceptional over the endpoint.
The endpoints in the two cases are the nef and Mori fiber endpoints supplied
by the terminating program theorem.
\end{proof}
==== END Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//05-projective-program-input.tex ====
==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//05-rank-one.tex ====
\section{Descent along a fibration}
\label{sec:rank-one}
Throughout this section we assume Assumption~\ref{asm:log-iitaka}.
We prove Proposition~\ref{rank:fibration-case}. Its geometric reduction
leads to a fibration whose very general fiber has logarithmic Kodaira
dimension zero. In sufficiently divisible degrees the log pluricanonical
systems on that fiber are one-dimensional, and a relative generating form
determines an adjoint line on the base. We prove the resulting rank-one
proposition by a secondary induction on the positive base dimension: after proving
pseudo-effectivity on the base, we either lower its dimension or lift a nef
line and its entire fixed divisor. The last step proves generation of the
remaining nef line.
\subsection{Reduction to the rank-one case}
\begin{proposition}[The rank-one case]
\label{rank:rank-one}
Assume Assumption~\ref{asm:log-iitaka}. Fix \(n>0\) and assume
\(\Ggood_d\) for every \(d<n\).
Let \(X\) be a smooth connected compact Kähler manifold of dimension \(n\),
let \(B\) be a rational SNC boundary, and suppose that
\(J=K_X+B\) is pseudo-effective. Suppose there are smooth compact Kähler
manifolds \(\widetilde X,W\), a modification
\(\pi:\widetilde X\to X\), and a proper surjective holomorphic map
\(g:\widetilde X\to W\) with connected fibers such that
\[
0<k:=\dim W<n,\qquad
W\text{ is projective or }a(W)=0.
\]
Assume that
\(\widetilde B=\pi_*^{-1}B+\operatorname{Exc}(\pi)_{\mathrm{red}}\)
has SNC support and that, on a very general smooth fiber \(F\) of \(g\),
\[
\kappa(F,K_F+\widetilde B_F)=0.
\]
Then \((X,B)\) satisfies \(\Ggood_n\).
\end{proposition}
\begin{proof}[Proof of Proposition~\ref{rank:fibration-case},
assuming Proposition~\ref{rank:rank-one}]
We use the reduced-exceptional boundary after every source modification.
The effective discrepancy identity in
Proposition~\ref{bir:resolution-transfer} preserves pseudo-effectivity,
and that proposition transfers the resulting decomposition back to the
original pair. A dominant meromorphic map to
a space in class \(\mathcal C\) can be resolved on smooth compact Kähler
models; taking Stein factorization gives connected fibers. These
operations preserve the positive dimensions of the base and the fiber.
We first suppose \(a(X)>0\). If \(a(X)=n\), then \(X\) is Moishezon and
Kähler, hence projective, and Proposition~\ref{proj:good-models} applies.
Otherwise resolve the algebraic reduction as a fibration
\(g:X'\to W\) to a smooth projective variety of dimension \(a(X)\).
Put \(J'=K_{X'}+B'\) for the modified log adjoint. Its restriction to
a very general smooth fiber is pseudo-effective. Indeed, one may choose
a sequence of positive currents in
\(c_1(J')+\varepsilon_j[\omega]\), with analytic singularities and
\(\varepsilon_j\downarrow0\), and restrict all of them outside a
countable union of proper analytic subsets of the base. By
\(\Ggood_{\dim F}\), the restricted adjoint has nonnegative Kodaira
dimension.
That dimension must be zero. If it were positive, generic coherent
base change would give a divisible \(q\) for which
\(\mathcal E=g_*\OO_{X'}(qJ')\) has a fiber system with a
positive-dimensional image on very general fibers. A sufficiently
large ample twist of the coherent sheaf \(\mathcal E\) is generated at
the generic point of the projective \(W\). Evaluation would therefore
give two sections of
\(\OO_{X'}(qJ')\otimes g^*\OO_W(A)\), for one ample \(A\), whose ratio
is nonconstant on a general \(g\)-fiber. This ratio is a meromorphic
function on \(X'\). Every meromorphic function on \(X'\) factors
through its algebraic reduction, a contradiction.
Proposition~\ref{rank:rank-one} now applies.
Suppose next that \(a(X)=0\), and resolve the fibration in the
hypothesis as \(f:(X',B')\to Y\). The base may be replaced by a smooth
compact Kähler model, and \(a(Y)=0\), since its meromorphic functions
pull back to \(X'\). Write \(d=\dim X'-\dim Y>0\).
The same restriction argument and \(\Ggood_d\) show that
\[
\kappa(F,K_F+B'_F)\geq0
\]
on very general smooth fibers. If this integer is \(j<d\), the relative
Iitaka construction of \cite[Lemma~2.6]{CI} gives, after the same
source modifications, a factorization
\[
X''\xrightarrow{g}W\xrightarrow{h}Y
\]
with connected fibers, \(\dim W=\dim Y+j\), and
\(\kappa(G,K_G+B''_G)=0\) on a very general \(g\)-fiber. The cited
lemma applies to rational SNC boundaries on compact manifolds in
class \(\mathcal C\); its conclusion concerns the restricted log
adjoint, and does not require finite generation. A Kähler model of
\(W\) has algebraic dimension zero because it is dominated by \(X''\).
Thus \(0<\dim W<n\), and Proposition~\ref{rank:rank-one} applies again.
It remains to exclude the possibility that \(K_F+B'_F\) is big.
The stable-family comparison below requires a horizontal boundary with
coefficients strictly less than one, so we first lower the boundary while
keeping the restricted adjoint big.
Let \(B'_{\mathrm{hor}}\) be the sum of the components of \(B'\)
dominating \(Y\). There is a rational \(0<\lambda<1\) such that
\(K_F+\lambda B'_{\mathrm{hor},F}\) is big for very general \(F\).
To justify the uniform choice, exclude at once the generic base-change
and image-dimension exceptional sets for all rational \(\lambda\)
and all divisible degrees. On a fiber outside this countable union,
openness of the big cone gives one such \(\lambda\), and a single
degree has full-dimensional image. Generic base change makes the
same choice work very generally. The boundary
\(\lambda B'_{\mathrm{hor}}\) is horizontal, rational SNC, and has
all coefficients strictly less than one. The stable-family
comparison \cite[Lemma~5.2]{CI} therefore supplies a proper
generically finite cover \(Y'\to Y\), a surjection \(Y'\to R\) to a
smooth projective variety, and a projective stable family
\(\mathcal V\to R\) with positive-dimensional general fiber, such
that the main transform of \(X'\) is bimeromorphic to
\(\mathcal V\times_RY'\).
The main transform still has algebraic dimension zero. Normalize it and
factor its proper generically finite map to \(X'\) through the normal Stein
space. A meromorphic function descends through the birational part.
Lemma~\ref{bir:finite-norm} gives its characteristic polynomial over the
finite part, with meromorphic coefficients on \(X'\). Those coefficients
are constant because \(a(X')=0\), so irreducibility makes the function
constant. Normalization and modifications preserve meromorphic functions.
It follows also that \(a(Y')=0\), so the surjection from \(Y'\) to
the projective \(R\) forces \(R\) to be a point. But then the main
transform is bimeromorphic to the product of \(Y'\) with a
positive-dimensional projective variety. Rational functions on that
factor contradict algebraic dimension zero. This excludes the big
case and completes the reduction.
\end{proof}
\subsection{The adjoint line on the base}
We prepare the rank-one fibration and record exactly what the relative
generating form supplies. The construction of the Hodge line and its
positivity are those of \cite[Proposition~2.7 and Theorem~3.1]{CI}.
The relative injection and the last assertion below are consequences of
the local order calculation in that proof; they will be needed after
the base is changed.
\begin{lemma}[Prepared comparison]
\label{rank:comparison}
Let \(g_0:(X_0,B_0)\to W_0\) be a fibration with connected fibers from
a smooth compact Kähler manifold to a smooth compact manifold in
class \(\mathcal C\). Assume \(B_0\) is a rational SNC boundary and
\(\kappa(F,K_F+B_{0,F})=0\) on very general smooth fibers.
After modifications there is a diagram
\[
\begin{tikzcd}
X \arrow[r,"\pi"] \arrow[d,"g"'] & X_0 \arrow[d,"g_0"]\\
W \arrow[r,"\rho"'] & W_0
\end{tikzcd}
\]
with \(X,W\) smooth compact Kähler, \(g\) a fibration with connected
fibers, and
\(B=\pi_*^{-1}B_0+\operatorname{Exc}(\pi)_{\mathrm{red}}\) SNC.
Every prime of \(X\) whose \(g\)-image has codimension at least two is
\(\pi\)-exceptional. There are a rational SNC boundary \(T\) on \(W\),
a surjection \(p:W\to S\) with connected fibers to a smooth projective
variety, and a nef rational line \(P_S\) on \(S\). Put
\[
M=p^*P_S,\qquad H=K_W+T+M,\qquad J=K_X+B.
\]
If \(\dim S>0\), then \(K_S+a_0P_S\) is big for some rational
\(a_0>0\); if \(S\) is a point, \(M\sim_{\Q}0\).
There is a rational divisor \(A_*\) on \(X\) giving an actual rational
line identity
\begin{equation}
J\sim_{\Q}g^*H+A_*.
\label{rank:line-comparison}
\end{equation}
These data satisfy the following properties.
\begin{enumerate}
\item The horizontal part \(A_{*,\mathrm{hor}}\) is effective.
On a very general smooth fiber it is the zero divisor of a relative
generating section of a divisible multiple of \(K_F+B_F\), divided
by that multiple.
\item For each prime \(D\subset W\), let \(E\) run over the primes of
\(X\) dominating \(D\), and put \(a_E=\ord_E(g^*D)\).
Then
\begin{equation}
(A_*)_E\geq0,\qquad
\min_{E\to D}\frac{(A_*)_E}{a_E}=0.
\label{rank:zero-minimum}
\end{equation}
There is no assertion here about the sign at a prime whose image has
codimension at least two.
\item For all degrees \(q\) divisible by one positive integer,
multiplication by the relative generating form identifies
\[
H^0(W,qH)\simeq H^0(X,qJ)
\]
and preserves ratios of sections. In the same degrees, for every
proper holomorphic \(h:W\to V\), there is a natural injection
\begin{equation}
(hg)_*\OO_X(qJ)\ \lhook\joinrel\longrightarrow\
h_*\OO_W(qH).
\label{rank:relative-injection}
\end{equation}
\item Let \(\tau:W'\to W\) be a further smooth modification included
in another diagram with the preceding properties over the same reference
\(X_0\), using the same source-boundary convention. Let \(H'\) be the
base line produced by that diagram. Then, as actual rational lines,
\begin{equation}
H'\sim_{\Q}\tau^*H+E_H,
\qquad E_H\geq0\ \text{\(\tau\)-exceptional}.
\label{rank:higher-base}
\end{equation}
\end{enumerate}
\end{lemma}
\begin{proof}
The flattening construction of \cite[Lemma~2.4]{CI} gives the stated
reference property, and preserves it after any finite sequence of
further base modifications. Its source boundary is the one in
\cite[Lemma~2.1]{CI}. We may resolve the discriminant and the
horizontal divisor of the relative generator so that all relevant
strata are smooth over the dense smooth log locus. Proposition~2.7
of \cite{CI} then gives \(T\) and a rational parabolic Hodge line
\(M\). A positive multiple of that line is the highest rank-one
Hodge step of a complex summand of a pure real-polarizable variation
with an integral lattice. Theorem~3.1 of \cite{CI} gives the
factorization \(M=p^*P_S\) and the stated positivity, after a further
modification and a positive rational rescaling. These are identities
in \(\Pic\otimes\Q\), not merely identities of classes.
Here is the order computation that gives
Equations~\eqref{rank:line-comparison}--\eqref{rank:relative-injection}.
Choose a degree \(m\) for which
\(g_*\OO_X(m(K_{X/W}+B))\) has generic rank one and all boundary
denominators are cleared. Its reflexive hull is a line on the smooth
\(W\). Let \(s\) be a meromorphic relative generating form in a
local frame of this line. At the generic point of a prime \(D\),
choose a nonvanishing ordinary base volume \(\omega_W\), and set
\[
l_E=\frac1m\ord_E(s\wedge g^*\omega_W^m),\qquad
\delta_E=B_E,\qquad
\alpha_D=\min_{E\to D}\frac{l_E+\delta_E}{a_E}.
\]
The relative-times-base identification in this formula is taken in
the \(m\)-th canonical tensor power. The source is resolved over
the generic point of \(D\), so all the primes needed for the
divisorial test occur in this minimum. In the notation of the
proof of \cite[Proposition~2.7]{CI}, the parabolic order is
\[
\beta_D=\min_{E\to D}\frac{l_E+1-a_E}{a_E},
\qquad T_D=\alpha_D-\beta_D.
\]
Thus the corresponding local frame of \(T+M\) has order
\(\alpha_D\). Comparing its pullback with the log adjoint form
defines \(A_*\), with
\begin{equation}
(A_*)_E=l_E+\delta_E-a_E\alpha_D .
\label{rank:vertical-order}
\end{equation}
Changing the generating form multiplies it by a meromorphic base
function. This changes both \(\alpha_D\) and \(\beta_D\) by its
normalized base order and leaves the comparison invariant. The local
comparisons therefore give the rational divisor and the line identity
globally. Equation~\eqref{rank:vertical-order} proves
Equation~\eqref{rank:zero-minimum}. Horizontally the comparison is
precisely the zero divisor of the fiberwise log section, so it is
effective and has the asserted restriction.
The same order computation proves the relative injection.
In a divisible degree \(q\), a meromorphic base coefficient
\(\varphi\) gives a regular base section at \(D\) exactly when
\[
\ord_D(\varphi)+q\alpha_D\geq0.
\]
Its corresponding upstairs orders at the primes \(E\to D\) are
\[
a_E\ord_D(\varphi)+q(l_E+\delta_E).
\]
Their simultaneous nonnegativity is equivalent to the preceding
inequality. Conversely, an upstairs section restricts to a multiple
of the generator on general connected fibers, so its ratio with the
generator descends meromorphically to \(W\). The order inequalities
make it regular outside codimension two on \(W\), and normality
extends it. Consequently
\[
g_*\OO_X(qJ)\ \lhook\joinrel\longrightarrow\
\OO_W(qH).
\]
Applying the left exact functor \(h_*\) proves
Equation~\eqref{rank:relative-injection}. The reverse global
comparison follows from the reference property: its only initially
untested poles are on \(g\)-contracted primes, all exceptional over
\(X_0\); pushing to \(X_0\), extending in codimension two, and pulling
back with the reduced-exceptional boundary removes them. This is
the global comparison in \cite[Proposition~2.7]{CI}.
Finally consider \(\tau:W'\to W\). The parabolic rational line pulls
back under SNC modifications, as in the construction preceding
\cite[Theorem~3.1]{CI}. The coefficient \(T_D\) at an old strict
transform is unchanged. Indeed the sheaf of regular relative
adjoint forms inside the common meromorphic generator line is
unchanged over the generic point of \(D\) by the log modification
formula. Equivalently, the old inequalities above already test all
vertical primes there; in SNC coordinates logarithmic pullback
preserves those inequalities, so new source exceptional primes
impose no stronger one. This keeps \(\alpha_D\) unchanged in
compatible frames, while parabolic pullback keeps \(\beta_D\)
unchanged. Over a new \(\tau\)-exceptional prime every source prime
is exceptional over \(X_0\): a pre-existing one had image of
codimension at least two on \(W\), and a new one is exceptional
by construction. All therefore have boundary coefficient one.
The formula for \(T_D=\alpha_D-\beta_D\) then gives coefficient one
on that new base prime. Thus
\[
T'=\tau_*^{-1}T+\operatorname{Exc}(\tau)_{\mathrm{red}}.
\]
The usual log discrepancy formula for the SNC pair \((W,T)\)
now gives \(K_{W'}+T'=\tau^*(K_W+T)+E_H\), with \(E_H\)
effective and exceptional. Adding the pulled-back parabolic line
proves Equation~\eqref{rank:higher-base}.
\end{proof}
\begin{lemma}[Pseudo-effectivity of the base line]
\label{rank:base-psef}
In the setting of Lemma~\ref{rank:comparison}, suppose
\(\dim X=n\), \(0<\dim W<n\), \(J\) is pseudo-effective, and
\(\Ggood_d\) holds for \(d<n\). Then
\[
A_{*,\mathrm{hor}}\leq N(J)
\quad\text{as divisors},\qquad H\ \text{is pseudo-effective}.
\]
\end{lemma}
\begin{proof}
On a very general smooth fiber \(F\), use the model supplied by
\(\Ggood_{\dim F}\) to write
\(\mu_F^*J_F\sim_{\Q}P_F+N(\mu_F^*J_F)\).
Lemma~\ref{bir:sections} identifies the divisible section spaces with
those of \(P_F\), so \(\kappa(P_F)=\kappa(F,J_F)=0\); a semiample line of Kodaira
dimension zero is torsion. Proposition~\ref{bir:finite-descent},
applied to the modification \(\mu_F\) with zero added divisor, gives
\(J_F\sim_{\Q}N(J_F)\) on \(F\), with \(N(J_F)\) rational.
Lemma~\ref{bir:sections} then identifies the normalized zero divisor of
the prepared relative generator with \(N(J_F)\). Thus
\[
A_{*,\mathrm{hor}}|_F=N(J_F).
\]
We compare these fiber multiplicities with those on \(X\).
For each of the finitely many components \(E\) of
\(A_{*,\mathrm{hor}}\), choose a countable sequence of
small-Kähler-perturbation currents with analytic singularities whose
generic orders at \(E\) approach \(\nu_E(J)\). Choose a common very
general \(F\) for these sequences and these components. The currents restrict
positively after the same perturbations, and analytic singularities
ensure that their orders along the components of \(E|_F\) are
their generic orders along \(E\). Each restricted order is at
least the corresponding perturbed minimal multiplicity on \(F\).
Passing to the limit gives
\((A_*)_E\leq\nu_E(J)\). This proves
\begin{equation}
A_{*,\mathrm{hor}}\leq N(J).
\label{rank:horizontal-negative}
\end{equation}
Choose a positive curvature current for the rational line \(J\).
Every such current contains its divisorial negative part, hence
subtracting \([A_{*,\mathrm{hor}}]\) leaves a positive current.
On the inverse image of a dense smooth open \(W^\circ\), with all
vertical data removed, Equation~\eqref{rank:line-comparison}
identifies the resulting singular metric with one on \(g^*H\).
In a frame pulled back from \(W^\circ\) its weight is plurisubharmonic.
It is constant on each compact connected smooth fiber, by the
maximum principle (with the value \(-\infty\) allowed). Local
holomorphic sections of the submersion show that these values form
a plurisubharmonic weight on \(H|_{W^\circ}\).
This weight extends across the missing divisors. At the generic
point of any such divisor \(D\), choose \(E\to D\) for which
\((A_*)_E=0\), using Equation~\eqref{rank:zero-minimum}.
At a general point of \(E\) away from the other comparison divisors,
the comparison after subtracting the horizontal part has neither
a zero nor a pole. We can choose local coordinates, with the remaining
vertical coordinates denoted by \(w\), in which
\[
g(u,z_2,\ldots,z_k,w)=(u^{a_E},z_2,\ldots,z_k).
\]
Indeed \(g|_E\) has maximal tangential rank generically and a local
equation of \(g^*D\) is a unit times \(u^{a_E}\); the unit has a
local \(a_E\)-th root. A smaller such chart covers a neighborhood
of the base point. The upstairs plurisubharmonic weight is locally
bounded above there, and the comparison unit is bounded. Hence
the descended weight is locally bounded above near the generic
point of \(D\). The removable singularity theorem for
plurisubharmonic functions extends it across \(D\) away from
codimension two, and the Hartogs extension for plurisubharmonic
functions extends it across the remaining analytic subset.
The extensions respect the line transitions because they agree on
the dense open. They give a positive singular metric on \(H\),
which proves pseudo-effectivity.
\end{proof}
\begin{lemma}[A base of algebraic dimension zero]
\label{rank:zero-base}
Under the assumptions of Lemma~\ref{rank:base-psef}, if \(a(W)=0\),
then, on \(W\),
\begin{equation}
H\sim_{\Q}E_W=N(H)
\quad\text{with }E_W\geq0.
\label{rank:zero-base-decomposition}
\end{equation}
\end{lemma}
\begin{proof}
The projective quotient \(S\) in Lemma~\ref{rank:comparison} is a
point: otherwise its rational functions pull back nontrivially to
\(W\). Thus \(M\sim_{\Q}0\) and \(H\sim_{\Q}K_W+T\).
By Lemma~\ref{rank:base-psef} and \(\Ggood_{\dim W}\), its pullback
to a smooth higher model is a semiample rational line plus its
negative divisor. Every semiample line on a space of algebraic
dimension zero is torsion, since its generated system has
zero-dimensional image. Proposition~\ref{bir:finite-descent}, applied
to this modification with zero added divisor, therefore gives
Equation~\eqref{rank:zero-base-decomposition}, including rationality
of the divisor.
\end{proof}
\input{sections/05-projective-program-input}
\subsection{Reducing a projective base}
The nef line \(M\) supplied by the Hodge construction need not be
semiample. A first projective program replaces \(H\) by a nef transform.
Positive Iitaka dimension gives either a smaller base or a big line. In
nonpositive Iitaka dimension, rationally trivial nef data give a trivial
line. In the remaining case,
decreasing \(M\) leads to a second program that lowers the base dimension
while preserving the first nef line.
\begin{lemma}[Projective base reduction]
\label{rank:base-reduction}
Assume Assumption~\ref{asm:log-iitaka} and the setting of
Lemma~\ref{rank:base-psef}, with \(W\) projective of dimension \(k\).
Then one of the following holds.
\begin{enumerate}
\item After a smooth source modification with the reduced-exceptional
boundary, there is a fibration to a smooth projective variety \(V\)
with \(0<\dim V<k\) and logarithmic Kodaira dimension zero on very
general smooth fibers.
\item There are a normal \(\Q\)-factorial projective variety \(W_m\),
a birational contraction \(W\dashrightarrow W_m\) extracting no
divisors, and a nef rational line \(H_m\) on \(W_m\), either big or
rationally trivial, such that on a common smooth resolution
\(\mu:W'\to W\), \(\nu:W'\to W_m\),
\begin{equation}
\mu^*H\sim_{\Q}\nu^*H_m+E,
\qquad E\geq0\ \text{\(\nu\)-exceptional}.
\label{rank:base-endpoint}
\end{equation}
\end{enumerate}
Here all the line comparisons are actual rational line identities.
\end{lemma}
\begin{proof}
The smooth pair with boundary \(T\) and nef data \(\boldsymbol M\)
determined by \(M\) on \(W\) is a rational \(\Q\)-factorial
generalized dlt pair. Its adjoint \(H\) is pseudo-effective by
Lemma~\ref{rank:base-psef}. Proposition~\ref{proj:generalized-mmp}
gives a chosen terminating program with scaling to a
\(\Q\)-factorial generalized dlt model \(W_{\mathrm{nef}}\) with nef transformed
adjoint
\[
H_{\mathrm{nef}}
=K_{W_{\mathrm{nef}}}+T_{\mathrm{nef}}+M_{\mathrm{nef}}.
\]
Here \(M_{\mathrm{nef}}\) is the trace of the nef b-divisor whose nef
representative is \(M\) on \(W\). On a common smooth resolution
\(\mu:W'\to W\), \(\sigma:W'\to W_{\mathrm{nef}}\), the actual comparison is
\begin{equation}
\mu^*H\sim_{\Q}\sigma^*H_{\mathrm{nef}}+E_{\mathrm{nef}},
\qquad E_{\mathrm{nef}}\geq0\ \text{\(\sigma\)-exceptional}.
\label{rank:first-nef-endpoint}
\end{equation}
The program extracts no divisors. We use this particular model also when
the nef data are rationally trivial.
If \(\kappa(H)>0\), Equation~\eqref{rank:first-nef-endpoint} preserves
the divisible section spaces. For \(\kappa(H)=k\) the nef
\(H_{\mathrm{nef}}\) is big, and the second alternative holds with
\(W_m=W_{\mathrm{nef}}\) and \(H_m=H_{\mathrm{nef}}\). If
\(0<\kappa(H)<k\), the exact global comparison in
Lemma~\ref{rank:comparison} gives \(\kappa(J)=\kappa(H)\).
Take the Iitaka fibration of \(J\) on the source. Its projective
image, followed by Stein factorization and a projective resolution,
has dimension \(\kappa(H)\). The log adjoint on very general
fibers has Kodaira dimension zero by the relative Iitaka construction
\cite[Lemma~2.6]{CI}, including its persistence under the
reduced-exceptional source convention. This is the first alternative.
Suppose \(\kappa(H)\leq0\) and \(M\sim_{\Q}0\). We may choose zero
nef data in this rational equivalence class. The same generalized
dlt model \(W_{\mathrm{nef}}\) is then an ordinary dlt model and
\(H_{\mathrm{nef}}\sim_{\Q}K_{W_{\mathrm{nef}}}+T_{\mathrm{nef}}\).
Its adjoint is nef, so
Proposition~\ref{proj:good-models} makes \(H_{\mathrm{nef}}\) semiample as an
actual rational line. Since
\(\kappa(H_{\mathrm{nef}})=\kappa(H)\leq0\), a generated multiple has
constant image and is trivial. Thus the second alternative holds with
\(W_m=W_{\mathrm{nef}}\) and \(H_m=H_{\mathrm{nef}}\sim_{\Q}0\).
It remains to suppose
\begin{equation}
\kappa(H)\leq0,\qquad M\not\sim_{\Q}0.
\label{rank:nontrivial-nef-data}
\end{equation}
Here \(\dim S>0\). For a rational \(c\in(0,1)\), put
\(H_c=K_W+T+cM\). We claim
\begin{equation}
H_c\ \text{is not pseudo-effective}
\quad(0<c<1,\ c\in\Q).
\label{rank:lowered-nonpsef}
\end{equation}
Suppose otherwise for one \(c\), and choose an ample rational line \(U\)
on \(S\). We will find \(\delta>0\) for which \(H-\delta p^*U\) has an
effective representative, forcing \(\kappa(H)\geq\dim S>0\).
Choose a rational \(a>\max\{1,a_0\}\) and a small
rational \(\lambda>0\) such that \(K_S+aP_S-\lambda U\) is big.
This is possible because \(P_S\) is nef and
\(K_S+a_0P_S\) is big. Choose
\[
0<\eta<\lambda\frac{1-c}{a-1}
\quad\text{rational}.
\]
The rational line \(cP_S+\eta U\) is ample. A sufficiently high
general member divided by its degree pulls back to a rational
boundary transverse to \(T\) with subunit coefficient. Applying
the nonvanishing part of Proposition~\ref{proj:good-models} to
this ordinary projective log pair gives
\[
E_c\sim_{\Q}H_c+\eta p^*U,\qquad E_c\geq0.
\]
A nonzero section defining \(E_c\) restricts nontrivially to very
general \(p\)-fibers in a fixed divisible degree. The hypotheses
of the weak-effectivity statement \cite[Lemma~3.3]{CI} are now
satisfied: the source is smooth in class \(\mathcal C\), \(T\)
is rational SNC, the base \(S\) is smooth projective, and a
relative log system is nonzero. Applied to the big line
\(K_S+aP_S-\lambda U\), it gives
\[
E_a\sim_{\Q}K_W+T+aM-\lambda p^*U,\qquad E_a\geq0.
\]
For
\[
\theta=\frac{1-c}{a-c},\qquad
\delta=\theta\lambda-(1-\theta)\eta>0,
\]
the rational interpolation is
\[
(1-\theta)E_c+\theta E_a
\sim_{\Q}H-\delta p^*U.
\]
Multiplying pullbacks of sections of \(U\) by a section of the
effective left-hand side yields
\(\kappa(H)\geq\kappa(p^*U)=\dim S>0\).
This contradicts Equation~\eqref{rank:nontrivial-nef-data} and
proves Equation~\eqref{rank:lowered-nonpsef}.
We next run a program to fiber type while descending the actual nef line
\(H_{\mathrm{nef}}\) through every contraction. Choose the common
resolution in Equation~\eqref{rank:first-nef-endpoint} to determine the nef
data. The negativity lemma gives
\[
F_M:=\sigma^*M_{\mathrm{nef}}-\mu^*M\geq0
\quad\text{\(\sigma\)-exceptional}.
\]
Indeed its negative is \(\sigma\)-nef and exceptional. For every
rational \(0<e<1\), Equation~\eqref{rank:first-nef-endpoint} gives
\begin{equation}
\mu^*H_{1-e}\sim_{\Q}
\sigma^*(H_{\mathrm{nef}}-eM_{\mathrm{nef}})
+E_{\mathrm{nef}}+eF_M .
\label{rank:lowered-comparison}
\end{equation}
If \(H_{\mathrm{nef}}-eM_{\mathrm{nef}}\) were pseudo-effective, this
equality and effectivity of the last two terms would make
\(\mu^*H_{1-e}\), and hence \(H_{1-e}\), pseudo-effective, contrary to
Equation~\eqref{rank:lowered-nonpsef}.
Fix rational \(0<\varepsilon<1\), choose an integer \(r>0\) with
\(rH_{\mathrm{nef}}\) Cartier, and choose an integer \(d_0>2kr\).
This coefficient will force every ray of the second program to have zero
degree for the descended nef line. Decreasing the nef data to
\((1-\varepsilon)\boldsymbol M\) preserves generalized dlt
singularities: on the displayed resolution the crepant boundary
changes by \(-\varepsilon F_M\), so discrepancies do not decrease.
Add \(d_0H_{\mathrm{nef}}\), determined as nef data on
\(W_{\mathrm{nef}}\). It changes no discrepancies, and the resulting
adjoint is
\[
(1+d_0)H_{\mathrm{nef}}-\varepsilon M_{\mathrm{nef}}
=(1+d_0)\left(
H_{\mathrm{nef}}-\frac{\varepsilon}{1+d_0}M_{\mathrm{nef}}\right).
\]
It is not pseudo-effective by the preceding paragraph. The
non-pseudo-effective case of Proposition~\ref{proj:generalized-mmp}
therefore gives a chosen terminating program to a Mori fiber
contraction in the \(\Q\)-factorial generalized dlt category.
At any stage of this second program, let \(H_i\) be the descended rational
line, assuming inductively that it is nef and \(rH_i\) is Cartier, and write
\[
G_i=K_{W_i}+T_i+(1-\varepsilon)M_i.
\]
An extremal ray negative for \(G_i+d_0H_i\) is negative for \(G_i\).
The pair with adjoint \(G_i\) is still generalized dlt: the added
nef b-divisor \(d_0H_i\) descends to the current model, so removing
it changes no discrepancy. The generalized length bound
\cite[Proposition~3.13]{HanLi} gives a curve \(C\)
on that ray with \(0<-G_i\cdot C\leq2k\). If \(H_i\cdot C>0\),
Cartier integrality would give
\[
\frac{d_0}{r}\leq d_0H_i\cdot C
<-G_i\cdot C\leq2k,
\]
a contradiction. Thus \(H_i\cdot C=0\).
The exact contraction statement \cite[Theorem~1.5]{Xie2022}
descends the Cartier line \(rH_i\) along this contraction as an
actual Cartier line. In a flip we pull that same line to the
flipped side. The descended line is nef: every curve on the
contraction base has a curve above it mapping with positive degree,
so its degree is nonnegative; pullback preserves nefness.
Consequently \(r\) and nefness persist, and the argument applies
inductively to all steps and to the final contraction. Each step
is crepant for \(H_i\).
Let \(W_m\) be the last birational model of this second program and let
\(H_m\) be the descended line on it. Crepancy for the \(H_i\), together
with the absence of extraction, turns
Equation~\eqref{rank:first-nef-endpoint} into
Equation~\eqref{rank:base-endpoint} on a common resolution, with an
effective divisor exceptional over \(W_m\).
The final contraction \(h:W_m\to V\) has connected fibers and,
by the same exact descent,
\[
H_m\sim_{\Q}h^*H_V
\]
for a rational line \(H_V\) on the normal projective \(V\).
If \(V\) is a point, the second alternative holds with
\(H_m\sim_{\Q}0\).
Assume \(\dim V>0\). To obtain the first alternative, it remains to show
that the log adjoint on the new very general fibers has Kodaira dimension
zero. Resolve the base program by
\(\mu:W''\to W\) and \(\nu:W''\to W_m\), and prepare the source
over \(W''\), calling it \(g'':(X'',B'')\to W''\). The diagram
we use is
\[
\begin{tikzcd}
X'' \arrow[r,"g''"] & W'' \arrow[r,"\nu"] \arrow[d,"\mu"'] &
W_m \arrow[r,"h"] & V\\
& W
\end{tikzcd}
\]
with composite \(f= h\nu g''\).
Equation~\eqref{rank:higher-base} and
Equation~\eqref{rank:base-endpoint} give
\[
H''\sim_{\Q}\nu^*H_m+E'',
\qquad E''\geq0\ \text{\(\nu\)-exceptional}.
\]
To see the exceptionality, the error in
Equation~\eqref{rank:base-endpoint} is already exceptional over
\(W_m\), and every \(\mu\)-exceptional prime is exceptional
over \(W_m\) as well: otherwise \(W_m\) would extract a divisor
over \(W\).
For every degree divisible by one fixed integer,
Equation~\eqref{rank:relative-injection}, exceptional descent,
and projection formula now yield
\begin{align*}
f_*\OO_{X''}(q(K_{X''}+B''))
&\lhook\joinrel\longrightarrow
(h\nu)_*\OO_{W''}(qH'')\\
&=h_*\OO_{W_m}(qH_m)=\OO_V(qH_V).
\end{align*}
Here \(\nu_*\OO_{W''}(qE'')=\OO_{W_m}\) by normality and
\(h_*\OO_{W_m}=\OO_V\). Generic coherent base change,
simultaneously for the countably many divisible degrees, bounds
the dimension of every such system on a very general \(f\)-fiber
by one. The log adjoint on that fiber is pseudo-effective by
restriction of a countable sequence of perturbed currents with
analytic singularities. Its dimension is less than \(n\), so
the lower-dimensional \(\Ggood\) supplies a nonzero section in
some divisible degree. Taking a common multiple shows that the
fiber log Kodaira dimension is zero. All maps in the composite
have connected fibers; resolving \(V\) and the source keeps this
property and gives a smooth projective base of dimension
\(0<\dim V<k\). This is the first alternative.
\end{proof}
\subsection{The negative divisor upstairs}
The next lemma lifts either base decomposition and determines the whole
negative divisor on the source, including primes over subsets of
codimension at least two in the base that the local comparison did not test.
\begin{lemma}[Lifting the decomposition]
\label{rank:lift}
Assume the setting of Lemma~\ref{rank:base-psef}. Suppose one of
the following additional conditions holds:
\begin{enumerate}
\item \(a(W)=0\) and \(H\sim_{\Q}E_W=N(H)\), with \(E_W\geq0\);
\item there is a birational morphism \(\nu:W\to W_m\) to a normal
\(\Q\)-factorial projective variety and an identity
\[
H\sim_{\Q}\nu^*H_m+E_W,
\qquad E_W\geq0\ \text{\(\nu\)-exceptional},
\]
where \(H_m\) is a nef rational line, either big or rationally
trivial.
\end{enumerate}
Then there is an effective rational divisor \(A\) on \(X\) and an actual
rational line identity
\begin{equation}
J\sim_{\Q}P_0+A,\qquad A=N(J)\geq0,
\label{rank:upstairs-decomposition}
\end{equation}
where \(P_0\sim_{\Q}0\) in the first case and in the rationally
trivial part of the second case, and
\(P_0=(\nu g)^*H_m\) in the big part of the second case.
\end{lemma}
\begin{proof}
Use the chosen base identity in Equation~\eqref{rank:line-comparison}.
In a rationally trivial case fix a rational trivialization of the positive
line and put \(P_0=0\); in the big case put \(P_0=(\nu g)^*H_m\). Define
the rational divisor
\[
A:=A_*+g^*E_W.
\]
The composed actual line identities give \(J\sim_{\Q}P_0+A\).
The divisor \(A\) may at first have signed coefficients. We first prove
\(A\geq0\).
In a rationally trivial case a divisible multiple of the base line
has a section whose normalized zero divisor is exactly \(E_W\).
Its section under the exact global comparison has normalized zero
divisor \(A\). Regularity of this section proves \(A\geq0\),
also on the initially untested primes.
In the big case choose a Kodaira decomposition
\[
H_m\sim_{\Q}A_0+G_0,
\qquad A_0\text{ ample},\quad G_0\geq0.
\]
For rational \(0<t<1\),
\[
H_m\sim_{\Q}
\underbrace{(1-t)H_m+tA_0}_{\text{ample}}+tG_0.
\]
Fix a prime \(\Gamma\subset X\). A sufficiently divisible
section of the ample summand can be chosen not to vanish at the
generic point of \((\nu g)(\Gamma)\); its pullback has zero order
at \(\Gamma\). Multiplication by the section of \(tG_0\) and
then the global comparison gives a regular upstairs section.
Its normalized order is
\[
A_\Gamma+
t\,\ord_\Gamma((\nu g)^*G_0)\geq0.
\]
Letting \(t\downarrow0\) proves \(A_\Gamma\geq0\). This applies
to every prime in \(A\).
Nefness of \(P_0\) now implies \(N(J)\leq A\).
Set \(Q=A-N(J)\geq0\), an effective real divisor. By
Equation~\eqref{rank:horizontal-negative}, \(Q\) is vertical.
Moreover
\[
\{P_0+Q\}=\{J\}-\{N(J)\}
\]
is modified nef. We prove \(Q=0\).
Let \(b:X\to Y\) denote \(g:X\to W\) in the first case and
\(\nu g:X\to W_m\) in the second. Thus \(\dim Y=k\), and \(b\)
has connected fibers. If \(Q\ne0\), let
\[
l=\max\{\dim b(\Gamma):\Gamma\text{ a component of }Q\}
\leq k-1.
\]
Choose a Kähler class \(\eta\) on \(Y\) (an ample class when \(Y\)
is projective) and a Kähler form \(\omega\) on \(X\). On
\(H^{1,1}(X,\R)\) define
\[
\langle \alpha,\beta\rangle_l
=\int_X \alpha\beta(b^*\eta)^l\omega^{\,n-l-2}.
\]
In this pairing a divisor or rational line denotes its cohomology class.
The exponent is nonnegative since \(l\leq k-1\leq n-2\).
The restriction of a modified-nef class to a resolution of every
prime is pseudo-effective. Applying this to \(\{P_0+Q\}\)
and pairing on each component of \(Q\) gives
\(\langle P_0+Q,\Gamma\rangle_l\geq0\).
The \(P_0\)-term vanishes by dimension, since it is a pullback
from \(Y\) and \(\dim b(\Gamma)\leq l\). Summing with the
coefficients of \(Q\) gives
\begin{equation}
\langle Q,Q\rangle_l\geq0.
\label{rank:nonnegative-square}
\end{equation}
We now separate images of codimension at least two, where the mixed Hodge
index theorem gives strict negativity, from divisorial images, where only
a multiple of the whole fiber can have square zero.
If \(l<k-1\), the mixed Hodge index theorem contradicts this
inequality. In this form \(b^*\eta\) has positive square because
\(l+2\leq k\), while
\(\langle Q,b^*\eta\rangle_l=0\) by image dimension. The form is
a limit of the mixed Hodge index forms obtained by replacing
\(b^*\eta\) with \(b^*\eta+\epsilon[\omega]\); it therefore has
at most one positive direction. In the orthogonal complement
of a positive-square vector it is negative semidefinite, with
equality only in the radical of the whole form. But
\(\langle Q,\omega\rangle_l>0\), since at least one component of
\(Q\) has image dimension \(l\). Thus \(Q\) is not in the radical,
and its square is strictly negative, contrary to
Equation~\eqref{rank:nonnegative-square}.
It remains to treat \(l=k-1\). Terms from primes whose image has
smaller dimension vanish in this form, as do cross terms from
different divisorial images. Fix a prime divisor \(D\subset Y\)
and take all the primes \(E_1,\ldots,E_s\subset X\) dominating it.
Write \(a_i=\ord_{E_i}(b^*D)>0\) and
\(C_{ij}=\langle E_i,E_j\rangle_{k-1}\).
The divisor \(D\) is rational Cartier, also when \(Y=W_m\).
Consequently
\[
C_{ij}\geq0\ (i\ne j),\qquad
\sum_j C_{ij}a_j=0.
\]
The first assertion is positivity of the proper intersection of
distinct effective primes. The second pairs \(E_i\) with the
pulled-back divisor \(D\); the additional base factor makes the
intersection vanish by dimension. Components of \(b^*D\)
mapping into a proper subset of \(D\) make no contribution.
The graph with edges \(C_{ij}>0\) is connected. Indeed over a
very general point of \(D\) the fiber is connected, all its points
lie in the components \(E_i\), and an intersection contributes a
positive entry precisely when it dominates \(D\).
For any real vector \(x=(x_i)\), the two displayed properties give
the exact identity
\[
x^{\mathsf T}Cx
=-\sum_{i<j}C_{ij}a_i a_j
\left(\frac{x_i}{a_i}-\frac{x_j}{a_j}\right)^2.
\]
Thus the block is negative semidefinite with kernel spanned by
\((a_i)\). Equation~\eqref{rank:nonnegative-square} forces every
nonzero block of \(Q\) to equal a positive scalar multiple of
\((a_i)\). In particular \(Q\) then contains every prime over
that divisorial image \(D\).
In the second case this is impossible. The morphism
\(\nu:W\to W_m\) is an isomorphism over the generic point of
\(D\), and \(E_W\) is exceptional.
Equation~\eqref{rank:zero-minimum} therefore supplies a component over
\(D\) whose \(A\)-coefficient is zero. Its \(Q\)-coefficient is
zero as well, contradicting the preceding conclusion.
In the first case, the same zero minimum shows that each
divisorial image of a nonzero block of \(Q\) is a component of
\(E_W=N(H)\). Here \(\{Q\}\) itself is modified nef. The class
\[
b_*(\{Q\}[\omega]^{\,n-k})
\]
is also modified nef on the smooth \(W\). To see this directly,
choose small-perturbation positive currents for \(\{Q\}\) with
zero generic divisorial Lelong numbers, wedge them with
\(\omega^{n-k}\), and push forward. Such a current has no trace
mass on a divisor, so its push has no mass on a base divisor:
the inverse image is a union of divisors and subsets of higher
codimension, all of zero trace mass. The pushed currents
therefore have zero generic divisorial Lelong numbers. Their
classes tend to the displayed class; a fixed smooth representative
for the vanishing error converts this into the defining
small-Kähler-perturbation condition for modified nefness.
On the other hand, pushing the actual divisor \(Q\) in this
formula kills the components with smaller image and gives a
nonzero positive linear combination of the components of \(N(H)\).
Boucksom's exceptionality of the components of a divisorial negative
part says that no such combination is modified nef
\cite[Definition~3.10 and Theorem~3.12]{Boucksom2004}.
This final contradiction proves \(Q=0\),
and hence Equation~\eqref{rank:upstairs-decomposition}.
\end{proof}
\subsection{Generation from a big nef line}
The remaining nef part is pulled back from a big nef line on a
projective variety. We use generation along the log canonical boundary
to prove that the adjoint itself is semiample.
\begin{proposition}[Generation from a big nef base line]
\label{rank:big-nef-generation}
Assume \(\Ggood_d\) for \(d<n\). Let \((Z,\Delta_Z)\) be a
globally \(\Q\)-factorial dlt compact Kähler pair of dimension \(n\)
satisfying the resolution condition of
Definition~\ref{def:lc-strata-resolution}, with effective rational boundary
and analytically nef adjoint
\(L=K_Z+\Delta_Z\). Suppose there are a smooth compact Kähler
manifold \(U\), a projective resolution \(r:U\to Z\), a proper
surjection \(f:U\to Y\) to a normal projective variety, and a
big nef rational line \(H_Y\) on \(Y\) such that
\begin{equation}
r^*L\sim_{\Q}f^*H_Y
\label{rank:big-pullback}
\end{equation}
as actual rational lines. Then \(L\) is semiample.
\end{proposition}
\begin{proof}
Put \(D_Z=\lfloor\Delta_Z\rfloor\). We first prove the following assertion
for every pair and diagram satisfying the hypotheses of this proposition:
\begin{equation}
\mathbf B(L)\cap D_Z=\varnothing.
\label{rank:floor-avoidance}
\end{equation}
Here \(\mathbf B(L)\) is the intersection of the base loci of all positive
Cartier multiples of \(L\). Once this is proved, a log canonical threshold
at any remaining stable base locus will create a new floor there and give a
contradiction.
Take a higher log resolution if needed and put
\[
K_U+F=r^*L=:\ell,\qquad C=F^{=1},\qquad
E=C-\lfloor F\rfloor,\qquad \Delta_F=F-\lfloor F\rfloor.
\]
The equality defining \(F\) uses the canonical meromorphic
identification. The divisor \(F\) is an SNC subboundary.
The integral divisor \(E\) is effective, \(r\)-exceptional, and
has no component in common with \(C\): a nonexceptional
coefficient of \(F\) is a coefficient of the effective
\(\Delta_Z\), while every coefficient of \(F\) is at most one.
The image of \(C\) is contained in \(D_Z\). By
Theorem~\ref{floor:generation}, a divisible multiple of
\(L|_{D_Z}\) is globally generated on the whole reduced floor.
Suppose first that a component \(C_i\) dominates \(Y\).
Pulling the floor sections to \(C_i\) shows that
\((f|_{C_i})^*H_Y\sim_{\Q}\ell|_{C_i}\) is semiample. Semiampleness
descends under a proper surjection to a normal space in this
situation. Factor \(C_i\to Y\) through its normal Stein factor
\(Y_i\). Projection formula descends generation
through the connected-fiber map. For the finite map \(Y_i\to Y\),
at each \(y\in Y\) choose a section of the generated pullback line
avoiding every point of the finite fiber; finitely many evaluation
kernels cannot cover the section space. Lemma~\ref{bir:finite-norm}
gives a norm section of a fixed power of the line on \(Y\), nonzero
at \(y\). These sections generate a fixed multiple of \(H_Y\).
Equation~\eqref{rank:big-pullback}
and normal birational descent then give semiampleness of \(L\).
We may therefore assume that \(C\) is vertical over \(Y\).
Choose a sufficiently divisible integer \(u>1\) for which
\(uL|_{D_Z}\) is generated, and consider the integral line
\begin{equation}
\mathscr N
:=\OO_U(u\ell+E-C)
=\OO_U\bigl(K_U+\Delta_F+(u-1)\ell\bigr).
\label{rank:vanishing-line}
\end{equation}
Although the expression on the right uses rational divisors, its
sum is the integral line on the left. The restriction sequence is
\[
0\longrightarrow\mathscr N
\longrightarrow\OO_U(u\ell+E)
\longrightarrow\OO_C(u\ell+E)\longrightarrow0.
\]
The direct image of the last sheaf is supported on the proper analytic
subset \(f(C)\), so it is torsion. Consequently, the two assertions
\begin{equation}
R^1f_*\mathscr N\ \text{torsion-free},
\qquad H^1(Y,f_*\mathscr N)=0
\label{rank:vanishing}
\end{equation}
will give surjectivity on global sections: torsion-freeness makes the
connecting map to \(R^1f_*\mathscr N\) zero, and the \(H^1\)-vanishing
then removes the obstruction to lifting a global section.
We prove Equation~\eqref{rank:vanishing} using analytic injectivity.
Equation~\eqref{rank:big-pullback} identifies
\(\mathscr N\otimes K_U^{-1}\) with
\(\Delta_F+(u-1)f^*H_Y\) in \(\Pic(U)\otimes\Q\); this comparison
need not be an isomorphism of the corresponding integral lines. Fix a
very ample line \(A_Y\) on \(Y\) and its Fubini--Study metric.
A Kodaira decomposition of the big nef \(H_Y\), given arbitrarily
small weight as in the proof of Lemma~\ref{rank:lift}, writes
\((u-1)H_Y\) as a positive rational multiple of \(A_Y\) plus
an effective rational divisor whose fixed singular contribution
is arbitrarily small and whose remaining contribution is a
general divided ample member. Choose the small weight below
the log canonical threshold of its pullback relative to the
klt SNC boundary \(\Delta_F\), and choose the ample member generally.
Then the resulting boundary on \(U\) is klt. Choose a positive
integer \(q_0\) clearing all its denominators and realizing both the
chosen Kodaira decomposition and the preceding rational-line comparison
as line isomorphisms, in particular
\[
(\mathscr N\otimes K_U^{-1})^{\otimes q_0}
\simeq \OO_U(q_0\Delta_F)\otimes
f^*\OO_Y\bigl(q_0(u-1)H_Y\bigr).
\]
The divisor metrics and the Fubini--Study metric
define a singular Hermitian metric on
\((\mathscr N\otimes K_U^{-1})^{\otimes q_0}\). Its \(q_0\)-th
root is a metric on the actual line \(\mathscr N\otimes K_U^{-1}\),
with multiplier ideal \(\OO_U\) and curvature dominating a positive
multiple of the pullback Fubini--Study form. The same statements hold after every
nonnegative twist by \(f^*A_Y\).
The injectivity theorem \cite[Theorem~A]{FujinoMatsumura} applies
on the compact Kähler \(U\): multiplication by the pullback of
any nonzero section of a positive power of \(A_Y\) is injective
on
\[
H^1(U,\mathscr N\otimes f^*A_Y^j)
\quad(j\geq0)
\]
into the correspondingly higher twist. Its assumptions are
exactly the semipositive smooth metric on the multiplying line,
the preceding positive lower curvature bound, and the trivial
multiplier ideal.
This injectivity implies Equation~\eqref{rank:vanishing}.
For the second assertion, the Leray edge map injects
\(H^1(Y,f_*\mathscr N)\) into \(H^1(U,\mathscr N)\).
Multiplication by a section of a sufficiently high \(A_Y\)-power
sends this subspace to zero, since Serre vanishing kills
\(H^1(Y,f_*\mathscr N\otimes A_Y^j)\) for large \(j\).
Injectivity upstairs forces the original subspace to be zero.
For the first assertion, suppose \(R^1f_*\mathscr N\) has a
nonzero torsion subsheaf. A section of a sufficiently high
power of \(A_Y\) annihilates a nonzero such subsheaf.
After a further sufficiently large twist it has a nonzero
global section, and Serre vanishing and Leray identify that
section with a nonzero class in
\(H^1(U,\mathscr N\otimes f^*A_Y^j)\).
Multiplication annihilates this class, again contradicting
injectivity. This proves both assertions.
The resulting surjectivity extends the floor sections as follows.
Pull any section of \(uL|_{D_Z}\)
to \(C\), multiply by the canonical section of \(E|_C\), and
extend by this surjectivity. The extended section descends to
\(Z\), since \(r_*\OO_U(E)=\OO_Z\) by exceptionality and
normality. On the strict transforms of the components of
\(D_Z\), division by the canonical section of \(E\) recovers
the prescribed section. The descended restriction therefore
equals it on the whole reduced \(D_Z\), since equality holds
generically on every component. Since \(uL|_{D_Z}\) is generated,
this proves Equation~\eqref{rank:floor-avoidance}.
We finish by showing that this stable base locus is empty. The
base loci of factorial Cartier multiples form a descending
sequence of compact analytic subsets and hence stabilize.
Sections exist because \(H_Y\) is big and
Equation~\eqref{rank:big-pullback} descends their pullbacks.
Choose a divisible degree \(v\) whose base locus is
\(\mathbf B(L)\), and let \(\mathfrak b\) be its base ideal.
If \(\mathbf B(L)\ne\varnothing\), the ideal is a unit near
\(D_Z\), while \((Z,\Delta_Z)\) is klt outside \(D_Z\).
Its log canonical threshold
\[
t=\operatorname{lct}_{(Z,\Delta_Z)}(\mathfrak b)
\]
is therefore a positive rational number, attained by a divisor
whose center is contained in \(\mathbf B(L)\).
This follows directly on a simultaneous log resolution of the
pair and the ideal: the threshold is the minimum of the positive
rational discrepancies divided by the positive integral ideal
orders.
Choose an integer \(s>t\) and general divisors
\(D_1,\ldots,D_s\in|vL|\). On the same resolution their fixed
part is the divisor of \(\mathfrak b\); their free transforms
are jointly transverse to the SNC data. Hence
\[
\left(Z,\ \Delta_Z+\frac{t}{s}\sum_{i=1}^s D_i\right)
\]
is log canonical: the fixed part is at its log canonical
threshold and the free components have coefficients \(t/s<1\).
There is a new lc place centered in \(\mathbf B(L)\), and the
new adjoint satisfies the actual identity
\[
L_{\mathrm{new}}\sim_{\Q}(1+tv)L.
\]
Apply \cite[Theorem~3.3]{HLX} to this compact Kähler lc rational
pair. It gives a projective strongly \(\Q\)-factorial dlt
modification \(d:Z'\to Z\). Since the pair is lc, all extracted
divisors have log discrepancy zero in our convention, so the
modification is crepant. The source is again compact Kähler.
Choose its defining dlt log resolution, with exceptional crepant
coefficients strictly below one and an isomorphism at the general
point of every lc stratum; resolving the remaining data away from
those points gives the resolution required by
Definition~\ref{def:lc-strata-resolution}. The floor on \(Z'\) has a point over the center
of this lc place, hence over \(\mathbf B(L)\).
The new nef adjoint is \(d^*L_{\mathrm{new}}\). On a common
resolution with \(U\), it satisfies
Equation~\eqref{rank:big-pullback} with the big nef line
\((1+tv)H_Y\). The previously proved assertion
\eqref{rank:floor-avoidance} therefore makes its stable base locus
disjoint from its floor. On the
other hand, normality and projection formula identify all
divisible sections under \(d\), and give
\[
\mathbf B(d^*L_{\mathrm{new}})
=d^{-1}\mathbf B(L).
\]
This locus contains the stated floor point, a contradiction.
Thus \(\mathbf B(L)=\varnothing\); factorial stabilization
provides an actual globally generated multiple of \(L\).
\end{proof}
\subsection{Completion of the rank-one case}
\begin{proof}[Proof of Proposition~\ref{rank:rank-one}]
By Proposition~\ref{bir:resolution-transfer} we can begin on the
resolved source in the statement. We use a secondary induction on
the positive base dimension \(k\). Apply
Lemma~\ref{rank:comparison} and then Lemma~\ref{rank:base-psef}
to obtain the prepared data and the pseudo-effective line \(H\).
If \(a(W)=0\), Lemma~\ref{rank:zero-base} and the first case of
Lemma~\ref{rank:lift} give
\(J\sim_{\Q}N(J)\), which is already the required decomposition
with trivial positive part. If \(W\) is projective, apply
Lemma~\ref{rank:base-reduction}. Its first alternative has a
strictly smaller positive base dimension, so the secondary induction
and birational transfer finish the proof. In its second alternative,
take a common smooth base resolution in
Equation~\eqref{rank:base-endpoint} and prepare the source over it.
Equation~\eqref{rank:higher-base} adds an effective divisor
exceptional over the old base. It is also exceptional over \(W_m\),
because the base program extracted no divisors. Thus the second
case of Lemma~\ref{rank:lift} applies.
We have obtained
\[
J\sim_{\Q}P_0+A,\qquad A=N(J)\geq0,
\]
with \(P_0\) nef. A rationally trivial \(P_0\) finishes the proof.
In the remaining case \(P_0=b^*H_m\), where \(b:X\to W_m\)
is proper and \(H_m\) is big and nef on the projective \(W_m\).
Apply Proposition~\ref{prog:contract-negative} to this
nef-plus-negative presentation. It gives a globally strongly
\(\Q\)-factorial dlt compact Kähler model \((Z,\Delta_Z)\) with nef
adjoint \(L_Z\). The ordinary dlt endpoint has the resolution property
of Definition~\ref{def:lc-strata-resolution}. On a common smooth resolution
\(\mu:U\to X\), \(r:U\to Z\) its exact positive-line comparison is
\[
r^*L_Z\sim_{\Q}\mu^*P_0
=(b\mu)^*H_m .
\]
Proposition~\ref{rank:big-nef-generation} makes \(L_Z\) semiample.
Hence \(\mu^*P_0\) is semiample. Lemma~\ref{bir:pullup} identifies
\(\mu^*A=N(\mu^*J)\), so this is precisely \(\Ggood_n\) on \(U\).
Finally Proposition~\ref{bir:resolution-transfer} transfers the
decomposition back through all the prepared source modifications.
\end{proof}
==== END Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//05-rank-one.tex ====
==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//06-simple-nonvanishing.tex ====
\section{Meromorphic nonvanishing on simple spaces}\label{sec:nonvanishing}
The simple case of the induction needs a divisor representing a multiple of
the canonical bundle; its coefficients need not be nonnegative. The
following result supplies precisely this starting point.
\begin{theorem}\label{nv:meromorphic}
Let \(X\) be a smooth connected compact Kähler manifold. Suppose that
\(a(X)=0\) and that no positive-dimensional proper compact analytic
subvariety passes through a very general point of \(X\). Then
\(K_X^{\otimes m}\) has a nonzero meromorphic section for some integer
\(m>0\).
\end{theorem}
The proof compares two ways of measuring poles. After normalizing a big
class on \(X\) to have volume one, its pole order at a very general point is
bounded. We construct a projective bundle over \(X^2\) whose volume grows
linearly with a parameter \(q\). Restriction to the exceptional divisor of
the diagonal in the square of this bundle then produces a high-rank
subsheaf of a symmetric cotangent power. A second diagonal forces
substantial vanishing of its determinant; descending that determinant
produces a point pole of order comparable to
\(q^{1/(2\dim X+1)}\), contradicting the point bound.
The argument uses only meromorphic sections of line bundles; it does not
require nonconstant meromorphic functions on \(X\).
For the proof, suppose that the conclusion fails. Write
\[
n=\dim X,\qquad \mathscr L=K_X,\qquad L=c_1(\mathscr L)
\in H^{1,1}_{\BC}(X,\R).
\]
We may assume \(n\ge2\): a compact complex curve is projective, and the
assertion for a point is immediate. Throughout this Section, an inequality
between real \((1,1)\)-classes is in pseudo-effective order:
\(\alpha\le\beta\) means that \(\beta-\alpha\) is pseudo-effective. For a
divisor \(D\), \(\{D\}\) denotes \(c_1(\OO(D))\). Pullbacks of classes will
occasionally be suppressed when the map is evident.
\subsection{Line subsheaves and point poles}
We first record the consequence of simplicity that controls all the
determinant lines used below.
\begin{lemma}\label{nv:slope}
Under the contrary hypothesis above, \(H^1(X,\OO_X)=0\), the group
\(\Pic(X)\) is countable, and \(L\ge0\). If \(\mathscr H\) is a holomorphic
line bundle and
\(\mathscr H\longrightarrow\Omega_X^{\otimes k}\) is nonzero, where
\(k\ge0\), then
\begin{equation}\label{nv:slope-X}
c_1(\mathscr H)\le kL .
\end{equation}
There is a complement of a countable union of proper analytic subsets of
\(X\) with the following further property. For every point \(x\) in this
complement, let
\[
a:Y=\Bl_xX\longrightarrow X,\qquad F=a^{-1}(x).
\]
Every nonzero line map
\(\mathscr H_Y\longrightarrow (a^*\Omega_X)^{\otimes k}\) satisfies
\begin{equation}\label{nv:slope-blowup}
c_1(\mathscr H_Y)\le k a^*L .
\end{equation}
\end{lemma}
\begin{proof}
If the Albanese map of \(X\) is nonconstant, simplicity makes its image
\(n\)-dimensional: a positive-dimensional general fiber of smaller
dimension would be a forbidden subvariety. At a point where the Albanese
map has rank \(n\), some \(n\) invariant one-forms have nonzero wedge after
pullback. This gives a nonzero holomorphic section of \(K_X\), contrary to
our assumption. Thus \(H^1(X,\OO_X)=0\). The exponential sequence embeds
\(\Pic(X)\) into the countable group \(H^2(X,\Z)\).
The manifold \(X\) is not uniruled, by simplicity. Ou's
non-pseudo-effectivity criterion for the canonical class therefore gives
\(L\ge0\) \cite[Theorem~1.1]{Ou2025}. We recall how the slope and foliation
results in the same paper give the more precise inequality in
Equation~\eqref{nv:slope-X}. Let \(\gamma\) be any class in the full dual of the
pseudo-effective cone, and use the slope
\[
\mu_\gamma(\mathscr E)
=\frac{c_1(\mathscr E)\cdot\gamma}{\rk\mathscr E}
\]
for torsion-free sheaves. If
\(\mu_{\gamma,\min}(\Omega_X)<0\), the first Harder--Narasimhan piece
\(\mathscr T\subset T_X\) has strictly positive slope. It is saturated and
semistable. The tensor slope inequality shows that the bracket
\(\bigwedge^2\mathscr T\to T_X/\mathscr T\) is zero: the minimum slope of
its source is at least \(2\mu_\gamma(\mathscr T)\), whereas the maximum
slope of its target is smaller than \(\mu_\gamma(\mathscr T)\).
Consequently \(\mathscr T\) is a foliation. Its dual has strictly negative
maximum slope and is not pseudo-effective by
\cite[Proposition~4.5]{Ou2025}. Ou's foliation theorem
\cite[Theorem~1.4]{Ou2025} makes this foliation the tangent foliation of a
meromorphic fibration with compact general leaf closures. Its rank is strictly
between zero and \(n\): rank \(n\) would give
\(\mu_\gamma(T_X)=-L\cdot\gamma/n\le0\). The general leaf closure is
therefore a positive-dimensional proper subvariety through a general
point, again contradicting simplicity.
We have proved \(\mu_{\gamma,\min}(\Omega_X)\ge0\). In its
Harder--Narasimhan filtration all quotient slopes are now nonnegative, so
\(\mu_{\gamma,\max}(\Omega_X)\le L\cdot\gamma\). The tensor slope
inequality \cite[Lemma~4.4]{Ou2025} gives
\[
c_1(\mathscr H)\cdot\gamma
\le \mu_{\gamma,\max}(\Omega_X^{\otimes k})
\le kL\cdot\gamma .
\]
Separation by the dual cone proves Equation~\eqref{nv:slope-X}. For \(k=0\) this is
also immediate from the effective zero divisor of a nonzero map
\(\mathscr H\to\OO_X\).
It remains to choose \(x\) uniformly. For any vector bundle
\(\mathscr V\), independent global sections are generically pointwise
independent. Indeed, choose a maximal pointwise independent subfamily on
a dense open set. Expressing any other section in that family by minors
gives meromorphic function coefficients. They are constant because
\(a(X)=0\), so maximality among a linearly independent family forces all
its members to be pointwise independent. Thus evaluation on
\(H^0(X,\mathscr V)\) is injective away from a proper analytic subset
(unless that vector space is zero, in which case there is no restriction).
Apply this to
\(\mathscr V=\mathscr H^{-1}\otimes\Omega_X^{\otimes k}\) for every
\(\mathscr H\in\Pic(X)\) and \(k\ge0\). There are countably many such
bundles.
For \(x\) outside the resulting exceptional set, write any line bundle on
\(Y=\Bl_xX\) uniquely as
\(a^*\mathscr H\otimes\OO_Y(mF)\), \(m\in\Z\). A line map from this bundle
to \((a^*\Omega_X)^{\otimes k}\), restricted away from \(F\), extends over
\(x\) to a section of
\(\mathscr H^{-1}\otimes\Omega_X^{\otimes k}\) by Hartogs' theorem. If
\(m>0\), this section vanishes at \(x\), since
\(a_*\OO_Y(-mF)=\mathcal I_x^m\). The choice of \(x\) rules this out.
Thus \(m\le0\), and Equation~\eqref{nv:slope-X} yields
\[
c_1(a^*\mathscr H\otimes\OO_Y(mF))
\le k a^*L+m\{F\}\le k a^*L.
\]
If a target has a finite filtration whose graded pieces are direct sums
of the indicated cotangent tensors, take the first nonzero associated
graded component of the line map and then a nonzero direct-sum component.
The preceding argument applies with the tensor order of that component.
\end{proof}
We will use several standard facts about volumes of real \((1,1)\)-classes.
Our normalization is \(\vol(\alpha)=\int\alpha^e\) for a nef class on an
\(e\)-fold. Volume is continuous, homogeneous, monotone in
pseudo-effective order, invariant under modification, and its \(e\)-th
root is concave on the big cone. Analytic Fujita approximation computes
it by Kähler parts on smooth projective modifications. Here a
\emph{smooth projective modification} means a projective modification
whose source is smooth; the sources used here are compact Kähler
manifolds obtained by resolving coherent analytic ideals. For a smooth
irreducible divisor \(D\) and a big class \(\alpha\), write
\(\operatorname{vol}_{\,|D}(\alpha)\) for the numerical restricted volume.
It is zero when \(D\) is contained in the non-Kähler locus
\cite[Theorem~1.1]{Vu2023}; otherwise it is
the supremum of the masses on \(D\) of restrictions of Kähler currents
with analytic singularities that are not generically singular on \(D\)
\cite[Lemma~2.7]{CollinsTosatti2018}. The divisorial derivative and its
continuity on the big cone are
\begin{equation}\label{nv:volume-derivative}
\frac{\dd}{\dd u}\vol(\alpha-u\{D\})
=-e\,\operatorname{vol}_{\,|D}(\alpha-u\{D\});
\end{equation}
see \cite[Theorem~1.1]{Vu2023}. We apply this formula only on smooth
manifolds while the varying class is big.
Here is a useful precise form of the approximation in the restricted
volume formula. Resolve the log-ideal singularities of a current
restricted to \(D\). Its pullback is an effective real divisor plus a
positive residual current with locally bounded potentials. The latter
dominates a positive multiple of the pulled-back Kähler form. A current
with locally bounded potentials has zero Lelong numbers, so Demailly
regularization makes its class, after subtracting that multiple, nef
\cite[Theorem~1.1]{DemaillyRegularization1992}.
On a projective modification there is an effective exceptional divisor
whose negative is relatively ample. Subtracting a sufficiently small
multiple of this divisor from the residual class therefore makes that
class Kähler; add the same multiple to the divisor part. Round all
divisor coefficients slightly upwards to rational numbers. Openness of
the Kähler cone preserves the Kähler property. These changes can be
arbitrarily small in top intersections, which compute the original mass
by the bounded-potential product formula. Thus we may approximate a
restricted mass by
\begin{equation}\label{nv:rational-fujita}
h^*(\alpha|_D)=\beta+\{D'\},\qquad
\beta\ \text{Kähler},\quad D'\ge0\ \text{a rational divisor}.
\end{equation}
Moreover, \(D'\) has at least the log-ideal orders of the original
restriction on every further resolution. We will use this last
property to turn poles into vanishing conditions. Only \(D'\) is made
rational; the class \(\beta\) and the horizontal part of \(\alpha\) remain
real.
\begin{lemma}[A point pole bound]\label{nv:pole-threshold}
Let \(W\) be a smooth compact Kähler manifold of dimension \(e\ge2\),
\(\alpha\) a big real \((1,1)\)-class, and \(z\in W\). Suppose a smooth
projective modification \(\mu:W'\to W\), which is an isomorphism near
\(z\), admits a decomposition
\[
\mu^*\alpha=K+\{D\},\qquad K\ \text{Kähler},\quad D\ge0,
\]
where \(D\) misses the point \(z'\) over \(z\). If the ordinary analytic
Seshadri constant \(\epsilon(K,z')\) is at least \(\eta>0\), then, on the
blowup \(b:\widehat W=\Bl_zW\to W\) with exceptional divisor \(G\),
\begin{equation}\label{nv:pole-formula}
\sup\{u\ge0:b^*\alpha-u\{G\}\ge0\}
\le \frac{\eta}{2}
+2^{e-1}\vol(\alpha)\eta^{-(e-1)} .
\end{equation}
\end{lemma}
\begin{proof}
Set \(u_0=\eta/2\). Blowing up \(z'\), the class
\(K-u_0\{G'\}\) is Kähler. The Fujita decomposition is unchanged near
\(G'\), so it supplies a Kähler current for
\(\alpha_{u_0}=b^*\alpha-u_0\{G\}\) that is smooth near \(G\).
Its restriction there has class \(u_0c_1(\OO_{\PP^{e-1}}(1))\).
The restricted volume is consequently \(u_0^{e-1}\): the current gives
this lower bound, and the volume of the restricted class gives the
opposite bound.
Let \(\tau\) denote the left side of Equation~\eqref{nv:pole-formula}. The classes
\(\alpha_u=b^*\alpha-u\{G\}\) are big for \(0\le u<\tau\), since
\(\alpha_0\) is big and the pseudo-effective cone is convex. The
concave function \(f(u)=\vol(\alpha_u)^{1/e}\) satisfies, by
Equation~\eqref{nv:volume-derivative},
\[
f'(u_0)=-\frac{u_0^{e-1}}{f(u_0)^{e-1}}.
\]
Its tangent line at \(u_0\) must remain positive up to \(\tau\).
Therefore
\[
\tau\le u_0+\frac{f(u_0)^e}{u_0^{e-1}}
\le u_0+\frac{\vol(\alpha)}{u_0^{e-1}},
\]
which is Equation~\eqref{nv:pole-formula}.
\end{proof}
Fix a Kähler class \(\omega\) on \(X\). The class \(L\) is not big:
a big holomorphic line bundle would make \(X\) Moishezon, contrary to
\(a(X)=0\). For \(t>0\) define
\begin{equation}\label{nv:normalization}
P=r(L+t\omega),\qquad
r=\vol(L+t\omega)^{-1/n}.
\end{equation}
Then \(P\) is a big real class, \(\vol(P)=1\), \(L\le P/r\), and
\(r\to\infty\) as \(t\downarrow0\). Choose a smooth Fujita model
\(\mu:Y_P\to X\) with Kähler part \(P'\) satisfying
\begin{equation}\label{nv:P-prime}
\mu^*P=P'+\{D_P\},\qquad v:=\int_{Y_P}(P')^n>\frac12.
\end{equation}
At a very general point \(y\) of this model there is no
positive-dimensional proper subvariety: its image would be one through a
very general point of \(X\), and \(y\) avoids the exceptional locus.
The ordinary Seshadri formula for a Kähler class,
\[
\epsilon(K,y)
=\inf_{\substack{W\ni y\\ \dim W>0}}
\left(\frac{\int_W K^{\dim W}}
{\operatorname{mult}_y W}\right)^{1/\dim W},
\]
therefore gives \(\epsilon(P',y)=v^{1/n}\ge2^{-1/n}\).
This is the ordinary nef/Kähler formula
\cite[Theorem~2.8]{Tosatti2016}; see also
\cite[Equation~(1.5)]{CollinsTosatti2018} and the
Kähler cone criterion of \cite{DemaillyPaun2004}. We use this ordinary
formula in both applications below. Lemma~\ref{nv:pole-threshold}, with
\(\vol(P)=1\), now yields a constant \(C_n\) depending only on \(n\)
such that, for a very general \(x\),
\begin{equation}\label{nv:point-bound}
\tau(P,x):=\sup\{u\ge0:a^*P-u\{F\}\ge0\}\le C_n,
\qquad a:\Bl_xX\to X.
\end{equation}
Here and below positive constants denoted \(c_n,C_n\) may be decreased or
increased from one occurrence to the next. They are independent of all
parameters and choices of currents and modifications.
\subsection{A projective bundle with controlled volume}
We now produce a class whose volume grows, while its point pole threshold
grows much more slowly. On \(X^2\), let \(\mathscr L_i\) and \(L_i\)
denote the pullbacks of \(\mathscr L\) and \(L\) from the \(i\)-th factor.
Use the quotient convention and put
\[
\pi:Z=\PP_{X^2}(\mathscr L_1\oplus\mathscr L_2)\longrightarrow X^2,
\qquad \xi=c_1(\OO_Z(1)),\qquad d=\dim Z=2n+1 .
\]
Thus \(\pi_*\OO_Z(m)=\Sym^m(\mathscr L_1\oplus\mathscr L_2)\) for
\(m\ge0\). The zero divisor \(A_i\) of
\(\pi^*\mathscr L_i\to\OO_Z(1)\) has class \(\xi-L_i\); it is the
section on which \(\xi\) restricts to \(L_{3-i}\). In particular
\(\xi=\{A_i\}+L_i\ge0\).
\begin{lemma}\label{nv:subvarieties}
For \(X\) and \(Z\) above, assume that \(K_X^{\otimes m}\) has no nonzero
meromorphic section for every integer \(m>0\).
Through a very general point of \(X^2\), the only positive-dimensional
proper compact irreducible analytic subvarieties are the two factor
slices. Through a very general point of \(Z\), the only
positive-dimensional compact irreducible analytic subvarieties are a
fiber of \(\pi\), the full inverse images of the two factor slices, and
\(Z\) itself.
\end{lemma}
\begin{proof}
Let \(C\subset X^2\) be irreducible through a pair whose coordinates are
very general in \(X\). Each projection image is a point or all of \(X\),
by properness and simplicity. If exactly one is a point, \(C\) is the
corresponding full slice. If both projections are surjective, test their
fibers at a general point of \(C\) whose coordinates are still very
general. A positive-dimensional fiber must be the full other factor.
Thus either \(C=X^2\), or \(\dim C=n\) and both projections are
generically finite.
We show that subvarieties of the latter kind cannot sweep \(X^2\).
The space of compact \(n\)-cycles on a compact Kähler manifold has
countably many irreducible components, each compact
\cite[Theorem~1.1]{LiebermanCycles}. On a component containing integral cycles
generically finite over both factors, integrality holds on a dense open
set and the two projection degrees remain positive, as can also be
tested by intersection with pulled-back Kähler forms. Suppose the
incidence over one such component dominates \(X^2\). Resolve the
parameter space and the dominating incidence component, obtaining maps
\[
h:\mathcal Y\longrightarrow\mathcal S,\qquad
F_i:\mathcal Y\longrightarrow X
\]
between smooth spaces. For general \(t\in\mathcal S\), the fiber
\(Y_t=h^{-1}(t)\) is smooth, compact, and bimeromorphic to the integral
cycle, hence irreducible. Its maps \(f_i=F_i|_{Y_t}:Y_t\to X\) are
generically finite. Choose \(t\) also so that \(Y_t\) meets the dense
open set where \(d(F_1,h)\) is invertible and \(d(F_1,F_2)\) has rank
\(2n\). The first assertion follows from generic finiteness of \(f_1\),
and the second from the assumed dominance in \(X^2\).
Our immediate aim is to turn this dominance into a meromorphic frame of
\(f_2^*T_X\). Its determinant would meromorphically trivialize
\(f_2^*K_X^{-1}\), and hence also \(f_2^*K_X\).
On \(Y_t\), the equal-rank bundle map
\[
d(F_1,h)|_{Y_t}:T_{\mathcal Y}|_{Y_t}
\longrightarrow f_1^*T_X\oplus
(\OO_{Y_t}\otimes T_t\mathcal S)
\]
has a meromorphic inverse, given by its adjugate and determinant.
For fixed \(v\in T_t\mathcal S\), apply this inverse to \((0,v)\)
and then apply \(dF_2\). The result is a meromorphic section of
\(f_2^*T_X\). At a general point where \(d(F_1,F_2)\) has rank
\(2n\), the resulting map \(T_t\mathcal S\to f_2^*T_X\) is
surjective: in local coordinates \((F_1,h)\), it is the derivative
of \(F_2\) in the parameter directions with \(F_1\) fixed. Choose
\(n\) fixed vectors \(v_i\) whose values there are independent.
Their wedge is a nonzero meromorphic section of \(f_2^*K_X^{-1}\)
on the irreducible \(Y_t\); its inverse trivializes \(f_2^*K_X\)
meromorphically.
Factor the proper generically finite map \(f_2\) as
\[
Y_t\longrightarrow \overline Y_t
\xrightarrow{\nu}X
\]
by Stein factorization. Here \(\overline Y_t\) is normal and irreducible, the first
map is a modification, and \(\nu\) is finite of degree \(N>0\).
Meromorphic sections descend across a modification of a normal
space, so the section descends to a nonzero meromorphic section of
\(\nu^*K_X\). Lemma~\ref{bir:finite-norm} gives a nonzero meromorphic
section of \(K_X^{\otimes N}\). This norm is defined in local finite
analytic fraction algebras, so it remains available even though the
global meromorphic function field of \(X\) is just \(\C\). It
contradicts our assumption.
Consequently the incidence image of each such cycle component is a
proper compact analytic subset of \(X^2\). There are only countably
many components, so their union misses a very general pair. This
proves the assertion for \(X^2\).
Now let \(W\subset Z\) be irreducible through a very general point.
Its image is a point, a full factor slice, or \(X^2\). If its generic
relative dimension is one, it is the full inverse image of that
image. Otherwise it is a multisection of the restricted
\(\PP^1\)-bundle. Over a slice or over \(X^2\), a multisection is a
divisor with line bundle
\(\OO(k)\otimes\pi^*\mathscr H\), \(k>0\). Its homogeneous equation
has coefficients in
\[
H^0\bigl(S,\mathscr H\otimes
\mathscr L_1^{\otimes i}
\otimes\mathscr L_2^{\otimes(k-i)}\bigr),
\qquad 0\le i\le k,
\]
where \(S\) is the relevant slice or \(X^2\), and a fixed-factor line is
understood as constant. A single nonzero monomial cuts out only the
axis sections and vertical divisors. A multisection not on an axis
therefore has two nonzero coefficients. Their quotient is a
meromorphic section of a nonzero power of
\(\mathscr L_1\otimes\mathscr L_2^{-1}\). Over a slice this is, up to
inversion and a constant line, a positive power of \(K_X\). Over
\(X^2\), restriction to a general factor slice gives the same
conclusion. This is impossible. The axes themselves miss a very
general point of \(Z\), proving the claim.
\end{proof}
Let \(q\) tend to infinity through positive integers. For each sufficiently
large \(q\), choose \(t=t(q)\) in Equation~\eqref{nv:normalization} so that
\(r=r(q)\ge q^2\), and use the resulting normalized class \(P=P(q)\).
Define the real class and the scale
\begin{equation}\label{nv:M-definition}
M=P_1+P_2+q\xi,\qquad A_q=q^{1/d}.
\end{equation}
\begin{lemma}\label{nv:two-slot-volume}
For these choices, \(M\) is big and
\begin{equation}\label{nv:M-bounds}
c_nq\le\vol(M)\le C_nq,\qquad
\tau(M,z):=\sup\{u\ge0:b_z^*M-u\{F_z\}\ge0\}\le C_nA_q
\end{equation}
at a very general point \(z\in Z\), where
\(b_z:\Bl_zZ\to Z\) has exceptional divisor \(F_z\).
\end{lemma}
\begin{proof}
Pull \(Z\) to the Fujita model \(Y_P^2\) from
Equation~\eqref{nv:P-prime}, and put \(H=P'_1+P'_2\) on this pulled-back bundle.
There is a further smooth projective modification
\(\nu:\widehat Z\to\PP_{Y_P^2}(\mu^*\mathscr L_1\oplus
\mu^*\mathscr L_2)\) and a Fujita decomposition of the pullback of \(M\)
whose Kähler part is exactly
\begin{equation}\label{nv:exact-K}
K=\frac12\nu^*H+\Theta ,
\end{equation}
where \(\Theta\) is Kähler and has degree \(q\) on a general vertical
line. The modification and the divisor part miss that line.
Here is the construction. It suffices to decompose \(H/2+q\xi\) as
\(\Theta\) plus an effective divisor, clean on a whole general vertical
line. Since \(\OO(1)\) is relatively ample, a small positive class of
the form \(\delta(\xi+cH)\) is Kähler for some \(c>0\). Choose
\(\delta>0\) so small that \(\delta<q\) and the remaining horizontal
part of \(H/2\) is Kähler. Represent the remaining \(\xi\) first as
\(\{A_1\}+L_1\) and then as \(\{A_2\}+L_2\). Regularize the
pseudo-effective classes \(L_i\) with analytic singularities, paying
their arbitrarily small negative errors from that horizontal Kähler
part. This gives two Kähler currents in \(H/2+q\xi\), each smooth off
its own axis and a proper horizontal analytic set. The maximum of
their potentials is locally bounded along an entire general vertical
line, since the two axes are disjoint. Analytic regularization
preserving a smaller Kähler lower bound
\cite[Theorem~2.1(ii)]{Boucksom2004} gives a Kähler current with
analytic singularities missing that line. Resolve its singularities
and make the small Kähler adjustment described before
Lemma~\ref{nv:pole-threshold}. The divisor still misses a general
vertical line, so the residual class \(\Theta\) has degree exactly
\(q\) there. Adding the other half of \(H\) and the pullbacks of the
divisor parts \(D_P\) gives Equation~\eqref{nv:exact-K}. Its two
summands are nef, with \(\Theta\) Kähler, so every mixed intersection
used in the following bounds is nonnegative.
At a very general point of \(\widehat Z\), Lemma~\ref{nv:subvarieties}
lists the possible positive-dimensional subvarieties: the vertical line,
the strict transforms of the two full bundles over slices, and
\(\widehat Z\). They have multiplicity one there. By
Equation~\eqref{nv:exact-K} and nonnegativity of mixed intersections of nef
classes, their top intersections are respectively bounded below by
\begin{align}
K\cdot(\text{vertical line})&=q, \notag\\
\int_{\text{slice}}K^{n+1}
&\ge \frac{n+1}{2^n}\,qv,\label{nv:K-intersections}\\
\int_{\widehat Z}K^d
&\ge \frac{d}{2^{2n}}\binom{2n}{n}\,qv^2. \notag
\end{align}
For example, the middle line is the term containing one factor
\(\Theta\) and \(n\) factors from the varying \(P'\); pushforward of
\(\Theta\) along a general vertical line is \(q\). The last line is the
term with one \(\Theta\) and \(2n\) horizontal factors. These
intersections give \(\vol(M)\ge c_nq\). The ordinary Seshadri formula
and \(q\ge1\) give
\(\epsilon(K,z')\ge c_nq^{1/d}=c_nA_q\) at a very general \(z'\):
each possible dimension is at most \(d\), and every numerator in that
formula is at least \(c_nq\).
For the upper bound, subtract the axis \(A_1\). For \(0\le u\le q\) put
\[
M_u=M-u\{A_1\}
=P_1+P_2+uL_1+(q-u)\xi .
\]
The endpoint \(M_q\) is pulled back from \(X^2\) and has zero volume on
the \(d\)-fold \(Z\). It is pseudo-effective, and \(M_0\) is big by
the preceding construction, so \(M_u\) is big for \(u<q\). The
restriction of \(M_u\) to \(A_1\simeq X^2\) is
\[
(P+uL)_1+(P+(q-u)L)_2.
\]
The restricted volume along \(A_1\) is at most the volume of this
restriction. For big classes \(\alpha_i\) on manifolds of dimensions
\(e_i\),
\begin{equation}\label{nv:product-volume}
\vol(\pr_1^*\alpha_1+\pr_2^*\alpha_2)
=\binom{e_1+e_2}{e_1}\vol(\alpha_1)\vol(\alpha_2).
\end{equation}
One can see this directly from the non-pluripolar product formula
\cite{BEGZ2010}: the
envelope with minimal singularities of a sum on a product is the sum
of the two envelopes, by testing the defining inequality on successive
slices. Its top product has only the indicated binomial term. Since
\(L\le P/r\) and \(r\ge q^2\), monotonicity and
Equation~\eqref{nv:product-volume} bound the restriction volume by
\[
\binom{2n}{n}(1+u/r)^n(1+(q-u)/r)^n\le C_n.
\]
Integrating Equation~\eqref{nv:volume-derivative} from \(0\) to \(q\) gives
\(\vol(M)\le C_nq\). Finally apply
Lemma~\ref{nv:pole-threshold} with \(e=d\),
\(\eta=c_nA_q\), and \(\vol(M)\le C_nq\).
Both terms on the right of Equation~\eqref{nv:pole-formula} are at most
\(C_nA_q\), since \(A_q^d=q\). This proves Equation~\eqref{nv:M-bounds}.
\end{proof}
\subsection{A direct-image estimate}
The next lemma bounds a Kähler volume upstairs in terms of a class on the
base. Its determinant formula will let Lemma~\ref{nv:slope} control that
base class. The base need not be projective, and the horizontal class is
allowed to be real.
\begin{lemma}\label{nv:direct-image}
Let \(f:U\to Z_0\) be a surjective projective morphism of smooth connected
compact Kähler manifolds, with
\(\dim Z_0=b_0\ge1\) and \(\dim U=b_0+e\), \(e\ge1\). Suppose
\[
\beta=f^*G+c_1(\Lambda)
\]
is a Kähler class, where \(G\in H^{1,1}_{\BC}(Z_0,\R)\) and
\(\Lambda\in\Pic(U)\otimes\Q\). Put
\[
w=\int_{U_z}\beta^e>0
\]
on a general fiber. For sufficiently large divisible integers \(j\),
let \(\mathcal F_j=f_*\OO_U(j\Lambda)\) and \(R_j=\rk\mathcal F_j\).
Here \(j\Lambda\) is an actual line bundle and
\(c_1(\mathcal F_j)\) means \(c_1(\det\mathcal F_j)\). Then
\begin{align}
R_j&=\frac{w}{e!}j^e+O(j^{e-1}),\label{nv:GRR-rank}\\
B_0:=G+\lim_j\frac{c_1(\mathcal F_j)}{jR_j}
&=\frac{f_*\beta^{e+1}}{(e+1)w}\ge0,\label{nv:GRR-base}\\
\int_U\beta^{b_0+e}
&\le (e+1)^{b_0}w\,\vol(B_0).\label{nv:mixed-bound}
\end{align}
The limit in Equation~\eqref{nv:GRR-base} is a limit of real Bott--Chern classes.
\end{lemma}
\begin{proof}
The restriction of \(c_1(\Lambda)\) to each fiber is represented by the
restriction of the Kähler form \(\beta\). The fiberwise criterion for
relative ampleness makes \(\Lambda\) relatively ample after clearing
denominators. Relative Serre vanishing then kills the higher direct
images for all sufficiently large divisible \(j\). Analytic
Grothendieck--Riemann--Roch \cite{LevyGRR}, in degrees zero and two, gives
\[
R_j=\frac{f_*c_1(\Lambda)^e}{e!}j^e+O(j^{e-1}),\qquad
c_1(\mathcal F_j)
=\frac{f_*c_1(\Lambda)^{e+1}}{(e+1)!}j^{e+1}+O(j^e).
\]
The degree-zero pushforward is \(w\). Expanding
\(\beta=f^*G+c_1(\Lambda)\) shows that
\[
f_*\beta^{e+1}
=f_*c_1(\Lambda)^{e+1}+(e+1)wG,
\]
because terms with at least two horizontal factors have negative
fiber degree after pushforward. This proves the equality in
Equation~\eqref{nv:GRR-base}; the cohomological GRR equality is an equality in
Bott--Chern cohomology by the \(\partial\bar\partial\)-lemma on compact
Kähler manifolds. Pushforward of the positive form
\(\beta^{e+1}\) is a positive closed \((1,1)\)-current, so \(B_0\) is
pseudo-effective.
To prove Equation~\eqref{nv:mixed-bound}, we need a nef class on the base.
The class \(B_0\) is presently only known to be pseudo-effective. We
flatten \(f\) to obtain a nef class above the base and compare it
with the pullback of \(B_0\). Take a smooth projective flattening
modification \(p:Z'_0\to Z_0\), the equidimensional main transform
\(\overline U\) of \(U\times_{Z_0}Z'_0\), and a resolution
\(U'\to\overline U\). Write \(f':U'\to Z'_0\) and \(q:U'\to U\) for
the maps and \(\beta'=q^*\beta\). The class
\[
B'_0=\frac{f'_*(\beta')^{e+1}}{(e+1)w}
\]
is nef. It is enough to show that the positive pushforward current
representing this class has zero Lelong numbers. This current can
be computed by integration on the cycle \(\overline U\) of the smooth
form pulled back from \(U\). At a base point, cover the compact fiber
by finitely many coordinate neighborhoods, each embedded in a product
of base coordinates and ambient coordinates. In the mass over a base
ball of radius \(\delta\), after wedging with a base Euclidean form to
power \(b_0-1\), the integrand is bounded by a finite sum of projection
volume forms using \(b_0-1\) base coordinates and \(e+1\) generic
linear combinations of all coordinates of the ambient product, including
the remaining base direction. These projections can be chosen finite on the
neighborhoods: fixing the \(b_0-1\) base coordinates leaves local
dimension at most \(e+1\), by equidimensionality, and generic ambient
coordinates finish a finite projection. After shrinking to compact
subneighborhoods their degrees are bounded. Change of variables
therefore bounds each integral by
\(O(\delta^{2(b_0-1)})\) times the measure of the remaining coordinate
range. On each compact subneighborhood these ranges, taken over closed
base balls, are nested compact sets whose intersection is the image of
the central fiber. That image has measure zero in the \(e+1\) coordinates,
because the fiber has dimension \(e\). Continuity of finite measure from
above shows that the range measures tend to zero. Thus the
mass is
\[
o\bigl(\delta^{2(b_0-1)}\bigr).
\]
This also covers \(b_0=1\), when it asserts absence of an atom. The
pushforward current has zero Lelong numbers at every point, and
Demailly regularization \cite[Theorem~1.1]{DemaillyRegularization1992}
proves that its class \(B'_0\) is nef.
Pushforward under \(p\) gives \(p_*B'_0=B_0\). For a modification between
smooth compact Kähler manifolds,
the kernel of pushforward on real Bott--Chern \((1,1)\)-classes is generated
by the classes of its exceptional prime divisors. Since
\(p_*p^*B_0=B_0\), it follows that \(B'_0-p^*B_0\) is an exceptional real
divisor class. It is
\(p\)-nef because \(B'_0\) is nef. Apply relative negativity to this
exceptional real divisor. Locally over the base, the usual proof for a
projective modification cuts by general hyperplanes to a surface and
uses the negative definite intersection matrix of exceptional curves;
it forces every coefficient of a relatively nef exceptional divisor to
be nonpositive. Thus
\[
B'_0=p^*B_0-\{D_0\},\qquad D_0\ge0\ \text{\(p\)-exceptional}.
\]
In particular
\begin{equation}\label{nv:flattened-volume}
\int_{Z'_0}(B'_0)^{b_0}
=\vol(B'_0)\le\vol(p^*B_0)=\vol(B_0).
\end{equation}
Choose a Kähler class \(\omega'\) on \(Z'_0\) and put
\(C=B'_0+\varepsilon\omega'\). For \(0\le k\le b_0\), set
\[
I_k=\int_{U'}(\beta')^{e+k}(f'^*C)^{b_0-k}.
\]
These are mixed intersections of nef classes. The first two satisfy
\[
I_0=w\int_{Z'_0}C^{b_0},\qquad
I_1=(e+1)w\int_{Z'_0}B'_0C^{b_0-1}
\le(e+1)w\int_{Z'_0}C^{b_0}.
\]
The mixed nef inequalities make the sequence \(I_k\) log-concave.
Every \(I_k\) is positive: on the dense open where \(q\) is a local
biholomorphism and \(f'\) is a submersion, \(\beta'\) is positive
definite and \(f'^*C\) has rank \(b_0\), so the defining top form
is strictly positive. The successive ratios are therefore at most
\(I_1/I_0\le e+1\). Thus
\[
\int_U\beta^{b_0+e}=I_{b_0}
\le(e+1)^{b_0}w\int_{Z'_0}C^{b_0}.
\]
Letting \(\varepsilon\downarrow0\) and using
Equation~\eqref{nv:flattened-volume} proves Equation~\eqref{nv:mixed-bound}.
\end{proof}
We will also use the determinant formula without its volume bound.
\begin{corollary}\label{nv:slope-descent}
Let \(Y\) be a smooth compact Kähler manifold and \(\mathscr E\) a
holomorphic vector bundle of rank at least two. Write
\(\pi_{\mathscr E}:\PP_Y(\mathscr E)\to Y\) and
\(\zeta_{\mathscr E}=c_1(\OO_{\PP(\mathscr E)}(1))\).
Suppose that a real class \(D\) on \(Y\) satisfies
\[
c_1(\mathscr H)\le kD
\quad\text{whenever}\quad
0\ne\bigl(\mathscr H\longrightarrow\mathscr E^{\otimes k}\bigr)
\]
for a line bundle \(\mathscr H\) and integer \(k\ge0\).
For a real class \(A\) on \(Y\) and a real number \(b\ge0\),
\begin{equation}\label{nv:slope-descent-equation}
\pi_{\mathscr E}^*A+b\zeta_{\mathscr E}\ge0
\quad\Longrightarrow\quad A+bD\ge0 .
\end{equation}
\end{corollary}
\begin{proof}
Choose a real class \(H_0\) on \(Y\) so that
\(\zeta_{\mathscr E}+\pi_{\mathscr E}^*H_0\) is Kähler; relative
ampleness of \(\OO(1)\) permits such a choice. Take numbers
\(\delta\downarrow0\) with \(b+\delta>0\) rational. Adding
\(\delta(\zeta_{\mathscr E}+\pi_{\mathscr E}^*H_0)\) to the
pseudo-effective class in Equation~\eqref{nv:slope-descent-equation} makes it
big. A Fujita decomposition on a smooth projective modification
\(h:U\to\PP_Y(\mathscr E)\), with the divisor coefficients rounded
upwards as in Equation~\eqref{nv:rational-fujita}, has the form
\[
\beta=f^*(A+\delta H_0)+c_1(\Lambda),\qquad
\Lambda=(b+\delta)h^*\OO(1)-\OO_U(D')
\ \text{in }\Pic(U)\otimes\Q,
\]
where \(f=\pi_{\mathscr E}h\), \(\beta\) is Kähler, and \(D'\ge0\)
is rational. For divisible \(j\),
\[
f_*\OO_U(j\Lambda)\ \subseteq\
\Sym^{j(b+\delta)}\mathscr E .
\]
Its determinant of rank \(R_j\) consequently maps into
\(\mathscr E^{\otimes j(b+\delta)R_j}\). The assumed line inequality
gives
\[
\frac{c_1(\mathcal F_j)}{jR_j}\le(b+\delta)D.
\]
Lemma~\ref{nv:direct-image} makes
\(A+\delta H_0+\lim c_1(\mathcal F_j)/(jR_j)\) pseudo-effective.
Adding the preceding pseudo-effective difference shows that
\(A+\delta H_0+(b+\delta)D\) is pseudo-effective. Let
\(\delta\downarrow0\). This proves the corollary. In this argument
only \(b+\delta\) and the coefficients of \(D'\) are rational; \(A\),
\(H_0\), and \(D\) remain real classes.
\end{proof}
\subsection{Restriction to the diagonal}
We now use the volume of \(M\) to obtain a high-rank subsheaf of a
symmetric cotangent power on \(Z\). Distinguish the two copies of \(Z\)
by bracketed indices and
blow up their diagonal:
\[
b:B=\Bl_{\Delta_Z}(Z\times Z)\longrightarrow Z\times Z,\qquad
E=b^{-1}(\Delta_Z).
\]
Thus \(E=\PP_Z(\Omega_Z)\); its points are normal lines in \(T_Z\).
The dimensions are \(\dim B=2d\) and \(\dim E=2d-1\), and
\(E\to Z\) has relative dimension \(d-1\).
Put \(\zeta=c_1(\OO_E(1))\), so that \(\{E\}|_E=-\zeta\).
For \(s\ge0\) define
\[
C_s=b^*(M_{[1]}+M_{[2]})-s\{E\},\qquad
s_{\max}=\sup\{s\ge0:C_s\ge0\}.
\]
The class \(C_0\) is big, so \(s_{\max}>0\) and \(C_s\) is big for
\(0\le s<s_{\max}\). Restricting a Kähler current with analytic
singularities to the fiber of the first projection at a very general
\(z\in Z\) gives the class \(b_z^*M-s\{F_z\}\). The current can be
restricted for a general such \(z\), and Lemma~\ref{nv:two-slot-volume}
therefore gives
\begin{equation}\label{nv:smax}
s_{\max}\le C_nA_q .
\end{equation}
Write \(v_E(s)=\operatorname{vol}_{\,|E}(C_s)\) for
\(0<s<s_{\max}\). The restriction class is
\begin{equation}\label{nv:restriction-class}
C_s|_E=2M+s\zeta .
\end{equation}
\begin{lemma}\label{nv:diagonal-mass}
There are constants \(c_n,C_n>0\) such that, for every sufficiently
large \(q\), one can choose a rational number
\[
c_nA_q\le s\le C_nA_q
\]
and a Kähler current \(T\) in \(C_s\) with analytic singularities,
not generically singular on \(E\), with the following property. On a
smooth projective modification \(h:U\to E\), its restricted mass has
a rational-divisor approximation
\begin{equation}\label{nv:selected-decomposition}
h^*(2M+s\zeta)=\beta+\{D'\},
\qquad \beta\ \text{Kähler},\quad D'\ge0\ \text{rational},
\end{equation}
which retains all log-ideal orders of \(T|_E\). If
\(f:U\to Z\) is the natural map and
\[
w=\int_{U_z}\beta^{d-1}
\]
on a general fiber, then
\begin{equation}\label{nv:selected-w}
w\ge c_ns^{d-1}.
\end{equation}
\end{lemma}
\begin{proof}
First let \(s\) be any rational number in \((0,s_{\max})\) with
\(v_E(s)>0\), and use the approximation in
Equation~\eqref{nv:rational-fujita} for any current used
to compute this restricted volume. In
Lemma~\ref{nv:direct-image}, the data for Equation~\eqref{nv:restriction-class}
are
\[
b_0=d,\quad e=d-1,\quad G=2M,\quad
\Lambda=s h^*\OO_E(1)-\OO_U(D')
\ \text{in }\Pic(U)\otimes\Q.
\]
For large divisible \(j\), the associated sheaves satisfy
\begin{equation}\label{nv:symmetric-inclusion}
\mathcal F_j=f_*\OO_U(j\Lambda)
\subseteq\Sym^{js}\Omega_Z,\qquad
R_j=\rk\mathcal F_j,\qquad 0<w\le s^{d-1}.
\end{equation}
The inclusion follows by pushing
\(\OO_U(-jD')\subseteq\OO_U\) through \(h\).
For the last inequality restrict the decomposition to a general
\(\PP^{d-1}\)-fiber: monotonicity of volume bounds the Kähler volume
there by \(\vol(s\zeta)=s^{d-1}\).
We claim that the class \(B_0\) of Equation~\eqref{nv:GRR-base} satisfies
\begin{equation}\label{nv:B0-bound}
0\le B_0\le C_nM .
\end{equation}
Here is the determinant calculation, including its vertical sign.
The equality \(H^1(X,\OO_X)=0\) and the Künneth decomposition give
\(\Pic(X^2)=\pr_1^*\Pic(X)\oplus\pr_2^*\Pic(X)\). Hence, for some
lines \(\mathscr Q_{j,i}\) on \(X\) and integer \(\ell_j\),
\[
\det\mathcal F_j
=\pi^*(\mathscr Q_{j,1}\boxtimes\mathscr Q_{j,2})
\otimes\OO_Z(\ell_j),\qquad
c_1(\det\mathcal F_j)=Q_j+\ell_j\xi,
\]
where \(Q_j=c_1(\mathscr Q_{j,1})_1+c_1(\mathscr Q_{j,2})_2\).
Taking the determinant of the inclusion in
Equation~\eqref{nv:symmetric-inclusion} gives a nonzero line
map into \(\Omega_Z^{\otimes k}\), \(k=jsR_j\). It is first defined
where \(\mathcal F_j\) is locally free and extends over the
codimension-two complement. Filter this tensor using
\[
0\longrightarrow\pi^*\Omega_{X^2}\longrightarrow\Omega_Z
\longrightarrow
\OO_Z(-2)\otimes\pi^*(\mathscr L_1\otimes\mathscr L_2)
\longrightarrow0 .
\]
A nonzero associated graded component has \(i\) relative factors and
\(k_1,k_2\) cotangent factors from the first and second factors of \(X^2\),
respectively, where
\(0\le i\le k\) and \(k_1+k_2=k-i\). Its relative degree is \(-2i\).
Pushing the component to \(X^2\) forces
\[
m_j:=-2i-\ell_j\ge0,\qquad \ell_j\le-2i\le0.
\]
A nonzero homogeneous monomial in
\(\Sym^{m_j}(\mathscr L_1\oplus\mathscr L_2)\), with exponent
\(0\le m_{j,1}\le m_j\) in the first factor, gives, on the two general slices,
\[
c_1(\mathscr Q_{j,1})\le(i+m_{j,1}+k_1)L,\qquad
c_1(\mathscr Q_{j,2})\le(i+m_j-m_{j,1}+k_2)L
\]
by Lemma~\ref{nv:slope}. Each coefficient on the right is at most
\(k-\ell_j\). Since \(L\ge0\), we obtain
\begin{equation}\label{nv:determinant-bound-Z}
\ell_j\le0,\qquad
Q_j\le(jsR_j-\ell_j)(L_1+L_2).
\end{equation}
Lemma~\ref{nv:direct-image} supplies a limit of the entire determinant
class divided by \(jR_j\). The projective-bundle decomposition of
Bott--Chern cohomology gives separate limits \(Q\) and \(\ell\) of
the two displayed components. Since
\[
B_0=2(P_1+P_2)+(2q+\ell)\xi+Q\ge0,
\]
testing on a general vertical line gives \(2q+\ell\ge0\).
Equation~\eqref{nv:determinant-bound-Z} gives \(\ell\le0\) and
\(Q\le(s-\ell)(L_1+L_2)\). Use \(\xi\ge0\), \(-\ell\le2q\),
\(L\le P/r\), \(r\ge q^2\), and \(s\le C_nA_q\le C_nq\).
They give
\[
B_0\le2(P_1+P_2)+2q\xi+(s+2q)(L_1+L_2)
\le C_nM,
\]
which proves Equation~\eqref{nv:B0-bound}.
Volume monotonicity, Lemma~\ref{nv:two-slot-volume}, and
Equation~\eqref{nv:mixed-bound} now imply
\begin{equation}\label{nv:restriction-mass-bound}
\int_U\beta^{2d-1}\le C_nwq,\qquad
v_E(s)\le C_nq\,s^{d-1}.
\end{equation}
For the second inequality, approximate the mass of each current
arbitrarily closely by \(\beta\) and use \(w\le s^{d-1}\), then take
the supremum over currents. This proves it for rational \(s\);
continuity of divisorial restricted volume extends it to every
\(s\in(0,s_{\max})\).
At \(s_{\max}\) the class is on the boundary of the pseudo-effective
cone, so its volume is zero. The product formula in
Equation~\eqref{nv:product-volume}, modification invariance, and the derivative
formula in Equation~\eqref{nv:volume-derivative} yield
\begin{equation}\label{nv:total-diagonal-mass}
2d\int_0^{s_{\max}}v_E(s)\,\dd s
=\vol(C_0)
=\binom{2d}{d}\vol(M)^2
\ge c_nq^2 .
\end{equation}
Choose a fixed small \(\theta_n>0\). The contribution of
\(0<s<\theta_nA_q\), by Equation~\eqref{nv:restriction-mass-bound}, is at
most \(C_n\theta_n^d q^2\). Fix \(\theta_n\) small enough that this
is less than half the last lower bound. The remaining interval has
length at most \(C_nA_q\), by Equation~\eqref{nv:smax}. Continuity therefore
permits a rational \(s\in[\theta_nA_q,s_{\max})\) such that
\[
v_E(s)\ge c_nq^2/A_q=c_nqA_q^{d-1}
\ge c_nq s^{d-1}.
\]
Choose a current with restricted mass at least three quarters of
\(v_E(s)\), and then an approximation as in
Equation~\eqref{nv:selected-decomposition} with
\(\int_U\beta^{2d-1}\ge v_E(s)/2\). The first inequality of
Equation~\eqref{nv:restriction-mass-bound} gives
\(w\ge c_ns^{d-1}\), as claimed.
\end{proof}
For the selected \(s,T,h,\beta,D'\) and \(w\), retain the natural map
\(f:U\to Z\) and put
\[
\Lambda=s h^*\OO_E(1)-\OO_U(D'),\qquad
\mathcal F_j=f_*\OO_U(j\Lambda),\qquad R_j=\rk\mathcal F_j,
\]
where \(j\) is sufficiently large and divisible. Here \(\Lambda\) is a
rational line, and Equation~\eqref{nv:symmetric-inclusion} gives
\(\mathcal F_j\subseteq\Sym^{js}\Omega_Z\). With \(q\) and these selected
data fixed, Equations~\eqref{nv:GRR-rank} and \eqref{nv:selected-w} give
\begin{equation}\label{nv:selected-rank-fraction}
\rk\Sym^{js}\Omega_Z=\binom{js+d-1}{d-1},\qquad
\lim_j\frac{R_j}{\rk\Sym^{js}\Omega_Z}
=\frac{w}{s^{d-1}}\ge c_n .
\end{equation}
The limit is through divisible integers. Thus the first diagonal has
produced a subsheaf occupying a fixed positive proportion of the symmetric
power for all sufficiently large divisible \(j\). The remaining proof
carries this rank bound to a modification of \(Z\) and converts it into a
large common order of vanishing for the determinant map on that model.
\subsection{A pole along the incidence}
We next locate a large pole of the selected current \(T\) along a smooth
incidence submanifold of \(B\). This will impose vanishing conditions on the
symmetric-power subsheaf in Equation~\eqref{nv:symmetric-inclusion}.
Let \(U_0=\pi^{-1}(\Delta_X)\subset Z\), where \(\Delta_X\) is the
diagonal in \(X^2\). Since
\(\mathscr L_1|_{\Delta_X}=\mathscr L_2|_{\Delta_X}=K_X\), there is a
canonical identification
\[
U_0=X\times\PP^1.
\]
For \(\lambda\in\PP^1\) write \(D_\lambda=X\times\{\lambda\}\subset Z\).
Inside \(Z\times Z\) consider
\[
\mathcal V_0
=\{((x,x,\lambda),(y,y,\lambda)):x,y\in X,\ \lambda\in\PP^1\}
\simeq X^2\times\PP^1.
\]
It meets \(\Delta_Z\) in \(\Delta_X\times\PP^1\). The strict transform
\(V\subset B\) is
\(\Bl_{\Delta_X\times\PP^1}\mathcal V_0\); it is smooth, of dimension
\(d=2n+1\). Its first projection is a smooth family over \(U_0\).
Over \(z=(x,x,\lambda)\) its fiber is
\[
Y=\Bl_xD_\lambda\simeq\Bl_xX,
\]
and its intersection with \(E\) in that fiber is the projective space
of lines in \(T_zD_\lambda\), inside the projective space of lines in
\(T_zZ\). These assertions follow either from the blowup of the
section \(y=x\) in this family or from the coordinates used below.
Let \(\rho_V:\Bl_VB\to B\) have exceptional divisor \(G_V\). Denote
by \(b_V\ge0\) the generic divisorial pole of \(\rho_V^*T\) along
\(G_V\). Equivalently, it is the generic log-ideal order of \(T\)
along \(V\), with the coefficient of the logarithm included.
\begin{lemma}\label{nv:incidence-pole}
The pole just defined satisfies
\begin{equation}\label{nv:pole-transfer}
b_V\ge s-C_n .
\end{equation}
\end{lemma}
\begin{proof}
Remove \(b_V[G_V]\) from \(\rho_V^*T\). Its residual positive current
can be restricted to \(G_V\): for analytic singularities removal of
the generic divisorial pole leaves a potential that is not identically
\(-\infty\) on that divisor. Choose \(z=(x,x,\lambda)\in U_0\)
general enough for all restrictions and for
Lemmas~\ref{nv:slope} and \ref{nv:pole-threshold}. On the bundle
\[
G_V|_Y=\PP_Y(N^*_{V/B}|_Y),\qquad Y=\Bl_xX,
\]
this restriction is a positive current in the class
\begin{equation}\label{nv:incidence-restriction}
a^*(2P+qL)-s\{F\}+b_V\zeta_V ,
\end{equation}
where \(\zeta_V=c_1(\OO_{\PP(N^*_{V/B}|_Y)}(1))\).
Indeed \(M|_{D_\lambda}=2P+qL\), the first projection to \(Z\) is constant on
\(Y\), and \(E|_Y=F\). Also \(G_V|_{G_V}=-\zeta_V\), explaining the
positive sign of the last term.
The conormal bundle in Equation~\eqref{nv:incidence-restriction} has exact
sequences
\begin{align}
0\longrightarrow\OO_Y^{\oplus n}
&\longrightarrow N^*_{V/B}|_Y
\longrightarrow a^*N^*_{D_\lambda/Z}\otimes\OO_Y(F)
\longrightarrow0,\label{nv:incidence-conormal}\\
0\longrightarrow\Omega_X
&\longrightarrow N^*_{D_\lambda/Z}
\longrightarrow\OO_X\longrightarrow0.\label{nv:small-conormal}
\end{align}
The first constant term is the conormal of \(U_0\) in the first copy of
\(Z\), evaluated at \(z\). For the last term in
Equation~\eqref{nv:incidence-conormal}, the conormal of the strict transform of
\(D_\lambda\) in \(\Bl_zZ\) is
\(a^*N^*_{D_\lambda/Z}\otimes\OO_Y(F)\): in a blowup chart, a normal
coordinate is divided by an exceptional coordinate, giving precisely
the twist by \(F\) for conormals. Equation~\eqref{nv:small-conormal}
is the conormal sequence for
\(D_\lambda\subset U_0\subset Z\); the diagonal in \(X^2\) has conormal
\(\Omega_X\), and the \(\lambda\)-normal direction is constant on
\(X\).
Filter a \(k\)-fold tensor of \(N^*_{V/B}|_Y\) by these sequences. A
graded term has the form
\[
\OO_Y(hF)\otimes(a^*\Omega_X)^{\otimes k'}
\otimes(\text{a constant vector space}),
\qquad 0\le k'\le h\le k.
\]
For any nonzero line map into that tensor, take a nonzero graded
component. Lemma~\ref{nv:slope} gives
\[
c_1(\mathscr H_Y)\le h\{F\}+k'a^*L
\le k(\{F\}+a^*L),
\]
since both \(\{F\}\) and \(a^*L\) are pseudo-effective. Apply
Corollary~\ref{nv:slope-descent} to
Equation~\eqref{nv:incidence-restriction}, with
\(D=\{F\}+a^*L\). We obtain
\begin{equation}\label{nv:incidence-base}
a^*(2P+(q+b_V)L)-(s-b_V)\{F\}\ge0.
\end{equation}
If \(b_V\ge s\), the conclusion is immediate. Otherwise
\(b_V<s\le C_nA_q\le C_nq\), and \(L\le P/r\) gives
\[
\left(2+\frac{q+b_V}{r}\right)a^*P-(s-b_V)\{F\}\ge0.
\]
Use the point bound in Equation~\eqref{nv:point-bound}. Since \(r\ge q^2\), its
consequence
\[
s-b_V\le C_n\left(2+\frac{q+b_V}{r}\right)
\]
is bounded by a dimensional constant. This proves
Equation~\eqref{nv:pole-transfer}.
\end{proof}
\subsection{Determinant vanishing and cancellation}
Blow up \(U_0\) in the base of \(E\):
\[
p_+:Z^+=\Bl_{U_0}Z\longrightarrow Z,\qquad J_+=p_+^{-1}(U_0).
\]
This is the bundle \(Z\) pulled to \(\Bl_{\Delta_X}(X^2)\), since
\(U_0=\pi^{-1}(\Delta_X)\). In particular \(Z^+\) is smooth. Put
\[
E^+=E\times_Z Z^+=\PP_{Z^+}(p_+^*\Omega_Z).
\]
Extend the incidence ideal from \(B\) to \(E\), and then to \(E^+\):
\[
\mathcal J=(\mathcal I_V\cdot\OO_E)\cdot\OO_{E^+}.
\]
The products denote extension of ideals under the indicated maps.
Figure~\ref{nv:two-diagonals} locates \(V\cap E\) and this extension of
the incidence ideal to \(E^+\).
\begin{figure}[ht]
\centering
\input{figures/two-diagonals}
\caption{Two geometric constructions used in the determinant estimate.
On the left, the strict transform \(V\subset B\) meets \(E_z\) in the
directions tangent to \(D_\lambda\). Under the quotient convention,
the displayed projectivized cotangent spaces parametrize tangent lines.
On the right, \(E^+\) is the Cartesian base change
over \(Z^+=\Bl_{U_0}Z\), whose exceptional divisor is \(J_+\).
The restricted incidence ideal extends to \(\mathcal J\) on \(E^+\).
The ensuing local calculation identifies the normal parameter of \(J_+\)
among its generators.}
\label{nv:two-diagonals}
\end{figure}
The next lemma turns the pole bound in
Equation~\eqref{nv:pole-transfer} into a common vanishing order for the
coefficients of each determinant map.
\begin{lemma}\label{nv:determinant-order}
On a smooth projective modification \(h_+:U^+\to E^+\) one can choose
an approximation
\begin{equation}\label{nv:plus-decomposition}
h_+^*(2p_+^*M+s\zeta)=\beta^++\{D^+\},
\qquad \beta^+\ \text{Kähler},\quad D^+\ge0\ \text{rational},
\end{equation}
retaining the log-ideal orders of \(T|_E\), such that the general
fiber volume \(w^+\) of \(\beta^+\) over \(Z^+\) satisfies
\(w^+\ge w/2\). Let \(f_+:U^+\to Z^+\) be the natural map and put
\[
\Lambda^+=s h_+^*\OO_{E^+}(1)-\OO_{U^+}(D^+),\qquad
\mathcal F_j^+=(f_+)_*\OO_{U^+}(j\Lambda^+),\qquad
R_j^+=\rk\mathcal F_j^+ .
\]
For every sufficiently large divisible \(j\), write
\[
\iota_j:\mathcal F_j^+\hookrightarrow p_+^*\Sym^{js}\Omega_Z,\qquad
\varphi_j:\det\mathcal F_j^+\longrightarrow
\bigwedge^{R_j^+}\!\bigl(p_+^*\Sym^{js}\Omega_Z\bigr)
\]
for the inclusion and its nonzero determinant map. Here
\(\det\mathcal F_j^+=(\bigwedge^{R_j^+}\mathcal F_j^+)^{**}\), and the
determinant map extends across the complement of the locally free locus,
which has codimension at least two.
Localize a coordinate local ring of \(Z^+\) at the height-one prime of
\(J_+\), obtaining a discrete valuation ring \(R\) with uniformizer \(u\).
In local frames over \(R\), define
\[
h_j=\min\{\ord_u(c):c\text{ is a nonzero coefficient of }\varphi_j\}.
\]
Equivalently, \(h_j\) is the minimum order of the maximal minors of
\(\iota_j\); it is independent of the frames. If \(\sigma_{J_+}\) is the
canonical section of \(\OO_{Z^+}(J_+)\), then \(\varphi_j\) factors as
\[
\det\mathcal F_j^+
\xrightarrow{\ \cdot\sigma_{J_+}^{h_j}\ }
\det\mathcal F_j^+(h_jJ_+)
\xrightarrow{\ \widetilde\varphi_j\ }
\bigwedge^{R_j^+}\!\bigl(p_+^*\Sym^{js}\Omega_Z\bigr),
\]
where \(\widetilde\varphi_j\) is holomorphic. The common order satisfies
\begin{equation}\label{nv:determinant-order-equation}
h_j\ge c_n\,jR_j^+s ,
\end{equation}
provided \(q\) is sufficiently large.
\end{lemma}
\begin{proof}
Blow up \(\mathcal J\) and take a common smooth projective resolution
with \(U\to E\) from Equation~\eqref{nv:selected-decomposition}. Pull back
\(\beta\) and \(D'\). Subtracting a sufficiently small rational
multiple of an effective exceptional divisor whose negative is
relatively ample makes the pulled-back Kähler class Kähler on this
resolution; add that multiple to the effective part. Further upward
rational rounding may be
arbitrarily small. This gives Equation~\eqref{nv:plus-decomposition} with all
original orders retained. Its general fiber intersection is as close
to \(w\) as desired, so arrange \(w^+\ge w/2\). Lemma~\ref{nv:direct-image}
and the effective divisor give
\begin{equation}\label{nv:plus-rank}
\mathcal F_j^+\subseteq p_+^*\Sym^{js}\Omega_Z,\qquad
R_j^+=\frac{w^+}{(d-1)!}j^{d-1}+O(j^{d-2}).
\end{equation}
We spell out the local order imposed on the polynomials in this
inclusion. Near \(z=(x,x,\lambda)\), use coordinates
\((\mathbf x,\mathbf a,\lambda)\) on the first \(Z\), where \(\mathbf x\)
and \(\mathbf a\) each have \(n\) components and \(U_0=(\mathbf a=0)\).
Write the second coordinates as
\((\mathbf x+\mathbf h,\mathbf a+\mathbf k,\lambda+\mu)\), with
\(\mathbf h,\mathbf k\) each having \(n\)
components. Then
\[
\mathcal V_0=(\mathbf a=\mathbf k=\mu=0),\qquad
\Delta_Z=(\mathbf h=\mathbf k=\mu=0).
\]
In a blowup chart meeting the lines tangent to \(D_\lambda\), take
\[
h_1=v,\quad h_i=v\alpha_i\ (2\le i\le n),\quad
k_i=vy_i\ (1\le i\le n),\quad \mu=vy_{n+1}.
\]
Here \(E=(v=0)\), while the strict transform is
\(V=(a_1=\cdots=a_n=y_1=\cdots=y_{n+1}=0)\).
In particular,
\begin{equation}\label{nv:ideal-before}
\mathcal I_V|_E=(a_1,\ldots,a_n,y_1,\ldots,y_{n+1}).
\end{equation}
On a chart of the blowup of \(U_0\), write
\(a_1=u\) and \(a_i=u\tau_i\) for \(i\ge2\). Along the general point
of \(J_+=(u=0)\), the ideal \(\mathcal J\) is exactly
\begin{equation}\label{nv:joint-ideal}
(u,y_1,\ldots,y_{n+1}).
\end{equation}
Here \(u\) measures vanishing along \(J_+\), while the \(y_i\) measure
transverse fiber directions. Thus a term of transverse degree \(\ell\)
can contribute at most \(\ell\) to the incidence order; the remaining
order must come from its coefficient in \(u\). We now make this statement
precise.
Suppose locally that the singularities of \(T\) are
\(c\log|\mathfrak a|+O(1)\), with the logarithmic normalization for
which a divisorial coefficient is \(c\) times the ideal order. If
\(k=\ord_V(\mathfrak a)\), then \(b_V=ck\).
The normal Taylor coefficients of degree less than \(k\) vanish on a
dense open set of \(V\), hence on \(V\) locally. Thus
\(\mathfrak a\subseteq\mathcal I_V^k\) also near a general point of
\(V\cap E\). Restricting to \(E\) and pulling to \(E^+\) gives the
corresponding order in Equation~\eqref{nv:joint-ideal}. At its general center
the displayed generators are regular coordinates. If \(H_{\mathcal J}\)
is the exceptional divisor of their blowup, its valuation is the
ordinary ideal-adic order:
\[
\ord_{H_{\mathcal J}}(g)
=\max\{k:g\in(u,y_1,\ldots,y_{n+1})^k\}.
\]
The common resolution dominates this blowup. Pullback preserves the
effective difference between \(D'\) and the resolved log divisor,
and the later adjustments only increase the effective part. Thus
\(D^+\) has coefficient at least \(b_V\) at this valuation.
Every polynomial image of a section of \(\mathcal F_j^+\) has
integral valuation at least \(\lceil jb_V\rceil\). Locally bounded
remainders have zero order at this valuation.
There are \(n\) homogeneous coordinates along \(T_zD_\lambda\) and
\(n+1\) transverse homogeneous coordinates in \(T_zZ\). Denote
these two groups by \(H_1,\ldots,H_n\) and
\(Y_1,\ldots,Y_{n+1}\). For \(m=js\), a local polynomial in
\(\Sym^m(p_+^*\Omega_Z)\) has the form
\[
\sum_{|\alpha|+|\gamma|=m}
c_{\alpha,\gamma}(u)\,H^\alpha Y^\gamma.
\]
Work over the discrete valuation ring \(R\) used to define \(h_j\), choosing
the chart parameter \(u\) as uniformizer, and let \(\kappa\) be its residue
field. The coefficients \(c_{\alpha,\gamma}\) lie in \(R\).
Equation~\eqref{nv:joint-ideal}
implies, monomial by monomial,
\begin{equation}\label{nv:coefficient-order}
\ord_u(c_{\alpha,\gamma})
\ge\max\{0,\lceil jb_V\rceil-|\gamma|\}.
\end{equation}
Indeed, on the chart \(H_1\ne0\), group the dehomogenized polynomial as
\[
\sum_\gamma
P_\gamma(H_2/H_1,\ldots,H_n/H_1)(Y/H_1)^\gamma,
\qquad P_\gamma\in R[H_2/H_1,\ldots,H_n/H_1].
\]
Put \(b_j=\lceil jb_V\rceil\). At the general center of
\((u,Y/H_1)\), order at least \(b_j\) forces
\(P_\gamma\) to be divisible by \(u^{b_j-|\gamma|}\) whenever
\(|\gamma|<b_j\). Otherwise its first nonzero reduction would be a
nonzero polynomial over \(\kappa\), which stays nonzero in
\(\kappa(H_2/H_1,\ldots,H_n/H_1)\); in the associated graded ring
it would give a nonzero term of total \((u,Y/H_1)\)-degree less than
\(b_j\). The transverse monomials there are independent. Comparing
the internal polynomial coefficients over the base residue field
\(\kappa\) now gives Equation~\eqref{nv:coefficient-order}.
We compare the number of monomials with the rank estimate in
Equation~\eqref{nv:plus-rank}. The
ambient rank and the number of monomials of transverse degree \(\ell\)
are
\[
N_m=\binom{m+2n}{2n},\qquad
N_{m,\ell}=\binom{\ell+n}{n}
\binom{m-\ell+n-1}{n-1}.
\]
For fixed \(\delta\in(0,1)\), let \(H_m(\delta)\) count the monomials
with \(\ell>(1-\delta)m\). Summing instead over their internal
degree gives
\[
H_m(\delta)
\le \binom{m+n}{n}\binom{\lceil\delta m\rceil+n}{n},\qquad
\limsup_{m\to\infty}\frac{H_m(\delta)}{N_m}
\le\binom{2n}{n}\delta^n.
\]
On the other hand, \(w^+\ge w/2\) and
Equation~\eqref{nv:plus-rank} show that the approximation on \(E^+\)
retains at least half the rank fraction in
Equation~\eqref{nv:selected-rank-fraction}:
\[
\lim_{j\to\infty}\frac{R_j^+}{N_{js}}
=\frac{w^+}{s^{d-1}}
\ge\frac{w}{2s^{d-1}}\ge c_n>0.
\]
Fix \(\delta>0\), depending only on \(n\), so small that, for each fixed
\(q\) and its selected data, fewer than \(R_j^+/2\) monomials have transverse
degree greater than \((1-\delta)js\) for all sufficiently large divisible
\(j\). For each
remaining monomial, Equation~\eqref{nv:coefficient-order} and
Lemma~\ref{nv:incidence-pole} give
\[
\ord_u(c_{\alpha,\gamma})
\ge j(s-C_n)-(1-\delta)js
=j(\delta s-C_n)\ge\frac{\delta js}{2}.
\]
The last inequality holds once \(q\) is large, since
\(s\ge c_nA_q\to\infty\).
Over the same discrete valuation ring \(R\), the torsion-free sheaf
\(\mathcal F_j^+\) becomes a free module of rank \(R_j^+\). Write its
inclusion into the symmetric power as a matrix with the monomials
as rows. Every maximal minor uses at least \(R_j^+/2\) of the rows
whose coefficients have order at least \(\delta js/2\). All maximal
minors therefore have order at least
\(\delta jR_j^+s/4\). By the definition of \(h_j\), this proves
Equation~\eqref{nv:determinant-order-equation}. Dividing the determinant
map by \(\sigma_{J_+}^{h_j}\) gives a holomorphic map at every point of
codimension one: the coefficients have the required order along \(J_+\),
and its local equation is a unit away from \(J_+\). Hartogs' theorem
extends the divided map across the remaining set of codimension at least
two, giving the stated factorization through
\(\det\mathcal F_j^+(h_jJ_+)\).
\end{proof}
\begin{proof}[Completion of the proof of Theorem~\ref{nv:meromorphic}]
Fix \(q\) sufficiently large for the preceding lemmas, with its chosen
\(P,s,T\) and modifications. Choose a very general point \(x\) in the first
factor of \(X^2\). The slice of \(Z^+\) above \(x\) is
\[
S=\PP_Y(\OO_Y\oplus a^*\mathscr L),\qquad
a:Y=\Bl_xX\longrightarrow X,
\]
and \(J_+|_S\) is the pullback of \(F\). We choose \(x\) so that
Equation~\eqref{nv:slope-blowup}, Equation~\eqref{nv:point-bound}, the restrictions of
the currents below, and all determinant maps for divisible \(j\)
can be tested on this slice. These exclude at most countably many
proper analytic sets and the measure-zero exceptional sets for
current restrictions.
Write the restricted determinant line as
\[
(\det\mathcal F_j^+)|_S
=\pi_S^*\mathscr Q_j^+\otimes\OO_S(\ell_j^+),\qquad
Q_j^+=c_1(\mathscr Q_j^+).
\]
Factoring the common zero of order \(h_j\) from the determinant
map gives a nonzero map whose source on \(S\) is
\[
\pi_S^*(\mathscr Q_j^+\otimes\OO_Y(h_jF))
\otimes\OO_S(\ell_j^+).
\]
The plus sign of \(h_jF\) follows because division of the map by
the local equation of \(J_+^{h_j}\) enlarges its source from
\(\det\mathcal F_j^+\) to
\(\det\mathcal F_j^+\otimes\OO(h_jJ_+)\).
Its target embeds into the \(jsR_j^+\)-fold tensor of the
restriction of \(p_+^*\Omega_Z\). In particular we use the
cotangent bundle of the original \(Z\), with its restricted
filtration
\[
0\longrightarrow
\pi_S^*(\OO_Y^{\oplus n}\oplus a^*\Omega_X)
\longrightarrow (p_+^*\Omega_Z)|_S
\longrightarrow
\OO_S(-2)\otimes\pi_S^*a^*\mathscr L
\longrightarrow0.
\]
The same homogeneous-monomial calculation as in
Equation~\eqref{nv:determinant-bound-Z}, now applying
Equation~\eqref{nv:slope-blowup} to the second factor of \(X^2\), gives
\begin{equation}\label{nv:determinant-bound-slice}
\ell_j^+\le0,\qquad
Q_j^++h_j\{F\}
\le(jsR_j^+-\ell_j^+)a^*L.
\end{equation}
More explicitly, if the chosen graded term has \(i\) relative
factors, the pushed polynomial has degree
\(-2i-\ell_j^+\ge0\). Its exponent in \(a^*\mathscr L\), together
with the \(i\) relative factors and the cotangent order from the second
factor of \(X^2\), is at most \(jsR_j^+-\ell_j^+\). The cotangent
factors from the first factor of \(X^2\) are constant. This proves the
displayed inequality using
the tensors of \(a^*\Omega_X\), without introducing \(\Omega_Y\)
or \(\Omega_{Z^+}\).
Apply Lemma~\ref{nv:direct-image} to the decomposition in
Equation~\eqref{nv:plus-decomposition}. Equation~\eqref{nv:GRR-base}
shows that the limits of the restricted determinant components divided by
\(jR_j^+\) exist; denote them by \(Q^+\) and \(\ell^+\). Divide
Equation~\eqref{nv:determinant-bound-slice} by \(jR_j^+\), use
Equation~\eqref{nv:determinant-order-equation}, and pass to the closed
pseudo-effective cone. We obtain
\begin{equation}\label{nv:slice-upper}
\ell^+\le0,\qquad
Q^++c_ns\{F\}\le(s-\ell^+)a^*L.
\end{equation}
The pseudo-effective class in Equation~\eqref{nv:GRR-base}, restricted to the
very general slice \(S\), is
\begin{equation}\label{nv:slice-psef}
\pi_S^*(Q^++2a^*P)+(\ell^++2q)\xi\ge0.
\end{equation}
Testing on a general vertical line gives \(\ell^++2q\ge0\).
Let \(A\subset S\) be the axis of
\(\PP_Y(\OO_Y\oplus a^*\mathscr L)\) with divisor class \(\{A\}=\xi\).
On \(A\simeq Y\), its normal class is \(\{A\}|_A=\xi|_A=a^*L\ge0\).
Add an arbitrarily small Kähler class to the class in
Equation~\eqref{nv:slice-psef}, and choose a Kähler current with analytic
singularities in the resulting big class. Remove its generic divisorial
pole along \(A\) and restrict
the residual current to \(A\). Adding back the removed nonnegative multiple
of \(a^*L\) preserves pseudo-effectivity. Passing to the limit gives
\begin{equation}\label{nv:slice-lower}
Q^++2a^*P+(\ell^++2q)a^*L\ge0.
\end{equation}
The slope bound in Equation~\eqref{nv:slice-upper} supplies the
pseudo-effective class \((s-\ell^+)a^*L-Q^+-c_ns\{F\}\). Adding it to
Equation~\eqref{nv:slice-lower} cancels the entire normalized determinant
class restricted to the axis, \(Q^++\ell^+a^*L\), leaving
\[
2a^*P+(s+2q)a^*L-c_ns\{F\}\ge0.
\]
Since \(L\le P/r\), the point bound in Equation~\eqref{nv:point-bound} implies
\begin{equation}\label{nv:contradiction}
\frac{c_ns}{\,2+(s+2q)/r\,}\le C_n.
\end{equation}
The denominator is bounded independently of \(q\), because
\(s\le C_nq^{1/d}\) and \(r\ge q^2\). The numerator tends to
infinity, because \(s\ge c_nq^{1/d}\). Equation~\eqref{nv:contradiction}
is impossible for arbitrarily large \(q\).
For each \(q\), the choice of \(t(q)\), the rational \(s\), the current
\(T\), and the modifications precedes the limit in \(j\). The monomial
cutoff \(\delta\) depends only on \(n\), while the required lower bound for
divisible \(j\) may depend on the fixed data. We choose the very general
points after those data are fixed.
Thus every simultaneous general-point test above involves at most
countably many conditions, and no bound depends on the point.
The contradiction disproves the contrary hypothesis and proves
Theorem~\ref{nv:meromorphic}.
\end{proof}
==== END Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//06-simple-nonvanishing.tex ====
==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//07-signed-rigidity.tex ====
\section{Signed rigidity on simple spaces}
\label{sec:signed}
The meromorphic section furnished by Theorem~\ref{nv:meromorphic} need
not have an effective divisor. We therefore need a boundary argument
that retains both its zeroes and its poles. The result below supplies
that argument.
\begin{theorem}[Signed rigidity]\label{signed:torsion}
Let \((X,D)\) be a dlt pair on a normal irreducible compact Kähler
space, and suppose that the pair has the resolution property of
Definition~\ref{def:lc-strata-resolution}. Suppose also that \(X\) is
globally \(\Q\)-factorial, that \(a(X)=0\), that \(X\) is simple,
and that \(D=\sum_{i=1}^sD_i\) is reduced. Put \(L=K_X+D\), and
assume the following.
\begin{enumerate}
\item The rational line \(L\) is analytically nef and has an actual
rational linear equivalence
\[
L\sim_{\Q}G=\sum_{i=1}^s a_iD_i,\qquad a_i\in\Q.
\]
\item For some real \(c>0\), the pullback of the class
\(\{L-cD\}\) to a projective resolution with smooth compact
Kähler source is pseudo-effective.
\item The rational holomorphic line \(L|_D\) is semiample on the
whole reduced analytic space \(D\): some Cartier multiple is
generated at every point of \(D\).
\end{enumerate}
Then \(L\) is torsion. More precisely, for some positive integer
\(q\), the holomorphic line \(\OO_X(qL)\) is isomorphic to
\(\OO_X\).
\end{theorem}
To apply this theorem in the proof of Proposition~\ref{simple:case},
Theorem~\ref{nv:meromorphic} supplies a signed representative of the
canonical bundle. After resolving, we enlarge the support of the
transformed boundary and this signed canonical divisor to a reduced SNC
divisor; the resulting adjoint remains pseudo-effective.
Corollary~\ref{prog:signed-nef} gives an ordinary dlt nef model
\((X_{\mathrm{nef}},D_{\mathrm{nef}})\) whose adjoint
\(L_{\mathrm{nef}}=K_{X_{\mathrm{nef}}}+D_{\mathrm{nef}}\) has a
signed representative supported on \(D_{\mathrm{nef}}\), supplying
the first hypothesis. Theorem~\ref{floor:generation} supplies
semiampleness on the whole reduced \(D_{\mathrm{nef}}\). On a common
smooth resolution, simplicity excludes uniruledness, so Ou's criterion
\cite[Theorem~1.1]{Ou2025} and exceptional translation for the canonical
comparison give pseudo-effectivity of the pullback of
\(K_{X_{\mathrm{nef}}}=L_{\mathrm{nef}}-D_{\mathrm{nef}}\). Thus the
second hypothesis holds with \(c=1\). Once signed rigidity makes the
actual line \(L_{\mathrm{nef}}\) torsion, the program comparison and
Lemma~\ref{bir:unique-current} let us subtract the added boundary and give
the required decomposition for the original adjoint. The final subsection
verifies these steps.
The signed construction in \cite[Proposition~3.3]{LI} is the source
of the root and separation operations used below. The lifting
method is that of \cite[Sections~10--12]{BL}.
We will establish the analytic form of the signed construction and
the change to the lifting argument caused by its residual poles.
The numerical argument first singles out positive-dimensional fibers in
the positive boundary. We then lift those fibers through every finite
boundary neighborhood and deform them to compact subspaces outside the
boundary. An argument using the cycle space and Baire's theorem turns these
deformations into a covering family, which simplicity excludes.
\subsection{The boundary detected by the nef class}
If \(D=0\), the signed equivalence already gives
\(L\sim_{\Q}0\). We may therefore assume in the constructions
below that \(D\ne0\); in particular \(\dim X>0\).
Fix the data of Theorem~\ref{signed:torsion}, and write
\[
G=G_+-G_-,\qquad D_0=\sum_{a_i=0}D_i,
\]
where \(G_+\) and \(G_-\) are effective with disjoint prime supports.
Choose a positive integer \(m\) so that \(mG_+\), \(mG_-\),
\(mD_0\), and \(mL\) are Cartier, the given rational equivalence
induces a fixed isomorphism
\[
N_0:=\OO_X(mG)\simeq\OO_X(mL),
\]
and \(N_0|_D\) is generated. Its sections give a morphism and an
actual line isomorphism
\begin{equation}\label{signed:boundary-map}
\phi:D\longrightarrow P=\PP^b,\qquad
N_0|_D\simeq\phi^*\OO_P(1).
\end{equation}
Let \(n=\dim X\). Choose a projective resolution
\(\mu:\widetilde X\to X\) as in the second hypothesis of the theorem,
and a Kähler class \(\omega\) on \(\widetilde X\). The numerical
dimension of the nef pullback is
\[
v=\nu(L):=\max\bigl\{k\in\{0,\ldots,n\}:
(\mu^*\{L\})^k\omega^{n-k}>0\bigr\}.
\]
The next argument is the Kähler form of
\cite[Lemma~3.2]{LI}.
\begin{lemma}\label{signed:boundary-geometry}
If \(v=0\), then \(D=0\) and \(L\sim_{\Q}0\). If \(v=n>0\),
then \(a(X)=n\). If \(0<v<n\) and \(r=v-1\), then
\[
\dim\phi(D_i)\leq r\quad\text{for every }i,
\qquad
\max_{a_i>0}\dim\phi(D_i)=r,
\]
and
\begin{equation}\label{signed:small-intersections}
\dim\phi\bigl(\Supp G_+\cap(\Supp G_-\cup D_0)\bigr)<r.
\end{equation}
For \(r=0\), the intersection in
Equation~\eqref{signed:small-intersections} is empty.
\end{lemma}
\begin{proof}
Choose a Kähler class \(\theta\) on \(X\). We use Cartier
intersection products downstairs, computed on a resolution by
projection formula. These products detect \(v\). Indeed, on the
chosen resolution the class \(h=\mu^*\theta\) is nef and big.
Choose \(C,\epsilon>0\) such that
\(C\omega-h\) is nef and \(h-\epsilon\omega\) is
pseudo-effective. Pairing these two inequalities successively with
products of the nef classes \(\mu^*\{L\}\), \(h\), and
\(\omega\) shows that
\[
(\mu^*\{L\})^k h^{n-k}>0
\quad\Longleftrightarrow\quad
(\mu^*\{L\})^k\omega^{n-k}>0.
\]
We may therefore use \(\theta\) in all the tests defining \(v\).
Suppose first that \(v<n\). The pseudo-effective pullback in the
theorem can be paired with nef products, so
\begin{equation}\label{signed:boundary-zero}
0\leq (\{L\}-c\{D\})\{L\}^{v}\theta^{n-v-1}
=-c\sum_i\{D_i\}\{L\}^{v}\theta^{n-v-1}.
\end{equation}
Each summand on the right is nonnegative. For example, pull the
generated line in Equation~\eqref{signed:boundary-map} and
\(\theta\) to a resolution of \(D_i\); their representatives are
semipositive and positive at general smooth points, respectively.
Thus every summand is zero. When \(v=0\), a nonzero effective
divisor has strictly positive \(\theta^{n-1}\)-degree. Hence
\(D=0\), and the signed equivalence gives \(L\sim_{\Q}0\).
Now let \(0<v<n\). For any irreducible effective cycle \(V\) in
\(D\), the mixed degree
\(\{L\}^k\theta^{\dim V-k}\cdot V\) is positive exactly when
\(\dim\phi(V)\geq k\). To see this, resolve \(V\) and use a
Fubini--Study representative for \(m\{L\}|_V\). Its generic rank
is \(\dim\phi(V)\), and its wedge with the remaining Kähler
factors is positive on a nonempty open set exactly in the asserted
range. The vanishing in Equation~\eqref{signed:boundary-zero}
therefore gives \(\dim\phi(D_i)\leq r\). On the other hand,
\begin{equation}\label{signed:positive-attainment}
0<\{L\}^{v}\theta^{n-v}
=\sum_i a_i\{D_i\}\{L\}^{r}\theta^{n-v}.
\end{equation}
All the component tests here are nonnegative, so a component with
\(a_i>0\) attains dimension \(r\).
It remains to separate its intersections from a general fiber.
On \(H^{1,1}_{\BC}(\widetilde X,\R)\) consider the form
\[
q(\alpha,\beta)=
\alpha\beta(\mu^*\{L\})^r h^{n-r-2},\qquad
Q_{ij}=q(\mu^*\{D_i\},\mu^*\{D_j\}).
\]
The mixed Kähler Hodge index theorem, followed by approximation of
the nef factors by Kähler classes, says that \(q\) has at most one
positive direction; see \cite[Theorems~A and C]{DinhNguyen}. Put
\(l=\mu^*\{L\}\). The definition of \(v\) and
Equation~\eqref{signed:boundary-zero} give
\[
q(l,l)=0,\qquad q(l,\mu^*\{D_i\})=0,
\qquad q(l,h)=\{L\}^{v}\theta^{n-v}>0.
\]
Linear algebra now makes \(q\) negative semidefinite on
\(l^\perp\): a positive vector in \(l^\perp\), together with a
suitable vector in the span of \(l,h\), would give two positive
directions. In particular \(Q\) is negative semidefinite.
For \(i\ne j\), one has \(Q_{ij}\geq0\). This assertion uses
the effective intersection cycle of the distinct \(\Q\)-Cartier
divisors \(D_i,D_j\) on \(X\), and projection formula. It does
not require their total transforms to intersect effectively. The
two Cartier equations have a proper intersection: the dlt ambient
space is klt, hence Cohen--Macaulay, after dropping the reduced
boundary, and the distinct divisor equations form a regular
sequence. The intersection cycle is consequently pure of
codimension two, and each of its mixed nef degrees is nonnegative.
Write the coefficient vector as
\(\mathbf a=\mathbf a^+-\mathbf a^-\), with nonnegative vectors
of disjoint supports. The signed equivalence and orthogonality to
\(l\) give \(Q\mathbf a=0\). Thus
\[
Q(\mathbf a^+,\mathbf a^+)
=Q(\mathbf a^+,\mathbf a^-)\geq0.
\]
Negative semidefiniteness forces equality and
\(Q\mathbf a^+=0\). For every \(j\) with \(a_j\leq0\), its
row is a sum of nonnegative off-diagonal terms:
\[
0=\sum_{a_i>0}a_iQ_{ji}.
\]
Each term is zero. Applying the generated-line degree test to the
effective intersection cycles proves
Equation~\eqref{signed:small-intersections}. If \(r=0\), any
nonempty intersection would have positive \(\theta^{n-2}\)-degree,
so these intersections are empty.
Finally, if \(v=n>0\), the nef volume criterion makes the rational
line on \(\widetilde X\) big
\cite[Theorem~0.5]{DemaillyPaun2004}. A big holomorphic line on a
compact Kähler manifold has maximal Iitaka dimension. Hence
\(a(\widetilde X)=n\), and birational invariance gives \(a(X)=n\).
\end{proof}
Thus, when \(0<v<n\), a positive component attaining image dimension
\(r=v-1\) has general fibers of dimension \((n-1)-r=n-v>0\).
Equation~\eqref{signed:small-intersections} makes those fibers disjoint
from \(\Supp G_-\cup D_0\). The next construction prepares their normal
directions and adjunction for infinitesimal lifting.
\subsection{Local roots and residue with residual poles}
We work for the rest of the signed argument in the range
\(0<v<n\), with \(r=v-1\). Choose \(\ell>1\) divisible by \(m\)
and by every nonzero integer \(|ma_i|\), and put
\(a=\ell/m\), a positive integer.
\begin{lemma}[Analytic signed charts]\label{signed:root-charts}
For each \(t\in P\), after shrinking an open neighborhood
\(U\subset P\), choose a line \(R_U\) together with an isomorphism
\(R_U^\ell\simeq\OO_P(1)|_U\). A comparison of two such roots on an
overlap is power-compatible if its \(\ell\)-th power commutes with these
isomorphisms. If
\(t\notin\phi(\Supp G_+)\), one may take \(U\) disjoint from this
closed image and set \(Z_U=S=T=\varnothing\); the assertions below are
then vacuous for the unique maps from the empty spaces. If
\(t\in\phi(\Supp G_+)\), there are a Hausdorff normal
analytic space \(Z_U\) of pure dimension \(n\), considered near a
reduced Cartier divisor \(S\) of pure dimension \(n-1\), a
reduced Weil divisor \(T\) with no component in common
with \(S\), and maps
\[
\pi:Z_U\longrightarrow X,\qquad g:S\longrightarrow U
\]
with the following properties.
\begin{enumerate}
\item The map \(g\) is proper, and \(\pi|_S\) is finite over
\(D_U=\phi^{-1}(U)\), with \(g=\phi\pi|_S\). Its image covers
the positive components over \(U\), and
\[
\OO_S(S)\simeq g^*R_U.
\]
The image of \(S\cap T\) lies in
\(\phi(\Supp G_+\cap(\Supp G_-\cup D_0))\).
Near \(S\), the underlying sets satisfy
\[
\pi^{-1}(|D|)=|S|\cup|T|.
\]
\item The pair \((Z_U,S+T)\) is log canonical, and there is a fixed
actual isomorphism of divisorial sheaves
\begin{equation}\label{signed:adjoint-chart}
\tau:\OO_{Z_U}(aS)\xrightarrow{\ \simeq\ }\omega_{Z_U}(S+T).
\end{equation}
The spaces \(Z_U\) and \(S\) are Cohen--Macaulay. The support
of \(T\) is locally set-theoretically principal.
Off \(T\), the space \(Z_U\) is klt and its canonical sheaf is
invertible. The map \(\pi\) is quasi-finite near \(S\setminus T\).
\item Projective log resolutions of the total divisor data can be
chosen with Kähler forms on neighborhoods of the compact sets over
each compact fiber of \(g\). Power-compatible changes of the
boundary root extend to isomorphisms of ambient germs that preserve
the full ideal \(\OO_{Z_U}(-S)\), the reduced divisor \(T\), and
Equation~\eqref{signed:adjoint-chart}. Resolutions may be chosen
functorially for these germ isomorphisms.
\end{enumerate}
Here the nonempty space \(Z_U\) is a neighborhood of the whole compact
fiber under consideration; no extension of \(g\) to \(Z_U\) is asserted.
\end{lemma}
\begin{proof}
The empty branch follows by shrinking away from the closed image
\(\phi(\Supp G_+)\). We construct the nonempty charts in four stages:
take roots and trivialize the remaining adjoint torsion, separate the
positive and negative root divisors, retain the root line that controls the
normal direction, and pass to an ordinary neighborhood of the compact fiber.
First consider an analytic open set on which the Cartier lines of
\(mG_+\), \(mG_-\), and \(mD_0\) are trivial, and let
\(f_+,f_-,f_0\) be the functions representing their sections. Normalize
the entire finite space
\[
\{y_+^\ell=f_+,\qquad y_-^\ell=f_-,\qquad y_0^\ell=f_0\},
\]
retaining all its components and the lifted action of \(\mu_\ell^3\)
that multiplies the three root coordinates. On an overlap, the functions
\(f_+,f_-,f_0\) change by holomorphic units. Local \(\ell\)-th roots of
those units give comparisons of the finite spaces that lift uniquely to
their normalizations; different choices differ by \(\mu_\ell^3\).
The quotient stacks of these normalized charts by \(\mu_\ell^3\) glue to
the simultaneous normalized root stack over \(X\), which we denote by
\(\mathcal X_{\mathrm{rt}}\). The zero divisors
of the three root sections are denoted by \(S_+,S_-,S_0\). At a
generic prime of multiplicity \(b\), a normalized chart of
\(y^\ell=x^b\) has ramification index \(\ell/b\) and
\(\ord(y)=1\). Here \(b=|ma_i|\) or \(b=m\), so it divides
\(\ell\). Thus each root divisor is generically reduced and
Cartier. A Cartier divisor on a normal space satisfies Serre's
condition \(S_1\), so it is reduced. Log ramification with the
full reduced boundary gives a log canonical pair and the rational
equivalence
\[
K_{\mathcal X_{\mathrm{rt}}}+S_++S_-+S_0
\sim_{\Q}a(S_+-S_-).
\]
The ambient charts of \(\mathcal X_{\mathrm{rt}}\) are klt. Off the
boundary this follows from
the klt complement downstairs and the unramified charts. For a
place centered in the Cartier boundary, dropping that boundary
increases its log discrepancy by its strictly positive order;
this also makes every zero-discrepancy place positive.
To turn the rational adjoint equivalence into an actual isomorphism,
consider the integral reflexive sheaf on \(\mathcal X_{\mathrm{rt}}\)
\[
\mathcal F=
\bigl(\omega_{\mathcal X_{\mathrm{rt}}}(S_++S_-+S_0)\otimes
\OO_{\mathcal X_{\mathrm{rt}}}(-a(S_+-S_-))\bigr)^{**}.
\]
It is torsion. Choose a periodicity
\(\mathcal F^{[q]}\simeq\OO\), take the relative spectrum of its
reflexive-power algebra, and normalize; write \(\rho_{\mathrm{ind}}\)
for this finite map, and retain \(S_+,S_-,S_0\) for the pulled-back
root divisors on the cover. At codimension one this
is a cover \(w^q=u\) for a unit \(u\), hence is unramified.
Log canonicity, the klt ambient property, and reduced Cartier
root divisors persist. The tautological evaluation of this
algebra trivializes the reflexive pullback
\((\rho_{\mathrm{ind}}^*\mathcal F)^{**}\): it does so in
codimension one, and the identity then extends reflexively.
The evaluation is a sheaf morphism on the cover stack itself,
so the resulting actual divisorial isomorphism
\[
\omega(S_++S_-+S_0)\simeq\OO\bigl(a(S_+-S_-)\bigr)
\]
on the cover is equivariant on every atlas.
Next separate the positive and negative root divisors. Let
\(\mathcal X_{\mathrm{sep}}\) be the normalization of the blowup of the
ideal generated by their equations on the cover, and write \(p_1\) for its
map. This blowup embeds in a relative
\(\PP^1\); its normalization still has fibers of dimension at
most one. Each exceptional prime therefore lies over a
codimension-two positive--negative intersection. At its generic
point the original dlt pair is a two-branch SNC pair. After
extracting roots of units the normalized Kummer chart there has
smooth coordinates
\(x=u^{\ell/b_+}\), \(z=w^{\ell/b_-}\), where \(b_+\) and
\(b_-\) are the two corresponding multiplicities. On this chart
\(\mathcal F\) is a line and its periodicity cover is unramified.
The two-coordinate blowup calculation consequently applies at
every exceptional generic point. If \(E\) denotes its
exceptional divisor, it gives reduced Cartier divisors
\[
E,\qquad S'_+=p_1^*S_+-E,\qquad S'_-=p_1^*S_--E,
\]
with \(S'_+\cap S'_-=\varnothing\). No such two-branch stratum
is contained in \(S_0\), so \(p_1^*S_0\) is reduced and has no
exceptional component. The full boundary
\(S'_++S'_-+E+p_1^*S_0\) is crepant and reduced. Its
codimension-one equality with the pullback adjoint gives the
discrepancy equality for every further valuation, hence log
canonicity. The ambient charts after this blowup are still
klt. Off the full boundary the blowup is an isomorphism to
the old klt complement. A place of zero pair discrepancy
centered in that boundary gains its strictly positive order
when the effective Cartier boundary is dropped, while a
place of positive pair discrepancy remains positive. The
strict divisor \(S'_+\) is finite over \(S_+\):
before normalization it lies in one section of the projective
ratio coordinate, so its proper map has finite fibers, and
normalization is finite.
On \(\mathcal X_{\mathrm{sep}}\), put \(S=S'_+\) and, temporarily,
\(T=S'_-+E+p_1^*S_0\). Write
\(\pi_{\mathrm{sep}}:\mathcal X_{\mathrm{sep}}\to X\) for the composite
map, and define the retained root line
\[
\mathcal R:=\OO_{\mathcal X_{\mathrm{sep}}}(S'_+-S'_-),
\qquad \mathcal R^\ell\simeq\pi_{\mathrm{sep}}^*N_0.
\]
Near \(S\), the negative strict root section is a unit. The fixed adjoint
isomorphism becomes
\(\omega_{\mathcal X_{\mathrm{sep}}}(S+T)
\simeq\OO_{\mathcal X_{\mathrm{sep}}}(aS)\), and the positive root
section divided by that negative unit section is a section of
\(\mathcal R\) with zero divisor \(S\). This line and section determine
the Cartier divisor \(S\) and its normal line; they must survive passage
to an ordinary chart.
The pair consisting of \(\mathcal R\) and its displayed power isomorphism
defines a map to the gerbe \(\sqrt[\ell]{N_0}\), which parametrizes
\(\ell\)-th roots of \(N_0\). Take the relative coarse space over this
gerbe. On an atlas this quotients by the finite relative inertia kernel,
the subgroup of each stabilizer acting trivially on \(\mathcal R\).
Thus the quotient retains the root character of \(\mathcal R\). The
deck transformations of the representable periodicity cover
are not this inertia kernel and are not removed.
Both \(\mathcal R\) and its section descend through the kernel, so the
reduced image \(S\) remains Cartier and its full ideal descends. We retain
\(S,T\) for the reduced images. A finite-group norm of
a local equation of the upstairs \(T\) has precisely its image
as zero set. Thus the reduced image \(T\) is locally
set-theoretically principal; it need not be Cartier. All
divisorial ramification lies on the full boundary. Log
ramification identifies the invariant log dualizing sheaf in
codimension one with the downstairs one. Taking reflexive
hulls and invariants of the equivariant adjoint isomorphism
gives an actual descended isomorphism \(\OO(aS)\simeq\omega(S+T)\).
Its ordinary pullback below is Equation~\eqref{signed:adjoint-chart}.
Outside \(S+T\), the local
finite quotient is quasi-étale, so its klt upstairs complement gives
a klt downstairs complement by finite discrepancy comparison. Away from
\(T\), dropping the Cartier divisor \(S\) from the lc pair
increases the log discrepancy of every place centered in \(S\) by
its strictly positive order. Thus the quotient charts are klt away from
\(T\), and the same isomorphism makes their canonical sheaves invertible
there. Off \(T\) the separating blowup is an
isomorphism, while the other operations are finite; hence
the composite map to \(X\) is quasi-finite near \(S\setminus T\).
The same quotient preserves Cohen--Macaulayness of both the
ambient space and \(S\). Before the quotient the ambient
space is klt and hence Cohen--Macaulay, and \(S\) is Cartier
there. Locally its finite quotient ring, and the quotient
ring of \(S\), are invariant direct summands of those finite
Cohen--Macaulay rings. A system of parameters downstairs is
one upstairs; the vanishing of lower local cohomology upstairs
and the invariant projection give its vanishing for the
downstairs direct summand. This proves the stated depth.
Removal of the relative inertia kernel also makes the strict
positive divisor finite and representable over
\[
D\times_P\sqrt[\ell]{\OO_P(1)}.
\]
Its root line is \(\OO_S(S)\), and its image covers
\(\Supp G_+\). The intersection \(S\cap T\) maps to the
positive--negative or positive--zero intersections: the only
new component is the separating exceptional divisor. Before the quotient,
the inverse image of \(|D|\) is the support of
\(S'_++S'_-+E+p_1^*S_0\). The finite quotient sends this full support
to \(|S|\cup|T|\), and the equality of underlying sets persists under the
ordinary base change below. The image of \(S\cap T\) in \(P\) consequently
has dimension \(<r\), by Lemma~\ref{signed:boundary-geometry}.
Finally we realize these stack data on an ordinary analytic neighborhood
of the whole compact fiber. We use the dimension-free root-neighborhood
result \cite[Lemma~10.1]{BL}.
For a compact analytic subspace \(F\) of a Hausdorff analytic
space, that lemma extends a specified \(\ell\)-th root of a
line on \(F\) to a root on a neighborhood. It also extends
a specified power-compatible comparison of two roots uniquely
as a germ about \(F\). Its proof is the analytic Kummer
sequence and continuity of the cohomology of \(\mu_\ell\)
over neighborhoods of a compact set, in degrees two, one,
and zero.
Apply it to \(F=\phi^{-1}(t)_{\mathrm{red}}\subset X\) with
the root prescribed by \(R_U\) in
Equation~\eqref{signed:boundary-map}. Apply the comparison
clause also inside \(D\). Properness of \(\phi\) allows a
shrink of \(U\) for which the comparison is defined along all
of \(D_U\): remove the closed image of its complement.
Pulling the relative coarse construction across this root
over the whole neighborhood of \(F\) gives the ordinary Hausdorff space
\(Z_U\). Here the chosen root defines a map from that neighborhood to
\(\sqrt[\ell]{N_0}\), and the relative quotient is representable over
the gerbe, so this pullback is an ordinary space. Finite invariant algebras
glue its local quotient charts. The preceding finite representable map
becomes the finite map \(S\to D_U\), so \(g\) is proper. The retained
line on \(S\) becomes \(g^*R_U\), giving \(\OO_S(S)\simeq g^*R_U\).
Normalization and the periodicity cover are finite also in the analytic
category. The separating blowup is projective. A common
power of its equivariant relative ample line kills the bounded
finite stabilizer characters and descends to the coarse
quotient; relative ampleness is checked on fibers after the
finite pullback. Thus these operations and a projective log
resolution admit relative ample metrics. Adding a sufficiently
large pullback of the ambient Kähler form gives Kähler forms
near the compact sets in question. Finally, power-compatible
root comparisons identify the full constructions as ambient
germs. Smooth-functorial projective resolution of the same
ordered divisor data preserves those identifications. This
proves all assertions.
\end{proof}
The residue assertions below are vacuous for an empty chart. Fix a
nonempty chart of Lemma~\ref{signed:root-charts}, write \(Z=Z_U\), and
choose a projective log resolution \(p:\widehat Z\to Z\). Write
\begin{equation}\label{signed:resolved-divisors}
\begin{gathered}
D_S=p^*S,\qquad H=(D_S)_{\mathrm{red}},\qquad
C=(p^{-1}T)_{\mathrm{red}},\\
A=\sum_{\substack{E\text{ component of }C\\E\text{ not a component of }H}}E.
\end{gathered}
\end{equation}
The reduced divisor \(H+A\) has simple normal crossings.
The divisor \(A\) may meet \(H\); it has no component in
common with \(H\). Thus \(A\) retains the components over \(T\)
whose poles are not already counted by the reduced divisor \(H\).
Let \(p_H=p|_H:H\to S\).
The maps that will be used in the residue and lifting arguments are
\[
\begin{tikzcd}[column sep=large,row sep=large]
H \arrow[r,hook] \arrow[d,"p_H"'] &
\widehat Z \arrow[d,"p"]\\
S \arrow[r,hook] \arrow[d,"\pi|_S"'] &
Z \arrow[d,"\pi"]\\
D_U \arrow[r,hook] \arrow[d,"\phi"'] & X\\
U &
\end{tikzcd}
\qquad g=\phi\pi|_S,\qquad h=g p_H.
\]
Only the support spaces in the left column map to \(U\).
\begin{lemma}[Residue with residual poles]\label{signed:residue}
For every \(i\geq0\), the fixed adjoint isomorphism gives a
split injection
\begin{equation}\label{signed:residue-injection}
R^ig_*\OO_S(aS)\lhook\joinrel\longrightarrow
R^ih_*\omega_H(A).
\end{equation}
The injection and its retraction commute with the root germ
comparisons. Under a product with a manifold they are the
relative canonical constructions; for absolute canonical
sheaves they are tensored with the canonical line of that
additional factor.
\end{lemma}
\begin{proof}
The quotient retraction follows the construction of
\cite[Proposition~10.4]{BL}; the residual divisor changes
the comparison sheaf, which we now compute.
Apply Equation~\eqref{signed:adjoint-chart} to the rational
section \(1\) of \(\OO_Z(aS)\), and denote the resulting
fixed meromorphic adjoint form again by \(\tau\). Its pullback has at
most a logarithmic pole along \(H+A\). This is the log
discrepancy inequality for \((Z,S+T)\); away from the inverse
boundary, the klt ambient property and integrality of the
orders remove a possible pole. We obtain
\[
p^*\OO_Z(aS)\longrightarrow\omega_{\widehat Z}(H+A).
\]
Residue on the reduced SNC divisor \(H\) gives a morphism
\begin{equation}\label{signed:derived-insertion}
\OO_S(aS)[0]\longrightarrow R(p_H)_*\omega_H(A)
\end{equation}
in the derived category of \(S\).
The insertion counts a component common to \(C\) and \(H\) in the
logarithmic divisor \(H\). The retraction compares with \(\omega_Z(T)\),
so it must retain every component over \(T\); it therefore uses the full
divisor \(C\). We claim
\begin{equation}\label{signed:full-residual-comparison}
Rp_*\omega_{\widehat Z}(C)=\omega_Z(T)[0].
\end{equation}
The right side is invertible, since it is
\(\omega_Z(S+T)(-S)\). A local frame pulls back with at
most simple poles over \(T\): subtracting the effective
Cartier \(S\) from the lc boundary leaves the required lc
order inequality. At a prime not over \(T\), the target is
klt with invertible canonical sheaf, so its integral order is
nonnegative. This proves the inclusion of the right side in
\(p_*\omega_{\widehat Z}(C)\). Conversely, orders at the
strict transforms give at most a simple pole on each prime of
\(T\) and none on any other prime. Normality and reflexivity
give the opposite inclusion.
For higher direct images this can be checked locally on \(Z\).
Choose an effective Cartier divisor \(V\) whose support is
\(T\), and a sufficiently small positive rational \(\epsilon\)
so that \(C=\lceil\epsilon p^*V\rceil\). There are only
finitely many relevant components after shrinking about a
compact fiber. The fractional support is SNC, and
\(\epsilon p^*V\) is \(p\)-nef and \(p\)-big. The latter
can be seen directly after shrinking the target to a Stein
open. Choose a \(p\)-ample Cartier divisor \(H_p\). The
coherent sheaf \(p_*\OO_{\widehat Z}(-H_p)\) has generic
rank one because \(p\) is birational. Cartan's Theorem~A
supplies a section nonzero at the generic points, giving an
effective divisor \(B_p\sim-H_p\) on the inverse image.
Thus \(\epsilon p^*V\sim H_p+(B_p+\epsilon p^*V)\), a
relatively ample divisor plus an effective one, which is
\(p\)-big.
Relative Kawamata--Viehweg vanishing for projective analytic
morphisms with smooth source
\cite[Theorem~5.1]{FujinoAnalyticBCHM} gives
\(R^ip_*\omega_{\widehat Z}(C)=0\) for \(i>0\). This
proves Equation~\eqref{signed:full-residual-comparison}.
Projection formula gives the same comparison after adding
\(D_S=p^*S\). The quotient sequence therefore identifies
\[
\begin{aligned}
R(p_D)_*\left(
\frac{\omega_{\widehat Z}(D_S+C)}{\omega_{\widehat Z}(C)}
\right)
&\simeq \omega_Z(S+T)|_S[0]\\
&\simeq \OO_S(aS)[0],
\end{aligned}
\]
where \(p_D:D_S\to S\) and the quotient is a sheaf on the
possibly nonreduced divisor \(D_S\). To justify the displayed
derived statement, first push it by the exact closed immersion
\(S\hookrightarrow Z\). There it is the quotient of the two
acyclic comparisons. Closed pushforward detects cohomology
sheaves, and canonical truncation yields the statement on \(S\).
There is a quotient morphism
\begin{equation}\label{signed:quotient-map}
\begin{aligned}
\omega_H(A)&=
\frac{\omega_{\widehat Z}(H+A)}{\omega_{\widehat Z}(A)}
\\
&\longrightarrow
\frac{\omega_{\widehat Z}(D_S+C)}{\omega_{\widehat Z}(C)}.
\end{aligned}
\end{equation}
Indeed \(H+A\leq D_S+C\) and \(A\leq C\). This morphism
need not be injective when \(C\) and \(H\) share a component.
Compose its derived pushforward with the preceding isomorphism.
The result retracts Equation~\eqref{signed:derived-insertion}:
at each generic point of \(S\), its composite is the identity
under the fixed residue convention. The composite is an
endomorphism of the invertible sheaf \(\OO_S(aS)\) in degree
zero; since \(S\) is reduced, agreement at every generic point
implies agreement everywhere. Applying \(Rg_*\) proves
Equation~\eqref{signed:residue-injection}. All maps use the
fixed adjoint isomorphism, divisor ideals, and residue, so they
commute with the germ comparisons. Relative canonical forms
under products, followed by wedging with the canonical frame
of the new factor, give the last assertion.
\end{proof}
\subsection{The localized Hodge argument}
The target in Equation~\eqref{signed:residue-injection} contains
the additional poles \(A\). The following version of the filtered
SNC calculation incorporates them. In its application, we embed \(H\)
by the graph of \(h\) in \(\widehat Z\times U\) and use the projection to
\(U\) as the ambient map. This projection is submersive, while its
restriction to the graph is the proper map \(h\); it requires no extension
of \(h\) to \(\widehat Z\). We will use the same graph construction after
a change of parameter. The parameter is projective only in the last
assertion below; the closed-stratum maps are proper Kähler maps.
\begin{proposition}\label{signed:localized-hodge}
Let \(q:\mathscr V\to Y\) be a holomorphic map of complex
manifolds, submersive near a reduced closed analytic subspace
\(i:H\hookrightarrow\mathscr V\) of pure dimension \(d\) and
codimension \(c_0>0\). Suppose \(h=q|_H\) is proper. Assume
that locally \(H\) is an SNC divisor in a smooth submanifold of
codimension \(c_0-1\), and that a reduced divisor \(A\) is
simultaneously SNC and transverse to those support equations.
Assume that the components \(H_i\) and \(A_j\) near \(H\) have
finite global smooth indexing, and that every closed stratum
\(H_I\cap A_J\), with \(I\ne\varnothing\), is smooth and proper
over \(Y\) and carries a global relative Kähler form. Empty or
disconnected strata are allowed.
Here \(H_I=\bigcap_{i\in I}H_i\) and
\(A_J=\bigcap_{j\in J}A_j\), with \(A_\varnothing=\mathscr V\).
Use right \(\mathscr D\)-modules and finite-pole local cohomology,
and put
\[
\mathcal K_A=\mathcal H^{c_0}_{[H]}(\omega_{\mathscr V})(*A).
\]
Its order filtration is generated from
\(F_0\mathcal K_A=i_*\omega_H(A)\), with \(F_p=0\) for \(p<0\).
For \(M^j=\mathcal H^j q_+\mathcal K_A\), the filtered direct
image is strict at every level, and \(M^j\) has a finite
filtration by submodules, strict for \(F\), whose quotients are
polarizable real pure Hodge modules. In particular,
\begin{equation}\label{signed:lowest-step}
F_0M^j=R^jh_*\omega_H(A)\lhook\joinrel\longrightarrow M^j,
\qquad F_pM^j=0\quad(p<0).
\end{equation}
Define the symbol morphism
\[
\sigma:F_0M^j\otimes T_Y\longrightarrow\gr^F_1M^j,
\qquad \sigma(m\otimes\xi)=[m\xi].
\]
If \(Y\) is smooth projective, then for every ample line \(N\) on \(Y\),
\begin{equation}\label{signed:symbol-vanishing}
\Hom_{\OO_Y}(N,\Ker\sigma)=0.
\end{equation}
This includes any coherent torsion in the kernel.
\end{proposition}
\begin{proof}
We verify the residual localization in the proof of
\cite[Proposition~11.1]{BL}. We first identify the lowest order step and
a filtration by the number of branch poles. Its graded pieces will be
dual constant modules on smooth strata, to which proper Kähler direct
image applies. The resulting strictness gives the injection at the lowest
filtration step,
and the graded de Rham complex on a projective base gives the symbol
vanishing.
At a point of \(H\), choose
simultaneous coordinates
\[
(z_1,\ldots,z_{c_0-1},x_1,\ldots,x_t,y_1,\ldots,y_s,\xi),
\quad
\mathscr I_H=(z_1,\ldots,z_{c_0-1},x_1\cdots x_t),
\quad A=(y_1\cdots y_s=0).
\]
Let \(\eta\) be the coordinate volume form. The localization
Čech complex for the displayed regular support equations has
cohomology only in degree \(c_0\). Localizing this cohomology
also in the \(y\)'s gives \(\mathcal K_A\). Successive finite
Taylor divisions give its unique additive polar normal forms:
finite sums of
\begin{equation}\label{signed:polar-form}
f(x_{I^c},y_{J^c},\xi)
\prod_{\nu=1}^{c_0-1}z_\nu^{-u_\nu}
\prod_{i\in I}x_i^{-v_i}
\prod_{j\in J}y_j^{-w_j}\eta,
\quad I\ne\varnothing,
\end{equation}
where every displayed order is positive and \(J\) is arbitrary,
including empty. The coefficient is independent of exactly
the variables occurring negatively. The normal form is an
additive statement, not an \(\OO_{\mathscr V}\)-linear
decomposition.
The simple fractions, with each displayed order equal to one,
are exactly the image of \(\omega_H(A)\). More explicitly,
their unreduced representatives are
\[
\frac{f\eta}
{z_1\cdots z_{c_0-1}(x_1\cdots x_t)(y_1\cdots y_s)}.
\]
The regular-equation residue identifies their numerators modulo
\(\mathscr I_H\) with the absolute dualizing sheaf of \(H\),
twisted by \(A\). This map is injective. If the fraction has
no remaining \(x\)-pole, Taylor division says that its
numerator modulo the \(z\)'s is divisible by \(x_1\cdots x_t\).
Localization in the \(y\)'s does not enlarge this kernel,
because each \(y_j\) is a nonzerodivisor modulo
\(\mathscr I_H\). This also proves the identification under
changes of coordinates, including the determinant of the
normal conormal frame.
Give a term of Equation~\eqref{signed:polar-form} the excess
\[
e=\sum_\nu(u_\nu-1)+\sum_{i\in I}(v_i-1)
+\sum_{j\in J}(w_j-1).
\]
The right canonical action of a coordinate derivative is minus
differentiation of the coefficient of \(\eta\). A derivative
in a denominator variable increases its order by one with a
nonzero scalar. Because the coefficient in the normal form
is independent of those variables, every term of excess
\(e\) is obtained from a simple term by \(e\) such derivatives.
Conversely, order \(p\) differentiations produce excess at
most \(p\). Thus \(F_p\mathcal K_A\) consists exactly of the
finite sums of excess at most \(p\). This is the good order
filtration generated from \(i_*\omega_H(A)\).
There is also an increasing filtration \(W\) by the total
number of branch poles. Intrinsically, its step of index
\(-d+k-1\) is the sum of the images of support modules for
subunions of the \(H_i\), localized along sublists of the
\(A_j\), for which the total number of selected branches is
at most \(k\). In the normal form it imposes
\(|I|+|J|\leq k\). The induced double filtration therefore
has
\begin{equation}\label{signed:localized-graded}
\gr^W_{-d+k-1}(\mathcal K_A,F)
\simeq
\bigoplus_{\substack{|I|+|J|=k\\ I\ne\varnothing}}
(i_{I,J})_+(\omega_{H_I\cap A_J},F^{(0)}).
\end{equation}
Here \(F^{(0)}_p\omega_{H_I\cap A_J}=0\) for \(p<0\) and is
\(\omega_{H_I\cap A_J}\) for \(p\geq0\).
Here \(A_\varnothing=\mathscr V\), and empty intersections
contribute zero. To check this formula, retain the polar type
using exactly \(I,J\). Its normal denominators are the
\(c_0-1\) equations \(z_\nu\), the \(|I|\) selected \(x\)'s,
and the \(|J|\) selected \(y\)'s. They are the regular
equations of the smooth stratum \(H_I\cap A_J\), whose
dimension is \(d-k+1\). Right closed transfer has no
codimension shift in \(F\), and the excess counts exactly its
normal derivative orders. Mayer--Vietoris boundary maps for
the \(x\)-subunions and the localization boundary for each
selected \(y\) identify these quotients intrinsically with
the support module of that intersection. For the latter
assertion, localization modulo the nonlocalized module in one
transverse \(y\) direction is its first local cohomology in
that direction. A fixed order of the component labels fixes
the residue signs, so the identifications glue. The smooth-
support graded modules are regular holonomic, and their finite
extension \(\mathcal K_A\) is regular holonomic as well.
The filtration \(W\) has a real realization. Tensor the
finite-pole de Rham comparisons for the regular support
coordinates with the open localization comparison for the
\(y\) coordinates, whose one-coordinate factor is the
cohomology of a punctured disk. These comparisons are
natural for all sublists. The resulting real support and
localization complexes complexify to the de Rham complexes
just computed. The latter are perverse by regular
holonomicity; exact and faithful complexification gives the
same assertion for the real complexes. Their real perverse
images then give \(W\).
On a graded stratum of dimension \(d-k+1\), the real object
is its dual constant object
\(\R^H[d-k+1](d-k+1)\). Its first right filtration step is
zero and its weight is \(-d+k-1\), in agreement with
Equation~\eqref{signed:localized-graded}. Residue signs and
constant normalizations preserve its real polarization. This
is the real comparison in \cite[Section~11.2]{BL}, with the
punctured-disk factors supplying the additional localization
boundaries.
The proper-support check also survives localization. If a
holomorphic germ \(f\) vanishes on reduced \(H\), then
\begin{equation}\label{signed:support-condition}
fF_p\mathcal K_A\subset F_{p-1}\mathcal K_A.
\end{equation}
For a generator \(z_\nu\) of \(\mathscr I_H\), multiplication
reduces its pole order or kills the class. Multiplication by
\(x_1\cdots x_t\) also lowers the excess or kills the class,
because the polar type always has \(I\ne\varnothing\).
The \(y\)-poles do not affect this argument. In particular
every graded stratum stays in the proper support \(H\), and
properness is needed only there.
We have now identified the order filtration, its real pure stratum pieces,
and the proper support of each level. We next pass these data to the base.
Apply Saito's constant-source Kähler direct-image theorem
\cite[Theorem~1 and Remark~1.2]{SaitoKahler} to each smooth
closed stratum in Equation~\eqref{signed:localized-graded},
using its stipulated relative Kähler form. The resulting
degree-\(j\) direct image of the step indexed by \(\lambda\)
is strict and polarizable real pure of weight \(\lambda+j\).
In the submersion neighborhood, the level-\(F_p\) relative
right de Rham complex has term
\[
F_{p-e}\mathcal K_A\otimes\bigwedge^e T_{\mathscr V/Y}
\quad\text{in degree }-e.
\]
The simultaneous excess and branch bounds show that passing
to a \(W\)-quotient commutes with taking each such level.
To prove strictness, use the finite \(W\) spectral sequence
\[
E_1^{-\lambda,j+\lambda}
=\mathcal H^j q_+\gr^W_\lambda\mathcal K_A
\quad\Longrightarrow\quad \mathcal H^j q_+\mathcal K_A.
\]
Its first-page terms and their level filtrations are strict
pure objects by the preceding application. The differentials
are real and filtration preserving, because the support and
localization comparisons are real and natural. The first
differential preserves weight, hence is a strict morphism of
pure Hodge modules; its kernels and cokernels remain pure.
A higher differential sends \((\lambda,j)\) to
\((\lambda-k,j+1)\) for \(k\geq2\) and strictly lowers
weight. Such a real filtered map is zero. On distinct strict
supports this follows from strict support; on the same dense
smooth support a real map preserves both Hodge filtrations,
and a source vector of type \((p,q)\) would have image in
\(F^p\cap\overline F^q\) of a pure object of smaller weight,
which is zero. Both the unfiltered and each level spectral
sequence therefore degenerate at the second page. Their
comparison is injective there. A least-\(W\)-step argument
on the finite abutment proves injection at every \(F\) level
and strictness of its induced \(W\) filtration. This is the
strict-direct-image argument of \cite[Proposition~11.1]{BL},
now justified for every graded term with \(A\)-poles.
For \(p<0\), the relative complex is zero. For \(p=0\), its
only term is \(F_0\mathcal K_A=i_*\omega_H(A)\) in degree
zero. The level injection gives
Equation~\eqref{signed:lowest-step}.
Suppose finally that \(Y\) is smooth projective. Each strict-
support pure quotient just obtained is of exactly the type in
\cite[Lemma~11.2]{BL}: a polarizable real pure direct-image
piece arising from a dual constant on a smooth Kähler source.
That lemma compares its actual right filtration with the
Sabbah--Schnell pure component extending the same generic
polarized real variation. Its proof uses uniqueness of the
intermediate extension and the reconstruction of \(F\) from
the canonical \(V\)-filtration, so it depends only on this
pure piece, not on the divisor presentation of its source.
It gives the negative-ample vanishing
\[
\mathbb H^u\bigl(Y,N^{-1}\otimes\gr^F_p\mathrm{DR}_Y Q\bigr)=0
\quad(u<0)
\]
for each pure quotient \(Q\), by
\cite[Theorem~16.3.10]{SabbahSchnell}. The finite strict
filtration extends this vanishing to \(M^j\). Since its
negative \(F\) levels vanish, its degree-one graded right
de Rham complex is exactly
\[
\gr^F_1\mathrm{DR}_Y M^j=
\left[F_0M^j\otimes T_Y\xrightarrow{\sigma}\gr^F_1M^j\right]
\quad\text{in degrees }-1,0.
\]
Its negative hypercohomology after tensoring by \(N^{-1}\)
is \(H^0(Y,N^{-1}\otimes\Ker\sigma)\). It vanishes, which
is Equation~\eqref{signed:symbol-vanishing}. This calculation
does not assume that the kernel is locally free.
\end{proof}
\subsection{Lifting every finite boundary neighborhood}
Return to the charts and resolution of
Lemma~\ref{signed:root-charts} and
Equation~\eqref{signed:resolved-divisors}. On an empty chart the sheaves
below are zero. On a nonempty chart put \(I=\OO_Z(-S)\), an invertible
ideal. For every integer \(j\)
and positive integer \(k\), put
\[
\mathcal A_j=I^j/I^{j+1}=\OO_S(-jS),\qquad
\mathcal T_{j,k}=I^j/I^{j+k}.
\]
These are sheaves on the underlying topological space of \(S\).
Pushforward by \(g\) has that meaning for \(\mathcal T_{j,k}\);
it does not require a morphism from a thickening to \(U\).
A local frame of \(R_U^{-1}\) gives a Laurent graded frame
\(u^j\) of \(\mathcal A_j\) for every \(j\). It is a frame
on the associated graded, not an extension of a frame to \(Z\).
\begin{proposition}[All finite lifting orders]\label{signed:all-orders}
For every root chart, every \(j\in\Z\), and every \(k\geq1\),
the map of sheaves of complex vector spaces on \(U\)
\begin{equation}\label{signed:all-order-surjection}
g_*\mathcal T_{j,k+1}\longrightarrow g_*\mathcal T_{j,k}
\end{equation}
is surjective. The assertion holds on every parameter germ,
including germs supported at special parameters.
\end{proposition}
\begin{proof}
The assertion is immediate for an empty chart. We induct on \(k\),
simultaneously for all integer \(j\) and all nonempty charts. The layer
algebra and its connecting maps are the
ones in \cite[Lemma~12.1]{BL}; we recall their construction to
specify exactly which obstruction is to be killed.
Assume all smaller orders. The connecting map for
\[
0\longrightarrow\mathcal A_{j+k}
\longrightarrow\mathcal T_{j,k+1}
\longrightarrow\mathcal T_{j,k}\longrightarrow0
\]
factors uniquely through a map
\begin{equation}\label{signed:obstruction}
\delta_k:g_*\mathcal A_j\longrightarrow R^1g_*\mathcal A_{j+k}.
\end{equation}
Indeed the smaller orders make
\(g_*\mathcal T_{j,k}\to g_*\mathcal A_j\) surjective. For
\(k>1\), the kernel comes from \(g_*\mathcal T_{j+1,k-1}\),
by the leading-layer sequence. The order \(k-1\) in degree
\(j+1\) lifts this kernel into \(g_*\mathcal T_{j,k+1}\), so
the current connecting map kills it. For \(k=1\) the leading
map is the identity. Vanishing of all maps in
Equation~\eqref{signed:obstruction} is equivalent to the
surjectivity at the current order.
The maps \(\delta_k\) form a degree-\(k\) derivation on the
Laurent graded algebra \(\bigoplus_jg_*\mathcal A_j\), with
values in its first-cohomology module. On a parameter germ, choose a lift
of a leading section \(x\) to \(g_*\mathcal T_{j,k}\), and represent that
lift locally by \(x_\alpha\in I^j\). Then
\(x_\beta-x_\alpha\in I^{j+k}\).
Their differences modulo \(I^{j+k+1}\) represent
\(\delta_k(x)\). For another degree-\(l\) section \(z\),
the product difference is
\[
x_\beta z_\beta-x_\alpha z_\alpha
=x_\alpha(z_\beta-z_\alpha)+z_\alpha(x_\beta-x_\alpha)
+(x_\beta-x_\alpha)(z_\beta-z_\alpha).
\]
The last term lies in \(I^{j+l+2k}\subset I^{j+l+k+1}\).
This proves the derivation rule. If \(t_1,\ldots,t_b\) are
coordinates on \(U\), choose simultaneous representatives
\(G_{i,\alpha}\) of their lifts to \(g_*\mathcal T_{0,k}\); their
differences lie in \(I^k\). Taylor's
formula modulo their squared differences gives, for any
holomorphic \(F\),
\begin{equation}\label{signed:chain-rule}
\delta_k(F(t))=\sum_i F_{t_i}(t)\delta_k(t_i).
\end{equation}
Both formulas hold on germs, without removing torsion in the
target, and commute with the prescribed germ comparisons.
Put \(m_k=a+k>0\). An order-\(k\) obstruction raises degree by \(k\),
whereas the residue injection accepts degree \(-a\). We therefore test
leading sections \(x\) of degree \(-m_k=-(a+k)\). Apply the split residue
injection to define
\[
c(x)=\operatorname{res}_*\delta_k(x),\qquad
e_i(x)=\operatorname{res}_*\bigl(x\delta_k(t_i)\bigr)
\quad\text{in }R^1h_*\omega_H(A).
\]
Both arguments have degree \(-a\), the degree of
\(\OO_S(aS)\) in Equation~\eqref{signed:residue-injection}.
The next construction places these classes in a filtered direct-image
module. For the identity graph, the calculation will give the undivided
relation \(c(x)+\sum_i e_i(x)\partial_{t_i}=0\). Taking its degree-one
symbol gives \(\sigma(\sum_i e_i(x)\otimes\partial_{t_i})=0\).
We will globalize that
symbol on a projective coordinate-power cover, where the boundary root is
an ample line. The adjugate formula expresses the symbol across the
ramification of this cover without dividing by its Jacobian determinant.
The symbol vanishing will then kill the obstructions of the parameter coordinates
on every stalk; the undivided relation will kill \(\delta_k(x)\), and the
derivation rule will recover the other degrees.
Let \(\psi:U'\to U\) be a holomorphic map between coordinate
opens of dimension \(b\), with coordinates \(w\) on \(U'\),
transverse to every smooth closed stratum \(H_I\cap A_J\).
The identity map is allowed. In \(\widehat Z\times U'\),
the graph support \(H^\flat=H\times_U U'\) is an SNC divisor
in a smooth complete intersection of codimension \(b\).
The residual divisor remains transverse. After shrinking about a
compact fiber, the finite-component Lemma~10.3 of \cite{BL}
gives finite smooth indexing of the graph strata, and their
relative Kähler forms restrict from the product neighborhood.
Complete-intersection adjunction gives the canonical factor
\[
\omega_{U'}\otimes\psi^*\omega_U^{-1},
\]
whose coordinate frame we denote by \(dw/dt\). This is a
ratio of canonical frames, with no inverse Jacobian. Pull
the Čech representatives of \(c(x),e_i(x)\) to this graph
and tensor by that frame. Proposition~\ref{signed:localized-hodge}
places the resulting classes \(\widetilde c,\widetilde e_i\)
in \(F_0M^\flat\), where \(M^\flat\) is the degree-one
direct-image module of the localized graph support.
Write \(J=(\partial\psi_i/\partial w_j)\) and
\(J^\#=\operatorname{adj}(J)\). Because the adjugate is polynomial in
\(J\), the following identity remains meaningful where \(J\) is singular:
\begin{equation}\label{signed:adjugate}
\widetilde c\det J+
\sum_{i,j}(\widetilde e_i\partial_{w_j})J^\#_{ji}=0
\quad\text{in }M^\flat.
\end{equation}
We verify the change from \cite[Lemma~12.2]{BL} explicitly.
Choose simultaneous representatives of the lifts to length \(k\)
\[
x_\alpha\in I^{-a-k},\quad x_\beta-x_\alpha\in I^{-a},
\qquad G_{i,\beta}-G_{i,\alpha}\in I^k.
\]
After pulling to \(\widehat Z\), put
\(\eta_\alpha=x_\alpha\tau\), using the meromorphic form
defined by Equation~\eqref{signed:adjoint-chart}, and put
\(Q_{i,\alpha}=\psi_i(w)-G_{i,\alpha}\),
\(P_\alpha=\prod_iQ_{i,\alpha}\). On an overlap write
\(\Delta_i=G_{i,\beta}-G_{i,\alpha}\). The insertion and
\(D_S\geq H\) give the precise pole bounds
\begin{align}
\eta_\alpha&\in\omega_{\widehat Z}(H+A+kD_S),\nonumber\\
\eta_\beta-\eta_\alpha,\quad \eta_\alpha\Delta_i
&\in\omega_{\widehat Z}(H+A),\label{signed:pole-bounds}\\
\eta_\alpha\Delta_i\Delta_l,\quad
(\eta_\beta-\eta_\alpha)\Delta_i
&\in\omega_{\widehat Z}(H+A-kD_S)
\subset\omega_{\widehat Z}(A).\nonumber
\end{align}
Thus every quadratic or cross difference has no \(H\)-pole,
although it may retain a pole on \(A\). Such a term is zero
in the support module localized along \(A\).
In that localized graph support module consider the finite
generalized fractions
\[
S_\alpha=\left[\frac{\eta_\alpha\wedge dw}{P_\alpha}\right].
\]
The \(H\)-pole is already in the numerator. Each polar class
is killed by a power of a reduced equation of \(H\), and each
\(\Delta_i\) is a multiple of that equation.
Changing \(Q_{i,\alpha}\) to \(Q_{i,\beta}\) therefore has a
finite geometric expansion on each class. The last line of
Equation~\eqref{signed:pole-bounds} kills all terms beyond the
linear terms, and gives the exact equality
\[
S_\beta-S_\alpha=
\left[\frac{(\eta_\beta-\eta_\alpha)\wedge dw}{P_\alpha}\right]
+\sum_i\left[
\frac{\eta_\alpha\Delta_i\wedge dw}
{Q_{i,\alpha}P_\alpha}\right].
\]
Let \(C_\bullet\) be the first cochain. Let \(E_{i,\bullet}\)
be the cochain with the numerator of the \(i\)-th summand and
only denominator \(P_\alpha\), and let \(E_{i,\bullet}^{(l)}\)
have its additional denominator \(Q_{l,\alpha}\). The simple
cochains \(C_\bullet,E_{i,\bullet}\) are cocycles representing
\(\widetilde c,\widetilde e_i\); their possible triple-overlap
errors are precisely the cross terms killed above. The normal
residue frame in this identification is \(dw/dt\).
The numerator of \(E_{i,\bullet}\), including its \(A\)-poles,
is independent of \(w\). Right differentiation therefore gives
\[
E_{i,\bullet}\partial_{w_j}=\sum_lJ_{lj}E_{i,\bullet}^{(l)}.
\]
The preceding fraction equality is
\(\check dS_\bullet=C_\bullet+\sum_iE_{i,\bullet}^{(i)}\).
Multiplying by \(\det J\) and using \(JJ^\#=(\det J)\id\)
gives at the cochain level
\[
\check d(S_\bullet\det J)=C_\bullet\det J+
\sum_{i,j}(E_{i,\bullet}\partial_{w_j})J^\#_{ji}.
\]
These cochains are in the end term of the relative right de
Rham complex, so a Čech coboundary there is a total
coboundary. Passing to the direct image proves
Equation~\eqref{signed:adjugate}, with \(J^\#_{ji}\) placed
after the right derivative as displayed. Passing to
\(\gr^F_1\) yields
\begin{equation}\label{signed:local-symbol}
\sigma\left(\sum_{j,i}J^\#_{ji}\widetilde e_i
\otimes\partial_{w_j}\right)=0.
\end{equation}
For \(\psi=\id\) the undivided identity is
\begin{equation}\label{signed:undivided}
c(x)+\sum_i e_i(x)\partial_{t_i}=0.
\end{equation}
For \(b=0\) there are no graph equations, and the same
undivided cochain computation gives \(c(x)=0\).
We now prove that the obstructions of the parameter coordinates vanish on every
stalk, including an obstruction supported only at a special parameter.
Suppose \(b>0\), and fix \(t_*\in P\). We will use one compact graph over
a projective parameter space, obtained by a parameter change unbranched
above \(t_*\). The vanishing of
\(\Hom\) in Equation~\eqref{signed:symbol-vanishing} applies to the entire
coherent symbol kernel, and the local isomorphism
above \(t_*\) will detect vanishing on the full germ there. The
construction of \cite[Proposition~10.5]{BL} gives a coordinate-
power map, here of exponent \(\ell\),
\[
\begin{gathered}
\psi:P'=\PP^b\longrightarrow P,\qquad
[w_0:\cdots:w_b]\longmapsto[w_0^\ell:\cdots:w_b^\ell],\\
R=\OO_{P'}(1),\qquad R^\ell\simeq\psi^*\OO_P(1),
\end{gathered}
\]
in suitably chosen coordinates, unbranched over \(t_*\).
We spell out why its geometry also accommodates \(A\).
Choose finitely many relatively compact parameter opens
covering \(\phi(D)\), with the chart resolutions defined on
slightly larger opens. Empty charts contribute no strata. Properness of
the nonempty supports leaves
only finitely many closed smooth strata \(H_I\cap A_J\) meeting
the corresponding compact sets. A general tuple of coordinate
hyperplanes is transverse to the maps of every such stratum,
for every subtuple, and avoids \(t_*\). This follows by Sard's
theorem applied to the incidence with variable hyperplanes;
varying each hyperplane supplies its normal direction. In
these coordinates \(d\psi\) has image the tangent space to
the intersection of its vanishing coordinate hyperplanes.
The transversality condition is consequently
\[
dh(T_z(H_I\cap A_J))+d\psi(T_wP')=T_{h(z)}P
\quad\text{whenever }h(z)=\psi(w).
\]
It gives the required simultaneous regular graph equations
and SNC transverse residual divisors.
To obtain a single compact graph support, first form the compact
graph \(\Xi=D\times_P P'\subset X\times P'\). On it the
pullback of \(N_0\) has the specified root \(R\). Lemma~10.1
of \cite{BL} extends that root to a neighborhood of \(\Xi\).
Perform the full construction of Lemma~\ref{signed:root-charts}
on this neighborhood, including the relative quotient and its ordinary
pullback, and use the canonical sheaf relative to \(P'\) in the periodicity
algebra; the absolute canonical factor is added only when
taking graph residues. Resolve the same ordered total divisor data before imposing
the graph equations. Near a compact graph slice, the root
comparison lemma identifies this construction with the ordinary
product of a local chart with a parameter open, as an ambient
germ preserving the ideal and the adjoint form. Uniqueness
of the comparison as a germ and properness of the graph allow
a base shrink on which the identification holds on the entire
slice. Normalization thus occurs before ramified graph base
change. Smooth-functorial resolution preserves the product
identification.
The graph cut \(H'\) in this resolved ambient space is compact,
has pure dimension \(n-1\), and is locally an SNC divisor in
a smooth complete intersection of codimension \(b\). Its
ambient projection to \(P'\) is submersive near \(H'\).
Let \(A'\) be the reduced ambient resolved residual divisor, restricted
to a neighborhood of \(H'\). Under each product-germ identification
it is \(A\times U'\); its restriction to the graph complete intersection
is the transverse residual cut. The chosen transversality makes \(A'\)
simultaneously SNC with the graph support equations.
All closed strata \(H'_I\cap A'_J\) are compact and smooth;
splitting their connected components leaves finitely many
indices. The relative metric construction in
Lemma~\ref{signed:root-charts} gives one Kähler form near
\(H'\), and hence the required forms on every stratum. Thus
Proposition~\ref{signed:localized-hodge} applies to this single
global graph. Write \(M'\) for its degree-one module.
Take \(x=u^{-m_k}\). On a graph chart choose a frame
\(\vartheta\) of \(R\) compatible with the downstairs frame
of \(\OO_P(1)\). The local rule
\begin{equation}\label{signed:global-symbol}
\vartheta^{m_k}\longmapsto
\sum_{j,i}J^\#_{ji}\widetilde e_i(u^{-m_k})
\otimes\partial_{w_j}
\end{equation}
defines a global morphism
\(R^{m_k}\to F_0M'\otimes T_{P'}\). Here is the transition
check, including the ramification locus. For changes of
coordinates let \(A_t=\partial t'/\partial t\) and
\(B_w=\partial w'/\partial w\), and let the root frame change
by \(\vartheta'=\lambda\vartheta\). Locally \(\lambda\)
descends from a holomorphic unit downstairs: its \(\ell\)-th
power does, and after choosing a local \(\ell\)-th root of
that unit the remaining ratio is a locally constant element
of \(\mu_\ell\). Root germ compatibility and
Equation~\eqref{signed:chain-rule} transform the downstairs
column of \(e_i\)'s by \(\lambda^{m_k}A_t\). No derivative
of \(\lambda\) appears, since \(x\) multiplies outside
\(\delta_k(t_i)\). The absolute canonical factor changes
by \(\det B_w/\det A_t\). Hence
\[
\begin{aligned}
\widetilde e'&=\frac{\det B_w}{\det A_t}\lambda^{m_k}A_t\widetilde e,
&J'&=A_tJB_w^{-1},\\
(J')^\#\widetilde e'&=\lambda^{m_k}B_wJ^\#\widetilde e.
\end{aligned}
\]
The adjugate equality is polynomial in \(J\), so holds also
when \(J\) is singular. The last formula, together with the
inverse change of the vector basis, is exactly the transition
of Equation~\eqref{signed:global-symbol}. The graph classes
here are natural Čech pullbacks through ambient germs; no
ramified base-change isomorphism for \(R^1h_*\) is used.
Equation~\eqref{signed:local-symbol} puts the image of this
morphism in the first-symbol kernel. Since
\(R^{m_k}=\OO_{\PP^b}(a+k)\) is ample,
Equation~\eqref{signed:symbol-vanishing} makes the morphism
zero. At a point above \(t_*\), the map \(\psi\) is locally
biholomorphic and \(J^\#\) is invertible. The ambient germ
comparison identifies the graph classes with the downstairs
classes there. Thus \(e_i(u^{-m_k})=0\) as germs at
\(t_*\), including any class supported there. The point was
arbitrary, so these classes vanish on every chart.
The split injection in Lemma~\ref{signed:residue} and the
invertible Laurent frame now give \(\delta_k(t_i)=0\).
For every leading section \(x\) of degree \(-m_k\), the
classes \(e_i(x)\) vanish. Equation~\eqref{signed:undivided},
the injection in Equation~\eqref{signed:lowest-step}, and
then the split residue injection give \(\delta_k(x)=0\).
When \(b=0\), the direct identity \(c(x)=0\) and the same
two injections give this conclusion without a graph test.
In particular, the derivation rule and characteristic zero give
\[
0=\delta_k(u^{-m_k})=-m_k u^{-m_k-1}\delta_k(u),
\qquad \delta_k(u)=0.
\]
For any local \(F\in g_*\OO_S\), the degree-\(-m_k\) section
\(Fu^{-m_k}\) then gives
\(0=\delta_k(Fu^{-m_k})=u^{-m_k}\delta_k(F)\).
Every graded section is \(Fu^j\), so \(\delta_k\) vanishes
in every degree. This proves the current order and completes
the induction.
\end{proof}
\subsection{Compact deformations and the signed theorem}
We now turn the infinitesimal lifting into actual compact
subspaces. The role of all the integer layers in
Proposition~\ref{signed:all-orders} is that both the equations
of a boundary fiber and a generator of its normal direction
can be lifted compatibly. This is the analytic counterpart of
the compact-family step in \cite[Proposition~5.1]{LI}.
\begin{lemma}\label{signed:escaping-fibers}
Assume \(0<v<n\), set \(d=n-v\), and choose a positive
component \(D_i\) with \(\dim\phi(D_i)=r\). There is a nonempty
open subset \(D_i^\circ\subset D_i\) with the following
property. For every \(x\in D_i^\circ\), there are a disk
\(\Delta\) about \(0\), a proper flat analytic family
\(\mathscr F\to\Delta\) of compact subspaces of pure
dimension \(d\), and a finite morphism
\[
\mathscr F\longrightarrow X\times\Delta
\]
over \(\Delta\), such that every nonzero fiber has image
disjoint from \(D\), while \(x\) is a limit of points in
those images as the parameter tends to zero. Its schematic
image is flat over the disk with pure \(d\)-dimensional fibers,
whose fundamental cycles form a bounded family in the cycle
space of \(X\) after shrinking the disk.
\end{lemma}
\begin{proof}
Choose a general point of the reduced image \(\phi(D_i)\),
outside the small image in
Equation~\eqref{signed:small-intersections} and the intersections
with distinct component images. Work on one ordinary root
chart over a neighborhood \(U\) of that point. Its finite
map \(S\to D_U\) has a component covering a nonempty open of
\(D_i\cap D_U\). Shrink to a smooth open \(Y\) of
\(\phi(D_i)\cap U\), remove the images of components that
do not dominate it, and remove the nonflat locus of the
proper map \(g\). The last removal is proper by analytic
generic flatness. The relevant \(S\) is now flat over the
smooth \(r\)-fold \(Y\), and its fibers avoid \(T\).
They are compact of pure dimension \(n-1-r=d>0\).
On the selected component the locus where it is singular or
where \(g\) has rank less than \(r\) is proper. Removing its
image, along with the preceding proper bad subsets, under the
finite map leaves a nonempty open \(D_i^\circ\subset D_i\).
Its points have preimages at smooth points of these fiber
components.
Fix \(x\in D_i^\circ\) and a point above it in the compact
fiber \(F=g^{-1}(t)\), for \(t\in Y\). Work on the single
ordinary chart neighborhood of that fiber from
Lemma~\ref{signed:root-charts}. Take regular coordinates
\(t_1,\ldots,t_r\) on \(Y\) centered at \(t\), extended to
local coordinates of its smooth embedding in \(P\). Their
pullbacks form a regular sequence on \(S\) along \(F\),
by flatness. In addition, \(S\) is Cohen--Macaulay, so
\(F\) is Cohen--Macaulay of pure dimension \(d\).
Choose once and for all a Hausdorff open neighborhood
\(W\subset Z\) of \(F\), disjoint from \(T\), on which
\(\pi\) is quasi-finite. Set \(I=\OO_Z(-S)\).
Proposition~\ref{signed:all-orders} lets us lift the finitely
many \(t_i\)'s to compatible germs of sections of \(\OO_Z/I^q\),
\(q\geq1\), along \(F\). It also lifts the conormal frame
\(u\in I/I^2\) to compatible germs
\(\widetilde y_q\in I/I^{q+1}\) along \(F\); write \(\widetilde y\)
for this system. At each order choose simultaneous
representatives on a neighborhood of \(F\) contained in
\(W\). These neighborhoods may shrink with the order.
The leading coefficient of \(\widetilde y\) is a unit
frame, so it generates \(I\) on each finite thickening.
For \(R_q=\C[s]/(s^q)\), cut the lifted \(r\) equations in
the Cartier thickening \(qS\), and map \(s\) to
\(\widetilde y\). Denote the resulting subspace by
\(F_q\). It is flat over the Artin analytic point
\(\Spec R_q\), with closed fiber \(F\). Before cutting,
multiplication by the Cartier generator identifies each
successive \(s\)-layer with \(\OO_S\); this is the
flatness criterion over \(R_q\). Quotienting by lifts of
the closed-fiber regular sequence preserves flatness, by
the local flatness criterion. Compatibility of the lifted
equations gives compatible reductions of all \(F_q\).
Each of these is an embedded compact deformation in the
same fixed neighborhood \(W\). To see why the shrinking
representative neighborhoods cause no difficulty, the
support of \(F_q\) is the fixed compact set \(F\). Its
ideal on its representative neighborhood glues with the
unit ideal on \(W\setminus F\), since the deformation is
empty on their overlap away from \(F\). This realizes it
as a closed subspace of \(W\times\Spec R_q\). The ideal
is coherent locally on both opens, and the family is proper
over the Artin point because its support is compact.
The Douady space of compact subspaces of \(W\) represents
proper flat embedded families \cite[Section~9]{Douady1966}.
The compatible \(F_q\)'s thus define a formal arc in its
analytic germ at \([F]\). Embed that germ in a finite
dimensional analytic coordinate space. The arc is a formal
solution of its convergent defining equations. Analytic
Artin approximation \cite[Theorem~1.2]{Artin1968} gives a
convergent arc agreeing modulo \(s^2\). Pull back the
universal family and shrink its disk to obtain
\(\mathscr F\subset W\times\Delta\), proper and flat over
\(\Delta\), with closed fiber \(F\) and the prescribed
first normal displacement.
The family may be taken to have pure \(d\)-dimensional
fibers throughout the disk. Here is a local justification.
The central fiber is Cohen--Macaulay of dimension \(d\).
Flatness over the regular one-dimensional base makes its
parameter a nonzerodivisor and gives the depth and dimension
of the total local ring as \(d+1\) along that fiber.
Cohen--Macaulayness is open for analytic local rings, and
properness allows a disk shrink for which it holds on the
whole family. Every total component then meets the central
fiber after shrinking, and flatness makes it dominate the
disk; the dimension formula gives it dimension \(d+1\).
Quotient by the nonzerodivisor at each
parameter gives Cohen--Macaulay fibers of dimension \(d\).
The analytic dimension formula and unmixedness of these
local rings give the asserted purity.
Let \(f\) be a local equation of \(S\). On \(\mathscr F\)
it vanishes on the central fiber, so flatness writes it as
\(f=s h\). Agreement modulo \(s^2\) with the formal
deformation says that \(h|_F\) is a unit: this is precisely
the lifted conormal generator. Finitely many such
neighborhoods cover the compact \(F\). Properness of
\(\mathscr F\to\Delta\) lets us shrink the disk so that
they cover the entire family and all their \(h\)'s are
units; remove the closed image of the complement. Thus a
nonzero fiber misses \(S\). The neighborhood \(W\) already
misses \(T\), and the set equality in Lemma~\ref{signed:root-charts} gives
\(\pi^{-1}(|D|)\cap W=|S|\cap W\). Hence the
nonzero fiber images miss \(D\).
The induced map \(\Pi:\mathscr F\to X\times\Delta\) is proper:
the source is proper over \(\Delta\) and the target is
Hausdorff over it, so the graph is closed and the projection
is proper. It is quasi-finite by the choice of \(W\),
hence finite. It therefore preserves the dimension of
every fiber component. Every total component through a central
point dominates the disk by flatness, and its punctured part is dense.
Thus every central point, including the point chosen above \(x\),
is a limit of points of nonzero fibers of dimension \(d\).
Let \(\mathscr Z\subset X\times\Delta\) be the schematic
image of this finite map. Its algebra is the coherent image
of \(\OO_{X\times\Delta}\to\Pi_*\OO_{\mathscr F}\).
Multiplication by a nonzero
germ from the disk is injective on \(\OO_{\mathscr F}\),
by flatness, and remains injective after the exact finite
pushforward. It is therefore injective on the image algebra.
That algebra is torsion-free, hence flat, over each local
discrete valuation ring of the disk. Thus \(\mathscr Z\)
is a proper flat family of subspaces of \(X\). The finite
map to its schematic image is surjective, also on each
fiber as a map of supports. Each \(\mathscr Z_s\) consequently
has pure dimension \(d\). The Douady-to-cycle morphism, in the
form recorded in \cite[Section~3.3]{Fujiki1978}, makes their
fundamental cycles an analytic family in the cycle space
of \(X\). The restriction to a smaller closed disk is
compact. The volumes of these cycles against a Kähler
form are bounded by continuity, proving the last assertion.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{signed:torsion}]
The assertion is immediate in dimension zero, and the case
\(D=0\) follows directly from the signed equivalence. By
Lemma~\ref{signed:boundary-geometry}, the case \(v=0\)
already gives \(L\sim_{\Q}0\), and \(v=n>0\) contradicts
\(a(X)=0\). Suppose, therefore, that \(0<v<n\). Choose
the positive \(D_i\) in that lemma and put \(d=n-v\), so
\(0<d<n\). We will contradict simplicity by using the
compact deformations in Lemma~\ref{signed:escaping-fibers}.
Let \(\mathcal C_d(X)\) be the Barlet space of nonzero
effective compact \(d\)-cycles on \(X\). It is a
second-countable analytic space, hence has only countably
many irreducible components \(B_\alpha\). For compact
Kähler \(X\) its connected components, and therefore its
irreducible components, are compact
\cite[Theorem~4.5]{Fujiki1978}. The universal support
\(\mathcal U_\alpha\subset B_\alpha\times X\) is a compact
analytic subspace. Take all of its full irreducible
components \(V_{\alpha\beta}\) that contain an incidence
point belonging to an integral \(d\)-cycle disjoint from
\(D\). There are countably many of these components,
since each compact analytic \(\mathcal U_\alpha\) has
finitely many. Their evaluation images
\[
W_{\alpha\beta}=\operatorname{pr}_X(V_{\alpha\beta})
\]
are closed irreducible analytic subspaces of \(X\), by
proper mapping. Each has a point outside \(D\). We use
the full compact components here, including their fibers
at limiting cycle parameters; the incidence over only the
open set of cycles avoiding \(D\) would not have a proper
evaluation map.
These countably many closed images cover \(D_i^\circ\).
Indeed, for \(x\in D_i^\circ\), choose the family in
Lemma~\ref{signed:escaping-fibers} and a sequence
\(x_j\to x\) in nonzero fiber images. Let \(C_j\) be an
integral component of the fundamental cycle of the schematic image fiber
containing \(x_j\), taken with coefficient one. The cycles
\(C_j\) avoid \(D\), and their volumes are bounded by the
volumes of the total image cycles. Bounded cycles
on compact \(X\) have relatively compact closure in
\(\mathcal C_d(X)\) \cite[Proposition~2.10]{Fujiki1978}.
The irreducible components of an analytic space are locally
finite, so this compact closure meets only finitely many
\(B_\alpha\). After passing to a subsequence, the cycles
lie in a fixed \(B_\alpha\); after a second subsequence,
the points \((C_j,x_j)\) lie in a fixed full component
\(V_{\alpha\beta}\). Its evaluation image is closed, so
it contains \(x\). This proves the covering assertion.
The analytic space \(D_i^\circ\) is locally compact and
Baire. If no \(W_{\alpha\beta}\) contained \(D_i\), each
intersection with \(D_i^\circ\) would be a proper closed
analytic subset and hence nowhere dense. The countable
cover just proved contradicts Baire. Consequently some
\(W_{\alpha\beta}\) contains \(D_i\). It also contains a
point outside \(D\). An irreducible proper analytic
subspace of the irreducible \(n\)-fold \(X\) that contains
the prime divisor \(D_i\) must equal \(D_i\), by dimension.
It follows that \(W_{\alpha\beta}=X\).
Every point of this full incidence component lies on the
support of the effective \(d\)-cycle at its parameter,
including limiting parameters. Thus its surjective
evaluation supplies through every point of \(X\) an
irreducible compact component of such a cycle. Each has
dimension \(d\), strictly between zero and \(n\). This
contradicts simplicity. The intermediate range for \(v\)
is impossible, leaving \(L\sim_{\Q}0\). By the meaning of
rational linear equivalence for the actual line \(L\), a
positive Cartier multiple is the trivial holomorphic line.
\end{proof}
\subsection{The simple case of the induction}
We now pass from the original pair to a reduced signed boundary and a nef
model, and then subtract the added boundary from the torsion comparison.
\begin{proof}[Proof of Proposition~\ref{simple:case}]
Let \((X,B)\) and \(J=K_X+B\) be as in that proposition.
A positive-dimensional compact complex curve is projective,
so \(a(X)=0\) implies \(n\geq2\). By
Theorem~\ref{nv:meromorphic}, a positive multiple of
\(K_X\) has a nonzero meromorphic section. It gives an
actual rational equivalence \(K_X\sim_{\Q}G_K\) for a
signed rational divisor \(G_K\).
Choose a projective log resolution \(p_0:W\to X\) of the
boundary and the support of this divisor. Let \(B_W\) be
the strict transform of \(B\) together with every new
exceptional prime with coefficient one. The resolution transfer in
Proposition~\ref{bir:resolution-transfer}
gives the actual rational identity
\[
K_W+B_W\sim_{\Q}p_0^*J+E_0,
\qquad E_0\geq0\quad\text{exceptional over }X.
\]
The canonical pullback formula transforms \(G_K\) to a
signed representative of \(K_W\). Enlarge the reduced
support of \(B_W\) and this representative to a reduced
SNC divisor \(D\). Then
\begin{equation}\label{signed:enlarged-adjoint}
J_D:=K_W+D\sim_{\Q}p_0^*J+E_0+(D-B_W),
\qquad E_0+(D-B_W)\geq0.
\end{equation}
In particular \(J_D\) is pseudo-effective and has a
signed rational representative supported on \(D\).
Simplicity and algebraic dimension zero persist under these
birational modifications.
Apply Corollary~\ref{prog:signed-nef} to \((W,D)\) and \(J_D\).
Equation~\eqref{signed:enlarged-adjoint} supplies pseudo-effectivity, and
the signed representative is supported on the floor \(D\).
Denote the resulting ordinary dlt nef model by
\((X_{\mathrm{nef}},D_{\mathrm{nef}})\), and put
\(L_{\mathrm{nef}}=K_{X_{\mathrm{nef}}}+D_{\mathrm{nef}}\).
As a working model of the program, \(X_{\mathrm{nef}}\) is globally
strongly \(\Q\)-factorial, and the pair has the resolution property of
Definition~\ref{def:lc-strata-resolution}. The space \(X_{\mathrm{nef}}\)
is simple and compact Kähler with \(a(X_{\mathrm{nef}})=0\);
\(D_{\mathrm{nef}}\) is reduced, \(L_{\mathrm{nef}}\) is nef, and its
actual signed representative is supported on \(D_{\mathrm{nef}}\).
Theorem~\ref{floor:generation} makes
\(L_{\mathrm{nef}}|_{D_{\mathrm{nef}}}\)
semiample on the whole reduced floor.
We check the remaining pseudo-effectivity hypothesis of
Theorem~\ref{signed:torsion}, with exactly \(c=1\).
Take a common projective resolution
\(\alpha:V\to W\), \(\beta:V\to X_{\mathrm{nef}}\), with \(V\) smooth
compact Kähler. The manifold \(V\) is still simple, hence
not uniruled. Ou's criterion
\cite[Theorem~1.1]{Ou2025} makes \(K_V\) pseudo-effective;
this implication uses no assumption about the absence of
canonical sections. Since \(X_{\mathrm{nef}}\) is globally strongly
\(\Q\)-factorial, \(K_{X_{\mathrm{nef}}}\) is a rational line. Write its
canonical comparison as an identity of actual rational lines
\[
K_V\sim_{\Q}\beta^*K_{X_{\mathrm{nef}}}+F_K,
\qquad F_K=F_K^+-F_K^-,
\]
where both \(F_K^+\) and \(F_K^-\) are effective and
\(\beta\)-exceptional. Thus
\(\beta^*\{K_{X_{\mathrm{nef}}}\}+\{F_K^+\}=\{K_V\}+\{F_K^-\}\) is
pseudo-effective. Exceptional translation in
Lemma~\ref{bir:exceptional} removes the effective
\(\beta\)-exceptional \(F_K^+\) and proves that
\(\beta^*\{K_{X_{\mathrm{nef}}}\}\) is pseudo-effective. Since
\(L_{\mathrm{nef}}-D_{\mathrm{nef}}=K_{X_{\mathrm{nef}}}\) as an actual
rational line, this is precisely the resolution hypothesis for
\(L_{\mathrm{nef}}-D_{\mathrm{nef}}\),
with \(c=1\). Theorem~\ref{signed:torsion} now gives
\(L_{\mathrm{nef}}\sim_{\Q}0\).
On the same common resolution, enlarging it if necessary,
the accumulated negative-step comparisons give
\begin{equation}\label{signed:program-decomposition}
\alpha^*J_D\sim_{\Q}\beta^*L_{\mathrm{nef}}+F,
\qquad F\geq0\quad\text{exceptional over }X_{\mathrm{nef}}.
\end{equation}
All identities here are identities of actual rational
lines. Because \(L_{\mathrm{nef}}\) is torsion, its pullback is nef
with zero class. Lemma~\ref{bir:exceptional} therefore
identifies \(F=N(\alpha^*J_D)\).
Put \(\mu=p_0\alpha:V\to X\) and
\(Q=\alpha^*(E_0+D-B_W)\geq0\). Pulling up
Equation~\eqref{signed:enlarged-adjoint} gives
\(\mu^*J\sim_{\Q}\alpha^*J_D-Q\), which is
pseudo-effective. Apply Lemma~\ref{bir:unique-current}
to Equation~\eqref{signed:program-decomposition}. A
positive current for \(\mu^*J\), plus \([Q]\), must be
the unique current \([F]\) in the larger adjoint class.
It follows that \(Q\leq F\), and negative-part
subtraction gives the actual identity
\[
\mu^*J\sim_{\Q}F-Q=N(\mu^*J).
\]
The divisor on the right is effective and rational. This
is exactly Proposition~\ref{simple:case} and the required
instance of \(\Ggood_n\) with torsion positive part.
\end{proof}
==== END Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//07-signed-rigidity.tex ====
==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//main.tex ====
\documentclass[11pt,letterpaper]{article}
\input{preamble}
\title{Log abundance for compact Kähler spaces\\
under logarithmic Iitaka subadditivity}
\author{OpenAI}
\date{October 6, 2026}
\begin{document}
\maketitle
\begin{abstract}
Assume logarithmic Iitaka subadditivity for surjective morphisms with
connected fibers between smooth projective complex varieties with compatible
reduced simple normal crossing boundaries. We prove log abundance for normal
irreducible compact Kähler spaces in every dimension: for a log canonical
pair \((X,\Delta)\) with effective rational boundary and \(K_X+\Delta\)
\(\Q\)-Cartier, analytic nefness of \(K_X+\Delta\) implies semiampleness.
\end{abstract}
\tableofcontents
\input{sections/01-introduction}
\input{sections/02-inductive-reduction}
\input{sections/03-programs}
\input{sections/04-boundary}
\input{sections/05-rank-one}
\input{sections/06-simple-nonvanishing}
\input{sections/07-signed-rigidity}
\bibliographystyle{plain}
\bibliography{references}
\end{document}
==== END Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//main.tex ====
==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//preamble.tex ====
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==== END Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//preamble.tex ====
==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//two-diagonals.tex ====
\begin{minipage}[c]{0.48\linewidth}
\centering
\small
\begin{tikzcd}[column sep=large,row sep=large]
V \arrow[r,hook] \arrow[d,"\Bl_{\Delta_X\times\PP^1}"']
& B \arrow[d,"\Bl_{\Delta_Z}"] \\
\mathcal V_0 \arrow[r,hook] & Z\times Z
\end{tikzcd}
\[
\begin{gathered}
V_z=\Bl_xX,\\
\begin{aligned}
(V\cap E)_z&=\PP(T_z^*D_\lambda)\\
&\subset E_z=\PP(T_z^*Z).
\end{aligned}
\end{gathered}
\]
\end{minipage}
\hfill
\begin{minipage}[c]{0.48\linewidth}
\centering
\small
\begin{tikzcd}[column sep=large,row sep=large]
E^+ \arrow[r] \arrow[d] & E=\PP_Z(\Omega_Z) \arrow[d] \\
Z^+ \arrow[r,"p_+=\Bl_{U_0}"] & Z
\end{tikzcd}
\[
\begin{gathered}
J_+=p_+^{-1}(U_0),\qquad U_0=X\times\PP^1,\\
\mathcal J=(\mathcal I_V\cdot\OO_E)\cdot\OO_{E^+}.
\end{gathered}
\]
\end{minipage}
==== END Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//two-diagonals.tex ====
==== BEGIN Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//references.bib ====
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==== END Log-abundance-for-compact-Kahler-spaces-under-logarithmic-Iitaka-subadditivity//references.bib ====