==== TABLE OF CONTENTS ====
Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//adjunction.tex
Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//analytic-extension.tex
Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//completion.tex
Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//contraction-inputs.tex
Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//conventions.tex
Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//covers.tex
Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//finite-null.tex
Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//gluing.tex
Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//hodge.tex
Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//introduction.tex
Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//lifting.tex
Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//links.tex
Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//main.tex
Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//outer-positivity.tex
Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//preamble.tex
Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//reduction.tex
Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//scalars.tex
Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//references.bib
===========================
==== BEGIN Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//adjunction.tex ====
\section{Adjunction on the entire dlt floor}\label{sec:adjunction}
We now prove the boundary-generation statement applied to the model
supplied by Proposition~\ref{prop:supported-model}. The adjoint line already exists
on the reduced floor as a restriction from the ambient space. Our task is
to construct enough sections of that line which agree through all of its
intersections. This section sets up the strata and the actual residue
identities; the next three sections construct the compatible sections.
\begin{theorem}[Generation on the whole dlt floor]\label{thm:floor-generation}
Let \((V,B)\) be a normal ordinary \(\Q\)-factorial compact K\"ahler dlt
fourfold with effective rational boundary and analytically nef actual
adjoint \(A=K_V+B\). Suppose that it has a projective log resolution
\(\pi:\widehat V\to V\) for which the strict boundary and exceptional
support have globally smooth distinct SNC components, every exceptional
crepant coefficient is below one, and \(\pi\) is generically an
isomorphism on the image of every irreducible component of an intersection
of distinct strict transforms of coefficient-one boundary primes.
Then the actual restriction \(A|_S\) to the entire reduced floor
\(S=\floor B\) is semiample.
\end{theorem}
If \(S\) is empty the conclusion is vacuous. Otherwise its components
are three-dimensional. We may work on each connected component of \(V\)
and take a common multiple over the finitely many components, so in the
proof we assume \(V\) connected and hence irreducible. The theorem needs no section of the ambient
adjoint. Its resolution hypothesis is precisely the output of
Lemma~\ref{lem:special-resolution}, so it does not add a hypothesis to
Theorem~\ref{thm:main}.
\subsection{Local adjunction calculations}
Lemma~\ref{lem:floor-connectedness} supplies the connectedness and
rationality properties used for successive strata. We next compare an
adjunction coefficient on a normal surface slice.
\begin{lemma}[A normal surface slice for an adjunction coefficient]
\label{lem:adjunction-slice}
Let \(Z\) be a normal Cohen--Macaulay irreducible analytic space of
dimension \(n\geq2\), let \(D\) be a normal prime divisor, and let
\(B=D+B'\) be an effective rational boundary with actual
\(\Q\)-Cartier adjoint. Fix a divisible degree \(m\), a local frame of
\(\OO_Z(m(K_Z+B))\), and its meromorphic \(m\)-residue on \(D\).
For a prime \(\Gamma\subset D\), a general point of \(\Gamma\) admits a
local slice by \(n-2\) holomorphic parameters with the following properties.
The slice \(T\) is a normal surface germ, the cut \(C=D\cap T\) is a
smooth curve germ transverse to \(\Gamma\), and the restricted line is
the actual \(m\)-adjoint line of the effective sliced boundary on \(T\).
After division by the base parameter volume, the order along \(\Gamma\)
of the residue on \(D\) equals the order at \(C\cap\Gamma\) of the
surface residue on \(C\). For \(n=2\), the slice is the surface germ
itself.
\end{lemma}
\begin{proof}
Work in a relatively compact local embedding in \(\C^N\), and refine
the regular loci of \(Z\), its singular locus, \(D\), \(\Gamma\), the
boundary primes, and their intersections into a locally finite collection
of smooth strata. After a neighborhood shrink only finitely many relevant
strata meet the chosen compact closure. Choose a linear map
\(\ell:\C^N\to\C^{n-2}\) whose differential is an isomorphism on
the tangent space of \(\Gamma\) at a general smooth point where \(D\)
is smooth. Such points exist because \(D\) is normal. For the restriction
of \(\ell\) to each smooth stratum of dimension at least \(n-2\),
Sard's theorem makes its critical values a measure-zero subset of the
parameter space; strata of smaller dimension have measure-zero image
\cite{Sard1942}. The image of \(\Gamma\) contains a neighborhood of the
chosen value. We may therefore choose a nearby value regular for all
these restrictions and still meeting \(\Gamma\) at a general point.
The corresponding level is transverse to every stratum it meets.
At regular points of \(Z\) the level is a smooth surface. The singular
locus of a normal \(n\)-fold has dimension at most \(n-2\); its strata
therefore meet this level only discretely. Every irreducible component
of the level has dimension at least two by the principal ideal theorem.
It cannot be contained in that discrete singular intersection, and hence
has dimension exactly two by its smooth dense part. The \(n-2\) level
equations have height \(n-2\) at every point. In the Cohen--Macaulay local
rings of \(Z\) they are a regular sequence. The quotient is consequently
Cohen--Macaulay of dimension two. It is generically reduced and regular
in codimension one, since its possible singular points are discrete;
Serre's criterion makes it normal. Transversality on \(D\) gives a smooth
curve at the cut point, and the fact that \(d\ell\) is an isomorphism on
\(T\Gamma\) makes this curve transverse to \(\Gamma\).
Each boundary prime cuts an effective curve with its original rational
coefficient, counted with its positive intersection multiplicity. At a
generic codimension-one point of the normal surface, the ambient space
and the slice are smooth and transverse to these primes. Dividing the
ambient canonical form by the parameter volume identifies the
restriction of the frame with the pluriadjoint frame of this sliced
effective boundary. Both are rank-one reflexive sheaves on the normal
surface, so the identification extends from codimension one and is an
actual line identity.
To compare the orders, on the smooth \(D\) near a general point of
\(\Gamma\) complete the parameters to coordinates \((z_1,\ldots,z_{n-2},u)\)
with \(\Gamma=(u=0)\). Write the meromorphic residue as
\[
u^c h(z,u)(du\wedge dz_1\wedge\cdots\wedge dz_{n-2})^{\otimes m},
\]
where \(h\) is a unit at a general point of \(\Gamma\). Choose the
regular level above also outside the proper zero set of its leading
coefficient. Division by the base volume and restriction to that level
then has order exactly \(c\) on the cut curve. On the ambient smooth
locus the two operations commute: in coordinates with \(D=(x=0)\),
both take the residue of
\(h(x,u,z)(dx/x\wedge du\wedge dz)^{\otimes m}\) to
\(h(0,u,z)(du)^{\otimes m}\) on the level, up to the ordering sign.
The preceding reflexive identification makes the surface term the residue
of the same actual frame. The restricted original residue is already a
meromorphic form on the smooth curve \(C\), so equality on its dense
smooth-ambient part extends across the cut point. This proves the order
assertion. In even degree the ordering sign is one.
\end{proof}
\subsection{The indexed strata and their actual adjoints}
Fix the resolution in Theorem~\ref{thm:floor-generation}, and let
\(\widehat S_i\), for \(i\) in a finite set \(I\), be the strict transforms
of the components of \(S\). For a nonempty subset \(J\subset I\), each
irreducible component \(\widehat Z\) of \(\bigcap_{i\in J}\widehat S_i\)
is smooth of dimension \(4-|J|\). Call its reduced image
\(Z=\pi(\widehat Z)\) a stratum and write \(\pi_Z:\widehat Z\to Z\).
We also include \(V\), with the empty index set and resolution \(\widehat V\).
The special resolution makes each \(\pi_Z\) bimeromorphic and generically
an isomorphism. The image determines its index set and its resolving
component: on the generic isomorphism locus, two different choices would
force an SNC intersection to have two different codimensions or local
branches. There are finitely many strata, since the intersections are
compact analytic spaces. Imposing one additional index \(j\notin J\)
cuts \(\widehat Z\) in a smooth, possibly disconnected divisor. Its
components give the incidences from \(Z\) to the next strata.
To perform adjunction inductively, allow each unused floor coefficient
either to remain one or to become \(1/2\). More precisely, for a stratum
indexed by \(J\), choose \(e_i=1\) for \(i\in J\) and
\(e_i\in\{1,1/2\}\) for \(i\notin J\), and put
\[
B^{\mathbf e}=B-\sum_{i\in I}(1-e_i)S_i.
\]
Ordinary global \(\Q\)-factoriality makes this an actual rational-line
operation. If \(\widehat G\) is the crepant boundary of \(B\), the new
one is
\[
\widehat G^{\mathbf e}=\widehat G-
\pi^*\sum_{i\in I}(1-e_i)S_i.
\]
The subtracted pullback is effective. Exceptional coefficients remain
below one, and the coefficient-one primes are exactly the retained strict
floor primes. Its support stays inside the resolved SNC support.
SNC adjunction along the indices in \(J\) defines a subboundary
\(\widehat B_{\widehat Z}^{\mathbf e}\) on \(\widehat Z\) and an actual
residue identity
\begin{equation}\label{eq:snc-stratum-adjunction}
(K_{\widehat V}+\widehat G^{\mathbf e})|_{\widehat Z}
=K_{\widehat Z}+\widehat B_{\widehat Z}^{\mathbf e}.
\end{equation}
Every coefficient is at most one. Its coefficient-one divisors are
precisely the one-index intersections for unused indices with \(e_i=1\).
The equality means equality of the meromorphic maps from the same
restricted pluriadjoint line in a sufficiently divisible even degree.
\begin{proposition}[Adjunction and full-floor descent]\label{prop:strata-adjunction}
Every stratum \(Z\) is normal and compact K\"ahler. For every allowed
choice \(\mathbf e\) there is an effective rational boundary
\(B_Z^{\mathbf e}\) such that
\begin{align}
\pi_Z^*(K_Z+B_Z^{\mathbf e})
&=K_{\widehat Z}+\widehat B_{\widehat Z}^{\mathbf e},
\label{eq:stratum-crepant}\\
\OO_Z\bigl(m(K_Z+B_Z^{\mathbf e})\bigr)
&\simeq\OO_V\bigl(m(K_V+B^{\mathbf e})\bigr)|_Z
\label{eq:stratum-line}
\end{align}
for all sufficiently divisible even \(m\). Both identities are the
actual meromorphic residue identifications. For the full weights write
\(B_Z=B_Z^{\mathbf 1}\) and \(C_Z=\floor{B_Z}\). The pair \((Z,B_Z)\)
is dlt; its floor components are exactly the next incident strata.
In a common degree, sections on those components that agree by residue
on every subordinate stratum descend uniquely to a section on the entire
reduced \(C_Z\). The same assertion holds for the floor \(S\subset V\).
\end{proposition}
\begin{proof}
We induct on the number of indices. The assertions for the empty index
set are the given normality, effective boundary, and crepant identity on
\(V\). Assume them for a stratum \(Z\) and all allowed weights there.
Lower all unused floor weights. The resulting crepant SNC boundary on
\(\widehat Z\) has every coefficient below one. The effective pair
\((Z,B_Z^{\mathbf e})\) is therefore klt. Lemma~\ref{lem:floor-connectedness}
shows that \(Z\) has rational singularities and is Cohen--Macaulay.
To add an index \(j\), retain only that unused weight at one and lower
the others. The coefficient-one locus on \(\widehat Z\) is
\(R=\widehat Z\cap\widehat S_j\), a disjoint union of smooth components.
Apply Lemma~\ref{lem:floor-connectedness} to \(\pi_Z\). Its connected
fibers prevent the images of two such components from meeting. On each
image \(D\), the equality \(\pi_{Z,*}\OO_R=\OO_{\pi_Z(R),\red}\)
identifies the proper bimeromorphic pushforward of the structure sheaf
of its smooth resolving component \(\widehat D\) with \(\OO_D\).
Factoring through the finite normalization shows that \(D\) is normal.
It is a compact analytic subspace of \(V\), so it inherits a K\"ahler
form by restricting local strictly plurisubharmonic potentials.
For arbitrary allowed weights retaining \(j\), define
\(B_D^{\mathbf e}=\pi_{D,*}\widehat B_{\widehat D}^{\mathbf e}\).
At this point it is a rational divisor whose effectivity remains to be
shown. In a divisible even degree, a local frame of the ambient adjoint
line pulls back and takes SNC residue to a meromorphic pluriform on
\(\widehat D\). The proper bimeromorphic map \(\widehat D\to D\) is an
isomorphism at the generic points of divisors of the normal \(D\).
There the same map identifies the divisorial adjoint for
\(B_D^{\mathbf e}\) with the restriction of the existing invertible line.
Normal reflexive extension gives \eqref{eq:stratum-line} on all of \(D\).
Pulling the line back to \(\widehat D\) recovers the original meromorphic
map, since the maps agree on a dense open. This proves
\eqref{eq:stratum-crepant}, including its meromorphic interpretation.
We verify that \(B_D^{\mathbf e}\) is effective. It is enough to test
the coefficient at a general point of each prime in \(D\), a codimension-two
locus in \(Z\). Apply Lemma~\ref{lem:adjunction-slice}, using the
Cohen--Macaulay property already proved for \(Z\). It reduces exactly
that coefficient, with the actual residue order preserved, to the smooth
cut curve in a normal surface germ with effective boundary. On a minimal
resolution of this surface germ, write its
crepant boundary as the effective strict transform plus an exceptional
divisor \(E\). For every exceptional curve \(C\),
\[
E\cdot C=-(K_{\widetilde T}+B^{\mathrm{str}})\cdot C\leq0.
\]
Indeed \(B^{\mathrm{str}}\cdot C\geq0\), while surface adjunction gives
\(K_{\widetilde T}\cdot C=2p_a(C)-2-C^2\geq0\): exceptional
self-intersections are negative and a smooth rational exceptional
\((-1)\)-curve is excluded by minimality. The negative-definite
exceptional intersection matrix, with nonnegative off-diagonal entries,
then implies \(E\geq0\). Explicitly, if \(E=P-N\) has disjoint
nonnegative parts and \(N\ne0\), then
\(E\cdot N=P\cdot N-N^2>0\), contradicting \(E\cdot C\leq0\) on
each component of \(N\). The strict
transform of the smooth cut curve is finite birational over it and hence
isomorphic. Adjunction to that smooth curve in a smooth surface with
effective remaining boundary has a nonnegative coefficient. This is
the coefficient originally tested. There is no coefficient to test when
the new stratum is a point. Effectivity follows and completes this part
of the induction.
For clarity, the full-weight pair on each stratum is dlt and has exactly
the claimed floor. The SNC subboundary has coefficients at most one, so
it is lc, and crepancy makes the pair on \(Z\) lc. On the smooth
\(\widehat Z\), let a divisorial
valuation have center of codimension \(c\), and choose local coordinates
\(x_1,\ldots,x_c,\ldots\) at a general point of that center, with the
SNC boundary components among the coordinate divisors. Give an unmarked
coordinate weight zero and write the boundary weights as \(b_i\leq1\).
The reduced coordinate arrangement is lc, so
\[
a(v;\widehat Z,\widehat B_{\widehat Z})
\geq\sum_{i=1}^c(1-b_i)v(x_i).
\]
All \(v(x_i)\) here are positive. A zero discrepancy therefore forces
every \(b_i=1\); its center is an intersection of coefficient-one
components. Each such intersection has image meeting the isomorphism
locus of the original special resolution. In particular each
coefficient-one divisor is nonexceptional for \(\pi_Z\); the others have
coefficients below one. Let \(F_Z\subset Z\) be the complement of the
largest open over which \(\pi_Z\) is an isomorphism. It has codimension
at least two on the normal target \(Z\), and contains no entire image of
a zero-discrepancy center. Its inverse image on \(\widehat Z\) may already
be divisorial. If necessary, principalize the pulled reduced ideal
\(\mathscr I(F_Z)\OO_{\widehat Z}\), and then apply the ordered-boundary
resolution as in Lemma~\ref{lem:special-resolution}, marking each distinct
prime in the prior SNC support, the newly principalized support, and every
divisorial exceptional prime. All new exceptional centers lie over
\(F_Z\). The displayed inequality makes their discrepancies positive.
This is a dlt resolution in the
standard sense. It also proves that the coefficient-one primes on \(Z\)
are exactly the images of the one-index intersections.
It remains to check the asserted descent of sections. With full weights,
let \(R=\widehat B_{\widehat Z}^{=1}\). The preceding floor identification
gives \((\pi_Z(R))_{\red}=C_Z\); write \(\pi_R:R\to C_Z\) for the induced
restriction of \(\pi_Z\). Sections on the incident strata
pull back to sections on its smooth components. Residue compatibility
means equality on the entire scheme-theoretic pairwise intersections,
which are reduced smooth SNC strata. The local equalizer for an SNC union
therefore glues them to a section on \(R\); it is the elementary equalizer
for the coordinate ideals of its components. Even-degree iterated
residues make all paths to deeper intersections identical. By
Lemma~\ref{lem:floor-connectedness} and projection formula,
\[
(\pi_R)_*\pi_R^*(L|_{C_Z})=(L|_{C_Z})
\]
for any ambient Cartier line \(L\) under consideration. The section
descends uniquely to all of \(C_Z\). This reasoning applies also to
\(Z=V\) and \(C_Z=S\). In particular it retains the conductor
coefficient at a general double crossing on each normalized component;
it does not posit an additional conductor boundary on the nonnormal
union itself.
\end{proof}
The proposition supplies a finite collection of normal compact K\"ahler
dlt strata, all carrying restrictions of the same actual ambient line.
It also specifies exactly what compatibility is needed to descend to a
whole floor. The next section proves that, after imposing a small list of
birational residue comparisons on the lower strata, these compatible
boundary sections extend to their parent stratum.
==== END Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//adjunction.tex ====
==== BEGIN Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//analytic-extension.tex ====
\subsection{Analytic realization of the floor thickenings}
\label{subsec:analytic-thickenings}
Retain the divisorial ideals \(\mathscr I_k\), the thickenings \(S_k\),
the map \(\phi:S\to W\) from Proposition~\ref{prop:conormal-layers},
and the sheaves \(\mathscr B_k\) in
\eqref{eq:thickening-ring-surjection}. We construct their analytic
structure from the already analytic source thickenings.
\begin{lemma}[Analytic base thickenings]\label{lem:analytic-thickenings}
For every \(k\geq1\), put
\[
W_k=(|W|,\mathscr B_k).
\]
Then \(W_k\) is a complex space with canonical reduction \(W\), and
the canonical sheaf evaluation defines a proper surjective holomorphic
map \(\phi_k:S_k\to W_k\) with
\[
(\phi_k)_*\OO_{S_k}=\OO_{W_k}.
\]
Its reduction is \(\phi\). The quotients
\(\mathscr B_{k+1}\twoheadrightarrow\mathscr B_k\) define closed
analytic embeddings \(W_k\hookrightarrow W_{k+1}\). They commute
with the embeddings \(S_k\hookrightarrow S_{k+1}\) and the maps
\(\phi_k\).
\end{lemma}
\begin{proof}
\par\medskip\noindent\textit{Canonical rings and their successive kernels.}\par
All direct images defining \(\mathscr B_k\) first use the continuous
map \(|\phi|:|S|\to|W|\). The equality
\(\mathscr B_1=\OO_W\) is the canonical equality for the holomorphic
map \(\phi\). Apply derived direct image for abelian sheaves to the
conormal sequence. Its derived image on an \(\OO_S\)-module is the
underlying sheaf of the analytic higher direct image. To see this
formally, an injective \(\OO_S\)-module is flasque: for open sets
\(V\subset U\), the extension-by-zero modules satisfy
\(j_{V!}\OO_V\hookrightarrow j_{U!}\OO_U\). Adjunction identifies
\(\operatorname{Hom}_{\OO_S}(j_{U!}\OO_U,J)\) with
\(\Gamma(U,J)\), so injectivity of \(J\) makes the restriction of
sections surjective. Flasque abelian sheaves are acyclic for continuous
direct image. An injective
module resolution therefore computes both the analytic module higher
images and their underlying abelian-sheaf higher images. Thus
\eqref{eq:conormal-vanishing} gives exact sequences
\begin{equation}\label{eq:analytic-thickening-sequence}
0\longrightarrow \mathscr F_k:=\phi_*\mathscr E_k
\longrightarrow\mathscr B_{k+1}
\longrightarrow\mathscr B_k\longrightarrow0\qquad(k\geq1).
\end{equation}
The displayed quotients are ring maps; exactness here concerns their
underlying abelian sheaves. Proper direct-image coherence for the
already holomorphic \(\phi\)
\cite[Theorem~1.1, p.~15-01]{Grothendieck1961TechniquesVIII}
makes \(\mathscr F_k\) a coherent
\(\OO_W\)-module.
In \(\OO_{S_{k+1}}\), the ideal \(\mathscr E_k\) is square-zero and
is annihilated by \(\mathscr I_1\): use
\(\mathscr I_k^2\subset\mathscr I_{2k}\subset\mathscr I_{k+1}\)
and \(\mathscr I_1\mathscr I_k\subset\mathscr I_{k+1}\).
Consequently the action of \(\mathscr B_{k+1}\) on the ideal
\(\mathscr F_k\) factors through \(\mathscr B_1=\OO_W\).
This is the \(\OO_W\)-module structure on the square-zero ideal used
in the following construction.
The composites of \eqref{eq:analytic-thickening-sequence} are
epimorphisms \(\mathscr B_k\to\OO_W\). Their kernels are
\[
\mathscr N_k=|\phi|_*(\mathscr I_1/\mathscr I_k),
\qquad \mathscr N_k^k=0,
\]
by left exactness and \(\mathscr I_1^k\subset\mathscr I_k\).
Since \(W\) is reduced, \(\mathscr N_k\) is exactly the nilradical.
Each stalk of \(\mathscr B_k\) is local: an element lifting a unit in
\(\OO_{W,w}\) has an inverse by a finite geometric series in the
nilpotent kernel, whereas an element mapping to the maximal ideal is
not a unit. Its residue field is \(\C\). Thus \(W_k\) is already
a locally ringed space, and it remains to construct local analytic
presentations for it.
\par\medskip\noindent\textit{A local analytic presentation from source functions.}\par
Fix \(k\) and \(w\in W\). Choose a local closed analytic embedding
of a neighborhood of \(w\) into an open polydisc. Lift the finitely
many germs of its coordinate restrictions through
\(\mathscr B_k\to\OO_W\), represent the lifts on a common
neighborhood, and shrink the polydisc. This gives a closed embedding
\(i:U\hookrightarrow D\subset\C^d\) and sections
\(b_1,\ldots,b_d\in\mathscr B_k(U)\) reducing to the coordinate
restrictions. Put \(Z=S_{k,U}\), the open subspace of \(S_k\) with
underlying set \(\phi^{-1}(U)\). By definition,
\(\mathscr B_k(U)=H^0(Z,\OO_Z)\).
For a possibly nonreduced complex space, a tuple of holomorphic
functions defines a holomorphic map to affine space, functorially on
structure sheaves \cite[Theorem~1.1, p.~10-02]{Grothendieck1961TechniquesIII}.
Apply this theorem to the \(b_j\). Their point values lie in \(D\),
so the map factors as
\[
g:Z\longrightarrow D.
\]
The same tuple theorem identifies its actual analytic restriction to
the reduction with \(i\phi\), because the reduced coordinate functions
are those of \(i\phi\). In particular \(|g|=i|\phi|\). The
restriction \(\phi^{-1}(U)\to U\) is proper and \(i\) is closed, so
\(g\) is proper; nilpotents do not change properness of the underlying
continuous map.
The holomorphic map \(g\) makes
\(\mathscr A=g_*\OO_Z\) a genuine \(\OO_D\)-algebra. Its underlying
module is coherent by the proper direct-image theorem
\cite[Theorem~1.1]{Grothendieck1961TechniquesVIII}. Directly from the
underlying maps, as sheaves of \(\C\)-algebras,
\[
\mathscr A=i_*(\mathscr B_k|_U).
\]
Reduction gives an epimorphism
\(\mathscr A\to i_*\OO_U\). It is \(\OO_D\)-linear because the
actual analytic restriction \(g|_S=i\phi\) identifies the reduced
action of every ambient holomorphic function with its restriction to
\(U\). The target is the coherent quotient of \(\OO_D\) by the
ideal of \(U\). Hence its kernel \(\mathscr N\) is a coherent
\(\OO_D\)-module. It is also a nilpotent ideal. Shrink \(D\) once
more and choose sections \(n_1,\ldots,n_r\) generating
\(\mathscr N\) as a module. They are actual sections of \(\OO_Z\)
whose reductions vanish. Since \(\OO_D\to i_*\OO_U\) is an
epimorphism, subtraction of an ambient lift of a reduction gives the
stalkwise module equality
\begin{equation}\label{eq:analytic-module-generation}
\mathscr A=\OO_D\cdot1+
\sum_{\nu=1}^r\OO_D\cdot n_\nu.
\end{equation}
No splitting of the reduction map is used here.
Apply the tuple theorem again to the \(b_j\) and \(n_\nu\). It gives
\[
H=(g,n_1,\ldots,n_r):Z\longrightarrow P=D\times\C^r.
\]
If \(j:U\hookrightarrow P\) is \(u\mapsto(i(u),0)\), then
\(|H|=j|\phi|\), since the \(n_\nu\) are nilpotent. Thus \(H\)
is proper. The sheaf \(\mathscr A'=H_*\OO_Z\) is a coherent
\(\OO_P\)-module and the unit is an algebra map
\[
\theta:\OO_P\longrightarrow\mathscr A'.
\]
As sheaves of rings \(\mathscr A'=j_*(\mathscr B_k|_U)\), so its
stalk at \(j(u)\) is \(\mathscr B_{k,u}=\mathscr A_{i(u)}\), and
its stalk outside \(j(U)\) is zero. The stalk identification uses
the cofinality of restrictions of ambient neighborhoods among
neighborhoods in the closed subspace \(U\).
The map \(\theta\) is surjective on every stalk. At \(j(u)\), any
element of the target has, by \eqref{eq:analytic-module-generation},
the form \(a_0+\sum a_\nu n_\nu\) with
\(a_\nu\in\OO_{D,i(u)}\). The projection of \(H\) to \(D\) is
\(g\), and the extra coordinate \(t_\nu\) pulls back to \(n_\nu\).
The element is therefore the image of the convergent germ
\[
a_0(z)+\sum_{\nu=1}^r a_\nu(z)t_\nu\in\OO_{P,j(u)}.
\]
At all other stalks the target is zero. The kernel
\(\mathscr Q=\ker\theta\) is a coherent analytic ideal
\cite[pp.~9-07--9-08]{Grothendieck1961TechniquesII}. It defines a closed
complex subspace \(Y=V(\mathscr Q)\subset P\) by
\cite[Definition~2.2 and adjacent construction, p.~9-10]{Grothendieck1961TechniquesII}.
Its support is exactly
\(j(U)\), since the target stalk there surjects onto the nonzero ring
\(\OO_{U,u}\). Under the homeomorphism with \(U\), its structure
sheaf is \(\mathscr B_k|_U\), and its reduction is \(U\).
Every germ of \(\mathscr Q\) pulls back to zero by the definition of
\(\theta\). The closed-subspace factorization criterion
\cite[pp.~9-04--9-05]{Grothendieck1961TechniquesII} therefore factors
\(H\) holomorphically through \(Y\). Its sheaf map, under the
identified structure sheaf, is the canonical evaluation
\[
|\phi|^{-1}(\mathscr B_k|_U)\longrightarrow\OO_Z.
\]
Indeed every germ of \(\mathscr B_k|_U\) lifts through the surjective
\(\theta\) to an ambient germ, whose pullback is precisely its value
as a section on the inverse image. This proves the assertion for every
germ, not only the chosen coordinates.
\par\medskip\noindent\textit{Gluing and closed transition maps.}\par
The ringed space \((|W|,\mathscr B_k)\) was fixed before any of the
local choices. The presentations just constructed make each of its
restrictions an analytic space by the local definition and adjacent
construction in \cite[Definitions~2.1--2.2 and adjacent prose]{Grothendieck1961TechniquesII}.
On overlaps the identifications are
the identity on the same sheaf of \(\C\)-algebras, and hence are
analytic isomorphisms satisfying the cocycle condition
\cite[Definition~2.4 and Proposition~2.5]{Grothendieck1961TechniquesII}.
This proves that \(W_k\) is a complex space. It retains the Hausdorff
topology of \(W\), and its reduction is canonically \(W\).
Write \(\varepsilon_k:|\phi|^{-1}\mathscr B_k\to\OO_{S_k}\) for
the global sheaf evaluation. It is locally the holomorphic factorization
above, so it defines \(\phi_k\). Its reduction is \(\phi\), and its
underlying map is \(|\phi|\); thus it is proper and surjective.
The direct-image equality is the definition of its target structure
sheaf, with the canonical unit as the identification.
For the closed transition, use the construction with \(H\) at level
\(k+1\) and restrict its coordinate functions to the closed subspace
\(S_{k,U}\). They give another proper holomorphic map to the same
\(P\), with underlying map \(j|\phi|\). Its unit is the composite
\[
\OO_P\twoheadrightarrow j_*(\mathscr B_{k+1}|_U)
\twoheadrightarrow j_*(\mathscr B_k|_U).
\]
The second quotient stays surjective under the closed embedding \(j\),
as seen on its stalks. The last sheaf with this ambient action is
coherent by proper direct image. The two units therefore have nested
coherent ideal kernels, so their analytic subspaces give the required
closed inclusion. These local inclusions glue because their ring maps
are the fixed quotients. Naturality of the evaluation maps gives
\[
\varepsilon_k\circ|\phi|^{-1}(\mathscr B_{k+1}\to\mathscr B_k)
=(\OO_{S_{k+1}}\to\OO_{S_k})\circ\varepsilon_{k+1},
\]
which proves commutativity with the source inclusions. No retraction
\(W_k\to W\), Cartesian square, or realization of the infinite tower
as one formal completion is asserted or needed.
\end{proof}
\subsection{Extension of a prime-floor contraction}
\label{subsec:prime-floor-extension}
\begin{proposition}[Extension from a prime floor]\label{prop:prime-floor-extension}
Under the hypotheses of Proposition~\ref{prop:conormal-layers}, there
are a normal compact analytic space \(Z\), a closed embedding
\(W\hookrightarrow Z\), and a proper bimeromorphic morphism
\(F:X\to Z\) such that \(F|_S\) is the given map \(\phi\) followed
by that embedding and
\begin{equation}\label{eq:extension-fibers}
X\setminus S\simeq Z\setminus W,\qquad
F^{-1}(|W|)=|S|,\qquad
|F^{-1}(w)|=|\phi^{-1}(w)|\quad(w\in W).
\end{equation}
The natural map \(\OO_Z\to F_*\OO_X\) is an isomorphism, and the
fibers are connected. For a sufficiently divisible global index
\(\ell\) of \(S\), the actual line \(\OO_X(-\ell S)\) is
\(F\)-ample; in particular \(F\) is projective. More generally an
actual rational line on \(X\) whose restriction to \(S\) is
\(\phi\)-ample is \(F\)-ample. The target is in Fujiki's class
\(\mathcal C\).
\end{proposition}
\begin{proof}
Choose a positive global index \(\ell\) such that \(\ell S\) is
Cartier and \(\OO_S(-\ell S)\) is \(\phi\)-ample. Put
\[
A=S_\ell=\ell S,\qquad A'=W_\ell,
\qquad f=\phi_\ell:A\longrightarrow A'.
\]
Here \(A\) is the full effective Cartier divisor, including its
nilpotents; its local equation on normal \(X\) is a nonzerodivisor.
Lemma~\ref{lem:analytic-thickenings} makes \(f\) a proper surjective
holomorphic map of complex spaces and gives the natural equality
\(f_*\OO_A=\OO_{A'}\).
Cartier multiplication gives
\(\mathscr I_\ell\mathscr I_j=\mathscr I_{j+\ell}\). Hence, for
every integer \(n>0\), on the full space \(A\) one has
\begin{equation}\label{eq:cartier-negative-layer}
\OO_A(-nA)
=\mathscr I_{n\ell}\otimes\OO_A
=\mathscr I_{n\ell}/\mathscr I_{(n+1)\ell}.
\end{equation}
Its finite filtration by
\(\mathscr I_j/\mathscr I_{(n+1)\ell}\), for
\(n\ell\leq j\leq(n+1)\ell\), has successive quotients
\(\mathscr E_j\) for \(n\ell\leq j<(n+1)\ell\). All indices are
at least one. These filtration terms are \(\OO_A\)-modules, because
\(\mathscr I_\ell\mathscr I_j\subset\mathscr I_{(n+1)\ell}\).
Let \(a:S\hookrightarrow A\) and \(i:W\hookrightarrow A'\) be the
closed reduction embeddings. The compatibility in
Lemma~\ref{lem:analytic-thickenings} gives \(fa=i\phi\).
Pushforward by a closed embedding is exact, so the composition identity
for derived direct images of abelian sheaves gives
\[
R^q f_*(a_*\mathscr E_j)=i_*R^q\phi_*\mathscr E_j=0\quad(q>0).
\]
The long exact sequences of the finite filtration in
\eqref{eq:cartier-negative-layer} therefore give
\begin{equation}\label{eq:cartier-negative-vanishing}
R^q f_*\OO_A(-nA)=0\qquad(q>0,\ n>0).
\end{equation}
This uses no vanishing for \(\mathscr E_0\) and keeps the full Cartier
thickening throughout.
The actual line \(\OO_A(-A)\) restricts to
\(\OO_S(-\ell S)\). The underlying map of \(f\) is \(\phi\),
so the reductions of their fibers, as closed subspaces of \(X\),
coincide: a reduced closed analytic subspace is determined by its support.
Thus \(\OO_A(-A)\) is ample on the reduction of every \(f\)-fiber.
For a line on a compact nonreduced complex space, ampleness on the
reduction implies ampleness on the whole space. Indeed the positive
metric from \cite[Corollary~1.12]{Fujino2026Fujiki} has local strictly
plurisubharmonic weights in common ambient embeddings, by its Lemma~2.4.
The same weights define a metric on the original line, since the
pointwise absolute values of transition units are unchanged by
nilpotents. The converse positive-metric criterion gives ampleness on
the full space. The proper fiberwise criterion
\cite[Definition~3.1 and Remark~3.2]{Fujino2026Fujiki} now makes
\(\OO_A(-A)\) \(f\)-ample.
We apply the analytic blowdown criterion stated in
\cite[Theorem~4.2, p.~14]{DasHacon2024OnMMP}, attributed there to
\cite[Theorem~2, p.~495]{Fujiki1975}. For a reduced complex space,
an effective Cartier divisor \(A\) with its full possibly nonreduced
structure, and a proper surjective holomorphic map \(f:A\to A'\),
this criterion assumes \(f\)-ampleness of \(\OO_A(-A)\) and
\[
R^1f_*\OO_A(-nA)=0\qquad\hbox{for every }n>0.
\]
It supplies a blowing down: a complex target \(Z\) containing \(A'\)
as an embedded subspace, a proper surjective holomorphic map
\(F:X\to Z\) whose restriction to \(A\) is \(f\), and an analytic
isomorphism of the complements. In its universal form it also supplies
the natural equality of sheaves of rings
\begin{equation}\label{eq:universal-blowdown-ring}
F_*\mathscr S_{X,A,f}=\OO_Z.
\end{equation}
Here \(\mathscr S_{X,A,f}\) consists along \(A\) of the germs in
\(\OO_X\) whose restrictions lie locally in
\(\operatorname{im}(f^{-1}\OO_{A'}\to\OO_A)\), and is \(\OO_X\)
away from \(A\). This is a subsheaf of rings; no module coherence of
this particular sheaf is used. The hypotheses of the stated criterion
follow from \eqref{eq:cartier-negative-vanishing} and the preceding
ampleness argument. No flatness or reducedness of \(A\) or \(A'\)
is required.
The embedded \(A'\) is closed because it is compact. Its reduction
embeds the original \(W\) as a closed subspace of \(Z\). The
compatible reductions give \(F|_S=\phi\) followed by this embedding.
Removing a subspace depends only on its support, so the complement
isomorphism is \(X\setminus S\simeq Z\setminus W\). It follows
that \(F^{-1}(|W|)=|S|\), and the prescribed restriction to \(A\)
gives the fiber-set equalities in \eqref{eq:extension-fibers}.
Outside \(W\) the fibers are single reduced points. These assertions
concern underlying fiber sets over \(W\); they require no Cartesian
square of the closed thickenings. The fibers are connected because
the floor fibers are connected. The target is compact as the image of
compact \(X\).
We next prove normality; it is not an extra conclusion assumed from the
blowdown criterion. For every open \(V\subset Z\), a section
\(s\in\OO_X(F^{-1}V)\) restricts to a section on the open space
\[
A\cap F^{-1}V=f^{-1}(V\cap A').
\]
The equality \(f_*\OO_A=\OO_{A'}\) makes this restriction the pullback
of a unique section on \(V\cap A'\). Thus every germ of \(s\) along
\(A\) satisfies the defining condition of \(\mathscr S_{X,A,f}\).
The reverse inclusion follows from \(\mathscr S_{X,A,f}\subset\OO_X\).
Consequently
\[
F_*\mathscr S_{X,A,f}=F_*\OO_X,
\qquad F_*\OO_X=\OO_Z
\]
as sheaves of rings, the second equality by
\eqref{eq:universal-blowdown-ring}. This reasoning uses equality of
the displayed open subspaces, not of any closed fiber products.
The equality first makes \(Z\) reduced. The complement is dense:
the inverse image of a nonempty open of \(Z\) is a nonempty open of
\(X\), and therefore meets \(X\setminus S\). The complement
isomorphism thus makes \(F\) bimeromorphic. Let
\(\nu:Z^\nu\to Z\) be the finite analytic normalization. The
finite normalization exists by
\cite[Part~B, Section~4, Corollaries~2--3]{Houzel1961}. The
dominant bimeromorphic map from normal \(X\) lifts to \(Z^\nu\).
Pullback along this lift gives inclusions, injective on the dense common
complement,
\[
\OO_Z\subset\nu_*\OO_{Z^\nu}\subset F_*\OO_X=\OO_Z.
\]
The normalization is consequently an isomorphism and \(Z\) is normal.
The natural direct-image equality also gives connected fibers by analytic
Stein factorization.
Finally consider the actual global line \(\OO_X(-\ell S)\). Its
restriction is ample on every reduced \(F\)-fiber: this is clear off
\(W\), and over \(W\) the reduction is the same embedded reduced
space as that of the corresponding \(\phi\)-fiber. The nonreduced
upgrade and the proper fiberwise criterion used above therefore make
this one line \(F\)-ample. The same proof works for any actual ambient
rational line with \(\phi\)-ample restriction after clearing its
global index. In particular it works for the negative full adjoint
in Proposition~\ref{prop:conormal-layers}. Thus \(F\) is projective.
A projective resolution with smooth source of the compact K\"ahler
\(X\) is again compact K\"ahler, and its composite with \(F\) is
bimeromorphic. Hence \(Z\) lies in Fujiki's class \(\mathcal C\).
\end{proof}
The proposition supplies the ambient step for floor data satisfying its
stated hypotheses, including the two actual ample-line conditions. In
the threefold argument below it extends divisor-to-curve contractions. In
Section~\ref{sec:reduction} it extends a selected floor contraction of
the supported fourfold program. When that floor contraction cuts out
the selected ray, the exact fiber description keeps the same contracted
curves, and the actual relatively ample lines make the ambient step
projective. Its target K\"ahler class is then established in the
supported-program argument.
==== END Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//analytic-extension.tex ====
==== BEGIN Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//completion.tex ====
\section{Proof of abundance after nonvanishing}\label{sec:completion}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
If \(\kappa(X,D)\geq1\), Proposition~\ref{prop:positive-kappa} gives
semiampleness on the original \(X\), by descent of the generated actual
Cartier multiple. It remains to treat \(\kappa(X,D)=0\).
Choose a positive integer \(m_0\) for which the actual line
\(\mathscr L=\OO_X(m_0D)\) is invertible and has a nonzero section
\(s_0\). A connected normal space is irreducible, so this section is
locally a nonzerodivisor. Let
\[
M=\frac1{m_0}\operatorname{div}_{\mathscr L}(s_0).
\]
It is the normalized effective rational \(\Q\)-Cartier divisor of the
plurisection, and the chosen line and section give the actual equivalence
\(M\sim_\Q D\).
Suppose \(M\ne0\). Proposition~\ref{prop:supported-model} produces a
normal ordinary \(\Q\)-factorial compact K\"ahler dlt fourfold \((V,B)\)
with effective rational boundary, analytically nef actual adjoint
\(A=K_V+B\), and a nonzero effective rational \(\Q\)-Cartier divisor
\(P\) such that
\[
A\sim_\Q P,\qquad \Supp P=\Supp\floor B,\qquad \kappa(V,A)=0.
\]
The space \(V\) is irreducible, being bimeromorphic to \(X\), and has
the special projective resolution in Lemma~\ref{lem:special-resolution}.
Theorem~\ref{thm:floor-generation} therefore makes the actual restricted
adjoint semiample on the entire reduced floor \(S=\floor B\). This
floor is nonempty because \(P\ne0\). Every hypothesis of
Theorem~\ref{thm:supported-lifting} is now satisfied, so it gives
\(\kappa(V,A)\geq1\), a contradiction.
Thus \(M=0\). The section \(s_0\) is nowhere zero. Indeed, if a local
representative were a nonunit at a point of the normal space, a minimal
prime over its nonzero principal ideal would have height one, and would
give a component of its zero divisor. Consequently \(s_0\) supplies
an isomorphism of holomorphic lines
\[
\OO_X\xrightarrow{\ \simeq\ }\OO_X(m_0D).
\]
This proves the asserted actual \(\Q\)-linear triviality in Iitaka
dimension zero, and in particular semiampleness there. Together with
the positive-Iitaka-dimensional case it proves the theorem.
\end{proof}
==== END Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//completion.tex ====
==== BEGIN Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//contraction-inputs.tex ====
\section{Conormal layers and the contraction interfaces}\label{sec:contraction-inputs}
We next prove the ordinary conormal statement used to extend a contraction
of one prime floor to an ambient contraction. It concerns divisorial
ideals and their actual reflexive powers. We then construct analytic
base thickenings from the resulting direct images and apply the analytic
blowdown criterion to obtain the ambient contraction.
\begin{proposition}[Conormal layers of a prime floor]\label{prop:conormal-layers}
Let \((X,S+B)\) be an ordinary plt pair, where \(X\) is a normal connected
compact K\"ahler space, \(S\) is an irreducible effective \(\Q\)-Cartier
prime, \(B\geq0\) is a rational \(\Q\)-Cartier divisor not containing
\(S\), and \(K_X+S+B\) is an actual \(\Q\)-Cartier adjoint. Suppose
\(\dim S\leq3\). Write the actual residue adjunctions as
\[
(K_X+S)|_S=K_S+\Theta,\qquad
(K_X+S+B)|_S=K_S+B_S .
\]
Let \(\phi:S\to W\) be a projective surjection to a normal compact
analytic space with \(\phi_*\OO_S=\OO_W\). Suppose that the actual
rational lines \(-(K_S+B_S)\) and \(-S|_S\) are \(\phi\)-ample.
For \(k\geq0\), put
\[
\mathscr I_k=\OO_X(-kS),\qquad
\mathscr E_k=\mathscr I_k/\mathscr I_{k+1}.
\]
Then \(S\) is normal, \((S,B_S)\) is klt, and each \(\mathscr E_k\)
is a rank-one reflexive sheaf on \(S\). For any positive global Cartier
index \(q\) of \(S\), there is a finite effective rational Weil divisor
\(\Xi_k\) on \(S\), with \(q\Xi_k\) integral, such that
\begin{equation}\label{eq:conormal-actual-identity}
\mathscr E_k^{[q]}
\simeq \bigl(\OO_X(-kqS)|_S\bigr)\otimes\OO_S(-q\Xi_k),
\qquad 0\leq\Xi_k\leq\Theta\leq B_S .
\end{equation}
Here \(\mathscr E_k^{[q]}=(\mathscr E_k^{\otimes q})^{**}\) is formed
on \(S\), and \(\OO_S(-q\Xi_k)\) is a divisorial reflexive sheaf,
which need not be invertible. Moreover
\begin{equation}\label{eq:conormal-vanishing}
R^i\phi_*\mathscr E_k=0\qquad(i>0,\ k\geq1).
\end{equation}
\end{proposition}
\begin{proof}
Choose an integer \(N>0\) clearing the actual adjoint index and the
Cartier indices of \(S+B\). On the common codimension-one locally free
open, and hence everywhere by reflexive extension,
\[
\omega_X^{[N]}
\simeq \OO_X\bigl(N(K_X+S+B)\bigr)\otimes\OO_X\bigl(-N(S+B)\bigr).
\]
Thus this reflexive canonical power is an actual line. In particular
\(K_X+S\) is an actual rational adjoint; no global canonical Weil
generator is being chosen. Subtracting the effective \(\Q\)-Cartier
boundary from discrepancies shows that \((X,S)\) is plt and \(X\) is
klt. Ordinary plt adjunction
\cite[Lemma~5.3]{DasHacon2024OnMMP} gives normal \(S\) and klt
\((S,B_S)\). With compatible residue embeddings, restriction of the
canonical section of a Cartier multiple of \(B\) gives
\[
B_S=\Theta+B|_S,\qquad B|_S\geq0.
\]
The coefficient computation below also proves \(\Theta\geq0\).
The notation \(S|_S\) always denotes the restriction of the actual
rational normal line.
\subsubsection*{The analytic quotient in codimension two}
Fix a prime \(P\) on \(S\). Near a general point choose a holomorphic
equation \(v\) for a Cartier multiple \(mS\), and normalize the full
finite root cover
\[
t^m=v.
\]
The root algebra is reduced before normalization. It is free with basis
\(1,t,\ldots,t^{m-1}\) over the normal local domain. Its generic
Kummer algebra is reduced in characteristic zero because \(v\ne0\),
and freeness makes the map to that generic algebra injective. A
nilpotent element must therefore be zero.
Finite analytic normalization
\cite[Part~B, Section~4, Corollaries~2--3]{Houzel1961}
gives a finite normal cover \(\tau:Y\to U\) with its complete
\(\mu_m\)-action. It can be disconnected, and we retain all components.
The invariant subalgebra of its normalization is \(\OO_U\):
in the total meromorphic algebra the invariants are the meromorphic
field of \(U\), and normality of \(U\) identifies its integral elements.
Thus the quotient is the original normal space.
This cover is étale at every codimension-one point. Off \(S\), the
equation is a root of a unit. At a general smooth point of \(S\), write
\(v=u x^m\) for a unit \(u\); after choosing a holomorphic root of \(u\),
normalization is a disjoint union of the branches
\(t=\zeta u^{1/m}x\). The lifted floor
\[
S_Y=\operatorname{div}(t)
\]
is reduced and Cartier. Its prime coefficients are one, and on a
normal space the principal divisorial ideal with these coefficients is
the radical ideal of their union.
The finite discrepancy formula is valid here in the analytic category.
Normalize a base change of a model carrying a tested divisorial valuation.
The Hurwitz formula for the corresponding discrete valuation rings and
the actual identity \(K_Y+S_Y=\tau^*(K_X+S)\) give
\begin{equation}\label{eq:cover-discrepancy}
a(F;Y,S_Y)=e(F/E)\,a(E;X,S).
\end{equation}
Extensions of the valuation of \(S\) are exactly the height-one
valuations of the lifted floor. All exceptional log discrepancies
remain positive, so the lifted pair is plt. Applying the same formula
without the floor shows that the components of \(Y\) are klt.
The lifted floor is normal. One can use the ordinary plt adjunction just
cited, or the following independent connectedness argument. On a
projective SNC resolution of the lifted pair, its coefficient-one locus
is the union of the strict floor components, with no exceptional
component. Two such components cannot meet: blowing up their
intersection would give an exceptional log discrepancy zero. They
are therefore disjoint and smooth. Lemma~\ref{lem:floor-connectedness}
gives the equality of their direct-image structure sheaf with that of
the reduced floor. Factoring the map through the finite normalization
of the floor, and applying normal bimeromorphic Hartogs extension on
that normalization, identifies this direct image with the normalization
sheaf. The equality forces the normalization to be an isomorphism.
This use of the connectedness lemma is independent of any contraction
existence assertion.
Choose a component \(P'\) above \(P\) and a general point on it.
The normal \(S_Y\) is smooth there, since \(P'\) has codimension one
in it, and \(P'\) is smooth after a further generic choice. The ambient
\(Y\) is smooth there as well. Indeed, in its local ring \(R'\) the
nonzerodivisor \(t\) has regular quotient \(R'/(t)\). Lifting its
minimal generators and adjoining \(t\) bounds the embedding dimension
of \(R'\) by \(\dim R'\), so \(R'\) is regular.
Remove the finitely many proper fixed loci on \(P'\). At a remaining
point its stabilizer \(G\subset\mu_m\) fixes \(P'\) pointwise. Holomorphic
linearization can preserve the smooth flag
\(P'\subset S_Y\subset Y\): average lifts of an eigenbasis of the
cotangent space inside the respective invariant ideals, and use their
independent differentials as local coordinates. The coordinates tangent
to \(P'\) are fixed. The stabilizer has no ineffective element: its
action sends \(t\) to \(\zeta t\), and the nonzerodivisor \(t\) forces
\(\zeta=1\) for an element acting trivially on the local germ. A
nonidentity stabilizer element cannot fix a transverse hyperplane,
because \(\tau\) is étale in codimension one. Both transverse
characters are consequently faithful. If \(r=|G|\), rescaling the
generator gives the analytic germ
\begin{equation}\label{eq:plt-analytic-quotient}
(X,S,P)\simeq
\bigl((\C^2,\{x=0\},0)/\mu_r(1,a)\bigr)
\times\C^{\dim X-2},\qquad \gcd(a,r)=1 .
\end{equation}
The smooth case is \(r=1\). The local Cartier index of \(S\) is \(r\):
\(x^r\) descends, whereas invariance of a unit times \(x^d\), evaluated
at the fixed point, forces \(r\mid d\).
On the quotient floor let \(z=y^r\). Residue of the \(r\)-th
log-canonical power changes \((dy)^{\otimes r}\) into a nonzero constant
times \(z^{-(r-1)}(dz)^{\otimes r}\); the tangential factors are
unchanged. The actual meromorphic residue embedding therefore gives
\begin{equation}\label{eq:bare-different-index}
\operatorname{coeff}_P\Theta=\frac{r-1}{r}.
\end{equation}
This derivation is on the analytic cover and does not infer a
higher-dimensional analytic quotient theorem from an algebraic slice.
\subsubsection*{Depth and the canonical multiplication}
The sheaves \(\mathscr I_k\) are maximal Cohen--Macaulay on \(X\).
Here is the index-cover argument at every point of \(X\). On a small
open about an arbitrary point, repeat the full normalized root construction
using a local equation for a Cartier multiple \(mS\). Codimension-one
étaleness and the discrepancy calculation above do not require the
generic choice on \(P\). The klt components of this full cover \(Y\)
are Cohen--Macaulay by the last assertion of
Lemma~\ref{lem:floor-connectedness}; its proof uses relative vanishing,
canonical torsion-freeness, and coherent duality. The total meromorphic
Kummer algebra has basis \(1,t,\ldots,t^{m-1}\), even when it splits.
Its character-\(j\) holomorphic summand on this open is
\[
\OO_X(jS)t^j\qquad(0\leq j0).
\]
The bijection of prime valuations for the small map, together with
normal divisorial extension, identifies
\[
p_*\OO_{S'}(G'_k)=\OO_S(G_k)=\mathscr E_k .
\]
This is an equality of meromorphic sections satisfying the same
codimension-one order conditions. Leray gives
\eqref{eq:conormal-vanishing} on \(U\), hence everywhere. This proves
the proposition.
\end{proof}
For \(k\geq1\), put \(S_k=V(\mathscr I_k)\). The inclusions
\(\mathscr I_1^k\subset\mathscr I_k\subset\mathscr I_1\) show that
\(\sqrt{\mathscr I_k}=\mathscr I_1\), so these thickenings have the
common underlying floor. Their conormal sequence is
\[
0\longrightarrow\mathscr E_k\longrightarrow\OO_{S_{k+1}}
\longrightarrow\OO_{S_k}\longrightarrow0 .
\]
Use the underlying continuous map \(|\phi|:|S|\to|W|\) to put
\(\mathscr B_k=|\phi|_*\OO_{S_k}\). Then
\(\mathscr B_1=\OO_W\), and \eqref{eq:conormal-vanishing} gives
epimorphisms of sheaves of rings on \(|W|\)
\begin{equation}\label{eq:thickening-ring-surjection}
\mathscr B_{k+1}\twoheadrightarrow\mathscr B_k\qquad(k\geq1).
\end{equation}
This is the immediate cohomological assertion used in
\cite[Claim~5.9]{DasHacon2024OnMMP}. Exactness is stalkwise, and no
splitting of the augmentation \(\mathscr B_k\to\OO_W\) is assumed.
The next construction realizes these ringed spaces analytically and
constructs the corresponding maps from \(S_k\).
\input{sections/analytic-extension}
\subsection{Descent of a dominating current}
\begin{lemma}[Descent of a dominating current]
\label{lem:dominating-current-descent}
Let \(p:Y\to Z\) be a proper bimeromorphic morphism between normal
compact complex spaces, with \(Y\) K\"ahler. Let \(\eta\) be a smooth
real closed \((1,1)\)-form with local smooth potentials on \(Z\).
Suppose that a positive closed current \(T\) with local potentials
represents the local-potential Bott--Chern class \(p^*[\eta]\), and that
\(T\geq\kappa\) for a K\"ahler form \(\kappa\) on \(Y\). For every
smooth positive Hermitian form \(g\) on \(Z\), understood in local
ambient embeddings, there are a constant \(c>0\) and a global
quasi-plurisubharmonic function \(u_Z\) such that
\begin{equation}\label{eq:dominating-current-descent}
\eta+i\partial\bar\partial u_Z\geq c g.
\end{equation}
This is a current in \([\eta]\) with local potentials, and it has the
bigness property preceding \cite[Theorem~2.29]{DasHaconPaun2024}.
\end{lemma}
\begin{proof}
We make the potential in the class equality explicit. Use the
normal-space potential description of Boucksom--Guedj
\cite[Section~4.6.1]{BoucksomGuedj2013}, specifically Lemma~4.6.1 and
the paragraph following Definition~4.6.2. The soft-sheaf descriptions
of Bott--Chern cohomology by smooth functions and distributions first
give a global real distribution \(u_Y\) for which
\[
T=p^*\eta+i\partial\bar\partial u_Y.
\]
Locally write \(T=i\partial\bar\partial v\) with \(v\) plurisubharmonic
and \(p^*\eta=i\partial\bar\partial h\) with \(h\) smooth. Then
\(u_Y+h-v\) is pluriharmonic as a distribution. By the cited lemma
it is a smooth real part of a holomorphic germ. Thus \(u_Y\) is locally
represented by a quasi-plurisubharmonic function. These representatives
agree as distributions on overlaps and hence, after taking their
upper-semicontinuous representatives, agree pointwise. They give one
global quasi-plurisubharmonic \(u_Y\), which is locally integrable and
locally bounded above.
In local ambient embeddings, \(p^*g\) is represented by a smooth
semipositive form and \(\kappa\) by a smooth positive definite form.
A finite relatively compact cover of \(Y\) therefore gives a constant
\(C>0\) with \(p^*g\leq C\kappa\). Consequently
\[
p^*\eta+i\partial\bar\partial u_Y\geq C^{-1}p^*g.
\]
The map \(p\) is a modification between normal spaces, and \(u_Y\)
has the local upper bound needed in the current descent argument.
\cite[Corollary~2.32 and its proof, p.~18]{DasHaconPaun2024} gives a global
quasi-plurisubharmonic \(u_Z\) with
\eqref{eq:dominating-current-descent} for \(c=C^{-1}\). It is an
\(L^1\) potential, and the formula keeps the descended current in the
chosen Bott--Chern class. The local potentials of \(\eta\) give local
potentials for that current. No K\"ahler form on \(Z\) has been used.
\end{proof}
\subsection{Threefold contractions used in the boundary argument}
\label{subsec:threefold-contraction-bridge}
We prove Proposition~\ref{prop:threefold-contraction} by applying the
cited threefold MMP with two analytic constructions supplied here:
Proposition~\ref{prop:finite-null} contracts the finite null loci, and
Proposition~\ref{prop:prime-floor-extension} extends the prime-floor
contractions. The projective relative MMP, the nonbig argument through
a surface contraction, and the terminal canonical threefold program
remain external inputs. We verify the hypotheses at each use of the
two constructions, then supply the required target positivity; neither
construction alone asserts that its target is K\"ahler. We first record
the cone observation used in the big and floor cases.
For a normal compact space \(Z\) in Fujiki's class \(\mathcal C\), put
\(N^1(Z)=H^{1,1}_{BC}(Z)\) in the local-potential convention. Let
\(N_1(Z)\) be the vector space of real closed currents of bidimension
\((1,1)\), modulo \(T_1\equiv T_2\) when \(T_1(\eta)=T_2(\eta)\) for every
real closed \((1,1)\)-form \(\eta\) with local potentials; the pairing is
evaluation. Following
\cite[Definitions~3.6 and~3.8, pp.~8--9]{HoeringPeternell2016}, let
\(\operatorname{NA}(Z)=\overline{\operatorname{NA}}(Z)\subset N_1(Z)\)
denote the closed cone generated by the numerical classes of positive
closed currents. Thus both notations used below refer to this same closed
cone.
\begin{lemma}[Positivity on the current cone]\label{lem:na-slice}
Let \(Z\) be a normal compact K\"ahler analytic variety with rational
singularities, and let \(\kappa\) be a K\"ahler class. In the
finite-dimensional local-potential numerical spaces put
\[
C=\overline{\operatorname{NA}}(Z),\qquad
\Sigma=\{z\in C:\kappa\cdot z=1\}.
\]
Then \(\Sigma\) is compact. If a real local-potential \((1,1)\)-class
\(\ell\) is strictly positive on \(\Sigma\), then \(\ell\) is
K\"ahler.
\end{lemma}
\begin{proof}
By \cite[Lemma~2.3 and Proposition~2.4]{DasHaconPaun2024}, the natural
pairing is perfect, \(\operatorname{Nef}(Z)=C^*\), and the K\"ahler
cone is open with closure the nef cone. In particular \(\kappa\) lies
in the interior of \(C^*\). A sequence in \(\Sigma\) with unbounded
norm would, after division by its norm and passage to a subsequence,
give a nonzero class \(z\in C\) with \(\kappa\cdot z=0\), contradicting
that interior property. The slice is closed and hence compact.
If it is nonempty, let \(\mu=\min_\Sigma\ell>0\). Homogeneity gives
\(\ell-\frac\mu2\kappa\in C^*=\operatorname{Nef}(Z)\). A smooth
local-potential nef approximation for this last class, bounded below by
\(-\mu\kappa/4\), becomes a positive representative after adding
\(\mu\kappa/2\). Thus \(\ell\) is K\"ahler. If \(\Sigma\) is
empty, then \(C=\{0\}\) by the interior property; the same conclusion
follows because every class is nef and a positive small multiple of
\(\kappa\) can be subtracted first. All cones here are the
local-potential cones in the cited perfect pairing.
\end{proof}
The spaces used below have rational singularities: this is Lemma~2.31
of \cite{DasHacon2024} for dlt pairs and its Remark~2.14 for klt varieties.
\begin{proof}[Proof of Proposition~\ref{prop:threefold-contraction}]
The proof of
\cite[Theorem~5.5]{DasHaconPaun2024} first uses a projective relative
small \(\Q\)-factorialization and then
\cite[Theorem~1.7]{DasHacon2024}. The latter proof passes to a small
strongly \(\Q\)-factorial model and treats a nef nonbig class by its
Theorem~5.5 and a nef big class by its Theorem~6.4. Here strong
\(\Q\)-factoriality means that every global coherent reflexive rank-one
sheaf has an invertible reflexive power. It implies ordinary global
\(\Q\)-factoriality; it does not assert the analogous property for
all analytic open subsets. The small models use the projective relative
MMP of \cite[Proposition~2.26 and Lemma~2.27]{DasHacon2024}, whose maps
are already projective. Compactness of the final bases is retained when
projective maps are composed, as required by
\cite[Remark~2.11]{DasHaconPaun2024}.
\paragraph{The imported programs and target descent.}
The nef nonbig branch follows the chain Theorem~5.5, Corollary~5.4,
and Theorem~5.2 of \cite{DasHacon2024}. It uses Step~4 in the proof of
\cite[Theorem~1.4, p.~242]{HoeringPeternell2015}, which contracts a
negative-definite collection of curves on a smooth compact K\"ahler
surface and descends through a graph. We use that nonbig argument as
an external input. The pseudo-effective cone lineage also uses the
terminal canonical threefold program: Assumption~10.1 of
\cite{DasOu2022} invokes its nonvanishing Theorem~9.1, whose proof at
p.~50 invokes \cite[Theorem~1.1]{HoeringPeternell2015} for a terminal
\(K\)-MMP and Mori fiber space. We retain that terminal program as
an external input. Together with the projective relative MMP just
specified, these are the birational inputs used below.
For an already constructed negative-ray contraction with ordinary
\(\Q\)-factorial compact K\"ahler dlt source and relatively ample
negative adjoint, \cite[Proposition~3.1]{CampanaHoeringPeternell2016}
gives rationality of the target and descent of the supporting class.
We use the proof of its Corollary~3.1 for target positivity. The
corollary assumes non-uniruledness to obtain a nef support; our ray
already has an exposed nef support. Its remaining argument descends
that class, proves positivity on the target current cone, and lifts
target curves through the projective morphism. These steps apply to
the data just stated. The proposition and corollary numbers here are
those of the journal edition. We now verify the finite-null and
prime-floor constructions to which this argument will be applied.
\paragraph{Nef and big classes with a finite null locus.}
Three geometric situations use Proposition~\ref{prop:finite-null}:
a small negative ray, the entire null locus at a stopping model, and a
small ray with pseudo-effective adjoint. The big threefold program uses
strongly \(\Q\)-factorial klt sources. The last situation can also
occur for ordinary dlt data, whose underlying space is ordinary
\(\Q\)-factorial and klt, as checked below. In every case the relevant
locus is the full top-intersection null locus of the proposition.
For a small negative ray, the small branch of
\cite[proof of Theorem~4.16, p.~40]{DasHacon2024} has, after its boundary
perturbation, a strongly \(\Q\)-factorial klt source and a nef big
supporting class \(a=(K+\Gamma')+\kappa\) with \(\kappa\) K\"ahler.
At this input \cite[Lemma~4.5]{DasHacon2024} says that an \(a\)-null
surface is covered by \(a\)-null curves. All those curves lie in the
exposed ray. Such a covering of a surface contradicts the smallness of
that ray. Lemma~\ref{lem:no-null-surface} therefore makes
\(\operatorname{Null}(a)\) a finite union of curves, and
Proposition~\ref{prop:finite-null} constructs its contraction.
This argument permits a non-pseudo-effective underlying adjoint.
For the entire null locus at the final model \(X_n\) of the program in
\cite[Claim~6.5 and proof of Theorem~6.4, pp.~50--51]{DasHacon2024},
the pair is strongly \(\Q\)-factorial klt, its class \(a_n\) is nef
and big, and every \(a_n\)-null curve has nonnegative degree for the
running adjoint \(A_n\). These are the stopping data from that program.
Its null-surface argument, using Lemma~4.5 of the same source, would
cover a null surface by \(a_n\)-null curves on which the remainder
\(a_n-c_1(A_n)\) has positive degree. Their \(A_n\)-degrees would
then be negative, contrary to the stopping data. There is therefore no
null surface. Lemma~\ref{lem:no-null-surface} again gives finitely many
null curves. Apply Proposition~\ref{prop:finite-null} to their entire
union. This use has no single-ray hypothesis and needs only the proper
bimeromorphic map to a normal compact analytic target supplied by that
proposition.
For a small ray with pseudo-effective adjoint, the branch in the proof of Theorem~6.4 at
p.~50 invokes the small case of Theorem~2.23(1) of the same source;
this case can also occur in its Theorem~3.1, Claim~3.2. Let \(A\) be
the negative adjoint and \(b\) an exposed nef support for the ray.
It may be scaled so that \(b-c_1(A)\) is K\"ahler. To check this
usual support step, normalize the closed cone of positive
bidimension-\((1,1)\) classes by a K\"ahler class. Its normalized slice
is compact. The continuous function \(-c_1(A)\) is strictly positive
on its intersection with the exposed ray, hence on a neighborhood of
that intersection. On the compact complement, the support has a
positive minimum. A large multiple of the support minus \(c_1(A)\)
is consequently strictly positive on the full slice.
Lemma~\ref{lem:na-slice} makes this difference K\"ahler. Since
\(A\) is pseudo-effective, the scaled
\(b\) is big. The same Lemma~4.5 and the smallness of the ray exclude
null surfaces, so Lemma~\ref{lem:no-null-surface} and
Proposition~\ref{prop:finite-null} apply. At the outer Theorem~6.4
use the pair is klt. If this step is made for the ordinary dlt data in
Claim~3.2, the underlying space is still klt, which is the singularity
condition of Proposition~\ref{prop:finite-null}; a small decrease of
the boundary preserves negativity when a klt adjoint is needed in a
subsequent result.
In each of the two single-ray situations just described, the constructed
map contracts exactly that ray. A positive-dimensional reduced fiber
is a finite connected union of compact curves, hence is projective.
For the corresponding negative adjoint \(A\), a positive global Cartier
multiple of the actual line \(-A\) has positive degree on every
irreducible component of that fiber, and is
therefore ample on the fiber. It is ample also on every zero-dimensional
fiber. Ampleness passes from a reduction to its nilpotent thickening by
the positive-metric criterion
\cite[Lemma~2.4 and Corollary~1.12]{Fujino2026Fujiki}. The proper
fiberwise criterion \cite[Definition~3.1 and Remark~3.2]{Fujino2026Fujiki}
now makes this one actual line relatively ample, so the contraction is
projective. Normality and proper bimeromorphicity give connected fibers.
The source is ordinary \(\Q\)-factorial klt, or dlt in the indicated
variant, and the exposed nef support is unchanged. Thus
\cite[Proposition~3.1 and proof of Corollary~3.1]{CampanaHoeringPeternell2016},
applied as specified above, gives the K\"ahler target and the descended
supporting class.
Thus Proposition~\ref{prop:finite-null} supplies the normal compact
analytic contraction, and this postprocessing supplies projectivity and
positivity in the single-ray cases. The entire-null cleanup uses the
different argument that follows.
In that cleanup, write \(\mu:X_n\to Z\) for the entire-null contraction
just constructed. The graph argument in
\cite[proof of Theorem~6.4, pp.~51--52]{DasHacon2024} descends its
composite with the program to a morphism \(\psi:X_0\to Z\), where
\(X_0\) is the original strongly \(\Q\)-factorial klt source of that
theorem. All steps of the program are trivial for the transported
supporting class. Before applying its Lemma~2.44 to \(\psi\), restore
the original boundary \(\Gamma^{\rm orig}\) and the original nef big
difference \(\omega^{\rm orig}\) on \(X_0\), before the internal
boundary replacement. The original supporting class is
numerically trivial on its contracted curves, and hence
\begin{equation}\label{eq:cleanup-original-boundary}
-(K+\Gamma^{\rm orig})\equiv_\psi\omega^{\rm orig}.
\end{equation}
The left side is \(\psi\)-nef and is \(\psi\)-big for the bimeromorphic
map. The rationality result of Lemma~2.44 thus applies to this original
Q-Gorenstein klt pair and makes the target \(Z\) rational. This step
uses the original nef big difference; the transported remainder
\(a_n-c_1(A_n)\) on the stopping model is not substituted into
\eqref{eq:cleanup-original-boundary}.
For the entire-null contraction \(\mu\), the source \(X_n\) is a
normal compact K\"ahler klt
threefold, so it has rational singularities. The target is normal and
compact, and is in Fujiki's class \(\mathcal C\): a projective resolution
of \(X_n\) with smooth source is compact K\"ahler, and its composite
with \(\mu\) is bimeromorphic. The exceptional image
\(E_Z=\mu(\operatorname{Null}(a_n))\) is finite. Every curve contracted
by \(\mu\) is \(a_n\)-null. Apply
\cite[Lemma~2.11]{DasHacon2024} to this map between normal compact
rational spaces in class \(\mathcal C\). It gives a smooth real closed
\((1,1)\)-form \(\eta_Z\) with local smooth potentials such that
\begin{equation}\label{eq:cleanup-descended-class}
a_n=\mu^*[\eta_Z]
\quad\text{in the local-potential Bott--Chern group.}
\end{equation}
The normalized graph constructed \(\psi\) earlier; this application of
Lemma~2.11 uses \(\mu\).
We check the three analytic positivity hypotheses on \(Z\). Fix a
smooth positive Hermitian form \(g_Z\) in local ambient embeddings.
The bimeromorphic nef descent theorem
\cite[Lemma~3.13]{HoeringPeternell2016} applies to the normal compact
threefold \(Z\) in class \(\mathcal C\) and the normal compact
threefold \(X_n\). It makes \([\eta_Z]\) analytically nef from
\eqref{eq:cleanup-descended-class}; it does not assume that \(Z\) is
K\"ahler. If its defining approximation uses a different positive
reference form, compact comparison with \(g_Z\) and rescaling the
approximation parameter give, for every \(\delta>0\), a smooth
function \(f_\delta\) with
\begin{equation}\label{eq:cleanup-nef}
\eta_Z+i\partial\bar\partial f_\delta\geq-\delta g_Z.
\end{equation}
Changing a smooth representative of the same local-potential class only
changes \(f_\delta\) by a global smooth potential.
The big class \(a_n\) on the compact K\"ahler source contains a positive
closed current \(T_n\) with local potentials and
\(T_n\geq\kappa_n\) for a K\"ahler form \(\kappa_n\). In view of
\eqref{eq:cleanup-descended-class}, Lemma~\ref{lem:dominating-current-descent}
applied to \(\mu\) gives a global quasi-plurisubharmonic, hence
\(L^1\), function \(u_Z\) and a constant \(c>0\) satisfying
\begin{equation}\label{eq:cleanup-big}
\eta_Z+i\partial\bar\partial u_Z\geq c g_Z.
\end{equation}
This establishes the required current bigness on the not yet K\"ahler
target in the selected class \([\eta_Z]\).
Finally let \(V\subset Z\) be an irreducible reduced analytic subspace
of dimension \(d>0\). Its strict transform \(V'\) is the closure
of the inverse image of \(V\setminus E_Z\). It has dimension \(d\),
maps bimeromorphically to \(V\), and is not contained in
\(\operatorname{Null}(a_n)\). Nef approximations on \(X_n\), their
restriction to the integration cycle of \(V'\), and Stokes' theorem
give \(a_n^d\cdot V'\geq0\). Equality would put \(V'\) into the
null locus by its definition, so the inequality is strict. The proper
cycle identity \(\mu_*[V']=[V]\) and projection formula therefore give
\begin{equation}\label{eq:cleanup-positive-intersections}
\int_{V_{\rm reg}}\eta_Z^d=a_n^d\cdot V'>0.
\end{equation}
This includes \(V=Z\); it requires neither \(V\) nor \(V'\) to be
normal. Additivity over the top-dimensional irreducible components
gives the same positivity for every positive-dimensional compact reduced
analytic subspace.
Equations \eqref{eq:cleanup-nef}, \eqref{eq:cleanup-big}, and
\eqref{eq:cleanup-positive-intersections} are the three hypotheses of
\cite[Theorem~2.29]{DasHaconPaun2024}. Apply that theorem directly to
\(\eta_Z\). It gives a smooth potential making \(\eta_Z\) positive
definite, so \([\eta_Z]\) is K\"ahler and \(Z\) is compact K\"ahler.
This verifies the positivity conclusion at the cleanup contraction.
\paragraph{Divisorial contractions.}
In the point case, let \(P\) be the selected prime surface and
\(\nu:P^\nu\to P\) its normalization. The nef-dimension-zero
condition for the supporting class is taken on \(P^\nu\); the whole
normalized fiber has zero restricted class by
\cite[Theorem~3.19(a) and its proof, p.~234]{HoeringPeternell2015}.
Use \cite[Corollary~4.4]{DasHacon2024OnMMP}, whose proof invokes its
Lemma~4.3, at the stated data: a \(\Q\)-factorial compact K\"ahler
source, a nef big exposed support, ray curves covering \(P\), and zero
restriction on \(P^\nu\). The corollary constructs the proper analytic
point contraction and its proof supplies an actual ample line
\(\OO_P(-mP)\) for some \(m>0\). Thus \(P\) is projective, and
ampleness persists on the full possibly nonreduced fiber by the
positive-metric criterion used above. The full-fiber criterion makes
\(-mP\) relatively ample.
Every curve of \(P\) lifts finitely to its normalization. The zero
restricted support therefore gives zero degree on every such curve, so
its ambient class lies in the exposed ray. The negative adjoint
restricted to \(P\) is consequently numerically a fixed positive
multiple of the negative normal line. Both have actual rational Cartier
multiples. Numerical invariance of ampleness on the projective \(P\)
and then the full-fiber criterion make the negative adjoint relatively
ample. The same
\cite[Proposition~3.1 and proof of Corollary~3.1]{CampanaHoeringPeternell2016}
then supplies rationality, class descent, and target positivity for this
negative-ray contraction. The point-contraction corollary was used
to construct the underlying proper analytic map.
For a divisor-to-curve step, Proposition~\ref{prop:prime-floor-extension}
supplies the ordinary extension used both in
\cite[Theorem~3.1, Claim~3.2]{DasHacon2024} and in its replay in
Theorem~4.16 of that source. We verify the two ample-line hypotheses
for this use. Write \((M,\Delta_M)\) for the strongly
\(\Q\)-factorial compact K\"ahler dlt threefold there,
\(A=K_M+\Delta_M\), \(P\) for the selected prime floor, and \(R\)
for its exposed ray. The selected ray satisfies
\[
A\cdot R<0,\qquad P\cdot R<0.
\]
Adjunction makes \(P\) a normal compact K\"ahler dlt surface and makes
\(A|_P\) its actual full adjoint. For its inclusion \(i:P\to M\), put
\[
F=\{z\in\overline{\operatorname{NA}}(P):i_*z\in R\}.
\]
Fix a K\"ahler class \(\kappa\) on \(M\) and a nonzero \(r\in R\).
For \(z\in F\) with \(\kappa|_P\cdot z=1\), positivity of
\(\kappa|_P\) gives \(i_*z=r/(\kappa\cdot r)\). Thus on the full
compact normalized vertical face
\begin{equation}\label{eq:inner-floor-positive-face}
-(A|_P)\cdot z=\frac{-A\cdot r}{\kappa\cdot r}>0,
\qquad
-(P|_P)\cdot z=\frac{-P\cdot r}{\kappa\cdot r}>0 .
\end{equation}
This uses the cone of positive current classes and includes its limits.
Let \(\zeta\) be the nef support exposing \(R\). Its restriction has
null cone \(F\). The compact-slice argument just used makes
\(\lambda\zeta|_P-c_1(A|_P)\) K\"ahler for sufficiently large
\(\lambda\): near \(F\) the negative adjoint has the uniform positive
bound in \eqref{eq:inner-floor-positive-face}, and away from \(F\)
the support has a positive minimum. Lemma~\ref{lem:na-slice} on the
rational compact K\"ahler surface turns this bound into a K\"ahler
class. The ordinary dlt case of the surface theorem
\cite[Theorem~2.32]{DasHaconYanez2026} therefore gives a projective
surjection \(\gamma:P\to W_P\) with connected fibers onto a normal
compact K\"ahler space, contracting exactly \(F\), with the support
pulled back from a K\"ahler class on \(W_P\). This use is of the
surface theorem.
Applying the same compact-slice estimate after adding a sufficiently
large pullback of that target K\"ahler class makes each of
\(-A|_P\) and \(-P|_P\) relatively K\"ahler over \(W_P\).
They have actual global rational Cartier multiples by the factoriality
of \(M\). The proper fiberwise positivity criterion makes both lines
\(\gamma\)-ample, including when a fiber is the whole surface.
Put \(H=\floor{\Delta_M}-P\). For small rational \(\varepsilon>0\),
the pair \((M,\Delta_M-\varepsilon H)\) is plt with sole floor \(P\).
Indeed on a dlt resolution the pullback of the effective globally
\(\Q\)-Cartier divisor \(H\) is effective. Subtracting it preserves
the strict inequality for every exceptional crepant coefficient and
lowers all the other strict floor coefficients below one. The negative
degree of \(A-\varepsilon H\) on \(R\) persists for small
\(\varepsilon\), so the first relative ample line persists by
\eqref{eq:inner-floor-positive-face}; the second is unchanged.
These are precisely the hypotheses of
Proposition~\ref{prop:prime-floor-extension}.
That proposition constructs the extension, whose fiber sets show that the ambient map
contracts exactly the original ray and is an isomorphism off \(P\).
The negative original adjoint is relatively ample by the same full-fiber
criterion. Return at once to \(\Delta_M\) and \(A\) for the running
program. For this projective negative-ray contraction,
\cite[Proposition~3.1 and proof of Corollary~3.1]{CampanaHoeringPeternell2016}
applies with the already exposed support, exactly as specified at the
start of the proof. It gives rationality, class descent, and K\"ahler
target positivity. Strong factoriality is preserved by
\cite[Lemma~2.5]{DasHacon2024}, and an
ordinary projective flip, when needed, is supplied by its Theorem~2.24.
Thus the original support and boundary are the ones transported to the
next step.
\paragraph{Return to the original threefold.}
The finite-null and divisorial constructions now supply the indicated
steps of the cited big-case program, with their required positivity.
Together with the external inputs stated at the start, they give the
asserted contraction for the data in
\eqref{eq:threefold-exposed-input} by the proof of
\cite[Theorem~1.7]{DasHacon2024}. The proof of
\cite[Theorem~5.5]{DasHaconPaun2024} descends the contraction from the
small model. A pulled-back K\"ahler class has degree zero on a constant
curve and positive degree on a nonconstant curve, as seen on its
normalization. This proves the asserted curve criterion and completes
Proposition~\ref{prop:threefold-contraction}.
\end{proof}
For the separate exposure input, the proof of Theorem~5.2 and
Corollary~5.3 of \cite{DasHaconPaun2024} uses the same Theorem~1.7 for
the supporting contractions, followed by the projective relative cone
theorem and convex separation. Its contraction inputs are therefore
the ones just established.
\subsection{The actual floor split}
\label{subsec:actual-floor-split}
We verify the use of that specialization for the dlt floor in the
fourfold supported program. Let \((X,S+B)\) be one of its compact
ordinary \(\Q\)-factorial K\"ahler dlt pairs, with reduced floor
\(S\), coefficients of \(B\) below one, and actual adjoint
\(A=K_X+S+B\). Fix a prime \(T\) of \(S\), and write
\[
H=S-T,\qquad A_0=A-H=K_X+T+B.
\]
The finite global divisor \(H\) is \(\Q\)-Cartier by ordinary global
factoriality, and \(A_0\) is an actual rational line. On a dlt
resolution, subtracting the effective pullback of \(H\) lowers the
exceptional crepant coefficients and removes the strict transforms of
the other floor components. Only the strict transform of \(T\) has
coefficient one. Hence \((X,T+B)\) is plt. Ordinary plt adjunction
\cite[Lemma~5.3]{DasHacon2024OnMMP} gives a normal \(T\), an effective
rational boundary \(B_0\) with \((T,B_0)\) klt, and the actual
restriction \(A_0|_T=K_T+B_0\).
Choose one positive integer \(m\) clearing the indices of \(A,A_0,H\).
The canonical section of \(\OO_X(mH)\) restricts nontrivially to \(T\)
because \(T\) is not a component of \(H\). Set
\[
B'=\frac1m\operatorname{div}(s_{mH}|_T)\geq0.
\]
It is an effective rational \(\Q\)-Cartier divisor. Use compatible
local meromorphic residue embeddings for the two adjunctions. Their
ratio in codimension one on \(T\) is multiplication by \(s_{mH}|_T\).
Reflexive extension on normal \(T\) then gives the divisor split and
actual line identities
\begin{equation}\label{eq:actual-floor-boundary-split}
\begin{aligned}
B_T&=B_0+B',\\
\OO_T(m(K_T+B_0))&\simeq\OO_X(mA_0)|_T,\\
\OO_T(mB')&\simeq\OO_X(mH)|_T.
\end{aligned}
\end{equation}
where \(A|_T=K_T+B_T\) is the full dlt adjunction. This proves the
split without assuming \(T\) to be \(\Q\)-factorial.
Suppose, as in the supported program, that
\(\alpha_T=c_1(A|_T)+[\omega_T]\) is nef and \(\omega_T\) is
K\"ahler. These properties hold by restricting the ambient smooth
metric nef approximations and local strictly plurisubharmonic potentials.
For a sufficiently small rational \(\varepsilon>0\), put
\[
B_\varepsilon=B_0+(1-\varepsilon)B',\qquad
\omega_\varepsilon=\omega_T+\varepsilon\theta_{B'},
\]
where \(\theta_{B'}\) is the normalized curvature of a smooth metric
on a Cartier multiple of \(B'\). The full adjunction is lc and the
lower adjunction is klt. Affineness of discrepancies in this convex
combination makes \((T,B_\varepsilon)\) klt. On finitely many compact
local embedding charts, the Hessian of \(\theta_{B'}\) is bounded
and that of \(\omega_T\) is uniformly positive. For the chosen small
\(\varepsilon\), \(\omega_\varepsilon\) is therefore K\"ahler and
\[
\alpha_T=c_1(K_T+B_\varepsilon)+[\omega_\varepsilon].
\]
If \(\dim T=3\), Proposition~\ref{prop:threefold-contraction} applies
and gives a projective
connected-fiber contraction \(T\to W\) to a normal compact K\"ahler
target with \(\alpha_T\) pulled back from a K\"ahler class. This is
the use of \cite[Corollary~5.6]{DasHaconPaun2024} in its Theorem~7.2.
The restricted class \(\alpha_T\) is not assumed big, and its null
face may have several rays. The contraction follows from the assembled
argument in the preceding subsection, using
Proposition~\ref{prop:prime-floor-extension} and the specified external
results.
==== END Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//contraction-inputs.tex ====
==== BEGIN Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//conventions.tex ====
\section{Actual adjoints and analytic conventions}\label{sec:conventions}
We collect the conventions that keep the argument on the given
holomorphic line bundles. All complex spaces are Hausdorff and countable
at infinity. A normal connected complex space is irreducible: its locally
irreducible components are open as well as closed. All divisors called
effective are Weil divisors with nonnegative coefficients.
\subsection{Adjoints, residues, and singularities}
For a normal complex space \(X\), let \(j:X_{\mathrm{reg}}\hookrightarrow X\)
be the regular locus and set \(\omega_X=j_*\omega_{X_{\mathrm{reg}}}\).
For an integral Weil divisor \(H\), \(\OO_X(H)\) is the rank-one reflexive
divisorial sheaf. If \(\Delta\) is rational and an integer \(r>0\)
clears its coefficients, the adjoint index condition is the invertibility
of
\begin{equation}
\mathscr L_r=\bigl(\omega_X^{[r]}\otimes\OO_X(r\Delta)\bigr)^{**}.
\label{eq:actual-adjoint}
\end{equation}
All later degrees are chosen divisible by such an index. Tensor powers
of \(\mathscr L_r\), rather than a separately chosen global canonical
divisor, define \(\OO_X(m(K_X+\Delta))\). Equality in \(\operatorname{Pic}(X)_\Q\)
means an isomorphism of holomorphic line bundles after a common positive
multiple. The notation \(P\sim_\Q K_X+\Delta\) for a rational
\(\Q\)-Cartier divisor means precisely
\(\OO_X(mP)\simeq\OO_X(m(K_X+\Delta))\) in such a degree.
A local frame of \(\mathscr L_r\) restricts on the regular locus to a
meromorphic \(r\)-pluricanonical form with the allowed boundary poles.
Pulling that form back differentially along a proper bimeromorphic model
\(p:W\to X\) with \(W\) smooth defines a rational crepant subboundary \(G_W\) by
\begin{equation}
K_W+G_W=p^*(K_X+\Delta).
\label{eq:crepant}
\end{equation}
This equality records both the rational divisor of the meromorphic map
and the actual isomorphism of the pulled-back line. It is independent
of the local frame: a change of frame is a holomorphic unit, and both
pullbacks change by its pullback. At a prime divisor \(E\) on \(W\),
\[
a(E;X,\Delta)=1-\operatorname{coeff}_E(G_W)
=1+\operatorname{coeff}_E\bigl(K_W-p^*(K_X+\Delta)\bigr)
\]
is the log discrepancy. We use the same definition on higher smooth
models. The pair is lc if all these numbers are nonnegative and klt if
they are positive. We use the standard Koll\'ar--Mori dlt notion, as
imported for analytic pairs in
\cite[Definition~2.7(4)]{DasHaconPaun2024}; in particular, a dlt pair is
klt away from its coefficient-one floor. Theorem~\ref{thm:floor-generation}
uses in addition the globally simple projective resolution supplied by
Lemma~\ref{lem:special-resolution} in the fourfold application.
Theorem~\ref{thm:supported-lifting} uses the standard dlt notion without
this additional resolution hypothesis.
On an SNC pair, adjunction to an intersection of distinct coefficient-one
components is the iterated meromorphic residue. In a degree divisible
by the adjoint index and by two, interchanging two residue operations
does not change the resulting pluriform. We always use such even degrees
when comparing paths through boundary strata. Equality of the pulled-back
invertible subsheaves of meromorphic pluriforms is stronger than an abstract
isomorphism of lines: the former fixes the residue transport of sections.
This stronger equality is what we mean by a \emph{crepant residue
comparison}. It will be proved for every comparison used below.
Reflexive extension is used in the following precise form. An isomorphism
between rank-one reflexive sheaves on a normal space, defined away from an
analytic subset of codimension at least two, extends uniquely if it is
given there by the same meromorphic identification. In particular, a
meromorphic pluriadjoint identity transported through a bimeromorphic map
that extracts no prime divisor can be tested in codimension one and then
extended. An isomorphism merely in numerical cohomology would not support
this operation.
We use ordinary global \(\Q\)-factoriality: every global Weil divisor has
a Cartier multiple, and a reflexive power of the canonical sheaf is
invertible. This is the convention of
\cite[Definition~2.7]{DasHaconPaun2024}. It does not assert factoriality
of all analytic germs. The input of Theorem~\ref{thm:main} has no such
factoriality assumption.
\subsection{K\"ahler positivity and section spaces}
A K\"ahler form on a complex space is given by smooth strictly
plurisubharmonic local potentials in local embeddings into complex
Euclidean spaces. Smooth metrics and their curvature are interpreted
in the same way. For a rational line represented by \(L_r\), we use the
metric criterion for analytic nefness stated in the introduction. It is
independent of the index and of the fixed K\"ahler form. Restriction to a
closed analytic subspace preserves the inequalities and hence nefness.
For a holomorphic map from a compact K\"ahler space, pullback also preserves
nefness: after pulling back the metric inequality, bound the pulled-back
fixed form by a constant multiple of a fixed K\"ahler form on the source.
This argument will be used for resolutions and for restrictions to
possibly singular strata. Curve intersection tests occur only in the
relative projective MMP calculations where their use is justified.
For an actual rational line \(D\), its Iitaka dimension is \(-\infty\)
if all positive Cartier powers have no sections; otherwise it is the
maximum dimension of the meromorphic images of their complete systems.
On an irreducible compact space, two linearly independent sections of one
line have a nonconstant meromorphic quotient. Consequently, if
\(\kappa(D)=0\), every Cartier degree has at most one independent section.
Conversely, unbounded dimensions of the section spaces of positive powers
force \(\kappa(D)\geq1\): one of those spaces has two independent sections.
We will repeatedly use two elementary consequences of normality. If
\(p:Y\to X\) is a proper bimeromorphic morphism between normal spaces, then
\(p_*\OO_Y=\OO_X\). Hence, for a line \(L\) on \(X\),
\[
H^0(Y,p^*L)=H^0(X,L).
\]
If \(p^*L\) is generated, then \(L\) is generated: at any \(x\in X\)
choose \(y\in p^{-1}(x)\) and a pulled-back section nonzero at \(y\).
The corresponding section of \(L\) is nonzero at \(x\). If a positive
multiple of \(D\) has a nowhere-vanishing section, that section is an
actual isomorphism \(\OO_X\simeq\OO_X(mD)\).
The second consequence concerns exceptional divisors. If \(E\geq0\)
is \(p\)-exceptional and integral, then
\begin{equation}
p_*\OO_Y(E)=\OO_X. \label{eq:exceptional-hartogs}
\end{equation}
Indeed, a section is a meromorphic function on \(X\) that is holomorphic
away from the codimension-at-least-two image of \(E\); normal Hartogs
extension makes it holomorphic everywhere. The same statement applies
to rational exceptional divisors after clearing their denominators.
Combined with projection formula, it identifies the section spaces in
the exceptional comparisons below.
\subsection{A connectedness and rationality lemma}
The following sheaf calculation is used both for the conormal layers of
a prime floor and for descent of compatible boundary sections. Its klt
case also supplies rationality and the Cohen--Macaulay property for the
strata used later.
\begin{lemma}\label{lem:floor-connectedness}
Let \((T,B_T)\) be a normal irreducible effective lc analytic pair with actual
\(\Q\)-Cartier adjoint, and let \(p:U\to T\) be a projective resolution
with smooth source whose crepant subboundary \(G\) has SNC support. Put
\[
R=G^{=1},\qquad H=\ceil{-G^{<1}}.
\]
Then \(H\) is effective and exceptional, has no component in common with
\(R\), and
\begin{equation}\label{eq:floor-pushforward}
p_*\OO_R=\OO_{p(R),\red}.
\end{equation}
In particular \(R\to p(R)\) has connected fibers. If the pair is klt,
then \(T\) has rational singularities and is Cohen--Macaulay.
\end{lemma}
\begin{proof}
The strict boundary coefficients belong to \([0,1]\), so positive
coefficients of \(H\) occur only on exceptional primes. Coefficient by
coefficient,
\[
H-R=-\floor G,
\qquad (H-R)-(K_U+\{G\})=-p^*(K_T+B_T).
\]
The fractional boundary \(\{G\}=G-\floor G\) is effective SNC with
coefficients below one. The displayed adjoint difference is relatively
nef and big for the bimeromorphic \(p\): it is relatively trivial, and
the generic relative dimension is zero. Analytic relative
Kawamata--Viehweg vanishing
\cite[Theorem~2.41]{DasHaconPaun2024} gives
\(R^ip_*\OO_U(H-R)=0\) for \(i>0\). Exceptional Hartogs extension gives
\(p_*\OO_U(H)=\OO_T\). The sequence for the Cartier divisor \(R\)
therefore yields a surjection
\[
\OO_T\longrightarrow p_*\OO_R(H).
\]
It factors through \(p_*\OO_R\). Multiplication by the canonical section
of \(H\) embeds \(\OO_R\) into \(\OO_R(H)\), because \(R\) is reduced
SNC and shares no component with \(H\). The surjection consequently
makes that embedding an isomorphism after pushforward and makes
\(\OO_T\to p_*\OO_R\) surjective. Its kernel is exactly the reduced
ideal of the proper image \(p(R)\): a holomorphic function vanishes after
restriction to \(R\) precisely when it vanishes on that image. This
proves \eqref{eq:floor-pushforward}; Stein factorization gives connected
fibers.
For the last assertion, take a klt resolution, so \(R=0\). The same
vanishing and Hartogs calculation identify
\(Rp_*\OO_U(H)\simeq\OO_T\). The factorization
\[
\OO_T\longrightarrow Rp_*\OO_U\longrightarrow Rp_*\OO_U(H)\simeq\OO_T
\]
is the identity, as can be checked on degree zero. It splits the first
map. Apply proper coherent duality. Canonical higher direct images of
the resolution vanish: the canonical torsion-freeness theorem
\cite[Theorem~2.9 and Proposition~2.11]{Fujino2023AnalyticVanishing}
applies locally on the base, and these higher images vanish generically
for a bimeromorphic map. The required local K\"ahlerness follows by
restricting to a small Stein base neighborhood and combining a relative
ample metric with a pulled-back strictly plurisubharmonic potential.
Thus the dual split surjection is the trace
\[
p_*\omega_U[\dim U]\longrightarrow\omega_T^\bullet,
\]
where \(\omega_T^\bullet\) is the dualizing complex. It follows that this
complex is concentrated in degree \(-\dim U\). On that cohomology sheaf
the trace is also injective: its source is torsion-free and the map is
generically an isomorphism. Hence it is an isomorphism. Biduality gives
\(Rp_*\OO_U=\OO_T\). This is rational singularity, and the concentration
of the dualizing complex is the Cohen--Macaulay property.
\end{proof}
==== END Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//conventions.tex ====
==== BEGIN Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//covers.tex ====
\section{Root neighborhoods and a split residue map}\label{sec:covers}
We begin the proof of Theorem~\ref{thm:supported-lifting}. A generated
multiple of the actual adjoint restriction constructs a morphism from
the reduced boundary to projective space. Near its compact fibers we
will replace the supported divisor by a reduced Cartier covering divisor
and study sections on its finite neighborhoods. The use of neighborhood
covers and successive finite thickenings has its threefold antecedents in
\cite[Sections~1--2 and~4]{Miyaoka1988} and
\cite[Section~4]{Kawamata1992}. Here no extension of the boundary map
off the reduced boundary is assumed.
Fix the data of Theorem~\ref{thm:supported-lifting}, and put
\(S=\floor B\), so \(\Supp P=\Supp S\). Choose a sufficiently divisible even integer
\(q>0\) which clears the actual adjoint and equivalence indices, the
boundary and \(P\) coefficients, and a generated degree of \(A|_S\).
There are then an actual line and section
\begin{equation}\label{eq:lifting-data}
L=\OO_V(qA),\qquad s\in H^0(V,L),\qquad
G=\operatorname{div}_L(s)=qP=\sum_i d_iS_i,
\end{equation}
where every \(d_i\) is a positive integer, and a morphism
\begin{equation}\label{eq:boundary-map}
f:S\longrightarrow T=\PP^b,\qquad L|_S\simeq f^*\OO_T(1).
\end{equation}
The morphism is constructed from a generating system on \(S\); it is
used only on that reduced subspace and is not required to be surjective.
Choose \(r>0\) divisible by \(q\) and
every \(d_i\), and put \(a=r/q\). In particular \(a\) is a positive
integer and \(d_i\leq r\).
\subsection{Root pairs near a compact analytic subspace}
\begin{lemma}[A root with a prescribed boundary comparison]
\label{lem:root-neighborhood}
Let \(A\) be a compact analytic subspace of a Hausdorff complex analytic
space \(X\), and let \(L\) be a holomorphic line on a neighborhood of
\(A\). Suppose a line \(Q\) on \(A\) and an isomorphism
\(\beta:Q^{\otimes r}\simeq L|_A\) are given, with \(r>0\).
There are a neighborhood \(U\) of \(A\), a line \(P\) on \(U\), and
isomorphisms
\[
\alpha:P^{\otimes r}\simeq L|_U,\qquad \gamma:P|_A\simeq Q,
\qquad \alpha|_A=\beta\circ\gamma^{\otimes r}.
\]
If two root pairs are already defined near \(A\), every prescribed
power-compatible isomorphism between their restrictions to \(A\) extends
to an isomorphism on a neighborhood, uniquely as a germ about \(A\).
The root pair itself is not asserted unique.
\end{lemma}
\begin{proof}
The analytic Kummer sequence
\[
1\longrightarrow\mu_r\longrightarrow\OO_X^\times
\xrightarrow{(\cdot)^r}\OO_X^\times\longrightarrow1
\]
is exact also for singular or nonreduced analytic spaces. A unit germ
has a local holomorphic root; the kernel is the locally constant sheaf
of \(r\)th roots of unity, since \(u^r-1\) has distinct roots, including
in a local ring with nilpotents. Line bundles are classified by
\(H^1(\OO^\times)\), and abelian torsors by \(H^1\) of the corresponding
sheaf \cite[Tags~09NU and~02FQ]{StacksProject}.
For an abelian sheaf \(F\) and a compact subset whose distinct points
have disjoint neighborhoods, SGA~4 continuity
\cite[Expos\'e~Vbis, Lemma~4.1.3]{SGA4Continuity} gives
\begin{equation}\label{eq:compact-continuity}
\varinjlim_{U\supset A}H^i(U,F)=H^i(A,F|_A)
\quad\text{for every }i.
\end{equation}
The separation hypothesis holds in the Hausdorff analytic space. Although
the intrinsic structure sheaf of \(A\) need not be the inverse image of
\(\OO_X\), the inverse image of \(\mu_r\) is exactly \(\mu_r\) on \(A\).
Naturality of Kummer therefore makes the obstruction to a root of \(L\)
restrict to zero in \(H^2(A,\mu_r)\), because \(Q\) is such a root there.
Continuity in degree two kills that obstruction on a smaller neighborhood,
giving a root pair \((P,\alpha)\).
The difference between \((P|_A,\alpha|_A)\) and \((Q,\beta)\) is a
\(\mu_r\)-torsor. By continuity in degree one it extends to a torsor on
a smaller neighborhood. The associated line with its trivialized
\(r\)th power twists \(P\) to give the prescribed pair on \(A\).
Finally, the sheaf of compatible isomorphisms of two root pairs is itself
a locally trivial finite \(\mu_r\)-torsor. A prescribed section on \(A\)
trivializes its restriction there. Continuity in degree one therefore
trivializes this torsor on a smaller neighborhood; choose a section there.
The ratio of its boundary restriction to the prescribed section is a
section of \(\mu_r\) on \(A\). Continuity in degree zero extends that
ratio uniquely as a germ, and correcting the chosen section gives the
prescribed extension. The same degree-zero injectivity gives uniqueness
as a germ. These sections are holomorphic, being locally solutions of
\(z^r=u\) for a holomorphic unit.
\end{proof}
Apply the lemma near a parameter \(t\in T\) to
\(A=(f^{-1}(t))_{\red}\), using a local frame \(\ell\) of \(\OO_T(1)\)
and the trivial root whose \(r\)th power is \(f^*\ell\). We obtain a
root pair \(P_1^r\simeq L\) on a neighborhood of that compact fiber.
The prescribed compatible frame on the fiber extends as a germ inside
\(S\), by the last part of the lemma. If it is defined on
\(N_S\subset S\) containing the fiber, properness of \(f\) gives a
smaller base neighborhood \(U\ni t\) with \(S_U=f^{-1}(U)\subset N_S\):
remove the closed set \(f(S\setminus N_S)\). Shrink once more so this
whole \(S_U\) lies in the ambient root neighborhood \(W_0\), and replace
\(W_0\) by \(W=W_0\setminus(S\setminus S_U)\). Thus
\begin{equation}\label{eq:local-root}
W\cap S=S_U,\qquad P_1^r\simeq L|_W,\qquad
p_1\text{ a frame of }P_1|_{S_U},\quad p_1^r=f^*\ell.
\end{equation}
Comparisons between two choices, with prescribed power-compatible
boundary frames, extend uniquely as ambient germs near a compact fiber.
Only this torsion comparison is extended; no arbitrary boundary
trivialization is extended off \(S\).
\subsection{The first normalized cover and the lifting target}
On a chart \eqref{eq:local-root}, take the full normalized cyclic cover
\(\pi:Z\to W\) defined by
\[
y^r=\pi^*s\quad\text{in }\pi^*P_1^r,
\]
retaining all components and the \(\mu_r\)-action. Finite analytic
normalization is available by
\cite[Part~B, Section~4, Corollaries~2--3]{Houzel1961}. At a general point of
\(S_i\), a root of a local unit reduces the equation to \(y^r=x^{d_i}\).
Because \(d_i\mid r\), each normalized branch is
\[
x=t^{r/d_i},\qquad y=\zeta t.
\]
The ramification index is \(r/d_i\) and \(y\) has order one. Outside
\(S\) the equation roots a unit and is \'etale. A finite map preserves
the dimension of a prime analytic subset, so no further divisor above a
codimension-two set is missed. The zero divisor
\[
E=(y=0)
\]
is Cartier: \(y\) is a nonzerodivisor on each normal component. A Cartier
divisor on a normal space is \(S_1\), and the calculation makes it
generically reduced; hence it is reduced. Its canonical line identity is
\begin{equation}\label{eq:cover-line}
\OO_Z(E)\simeq\pi^*P_1,
\end{equation}
sending the canonical section of \(\OO_Z(E)\) to \(y\).
Put
\begin{equation}\label{eq:cover-layers}
I=\OO_Z(-E),\quad g=f\circ\pi|_E:E\to U,\quad
\mathcal A_j=I^j/I^{j+1}=\OO_E(-jE)\quad(j\in\Z).
\end{equation}
The map \(E\to S_U\) is finite and \(g\) is proper. Each \(\mathcal A_j\)
is invertible on the possibly singular reduced \(E\). In the Laurent
graded algebra of these layers, \(u=y/p_1\) denotes the degree-one frame.
This is intrinsic on the associated graded: locally divide the equation
of \(E\) by the boundary value of a frame of \(P_1\). It does not assert
that \(p_1\) is an ambient frame.
For \(j\in\Z\) and \(k\geq1\), put
\begin{equation}\label{eq:truncations}
\mathcal T_{j,k}=I^j/I^{j+k}.
\end{equation}
We view this as a sheaf of complex vector spaces on the underlying
topological space of \(E\). A sheaf on \(Z\) supported on the closed
subset \(E\) is canonically such a sheaf. Thus \(g_*\) and its derived
functors are defined for \(\mathcal T_{j,k}\), although there is no
holomorphic map from its thickening to \(U\). For a coherent layer
\(\mathcal A_j\), this is the usual coherent analytic direct image.
The finite-neighborhood assertion we will prove is the following.
\begin{proposition}[All finite lifting orders]\label{prop:all-orders}
For every local root chart \eqref{eq:local-root}, every integer
\(j\), and every \(k\geq1\), the map
\begin{equation}\label{eq:all-order-surjection}
g_*\mathcal T_{j,k+1}\longrightarrow g_*\mathcal T_{j,k}
\end{equation}
is an epimorphism of sheaves of complex vector spaces on \(U\).
The neighborhoods used to lift a particular germ may depend on \(j\)
and \(k\).
\end{proposition}
The proof is in Section~\ref{sec:lifting}. To see what geometry it
requires, consider the exact sequence
\begin{equation}\label{eq:current-truncation-sequence}
0\longrightarrow\mathcal A_{j+k}\longrightarrow\mathcal T_{j,k+1}
\longrightarrow\mathcal T_{j,k}\longrightarrow0.
\end{equation}
Its connecting map takes a section of \(g_*\mathcal T_{j,k}\) to an
obstruction in \(R^1g_*\mathcal A_{j+k}\). In order \(k\), choosing
\(j=-a-k\) puts that obstruction in the fixed sheaf
\(R^1g_*\mathcal A_{-a}\). We next construct a split insertion of this
sheaf into the first cohomology of an SNC dualizing sheaf. The Hodge
argument will apply there, and the induction will return from these
negative degrees to all integer degrees.
\subsection{A canonical root and its resolved support}
Take a second full normalized cover, this time adjoining a \(q\)th root
of \(\pi^*s\) as a meromorphic \(q\)-pluricanonical form. The local
construction in Lemma~\ref{lem:floor-torsion-free} also proves its
existence on the normal analytic \(Z\): if its coefficient in a
meromorphic canonical generator is \(a_0/b_0\), normalize the finite
reduced monic algebra
\[
A[w]/(w^q-a_0b_0^{q-1})
\subset \operatorname{Frac}(A)[t]/(t^q-a_0/b_0).
\]
The total meromorphic algebra is separable and its components all dominate.
A change of generator identifies the total meromorphic algebras by
multiplying the root by a meromorphic unit. The displayed finite monic
orders need not coincide under this identification, but their integral
closures do: each is the integral closure of the base ring in that same
total algebra. These closures therefore glue. The forms
\(t\eta\) glue to the tautological meromorphic top form \(\tau\).
For the total meromorphic field \(K\) of a normal base component, the
full \(\mu_q\)-action has invariant algebra \(K\) and root-character
space \(Kt\); the corresponding meromorphic form space is \(K\tau\).
The invariant subalgebra of the normalized holomorphic algebra is the
base ring \(A\), by normality. These statements are read componentwise
on a disconnected base.
The earlier \(\mu_r\)-action lifts functorially, preserving the pulled-back
pluriform, and commutes with \(\mu_q\). Choosing one component instead
would in general destroy these assertions.
Resolve this second cover and principalize the pulled ideal of \(E\),
equivariantly for these finite actions. We use the analytic
smooth-functorial resolution and principalization of
\cite[Theorems~1.1.11 and~1.1.13]{Temkin2017}, together with its nonembedded
analytic construction \cite[Sections~5.2.2 and~5.3.2]{TemkinNonembedded}.
We retain the final indexed complete boundary in the construction of
\(F_{\mathrm{princ}}\): after the blowup of the ideal, boundary
desingularization makes the final strictly monomial support a union of
components of this SNC boundary. Its closed indexed intersections are
regular by \cite[Lemma~2.1.10]{Temkin2017}. Splitting the disjoint
connected components of each boundary component near the compact fiber
gives globally smooth component labels; their intersections remain
smooth, and the finite group may permute the labels. On a neighborhood of a compact
parameter fiber only finitely many stages of the locally finite blowup
hypersequence occur; a finite-group invariant shrink preserves equivariance.
Write the composite as \(\rho:\widehat Z\to Z\), with \(\widehat Z\)
smooth, and put
\begin{equation}\label{eq:resolved-boundary}
D_E=\rho^*E,\qquad H=(D_E)_{\red},\qquad
\rho_H:H\to E,\qquad h=g\rho_H.
\end{equation}
Here \(D_E\) is the scheme inverse Cartier divisor and \(H\) is its
globally simple SNC reduction. The tautological form satisfies
\(\tau^{\otimes q}=(\pi\rho)^*s\) with differential pullback understood.
These constructions commute with restrictions to opens and ordinary
products by a complex manifold. For normalization, the product of the
normal normalization with a manifold is finite, normal, and bimeromorphic
to the reduced product, hence is its normalization by uniqueness;
normality of these analytic products is part of
\cite[Part~C, Theorem~4(b)]{Houzel1961}.
The resolution is functorial for smooth morphisms, including products.
We make no assertion about normalization or resolution under a ramified
base change.
We will also need the K\"ahler property near the resolved support. A finite
analytic map is projective locally: generators of its finite algebra embed
it in a relative projective space, and the affine coordinate \(1\) makes
the trivial line relatively ample. The finite maps and the finitely many
blowups above are therefore projective near the preimage of a compact set.
Patch local positive metrics of a relatively ample line by partitions
pulled from the base. Their curvature remains positive on vertical tangent
directions. A sufficiently large multiple of the pulled-back K\"ahler
form of \(W\) makes it positive in all directions near the compact
preimage; compactness bounds the mixed terms uniformly. This gives a
closed K\"ahler form on an ambient neighborhood \(\mathcal N\) of the
compact resolved fiber under consideration. Properness of \(h:H\to U\)
lets us shrink \(U\) so that all of \(H\) above the smaller \(U\) lies in
\(\mathcal N\): remove the closed image of \(H\setminus\mathcal N\).
We make this shrink in each root chart. Every closed intersection of
components of \(H\) is then smooth and proper over \(U\), and inherits
one global relative K\"ahler form there. A connected component can split
after a further restriction of the base, so we make the indexing choice
only after these neighborhood restrictions.
\subsection{Finite indexing near a parameter fiber}
The filtered direct-image construction will use the closed intersections
of globally indexed smooth components. The following lemma makes that
indexing finite after a base restriction, even when some strata map only
to special parameter loci.
\begin{lemma}[Components near a compact fiber]\label{lem:finite-components}
Let \(h:H\to T\) be a proper holomorphic map to a complex manifold, and
fix \(t_0\in T\). Suppose that near \(h^{-1}(t_0)\) the reduced support
\(H\) has a locally finite family of closed smooth labels \(D_\lambda\)
which locally are its distinct SNC branches, and that every closed
intersection of these labels is smooth. After restricting to a suitable
open neighborhood \(U\) of \(t_0\), the support has finitely many globally
smooth indexed irreducible components. Every closed intersection of the
new components has finitely many connected components, all smooth and
proper over \(U\). A global relative K\"ahler form already given on the
old strata restricts to such a form on the new ones. The open \(U\) can
be chosen inside any previously prescribed neighborhood of \(t_0\).
\end{lemma}
\begin{proof}
Local finiteness and compactness give an open neighborhood
\(\mathcal N_1\) of the central fiber which meets only finitely many
labels. Remove from the base the closed image of
\(H\setminus\mathcal N_1\); properness then puts the entire restricted
support in \(\mathcal N_1\). There are now finitely many closed indexed
intersections to consider. The connected components of each form a
locally finite family in \(H\): the intersection is closed and smooth,
so near one of its points a small smooth neighborhood meets only its own
component, and near a point outside it a neighborhood avoids it.
Compactness, followed by the same properness argument, gives a second
neighborhood \(\mathcal N_2\) meeting only finitely many such components
and a base restriction on which the whole support lies in
\(\mathcal N_2\).
Let \(C_1,\ldots,C_N\) be the whole connected components of the
pre-restriction closed intersections which meet \(\mathcal N_2\).
We retain the whole components, not just their intersections with
\(\mathcal N_2\). Each \(C_a\) is closed in a closed stratum and is a
connected complex manifold. Thus \(f_a=h|_{C_a}\) is proper over the
preceding base and \(C_a\) is irreducible. Their restrictions cover
every closed intersection after the current restriction, including the
singleton intersections.
Put \(S_a=f_a(C_a)\) with its reduced structure. Remmert's proper
mapping theorem makes \(S_a\) closed analytic
\cite[Section~7, no.~1, Satz~1, p.~60]{Grauert1960ProperMapping}, and it
is irreducible because \(C_a\) is. Inside any prescribed open coordinate
neighborhood \(V\ni t_0\) contained in the preceding restrictions,
take the locally finite irreducible decompositions of all \(S_a\cap V\).
Let \(B\) be the union of those components which do not contain \(t_0\).
A union of a subfamily of the locally finite irreducible components of
an analytic set is closed analytic. Since the list of \(a\)'s is finite,
\(B\) is closed analytic in \(V\) and misses \(t_0\). Set \(U=V\setminus B\).
This removal does not split any retained irreducible component \(A\) of
an \(S_a\cap V\). Its regular locus is connected, and its intersection
with \(B\) is a proper analytic subset: otherwise closedness and density
would imply \(A\subset B\), contrary to \(t_0\in A\setminus B\).
The complement of a proper analytic subset in a connected complex
manifold is connected. One may see this by perturbing a path, in a
finite chain of coordinate balls, off a subset of positive complex
codimension. Hence \(\operatorname{Reg}(A)\setminus B\) is connected
and \(A\cap U\) is irreducible. Only finitely many components of
\(S_a\cap V\) contain \(t_0\), by local finiteness. Their restrictions
give a finite cover of \(S_a\cap U\) by irreducible closed analytic
subsets, all containing \(t_0\). Therefore every irreducible component
of \(S_a\cap U\) contains \(t_0\); if \(S_a\) misses \(t_0\), its
restriction is empty.
Let \(W\) be a connected component of \(f_a^{-1}(U)\). It is open in
the connected smooth \(C_a\), is closed in \(f_a^{-1}(U)\), and is
smooth and irreducible. Its map to \(U\) is proper. Put
\(r_a=\dim S_a\). The maximal-rank locus of \(f_a\) is dense in \(C_a\):
the generic rank theorem gives maximal rank \(r_a\), and the lower-rank
locus is a proper analytic subset defined by the maximal minors.
The nonempty open \(W\) meets the maximal-rank locus. Its proper image is consequently
an irreducible closed analytic subset of \(S_a\cap U\) of dimension
\(r_a\). Every component of \(S_a\cap U\) has this dimension because
\(S_a\) is irreducible and hence pure-dimensional. The image is therefore
an irreducible component, since a proper analytic subset of such a
component has smaller dimension.
It therefore contains \(t_0\), and \(W\) meets \(f_a^{-1}(t_0)\).
The compact analytic fiber has finitely many connected components:
on its reduction, irreducible components are locally finite and
compactness makes their number finite. Each
\(W\cap f_a^{-1}(t_0)\) is a nonempty open-and-closed subset of this
fiber. Different \(W\)'s give disjoint such subsets, so
\begin{equation}\label{eq:finite-fiber-components}
\#\pi_0(f_a^{-1}(U))
\leq \#\pi_0\bigl((f_a^{-1}(t_0))_{\red}\bigr)<\infty.
\end{equation}
This includes zero-dimensional and other vertical images and permits
nonreduced scheme fibers.
Reindex the singleton strata by these finitely many components \(W\).
An intersection of the new labels is an open-and-closed part of the
restriction of the corresponding old closed intersection, hence is a
union of some of its finitely many connected components. It remains
smooth and proper, and the relative form restricts. Distinct new pieces
from one old label are disjoint, so the local SNC branches and their
ordering are unchanged. This proves the assertion.
\end{proof}
Apply Lemma~\ref{lem:finite-components} after the preceding K\"ahler
neighborhood and properness restrictions, to all regular labels and their
closed intersections. The resulting local support has the finite global
component indexing used in Section~\ref{sec:hodge}. The final open need
not be Stein or a polydisc: the roots and covers have already been
constructed, and their remaining local uses only restrict the existing
coordinate charts.
\subsection{A residue insertion that retains all cohomology}
\begin{proposition}[Split residue insertion]\label{prop:residue-insertion}
For the data \eqref{eq:cover-layers}--\eqref{eq:resolved-boundary},
multiplication by \(\tau\) gives a morphism
\[
\rho^*\OO_Z(aE)\longrightarrow\omega_{\widehat Z}(H),
\]
and residue gives \(\rho_H^*\mathcal A_{-a}\to\omega_H\). The resulting
map in the derived category of \(E\) has a retraction in the
\(\tau\)-character. Consequently, for every \(j\), and in particular
for \(j=1\), it gives a split injection
\begin{equation}\label{eq:residue-injection}
\tau_*:R^jg_*\mathcal A_{-a}\lhook\joinrel\longrightarrow R^jh_*\omega_H.
\end{equation}
The construction is natural under the power-compatible root comparisons
of Lemma~\ref{lem:root-neighborhood}. Under an ordinary product with a
complex manifold \(B_0\), it is the pullback construction for canonical
sheaves relative to \(B_0\). For absolute canonical sheaves the insertion
and its retraction are tensored with \(\omega_{B_0}\); in particular the
product insertion has source \(\mathcal A_{-a}\boxtimes\omega_{B_0}\)
and target \(R(\rho_H\times\id)_*\omega_{H\times B_0}\).
\end{proposition}
\begin{proof}
Let \(v\) be a local meromorphic frame of \(\OO_Z(aE)\). By
\eqref{eq:cover-line} and \(aq=r\), the \(q\)-pluriform
\(v^q\pi^*s\) is, up to a holomorphic unit, the pullback of a local frame
\(\ell_V\) of \(L\). On a log resolution of \((V,B,G)\), write \(b_F\)
for a crepant boundary coefficient and
\(m_F=\ord_F(s/\ell_V)\). Then \(m_F\geq0\), \(b_F\leq1\), and
\(b_F=1\) implies \(m_F>0\): the dlt pair is klt off \(S\), and the
section vanishes exactly on \(S\). Thus the local density
\[
|\ell_V|^{2/q}|s/\ell_V|^{2\epsilon}
\]
has exponent \(-b_F+\epsilon m_F>-1\) at each prime for every
\(\epsilon>0\), and is locally integrable. Change of variables under
the proper generically finite map \(\pi\rho\) preserves this integrability.
For \(\alpha=(\rho^*v)\tau\), let \(k_F\) be its integral order at a
prime upstairs and \(n_F\) the order of the pulled section ratio.
Integrability says \(k_F+\epsilon n_F>-1\) for all \(\epsilon>0\).
The order \(n_F\) is positive exactly on \(H\). Hence \(k_F\geq-1\)
on \(H\) and \(k_F\geq0\) elsewhere. This proves the logarithmic map.
Cartier adjunction on the reduced SNC divisor gives its residue map on
\(H\), with \(\rho_H^*\mathcal A_{-a}=\rho^*\OO_Z(aE)|_H\).
Let \(\chi\) be the \(\tau\)-character of the second cover. At a general
point of \(E\) over \(S_i\), differential pullback gives
\[
\ord_E(\pi^*s)
=\frac r{d_i}(d_i-q)+q\left(\frac r{d_i}-1\right)
=r-q=q(a-1).
\]
The second root is unramified there and \(\ord_E\tau=a-1\). At any
other codimension-one prime the first cover is \'etale and the original
boundary coefficient is \(\beta\in[0,1)\). For
\(e=q/\gcd(q,q\beta)\), the second-cover order is
\(e(1-\beta)-1\in\{0,\ldots,e-1\}\), by
\eqref{eq:canonical-root-order}. A meromorphic form in character
\(\chi\) is \(f\tau\) for an invariant meromorphic \(f\) descended
to \(Z\). The orders force \(\ord_E f\geq-(a-1)\) and nonnegative
orders at all other primes. Normality extends it as a section of the
indicated invertible line. Conversely, the logarithmic map multiplied
by the equation of \(E\) has no pole because \(D_E\geq H\). We obtain
the exact sheaf identities
\begin{align}
(\rho_*\omega_{\widehat Z})_\chi
&=\OO_Z((a-1)E)\tau,\label{eq:residue-character-one}\\
(\rho_*\omega_{\widehat Z}(D_E))_\chi
&=\OO_Z(aE)\tau.\label{eq:residue-character-two}
\end{align}
The second is projection formula for the invertible line \(\OO_Z(E)\).
Canonical torsion-freeness
\cite[Theorem~2.9 and Proposition~2.11]{Fujino2023AnalyticVanishing}
applies locally near \(E\) to each dominating resolved component. More
precisely, around a compact fiber choose the K\"ahler neighborhood just
constructed. Properness of \(\rho\) supplies a smaller normal target
neighborhood whose entire inverse image lies in it, by removing the
closed image of its complement. After a base shrink this neighborhood
contains all of \(E\) in the root chart. The restricted morphism is
proper with K\"ahler smooth source, so the cited theorem applies there.
These neighborhoods cover \(E\). On each, the higher direct images for
the generically finite resolution vanish because they vanish generically.
Finite pushforward is exact. Thus on their union \(Z^\circ\subset Z\),
an open neighborhood of \(E\),
\[
(R^i\rho_*\omega_{\widehat Z})|_{Z^\circ}=0\quad(i>0).
\]
Projection formula gives the same restricted vanishing for
\(\omega_{\widehat Z}(D_E)\).
Put \(A_E=\mathcal A_{-a}=\OO_E(aE)\). Cartier adjunction for the
possibly nonreduced divisor \(D_E\) is the exact sequence
\[
0\longrightarrow\omega_{\widehat Z}\longrightarrow
\omega_{\widehat Z}(D_E)\longrightarrow\omega_{D_E}\longrightarrow0.
\]
Since \(D_E\) is the scheme inverse image of \(E\), it has a morphism
\(\rho_D:D_E\to E\). The preceding vanishings and character identities
identify
\begin{equation}\label{eq:residue-quotient}
\bigl(R(\rho_D)_*\omega_{D_E}\bigr)_\chi\simeq A_E\tau[0]
\quad\text{in }D(\OO_E).
\end{equation}
One can see this first after exact closed pushforward from \(E\) to
\(Z^\circ\), where it
is the quotient of \eqref{eq:residue-character-two} by
\eqref{eq:residue-character-one}. Closed pushforward detects its
cohomology sheaves, and the canonical truncation of a complex concentrated
in degree zero gives \eqref{eq:residue-quotient} on \(E\). No derived
full-faithfulness assertion is needed.
Because \(A_E\) is invertible, its pullback followed by the residue map
defines \(A_E[0]\to R(\rho_H)_*\omega_H\). The inclusion
\(\omega_H\hookrightarrow\omega_{D_E}\), followed by the character
projection and \eqref{eq:residue-quotient}, gives a map back to
\(A_E\tau[0]\). On degree zero their composition sends \(f\) to
\(f\tau\) in the quotient of the two character lines. Under the chosen
identification it is the identity. An endomorphism of the sheaf
\(A_E[0]\) in degree zero is determined by its sheaf map, so this is a
derived retraction. Applying \(Rg_*\) proves the split injections
\eqref{eq:residue-injection}. All maps were defined by the tautological
form, the actual divisor ideal, and residue, so they respect root
comparisons. Under an ordinary product the tautological form is relative
to the new factor; wedging with a local frame of its canonical line gives
the absolute map and the stated \(\omega_{B_0}\) factor.
\end{proof}
The derived retraction is stronger than an injection only on functions or
at general fibers. It is this strength that allows the later argument
to retain obstruction classes supported on special parameters.
\subsection{One compact graph on a parameter cover}
We next prepare the geometric setting for a global symbol test. For any
chosen parameter \(t_*\), the construction gives a projective parameter
cover unbranched over \(t_*\) and one compact SNC graph over the entire
cover. Global vanishing can then be tested back at \(t_*\) without
discarding a class supported there. The cover may change with \(t_*\).
We form and resolve the covers before imposing the graph equations; this
avoids assuming that normalization commutes with ramified base change.
\begin{proposition}[Compact transverse graph]\label{prop:global-graph}
Assume \(b>0\), and fix any \(t_*\in T=\PP^b\). There is a coordinate-power
map \(\psi:T'=\PP^b\to T\) of degree \(r\) in each coordinate, with
\(R=\OO_{T'}(1)\) and an actual identity \(R^r\simeq\psi^*\OO_T(1)\),
which is unbranched over \(t_*\), and the following data. On a neighborhood
of the compact graph \(\Xi=S\times_T T'\subset V\times T'\) there are
the two full root covers and a projective resolution with smooth ambient
source \(\mathscr W\). The graph cut \(H'\) inside the reduced inverse
section divisor is compact and reduced SNC of pure dimension \(\dim V-1\),
with finitely many globally smooth components. The projection
\(p':\mathscr W\to T'\) is a submersion near \(H'\), and every nonempty
closed component intersection is smooth and proper over \(T'\), carrying
the restriction of one K\"ahler form on a neighborhood of \(H'\).
Locally near every compact graph slice over \(w\in T'\), the resolved
ambient data, including their projection and full tautological divisor
ideal, are isomorphic to an open restriction of the ordinary product
\(\widehat Z\times U'\) for a local root chart. There the cut is
\(H^\flat=H\times_U U'\), and its actual dualizing line is
\begin{equation}\label{eq:graph-adjunction}
\omega_{H^\flat}\simeq\operatorname{pr}_H^*\omega_H\otimes
\operatorname{pr}_{U'}^*(\omega_{U'}\otimes\psi^*\omega_U^{-1}).
\end{equation}
\end{proposition}
\begin{proof}
Choose finitely many pairs of parameter opens \(U_i'\Subset U_i\), with
the \(U_i'\) covering \(f(S)\), such that a local root chart and its
proper map \(h_i:H_i\to U_i\) are defined on \(U_i\). Each
\(h_i^{-1}(\overline{U_i'})\) is compact. Local finiteness therefore
leaves only finitely many smooth component intersections meeting these
compact sets. For each such stratum \(C\) and each subset of
variable hyperplanes in \(T\), consider the incidence
\[
\{(z,H_1,\ldots,H_v):h(z)\in H_1\cap\cdots\cap H_v\}.
\]
It is smooth, since varying each hyperplane independently supplies its
normal direction. Sard's theorem for the projection to the hyperplane
parameter space gives transversality for a general tuple; a countable
atlas handles any noncompact stratum. Choose a tuple satisfying all
the finitely many conditions and also the open conditions that it forms
a coordinate basis and that none of its hyperplanes contains \(t_*\).
Use those coordinates for
\[
\psi([w_0:\cdots:w_b])=[w_0^r:\cdots:w_b^r].
\]
Its derivative at a point has image the tangent space to the intersection
of exactly the coordinate hyperplanes corresponding to vanishing
coordinates, or is invertible when none vanish. The chosen transversality
therefore gives, at \(h(z)=\psi(w)\),
\begin{equation}\label{eq:graph-transversality}
dh(T_zC)+d\psi(T_wT')=T_{h(z)}T
\end{equation}
for every smooth SNC stratum. The map is unbranched over \(t_*\).
In local coordinates \(t_i\) on \(U\), extend the coordinate functions
of \(h\) holomorphically off \(H\) to functions \(G_i\) on local opens
of \(\widehat Z\). In \(\widehat Z\times U'\), the equations
\(\psi_i(w)-G_i=0\) have surjective differential on each \(H\)-stratum
by \eqref{eq:graph-transversality}. They cut a smooth submanifold of
codimension \(b\) transverse to the SNC divisor. The cut
\(H^\flat=H\times_U U'\) is therefore reduced SNC of pure dimension
\(\dim V-1\), and has codimension \(b+1\) in the product ambient.
Complete-intersection adjunction gives \eqref{eq:graph-adjunction}.
In coordinates its last factor has the wedge frame \(dw/dt\); this means
a frame of \(\omega_{U'}\otimes\psi^*\omega_U^{-1}\), not an inverse
Jacobian.
The graph \(\Xi\) is a compact closed analytic subspace. Its actual
line identity is
\[
\operatorname{pr}_V^*L|_\Xi\simeq
\operatorname{pr}_{T'}^*R^r|_\Xi.
\]
Lemma~\ref{lem:root-neighborhood} gives a root \(P_*\) of
\(\operatorname{pr}_V^*L\) on a neighborhood of \(\Xi\), identified
with \(\operatorname{pr}_{T'}^*R\) along \(\Xi\). Form the same two
full normalized covers there. For the second cover use coefficients
of \(s\) in meromorphic canonical generators from the first factor
\(V\). This defines its relative tautological form without requiring a
Cartier relative canonical line on the singular cover; changing to a
generator after differential pullback changes the coefficient by a
\(q\)th power and identifies the total root algebra and its normalization.
Resolve and principalize the full tautological ideal functorially on this neighborhood, before
cutting by the graph.
Near \(w_0\), choose a frame \(\bar p\) of \(R\) whose \(r\)th power
is \(\psi^*\ell\) for a downstairs frame \(\ell\); a local root of a
unit gives such a choice. Along the compact graph slice, compare the
pullback of \(P_1\) with \(P_*\) by sending the boundary frame \(p_1\)
to \(\bar p\). Lemma~\ref{lem:root-neighborhood} extends this to an
ambient root isomorphism as a germ. On the graph, its ratio with the
prescribed comparison is a \(\mu_r\)-section equal to one on the compact
slice, hence equal to one on an open graph neighborhood of that slice.
Properness of \(\Xi\to T'\) permits a base shrink on which the ambient
comparison is defined and agrees with the prescribed one along the entire
graph: remove the closed image of the complement of that neighborhood.
This argument also applies to a nonreduced graph. The comparison identifies
the first cover with the local ordinary product cover, and product
normality identifies the second cover as well. Apply the same fixed
smooth-functorial principalization to the same full ideal and ordered
boundary data on both sides; functoriality identifies the results. These are isomorphisms of resolved
ambient germs over the product base preserving the whole divisor ideal,
not only isomorphisms of their supports.
Let \(H'\) be the graph cut inside the reduced inverse section divisor
on this global resolution. Locally it is exactly \(H^\flat\) above,
so it is reduced SNC with the asserted pure dimension. It is compact,
being over the compact \(\Xi\) by proper maps. The ambient projection
is a submersion near it by the local product comparison. The transverse
cuts of the globally smooth inverse-divisor components are smooth; split
them into their disjoint connected components. Compactness and local
finiteness leave only finitely many. Their closed intersections are
smooth and compact, hence proper over \(T'\). Finally, the relative
metric construction above, applied over the K\"ahler product neighborhood
of \(\Xi\), gives one K\"ahler form near \(H'\). Its restrictions supply
all the required relative K\"ahler forms.
\end{proof}
The local and global graph supports in this proposition are locally a
reduced SNC divisor in a smooth complete intersection, with submersive
ambient projection. The next section develops the filtered direct-image
statement for precisely that geometry.
==== END Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//covers.tex ====
==== BEGIN Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//finite-null.tex ====
\section{Threefold contractions for the boundary argument}\label{sec:finite-null}
The boundary argument uses the following specialization of
Das--Hacon--P\u{a}un's threefold contraction theorem
\cite[Theorem~5.5]{DasHaconPaun2024}. It supplies contractions both on
the floors of the supported fourfold program and on lower strata in the
restriction argument. Throughout these preparations, a real
\((1,1)\)-class on a singular space is a Bott--Chern class defined by
smooth local potentials.
\begin{proposition}[Threefold contraction of an adjoint plus a K\"ahler class]
\label{prop:threefold-contraction}
Let \((Z,\Gamma)\) be a normal connected compact K\"ahler klt threefold
with effective rational boundary and actual \(\Q\)-Cartier adjoint
\(A=K_Z+\Gamma\). Suppose
\begin{equation}\label{eq:threefold-exposed-input}
\alpha=c_1(A)+[\omega]\quad\hbox{is nef},\qquad
\omega\quad\hbox{is K\"ahler}.
\end{equation}
Then there is a projective surjection \(f:Z\to Y\) with
\(f_*\OO_Z=\OO_Y\), where \(Y\) is normal compact K\"ahler with
rational singularities, and a K\"ahler class \([\omega_Y]\) satisfying
\(\alpha=f^*[\omega_Y]\). The actual rational line \(-A\) is
\(f\)-ample. A compact curve is contracted exactly when its
\(\alpha\)-degree is zero.
\end{proposition}
The proposition assumes no bigness of \(\alpha\), pseudo-effectivity
of \(A\), or factoriality of \(Z\). Its proof is given in
Subsection~\ref{subsec:threefold-contraction-bridge}, after the required
constructions. The separate exposure input
\cite[Corollary~5.3]{DasHaconPaun2024} supplies a class of the form
\eqref{eq:threefold-exposed-input} exposing any negative extremal ray in
the cited threefold cone theorem. In that case the proposition contracts
exactly the curves in that ray.
The preparation has three parts. First we contract the entire finite
curve null locus of a nef and big class, obtaining a normal compact
analytic target. Next, conormal direct-image vanishings and analytic
thickenings extend a contraction of a prime floor to its ambient space.
Finally, these two constructions combine with the named projective MMP,
descent, positivity, and nonbig inputs to prove the proposition and its
floor restriction. The K\"ahler target in the proposition is part of
this combined argument, beyond the finite-null construction alone.
\subsection{Contraction of the full finite null locus}
We first prove the finite-null contraction assertion of
\cite[Proposition~6.2]{DasHacon2024} used in the nef and big cases.
For an irreducible positive-dimensional analytic subspace \(V\), its top
intersection with such a class is computed on a resolution of \(V\).
Write
\[
\operatorname{Null}(\alpha)=
\bigcup_{\substack{V\subset X\ {\rm irreducible}\\
\dim V>0,\ \alpha^{\dim V}\cdot V=0}} V .
\]
\begin{proposition}[A finite null locus]\label{prop:finite-null}
Let \(X\) be a normal connected ordinary \(\Q\)-factorial compact
K\"ahler threefold with klt singularities. Let \(\alpha\) be a nef and
big real \((1,1)\)-class. Suppose that
\(C=\operatorname{Null}(\alpha)\) is a finite union of curves. There is
a proper bimeromorphic morphism \(\phi:X\to X'\) to a normal compact
analytic space such that \(\phi\) is an isomorphism on \(X\setminus C\),
and the underlying sets of its nontrivial fibers are exactly the
connected components of \(C\).
\end{proposition}
\begin{proof}
The assertion is the identity map if \(C\) is empty. Otherwise give
\(C\) its reduced structure and write \(C_1,\ldots,C_s\) for its
irreducible components.
We will construct a positive line on a resolution and use it to give
the conormal of the full analytic inverse image a positive presentation.
Grauert's contraction criterion will then apply to that conormal.
\subsubsection*{An exceptional divisor and one actual positive line}
Choose a projective resolution \(\pi:\widehat X\to X\) with smooth
source which is an isomorphism over \(X_{\rm reg}\) and has pure
divisorial exceptional locus
\cite[Theorem~2.13]{DasHaconPaun2024}. Compactness makes the resolution
a finite composition of blowups. Its source is compact K\"ahler:
metrics of a relatively ample line can be patched using a partition of
unity from the base, preserving positivity on the vertical tangent
spaces, and a sufficiently large pulled-back K\"ahler form makes the
curvature positive in every direction. Compactness supplies one
constant for this last step.
Put \(\beta=\pi^*\alpha\). Pullback of the metric nef approximations
shows that \(\beta\) is nef. We check the positive top intersection
needed on the smooth source. By bigness choose a K\"ahler current
\(T_X\in\alpha\) with \(T_X\geq\omega_X\), where \(\omega_X\) is a
K\"ahler form on \(X\). Pulling back its local plurisubharmonic
potentials gives
\[
T=\pi^*T_X\in\beta,\qquad T\geq\eta:=\pi^*\omega_X.
\]
The pulled potentials are not identically minus infinity, since
\(\widehat X\) dominates \(X\). The form \(\eta\) is smooth and
semipositive. Fix a K\"ahler form \(\kappa_{\widehat X}\), and for
\(\delta>0\) choose a positive form
\(b_\delta\in\beta+\delta[\kappa_{\widehat X}]\). Wedge the positive
current \(T-\eta\), successively, with \(b_\delta^2\),
\(\eta\wedge b_\delta\), and \(\eta^2\). Integration and
\(\delta\downarrow0\) give
\begin{equation}\label{eq:null-positive-cube}
\beta^3\geq\beta^2[\eta]\geq\beta[\eta]^2\geq[\eta]^3>0 .
\end{equation}
The last inequality holds because \(\eta\) is positive on a nonempty
open set. All these products are on the smooth compact \(\widehat X\);
only a current wedged with smooth semipositive forms was used.
For an irreducible \(d\)-dimensional positive-dimensional subspace
\(V\subset\widehat X\), projection of its fundamental cycle gives
\begin{equation}\label{eq:null-cycle-projection}
\beta^d\cdot V=
\begin{cases}
0,&\dim\pi(V)0\), with
\(E_+(S)=V(\mathfrak a)=\operatorname{Null}(\beta)\) as sets.
Principalize this global ideal by
a finite sequence \(g:Y\to\widehat X\) of blowups with smooth centers,
using \cite[Remark~2.14]{DasHaconPaun2024}. The source \(Y\) is smooth
and compact K\"ahler. If
\(\mathfrak a\OO_Y=\OO_Y(-D_{\mathfrak a})\), the logarithmic
presentation and Poincaré--Lelong, in the cited normalization, give
\[
g^*S=\theta+[G],\qquad G=cD_{\mathfrak a}\geq0,
\]
where \(G\) is a finite real divisor supported over the singular set and
\(\theta\) is a global smooth closed form.
If \(S\geq\epsilon\kappa_{\widehat X}\), the equality off \(G\) and
continuity show that \(\theta\geq\epsilon g^*\kappa_{\widehat X}\)
everywhere.
Discarding blowups along Cartier centers, which are isomorphisms, choose
for this finite composition an effective
\(g\)-exceptional integral divisor \(F\) with \(\OO_Y(-F)\)
\(g\)-ample. Such a divisor is obtained from the exceptional
tautological divisors, giving earlier pullbacks sufficiently large
positive weights. A smooth representative \(v\) of \(c_1(F)\) can be
chosen so that \(g^*\kappa_{\widehat X}-\delta v\) is K\"ahler for some
\(\delta>0\): the curvature of \(-F\) is positive on the vertical
kernels, and compactness bounds the horizontal and mixed terms. If
\(g\) is an isomorphism, take \(F=0\). Consequently
\(\theta-\epsilon\delta v\) is K\"ahler. With \(p=\pi g\), we obtain
\begin{equation}\label{eq:null-exceptional-decomposition}
p^*\alpha=\kappa_0+[E],\qquad
\kappa_0=[\theta-\epsilon\delta v]\ \text{K\"ahler},\qquad
E=G+\epsilon\delta F\geq0 .
\end{equation}
Every prime of \(G\) maps into \(C\cup\operatorname{Sing}X\), and every
prime of \(F\) maps to a set of codimension at least two in
\(\widehat X\). Hence every prime of \(E\) is \(p\)-exceptional.
The morphism \(p\) is projective: projective morphisms compose over the
compact base \(X\) by \cite[Remark~2.11]{DasHaconPaun2024}.
Openness of the K\"ahler cone on the smooth \(Y\) permits a nonnegative
rational divisor \(E'\) with the same exceptional prime support and
coefficients sufficiently close to those of \(E\) so that
\[
\kappa'=p^*\alpha-[E']\quad\hbox{is K\"ahler}.
\]
Choose \(r>0\) clearing its denominators, and put
\begin{equation}\label{eq:null-positive-line}
H=rE'\geq0,\qquad L=\OO_Y(-H).
\end{equation}
Thus \(H\) is an integral exceptional Cartier divisor and \(L\) is an
actual holomorphic invertible sheaf.
We verify relative ampleness on every full fiber of \(p\). Let \(a\)
be a smooth representative of \(\alpha\) with local potentials on \(X\),
let \(\Theta_L\) be the curvature of a smooth metric on \(L\), and choose
a K\"ahler representative for \(\kappa'\). Equality of Bott--Chern
classes on \(Y\) gives a global smooth function \(\varphi\) with
\[
r\kappa'=r p^*a+\Theta_L+dd^c\varphi .
\]
Adjust the metric of \(L\) by \(\varphi\). On a base chart
\(a=dd^c\rho\), adding \(r\rho\circ p\) to its local weights makes their
curvature \(r\kappa'\). On a fiber, that added function is constant.
The restricted weights therefore have strictly plurisubharmonic ambient
extensions. The same weights define positivity on the full possibly
nonreduced fiber: nilpotents do not change their values or the absolute
values of transition units. The compact analytic positivity criterion
and the proper fiberwise criterion
\cite[Definition~2.8, Corollary~1.12, and Remark~3.2]{Fujino2026Fujiki}
now show that \(L\) is \(p\)-ample.
\subsubsection*{Numerical ampleness on the projective inverse image}
The compact reduced curve \(C\) is projective, even if it is reducible
or disconnected. Indeed, choose one point on each irreducible component
which is smooth on all of \(C\) and is outside the other components.
Their sum is an effective Cartier divisor. Its canonical section is
nonzero and has a zero on each positive-dimensional irreducible subspace
of \(C\). The positivity lemma in Section~3.4 of
\cite{Grauert1962}, followed by Section~3.2, Satz~2, makes its line
positive and ample. Fix a very ample line \(A\) on \(C\).
Put
\[
D=(p^{-1}C)_{\red},\qquad f_D:D\to C,\qquad L_D=L|_D .
\]
The reduced base change of \(p\) is projective over \(C\). Since \(C\)
is compact and projective and the final base is a point, the
compact-base projective composition assertion makes \(D\) projective
\cite[Remark~2.11]{DasHaconPaun2024}. Chow and GAGA allow \(D\) and
its holomorphic invertible sheaves to be treated as a projective scheme
over \(\C\) \cite{Serre1956GAGA}. The space \(D\) can be reducible,
nonnormal, and of mixed dimension. The restriction \(L_D\) is the
restriction of the actual line \(L\); a component of \(D\) can lie
inside \(H\).
Choose an ample line \(A_D\) on \(D\) with a smooth curvature form
\(\Theta_{A_D}\). For one integer \(q>0\) sufficiently large,
\[
qr\kappa'|_D-\Theta_{A_D}
\]
is positive. To see this on the singular space, extend the local
weights of \(A_D\) to finitely many relatively compact ambient charts
covering \(D\), and bound their Hessians by the ambient K\"ahler form.
This is the local-potential positivity convention. If
\(\Gamma\subset D\) is an integral curve, its image is a point or one
of the \(C_l\); projection on the normalizations gives
\((p^*\alpha)\cdot\Gamma=0\). Therefore
\begin{equation}\label{eq:null-numerical-margin}
\deg_\Gamma(L_D^q\otimes A_D^{-1})
=qr\kappa'\cdot\Gamma-\Theta_{A_D}\cdot\Gamma>0 .
\end{equation}
The class of \(L_D^q\otimes A_D^{-1}\) is consequently in the dual of
the closed cone of curves of the projective scheme \(D\). The numerical
space is finite-dimensional. Kleiman's line-bundle criterion
\cite[Section~4.8 and Theorem~4.10]{Fujino2009Fundamental} says that the
ample \(A_D\) is positive on every nonzero class of the closed curve
cone. Its minimum on a compact unit slice is positive, so its class
is in the interior of the dual cone.
The sum of an element of the dual cone and an element of its interior
is again in the interior. Thus \(L_D^q\), and hence \(L_D\), is ample.
The ample subtraction in \eqref{eq:null-numerical-margin} supplies the
uniform margin; strict positivity on individual curves alone would not
give this conclusion.
The ample line \(L_D\) is the positivity needed for the contraction.
The remaining ideal calculations transfer it to a positive presentation
of the conormal of the full analytic inverse image, including its
nonreduced structure.
\subsubsection*{The full scheme inverse image}
Let \(\mathfrak c=\mathcal I_C\), and define
\begin{equation}\label{eq:null-full-inverse}
J=\mathfrak c\OO_Y=\im(p^*\mathfrak c\to\OO_Y),\qquad
Z=V(J)=Y\times_X C,\qquad f:Z\to C.
\end{equation}
The space \(Z\) has its full analytic subspace structure, including
possible multiplicities and embedded components, and \(Z_{\red}=D\).
Put \(K_m=p_*L^m\) for \(m\geq1\). It is a coherent ideal of
\(\OO_X\): effectivity of \(H\) gives \(L^m\subset\OO_Y\), and
\(p_*\OO_Y=\OO_X\) by normality and proper bimeromorphicity.
We use relative Serre vanishing in the following exact form. For a
proper holomorphic map, a relatively ample invertible sheaf, a fixed
coherent twist, and a compact part of the base, its positive higher
direct images vanish there for every sufficiently large tensor power.
This is \cite[Definitions~2.1--2.2 and Proposition~2.5]{Ancona1982};
it permits nonreduced complex spaces and assumes they are separated
and countable at infinity. Those conditions hold for the present spaces and the
relatively compact opens used next.
On a base open \(U\) choose generators \(g_1,\ldots,g_t\) of
\(\mathfrak c\). Let \(Q_U\) be the coherent kernel in
\[
0\longrightarrow Q_U\longrightarrow\OO_{p^{-1}U}^{\oplus t}
\xrightarrow{(p^*g_1,\ldots,p^*g_t)}
J|_{p^{-1}U}\longrightarrow0 .
\]
Choose finitely many such opens with compact subsets whose interiors
cover \(C\). Relative Serre vanishing for
\(Q_U\otimes L^m\) on these compact subsets and for \(J\otimes L^m\)
over \(C\) gives one integer \(M\) for which the needed \(R^1p_*\)'s
vanish for every \(m\geq M\). Tensor the displayed presentation by
\(L^m\) and push forward. Inside \(K_m\), the image of \(K_m^{\oplus t}\)
is exactly multiplication by the \(g_i\). Hence, at every stalk over
\(C\),
\begin{equation}\label{eq:null-ideal-product}
p_*(J\otimes L^m)=\mathfrak cK_m\qquad(m\geq M).
\end{equation}
This is an image computation; it uses no projection formula for the
possibly nonflat ideal \(\mathfrak c\).
Push forward the exact sequence
\[
0\longrightarrow J\otimes L^m\longrightarrow L^m
\longrightarrow L^m|_Z\longrightarrow0 .
\]
The other \(R^1\) vanishing and \eqref{eq:null-ideal-product} give the
canonical isomorphism of coherent \(\OO_C\)-modules
\begin{equation}\label{eq:null-full-quotient}
P_m:=K_m/\mathfrak cK_m
=K_m\otimes_{\OO_X}\OO_C
\xrightarrow{\ \sim\ } f_*(L^m|_Z),\qquad m\geq M .
\end{equation}
In particular the full inverse \(Z\), rather than only its reduction,
occurs in this equality. No flatness or Cartier assumption on \(J\)
has been used.
\subsubsection*{A positive presentation of the conormal}
Let \(\mathfrak n=\ker(\OO_Z\to\OO_D)\). This coherent nilradical
satisfies \(\mathfrak n^e=0\) for one \(e\), by local Noetherianity and
compactness of \(Z\). For
\[
T_m=L^m|_Z\otimes f^*A^{-2},
\]
the filtration \(\mathfrak n^jT_m\) has quotients
\[
(\mathfrak n^j/\mathfrak n^{j+1})\otimes
L_D^m\otimes f_D^*A^{-2},\qquad 0\leq j0\) we obtain
\(E'\cdot\Gamma<0\). If \(\Gamma\) were not contained in the support
of the effective Cartier divisor \(H\), its canonical section on the
normalization of \(\Gamma\) would give nonnegative degree, a contradiction.
Thus an irreducible component of \(H\) contains \(\Gamma\). Its image
contains \(C_l\) and has dimension at most one by exceptionality, so
that image equals \(C_l\). At every
point of \(C_l\) a unit germ on \(X\) pulls back to a unit and cannot
vanish along that component. Hence \(K_m\) is a proper ideal at every
point of \(C\).
Outside \(p(H)\), the ideal \(K_m\) equals \(\OO_X\). The image of each
exceptional prime has codimension at least two in the normal threefold.
It follows that \(V(K_m)_{\red}\) is a finite union of curves and points.
Since it contains every \(C_l\), each \(C_l\) is one of its irreducible
curve components.
Let \(B\) be the union of its irreducible components other than the
selected curves \(C_l\), and put \(\mathfrak b=\mathcal I_B\). Define
\[
I=(K_m:\mathfrak b^\infty).
\]
This is a coherent ideal with one global finite saturation exponent.
Indeed, for each integer \(j\geq0\), the ideal
\((K_m:\mathfrak b^j)\) is the kernel of the coherent
morphism
\(\OO_X\to\mathcal Hom(\mathfrak b^j,\OO_X/K_m)\).
The ascending chain stabilizes at each stalk by Noetherianity. Equality
of two successive coherent ideals then holds on a neighborhood, and the
colon recursion gives all later equalities there. Compactness supplies
one exponent for all of \(X\).
On \(X\setminus B\) we have \(I=K_m\). On \(X\setminus C\), the analytic
Nullstellensatz puts a local power of \(\mathfrak b\) inside \(K_m\),
so \(I=\OO_X\). Closure of the dense parts \(C_l\setminus B\) now gives
\[
V(I)_{\red}=C,\qquad \sqrt I=\mathfrak c .
\]
The set \(B\cap C\) is finite. The induced map
\[
P_m=K_m/\mathfrak cK_m\longrightarrow Q:=I/\mathfrak cI
\]
is therefore an isomorphism away from finitely many points of \(C\).
After tensoring by \(A^{-2}\), its kernel and cokernel are still
point-supported. Splitting through its image, the two long exact
sequences, \eqref{eq:null-base-vanishing}, and the vanishing of positive
cohomology of point-supported sheaves give
\(H^1(C,Q\otimes A^{-2})=0\). Here a coherent sheaf on the projective
curve has no \(H^2\). The same regularity argument as above gives
\begin{equation}\label{eq:null-conormal-presentation}
A^{\oplus N_Q}\longrightarrow Q\longrightarrow0 .
\end{equation}
Since \(I\subset\mathfrak c\), right exactness of restriction gives the
actual conormal identity
\begin{equation}\label{eq:null-conormal}
Q=I/\mathfrak cI=(I/I^2)\otimes_{\OO_X}\OO_C .
\end{equation}
Its support is all of \(C\). At each point there, \(I\) is a nonzero
proper finitely generated ideal and \(\mathfrak c\) lies in the maximal ideal, so
Nakayama gives \(I/\mathfrak cI\ne0\). Nonzeroness of the ideal also
follows from its one-dimensional zero set in the normal threefold.
\subsubsection*{The coherent conormal contraction criterion}
For a coherent ideal, Grauert's associated normal linear space is
defined by its local linear relations. Restricting a finite presentation
of \(I\) to \(C\) presents the module in \eqref{eq:null-conormal}.
Thus the reduced associated space in Sections~3.6--3.7 of
\cite{Grauert1962} is
\[
N_I=\bigl(\operatorname{Specan}_C\operatorname{Sym}Q\bigr)_{\red}.
\]
This definition does not require \(Q\) to be locally free. The
surjection \eqref{eq:null-conormal-presentation} embeds \(N_I\) as a
closed reduced linear subspace of
\(\operatorname{Tot}((A^{-1})^{\oplus N_Q})\).
Put \(V_A=H^0(C,A)^*\). The very ample embedding
\(C\hookrightarrow\PP(V_A)\), where \(\PP(V_A)\) parametrizes lines,
identifies \(A^{-1}\) with the restricted
tautological line. Consequently
\(\operatorname{Tot}((A^{-1})^{\oplus N_Q})\) is closed in
\(C\times V_A^{N_Q}\). Its projection to \(V_A^{N_Q}\) is proper because \(C\)
is compact, and it is an analytic embedding off the zero section:
on the open set where its \(j\)-th vector is nonzero, projectivization
to that vector's line, followed by the inverse of the embedding of
\(C\) on its image, recovers the base point holomorphically. The
squared Euclidean norm of this projection
is strictly plurisubharmonic off the zero section. Its positive
sublevels are relatively compact neighborhoods of that section with
strongly pseudoconvex boundaries; scalar multiplication gives a nonzero
radial derivative at every positive level. Restricting them to \(N_I\) proves
exactly the weak negativity of this reduced linear space, including
when \(C\) is singular or disconnected and \(Q\) has torsion.
Grauert's Section~3.7, Satz~8, applies to any coherent ideal with the
given compact zero set and with weakly negative associated normal
linear space \cite{Grauert1962}. It does not require the ideal to be
radical or locally principal. The ambient \(X\) is reduced and
normal, and its compact zero set \(C\) is a union of curves with no
isolated points. The criterion therefore makes \(C\) exceptional.
Section~2, Definition~3 and Satz~5, of that source give a global proper
surjective holomorphic map \(\phi:X\to X'\) with \(\phi(C)\) discrete,
\(\phi^{-1}(\phi(C))=C\) as underlying sets, and
\[
X\setminus C\simeq X'\setminus\phi(C).
\]
The Remmert reduction used there has connected fibers. Hence each
connected component of \(C\) has one image point and two components
cannot have the same image. Its normality statement for a normal source
(p.~337) makes the Remmert-reduced target neighborhood normal; gluing it to the
unchanged normal complement makes \(X'\) normal. Compactness of \(X\)
makes \(X'\) compact, and the complement isomorphism makes \(\phi\)
bimeromorphic. This proves the proposition.
\end{proof}
The proposition supplies the normal compact analytic contraction and
its fiber sets. The later assertions that a target is K\"ahler and that
a class descends to a K\"ahler class are separate inputs.
\begin{lemma}[Finiteness when there is no null surface]\label{lem:no-null-surface}
Let \(X\) be a normal connected compact K\"ahler threefold, and let
\(\alpha\) be nef and big. If no irreducible surface has zero top
intersection with \(\alpha\), then \(\operatorname{Null}(\alpha)\)
is a finite union of curves, possibly empty.
\end{lemma}
\begin{proof}
Use a projective resolution \(\pi:\widehat X\to X\) with smooth compact
K\"ahler source as above. The class \(\beta=\pi^*\alpha\) is nef with
positive cube by \eqref{eq:null-positive-cube}. Collins--Tosatti makes
\(\operatorname{Null}(\beta)\) a proper analytic set.
Each irreducible component \(Z\) of this analytic set is itself null.
It has positive dimension, since each of its points lies on a
positive-dimensional null subspace.
Suppose instead that \(\beta^{\dim Z}\cdot Z>0\). Resolve \(Z\)
projectively. The restricted pulled class on the smooth compact
K\"ahler resolution is nef with positive top intersection, so
Collins--Tosatti gives it a proper null set. Choose a point of \(Z\)
outside the image of that set, the target nonisomorphism locus of the resolution,
the singular locus of \(Z\), and the other components of
\(\operatorname{Null}(\beta)\). Such a point exists because these are
proper analytic subsets of \(Z\). By the definition of
\(\operatorname{Null}(\beta)\), a null irreducible subspace passes through
it. That subspace lies in \(Z\), since the point is on no other
component. Its strict transform is null for the restricted class,
contradicting the choice of the point.
There are finitely many components of the compact analytic null set.
By \eqref{eq:null-cycle-projection}, each of its surface components is
exceptional, since a nonexceptional one would map to a null surface on
\(X\). Thus their images, as well as those of its curve components,
have dimension at most one. For any downstairs null curve \(C_0\),
choose an irreducible component \(V\) of \(\pi^{-1}C_0\) dominating
\(C_0\). Proper surjectivity supplies such a component. If
\(\dim V=1\), cycle projection gives
\(\beta\cdot V=\deg(V/C_0)\,\alpha\cdot C_0=0\); if \(\dim V>1\),
it gives zero by dimension drop. Thus \(V\) lies in an upstairs null
component whose image contains \(C_0\). Every downstairs null curve
is therefore contained in the finite union of images. An irreducible
curve in a finite union of curves and
points is one of the curve components. There are therefore only
finitely many downstairs null curves, as asserted.
\end{proof}
==== END Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//finite-null.tex ====
==== BEGIN Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//gluing.tex ====
\section{Compatible sections on the whole floor}\label{sec:gluing}
We finish Theorem~\ref{thm:floor-generation} by constructing sections on
all strata at one common degree. The construction follows the
pre-admissible and admissible section induction of Fujino
\cite[Section~4]{Fujino2000}, using the restriction criterion and finite
images proved in the preceding sections. The point to retain is that
compatibility holds on every intersection before the sections descend to
the existing line on the reduced floor.
We proceed from lower strata to higher ones. At each stage the restriction
criterion lets us choose all extensions of an already compatible lower
collection. Those extensions need not be invariant under comparisons.
Finite products of their transports will impose invariance while raising
every prescribed lower section to the same power. To make this possible,
we first show that transport preserves the prescribed lower restrictions,
including when a surface comparison contracts a floor curve.
For a positive degree \(k\) and \(d\in\{0,1,2,3\}\), a \emph{system of
tuples through dimension \(d\)} is a vector subspace
\[
\mathcal V_d(k)\subset
\bigoplus_{\dim Z\leq d} H^0(Z,L_Z^k)
\]
whose tuples satisfy the residue restriction equality along every
incidence among these strata. We call the system invariant if, for each
\(e\in\{0,\ldots,\min(d,2)\}\), each tuple is compatible with every
arrow of \(\mathcal G_e\): the arrow carries its component at the source
to its component at the target. It \emph{generates} if for every such
stratum \(Z\) and every \(z\in Z\), some tuple has its \(Z\)-component
nonzero at \(z\). These are linear conditions except for generation.
The vector space is finite-dimensional because there are finitely many
strata and each is compact.
If a system generates, the span of the componentwise \(v\)th powers
of its tuples generates in degree \(kv\). Restriction and transport
commute with products, so compatibility and invariance persist. We may
therefore enlarge a successful degree to any sufficiently divisible later
degree. Empty collections of strata impose no condition.
\begin{lemma}[Transport preserves lower restrictions]\label{lem:transport-lower-restrictions}
Fix \(k>0\) and \(d\in\{1,2\}\). Let a tuple through dimension \(d\)
be compatible along all incidences, and suppose that its part through
dimension \(d-1\) is invariant. For any arrow of \(\mathcal G_d\),
transport of the tuple's source component has the assigned lower
restriction on every floor stratum of its target.
\end{lemma}
\begin{proof}
For curves, the comparison is an isomorphism preserving the marked points
and their residue generators. Invariance makes the lower point values
equal, so the assertion follows.
For surfaces, take a target floor curve and view its valuation on the
source of the comparison. If its center is a floor curve there,
adjunction of the equality of meromorphic adjoint subsheaves on the graph
gives a crepant residue comparison of the two curve pairs. This is an
arrow of \(\mathcal G_1\), and the assigned curve tuple is invariant.
If the center is a point, it is a zero-dimensional lc center of the
source dlt surface, by crepancy. It is one of the point strata and is an
actual smooth crossing of two coefficient-one curves: the resolution
used in Proposition~\ref{prop:strata-adjunction} is an isomorphism at
such a point. In coordinates \((x,z)\) for the crossing, a divisorial
lc place above it arises by successive blowups of crossings of two
coefficient-one branches. To verify this, factor a surface resolution
into point blowups. In an SNC surface boundary the new crepant
coefficient at the blowup of a crossing is the sum of the two coefficients
minus one; at a point on just one branch it is that coefficient minus one.
Starting with coefficients at most one, a new coefficient one can arise
only at a crossing of two coefficient-one branches. This remains true
at every subsequent step.
The corresponding exceptional curve is a \(\PP^1\) whose different has
exactly the two adjacent coefficient-one points. Residue of
\((dx/x\wedge dz/z)^{\otimes qk}\) along it is
\((dt/t)^{\otimes qk}\), up to a sign removed in the even degree: at each
crossing blowup the logarithmic change-of-coordinate determinant is
\(\pm1\). Further point blowups do not change this meromorphic residue
on its strict transform. The normal target floor curve is bimeromorphic,
hence isomorphic, to this \(\PP^1\), and its actual log line is trivial
with that generator. Its section is determined by the common residue at
the two point strata. Pulling a local surface section through the crossing
restricts on the exceptional curve to its value at the point times that
generator. Compatibility of the original tuple identifies this value
with its assigned point component, and invariance makes it the assigned
value at both target point strata. Thus the entire target curve receives
exactly the prescribed lower section.
The two alternatives apply to every composite surface comparison, so the
assertion holds for the whole groupoid, not only for a generating link.
\end{proof}
\begin{proposition}\label{prop:generating-tuples}
There is a degree \(k>0\) and a compatible generating system of tuples
through dimension three which is invariant in dimensions zero, one, and
two.
\end{proposition}
\begin{proof}
For point strata take the same scalar in the canonical zero-form generator
\(1\) on every point. The even iterated residue convention identifies
these generators along all paths. This gives an invariant generating
system in dimension zero.
Suppose a system has been constructed through dimension \(d-1\), for
\(1\leq d\leq3\). Replace it by powers and their span in a sufficiently
large common degree, still denoted \(k\). For a \(d\)-stratum \(Z\),
the components of any lower tuple glue to a section on its entire floor
\(C_Z\) by Proposition~\ref{prop:strata-adjunction}. They satisfy its
link comparison: for \(d=1\) the link is a point comparison, for \(d=2\)
a curve comparison, and for \(d=3\) one of the generating surface links.
Lower invariance supplies each equality. Proposition~\ref{prop:restriction-link}
therefore extends every such boundary section to \(Z\), once the degree
is sufficiently large. There are only finitely many strata, so the degree
can be chosen uniformly.
Let \(\mathcal P_d(k)\) be the vector space of all tuples through
dimension \(d\) whose lower part belongs to the chosen lower system and
whose \(d\)-components restrict to those glued floor sections. It maps
surjectively to the lower system, since the finitely many extensions can
be chosen independently. This is a linear space of lifts; no choice of
a nonlinear extension operator is involved.
The pre-system \(\mathcal P_d(k)\) generates. For \(z\in Z\) with
\(y=f_Z(z)\in f_Z(C_Z)\), choose a floor point above \(y\) and a lower
tuple nonzero there. Any extension to \(Z\) is the pullback of a section
of \(N_Z^k\), by \eqref{eq:stratum-semiample-map}. Its value at \(y\)
is nonzero, and hence it is nonzero at \(z\). If \(y\notin f_Z(C_Z)\),
the sheaf \(\mathscr I_{f_Z(C_Z)}\otimes N_Z^k\) is generated for large
\(k\) by Serre's theorem. A section nonzero at \(y\) pulls back to a
section vanishing on \(C_Z\); with zero lower tuple and all other new
components zero, it belongs to \(\mathcal P_d(k)\). This also covers an
empty floor. Lower points remain generated because the projection to
the lower system is surjective.
For \(d=3\) this pre-system is the required final system. For \(d=1,2\)
we impose invariance by norm products. Lemma~\ref{lem:transport-lower-restrictions}
applies to every pre-tuple: all transported top components have the same
assigned lower restrictions. A product of \(h\) such factors will
therefore have the original lower tuple raised componentwise to \(h\).
For each orbit of \(d\)-strata choose a representative \(Z\), and let
\(G_Z\) be the finite isotropy image on \(H^0(Z,L_Z^k)\) from
Proposition~\ref{prop:finite-isotropy}. Given a pre-tuple with component
\(s_Z\), form its 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 integer \(h\) divisible by every \(|G_Z|\), and use
\(P_Z(s_Z)^{h/|G_Z|}\). Transport this section to every member of its
orbit. This is well defined: a change of the chosen transport by an
isotropy arrow permutes the factors in degree \(k\), and transport
commutes with multiplication. No finiteness statement about the
possibly larger representation in degree \(kh\) is needed. Each factor
has the assigned lower restriction proved above, so the resulting lower
tuple is the original lower tuple raised componentwise to \(h\).
Thus all these sections are compatible across different orbits and all
deeper intersections, and they are invariant in dimension \(d\).
They still generate. Fix a point \(z\) of 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 subsheaves identifies the line fibers. Nonvanishing
of that factor at \(z\) is therefore the nonvanishing of the representative
pre-section at a definite point of \(Z\). Each such condition is a
nonzero linear evaluation functional on the generating vector space
\(\mathcal P_d(k)\). A finite union of the proper kernels of these
functionals cannot cover a complex vector space. One pre-tuple satisfies
all of them, and its norm is nonzero at \(z\). At a lower point choose
a pre-tuple whose assigned lower value is nonzero; its \(h\)th power
remains nonzero. Finally take the linear span of all the constructed
norm tuples. Compatibility and invariance are linear conditions, so this
span is an invariant generating system in degree \(kh\).
This completes the induction through dimensions one and two; the
pre-system at dimension three then has all the required properties.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:floor-generation}]
Apply Proposition~\ref{prop:generating-tuples}. Its three-dimensional
components agree on every subordinate stratum, so
Proposition~\ref{prop:strata-adjunction} descends each tuple uniquely to
a section of \(L^k|_S\). For any \(x\in S\), choose a component through
\(x\). The generating system has a tuple whose value there is nonzero.
It is the pullback of the descended value in the actual line fiber at
\(x\), so that descended value is nonzero. Nakayama's lemma makes the
evaluation map onto \(L^k|_S\) surjective at \(x\). The finite-dimensional
span of these global sections therefore generates at every point, proving
semiampleness on the whole reduced floor.
\end{proof}
For the supported model in Proposition~\ref{prop:supported-model}, choose
a common multiple of the actual adjoint index, the coefficients and actual
equivalence of \(P\), and the generated boundary degree just obtained.
It gives a line \(L=\OO_V(qA)\), a section with divisor \(qP\), and a
generated restriction on \(S\). Its boundary sections define a holomorphic
map \(S\to\PP^b\) with the actual pullback identity for \(L|_S\).
The gluing argument has not extended these sections to \(V\); that is
the distinct lifting task to which we now turn.
==== END Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//gluing.tex ====
==== BEGIN Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//hodge.tex ====
\section{Filtered direct images on simple normal-crossing supports}\label{sec:hodge}
The residue insertion has placed the finite lifting obstruction in
\(R^1h_*\omega_H\). We need two facts about this absolute dualizing
direct image: it embeds as the lowest step of a filtered right
\(\Dcal\)-module, and on a projective base the kernel of first-symbol
multiplication admits no map from an ample line. In
Section~\ref{sec:lifting}, differentiation of representatives of the
finite obstruction will produce just such a symbol relation. We prove
the two facts here at the geometric scope of the graph supports.
Let \(p:A\to T\) be a holomorphic map of complex manifolds, where \(A\)
has pure dimension \(n\). Let \(i:H\hookrightarrow A\) be a reduced
closed analytic subspace of pure dimension \(d\), and put \(c=n-d>0\).
Assume the following.
\begin{enumerate}
\item The map \(p\) is a submersion near \(H\), and \(h=p|_H:H\to T\)
is proper.
\item Locally in \(A\), the space \(H\) is a reduced SNC divisor in a
smooth submanifold of dimension \(d+1\). It has finitely many globally
smooth indexed irreducible components \(H_1,\ldots,H_s\), and every
closed intersection \(H_I=\bigcap_{i\in I}H_i\) is smooth, possibly
empty or disconnected.
\item Every connected component of every nonempty \(H_I\) is proper
over \(T\) and has a global closed real \((1,1)\)-form which, locally
on \(T\), becomes K\"ahler after adding the pullback of a K\"ahler form
from \(T\).
\end{enumerate}
The last condition is the relative-form convention for a proper K\"ahler
map. A K\"ahler form on a neighborhood of \(H\), as constructed in the
preceding section, implies it. We may replace \(A\) by the submersion
neighborhood of \(H\). All direct images below are proper on their
support; the ambient map \(p\) itself need not be proper.
We use right \(\Dcal\)-modules with the increasing order filtration.
Set
\begin{equation}\label{eq:support-module}
K=\mathcal H^c_{[H]}(\omega_A),\qquad
F_0K=\im\bigl(\mathcal{E}xt^c_A(\OO_H,\omega_A)\longrightarrow K\bigr),
\qquad F_pK=(F_0K)F_p\Dcal_A\quad(p\geq0),
\end{equation}
and \(F_pK=0\) for \(p<0\). The brackets mean algebraic local
cohomology with finite pole orders. We will show that the first map is
injective and its image is the absolute dualizing sheaf \(\omega_H\).
For filtered right \(\Dcal\)-modules, \(p_+\) denotes the direct image;
in this submersion setting it is computed by the relative right de Rham
(Spencer) complex followed by \(Rp_*\).
\begin{proposition}[The filtered SNC direct image]\label{prop:filtered-snc}
For the preceding data, \((K,F)\) is a good filtered regular holonomic
right module. Put \(M^j=\mathcal H^j p_+K\) for any integer \(j\), and
give it the filtration induced by the filtered direct image. Every
level cohomology map
\[
\mathcal H^j F_p p_+(K,F)\longrightarrow M^j
\]
is injective, with image \(F_pM^j\). The module \((M^j,F)\) has a finite
filtration by \(\Dcal_T\)-submodules, strict at every \(F_p\), whose
quotients are polarizable real pure Hodge modules. More precisely, the
quotient with support-filtration index \(\ell\) has weight \(\ell+j\).
There are canonical identities, with the first realized by that injection,
\begin{equation}\label{eq:lowest-direct-image}
F_0M^j=R^jh_*\omega_H\lhook\joinrel\longrightarrow M^j,
\qquad F_pM^j=0\quad(p<0).
\end{equation}
Here \(R^j=0\) for \(j<0\), and \(\omega_H\) is the absolute dualizing
sheaf.
\end{proposition}
For the later use with \(j=1\), define the first-symbol map for the right
action by
\[
\sigma:F_0M^1\otimes T_T\longrightarrow\gr^F_1M^1,
\qquad m\otimes\xi\longmapsto[m\xi].
\]
When \(T\) is smooth projective, Corollary~\ref{cor:symbol-kernel} will
prove \(\operatorname{Hom}_{\OO_T}(N,\ker\sigma)=0\) for every ample line
\(N\) on \(T\).
\subsection{Finite poles and the two filtrations}
At a point of \(H\), choose analytic coordinates
\[
(z_1,\ldots,z_{c-1},x_1,\ldots,x_t,y),\qquad
\mathscr I_H=(z_1,\ldots,z_{c-1},f),\quad f=x_1\cdots x_t.
\]
The \(z\)-list is empty when \(c=1\). Write \(R\) for the local analytic
ring and \(\eta\) for its coordinate volume form. These equations are
a regular sequence. Their localization \v Cech complex, equivalently
the direct limit of their Koszul complexes, has cohomology only in degree
\(c\). In that degree it is
\begin{equation}\label{eq:polar-cech}
\frac{R[z_1^{-1},\ldots,z_{c-1}^{-1},f^{-1}]\eta}
{\displaystyle\sum_{j=1}^{c-1}
R[z_1^{-1},\ldots,\widehat{z_j^{-1}},\ldots,z_{c-1}^{-1},f^{-1}]\eta
+R[z_1^{-1},\ldots,z_{c-1}^{-1}]\eta}.
\end{equation}
The same assertion holds for a subunion of the branches, with the product
of its selected \(x_i\)'s. An intersection of \(k\) branches instead
has \(c+k-1\) regular equations.
Successive finite Taylor divisions give a unique additive polar normal
form in \eqref{eq:polar-cech}: a finite sum of terms
\begin{equation}\label{eq:polar-normal-form}
a_{\mathbf u,\mathbf v,J}(x_{J^c},y)
\prod_{j=1}^{c-1}z_j^{-u_j}\prod_{i\in J}x_i^{-v_i}\eta,
\qquad u_j,v_i\geq1,\quad \varnothing\ne J\subset\{1,\ldots,t\}.
\end{equation}
The coefficient is a holomorphic germ in exactly the displayed
variables. Each fraction has bounded negative orders, so only finitely
many Taylor coefficients in its denominator variables enter this
description. It is an additive normal form, not an assertion that the
polar types form an \(R\)-linear direct sum.
The Koszul generator of \(\mathcal{E}xt^c_A(\OO_H,\omega_A)\) maps to
the class
\begin{equation}\label{eq:simple-fraction}
\frac{a\eta}{z_1\cdots z_{c-1}f}.
\end{equation}
If it is zero in \eqref{eq:polar-cech}, the expansion of \(a|_{z=0}/f\)
has no negative \(x_i\)-power. Every Taylor monomial of \(a|_{z=0}\)
is then divisible by every \(x_i\), hence by \(f\). Thus
\(a\in(z_1,\ldots,z_{c-1},f)\), proving injectivity. Complete-intersection
adjunction, including its determinant of the conormal, identifies its
image intrinsically with \(\omega_H\).
For a polar term define the excess order
\[
e(\mathbf u,\mathbf v,J)=
\sum_{j=1}^{c-1}(u_j-1)+\sum_{i\in J}(v_i-1).
\]
The simple fractions \eqref{eq:simple-fraction} are exactly the terms
of excess zero. On a right canonical module a coordinate derivative
acts by minus differentiation of the coefficient of \(\eta\). A
derivative in a denominator variable raises its pole order by one with
a nonzero scalar. Conversely, in \eqref{eq:polar-normal-form} the
coefficient is independent of those variables, so these derivatives
produce each desired term from an excess-zero term without a tangential
error. It follows that
\begin{equation}\label{eq:pole-filtration}
F_pK=\{\text{finite sums of terms \eqref{eq:polar-normal-form} of excess
at most }p\},\qquad F_pK=0\ (p<0).
\end{equation}
This proves the generated formula in \eqref{eq:support-module} locally,
and proves that it is a good filtration. The generated definition
makes it independent of the coordinates.
For a closed immersion of smooth manifolds \(j:Y\hookrightarrow A\),
the right transfer convention is
\begin{equation}\label{eq:right-transfer}
F_pj_+(N,F)=\sum_{\nu}F_{p-|\nu|}N\,\partial_{\perp}^{\nu}.
\end{equation}
There is no codimension shift. For example, the codimension shift for
left transfer cancels the two dimension shifts in changing sides. These
right conventions are recorded in
\cite[Section~1.1, equations~(1.1.2)--(1.1.4)]{FujinoFujisawaSaito2014}.
We define an increasing support filtration on \(K\) by summing the
images of the local-cohomology modules of subunions of at most \(k\)
of the indexed components:
\begin{equation}\label{eq:support-weight}
W_{-d+k-1}K=
\sum_{|I|\leq k}\im\left(
\mathcal H^c_{[\bigcup_{i\in I}H_i]}(\omega_A)\longrightarrow K\right)
\quad(1\leq k\leq s).
\end{equation}
Set the lower steps to zero and the upper steps to \(K\). In the polar
normal form the summand for \(I\) consists of the types \(J\subset I\).
The support maps are locally injective, and hence
\[
W_{-d+k-1}K=\{\text{terms with }|J|\leq k\}.
\]
The intersection \(F_pK\cap W_{-d+k-1}K\) imposes this bound and the
excess bound simultaneously. Keeping exactly the polar type \(J=I\),
\(|I|=k\), leaves \(c+k-1\) normal denominators. By
\eqref{eq:right-transfer} the resulting filtered quotient is
\begin{equation}\label{eq:weight-gradeds}
\gr^W_{-d+k-1}(K,F)
\simeq\bigoplus_{|I|=k}(i_I)_+(\omega_{H_I},F^{(0)}),
\qquad
F^{(0)}_p\omega_{H_I}=
\begin{cases}0,&p<0,\\ \omega_{H_I},&p\geq0.
\end{cases}
\end{equation}
Here \(i_I:H_I\hookrightarrow A\), empty intersections contribute zero,
and \(d_I=\dim H_I=d-k+1\). Distinct sets \(I\) give distinct polar
types even at a point incident to further components. This proves the
exact induced \(F\) in \eqref{eq:weight-gradeds}, not just the unfiltered
quotient.
The identifications glue intrinsically. For an ordered \(I\) of size
\(k\), the iterated connecting maps for Mayer--Vietoris triangles of
supports give
\begin{equation}\label{eq:support-mayer-vietoris}
\frac{\mathcal H^c_{[\bigcup_{i\in I}H_i]}(\omega_A)}
{\displaystyle\sum_{J\subsetneq I}\im
\mathcal H^c_{[\bigcup_{j\in J}H_j]}(\omega_A)}
\longrightarrow \mathcal H^{c+k-1}_{[H_I]}(\omega_A).
\end{equation}
Locally this map selects the term with every \(i\in I\) in the polar
type, up to the sign of the order, and is therefore an isomorphism.
The one fixed ordering of the global smooth components fixes those signs
on overlaps. Disconnected intersections are treated componentwise.
This identifies \eqref{eq:support-mayer-vietoris} with
\eqref{eq:weight-gradeds} globally. The smooth-support modules on the
right are regular holonomic; their finite extension \(K\) is regular
holonomic as well.
\subsection{The real structure and strict direct image}
The filtration \(W\) has a real realization. With the absolute right
de Rham complex ending in degree zero, \(\DR_A\omega_A\simeq\C_A[n]\).
The ordinary unshifted finite-pole de Rham complex along one coordinate
has the degree-zero constant and the degree-one logarithmic class of a punctured disk.
Tensoring these local comparisons and then taking the localization
\v Cech complexes gives, naturally for all inclusions of subunions,
\begin{equation}\label{eq:real-support}
\DR_A K\simeq R\Gamma_H\C_A[n+c]
\simeq i_*\mathbb D_H(\C_H[d]).
\end{equation}
Here \(R\Gamma_H\) is the sheaf-valued support functor on \(A\).
The shift in the second identity is fixed by the complex orientation:
\(n+c=2n-d\). The same support complexes with \(\R\) supply a real
perverse sheaf, since complexification is exact and faithful and its
complexification is the de Rham realization of the regular holonomic
module. Taking real perverse images of the subunion maps defines a real
filtration whose complexification is \eqref{eq:support-weight}.
The real version of \eqref{eq:support-mayer-vietoris} is an isomorphism
because its complexification is. Apply the support comparison anew to
its graded target \(\mathcal H^{c+k-1}_{[H_I]}(\omega_A)\), now with
support dimension \(d_I\). Its real realization is
\[
(i_I)_*\mathbb D_{H_I}(\R_{H_I}[d_I])
\simeq (i_I)_*\R_{H_I}[d_I],
\]
using smoothness and the complex orientation of \(H_I\). This is the
shifted rank-one constant real local system of the graded target.
The residue normalization may multiply its complex realization
by a nonzero constant on a connected component; it introduces neither
a varying factor nor monodromy. Such a real rank-one form is polarizable
for its single Hodge type. With the filtration in
\eqref{eq:weight-gradeds}, the right module on \(H_I\) is the dual
constant
\[
\R^H_{H_I}[d_I](d_I).
\]
The untwisted constant has weight \(d_I\) and first right step
\(F_{-d_I}\); the twist by \(d_I\) moves that step to zero and changes
the weight by \(-2d_I\). Thus its weight is
\(-d_I=-d+k-1\). These dual-constant conventions agree with
\cite[equations~(1.2.1)--(1.2.6) and Theorem~2.3]{FujinoFujisawaSaito2014}.
There is also a useful support check. If a holomorphic germ \(g\)
vanishes on reduced \(H\), then
\begin{equation}\label{eq:filtered-support-condition}
gF_pK\subset F_{p-1}K.
\end{equation}
Indeed \(g\in(z_1,\ldots,z_{c-1},f)\). Multiplication by \(z_j\)
lowers the excess or kills the polar class; multiplication by \(f\)
does the same, since a surviving class has some remaining \(x_i\)-pole.
This is the filtered support condition for treating the object on \(H\)
through its local embeddings \cite[Section~1.17]{Saito1990Kahler}.
In particular properness on \(H\), rather than properness of \(A\),
is the relevant properness here.
We now prove Proposition~\ref{prop:filtered-snc}. Apply the proper
K\"ahler direct-image theorem
\cite[Theorem~1 and Remark~1.2]{Saito2022Kahler} after restricting to
each connected component of \(T\), and then separately to the constant
real Hodge module on each connected smooth component of \(H_I\) over it,
using the stipulated global relative form. These componentwise sums are
locally finite on \(T\): near any parameter, apply
Lemma~\ref{lem:finite-components} to the proper support and its smooth
closed intersections. Thus the componentwise direct images and their
polarizations give those for the full \(H_I\); in the cover and graph
applications the final indexing is already finite. The theorem supplies
strict filtered direct image, Lefschetz decomposition, and real primitive
polarizations.
The latter give a polarization on each full cohomology module. After
twisting by \(d_I\) and using filtered closed-embedding transitivity,
the degree-\(v\) direct image of a summand of
\(\gr^W_\ell K\) is polarizable real pure of weight
\begin{equation}\label{eq:direct-image-weight}
d_I+v-2d_I=-d_I+v=\ell+v.
\end{equation}
We use the convention fixed by the constant input in Section~1.1 and
the direct image in Section~2.2 of that source. The opposite sign for
the untwisted dimension in its displayed theorem statement is a
typographical error, also ruled out by applying it to the identity map.
This application is only to constant modules on smooth sources.
Write \(C=p_+(K,F)\) for the filtered direct-image complex. Since
\(p\) is a submersion near its support, before applying \(Rp_*\) its
right relative de Rham complex has at level \(F_p\) and degree \(-v\)
the term
\begin{equation}\label{eq:relative-right-dr}
F_{p-v}K\otimes_{\OO_A}\bigwedge^v T_{A/T}.
\end{equation}
The tangent wedges are locally free. The simultaneous pole and branch
bounds show that the quotient of this level complex by \(W\) is exactly
the same level of the relative complex for \(\gr^W K\).
Use the increasing \(W\) spectral sequence, written as
\begin{equation}\label{eq:weight-spectral-sequence}
E_1^{-\ell,v+\ell}=\mathcal H^v p_+\gr^W_\ell K
\quad\Longrightarrow\quad \mathcal H^v p_+K.
\end{equation}
Each term on the first page is a strict filtered direct image and a
polarizable real pure module of weight \(\ell+v\), by
\eqref{eq:direct-image-weight}. Strictness says that the corresponding
level-\(F_p\) first page injects into it with image \(F_p\).
The real support realization \eqref{eq:real-support}, its proper direct
image on the support, and the de Rham comparison make the unfiltered
differentials real. Comparison with the level spectral sequences makes
them filtration preserving.
The differential \(d_r\) sends \((\ell,v)\) to
\((\ell-r,v+1)\). For \(r=1\) these weights are equal. The differential
is therefore a morphism of real pure Hodge modules, hence is strict for
\(F\), and its kernels and cokernels remain polarizable pure modules
\cite[Propositions~1.10 and~1.14]{Saito1990Kahler}. Consequently the
level second page injects into the unfiltered second page with image
\(F_p\), and the latter consists of pure pieces of the indicated weights.
For \(r\geq2\) the target weight \(\ell+v-r+1\) is smaller than the
source weight \(\ell+v\). Here is why a real filtered differential
between those pieces is zero. Decompose both into their locally finite
strict-support summands. A perverse map between distinct strict supports
has image supported properly in at least one of them, and hence is zero.
On a common smooth dense support of dimension \(e\), the map is a real
map of local systems preserving their Hodge filtrations; this follows
also directly from \eqref{eq:right-transfer}, which recovers the intrinsic
filtered module as the sections annihilated by the normal ideal. The
two variation weights are the module weights minus \(e\). A real map
preserving \(F^\bullet\) preserves \(\overline F^\bullet\) as well.
The image of a vector of source type \((a,b)\) lies in
\(F^a\cap\overline F^b\) of the target. Since \(a+b\) is the larger
source weight, that intersection in the pure target is zero. Thus the
generic map is zero, and strict support makes the map zero everywhere.
Inductively, injection of a level page implies its differential is zero
when the unfiltered differential is zero. Hence both sequences
degenerate at the second page and the level infinity page injects with
image \(F_p\). The filtration \(W\) is finite. If the abutment map
\(\mathcal H^j(F_pC)\to\mathcal H^j(C)\) had a nonzero kernel element,
choose the least \(\ell\) containing it. Its nonzero leading class in
\(\gr^W_\ell\) would contradict the injection on the infinity page.
This proves the asserted injection of every level.
The same leading-class argument proves strictness of the induced \(W\):
if an element of \(\mathcal H^j(F_pC)\) maps into \(W_\ell M^j\), it
already lies in \(W_\ell\mathcal H^j(F_pC)\). The resulting finite
filtration of \((M^j,F)\) is therefore strict for every level and has
exactly the pure second-page quotients, of weights \(\ell+j\).
It follows as well that \(M^j\) is regular holonomic and its filtration
is good: these properties are preserved under this finite filtered
extension. Taking \(\gr^F\), and then \(\gr^F\DR_T\), preserves its
short exact sequences. If indexed by their pure weights, the output
weight filtration is the shifted filtration \(W[j]\).
For \(p<0\), every term in \eqref{eq:relative-right-dr} vanishes.
For \(p=0\), its only term is \(F_0K=i_*\omega_H\) in degree zero.
The just-proved level injection consequently gives exactly
\eqref{eq:lowest-direct-image}. This completes the proof of
Proposition~\ref{prop:filtered-snc}. A relative dualizing sheaf would
appear only after tensoring by \(\omega_T^{-1}\); the line in
\eqref{eq:lowest-direct-image} is absolute.
The first residue cohomology \(R^1h_*\omega_H\) is therefore embedded
as \(F_0M^1\). For lifting, it remains to prove the symbol-kernel
vanishing when \(T\) is smooth projective.
\subsection{Projective vanishing for the pure quotients}
The projective vanishing we will use is stated for the filtered components
of the real or complex theory of Sabbah--Schnell. The K\"ahler
direct-image theorem above supplied real polarizations; a rational
polarization was not part of its output. We therefore need vanishing
in the real theory and must identify the actual filtration, not merely
the underlying regular holonomic module. The following comparison does
this for each strict-support pure summand.
\begin{lemma}[Pure-component comparison and vanishing]\label{lem:real-pure-vanishing}
Let \(T\) be smooth projective, and let \((Q,F)\) be the right filtered
module of one strict-support summand of a polarizable real pure quotient
in Proposition~\ref{prop:filtered-snc}. Write \(Z\) for its support and
\(w\) for its weight. It is isomorphic, as a right filtered module, to
the prime holomorphic component of a pure object in
\(\mathrm{pHM}_Z(T,\R,w)\) of Sabbah--Schnell, attached to the same
generic polarized real variation. In particular, for every ample line
\(N\) on \(T\), every integer \(p\), and every \(v<0\),
\begin{equation}\label{eq:real-negative-vanishing}
\mathbb H^v\bigl(T,N^{-1}\otimes\gr^F_p\DR_T Q\bigr)=0.
\end{equation}
\end{lemma}
\begin{proof}
We use Saito's analytic category of pure real Hodge modules
\cite[Sections~1.4 and~1.8]{Saito1990Kahler}, which permits discrete real
indices for the \(V\)-filtration. The pure properties supplied above
are in that category. If the constant-source theorem is read with the
stronger rationally indexed pure conditions, those imply the broad real
specializability conditions, and induction on support dimension gives
the pure clauses of Definition~1.8 as well.
On a smooth dense analytic Zariski open \(Z^\circ\) of \(Z\), the piece
\(Q\) gives a polarized real variation \(\mathbb H\) of weight
\(w-\dim Z\), by Lemmas~1.9 and~1.13 of that source.
Theorem~16.2.1 and Corollary~16.3.6 of
\cite{SabbahSchnell2026} give a pure object
\(B\in\mathrm{pHM}_Z(T,\R,w)\) extending this same variation with its
real polarization. Use its prime holomorphic filtered component.
Both underlying complex regular holonomic modules are the intermediate
extension \(\operatorname{IC}_Z(\mathbb H_\C)\). For the Saito filtered
module use strict support and regularity in Remark~1.11 of
\cite{Saito1990Kahler}; for the Sabbah--Schnell component use
\cite[Section~14.2.13 and Theorem~14.7.1]{SabbahSchnell2026}.
Regular Riemann--Hilbert and uniqueness of intermediate extension
\cite[Theorem~13.2.11 and Section~13.2.12]{SabbahSchnell2026}
give the unique unfiltered identity extending the chosen identity on
\(Z^\circ\). It remains to prove that this identity preserves \(F\)
at the boundary.
On a smooth support \(U\) of dimension \(e\), write \(\mathscr H\) for
the flat holomorphic bundle of the variation, with decreasing Hodge
filtration. Its right filtration is
\begin{equation}\label{eq:right-generic-filtration}
F_p(\omega_U\otimes\mathscr H)=\omega_U\otimes F^{-p-e}\mathscr H.
\end{equation}
Indeed the left filtration is \(F_p^L=F^{-p}\), and the side change is
\(F_p^R=\omega_U\otimes F_{p+e}^L\). The same formula is recorded in
\cite[Section~16.3.12]{SabbahSchnell2026}. The right closed transfer
\eqref{eq:right-transfer} introduces no further shift. Thus the two
right filtrations agree away from the boundary of \(Z^\circ\).
Work locally on \(T\). Choose a holomorphic \(g\) whose zero set
contains that boundary but no local component of \(Z\), and split into
the finitely many local strict-support factors if necessary. Such a
\(g\) exists by prime avoidance: the boundary has smaller dimension
than each local support branch. If the boundary is empty there is
nothing to prove. Use the graph embedding in \(T\times\C_t\), and let
\(Q_g\) denote the transferred module. Put \(b=\dim T\). Saito's
left convention uses increasing \(F^L\) and decreasing \(V_L^\alpha\)
with \(t\partial_t-\alpha\) nilpotent on its \(\alpha\)-grade. On the
graph ambient the conversion is
\begin{equation}\label{eq:graph-side-change}
F_p^R=\omega_{T\times\C}\otimes F^L_{p+b+1},
\qquad V_L^\alpha\longleftrightarrow V^R_{-\alpha-1}.
\end{equation}
The second rule follows because transposition sends
\(t\partial_t\) to \(-t\partial_t-1\). In particular
\(V_L^{>-1}\) becomes \(V^R_{<0}\). The sign change of a normal
\(\partial_t\) does not change a generated sum.
We verify the filtered surjectivity needed in Saito's reconstruction.
Put \(\Psi=\psi_{g,1}Q\) and \(\Phi=\phi_{g,1}Q\). Strict support
gives a surjective \(\mathrm{can}:\Psi\to\Phi\) and an injective
\(\mathrm{Var}\), by \cite[Proposition~1.5]{Saito1990Kahler}. Forgetting
\(F\), this is the nilpotent middle-extension monodromy quiver of
holonomic modules. Its
monodromy filtrations centered at zero obey
\begin{equation}\label{eq:can-monodromy}
\mathrm{can}(M_k\Psi)=M_{k-1}\Phi
\end{equation}
by \cite[Lemma~3.3.13]{SabbahSchnell2026}. The centers of the weight
filtrations in Saito's pure definition are \(w-1\) for \(\Psi\) and
\(w\) for \(\Phi\). Thus \eqref{eq:can-monodromy} says
\(\mathrm{can}(W_i\Psi)=W_i\Phi\). Condition~(1.8.2) of Definition~1.8 and the
direct-factor Lemma~1.15 of \cite{Saito1990Kahler} put both
\(\gr_i^W\Psi\) and \(\gr_i^W\Phi\) in Saito's pure real category of
weight \(i\). The induced surjection on each grade is a filtered pure
morphism, hence is \(F\)-strict by its Proposition~1.10.
At a stalk, take \(y\in F_p\Phi\) and choose \(i\) with \(y\in W_i\Phi\).
Strictness on the \(i\)th grade lifts its class by an element of
\(F_p\Psi\cap W_i\Psi\). Subtract its image and repeat on the next
lower weight. The finite monodromy filtration terminates this procedure.
It proves
\[
\mathrm{can}(F_p\Psi)=F_p\Phi\quad\text{for all }p.
\]
The finite lifting argument obtains filtered surjectivity from pure
strictness on each weight grade.
Saito's filtered middle-extension formula
\cite[equation~(1.7.2)]{Saito1990Kahler} now applies: the local pure
factor satisfies its specializability conditions, the zero set of \(g\)
does not contain its support, and \(\mathrm{can}\) is filtered
surjective. After \eqref{eq:graph-side-change}, that formula is
\begin{equation}\label{eq:common-middle-extension}
F_pQ_g^R=
\sum_{v\geq0}
\left(V^R_{<0}Q_g^R\cap
(j_{\mathrm{op}})_*F_{p-v}(Q_g^R|_{t\ne0})\right)\partial_t^v,
\end{equation}
where \(j_{\mathrm{op}}:T\times\C^*\hookrightarrow T\times\C\) and the
intersection is inside \((j_{\mathrm{op}})_*j_{\mathrm{op}}^{-1}Q_g^R\).
The strict-support module and its \(V\)-pieces embed there by restriction.
For the Sabbah--Schnell component, strict real specializability, pure
support, and strictness of \(\mathrm{can}\) follow respectively from
Definition~14.2.2, Theorem~14.2.19, and Corollary~14.2.23 of
\cite{SabbahSchnell2026}. Its unfiltered intermediate-extension property
makes \(\mathrm{can}\) surjective, so it is a filtered middle extension.
Definition~10.6.1, Remark~10.6.2, and Proposition~10.6.5 of the same
source give precisely \eqref{eq:common-middle-extension} for its right
component: for \(\alpha<0\), the filtered \(V_\alpha\) is the
intersection with the filtration off the divisor, and the full module
is generated from \(V_{<0}\) by normal derivatives.
The unfiltered intermediate-extension identity preserves the canonical
\(V\)-filtration and, by \eqref{eq:right-generic-filtration}, preserves
\(F\) off the divisor. The two right sides of
\eqref{eq:common-middle-extension} are therefore identical. This proves
equality of \(F\) on the graph. Formula \eqref{eq:right-transfer}
recovers the module before graph transfer, and its filtration, as the
sections annihilated by the normal ideal. Hence the identity is filtered
on \(T\). The local identities glue by uniqueness of the unfiltered
identity. The comparison uses the actual generic variation
and its filtration; the two theories' internal twist symbols need not
have the same weight convention.
A pure object of Sabbah--Schnell is an object of \(\mathrm{WHM}(T)\)
with its one-step weight filtration, by Section~14.2.14 of
\cite{SabbahSchnell2026}. Its Theorem~16.3.10, on the smooth projective
\(T\), gives negative-ample graded de Rham vanishing for either filtered
component. Sections~8.4.1--8.4.9 of that source identify the perverse
right de Rham convention with the Spencer complex ending in the module
in degree zero, with its filtration grading preserved. The theorem
therefore gives \eqref{eq:real-negative-vanishing} for every homogeneous
degree \(p\) in our convention. This proves the lemma.
\end{proof}
\begin{corollary}[No ample line in the first symbol kernel]\label{cor:symbol-kernel}
In Proposition~\ref{prop:filtered-snc}, assume in addition that \(T\)
is smooth projective, and fix \(j\). Put \(M=M^j\), and let
\[
\sigma:F_0M\otimes T_T\longrightarrow\gr^F_1M,
\qquad m\otimes\xi\longmapsto[m\xi]
\]
be principal-symbol multiplication for the right action. For every
ample line \(N\) on \(T\),
\begin{equation}\label{eq:symbol-hom-vanishing}
\operatorname{Hom}_{\OO_T}(N,\ker\sigma)=0.
\end{equation}
This assertion includes a coherent torsion subsheaf of the kernel.
\end{corollary}
\begin{proof}
On compact \(T\) the locally finite strict-support decomposition of
each pure quotient of Proposition~\ref{prop:filtered-snc} is finite.
Apply Lemma~\ref{lem:real-pure-vanishing} to each factor. The finite
filtration of \(M\) is \(F\)-strict, so its short exact sequences remain
exact after \(\gr^F\DR_T\). Their hypercohomology long exact sequences
give \eqref{eq:real-negative-vanishing} for \(M\) itself. This deduction
uses the comparison for the pure quotients and the strict filtration of
\(M\).
By \eqref{eq:lowest-direct-image}, the \(p=1\) right graded de Rham
complex has exactly two terms:
\begin{equation}\label{eq:first-symbol-complex}
\gr^F_1\DR_T M=
\left[F_0M\otimes T_T\xrightarrow{\ \sigma\ }\gr^F_1M\right]
\quad\text{in degrees }-1,0.
\end{equation}
Since \(N\) is invertible, its tensor is exact, and negative
hypercohomology gives
\[
0=\mathbb H^{-1}\bigl(T,N^{-1}\otimes\gr^F_1\DR_T M\bigr)
=H^0(T,N^{-1}\otimes\ker\sigma)
=\operatorname{Hom}_{\OO_T}(N,\ker\sigma).
\]
The two-term calculation uses no local freeness of \(\ker\sigma\), so
it applies to torsion as well.
\end{proof}
==== END Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//hodge.tex ====
==== BEGIN Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//introduction.tex ====
\section{Introduction}\label{sec:introduction}
A rational holomorphic line is \emph{semiample} if some positive Cartier
multiple is generated by global sections. Such a multiple defines a
holomorphic map to projective space. The abundance question considered
here is whether analytic nefness of a klt adjoint on a compact K\"ahler
fourfold, together with one nonzero plurisection, forces this conclusion.
Let \(\Delta\) be an effective rational Weil divisor on a normal complex
space \(X\). We say that \(D=K_X+\Delta\) is an \emph{actual
\(\Q\)-Cartier adjoint} when an appropriate reflexive adjoint power is an
invertible holomorphic sheaf. All multiples and sections of \(D\) then
refer to tensor powers of that line. This convention allows \(K_X\) and
\(\Delta\) to fail to be separately \(\Q\)-Cartier. The line is
\emph{analytically nef} if, for a fixed K\"ahler form and every
\(\varepsilon>0\), a fixed Cartier multiple has a smooth Hermitian metric
whose normalized curvature is bounded below by minus \(\varepsilon\)
times that form, using local smooth potentials on \(X\). Section
\ref{sec:conventions} gives the full sheaf and metric conventions.
The Iitaka condition \(\kappa(X,D)\geq0\) means that a positive Cartier
multiple has a nonzero holomorphic section.
\begin{theorem}\label{thm:main}
Let \(X\) be a normal connected compact K\"ahler complex space of
dimension four. Let \(\Delta\) be an effective rational Weil divisor such
that \((X,\Delta)\) is Kawamata log terminal and the actual adjoint
\(D=K_X+\Delta\) is \(\Q\)-Cartier. If \(D\) is analytically nef and
\(\kappa(X,D)\geq0\), then there is an integer \(m>0\) for which
\(mD\) is Cartier and
\[
H^0(X,\OO_X(mD))\otimes_\C\OO_X\longrightarrow\OO_X(mD)
\]
is surjective at every point. If \(\kappa(X,D)=0\), then one may choose
\(m\) so that \(\OO_X(mD)\simeq\OO_X\).
\end{theorem}
Nonvanishing is a hypothesis. The conclusion concerns the given
holomorphic adjoint line on \(X\), without projectivity or
\(\Q\)-factoriality of \(X\) and without a numerical-dimension
restriction. When the Iitaka dimension is zero, the conclusion is an
actual rational-linear trivialization.
The threefold abundance theorems provide both the lower-dimensional
input and the methods behind this problem. In the projective terminal
case, Miyaoka treated numerical dimension one
\cite{Miyaoka1988}, and Kawamata proved abundance for minimal
threefolds \cite{Kawamata1992}. Passing to compact K\"ahler spaces
requires contraction and positivity arguments which cannot be obtained
by choosing a global ample divisor. H\"oring--Peternell developed the
minimal-model and Mori-fiber-space theory for compact K\"ahler
threefolds \cite{HoeringPeternell2016,HoeringPeternell2015}.
Campana--H\"oring--Peternell established canonical abundance for normal
ordinary \(\Q\)-factorial compact K\"ahler threefolds with terminal
singularities \cite{CampanaHoeringPeternell2016}; the corrected
Chern-class argument is given in their appendix to
Guenancia--P\u{a}un's orbifold Bogomolov--Gieseker theorem
\cite[Appendix, Theorem~A.2]{GuenanciaPaun2025}.
For lc compact K\"ahler threefold pairs with rational boundary and nef
adjoint, Das--Ou proved semiampleness when the numerical dimension is
different from two or the Iitaka dimension is positive
\cite[Theorem~1.1]{DasOu2022}. Their sequel treats numerical dimension
two and obtains abundance for lc compact K\"ahler threefolds
\cite[Theorem~1.2 and Corollary~1.3]{DasOu2025}.
These results generate the adjoint lines on the normal three-dimensional
components of a fourfold boundary. They do not by themselves choose
sections agreeing on its intersections or extend those sections to the
fourfold. Those are the two boundary problems treated here.
For positive Iitaka dimension, H\"oring--Lazi\'c--Lehn prove
semiampleness for nef klt adjoints on ordinary \(\Q\)-factorial compact
K\"ahler spaces through dimension four
\cite[Theorem~4.1]{HoeringLazicLehn2025}. A crepant ordinary
\(\Q\)-factorial K\"ahler model projective over \(X\) places our pair
in that scope. The sections of the pulled-back Cartier line are exactly
the sections from \(X\), so generation descends to the original line.
It remains to treat Iitaka dimension zero.
\subsection*{From a plurisection to a supported model}
Choose \(s\ne0\) in \(H^0(X,\OO_X(mD))\), and let
\[
M=\frac1m\operatorname{div}(s)
\]
be its normalized effective rational divisor. If \(M=0\), the section
already trivializes \(\OO_X(mD)\). Suppose that \(M\ne0\).
The birational reduction requires projective contractions with
K\"ahler targets. We prepare them by two constructions. First,
Proposition~\ref{prop:finite-null} contracts the entire null locus of a
nef and big threefold class when that locus is a finite union of curves;
its initial target is normal compact analytic. Second, conormal
direct-image vanishings and analytic thickenings extend a contraction of
a prime floor to the ambient space in
Proposition~\ref{prop:prime-floor-extension}. These two constructions,
combined with the specified relative MMP, rationality, descent, and
positivity inputs, give the threefold contraction statement
Proposition~\ref{prop:threefold-contraction}. It is used first on the
floors of the fourfold program and later on lower strata in the boundary
comparison argument.
The surrounding birational results are the pseudoeffective terminal
canonical threefold program of H\"oring--Peternell, the logarithmic
threefold program of Das--Hacon, and the crepant dlt models and supported
fourfold construction of Das--Hacon--P\u{a}un
\cite{HoeringPeternell2016,DasHacon2024,DasHaconPaun2024}.
The proof below states the precise imported inputs alongside the
constructions just described.
On a log resolution, we raise to one the coefficients of the strict
transforms of \(\Supp M\) and of the exceptional primes. The resulting
adjoint has an effective representative supported on its entire reduced
floor. The supported program then gives the model of
Proposition~\ref{prop:supported-model}: a normal ordinary
\(\Q\)-factorial compact K\"ahler dlt fourfold \((V,B)\) with effective
rational boundary and
\[
A=K_V+B,\qquad P\sim_\Q A,\qquad S=\floor B,\qquad
\Supp P=\Supp S,
\]
where \(A\) is an analytically nef actual \(\Q\)-Cartier adjoint,
\(P\) is an effective nonzero rational
\(\Q\)-Cartier divisor, and \(\kappa(V,A)=0\). The equivalence is an
isomorphism of actual rational holomorphic lines. Nonvanishing of \(P\)
follows from exceptional negativity: if its support disappeared, the
pullback of \(M\) would be a nonzero effective nef exceptional divisor.
The model also has the special projective resolution in
Lemma~\ref{lem:special-resolution}. Its strict boundary and exceptional
support have globally smooth distinct SNC components, its exceptional
crepant coefficients are below one, and it is generically an isomorphism
on the image of every irreducible component of an intersection of distinct
strict floor components.
The first boundary result, Theorem~\ref{thm:floor-generation}, proves
that the already existing restriction \(A|_S\) is semiample for this
model. It uses this resolution condition and requires no ambient
section.
\subsection*{Compatible sections on the whole floor}
The normal components and lower strata carry actual adjoint lines.
Lower-dimensional abundance, applied after the small models specified
below, makes these lines semiample; the threefold input is Das--Ou's
lc abundance theorem \cite[Corollary~1.3]{DasOu2025}. We must choose
sections whose iterated residue restrictions agree through every
intersection, so that they descend to sections on the reduced space
\(S\) itself.
The restriction criterion sometimes extends a section without a
comparison. When it requires a comparison, a perturbed program on the
lower stratum reduces the condition, in a sufficiently divisible common
degree, to the two coefficient-one markings on a projective-line fiber
of a Mori contraction. They give a birational residue comparison,
called a link. On a common projective graph, a link identifies the
compatible invertible meromorphic adjoint subsheaves; this equality
transports the actual residue sections and their products. The two
markings can belong to distinct primes or to one degree-two branch, whose
normal finite Stein space carries the exchange involution.
Fujino's induction by pre-admissible and admissible sections and finite
symmetrization is the ancestry of this compatibility construction
\cite[Sections~2--4]{Fujino2000}. Koll\'ar's sources and links describe
the related algebraic configuration with two disjoint distinguished
sections of a projective-line fiber
\cite[Definition~9, Theorem~10, and Proposition~14]{Kollar2013Sources};
the last proposition records the compatibility of even iterated residues
along these links.
For nonprojective surface strata, the links carry an additional geometric
constraint. Restrictions of one K\"ahler form on \(V\) induce
positive-square classes on their smooth minimal surface models. A link
from a three-dimensional stratum transports these classes modulo the
real span of divisor classes. A returning chain begins and ends on the
same surface stratum. After a common Cartier degree \(q\) is chosen,
the image of all returning chains on the space of \(qk\)-plurisections
is finite for each fixed integer \(k>0\). The groupoid may have
infinitely many arrows: its surface arrows are generated only by these
links and their inverses, while point and curve comparisons are treated
separately. Finite norm products impose the required invariance, and
finite hyperplane avoidance chooses sections nonzero at prescribed
points. Their residue agreement through all intersections gives
generators on the whole floor.
This common-class constraint is essential. Nonprojective K3 surfaces
can have automorphisms acting by a non-root-of-unity scalar on the
holomorphic two-form \cite[Theorems~3.5 and~4.1]{McMullen2002}.
Swapnajit Das's recent preprint states abundance for compact K\"ahler
semi-log-canonical threefolds \cite[Theorem~1.3]{Das2026}. Its
Lemmas~7.2--7.3 and Corollary~7.4 contain a closely related
positive-class and ruling mechanism. We give the explicit residue
construction here, including the actual meromorphic adjoint equality,
the normalized degree-two branch, and the uniform fixed-degree bound for
the link action. The cited surface mechanism is prior art; the stated
semi-log-canonical theorem is not an input to this proof.
\subsection*{Lifting from the supported boundary}
The second boundary result is a dimension-free statement for an arbitrary
standard dlt pair. The special resolution used in the fourfold
floor-generation application is not part of its data.
\begin{theorem}[Supported lifting]\label{thm:supported-lifting}
Let \((V,B)\) be a normal irreducible compact K\"ahler dlt pair of positive
dimension, with effective rational boundary and actual \(\Q\)-Cartier
adjoint \(A=K_V+B\). Suppose that an effective nonzero rational
\(\Q\)-Cartier divisor \(P\) satisfies
\[
P\sim_\Q A,\qquad \Supp P=\Supp\floor B.
\]
If the actual restriction \(A|_S\) to the whole reduced subspace
\(S=\floor B\) is semiample, then \(\kappa(V,A)\geq1\).
\end{theorem}
The support equality places the divisor of the initial section exactly
on the boundary where generation is known, and makes the pair klt away
from that divisor. The theorem assumes generation on the entire reduced
\(S\), including its intersections; it does not assume nefness of \(A\).
Choose an effective Cartier multiple \(G=qP\) and an actual identity
\(\OO_V(G)\simeq\OO_V(qA)\) such that its restriction to \(S\) is
generated. A generating system constructs a morphism
\(f:S\to T=\PP^b\) and an actual identity
\[
\OO_V(G)|_S\simeq f^*\OO_T(1).
\]
On an ambient neighborhood \(W\subset V\) of each compact fiber, a
root pair and a normalized cyclic cover \(\pi:Z\to W\) replace the
supported divisor by a reduced Cartier
divisor \(E\). Put \(I=\OO_Z(-E)\), and let \(g:E\to U\) be the
induced map over a parameter neighborhood \(U\subset T\). The finite
lifting target is that, for every integer \(j\) and every \(k\geq1\),
\[
g_*(I^j/I^{j+k+1})\longrightarrow g_*(I^j/I^{j+k})
\]
is an epimorphism of sheaves of complex vector spaces on \(U\).
The quotients are viewed on the underlying space of \(E\), and the
neighborhood used to lift a particular germ may depend on \(j\) and
\(k\). Thus the assertion lifts section germs by one finite order using
the boundary map \(g\).
The obstruction to each order is a graded derivation with values in the
first cohomology of a normal layer. A second root and a split residue
map embed that cohomology as a direct summand of the residue cohomology
of a resolved SNC divisor, retaining classes supported at special
parameters. A local differentiation identity turns any nonzero value
of the derivation on a base coordinate into a map excluded by the Hodge
vanishing on a smooth projective parameter cover. The same identity
then kills the remaining derivation. Finally, cyclic invariants give
recurring divisorial layers on \(V\) with positive twists from
\(\OO_T(1)\). Their sections grow while their higher cohomology and
the final loss from \(H^1(V,\OO_V)\) stay bounded. This forces
\(\kappa(V,A)\geq1\).
The direct ancestors of this lifting method are the threefold arguments
of Miyaoka and Kawamata. Miyaoka studies an effective pluricanonical
divisor on a minimal projective terminal threefold through neighborhood
covers and successive thickenings in the numerical-dimension-one case
\cite[Section~4]{Miyaoka1988}; he credits Reid with the suggestion to
analyze that divisor \cite[p.~220]{Miyaoka1988}. Kawamata's alternative
proof of that case uses neighborhood roots, residues, compatible
thickenings, and a mixed Hodge complex \cite[Section~4]{Kawamata1992}.
The proof here establishes the stated compact K\"ahler dlt lifting
theorem.
Apply the two boundary results to the supported model. Generation of
\(A|_S\) and Theorem~\ref{thm:supported-lifting} force positive Iitaka
dimension because \(P\ne0\), contradicting \(\kappa(V,A)=0\).
Therefore \(M=0\), and the original section \(s\) trivializes the
actual Cartier line \(\OO_X(mD)\).
Related manuscripts by OpenAI treat log abundance in the projective
characteristic-zero setting \cite{OpenAILogAbundance2026} and supported
boundary lifting \cite{OpenAILifting2026}. The projective supplement
also states rational-boundary semi-log-canonical abundance over \(\C\)
and a dlt restriction result for a nef adjoint with an effective rational
representative whose support contains the floor, in every Cartier degree
\(m\geq2\) \cite[Corollaries~11.5--11.6]{OpenAILogAbundance2026}.
Those projective statements are separate context; both analytic boundary
arguments needed here are proved in this manuscript.
Section~\ref{sec:conventions} fixes the actual-line conventions and a
connectedness lemma. Sections~\ref{sec:finite-null} and
\ref{sec:contraction-inputs} establish the contraction preparations used
on fourfold floors and on lower strata. Section~\ref{sec:reduction}
constructs the supported model. Sections~\ref{sec:adjunction}--\ref{sec:gluing}
construct compatible sections and prove generation of the existing
adjoint restriction on the entire reduced floor.
Section~\ref{sec:covers} constructs root neighborhoods and the split
residue map; Section~\ref{sec:hodge} proves the vanishing used for lifting;
and Section~\ref{sec:lifting} proves the finite-order surjections and
section growth. Section~\ref{sec:completion} assembles the main theorem.
==== END Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//introduction.tex ====
==== BEGIN Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//lifting.tex ====
\section{Lifting through every boundary neighborhood}\label{sec:lifting}
We retain the data \(L,s,G,q,r,a\), the local root charts, and the maps
\(g:E\to U\), \(h:H\to U\) of Section~\ref{sec:covers}. In particular
\(E\) is reduced Cartier, \(I=\OO_Z(-E)\) is invertible, and
\(\mathcal A_j=I^j/I^{j+1}\) has the Laurent graded frame \(u^j\) for
every integer \(j\). The frame uses the prescribed boundary root
comparison; it is a frame on the associated graded, not an extension
of that boundary frame to \(Z\).
The finite quotients \(\mathcal T_{j,k}=I^j/I^{j+k}\) have the lifting
target in Proposition~\ref{prop:all-orders}. The intervening
constructions supplied the split residue insertion and the projective
symbol-kernel vanishing. We now use them to kill the connecting maps of
\eqref{eq:current-truncation-sequence} at every finite order. The
quotients remain sheaves on the underlying space of \(E\); no map from
an infinitesimal thickening to the parameter space is required.
\subsection{The obstruction is a graded derivation}
We prove the proposition by induction on \(k\), simultaneously in all
integer degrees and all root charts. The following description uses
only the orders strictly smaller than the current one.
\begin{lemma}\label{lem:obstruction-derivation}
Fix \(k\geq1\), and assume \eqref{eq:all-order-surjection} for every
smaller order, every integer degree, and every root chart. The connecting
maps for the current order factor uniquely as maps of complex vector
space sheaves
\begin{equation}\label{eq:obstruction-map}
\delta_k:g_*\mathcal A_j\longrightarrow R^1g_*\mathcal A_{j+k}
\quad(j\in\Z).
\end{equation}
They form a degree-\(k\) derivation from the Laurent graded algebra
\(\bigoplus_jg_*\mathcal A_j\) to its graded cohomology module
\(\bigoplus_jR^1g_*\mathcal A_j\). If \(t_1,\ldots,t_b\) are local
coordinates on \(U\) and \(F\) is holomorphic in those coordinates, then
\begin{equation}\label{eq:obstruction-chain-rule}
\delta_k(F(t))=\sum_{i=1}^b F_{t_i}(t)\,\delta_k(t_i).
\end{equation}
These maps and formulas commute with restriction and with the ambient
power-compatible root comparisons. Vanishing of \(\delta_k\) in all
degrees proves the current order of \eqref{eq:all-order-surjection}.
\end{lemma}
\begin{proof}
Use \eqref{eq:current-truncation-sequence} together with
\begin{equation}
0\longrightarrow\mathcal T_{j+1,k-1}\longrightarrow\mathcal T_{j,k}
\longrightarrow\mathcal A_j\longrightarrow0\quad(k>1).
\label{eq:leading-truncation-sequence}
\end{equation}
All are valid for negative \(j\) because \(I\) is invertible. Write
\(\partial_{j,k}:g_*\mathcal T_{j,k}\to R^1g_*\mathcal A_{j+k}\) for
the connecting map of the first sequence. The shorter orders make
\(g_*\mathcal T_{j,k}\to g_*\mathcal A_j\) surjective. For \(k>1\),
its kernel is the image of \(g_*\mathcal T_{j+1,k-1}\), by left exactness
in the second sequence. The shorter order \(k-1\) in degree \(j+1\)
lifts that image through \(g_*\mathcal T_{j+1,k}\), which maps into
\(g_*\mathcal T_{j,k+1}\). The connecting map vanishes on the kernel.
For \(k=1\), the leading map is the identity and its kernel is zero.
This proves the unique factorization \eqref{eq:obstruction-map}.
If it is zero, exactness of the first sequence gives the desired
epimorphism.
Here is the derivation calculation on germs. A leading section in
degree \(j\) lifts to a section of \(\mathcal T_{j,k}\) after a base
shrink. Choose local representatives \(x_\alpha\in I^j\) on an ambient
open cover near \(E\). Their differences belong to \(I^{j+k}\), and
their classes modulo \(I^{j+k+1}\) represent \(\delta_k(x)\). For a
second section \(z\) of degree \(l\), take simultaneous representatives.
On an overlap,
\[
x_\beta z_\beta-x_\alpha z_\alpha
=x_\alpha(z_\beta-z_\alpha)+z_\alpha(x_\beta-x_\alpha)
+(x_\beta-x_\alpha)(z_\beta-z_\alpha).
\]
The last term lies in \(I^{j+l+2k}\subset I^{j+l+k+1}\). The first two
terms reduce to multiplication by the leading coefficients. Their
\v Cech classes therefore give
\[
\delta_k(xz)=x\delta_k(z)+z\delta_k(x).
\]
This uses the ordinary multiplication action of layer sections on their
first cohomology, and gives the asserted graded derivation.
For the chain rule, choose simultaneous shorter representatives
\(G_{i,\alpha}\in\OO_Z\) of the degree-zero sections \(t_i\). Their
differences lie in \(I^k\). Shrink the local opens so that
\(F(G_{1,\alpha},\ldots,G_{b,\alpha})\) is defined. Its holomorphic
Taylor difference, modulo the square of the differences, is
\(\sum_iF_{t_i}(G_\alpha)(G_{i,\beta}-G_{i,\alpha})\).
The square lies in \(I^{2k}\subset I^{k+1}\), and the coefficient
restricts to \(F_{t_i}(t)\) on \(E\). This gives
\eqref{eq:obstruction-chain-rule} before any restriction to a generic
parameter. It is valid when the target contains torsion. The connecting
map and all representative calculations are natural for restrictions
and the prescribed ambient root isomorphisms.
\end{proof}
\subsection{The local differential-operator identity}
Fix the current order \(k\), and put \(m=a+k>0\). For a local leading
section \(x\in g_*\mathcal A_{-m}\), define, in coordinates
\(t=(t_1,\ldots,t_b)\) on \(U\),
\begin{equation}\label{eq:residue-obstructions}
c(x)=\tau_*\delta_k(x),\qquad
e_i(x)=\tau_*\bigl(x\delta_k(t_i)\bigr)
\quad\text{in }R^1h_*\omega_H.
\end{equation}
Both arguments of \(\tau_*\) have layer degree \(-a\), since
\(-m+k=-a\). The map is the residue insertion of
Proposition~\ref{prop:residue-insertion}; it is injective on every
parameter germ.
Consider a holomorphic \(\psi:U'\to U\) between manifolds of the same
dimension \(b\), in coordinates \(w=(w_1,\ldots,w_b)\), satisfying
the transversality \eqref{eq:graph-transversality} for every smooth
component intersection of \(H\). The identity map is one such map.
As in Proposition~\ref{prop:global-graph}, after a shrink about a compact
fiber the support
\[
H^\flat=H\times_U U'\subset\widehat Z\times U'
\]
is reduced SNC, locally a divisor in a smooth complete intersection,
of codimension \(b+1\) in the product. Its ambient projection
\(p^\flat\) is a submersion and its closed strata are proper with global
relative K\"ahler forms. After the restrictions needed for the root
comparisons and the K\"ahler neighborhood, apply
Lemma~\ref{lem:finite-components} anew to the smooth closed strata of
the graph cut over \(U'\), and use its final open neighborhood. Properness
survives this base change, and the lemma gives finitely many globally
smooth irreducible component labels and finitely many smooth proper
connected pieces of each closed intersection. The existing coordinates
and relative forms restrict to that open. Set
\[
K^\flat=\mathcal H^{b+1}_{[H^\flat]}(\omega_{\widehat Z\times U'}),
\qquad M^\flat=\mathcal H^1p^\flat_+K^\flat.
\]
Proposition~\ref{prop:filtered-snc} applies with no additional
codimension shift and gives
\(F_0M^\flat=R^1h^\flat_*\omega_{H^\flat}\hookrightarrow M^\flat\).
The adjunction identity \eqref{eq:graph-adjunction} supplies the factor
\(\omega_{U'}\otimes\psi^*\omega_U^{-1}\). We denote its wedge frame
by \(dw/dt\); this notation is a ratio of canonical frames, not an
inverse Jacobian. Pull the \v Cech classes representing \(c(x),e_i(x)\)
to \(H^\flat\) and tensor with this frame. Write their images in
\(F_0M^\flat\) as \(\widetilde c,\widetilde e_i\). These are the natural
pullbacks of these classes; no base-change isomorphism is asserted for
a ramified \(\psi\).
\begin{lemma}[The adjugate identity]\label{lem:local-symbol}
Let \(J=(\partial\psi_i/\partial w_j)\) and
\(J^\#=\operatorname{adj}(J)\). Under the preceding assumptions, the
following identity holds in the right module \(M^\flat\):
\begin{equation}\label{eq:adjugate-operator}
\widetilde c\,\det J+
\sum_{i,j}(\widetilde e_i\,\partial_{w_j})J^\#_{ji}=0.
\end{equation}
In particular,
\begin{equation}\label{eq:local-symbol-zero}
\sigma\left(\sum_{j,i}J^\#_{ji}\widetilde e_i\otimes\partial_{w_j}\right)=0
\quad\text{in }\gr^F_1M^\flat.
\end{equation}
For \(\psi=\id\), after the natural graph identification, the exact
identity is
\begin{equation}\label{eq:identity-operator}
c(x)+\sum_i e_i(x)\partial_{t_i}=0\quad\text{in }M^\flat.
\end{equation}
When \(b=0\), the same construction without graph equations gives
\(c(x)=0\) in its degree-one direct-image module.
\end{lemma}
\begin{proof}
All statements are on germs, so choose a common parameter shrink and
simultaneous shorter lifts of the finitely many sections \(x,t_i\).
Choose local ambient representatives near \(E\), indexed by \(\alpha\),
with
\begin{equation}\label{eq:shorter-representatives}
x_\alpha\in I^{-a-k},\quad x_\beta-x_\alpha\in I^{-a},\qquad
G_{i,\alpha}\in\OO_Z,\quad G_{i,\beta}-G_{i,\alpha}\in I^k.
\end{equation}
They exist by the smaller orders in Lemma~\ref{lem:obstruction-derivation};
for \(k=1\) the shorter truncation is already the layer. Pull them to
\(\widehat Z\), suppressing pullback symbols, and put
\[
\eta_\alpha=x_\alpha\tau,\qquad
Q_{i,\alpha}=\psi_i(w)-G_{i,\alpha},\qquad
P_\alpha=\prod_i Q_{i,\alpha}.
\]
On \(H\) the \(G_{i,\alpha}\) restrict to the coordinate functions of
\(h\). The \(Q_i\)'s and a reduced equation of \(H\) are therefore
the regular graph-support equations. The logarithmic insertion and
\(D_E=\rho^*E\geq H\) give the precise pole bounds
\begin{align}
\eta_\alpha&\in\omega_{\widehat Z}(H+kD_E),\nonumber\\
\eta_\beta-\eta_\alpha,\quad
\eta_\alpha(G_{i,\beta}-G_{i,\alpha})
&\in\omega_{\widehat Z}(H),\label{eq:fraction-pole-bounds}\\
\eta_\alpha(G_{i,\beta}-G_{i,\alpha})(G_{l,\beta}-G_{l,\alpha}),
\quad(\eta_\beta-\eta_\alpha)(G_{i,\beta}-G_{i,\alpha})
&\in\omega_{\widehat Z}(H-kD_E)\subset\omega_{\widehat Z}.
\nonumber
\end{align}
The last inclusion uses \(k\geq1\). Thus the quadratic and cross
difference terms have no pole on the first-factor support \(H\).
In algebraic local cohomology consider the local sections
\[
S_\alpha=\left[\frac{\eta_\alpha\wedge dw}{P_\alpha}\right]
\quad\text{of }K^\flat.
\]
The \(H\)-pole is already in \(\eta_\alpha\). These are finite-pole
generalized fractions with a fixed ordering of the graph equations.
To expand their differences, first quotient the localization along
\(H\) by forms regular there, and then localize for the \(Q_i\)'s.
Each polar class is killed by a power of a reduced equation of \(H\),
and every \(G_{i,\beta}-G_{i,\alpha}\) is a multiple of that equation.
The change from \(Q_{i,\alpha}\) to
\(Q_{i,\beta}=Q_{i,\alpha}-(G_{i,\beta}-G_{i,\alpha})\) therefore has
a finite geometric expansion on each polar class. By
\eqref{eq:fraction-pole-bounds} only its linear terms survive. This
proves the exact generalized-fraction equality on an overlap
\begin{equation}\label{eq:fraction-difference}
S_\beta-S_\alpha=
\left[\frac{(\eta_\beta-\eta_\alpha)\wedge dw}{P_\alpha}\right]
+\sum_i\left[
\frac{\eta_\alpha(G_{i,\beta}-G_{i,\alpha})\wedge dw}
{Q_{i,\alpha}P_\alpha}\right].
\end{equation}
There is no infinite series or division by a Jacobian in this identity.
Let \(C\) be the first \v Cech cochain in
\eqref{eq:fraction-difference}. Let \(E_i\) have the numerator of its
\(i\)th summand but only the denominator \(P_\alpha\), and let
\(E_i^{(l)}\) denote that cochain with the additional denominator
\(Q_{l,\alpha}\). The simple fractions \(C,E_i\) are cocycles: their
reductions are respectively the cocycles for \(\delta_k(x)\) and
\(x\delta_k(t_i)\); on a triple overlap the possible change of the
representative of \(x\) is a cross term with no \(H\)-pole by
\eqref{eq:fraction-pole-bounds}. The description of the residue insertion on representatives and
complete-intersection adjunction identify their classes with
\(\widetilde c,\widetilde e_i\) in \(F_0M^\flat\). More explicitly,
the residue of a log form \(\eta\) on \(H\), pulled with \(dw/dt\), is
represented by \([\eta\wedge dw/P_\alpha]\); the determinant of the
graph conormal gives exactly that frame. Thus this identification holds
also at ramification. The cochains \(E_i^{(l)}\) need not individually
be cocycles.
The numerator of \(E_i\) is independent of \(w\). The right canonical
action is minus differentiation of the coefficient of the volume form,
so at the cochain level
\begin{equation}\label{eq:right-fraction-action}
E_i\partial_{w_j}=\sum_l J_{lj}E_i^{(l)}.
\end{equation}
Write \(\check d\) for the \v Cech differential. Equation
\eqref{eq:fraction-difference} is
\(\check dS=C+\sum_iE_i^{(i)}\). Multiply it by \(\det J\) and use
\(JJ^\#=(\det J)\id\) together with
\eqref{eq:right-fraction-action}. The result, still at the cochain
level, is
\begin{equation}\label{eq:cochain-adjugate}
\check d(S\det J)=C\det J+
\sum_{i,j}(E_i\partial_{w_j})J^\#_{ji}.
\end{equation}
The placement of \(J^\#_{ji}\) after the right derivative is part of
this exact formula.
These cochains lie in the end term of the relative right de Rham complex
for the product projection, with first-factor coordinates held fixed.
That term has no outgoing relative differential, so a \v Cech coboundary
there is a total coboundary in direct image. The action in
\eqref{eq:right-fraction-action} induces the base right action there.
Passing \eqref{eq:cochain-adjugate} to \(M^\flat\) gives
\eqref{eq:adjugate-operator}. This uses the natural map from these
\v Cech classes to direct image, and does not require that all direct-image
classes be computed by this cover.
The term \(\widetilde c\det J\) is in \(F_0\). Moving a holomorphic
function past a right vector field changes an operator by order zero.
Taking \(\gr^F_1\) of \eqref{eq:adjugate-operator} gives
\eqref{eq:local-symbol-zero}. With unchanged coordinates \(J=\id\),
the exact formula itself gives \eqref{eq:identity-operator}. If \(b=0\),
there are no \(Q_i\); the undivided difference of the \(\eta_\alpha\)
in the support module of \(H\) is the cochain for \(c(x)\) and is a total
coboundary. This proves the last assertion as well.
\end{proof}
\subsection{One global symbol kills the current obstruction}
\begin{proof}[Proof of Proposition~\ref{prop:all-orders}]
Assume the shorter orders and use Lemma~\ref{lem:obstruction-derivation}.
First suppose \(b>0\), and fix an arbitrary point \(t_*\in T=\PP^b\).
Proposition~\ref{prop:global-graph} gives the coordinate-power map
\(\psi:T'=\PP^b\to T\), the line \(R=\OO_{T'}(1)\) with
\(R^r\simeq\psi^*\OO_T(1)\), and a single compact SNC graph support
\(H'\subset\mathscr W\) with submersive projection \(p':\mathscr W\to T'\).
It is unbranched over \(t_*\). Put
\[
K'=\mathcal H^{b+1}_{[H']}(\omega_{\mathscr W}),\qquad
M'=\mathcal H^1p'_+K'.
\]
The global graph proposition verifies every hypothesis of
Proposition~\ref{prop:filtered-snc}: the support has pure dimension
\(\dim V-1\), globally smooth finitely many components, and proper
smooth closed intersections with one ambient K\"ahler form. Thus
\(F_0M'=R^1h'_*\omega_{H'}\hookrightarrow M'\), in the absolute
canonical convention, and Corollary~\ref{cor:symbol-kernel} applies.
Take the local leading section \(x=u^{-m}\). On a local graph chart
choose a frame \(\bar p\) of \(R\) with \(\bar p^r=\psi^*\ell\) for a
downstairs frame \(\ell\). The local formula
\begin{equation}\label{eq:global-symbol-local}
\bar p^m\longmapsto
\sum_{j,i}J^\#_{ji}\widetilde e_i(u^{-m})\otimes\partial_{w_j}
\end{equation}
defines a global morphism
\begin{equation}\label{eq:global-symbol-map}
R^m\longrightarrow F_0M'\otimes T_{T'}.
\end{equation}
We verify the transition, including on the branch locus.
Let \(t'=t'(t)\) and \(w'=w'(w)\) be changes of coordinates, and put
\(A_t=\partial t'/\partial t\), \(B_w=\partial w'/\partial w\). These
matrices are invertible. If \(\bar p'=\lambda\bar p\), then
\(\lambda^r=\psi^*\mu\) for the downstairs unit \(\mu\) changing the
frame of \(\OO_T(1)\). Locally even at a branch point, \(\lambda\)
descends as a holomorphic unit: take an \(r\)th root of \(\mu\) on a
small downstairs neighborhood, and observe that the remaining ratio has
\(r\)th power one and hence is locally constant. Denote the resulting
downstairs unit also by \(\lambda\).
Compare the local root pairs by sending their new boundary frame to
\(\lambda p_1\). Lemma~\ref{lem:root-neighborhood} extends this
power-compatible comparison to the ambient germs. Its pullback agrees
with the transition of the global root near the graph, by the uniqueness
and properness argument in Proposition~\ref{prop:global-graph}. It
therefore identifies both full covers, their tautological forms, and
the fixed functorial principalizations. Under this identification
\(u'^{-m}=\lambda^m u^{-m}\). In the definition of \(e_i(x)\), the
factor \(x\) multiplies outside the derivation. Hence no derivative of
\(\lambda\) occurs. Naturality and the chain rule
\eqref{eq:obstruction-chain-rule} transform the downstairs column of
\(e_i\)'s by \(\lambda^m A_t\).
The absolute graph factor changes by
\(dw'/dt'=(\det B_w/\det A_t)\,dw/dt\). Consequently, for the pulled
columns and Jacobians,
\begin{align}
\widetilde e'&=\frac{\det B_w}{\det A_t}\lambda^m A_t\widetilde e,
&J'&=A_tJB_w^{-1},\label{eq:symbol-transition-one}\\
(J')^\#&=\frac{\det A_t}{\det B_w}\,B_wJ^\#A_t^{-1},
&(J')^\#\widetilde e'&=\lambda^m B_wJ^\#\widetilde e.
\label{eq:symbol-transition-two}
\end{align}
The adjugate identity holds for every \(J\): it follows for invertible
\(J\) from the inverse formula and then for all \(J\) as a polynomial
identity. Only the coordinate changes \(A_t,B_w\) are inverted.
Since the vector basis changes by \(B_w^{-1}\), the last equality is
exactly the transition for the image of the frame \(\bar p^m\) in
\eqref{eq:global-symbol-local}. This proves \eqref{eq:global-symbol-map}
on all of \(T'\). At ramification the classes are the natural \v Cech
pullbacks under the resolved ambient germ comparisons; a flat base-change
isomorphism for \(R^1h_*\omega_H\) is not used.
By Lemma~\ref{lem:local-symbol}, the image of
\eqref{eq:global-symbol-map} lies in \(\ker\sigma\). The line
\(R^m=\OO_{\PP^b}(a+k)\) is ample because \(a+k>0\). Corollary
\ref{cor:symbol-kernel} makes the global map zero. At a point above
\(t_*\), the map \(\psi\) is locally biholomorphic and \(J^\#\) is
invertible. The ambient product comparison then identifies the graph
family and its \v Cech classes with the downstairs family. Thus the
zero map implies \(e_i(u^{-m})=0\) as a downstairs germ near \(t_*\),
including a class supported at that parameter. The point \(t_*\) was
arbitrary, and the root comparisons allow any chart at it. It follows
that these classes vanish locally everywhere.
The injection \(\tau_*\) and the invertible layer frame \(u^{-m}\)
now imply \(\delta_k(t_i)=0\) on every chart: multiplication by that
frame is an isomorphism from the layer \(\mathcal A_k\) to
\(\mathcal A_{-a}\), including on first direct images. For any local
\(x\) of degree \(-m\), all \(e_i(x)\) vanish. The exact unchanged
coordinate identity \eqref{eq:identity-operator} gives \(c(x)=0\) in
\(M^\flat\). The injection \eqref{eq:lowest-direct-image} brings this
back to \(R^1h_*\omega_H\), and the residue injection gives
\(\delta_k(x)=0\). When \(b=0\), Lemma~\ref{lem:local-symbol} gives
\(c(x)=0\) directly, and the same two injections give this conclusion;
there are no parameter coordinates to treat.
In particular \(\delta_k(u^{-m})=0\). The derivation rule for the
invertible Laurent frame gives
\[
0=\delta_k(u^{-m})=-m u^{-m-1}\delta_k(u),
\]
so \(\delta_k(u)=0\). For any local \(F\in g_*\OO_E\), the section
\(Fu^{-m}\) also has degree \(-m\), and
\[
0=\delta_k(Fu^{-m})=u^{-m}\delta_k(F).
\]
Thus \(\delta_k(F)=0\). Every local graded section is \(Fu^j\), so
\(\delta_k\) is zero in every integer degree. Lemma
\ref{lem:obstruction-derivation} closes the current order. Induction
from \(k=1\) proves Proposition~\ref{prop:all-orders}.
\end{proof}
\subsection{Invariant layers and the growth of sections}
Finite lifting is now available in every degree. To obtain ambient
sections, we pass to cyclic invariants and count the resulting finite
quotients downstairs. Translation by the Cartier divisor \(G\) will
make the same finitely many coherent layers recur with increasing
positive twists on \(T\). Those twists supply the section growth and
bound the higher-cohomology loss.
\begin{proof}[Completion of Theorem~\ref{thm:supported-lifting}]
Define global divisorial subsheaves of meromorphic functions on the
normal space \(V\) by
\begin{equation}\label{eq:downstairs-ideals}
J_j=\OO_V\left(-\ceil{jG/r}\right)\quad(j\in\Z).
\end{equation}
They agree in each root chart with \((\pi_*I^j)^{\mu_r}\). Indeed, at
a prime over \(S_i\), an invariant meromorphic function of downstairs
order \(v_i\) has order \((r/d_i)v_i\). Membership in \(I^j\) is
equivalent to
\[
v_i\geq\ceil{jd_i/r}.
\]
At all other primes it is regular. The full invariant meromorphic
algebra is the downstairs algebra, and normality extends the
codimension-one test. Finite coherent pushforward and averaging by
\(\mu_r\) are exact. This also proves coherence of the divisorial
sheaves in \eqref{eq:downstairs-ideals} and of their quotients locally,
and their valuation descriptions glue globally.
Since \(00\), each \(-Nr\leq j<0\) is uniquely \(j=l-tr\) with
\(0\leq l0\),
\begin{equation}\label{eq:higher-layer-bound}
\sum_{j=-Nr}^{-1}h^i(T,F_j)
\leq C_i:=\sum_{l=0}^{r-1}\sum_{t\geq1}h^i(T,F_l(t))<\infty.
\end{equation}
These constants are independent of \(N\), and they are zero for
\(i>\dim T\). The long exact sequences of the finite filtration
\eqref{eq:downstairs-filtration} give the same bounds for
\(h^i(T,\mathscr F_{-Nr,0})\), as well as finiteness and vanishing above
that fixed dimension. Serre vanishing is being applied only to the
coherent quotients \(F_l(t)\).
There is at least one section of \(F_0(t)\) for every \(t\geq1\).
Choose a point of the nonempty \(S\) and a section of \(\OO_T(t)\)
nonzero at its image; its pullback is nonzero there. Thus
\(\sum_{j=-Nr}^{-1}h^0(T,F_j)\geq N\). Additivity of the finite Euler
characteristics in \eqref{eq:downstairs-filtration}, and the uniform
higher bounds for both the quotients and \(\mathscr F\), give a constant
\(C\) independent of \(N\) with
\begin{equation}\label{eq:thickening-growth}
h^0(T,\mathscr F_{-Nr,0})\geq N-C.
\end{equation}
For example one can take \(C=2\sum_{i=1}^{\dim T}C_i\).
Finally, \(J_{-Nr}=\OO_V(NG)\) and \(J_0=\OO_V\). Degree-zero direct
image on the underlying support gives
\[
H^0(T,\mathscr F_{-Nr,0})=
H^0(V,\OO_V(NG)/\OO_V).
\]
The exact sequence on \(V\) loses at most the fixed finite dimension
\(h^1(V,\OO_V)\) when lifting these quotient sections to
\(H^0(V,\OO_V(NG))\); finiteness follows from proper coherent direct
image to a point. Equation \eqref{eq:thickening-growth} therefore makes
these section spaces unbounded. For some \(N\) the line \(\OO_V(NG)\)
has two independent sections. Their quotient is a nonconstant meromorphic
function on the irreducible normal \(V\), so its Iitaka dimension is at
least one. Since \(\OO_V(G)=L=\OO_V(qA)\) is the actual adjoint
identity, this proves \(\kappa(V,A)\geq1\), as claimed.
\end{proof}
==== END Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//lifting.tex ====
==== BEGIN Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//links.tex ====
\section{Extension from a stratum and its residue link}\label{sec:links}
Proposition~\ref{prop:strata-adjunction} gives normal dlt pairs on the
strata and a single actual adjoint line along every incidence. We now
show that a section on the floor of a stratum extends after it satisfies
at most one birational residue comparison. The comparison comes from
two markings of a projective-line fiber on a Mori model. In the next
section we will control the actions generated by these comparisons.
\subsection{Semiample lines on the normal strata}
We use a small model both to place lower-dimensional abundance inside its
explicit sheaf conventions and to run the later perturbed program. Let
\(Z\) be a positive-dimensional stratum of dimension at most three.
Lower its remaining floor coefficients as in the preceding section,
obtaining an effective klt boundary \(B_Z^-\) with actual adjoint.
There is a projective small ordinary \(\Q\)-factorialization
\(q_Z:Z_0\to Z\) of this pair. Here is a construction that also checks
the actual line used on it.
On a projective log resolution \(r:W\to Z\), keep the effective strict
boundary and assign to each exceptional prime a rational weight just below
one but strictly above its crepant coefficient. This produces an
effective SNC klt boundary \(C_W\) with
\[
K_W+C_W=r^*(K_Z+B_Z^-)+F,
\]
where \(F\) is effective, exceptional, and positive on every divisorial
exceptional component. The equality is the meromorphic actual-line
identity. The adjoint is relatively pseudo-effective. Apply the
relative MMP for a projective morphism of compact analytic spaces in
dimension at most three \cite[Proposition~2.26]{DasHacon2024}. Its source
is the ordinary \(\Q\)-factorial smooth \(W\), its pair is effective SNC
klt, and its morphism is a projective surjection of normal compact
analytic spaces. On the
relative ordinary \(\Q\)-factorial minimal model over \(Z\), the transform
of \(F\) is exceptional and relatively nef, since the pulled-back adjoint
is relatively trivial. Negativity makes that transform zero. No step
extracts a divisor, so the resulting morphism \(q_Z:Z_0\to Z\) is small.
It is projective, hence \(Z_0\) is compact K\"ahler. Codimension-one
comparison and reflexive extension identify its perturbed adjoint with
the actual pullback from \(Z\).
The same smallness gives the crepant actual pullback for the full boundary.
Moreover \(q_Z\) is an isomorphism over a smooth germ of \(Z\). To see
this, take a relatively very ample line over a small Stein neighborhood
of the germ. It has a meromorphic section there: after large relative
ample twists, relative Serre generation and Cartan generation supply
sections whose quotient is such a section. Its divisor pushes to a
Cartier divisor on the smooth factorial target. Smallness leaves no
exceptional prime, so the original divisor and line are its pullback.
A pulled-back line cannot be relatively ample on a positive-dimensional
fiber. Any nonisomorphism fiber here would have positive dimension:
otherwise properness and the fiber-dimension theorem make the morphism
finite near that fiber, and a finite bimeromorphic morphism to the normal
target is an isomorphism. Thus the small model is unchanged over this
smooth germ. The lc centers of the full pair meet the
smooth crossing locus established in
Proposition~\ref{prop:strata-adjunction}; the small model is generically
unchanged there. The full transformed pair \((Z_0,B_0)\) is therefore
dlt, and its floor \(C_0\) is the strict transform of \(C_Z\).
For a three-dimensional stratum, apply Das--Ou's lc threefold abundance
theorem \cite[Corollary~1.3]{DasOu2025} to this full pair. Its adjoint is
the analytically nef actual pullback of the ambient restriction: the
metric lower bound restricts to \(Z\) and then pulls back. The detailed
conventions in \cite[Section~2.1]{DasOu2025} use the reflexive canonical
sheaf, the invertible reflexive pluriadjoint, actual restriction
isomorphisms, and smooth-potential nefness. Thus this application does not
choose a global canonical Weil divisor. Generation descends to \(Z\) by
normality and projection formula. Lower strata inherit generation by
restriction from a three-dimensional stratum containing them.
Choose a common sufficiently divisible even integer \(q\) for the finite
list of strata and set
\[
L=\OO_V(q(K_V+B)),\qquad
L_Z=L|_Z=\OO_Z(q(K_Z+B_Z)).
\]
Enlarge \(q\) so all \(L_Z\) in dimension at most three are generated.
Their complete systems followed by Stein factorization give
\begin{equation}\label{eq:stratum-semiample-map}
f_Z:Z\longrightarrow Y_Z,\qquad L_Z\simeq f_Z^*N_Z,
\end{equation}
where \(f_Z\) has connected fibers, \(Y_Z\) is normal projective, and
\(N_Z\) is ample. Indeed the image in projective space is projective by
Chow, and the finite analytic Stein space is projective by algebraizing
its finite coherent algebra with GAGA. Its line is the pullback of the
tautological ample line. In particular, for every \(k\geq0\), all
sections of \(L_Z^k\) come from \(N_Z^k\).
\subsection{A torsion-free obstruction to restriction}
The following lemma is what lets general-fiber matching control all
parameters, including those where the floor has a singular or nonreduced
scheme fiber. Its hypothesis that the pair is klt off the floor is
essential to the canonical character calculation.
\begin{lemma}\label{lem:floor-torsion-free}
Let \((T,G)\) be a normal irreducible compact K\"ahler lc pair with effective rational
boundary, and put \(C=\floor G\) with its reduced structure. Assume that
the pair is klt on \(T\setminus C\). Let \(f:T\to P\) be a surjective
holomorphic map with connected fibers to a normal irreducible compact complex space.
Suppose an actual positive adjoint multiple is pulled back from a line on
\(P\). Then
\[
R^1f_*\OO_T(-C)\quad\text{is torsion-free on }P.
\]
Here \(\OO_T(-C)\) is the ideal of the reduced floor.
\end{lemma}
\begin{proof}
The equality with the ideal holds because on the normal \(T\) both sheaves
consist of holomorphic functions vanishing at every height-one prime in
\(C\). Work on a small connected base open trivializing the line whose
pullback is \(m(K_T+G)\). Its nonvanishing pulled-back frame is a local
meromorphic \(m\)-pluricanonical form \(\theta\).
Adjoin the full set of \(m\)th roots of this form. This construction is
local even without a canonical Cartier frame. At a normal local domain
\(A\), choose a meromorphic canonical generator \(\eta\) and write
\(\theta/\eta^m=a/b\). The monic algebra
\[
A[w]/(w^m-a b^{m-1})\ \subset\
\operatorname{Frac}(A)[t]/(t^m-a/b),\qquad w=bt,
\]
is finite and reduced; its analytic normalization is finite
\cite[Part~B, Section~4, Corollaries~2--3]{Houzel1961}. Changes of meromorphic canonical generator multiply
the root coefficient by an \(m\)th power, and identify the total algebras
and their integral closures. They therefore glue to the normalized cover.
Keep every component and the full \(\mu_m\)-action. The tautological
form \(\tau=t\eta\) is a meromorphic top form on a projective resolution
with smooth source; take this resolution functorially and equivariantly, after
shrinking near the compact fibers. Write \(l:H\to T\) for the composite.
At a general prime of \(T\) with boundary coefficient \(b\), the order of
\(\tau\) upstairs is
\begin{equation}\label{eq:canonical-root-order}
e(1-b)-1,
\end{equation}
where \(e\) is the ramification index. This follows from the pole order
\(-eb\) of the root coefficient and the differential ramification order
\(e-1\). It is \(-1\) when \(b=1\) and is an integer between zero and
\(e-1\) when \(0\leq b<1\). A meromorphic form in the \(\tau\)-character
has the form \(h\tau\), where \(h\) is an invariant meromorphic function
and hence descends to \(T\); this uses the entire root algebra, whose
invariants are the normal base. Regularity in codimension one forces
\(h\) to be holomorphic and to vanish on \(C\).
Conversely, suppose \(h\in\OO_T(-C)\). On a log resolution of the pair,
every crepant coefficient is at most one. A coefficient-one exceptional
prime has center in \(C\), because the pair is klt away from \(C\); the
pullback of \(h\) has positive order there. Thus in SNC coordinates
the density \(|h|^2|\theta|^{2/m}\) has every exponent strictly above
the integrability threshold. It is locally integrable. Change of
variables makes \(l^*h\tau\) locally square integrable on the smooth
resolution, and a square-integrable meromorphic top form is holomorphic.
This proves the exact character equality
\begin{equation}\label{eq:canonical-character}
(l_*\omega_H)_{\tau}=\OO_T(-C)\tau.
\end{equation}
Each component of the full normalized cover dominates the normal base,
and each resolved component does so as well. The inverse image in \(T\)
of the connected base open is connected, because \(f\) is proper with
connected fibers; normality makes it irreducible. Finite maps followed by
projective resolutions have K\"ahler sources near the whole inverse image
of a sufficiently small base neighborhood: patch relative ample metrics
and add a large pulled-back K\"ahler form. Canonical torsion-freeness
\cite[Theorem~2.9 and Proposition~2.11]{Fujino2023AnalyticVanishing}
applies componentwise. For the generically finite map to \(T\), its
higher canonical images vanish generically and hence vanish. For the
map to \(P\), its first canonical image is torsion-free. Leray and the
character projector in \eqref{eq:canonical-character} make
\(R^1f_*\OO_T(-C)\) a direct summand of that torsion-free sheaf. This
proves the assertion locally, and hence on \(P\).
\end{proof}
Here is its restriction consequence. Suppose \(L_T=f^*N\) in the lemma,
where \(P\) is projective and \(N\) ample. The ideal sequence gives
\begin{equation}\label{eq:restriction-cokernel}
\mathscr Q:=\coker\bigl(\OO_P\longrightarrow f_*\OO_C\bigr)
\lhook\joinrel\longrightarrow R^1f_*\OO_T(-C).
\end{equation}
If \(C\) is vertical, meaning that it does not dominate \(P\), then
\(\mathscr Q\) is supported on a proper analytic subset and is zero by
torsion-freeness. Serre vanishing for the kernel of
\(\OO_P\to f_*\OO_C\), after tensoring by \(N^k\), shows that every
section of \(L_T^k|_C\) extends to \(T\) for all sufficiently large \(k\).
The bound is uniform over the sections.
If \(C\) dominates \(P\), the left arrow into \(f_*\OO_C\) is injective,
and \(f_*\OO_C\) is torsion-free by \eqref{eq:restriction-cokernel}.
The finite Stein space of the reduced \(C\to P\) is reduced and every
irreducible component dominates \(P\): a vertical minimal prime of a
finite reduced algebra over a domain would supply a nonzero torsion
element. Off a proper analytic subset this finite map is \'etale. A
section of \(L_T^k|_C\) whose values agree throughout the reduced floor
fiber over every point of some dense open of \(P\) has zero image in
\(\mathscr Q\otimes N^k\) on that open,
and hence everywhere by torsion-freeness. Its local lifts from \(N^k\)
are unique and glue. Thus it extends. This argument neither asserts
that special scheme fibers are reduced nor uses cohomology base change at
a special point.
\subsection{A Mori contraction over the semiample target}
The torsion-free argument has settled extension when the floor is
vertical over the semiample target. For a horizontal floor, it remains
to make a section take the same value on the connected components of a
general floor fiber. We first construct a Mori model on which those
components can be counted.
Fix a stratum \(Z\) of dimension one, two, or three and its small model
\(q_Z:(Z_0,B_0)\to(Z,B_Z)\). Put \(C_0=\floor{B_0}\) and
\(f_0:Z_0\to Y=Y_Z\). The full adjoint is the actual pullback of the
semiample line on \(Y\).
\begin{lemma}[A Mori model for a horizontal floor]\label{lem:horizontal-mori-model}
Assume that \(C_0\) dominates \(Y\). There is a finite sequence of
projective divisorial contractions and flips from \(Z_0\) to an ordinary
\(\Q\)-factorial compact K\"ahler model \(T\), followed by a projective
Mori contraction
\[
u:T\longrightarrow W,\qquad g:W\longrightarrow Y,
\]
where \(W\) is normal compact K\"ahler and both maps have connected
fibers. Write \(G\) for the transform of \(B_0\) and
\(C^+=\floor G\). The pair \((T,G)\) is effective lc and klt off
\(C^+\); the divisor \(C^+\) is relatively ample for \(u\) and still
dominates \(Y\). All steps factor over \(Y\) and extract no divisor.
On a common projective resolution of the sequence, the full adjoints
are equal as meromorphic pullbacks of the same actual line from \(Y\).
\end{lemma}
\begin{proof}
Choose a small rational \(\varepsilon>0\) for which
\(B_0-\varepsilon C_0\) is effective klt. This uses dlt and the fact
that all its lc centers are contained in the floor. Let \(H\) be an
integral very ample divisor on \(Y\) and choose a rational effective
\(H'\sim_\Q dH\) with \(d>2\dim Z\) such that
\[
B_0-\varepsilon C_0+f_0^*H'
\]
is still klt. A sum of sufficiently many general members of \(|H|\)
with small rational weights does this by Bertini on a log resolution.
If \(Y\) is a point take \(H'=0\).
The adjoint of this perturbed pair is not pseudo-effective. On a smooth
general fiber of a resolution of \(f_0\), its class is the negative of
the nonzero effective divisor \(\varepsilon C_0\) restricted to that
fiber: the full adjoint and \(H'\) are pulled back from \(Y\), and the
floor is horizontal. Its integral against a K\"ahler power is strictly
negative. If the ambient class were pseudo-effective, the potential of
a positive current representing its pullback would restrict to a positive
current on almost every such smooth fiber, contradicting this integral.
The smooth horizontal-fiber conditions hold on a nonempty open, so the
almost-everywhere restriction is sufficient.
This same integral also produces a negative class in the cone used by the
K\"ahler cone theorem. If the smooth fiber has dimension \(v>0\), push
its positive closed bidimension-\((1,1)\) current \(\omega_F^{v-1}\)
through the fiber inclusion and the resolution to \(Z_0\). It represents
a class in \(\overline{\operatorname{NA}}(Z_0)\), and its pairing with
the perturbed adjoint is exactly that strictly negative integral. The
cone decomposition therefore supplies a negative extremal ray.
Run the klt K\"ahler program for this perturbed adjoint. In dimension
three, the negative extremal-ray and contraction theorems
\cite[Corollary~5.3 and Theorem~5.5]{DasHaconPaun2024} are used through
Subsection~\ref{subsec:threefold-contraction-bridge}, which incorporates
Proposition~\ref{prop:prime-floor-extension} and the specified external
results. They give
connected projective contractions to normal compact K\"ahler targets.
The supporting class differs from the klt adjoint by a K\"ahler class,
which is the projectivity condition in that theorem. The supporting
class need not be big, and the perturbed adjoint here is not
pseudo-effective. Flips and termination
of flip sequences are provided by
\cite[Theorems~2.24--2.25]{DasHacon2024}. In dimensions at most two the
non-pseudo-effective pair is projective and the ordinary projective program
applies. For a surface, otherwise a non-Moishezon K\"ahler resolution
has a nonzero holomorphic two-form: if \(H^{2,0}=0\), rational approximation
of a K\"ahler class would make it projective. That form descends
reflexively and, with the effective boundary, gives a section of a positive
adjoint multiple, contradicting non-pseudo-effectivity. A Moishezon
K\"ahler klt surface is projective by Namikawa's criterion
\cite[Corollary~6']{Namikawa2002}. Curves are projective.
We specify the ordinary \(\Q\)-factorial and finiteness details used by
this program. For a projective negative extremal contraction, a global
rational line of degree zero on all contracted curves descends in a
multiple by \cite[Theorem~2.44(3c)]{DasHaconPaun2024}. Here we apply its
projective relative theorem to the effective klt pair, with the compact
set equal to the entire compact target. For this projective surjection,
property~Q of \cite[Definition~2.42]{DasHaconPaun2024} requires a normal
source and compact source and target, as here. All fiber curves span the one relative
ray, so the theorem's contraction is the given contraction. Indeed its
map is constant on each connected projective fiber of the given contraction,
and rigidity factors it through that contraction. Its structure map back
to the relative base supplies the reverse factorization; surjectivity makes
the two factorizations inverse. This use of Theorem~2.44 does
not assert a factoriality hypothesis for that theorem. At a divisorial step, adjust the
transform of a destination Weil divisor by a \(\Q\)-Cartier exceptional
prime of nonzero degree on the ray, and descend the resulting degree-zero
line. Comparison in codimension one makes the destination divisor
\(\Q\)-Cartier. At a flip, adjust on the negative side by the negative
adjoint, descend, and pull to the positive side; the positive adjoint is
\(\Q\)-Cartier and the same comparison applies. Boundary components
and the adjoint then also give the canonical reflexive power. This
preserves ordinary \(\Q\)-factoriality.
The rank of global rational divisor classes modulo curve numerical
equivalence is finite. Each running model is a connected compact complex
analytic space. Its reduced underlying real analytic space has bounded
Zariski tangent dimension by a finite analytic chart cover.
Acquistapace--Broglia--Tognoli \cite[Theorem~1]{AcquistapaceBrogliaTognoli1979}
embed it closedly in Euclidean space, and \L{}ojasiewicz
\cite[Theorem~1]{Lojasiewicz1964} gives a compatible locally finite
triangulation; the subcomplex corresponding to the compact model is
finite. Hence its rational \(H^2\) is finite-dimensional. The first
Chern class sends global rational Cartier divisors to that group. Every
divisor in its kernel has degree zero on each compact curve, by restriction
to the curve's normalization. The rational divisor space modulo curve
numerical equivalence is consequently a quotient of the Chern-class image,
and has finite rank. The rank drops at a divisorial contraction, by the
nonzero exceptional ray degree, and does not increase at a flip. For
the latter assertion a numerically trivial line on the negative side
descends as above; the descended line is curve-numerically trivial because
every curve downstairs is covered by a curve upstairs, using projectivity
over that curve. Its pullback is trivial on curves on the positive side,
and every divisor there is a transform by smallness. There can therefore
be only finitely many divisorial steps. Together with termination of flip
sequences this gives a finite program to a nef model or a Mori contraction.
All steps extract no divisors.
Every step factors over \(Y\). Inductively write the running perturbed
adjoint as
\[
A_i^{\varepsilon}=K_{X_i}+B_i-\varepsilon C_i+f_i^*H'.
\]
For a negative contraction apply the relative cone and length theorem
\cite[Theorem~2.45]{DasHaconPaun2024} to \(K_{X_i}+B_i-\varepsilon C_i\)
over its entire compact contraction target. The source is ordinary
\(\Q\)-factorial and the pair is klt. Property~Q holds as above; because
the compact set is the whole target, Theorem~2.45's factoriality over
that set is precisely the global factoriality of the source, including
its canonical reflexive power. A negative generator \(\Gamma\) has
\[
-(K_{X_i}+B_i-\varepsilon C_i)\cdot\Gamma\leq2\dim Z.
\]
If \(f_i(\Gamma)\) were a curve, then \(f_i^*H'\cdot\Gamma\geq d\),
contradicting the negativity of \(A_i^{\varepsilon}\). On the nonnegative
part of that relative cone, both \(K_{X_i}+B_i-\varepsilon C_i\) and the nef
\(f_i^*H'\) are nonnegative, while their sum is nonpositive on the
contracted cone; its \(H'\)-degree is zero there as well. All contracted
curves are vertical over \(Y\). A holomorphic map from a connected
projective fiber that is nonconstant would map a curve nontrivially, so
\(f_i\) is constant on each fiber. Rigidity and normality of the target
give the factorization, and the positive side of a flip factors through
the same base.
We check the boundary properties needed to repeat the step. No prime is
extracted, so the transformed full boundary is effective and its floor
is exactly the transform \(C_{i+1}\) of \(C_i\). The running MMP
preserves the klt pair
\(B_i-\varepsilon C_i+f_i^*H'\). Removing the effective divisor
\(f_{i+1}^*H'\) shows that \(B_{i+1}-\varepsilon C_{i+1}\) is klt.
In particular the full pair is klt away from its floor.
The full adjoints remain actually crepant pullbacks from \(Y\).
Indeed the chosen meromorphic pluriadjoint identity extends to the new
model in codimension one, since no prime is extracted, and then
reflexively. On a common graph both meromorphic maps start from the same
line pulled back from \(Y\) and agree on the dense isomorphism open.
They agree everywhere as meromorphic maps. Equality of discrepancies
therefore preserves log canonicity of the full pair. If \(r,s\) are
the graph projections, negativity for the perturbed step gives
\begin{equation}\label{eq:horizontal-persistence}
0\leq r^*A_i^{\varepsilon}-s^*A_{i+1}^{\varepsilon}
=\varepsilon(s^*C_{i+1}-r^*C_i).
\end{equation}
A horizontal component of \(C_i\) cannot disappear into an entirely
vertical \(C_{i+1}\), since the coefficient of the right side along its
strict valuation would then be negative. Thus the floor remains horizontal.
On every subsequent model the full adjoint and \(H'\) still come from
\(Y\). Repeating the resolved smooth-fiber integral and pushing its
K\"ahler-power current as above gives a negative class in its
\(\overline{\operatorname{NA}}\) cone and hence a negative extremal ray.
In particular a nef endpoint is impossible.
We have obtained a Mori contraction
\[
u:T\longrightarrow W,\qquad g:W\longrightarrow Y,
\]
with \(T\) normal compact K\"ahler, \(u\) projective of fiber type, and
both maps having connected fibers. For \(g\), this follows also from
\((g u)_*\OO_T=\OO_Y\) and \(u_*\OO_T=\OO_W\). The full boundary
\(G\) is effective lc, its adjoint is an actual pullback from \(Y\), and
it is klt away from \(C^+=\floor G\): subtracting \(\varepsilon C^+\)
is klt. The running perturbed adjoint satisfies
\[
A^{\varepsilon}\equiv_u-\varepsilon C^+,
\qquad -A^{\varepsilon}\equiv_u+\varepsilon C^+.
\]
The left side of the second equivalence is relatively ample for the
Mori contraction. Thus \(C^+\) is relatively ample for \(u\).
The graphs of the divisorial steps and flips are projective over both
sides. A main component of their iterated fiber product, followed by
normalization and projective resolution, gives the common projective
resolution in the statement. Its smooth source is compact K\"ahler.
\end{proof}
\subsection{The restriction criterion}
We now use the Mori model only to compare the values of a boundary
section on general fibers. The torsion-free obstruction then extends
that comparison over every parameter.
\begin{proposition}[Restriction criterion]\label{prop:restriction-link}
For every stratum \(Z\) of dimension one, two, or three, there is either
no comparison or one proper bimeromorphic comparison between two normal
components of \(C_Z\), allowing a component to be compared with itself,
with the following properties.
\begin{enumerate}
\item On a resolution of its graph with smooth source, the two pulled-back actual
boundary adjoint lines are equal as invertible subsheaves of meromorphic
pluricanonical forms.
\item For all sufficiently large divisible \(k\), uniformly over
sections, a section of \(L_Z^k|_{C_Z}\) extends to \(Z\) if its two
pullbacks agree under this comparison. With no comparison there is no
matching condition.
\item When present, the comparison pairs two coefficient-one markings
on general projective-line fibers of a projective Mori contraction on
a bimeromorphic compact K\"ahler model of \(Z\).
\end{enumerate}
\end{proposition}
\begin{proof}
If \(C_Z\) is vertical over \(Y_Z\), the consequence of
\eqref{eq:restriction-cokernel} gives extension in a uniform large-degree
tail without a comparison. Suppose therefore that it is horizontal,
and use the small model \(q_Z\) and the Mori model
\(u:T\to W\), \(g:W\to Y\) of Lemma~\ref{lem:horizontal-mori-model}.
Extending the pulled-back section on \(C_0\) suffices: a section on
\(Z_0\) descends to \(Z\), and equality of its restriction after
pullback implies equality on the reduced \(C_Z\), since every component
is covered birationally.
\subsection*{The floor on general Mori fibers}
We now compare the connected components of a general floor fiber with
the markings of one Mori fiber. On a common projective log resolution
of the program, the full crepant boundary is the same on both sides.
Its coefficient-one union maps to both \(C_0\) and \(C^+\) with connected
fibers by Lemma~\ref{lem:floor-connectedness}. After restriction over
any \(y\in Y\), proper closed maps with connected fibers preserve
connected components. Thus the connected components of the underlying
spaces of \(C_{0,y}\) and \(C^+_y\) correspond. This is a topological
statement, not a reducedness claim for their scheme fibers.
The finite Stein space of \(C^+\to W\) has every component dominating
\(W\). Indeed relative ampleness makes \(C^+\to W\) surjective, and
Lemma~\ref{lem:floor-torsion-free} and
\eqref{eq:restriction-cokernel} give the torsion-free property used in
the preceding discussion. For general \(y\), every irreducible component
of the finite Stein fiber has dimension \(\dim W-\dim Y\), by the
dimension theorem. The fiber \(W_y\) is irreducible of that dimension:
a resolution of \(W\) has connected fibers over \(Y\), and a general
one is smooth and connected, hence irreducible, and surjects onto \(W_y\).
Each component of the finite Stein fiber therefore dominates \(W_y\).
It follows that every connected component of \(C^+_y\) meets
\(u^{-1}(w)\) for a common general \(w\in W_y\).
The underlying space of a general fiber \(F\) of \(u\) is irreducible
by the same resolution argument. If \(\dim F\geq2\), a Cartier multiple
of \(C^+\) restricts to a nonzero effective ample Cartier divisor on
\(F_{\red}\). Its support is connected. To recall the reason, if an
ample effective divisor split into two disjoint nonzero parts, general
hyperplane sections would reduce to an irreducible projective surface
\(F_2\). Write the two induced nonzero effective Cartier parts on it as
\(D_1,D_2\), and let \(v:\widetilde F\to F_2\) be a projective resolution
with smooth source. Their pullbacks are effective and orthogonal. Although
\(v^*(D_1+D_2)\) is only nef and big, the projection formula gives
\[
v^*(D_1+D_2)\cdot v^*D_i
=(D_1+D_2)\cdot v_*[v^*D_i]>0\qquad(i=1,2):
\]
the pushforward is a nonzero effective curve cycle and \(D_1+D_2\) is
ample on the integral surface. Orthogonality then gives
\((v^*D_i)^2>0\) for both \(i\). The surface Hodge index theorem forbids
two such orthogonal positive classes. This argument does not require
normality of \(F_{\red}\).
If \(\dim F=1\), choose \(F\) also in the generic-smoothness open.
Generality avoids the singular locus of \(T\), the
singularities and intersections of the boundary, and the ramification of
its horizontal primes. The fiber is a smooth connected curve, and the
full pullback identity gives
\[
0=\deg(K_F+G|_F)=2g(F)-2+\deg(G|_F).
\]
There is at least one coefficient-one point. Hence \(F\simeq\PP^1\),
and its floor has one or two points. In the two-point case these points
exhaust the horizontal boundary on \(F\). In all connected cases,
every connected component of \(C^+_y\) meets the same connected subset
\(C^+\cap F\), so \(C^+_y\), and therefore \(C_{0,y}\), is connected.
Then no comparison is needed. In the two-point case each connected
component of \(C_{0,y}\) is represented by at least one of these two
markings, after their transfer through the common graph.
\subsection*{The two-marking comparison}
Normalize each surviving horizontal prime \(D\subset C^+\) and take the
Stein factorization
\[
D^\nu\longrightarrow E_D\longrightarrow W.
\]
The first map is projective bimeromorphic and \(E_D\) is normal; the
second is finite. If there are two horizontal primes, each finite map
has degree one and is an isomorphism over the normal \(W\). The main
component of the fiber product of their normalizations over \(W\) gives
a proper bimeromorphic comparison. If there is one horizontal prime,
\(E_D\to W\) has degree two. On a dense open it is \'etale and has an
exchange. The reduced horizontal non-diagonal component of
\(E_D\times_W E_D\) has finite generically one-to-one projections to
the normal \(E_D\); both are isomorphisms. It defines the exchange
involution globally, including across the branch. The main component
of \(D^\nu\times_{E_D,\mathrm{exchange}}D^\nu\) lifts it to a proper
bimeromorphic graph. It is the Stein space of the normalized prime
that is used here, not the possibly nonnormal Stein space of the entire
floor.
Compose this graph with the common graphs of the program and with the
small model. It gives the asserted comparison between normal primes of
\(C_Z\), possibly a self-comparison. Each surviving prime is generically
finite over \(W\); the image in \(W\) of the proper subset where its
transfer is not an isomorphism is proper. Thus both markings on a general
Mori fiber lie in the common transfer isomorphism loci.
These graph projections are projective. The maps \(D^\nu\to E_D\)
are projective and the maps \(E_D\to W\) are finite; their fiber products
and closed main components are therefore projective over the branches.
The small-model and MMP graphs have the same property. Iterated main
components and finite normalization preserve it, so the comparison admits
a projective resolution with smooth compact K\"ahler source.
We verify the actual meromorphic identity required by property (1).
The two residues of a logarithmic fiber form give the same comparison
as the even Poincar\'e-residue diagrams of
\cite[Section~3, Definition~13 and Proposition~14]{Kollar2013Sources};
we include the calculation for the normalized two-branch construction
above.
On a smooth ruled open of \(W\), order the two markings after an \'etale
local cover when necessary, choose a base volume form \(\xi\), and a
fiber coordinate with markings at zero and infinity. The full
\(q\)-pluriadjoint frame coming from \(Y\) has the form
\[
a(w)(dz/z)^{\otimes q}\otimes\xi^{\otimes q}.
\]
There is no other horizontal divisor on the general fiber. Its two
residues are \(a(w)\xi^{\otimes q}\) and
\((-1)^q a(w)\xi^{\otimes q}\); they agree because \(q\) is even.
The calculation is invariant under exchanging the two local markings.
Full crepancy transfers it to the original primes. On a resolution of
their comparison graph, both actual adjoint lines are the same pullback
of \(N_Z\) from \(Y\). Their meromorphic embeddings agree on this dense
ruled open, hence agree everywhere. Thus the invertible subsheaves of
meromorphic pluriforms are equal even at exceptional primes. An abstract
Q-linear equivalence without this residue calculation would not suffice.
A section satisfying this comparison has equal values at the two
representatives over a general \(w\). On every connected component of
a compact reduced floor fiber it is constant in the trivialized pulled-back
line: holomorphic functions on a compact connected reduced complex space
are constant. The representatives meet every component, so its values
agree on \(C_{0,y}\) for every \(y\) in a dense base open. The horizontal consequence of
\eqref{eq:restriction-cokernel} extends it to \(Z_0\), and then it descends
to \(Z\). The only asymptotic bound on \(k\) came from Serre vanishing in
the vertical case. Finitely many strata admit a common sufficiently
divisible tail. This proves all three properties.
\end{proof}
For the remainder of the boundary proof, call each comparison supplied
by Proposition~\ref{prop:restriction-link} a \emph{link}. This word
refers to its proper bimeromorphic graph together with the proved equality
of meromorphic adjoint subsheaves. It does not mean only an abstract
linear equivalence. We will use all curve comparisons, but on surfaces
only the groupoid generated by links of three-dimensional strata. That
restriction is what allows the common ambient K\"ahler class to control
their scalar action.
==== END Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//links.tex ====
==== BEGIN Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//main.tex ====
\documentclass[11pt,letterpaper]{article}
\input{preamble}
\title{Abundance after nonvanishing for compact K\"ahler fourfolds}
\author{OpenAI}
\hypersetup{pdftitle={Abundance after nonvanishing for compact K\"ahler fourfolds},pdfauthor={OpenAI}}
\date{September 27, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove semiampleness of the actual \(\Q\)-Cartier adjoint
\(K_X+\Delta\) of a normal connected compact K\"ahler klt fourfold
whenever it is analytically nef and some positive Cartier multiple has
a nonzero section. The boundary is effective and rational; the fourfold
need not be projective or \(\Q\)-factorial. In Iitaka dimension zero,
a positive Cartier multiple is the trivial holomorphic line bundle.
\end{abstract}
\tableofcontents
\input{sections/introduction}
\input{sections/conventions}
\input{sections/finite-null}
\input{sections/contraction-inputs}
\input{sections/reduction}
\input{sections/adjunction}
\input{sections/links}
\input{sections/scalars}
\input{sections/gluing}
\input{sections/covers}
\input{sections/hodge}
\input{sections/lifting}
\input{sections/completion}
\bibliographystyle{plain}
\phantomsection
\addcontentsline{toc}{section}{References}
\bibliography{references}
\end{document}
==== END Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//main.tex ====
==== BEGIN Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//outer-positivity.tex ====
\paragraph{The descended class and its floor restriction.}
In this paragraph and the next three, write \(X=X_i\), \(Z=Z_i\), and
\(\alpha=\alpha_i\). We use
\(dd^c=\frac{i}{2\pi}\partial\bar\partial\), the normalization for
which \(dd^c\phi\) is the normalized Chern curvature of a metric
locally written as \(e^{-\phi}\). A fixed positive rescaling of potentials gives the
\(i\partial\bar\partial\) convention in the cited analytic inequalities.
Both ends of \(c\) are normal compact spaces in class \(\mathcal C\)
with rational singularities, and \(\alpha\) is zero on every contracted
curve. The Bott--Chern descent
\cite[Lemma~2.11]{DasHacon2024} therefore gives a class \(\beta\) on
\(Z\) and a smooth real closed representative \(\beta_0\), with local
smooth potentials, such that
\begin{equation}\label{eq:outer-descended-class}
c^*\beta=\alpha\quad\text{in }H^{1,1}_{BC}(X).
\end{equation}
Here and below the group is the local-potential Bott--Chern group.
Its smooth representative description, including adjustment by a global
smooth \(dd^c\)-potential, is
\cite[Definition~3.1 and Remark~3.2]{HoeringPeternell2016}.
Let \(i_W:W\hookrightarrow Z\) be the embedded floor image and put
\(\delta=i_W^*\beta\). The actual identity
\(c|_T=i_W\circ\varphi\), \eqref{eq:floor-contraction-class}, and
\eqref{eq:outer-descended-class} give
\begin{equation}\label{eq:outer-floor-image-class}
\varphi^*\delta=\alpha|_T=\varphi^*[\omega_W]
\quad\text{in }H^{1,1}_{BC}(T).
\end{equation}
Thus the pullback of \(\delta-[\omega_W]\) is zero and hence nef.
Apply \cite[Lemma~2.38]{DasHaconPaun2024} only to
\(\varphi:T\to W\): it is proper and surjective, and both spaces are
normal compact K\"ahler. The lemma makes \(\delta-[\omega_W]\) nef.
For a smooth representative \(e\) of that difference, choose a nef
approximation \(e+dd^c u\geq-\omega_W/2\). Then
\(\omega_W+e+dd^c u\) is a K\"ahler representative of \(\delta\).
In particular its restriction gives a positive current in the class
\(\beta|_V\) and a positive top integral for every positive-dimensional
irreducible \(V\subset W\). Only the nefness of
\(\delta-[\omega_W]\) enters this deduction.
\paragraph{Nefness on the target.}
For every positive-dimensional irreducible reduced compact subspace
\(V\subset Z\), we first produce a current
\begin{equation}\label{eq:outer-restriction-current}
T_V=\beta_0|_V+dd^c q_V\geq0
\end{equation}
with a global real distribution \(q_V\) on \(V\). Currents and their
positivity on the pure-dimensional reduced \(V\) are defined by duality
with smooth test forms from local ambient embeddings; see
\cite[Section~1, Definitions~1.1--1.2, pp.~14--15]{Demailly1985}.
We will use exactly this positive-current meaning of pseudoeffectivity
on a possibly nonnormal subspace. The preceding K\"ahler representative
of \(\delta\) gives \eqref{eq:outer-restriction-current} for
\(V\subset W\), with a smooth potential.
Suppose \(V\not\subset W\), and let \(V'\subset X\) be the closure
of the inverse image of \(V\setminus W\). The isomorphism
\(X\setminus T\simeq Z\setminus W\) makes \(V'\) irreducible and
its map to \(V\) proper and bimeromorphic. Choose a projective
resolution \(r:U\to V'\) with smooth source. The restricted K\"ahler
form makes \(V'\) a compact K\"ahler space. A relatively positive
metric for the projective resolution, plus a sufficiently large pullback
of this form, makes \(U\) a compact K\"ahler manifold. This metric
construction applies to the possibly nonnormal \(V'\).
Let \(\nu:V^\nu\to V\) be normalization. The dominant map from the
normal \(U\) factors through \(\nu\), giving a proper bimeromorphic
map \(p:U\to V^\nu\), hence a modification between normal compact
spaces. For the other map \(j:U\to X\), functoriality of the actual
maps and \eqref{eq:outer-descended-class} gives
\[
p^*\nu^*(\beta|_V)=j^*\alpha.
\]
The class on the right is nef. Indeed, pull back the smooth nef
approximations from \(X\); on compact \(U\), the pullback of their
reference form is bounded above by a fixed multiple of a K\"ahler form
\(\eta\) on \(U\), so rescaling the error gives the nef inequalities.
Set \(\gamma=p^*\nu^*(\beta_0|_V)\). Choose smooth approximations
\(\gamma+dd^c f_n\geq-n^{-1}\eta\). The closed positive currents
\(\gamma+dd^c f_n+n^{-1}\eta\) have bounded \(\eta\)-mass by
Stokes' theorem. Weak compactness gives a positive closed limit in
\([\gamma]\). The \(\partial\bar\partial\)-lemma on the compact
K\"ahler manifold \(U\), and the global potential description of a
positive current, give a global quasi-plurisubharmonic \(q_U\) with
\[
\gamma+dd^c q_U\geq0.
\]
These are also the closed-cone statement following Definition~1.6 and
the potential description in (3.1) and its following paragraph in
\cite[pp.~1253 and 1260]{DemaillyPaun2004}. Apply
\cite[Corollary~2.32 and its proof, p.~18]{DasHaconPaun2024} to \(p\), with target form
\(\nu^*(\beta_0|_V)\) and constant zero in its current inequality.
The locally integrable, locally bounded above \(q_U\) supplies the
required potential. We obtain a global quasi-plurisubharmonic
\(q_\nu\) satisfying
\begin{equation}\label{eq:outer-normalized-current}
T_\nu=\nu^*(\beta_0|_V)+dd^c q_\nu\geq0.
\end{equation}
This applies the normal-modification descent to the displayed
local-potential inequality on the still general normal compact target.
It remains to push through the finite normalization. Locally write
\(\beta_0|_V=dd^c h\) with \(h\) smooth. The function
\(\nu^*h+q_\nu\) is locally integrable and locally bounded above, and
its Hessian is \(T_\nu\geq0\). On normal \(V^\nu\), its upper
regularization is plurisubharmonic. The finite trace is weakly
plurisubharmonic and its Hessian is the positive current \(\nu_*T_\nu\)
by \cite[Corollary~1.11 and Proposition~1.13(a), p.~22]{Demailly1985}.
Since \(\nu\) has generic degree one, change of variables off the
proper analytic exceptional sets gives
\(\nu_*\nu^*(\beta_0|_V)=\beta_0|_V\) as currents; those sets have
measure zero for the smooth top-degree integrands. Proper push of
distributions commutes with \(dd^c\). Hence
\begin{equation}\label{eq:outer-finite-trace}
T_V=\nu_*T_\nu
=\beta_0|_V+dd^c(\nu_*q_\nu)\geq0.
\end{equation}
The traced distributions are locally integrable and glue because
\(q_\nu\) is global. On a locally reducible \(V\), these local
potentials may be only weakly plurisubharmonic; the argument below uses
their associated positive currents.
We now apply the sufficient direction of the restriction criterion
\cite[Theorem~2.36 and Remark~2.37]{DasHaconPaun2024}, spelling out its
singular-current input. Proposition~3.3(iv), p.~1262, and the following
paragraph, pp.~1262--1263, of \cite{DemaillyPaun2004}
applies to a compact complex space and a smooth class containing a
closed positive current. It gives nefness if the restrictions to the
irreducible components of every positive Lelong level set are nef.
Induct on the dimension of each irreducible \(V\subset Z\); points are
automatic. For a positive-dimensional \(V\), use
\eqref{eq:outer-restriction-current}. The Siu analyticity assertion
accompanying that proposition makes each positive Lelong level set
analytic, and it is proper. Indeed, if a fixed positive level filled \(V\), pack
disjoint radius-\(r\) balls in a coordinate ball of \(V_{\rm reg}\).
The Lelong mass lower bound on each ball would force the locally finite
mass of the \((1,1)\)-current to grow at least as a positive multiple
of \(r^{-2}\) when \(r\) tends to zero. This is impossible. The level
set components therefore have smaller dimension, and their restricted
classes are nef by induction. The proposition makes \(\beta|_V\) nef.
Taking \(V=Z\) proves analytic nefness of \(\beta\). This uses the
positive-current formulation above also on nonnormal subspaces, where
the local potentials may be only weakly plurisubharmonic.
\paragraph{Bigness from the supported section.}
Use the effective actual representative already present in
\eqref{eq:running-supported-representative}. Choose \(m>0\) clearing
the indices and an isomorphism of holomorphic lines
\[
L=\OO_X(mA_i)\simeq\OO_X(mP_i).
\]
The canonical section of the right side gives a nonzero holomorphic
section \(s\) of \(L\) with divisor \(mP_i\). Choose a smooth metric
\(h\) on \(L\). In a local frame write \(h=e^{-\phi}\), \(s=f\),
and put
\[
\vartheta=\frac1m dd^c\phi,\qquad
\ell=\frac1m\log|s|_h^2.
\]
The first formula defines the normalized global curvature form.
The second is a global quasi-plurisubharmonic function, locally
\(m^{-1}(\log|f|^2-\phi)\). The holomorphic \(f\) extends in a local
ambient embedding and is not identically zero on any nonempty open
subset of that chart in \(X\). Local integrability is Proposition~1.8
of \cite[p.~19]{Demailly1985}, and
the unnumbered prose after its proof gives the current positivity
\[
\vartheta+dd^c\ell=\frac1m dd^c\log|f|^2\geq0.
\]
The smooth form \(\vartheta+\omega_i\) represents
\(\alpha=c^*\beta\), so the smooth local-potential representative
description supplies a global smooth \(v\) with
\(\vartheta+\omega_i=c^*\beta_0+dd^c v\). Thus
\begin{equation}\label{eq:outer-supported-current}
c^*\beta_0+dd^c(v+\ell)\geq\omega_i.
\end{equation}
This is a global quasi-plurisubharmonic potential obtained from the
supported section at this running step.
Fix any smooth positive Hermitian form \(g_Z\) on \(Z\) in local
ambient embeddings. Local holomorphic coordinate lifts of \(c\) make
\(c^*g_Z\) semipositive in those embeddings. A finite relatively
compact cover of \(X\) and comparison with the positive definite
ambient representatives of \(\omega_i\) give one \(C>0\) with
\(c^*g_Z\leq C\omega_i\). Corollary~2.32 of
\cite{DasHaconPaun2024} applies to the normal modification \(c\) and
\eqref{eq:outer-supported-current}. It gives a global
quasi-plurisubharmonic, hence \(L^1\), function \(v_Z\) with
\begin{equation}\label{eq:outer-current-big}
\beta_0+dd^c v_Z\geq C^{-1}g_Z.
\end{equation}
This is the required dominating current in the selected target class.
\paragraph{Top intersections and the other floor.}
Let \(V\subset Z\) be irreducible of dimension \(k>0\). The case
\(V\subset W\) was proved using the K\"ahler class \(\delta\).
Otherwise use its strict transform \(V'\) above. The proper integration
cycle identity \(c_*[V']=[V]\), projection formula, and
\eqref{eq:outer-descended-class} give
\begin{equation}\label{eq:outer-strict-transform}
\int_{V_{\rm reg}}\beta_0^k
=\int_{V'_{\rm reg}}\alpha^k.
\end{equation}
The right side denotes the integral of any smooth representative of
\(\alpha\); Stokes' theorem makes it independent of that choice.
Neither subspace needs to be normal.
If \(V'\not\subset\Supp P_i\), take the projective resolution
\(r:U\to V'\) with smooth K\"ahler source used above. The pullback of
\(s\) is nonzero and defines
the effective rational divisor \(E=m^{-1}\operatorname{div}(r^*s)\)
on \(U\). Put \(a=r^*(\alpha|_{V'})\) and
\(w=r^*[\omega_i|_{V'}]\). The class \(a\) is nef, \(w\) has a
semipositive smooth representative, and the actual section gives
\(a-w=c_1(E)\). Therefore
\begin{equation}\label{eq:outer-outside-support}
\int_U a^k-\int_U w^k
=\sum_{j=0}^{k-1}c_1(E)\cdot a^{k-1-j}w^j\geq0.
\end{equation}
For each summand, choose a smooth representative of
\(a+\varepsilon[\eta]\) which is semipositive, where \(\eta\) is a
fixed K\"ahler form on \(U\). Restrict it and the semipositive
representative of \(w\) to every effective divisor component, integrate,
and let \(\varepsilon\) tend to zero.
This proves its nonnegativity, including the degree statement when
\(k=1\). Projection formula gives
\(\int_U w^k=\int_{V'_{\rm reg}}\omega_i^k>0\). Equations
\eqref{eq:outer-strict-transform}--\eqref{eq:outer-outside-support}
give the desired strict positivity.
It remains that \(V'\) lies in a component \(T'\) of \(\Supp P_i\).
It meets \(X\setminus T\), so \(T'\neq T\). The actual floor
specialization in Subsection~\ref{subsec:actual-floor-split} gives
\begin{equation}\label{eq:outer-other-floor}
g:T'\to\overline W,\qquad \alpha|_{T'}=g^*\kappa,
\end{equation}
where \(g\) is projective with connected fibers, its source and target
are normal compact K\"ahler, and \(\kappa\) is a K\"ahler class.
The equality is in the local-potential Bott--Chern group.
Every irreducible curve in a \(g\)-fiber has \(\alpha\)-degree zero,
so \eqref{eq:global-support-zero-ray} puts its class in \(R_i\), and
\(c\) contracts it.
It follows that \(h=c|_{T'}\) is pointwise constant on every full
\(g\)-fiber. To see this also for nonreduced fibers, their reductions
are connected projective varieties, possibly reducible. If the image
of such a reduction were
not a point, one irreducible component \(Q\) would have nonconstant
image; otherwise its image would be a connected finite set. At a smooth
point of \(Q\) where a local coordinate of \(h\) has nonzero
differential, general hyperplanes through that point cut an irreducible
projective curve component whose tangent is not in the differential's
kernel.
That curve has nonconstant image, a contradiction. If \(\dim Q=1\),
take \(Q\) itself.
There is consequently a holomorphic factorization
\begin{equation}\label{eq:outer-factorization}
h=b\circ g,\qquad b:\overline W\to Z.
\end{equation}
For clarity, the specialization gives
\(g_*\OO_{T'}=\OO_{\overline W}\); this also follows from analytic
Stein factorization, connected fibers, and the normality of the two
spaces. Proper surjectivity makes \(g\) a closed quotient map, so the
pointwise constancy first defines a continuous \(b\). For an open
\(O\subset Z\) embedded in a polydisc, restrict to the open
\(b^{-1}(O)\). The coordinate functions of \(h\) on its full inverse
image descend through \(g_*\OO_{T'}=\OO_{\overline W}\) to a
holomorphic tuple. It lies pointwise in the embedded \(O\);
its defining ideal therefore vanishes on the reduced open \(b^{-1}(O)\).
The resulting local holomorphic maps glue to
\eqref{eq:outer-factorization}. This factorization uses pointwise
constancy on the underlying full fibers.
Put \(H=g(V')\), a closed irreducible analytic subspace of
\(\overline W\). Since \(c|_{V'}\) is bimeromorphic onto \(V\),
\eqref{eq:outer-factorization} yields
\[
k=\dim c(V')=\dim b(H)\leq\dim H\leq\dim V'=k.
\]
Thus \(g|_{V'}\) is generically finite onto \(H\), of some integer
degree \(d>0\). The projection formula using
\eqref{eq:outer-other-floor} now gives
\begin{equation}\label{eq:outer-positive-intersections}
\int_{V_{\rm reg}}\beta_0^k
=\int_{V'_{\rm reg}}(g^*\kappa)^k
=d\int_{H_{\rm reg}}\kappa^k>0.
\end{equation}
The last integral is a positive K\"ahler volume. The one-way
factorization \(h=b\circ g\) and this degree calculation give the needed
implication in \cite[proof of Theorem~7.2, p.~48]{DasHaconPaun2024}.
These cases cover every irreducible positive-dimensional \(V\).
Additivity over top-dimensional irreducible components gives strict
positivity for every positive-dimensional compact reduced subspace.
We have proved analytic nefness of \(\beta\), the dominating current
\eqref{eq:outer-current-big}, and all positive top intersections.
The normal compact \(Z\) and the smooth real closed \(\beta_0\)
therefore satisfy the three hypotheses of
\cite[Theorem~2.29]{DasHaconPaun2024}. That theorem gives a positive
smooth representative of \(\beta\). Its local smooth potentials make
this a K\"ahler form \(\omega_{Z_i}\). Equation
\eqref{eq:outer-descended-class} gives
\eqref{eq:outer-target-kahler}, and \(Z_i\) is compact K\"ahler.
==== END Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//outer-positivity.tex ====
==== BEGIN Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//preamble.tex ====
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage{microtype}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{mathrsfs}
\usepackage{enumitem}
\usepackage{geometry}
\geometry{margin=1.05in}
\usepackage{hyperref}
\hypersetup{colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue}
\usepackage{bookmark}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{claim}[theorem]{Claim}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\newcommand{\C}{\mathbb C}
\newcommand{\Q}{\mathbb Q}
\newcommand{\R}{\mathbb R}
\newcommand{\Z}{\mathbb Z}
\newcommand{\PP}{\mathbb P}
\newcommand{\OO}{\mathcal O}
\newcommand{\Dcal}{\mathcal D}
\newcommand{\Supp}{\operatorname{Supp}}
\newcommand{\ord}{\operatorname{ord}}
\newcommand{\gr}{\operatorname{gr}}
\newcommand{\NS}{\operatorname{NS}}
\newcommand{\DR}{\operatorname{DR}}
\newcommand{\Ext}{\operatorname{Ext}}
\newcommand{\im}{\operatorname{im}}
\newcommand{\coker}{\operatorname{coker}}
\newcommand{\red}{\mathrm{red}}
\newcommand{\an}{\mathrm{an}}
\newcommand{\id}{\mathrm{id}}
\newcommand{\ceil}[1]{\left\lceil #1\right\rceil}
\newcommand{\floor}[1]{\left\lfloor #1\right\rfloor}
\newcommand{\Sha}{\mathscr}
\setlist[enumerate]{itemsep=0.35em,topsep=0.5em}
\setlist[itemize]{itemsep=0.25em,topsep=0.5em}
\numberwithin{equation}{section}
==== END Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//preamble.tex ====
==== BEGIN Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//reduction.tex ====
\section{Reduction to a supported nef boundary}\label{sec:reduction}
The positive-Iitaka-dimensional case is an application of a known
K\"ahler abundance theorem. The purpose of this section is to prepare
the remaining case for the two boundary arguments: from a nonzero divisor
of a section in Iitaka dimension zero, we construct a nef dlt fourfold
whose adjoint has a nonzero effective representative supported on exactly
its reduced floor. We also construct the particular log resolution used
to index and compare all strata of that floor.
\begin{proposition}[Supported nef model]\label{prop:supported-model}
Under the hypotheses of Theorem~\ref{thm:main}, suppose
\(\kappa(X,D)=0\) and the normalized rational divisor \(M\) of a nonzero plurisection of
\(D\) is nonzero. Then there are a normal ordinary \(\Q\)-factorial
compact K\"ahler dlt fourfold \((V,B)\), with effective rational boundary,
and a nonzero effective rational \(\Q\)-Cartier divisor \(P\) such that
\[
A=K_V+B\text{ is analytically nef},\qquad
A\sim_\Q P,\qquad \Supp P=\Supp\floor B,\qquad \kappa(V,A)=0.
\]
The pair has the projective resolution described in
Lemma~\ref{lem:special-resolution}.
\end{proposition}
The resolution has globally smooth distinct SNC strict boundary and
exceptional components, all exceptional crepant coefficients are below
one, and it is generically an isomorphism on the image of every
intersection component of distinct strict floor primes. These properties
let the floor argument index its strata by actual boundary intersections.
The proposition will follow from the construction below; the final step
uses the original nef adjoint to show that \(P\) cannot disappear.
We use the following analytic negativity theorem at several points.
If \(h:W\to Z\) is a proper bimeromorphic morphism of normal irreducible
analytic spaces and \(E\) is a rational \(\Q\)-Cartier divisor with \(-E\)
relatively nef, then
\[
E\geq0\quad\Longleftrightarrow\quad h_*E\geq0.
\]
This is \cite[Lemma~1.3]{Wang2021}, after clearing an index. In particular,
an exceptional relatively nef divisor is nonpositive. When \(h\) is
projective, nonnegative degrees on its contracted curves give the relative
nefness used in this theorem; see \cite[Appendix~B]{Wang2021}. This relative
curve test will not be used as a definition of nefness on a nonprojective
compact K\"ahler space.
\subsection{Positive Iitaka dimension}
\begin{proposition}\label{prop:positive-kappa}
Under the hypotheses of Theorem~\ref{thm:main}, if \(\kappa(X,D)\geq1\),
then \(D\) is semiample on \(X\).
\end{proposition}
\begin{proof}
Das--Hacon--P\u{a}un's dlt modification theorem
\cite[Theorem~6.1]{DasHaconPaun2024} applies to a compact K\"ahler lc
fourfold with effective rational boundary and \(\Q\)-Cartier adjoint.
It gives a projective bimeromorphic morphism
\[
g:(X',\Delta')\longrightarrow(X,\Delta)
\]
with \(X'\) compact K\"ahler and ordinary \(\Q\)-factorial, the pair
\((X',\Delta')\) dlt, and the adjoint crepant. The input is klt, so equality
of discrepancies makes the output klt as well. The crepant equality is
an equality of actual rational lines: pull a local pluriadjoint frame to
a common resolution with smooth source as in \eqref{eq:crepant}, compare the crepant
coefficients, and extend the identical meromorphic map across codimension
two on the normal \(X'\). Thus for a common index
\[
\OO_{X'}(m(K_{X'}+\Delta'))\simeq g^*\OO_X(mD).
\]
There is no possible undetected flat-line difference in this comparison.
The pulled-back line is analytically nef by the metric criterion. Also,
normality and projection formula identify all its sections with those of
\(\OO_X(mD)\). The meromorphic maps agree on the common dense open, so
their Iitaka dimensions agree. Theorem~4.1 of
H\"oring--Lazi\'c--Lehn \cite{HoeringLazicLehn2025} states that a nef
adjoint of a compact ordinary \(\Q\)-factorial K\"ahler klt pair of
positive Iitaka dimension is semiample, unconditionally in dimension at
most four. Its analytic positivity and canonical-sheaf conventions are
the ones in use here. Apply it to \((X',\Delta')\), then enlarge the
generated degree to a multiple of the original index. All sections are
pullbacks. A section nonzero at a chosen point above \(x\in X\) descends
to a section nonzero at \(x\); Nakayama's lemma gives the evaluation
surjectivity on \(X\).
\end{proof}
\subsection{A log-smooth supported representative}
We next begin with a divisor of a section. No nefness is needed for the
preparation in this subsection. The later proof that the support survives
the minimal model program will use the nefness of the original adjoint.
\begin{lemma}\label{lem:supported-preparation}
Let \((X,\Delta)\) be a normal connected compact K\"ahler klt pair with
effective rational boundary and actual \(\Q\)-Cartier adjoint \(D\).
Assume \(\kappa(X,D)=0\), and let \(M\geq0\) be a rational
\(\Q\)-Cartier divisor with \(M\sim_\Q D\). There are a projective
modification \(p:Y\to X\) with \(Y\) smooth compact K\"ahler, an effective rational
SNC boundary \(B_Y\), and an effective rational divisor \(P_Y\) such that
\[
K_Y+B_Y\sim_\Q P_Y,\qquad
\Supp P_Y=\Supp\floor{B_Y},\qquad
\kappa(Y,K_Y+B_Y)=0.
\]
The SNC support has globally smooth distinct components, and \(B_Y\) has
coefficient one on every \(p\)-exceptional prime and on every strict
transform of a prime in \(M\).
\end{lemma}
\begin{proof}
Principalize the reduced coherent ideal of \(\Supp(\Delta+M)\), using
\cite[Theorem~2.13 and Remark~2.14]{DasHaconPaun2024}. This use of an
ideal is important: the original boundary need not be \(\Q\)-Cartier.
The resulting modification has smooth source, is projective, and has locally normal
crossing strict and exceptional support. Mark each of its finitely many
global prime components separately as an ordered Cartier boundary and
apply the ordered-boundary resolution of
\cite[Theorems~1.1.3 and~1.1.13]{Temkin2017}. Its final indexed
components and their intersections are smooth, and its complete support
consists of strict transforms and new exceptional components
\cite[Lemmas~2.1.10 and~2.2.9(iv)]{Temkin2017}. Denote the composite by
\(p:Y\to X\). It is projective over the compact base. A relative ample
metric plus a sufficiently large pullback of a K\"ahler form makes \(Y\)
compact K\"ahler.
Let \(G_Y\) be the crepant subboundary for \(D\), defined by the actual
meromorphic pluriadjoint pullback. Then \(p_*G_Y=\Delta\), and klt gives
\(\operatorname{coeff}_E(G_Y)<1\) at every prime. Define \(B_Y\) only
after the preceding resolution: assign coefficient one to the actual
\(p\)-exceptional primes and the strict transforms of primes in \(M\),
and retain the coefficients of \(\Delta\) on the other strict boundary
primes. An exceptional label in the resolution bookkeeping that is not
an actual exceptional divisor does not change this rule. This is an
effective rational SNC boundary, hence dlt.
Set
\begin{equation}\label{eq:prepared-P}
P_Y=p^*M+(B_Y-G_Y).
\end{equation}
Pullback of the holomorphic equation of a Cartier multiple of \(M\)
makes \(p^*M\) effective. On a strict prime of \(M\), the second summand
has coefficient \(1-\operatorname{coeff}(\Delta)>0\); on an exceptional
prime it has coefficient \(1-\operatorname{coeff}(G_Y)>0\); on any other
strict boundary prime it is zero. Thus \(P_Y\) is effective and its
reduced support is exactly the floor of \(B_Y\). The actual identity
\(p^*D\sim_\Q p^*M\), together with \eqref{eq:crepant}, gives
\(K_Y+B_Y\sim_\Q P_Y\).
Choose an integer \(t>0\) at least the ratios of the coefficients of
\(P_Y\) to those of \(p^*M\) on every nonexceptional prime of \(M\).
The positive part of \(P_Y-tp^*M\) is then an effective rational
exceptional divisor \(E\), and
\[
P_Y\leq tp^*M+E.
\]
For every sufficiently divisible \(n\), exceptional Hartogs extension
and projection formula give
\[
H^0\bigl(Y,\OO_Y(n(tp^*M+E))\bigr)
=H^0\bigl(X,\OO_X(ntM)\bigr).
\]
The space on the right has dimension one: it is nonzero and
\(\kappa(X,D)=0\). The displayed divisor inequality bounds
\(h^0(Y,\OO_Y(nP_Y))\) by one, while effectivity makes it nonzero.
This controls every sufficiently divisible degree. If two independent
sections existed in another Cartier degree, their powers \(s_0^v\) and
\(s_0^{v-1}s_1\) would remain independent in a degree divisible by the
fixed common index, a contradiction. Therefore \(\kappa(Y,P_Y)=0\).
\end{proof}
\subsection{Projective steps with K\"ahler targets}
Apply the preceding lemma in dimension four. We follow the supported
construction of Das--Hacon--P\u{a}un
\cite[Theorem~7.2]{DasHaconPaun2024} from the compact ordinary
\(\Q\)-factorial K\"ahler dlt pair \((Y,B_Y)\). We isolate its
three-dimensional floor-contraction input through
Subsection~\ref{subsec:actual-floor-split} and use
Proposition~\ref{prop:prime-floor-extension} for the ambient contraction.
The construction below gives projective steps with K\"ahler targets and
supplies the graphs needed for the discrepancy comparison.
Write \((X_i,B_i)\) for a nonnef running pair and
\(A_i=K_{X_i}+B_i\). At the start of this step, induction from
\(P_Y\) gives an effective rational divisor \(P_i\) with
\begin{equation}\label{eq:running-supported-representative}
P_i\sim_\Q A_i,\qquad
\Supp P_i=\Supp\floor{B_i}.
\end{equation}
The equivalence is an isomorphism of actual rational holomorphic lines.
It holds initially by the supported-model construction, and the
one-step transform identity proved below supplies it at the next step
only after the current step has been completed. Thus the representative
used now is already available before constructing the current target.
The floor-contraction input in the proof of
Theorem~7.2, using its Corollary~5.6, is the following assertion.
There are a global irreducible floor prime \(T\), a K\"ahler form
\(\omega_i\) on \(X_i\), a projective contraction
\(\varphi:T\to W\) to a normal compact K\"ahler space with
\(\varphi_*\OO_T=\OO_W\), and a K\"ahler form \(\omega_W\).
The class \(\alpha_i=c_1(A_i)+[\omega_i]\) is nef and big but not
K\"ahler, and
\begin{equation}\label{eq:floor-contraction-class}
c_1(A_i|_T)+[\omega_i|_T]=\varphi^*[\omega_W].
\end{equation}
Here the equality is in the local-potential Bott--Chern group. The
contracted compact curves are exactly those whose classes on \(T\) lie in
\[
F_T=\bigl(c_1(A_i|_T)+[\omega_i|_T]\bigr)^\perp
\cap\operatorname{NA}(T).
\]
This face is generated by finitely many curve classes whose images in
\(X_i\) lie in one \(A_i\)-negative ray
\(R_i\subset\alpha_i^\perp\), with \(T\cdot R_i<0\).
For the global form of this selection, apply Lemma~7.1 in the setup of
Theorem~7.2 together with Claim~7.3 and its proof. We choose one
representative per proportionality ray among the countably many compact
curve classes to meet the lemma's nonproportionality hypothesis. A
K\"ahler degree is positive on each curve, so proportional effective
representatives differ by a positive scalar. Lemma~7.1 permits at most
one representative on which the selected class vanishes, and Claim~7.3
and its proof supply it. Consequently the selection gives the global
implication
\begin{equation}\label{eq:global-support-zero-ray}
\alpha_i\cdot C=0\quad\Longrightarrow\quad [C]\in R_i
\quad\text{for every compact irreducible curve }C\subset X_i.
\end{equation}
For a positive global Cartier multiple
\(mT\), the line \(\OO_T(-mT)\) is \(\varphi\)-ample.
The use of Corollary~5.6 here is the actual-line specialization proved
in Subsection~\ref{subsec:actual-floor-split}, with the inputs stated in
Subsection~\ref{subsec:threefold-contraction-bridge}.
In particular, the restriction to \(T\) need not be big and \(T\)
need not be \(\Q\)-factorial. The numerical selection of \(T\),
the ray, and the negative normal line is the one in the supported
construction. Its threefold contraction is supplied by the bridge,
using Proposition~\ref{prop:prime-floor-extension} and the external
results stated there.
Write \(B_i=T+C\), where \(C\geq0\) has no component \(T\).
Ordinary \(\Q\)-factoriality makes \(C\) a global rational
\(\Q\)-Cartier divisor. For a positive rational
\(\varepsilon<1\), put
\[
B_i(\varepsilon)=T+(1-\varepsilon)C,\qquad
A_i(\varepsilon)=A_i-\varepsilon C
=K_{X_i}+B_i(\varepsilon).
\]
The second identity is an identity of actual rational adjoint lines.
The perturbed pair is plt. Indeed, choose a log resolution
\(f:U\to X_i\) with smooth source that witnesses the standard dlt
criterion for \((X_i,B_i)\).
Its strict boundary and exceptional primes have simple normal crossings,
and all exceptional crepant coefficients are below one. If \(G\) is its
crepant boundary, the new boundary on this resolution is
\(G-\varepsilon f^*C\). The divisor \(f^*C\) is effective: pull back
a local holomorphic equation for an effective Cartier multiple of \(C\).
It follows that all exceptional coefficients remain below one. The
strict transform of \(T\) is the only coefficient-one prime, and all
other strict coefficients are below one. The plt criterion on this
resolution now applies. This is the ordinary Koll\'ar--Mori convention
used in Section~2 of \cite{DasHacon2024OnMMP}. If \(C=0\), the same
argument applies to the unchanged pair \((X_i,T)\).
Choose a smooth Hermitian metric on an actual Cartier multiple of \(C\),
and let \(\theta_C\) be its normalized curvature form. On a finite
relatively compact local embedding cover of the compact \(X_i\), choose
smooth ambient extensions of the local potentials and metric weights.
After shrinking the charts, the Levi forms for \(\omega_i\) have a
uniform positive lower bound on their compact closures, while those for
\(\theta_C\) are bounded. Thus \(\omega_i+\varepsilon\theta_C\) is K\"ahler for
one sufficiently small positive rational \(\varepsilon\), chosen for
this running step. No positivity of \(C|_T\) is asserted. Equation
\eqref{eq:floor-contraction-class} gives
\begin{equation}\label{eq:plt-contraction-class}
-c_1(A_i(\varepsilon)|_T)+\varphi^*[\omega_W]
=[\omega_i|_T+\varepsilon\theta_C|_T].
\end{equation}
The right side is K\"ahler. Locally on \(W\), a potential for
\(\omega_W\) can be pulled back, so this equality says precisely that
\(-A_i(\varepsilon)|_T\) is relatively K\"ahler over \(W\).
After clearing its index, the same local metric weights restrict to
positive weights on every full \(\varphi\)-fiber. The proper fiberwise
criterion \cite[Corollary~1.12, Definition~3.1 and Remark~3.2]{Fujino2026Fujiki}
therefore makes this actual rational line \(\varphi\)-ample. The other
required positivity, that of \(-T|_T\), is unchanged. With the boundary
\((1-\varepsilon)C\), these are the hypotheses of
Proposition~\ref{prop:conormal-layers}; its conormal direct-image
vanishings hold for the present floor contraction.
Proposition~\ref{prop:prime-floor-extension} now constructs a proper
bimeromorphic morphism \(c:X_i\to Z_i\) to a normal compact analytic
space. It restricts to \(\varphi\) on \(T\), embeds \(W\) as the
reduced image of \(T\), has exactly the \(\varphi\)-fiber sets over
\(W\), and is an isomorphism away from \(T\). Its fibers are
connected. A sufficiently divisible global line \(\OO_{X_i}(-\ell T)\)
is \(c\)-ample, and the actual rational line
\(-A_i(\varepsilon)\) is \(c\)-ample as well. The target is in
Fujiki's class \(\mathcal C\). The proposition proves these assertions
through the analytic thickenings and the full-fiber criterion. For this
constructed contraction, we use the rationality assertion of
\cite[Remark~5.11]{DasHacon2024OnMMP}. The target K\"ahler class is
proved below.
Every curve contracted by \(c\) is a \(\varphi\)-contracted curve in
\(T\), and therefore has class in \(R_i\). Conversely a curve in
\(R_i\) lies in \(T\), since \(T\cdot R_i<0\) whereas the effective
Cartier section of a multiple of \(T\) has nonnegative degree on the
normalization of any curve not contained in \(T\). The floor assertion
then contracts it. The constructed morphism thus contracts precisely
the compact curves whose classes lie in the original negative ray.
The boundary transported in the supported
program remains \(B_i\); the plt boundary is used only to construct \(c\).
\begin{claim}[K\"ahler positivity on the current target]
\label{claim:outer-target-positivity}
Retain the current pair \((X_i,B_i)\), its effective rational divisor
\(P_i\) with the actual line identity in
\eqref{eq:running-supported-representative}, the nef and big class
\(\alpha_i=c_1(A_i)+[\omega_i]\) with \(\omega_i\) K\"ahler, and the
selected ray and floor in
\eqref{eq:floor-contraction-class}--\eqref{eq:global-support-zero-ray}.
Let \(c:X_i\to Z_i\) be the projective contraction with connected fibers
just constructed. It is an isomorphism away from \(T\), has the embedded
floor image \(W\) and the exact \(\varphi\)-fiber sets, and contracts
precisely the curves in \(R_i\). Its target is a normal compact analytic
space in class \(\mathcal C\) with rational singularities. For every other floor prime,
retain the projective contraction with connected fibers supplied by
Subsection~\ref{subsec:actual-floor-split}, whose target is normal compact
K\"ahler and whose pulled-back K\"ahler class is the restricted
\(\alpha_i\).
Then there is a K\"ahler form \(\omega_{Z_i}\) on \(Z_i\) such that
\begin{equation}\label{eq:outer-target-kahler}
\alpha_i=c^*[\omega_{Z_i}]
\quad\text{in }H^{1,1}_{BC}(X_i),
\end{equation}
where the group is defined by local smooth potentials.
\end{claim}
\begin{proof}
Bott--Chern descent first supplies a target class whose pullback is
\(\alpha_i\). We prove that this class is nef, contains a current
dominating a positive Hermitian form, and has positive top intersection
on every positive-dimensional compact reduced subspace. These are the
three conditions in \cite[Theorem~2.29]{DasHaconPaun2024} that make the
target class K\"ahler.
\input{sections/outer-positivity}
\end{proof}
For a small step, the construction of its flipped morphism also has to
respect the hypotheses of the relative canonical-model theorem.
Use the current representative in
\eqref{eq:running-supported-representative}, and write
\(P_i=\sum p_jS_j\) with every \(p_j>0\) and \(S_j\) the floor primes.
At a nontrivial step this list is nonempty, since otherwise
\(A_i\sim_\Q0\) is nef. Independently choose a rational
\(0<\varepsilon<1\) small
enough that
\[
B_i^\varepsilon=B_i-\varepsilon P_i
\]
is effective. Its floor coefficients are below one. On a dlt resolution,
subtracting the pullback of the effective \(\Q\)-Cartier divisor
\(\varepsilon P_i\) can only lower the exceptional crepant coefficients,
which were already below one. Hence \((X_i,B_i^\varepsilon)\) is klt.
Its actual adjoint satisfies
\begin{equation}\label{eq:flip-perturbation}
K_{X_i}+B_i^\varepsilon=A_i-\varepsilon P_i
\sim_\Q(1-\varepsilon)A_i.
\end{equation}
The projective contraction is bimeromorphic, so this adjoint is relatively
big. Corollary~3.7 of \cite{DasHaconPaun2024} therefore applies at its
stated klt scope. The actual isomorphism in \eqref{eq:flip-perturbation}
identifies common Cartier Veroneses of its relative algebra and of the
relative \(A_i\)-algebra. On the compact base the latter is thus finitely
generated. A further Veronese is generated in degree one, and its coherent
degree-one piece embeds its relative \(\operatorname{Projan}\) in the
associated relative projective space. The tautological line is globally
relatively ample. The ordinary small flip supplied by Theorem~7.2 is
this relative canonical model; in a common degree the tautological line
agrees with the flipped adjoint line on the common codimension-one open
and hence everywhere by reflexive extension. The flipped morphism is
projective and the flipped adjoint is positive on its contracted curves.
Thus Corollary~3.7 is applied to the klt perturbation, whose common
Veronese is identified with that of the original adjoint.
The transform identities follow by induction from \(P_Y\), one completed
step at a time. Divisorial contractions
remove their contracted primes and flips change no prime valuation. On
the open containing all codimension-one points of the destination, the
fixed meromorphic pluriadjoint section identifies its line with the
strict transform \(P_{i+1}\). After clearing the new indices, reflexive
extension gives
\[
P_{i+1}\sim_\Q A_{i+1},\qquad
\Supp P_{i+1}=\Supp\floor{B_{i+1}}.
\]
This establishes the identity needed for the perturbation at the next
step. It uses ordinary \(\Q\)-factoriality only for the finitely many
global prime transforms.
We have now constructed a projective step, proved that its target is
K\"ahler, and supplied the effective actual representative needed to
repeat the construction. The factorial/dlt preservation and the
special-termination argument in
\cite[proof of Theorem~7.2]{DasHaconPaun2024} apply to the unchanged
original boundary. That construction, with
Proposition~\ref{prop:prime-floor-extension} at its ambient extension
steps and the target positivity proved above, therefore gives a finite
supported sequence to an analytically nef adjoint.
For a flip, take the main component of
\(X_i\times_{Z_i}X_{i+1}\) as its graph; its projections are projective.
For a divisorial contraction the source is its graph. The main component
of the iterated fiber product of these finitely many graphs, followed by
normalization, is a normal common graph \(\Gamma\). Normalization is
finite and projective, and projective morphisms compose over a compact
base \cite[Remark~2.11]{DasHaconPaun2024}. Thus \(\Gamma\) is projective
over every running model. Projective resolutions of it supply all common
resolutions with smooth source used below. Their sources are compact K\"ahler by
the relative metric construction.
On a common projective resolution of one step with smooth source \(W_i\),
let \(r:W_i\to X_i\) and \(s:W_i\to X_{i+1}\) be the maps, and put
\begin{equation}\label{eq:step-difference}
E_i=r^*P_i-s^*P_{i+1}.
\end{equation}
It is \(s\)-exceptional by the codimension-one comparison. For an
\(s\)-contracted curve, its image under \(r\) is a point or a curve in
the negative source contraction fiber, so \(E_i\) has nonpositive degree.
Negativity gives \(E_i\geq0\). In any common Cartier degree,
\begin{align*}
H^0(X_i,\OO_{X_i}(mP_i))
&=H^0(W_i,\OO_{W_i}(mr^*P_i))\\
&=H^0(W_i,\OO_{W_i}(ms^*P_{i+1}+mE_i))\\
&=H^0(X_{i+1},\OO_{X_{i+1}}(mP_{i+1})).
\end{align*}
The last equality is exceptional Hartogs extension for \(s\).
The identifications agree with the fixed section on the common open and
therefore respect multiplication. A common Veronese section ring is
unchanged through the finite chain. The power argument in
Lemma~\ref{lem:supported-preparation} handles other degrees, so the final
pair \((V,B)\), with \(A=K_V+B\) and \(P\) the final transform, satisfies
\begin{equation}\label{eq:final-supported}
A\sim_\Q P\geq0,\qquad
\Supp P=\Supp\floor B,\qquad \kappa(V,A)=0.
\end{equation}
\subsection{Discrepancy increase and a resolution of the floor}
We now show that the final pair has a resolution which is generically
unchanged along every floor intersection. The point is that a negative
step strictly increases discrepancies over all its nontrivial fibers,
including those on the positive side of a flip.
\begin{lemma}[Support on a projective fiber]\label{lem:fiber-support}
Let \(h:W\to Z\) be a projective morphism between normal analytic spaces
with connected fibers, let
\(F\) be a scheme fiber, and let \(E\geq0\) be a rational
\(\Q\)-Cartier divisor on \(W\). Suppose that \(E\cdot C\leq0\) for
every irreducible curve \(C\) in \(F\). Then either
\(\Supp E\cap F=\varnothing\) or \(F\subseteq\Supp E\).
\end{lemma}
\begin{proof}
Clear the index and pass to the reduction of \(F\). If an irreducible
component \(F_\alpha\) is not contained in \(\Supp E\) but meets it,
the restricted canonical section cuts a nonempty effective Cartier
divisor on the integral projective \(F_\alpha\). This component has
positive dimension: a point that is an irreducible component cannot meet
another component, and a connected zero-dimensional fiber is a point.
Cutting by sufficiently general hyperplanes of a very ample line on
\(F_\alpha\) gives a curve on which \(E\) has strictly positive degree,
a contradiction. If some components are contained in the support and
others are not, connectedness gives a noncontained component meeting a
contained one and the same contradiction. These alternatives prove the
claim, with no reducedness assumption on the original fiber.
\end{proof}
Apply the lemma to a common resolution over the contraction base of one
step and to \(E_i\) in \eqref{eq:step-difference}. The maps to the
contraction base have connected fibers: they are proper bimeromorphic
over a normal space. On a base-fiber curve \(C\), a divisorial step has
\(E_i\cdot C=(r^*A_i)\cdot C\leq0\). For a flip,
\[
E_i\cdot C=(r^*A_i)\cdot C-(s^*A_{i+1})\cdot C\leq0,
\]
because the source adjoint is negative on its contracted curves and the
flipped adjoint is positive on its contracted curves. Above any point
where either side has a nontrivial fiber, a curve in the common fiber can
be chosen to dominate a curve of that side: take a component over the
curve and cut by relative ample hyperplanes. The corresponding term in
the preceding degree is then strictly negative. Thus \(E_i\) meets the
common fiber, and Lemma~\ref{lem:fiber-support} puts the entire fiber in
\(\Supp E_i\).
Use the fixed meromorphic pluriadjoint section to calculate discrepancies
on the resolution. Its divisor as a section of the old pulled-back line
is \(r^*P_i\), and as a section of the new line is \(s^*P_{i+1}\).
The two meromorphic pluriforms agree. Their difference is consequently
the difference of the two crepant boundary divisors, so for every prime
\(F\)
\begin{equation}\label{eq:discrepancy-increase}
\operatorname{coeff}_F(E_i)
=a(F;X_{i+1},B_{i+1})-a(F;X_i,B_i).
\end{equation}
The same formula applies on higher projective resolutions after pullback.
For each contraction base \(Z_i\), let \(C_i\subset Z_i\) consist of
the points whose fiber on the source or, for a flip, on the destination
has positive dimension. This is a closed analytic subset of the base
by properness and the fiber-dimension theorem. A proper bimeromorphic
morphism to a normal space is an
isomorphism near any zero-dimensional fiber, since it is finite there.
Write \(v:\Gamma\to V\) and \(q_i:\Gamma\to Z_i\) for the projections
from the normal common graph to the final model and the contraction bases.
Set
\begin{equation}\label{eq:graph-bad-set}
D_\Gamma=\bigcup_i q_i^{-1}(C_i),\qquad
B_V=v(D_\Gamma),\qquad U_V=V\setminus B_V.
\end{equation}
The sets \(D_\Gamma\) and \(B_V\) are closed analytic, the latter by
properness. We claim that
\begin{equation}\label{eq:graph-saturation}
v^{-1}(B_V)=D_\Gamma.
\end{equation}
Indeed, at a point \(\gamma\notin D_\Gamma\), both sides of every step
are isomorphisms near the corresponding point of \(Z_i\). Restricting
to these neighborhoods makes their iterated graph a common normal open,
so normalization does not change it and \(v\) is an isomorphism near
\(\gamma\). Thus \(\{\gamma\}\) is open in its \(v\)-fiber, and it is
closed because the spaces are Hausdorff. The fibers of \(v\) are
connected: a proper bimeromorphic morphism between normal spaces has
\(v_*\OO_\Gamma=\OO_V\), and its Stein factorization has connected
fibers. That fiber is therefore \(\{\gamma\}\), so it cannot meet
\(D_\Gamma\). This proves \eqref{eq:graph-saturation}.
The corresponding open \(U_Y\subset Y\) is isomorphic to \(U_V\)
through every step, because every successive lift there is unique.
No step extracts a divisor, so the
generic point of any prime on \(V\) follows a surviving prime through
the chain and avoids the contraction centers. Therefore
\begin{equation}\label{eq:bad-codim}
\operatorname{codim}_V(V\setminus U_V)\geq2.
\end{equation}
Every final log-canonical center tested by a prime on a projective
resolution meets \(U_V\). To see this, suppose the final discrepancy
of that prime is zero. Initial log canonicity, \(E_i\geq0\), and
\eqref{eq:discrepancy-increase} force all intermediate discrepancies
and all coefficients in the \(E_i\) to be zero. If its center lay over
some \(C_i\), the fiber-support conclusion would place that center in
\(\Supp E_i\). A local equation of a positive Cartier multiple of
\(E_i\) would then have positive order in the valuation, a contradiction.
If its center on \(V\) were contained in \(B_V\), its center on the
proper graph \(\Gamma\) would be contained in
\(v^{-1}(B_V)=D_\Gamma\). Irreducibility would then put that entire
center in one \(q_i^{-1}(C_i)\). On a higher projective resolution
carrying the prime, the center would lie over \(C_i\), giving the
contradiction just proved. Thus it meets
\(U_V\), as claimed. We only need this statement for primes on the
projective resolutions constructed here and for the blowups of their
floor strata.
\begin{lemma}[A globally simple dlt resolution]\label{lem:special-resolution}
The final pair \((V,B)\) has a projective log resolution
\(\pi:\widehat V\to V\) with the following properties:
\begin{enumerate}
\item the strict boundary and exceptional support have globally smooth
distinct SNC components;
\item every \(\pi\)-exceptional crepant coefficient is strictly below one;
\item if \(\widehat Z\) is an irreducible component of the nonempty
intersection of distinct strict transforms of floor primes, then
\(\pi\) is generically an isomorphism on \(\pi(\widehat Z)\).
\end{enumerate}
Moreover \(\pi\) is an isomorphism over \(U_V\).
\end{lemma}
\begin{proof}
Write \(b:\Gamma\to Y\) for the normal common graph. Choose a relatively
very ample and generated line \(\mathscr L\) for \(b\); compactness permits
one global power. Its evaluation embeds \(\Gamma\) in
\(\PP_Y(\mathscr E)\), where \(\mathscr E=b_*\mathscr L\). This coherent
sheaf is torsion-free of rank one: a section killed by a nonzero function
vanishes on the dense isomorphism locus and hence everywhere. Since \(Y\)
is smooth, \(\mathscr E^{**}\) is a line, and
\[
\mathscr J=\mathscr E\otimes(\mathscr E^{**})^{-1}\subset\OO_Y
\]
is a coherent ideal equal to \(\OO_Y\) on \(U_Y\).
Apply the ideal principalization functor of
\cite[Theorems~1.1.11 and~1.1.13]{Temkin2017} to \(\mathscr J\).
For smooth input it is supported on the cosupport of the ideal, so it
avoids \(U_Y\); its total transform is invertible on a smooth space
\(Y_1\). The resulting invertible quotient of the pulled-back
\(\mathscr E\) defines a map \(Y_1\to\PP_Y(\mathscr E)\). It lands
in the closed subspace \(\Gamma\) on a dense open, and hence everywhere
because \(Y_1\) is reduced. The map \(Y_1\to\Gamma\) is projective:
its graph is a closed immersion into the base change of the projective
map \(Y_1\to Y\). Its composite to \(V\) is projective and is an
isomorphism over \(U_V\).
This principalization is not required to have transverse intermediate
centers. Instead, after it is complete, mark as separate ordered Cartier
boundary components all strict initial boundary primes, all primes in the
principalized support, and all divisorial exceptional primes. On \(U_Y\)
this is the original globally simple SNC list. Apply the ordered-boundary
functor of \cite[Theorems~1.1.3 and~1.1.13]{Temkin2017}. Its centers
lie over the original labeled bad locus and therefore avoid \(U_Y\).
The final indexed components and intersections are smooth, and the total
support consists of the strict support and new exceptional support.
The already invertible ideal remains invertible, so the map to \(\Gamma\)
persists.
Finally apply the same two stages to the pullback of the reduced coherent
ideal of \(V\setminus U_V\). In the second stage mark, each distinct
prime once, every component of the prior resolved support, every component
of the newly principalized support, and every divisorial exceptional
prime. Its total transform has divisorial support equal to
the entire inverse image of \(V\setminus U_V\); every such divisor maps
to a subset of codimension at least two by \eqref{eq:bad-codim}. Outside
this support the resulting map \(\pi:\widehat V\to V\) is an
isomorphism. Conversely a point on this support cannot be a local
isomorphism point, because the divisor germ through it would map to a
hypersurface contained in \(V\setminus U_V\). Thus this is exactly the
nonisomorphism locus. The complete support is globally simple SNC and
includes every strict boundary component of \(V\).
Each \(\pi\)-exceptional prime has center in \(V\setminus U_V\). The
preceding discrepancy argument shows that its log discrepancy is positive,
proving property (2). Let \(\widehat Z\) be an irreducible intersection
of \(k\) distinct transformed floor components. At its general point
exactly those components occur, by SNC. If \(k=1\), its prime valuation
has log discrepancy zero. If \(k\geq2\), the blowup of the smooth stratum
has log discrepancy \(\operatorname{codim}(\widehat Z)-k=k-k=0\).
This is a prime on a projective resolution, so its image meets \(U_V\).
Since \(\pi\) is an isomorphism there, it is generically an isomorphism
on that image. This proves property (3).
\end{proof}
\subsection{The support survives}
\begin{proof}[Proof of Proposition~\ref{prop:supported-model}]
All statements except \(P\ne0\) have been established. Suppose that
\(P=0\). Every component of \(P_Y\), and hence every component of
\(p^*M\), has disappeared. On a common projective resolution with smooth source
\(r:W\to Y\), \(s:W\to V\), set \(Q=r^*p^*M\). It is effective and
nonzero: the pullback of a nonzero effective Cartier multiple has a
nonzero strict transform. It is \(s\)-exceptional. Indeed, a prime of
\(W\) mapping onto a prime of \(V\) corresponds, by no extraction, to
a surviving prime of \(Y\); its coefficient in \(Q\) is zero under the
supposition that every component of \(p^*M\) disappeared.
The actual rational line of \(Q\) is \((pr)^*D\). It is analytically
nef on the compact K\"ahler \(W\), by pullback of the metric inequalities
for the original \(D\). In particular it is relatively nef for the
projective \(s\). Exceptional negativity gives \(Q\leq0\), contradicting
its effectivity and nonvanishing. Thus \(P\ne0\).
\end{proof}
The construction uses the original section but supplies no new ambient
section. Its floor line is already defined by restricting a Cartier
multiple of \(A\) to the reduced subspace \(S=\floor B\). The next
sections will generate that line on all of \(S\), after which supported
lifting contradicts the last equality in
Proposition~\ref{prop:supported-model}.
==== END Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//reduction.tex ====
==== BEGIN Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//scalars.tex ====
\section{Finite residue actions from one K\"ahler class}\label{sec:scalars}
The restriction criterion asks for equality under links on lower strata.
To build enough sections with all these equalities, we need finiteness of
the induced actions on each fixed section space. On projective strata this
is the usual log pluricanonical representation theorem. On nonprojective
surfaces it holds here because the links all come from the boundary of one
compact K\"ahler fourfold and therefore compare restrictions of one
ambient K\"ahler class.
Fix the common even degree \(q\) and lines \(L_Z\) from
\eqref{eq:stratum-semiample-map}. For \(d=0,1,2\), form a groupoid
\(\mathcal G_d\) whose objects are the finitely many \(d\)-dimensional
strata. In dimension zero use all identifications of points, with the
canonical zero-form generator \(1\). In dimension one use all crepant
residue comparisons between the indicated curve pairs. In dimension two
use only the groupoid generated by the links of three-dimensional strata
from Proposition~\ref{prop:restriction-link}, together with their inverses.
Here a comparison includes equality of the pulled-back invertible
meromorphic adjoint subsheaves, as specified there.
Such a comparison induces an isomorphism
\[
H^0(Z_1,L_{Z_1}^k)\longrightarrow H^0(Z_2,L_{Z_2}^k)
\]
for every \(k>0\). Pull a section to a resolution of the graph, use the
equality of invertible meromorphic subsheaves, and descend to the other
normal stratum by projection formula. This transport is compatible with
composition, products of sections, and residue restriction along boundary
divisors whose centers are birational on both strata. Lemma
\ref{lem:transport-lower-restrictions} proves the compatibility also when
a floor curve is contracted, before that case enters the construction.
The same
argument shows that evaluation after a graph pullback is evaluation in
the identified actual line fibers, a fact needed for generation later.
\begin{proposition}[Finite isotropy images]\label{prop:finite-isotropy}
For every \(d\in\{0,1,2\}\), every \(d\)-stratum \(Z\), and every fixed
integer \(k>0\), the image of the self-comparisons in \(\mathcal G_d\)
on \(H^0(Z,L_Z^k)\) is a finite group.
\end{proposition}
The restriction on \(\mathcal G_2\) is essential. A nonprojective K3
surface with trivial canonical line can have an automorphism acting by
a non-root-of-unity scalar on its holomorphic two-form
\cite[Theorems~3.5 and~4.1]{McMullen2002}. Thus the proposition would be false
for arbitrary bimeromorphic self-comparisons of nonprojective surfaces.
\subsection{Points, curves, and projective surfaces}
On a point the zero-form generator is \(1\) and every comparison acts
as the identity. A normal compact curve is smooth, and a bimeromorphic
map is an isomorphism preserving the boundary coefficients. For genus
at least two the automorphism group is finite. For genus one, the
stabilizer of a nonempty finite boundary support is finite; with empty
boundary translations act trivially on holomorphic forms and the linear
automorphism group is finite. For genus zero, nefness of the adjoint and
coefficients at most one require at least two marked points. If there
are exactly two, both coefficients are one and the log line is trivial
with generator \((dt/t)^{\otimes qk}\). Scaling fixes it, and exchange
has sign one in the even degree. With at least three marked points the
stabilizer is finite. This proves Proposition~\ref{prop:finite-isotropy}
in dimensions zero and one.
Suppose a surface stratum \(Z\) is Moishezon. Its klt perturbation in
Proposition~\ref{prop:strata-adjunction} gives rational singularities.
Namikawa's projectivity criterion \cite[Corollary~6']{Namikawa2002}
makes the compact K\"ahler \(Z\) projective. Its effective full boundary
is dlt and its adjoint is semiample. The analytic comparison graphs are
algebraic by Chow, and their meromorphic equality is the usual crepant
B-birational equality for compatible canonical identifications.
Fujino--Gongyo's log pluricanonical representation theorem
\cite[Theorem~1.1]{FujinoGongyo2014} gives finite image in every fixed
Cartier degree. It applies exactly in this projective case.
\subsection{The class attached to a nonprojective surface}
Let \(Z\) now be a non-Moishezon surface stratum, and let \(P\) be the
minimal smooth surface of a compact K\"ahler resolution of \(Z\).
Point blowdowns preserve K\"ahlerness, for example by the even-\(b_1\)
criterion \cite[Theorem~11]{Buchdahl1999}. Algebraic dimension is
bimeromorphically invariant, so \(P\) is nonprojective and has algebraic
dimension zero or one. It has a nonzero holomorphic two-form: otherwise
\(H^2(P,\R)=H^{1,1}(P,\R)\), and rational approximation of a K\"ahler
class followed by Kodaira embedding would make \(P\) projective. Thus
\(\kappa(P)\geq0\). The minimal model is neither rational nor ruled;
its bimeromorphic maps are automorphisms, and bimeromorphic maps between
such minimal surfaces are isomorphisms
\cite[Proposition~3.5]{ProkhorovShramov2017}. Under a common resolution,
\(H^0(Z,L_Z^k)\) embeds as a finite-dimensional space of meromorphic
\(qk\)-pluricanonical forms on \(P\), compatibly with all transports.
Fix once and for all a K\"ahler form \(\omega_V\) on the ambient fourfold.
For a common resolution \(h:R\to P\), \(\rho:R\to Z\), define
\begin{equation}\label{eq:surface-class}
c_Z=h_*[\rho^*(\omega_V|_Z)]\in H^{1,1}(P,\R).
\end{equation}
The brackets denote the de Rham class of the pulled-back local-potential
form on the smooth resolution. All cohomology in the comparison below
is taken on smooth manifolds.
This is independent of further resolutions, since \(v_*v^*=1\) for a
modification \(v\). Put \(\beta=[\rho^*(\omega_V|_Z)]\). Its smooth
representative is semipositive and is strictly positive on a nonempty
open: the original stratum is generically immersed in the smooth
isomorphism locus of the special resolution. Hence \(\beta^2>0\).
The blowup orthogonal decomposition has
\(\beta=h^*c_Z+e\), with \(e\) in the negative-definite exceptional
subspace. Consequently
\begin{equation}\label{eq:surface-class-positive}
c_Z^2=\beta^2-e^2>0.
\end{equation}
\begin{lemma}[The same ambient class across a link]\label{lem:class-link}
Let a link of a three-dimensional stratum compare non-Moishezon surface
strata \(Z_1,Z_2\), allowing \(Z_1=Z_2\). Let \(P_i\) be their smooth
minimal models and \(\tau:P_1\simeq P_2\) the induced isomorphism. For
the classes in \eqref{eq:surface-class},
\begin{equation}\label{eq:class-mod-ns}
c_{Z_1}-\tau^*c_{Z_2}\in\NS(P_1)_\R.
\end{equation}
Here \(\NS(P)_\R\) is the real span of the first Chern classes of
holomorphic line bundles in \(H^{1,1}(P,\R)\).
\end{lemma}
\begin{proof}
Let \(i:Z\hookrightarrow V\) be the parent stratum, and choose a common
projective resolution with smooth compact K\"ahler source \(U\), with
maps \(a:U\to Z\) and \(b:U\to T\) to its original and Mori models.
Use the single class
\[
\alpha=[(i\circ a)^*\omega_V]\in H^{1,1}(U,\R).
\]
Resolve the marked-branch comparison graph and its maps to \(U\) and
the minimal surfaces. We obtain a smooth compact K\"ahler surface \(D\)
with maps \(r_i:D\to U\), bimeromorphic maps \(d_i:D\to P_i\), and
bimeromorphic maps to the original \(Z_i\), such that \(a r_i\) is the
inclusion of that original branch after its resolution and
\(d_2=\tau d_1\). These identities hold on the
common marked open and hence everywhere. This construction includes a
self-link: its two maps to the same original prime may differ by the
normalized exchange. Resolution independence and projection formula give
\begin{equation}\label{eq:class-correspondence}
c_{Z_1}-\tau^*c_{Z_2}
=d_{1,*}(r_1^*-r_2^*)\alpha.
\end{equation}
The exceptional image of the modification \(b:U\to T\) has codimension
at least two in the normal threefold, and hence dimension at most one.
The Mori base in this two-marking case has dimension two. After removing
its image, the discriminant, and the singular base locus, \(U\) is
therefore a smooth proper \(\PP^1\)-fibration over a dense open of the
base. For such a fibration \(\pi:U^\circ\to W^\circ\), the differential
sequence and \(H^0(\PP^1,\Omega^1)=0\) give
\[
\pi_*\Omega^2_{U^\circ}=\Omega^2_{W^\circ}.
\]
Indeed the quotient of \(\Omega^2_{U^\circ}\) by
\(\pi^*\Omega^2_{W^\circ}\) is
\(\pi^*\Omega^1_{W^\circ}\otimes\Omega^1_{U^\circ/W^\circ}\), whose
direct image is zero, while \(\pi_*\OO_{U^\circ}=\OO_{W^\circ}\).
Thus the two restrictions of a holomorphic two-form on \(U\) agree on
the dense paired-branch open of \(D\), even for the double branch after
an \'etale local ordering. They agree on all of \(D\).
Holomorphic pullback and proper pushforward on compact K\"ahler manifolds
preserve Hodge type. The map
\[
d_{1,*}(r_1^*-r_2^*):H^2(U,\Q)\longrightarrow H^2(P_1,\Q)
\]
is therefore a rational Hodge morphism; pushforward here is between
equal-dimensional surfaces. The preceding equality kills its
\((2,0)\) part and, by conjugation, its \((0,2)\) part. Its rational
image is of type \((1,1)\), hence belongs to \(\NS(P_1)_\Q\) by
Lefschetz \((1,1)\). Extend scalars to \(\R\) in
\eqref{eq:class-correspondence} to obtain \eqref{eq:class-mod-ns}.
\end{proof}
Swapnajit Das's positive-class and ruled-branch arguments
\cite[Lemmas~7.2--7.3 and Corollary~7.4]{Das2026} are close predecessors
of this mechanism for two disjoint degree-one branches. The proof above
uses the same ambient class on both restrictions and includes the single
normalized degree-two branch. It requires no rational polarization.
\subsection{A uniform scalar bound}
We spell out why the class congruence gives finite image for a whole group,
not just a finite-order scalar for each individually chosen comparison.
\begin{lemma}\label{lem:lattice-scalar}
Let \(P\) be a smooth connected nonprojective compact K\"ahler surface
with a nowhere-vanishing holomorphic two-form \(\eta\). Put
\(N=\NS(P)_\R\). Let \(G\subset\operatorname{Aut}(P)\) be a subgroup.
Assume either that \(N\) is degenerate, or that there exists
\(c\in H^{1,1}(P,\R)\) with \(c^2>0\) and
\(\sigma^*c-c\in N\) for every \(\sigma\in G\). Then the image of
\(G\) on \(\C\eta^{\otimes m}\) is finite for every \(m>0\).
More precisely, every volume scalar has order in the finite set of
integers \(n\) with \(\varphi(n)\leq b_2(P)\).
\end{lemma}
\begin{proof}
The intersection form on \(H^{1,1}(P,\R)\) has Lorentz signature
\((1,h^{1,1}-1)\). Its restriction to \(N\) is nonpositive. To see
this, a rational positive-square class in \(N\), after scaling and
choosing its sign, is \(c_1(L)\) with positive K\"ahler degree.
Riemann--Roch and Serre duality give \(h^0(P,L^v)\gg v^2\): the Euler
characteristic has positive quadratic leading term and
\(H^0(P,K_P\otimes L^{-v})=0\) for large \(v\) by its negative K\"ahler
degree. This would make \(P\) Moishezon and hence projective. Rational
approximation in \(N\) rules out a positive real class as well.
If \(N\) is negative definite, orthogonally project \(c\) to \(N^\perp\).
The projection \(c_\perp\) has positive square and is fixed by every
\(\sigma^*\): the congruence kills its projected difference, and each
automorphism preserves \(N\) and the intersection form. The perpendicular
of \(c_\perp\) is negative definite, so all eigenvalues on
\(H^{1,1}(P,\R)\) have modulus one.
If \(N\) is degenerate, its radical is a rational isotropic line
\(\ell\); nonpositivity in a Lorentz space permits no larger radical.
An integral automorphism preserves a primitive integral generator of
\(\ell\) up to sign. The invariant flag
\[
\ell\subset\ell^\perp\subset H^{1,1}(P,\R)
\]
has eigenvalues \(\pm1\) on the first and last quotients, which pair
dually, and a negative-definite middle quotient \(\ell^\perp/\ell\).
All \((1,1)\) eigenvalues again have modulus one. This argument permits
unipotent action and makes no finiteness assertion for all cohomology.
Write \(\sigma^*\eta=\delta_\sigma\eta\). Integration of
\(\eta\wedge\bar\eta\) gives \(|\delta_\sigma|=1\), and the conjugate
eigenvalue has the same modulus. Since a nowhere-vanishing canonical form
spans \(H^{2,0}(P)\), all eigenvalues on \(H^2(P,\C)\) now have modulus
one. The action on \(H^2(P,\Z)/\mathrm{torsion}\) is integral. Kronecker's
theorem makes each eigenvalue a root of unity; if its order is \(n\), its
cyclotomic polynomial has degree \(\varphi(n)\leq b_2(P)\). There are
only finitely many such \(n\). The scalars of every element of \(G\)
therefore lie in one finite set of roots of unity, which proves finite
image on \(\C\eta^{\otimes m}\).
\end{proof}
If \(P\) has algebraic dimension zero, its minimality and
\cite[Theorem~4]{KodairaCompactAnalytic} imply that its canonical line
is trivial, so it has a nowhere-vanishing
volume \(\eta\). Every meromorphic pluriform is
a constant multiple of a power of \(\eta\), since the quotient is a
meromorphic function and \(a(P)=0\). In particular the section space
under consideration has dimension at most one. Composing
\eqref{eq:class-mod-ns} along a returning sequence of links gives
\(\sigma^*c_Z-c_Z\in\NS(P)_\R\); each intervening isomorphism preserves
N\'eron--Severi. Lemma~\ref{lem:lattice-scalar} applies, with the fixed
cohomology rank of this \(P\). It gives a finite scalar image for all
returning compositions at once.
\subsection{Algebraic dimension one}
Suppose \(a(P)=1\). Its holomorphic algebraic reduction is an elliptic
fibration \(v:P\to J\) with connected fibers over a smooth compact curve,
and every curve on \(P\) is vertical
\cite[Theorem~2]{KodairaCompactAnalytic}. Every automorphism preserves
this reduction, which is determined by the meromorphic function field.
Let \(G\) be the group of returning link automorphisms and let \(H\) be
the corresponding finite-dimensional invariant space of meromorphic
\(qk\)-pluriforms on \(P\).
On a smooth base open avoiding the finitely many polar fibers of a basis
of \(H\), division by a base differential makes these forms holomorphic
powers of the elliptic differential on each fiber. An automorphism over
the base multiplies that differential by a root of unity of order
\(1,2,3,4\), or \(6\); its multiplier is holomorphic with values in a
finite set, hence locally constant. It acts on all of \(H\) by the
corresponding common scalar. Thus the kernel of \(G\to\operatorname{Aut}(J)\)
has finite image on \(H\). If the image on \(J\) is finite, its finitely
many cosets make the full image finite as well.
It remains to consider an infinite base image. A curve of genus at
least two has finite automorphism group. If \(J=\PP^1\), the finite set
of nonsmooth fibers, including multiple fibers, is invariant, so it would
have at most two points. This contradicts the theorem that a nonalgebraic compact
K\"ahler elliptic surface over \(\PP^1\) has at least three singular
fibers \cite[Proposition~A.1]{ClaudonHoeringLin2019}. This theorem is a
short form of the elliptic canonical-degree exclusion needed here and
does not assume a section of the fibration.
If \(J\) has genus one, the infinite base group contains infinitely many
translations. Its invariant finite special-fiber set must be empty.
Choose a nonzero holomorphic two-form \(\eta\) on \(P\), whose existence
was proved above, and write its effective integral zero divisor as
\(D_P\). Every component is vertical. Each fiber is now smooth and
connected, hence an irreducible reduced elliptic curve; a vertical prime
is that entire fiber, and smoothness gives it multiplicity one in the
pullback of its base point. Thus \(D_P=v^*D_J\) for an effective integral
divisor \(D_J\) on \(J\), and the actual section gives
\[
\omega_P\simeq\OO_P(D_P)\simeq v^*\OO_J(D_J).
\]
Pullback \(v^*:\operatorname{Pic}(J)\to\operatorname{Pic}(P)\) is
injective. Indeed, if \(v^*M\simeq\OO_P\), connected fibers and
projection formula give
\[
M\simeq M\otimes v_*\OO_P\simeq v_*v^*M\simeq v_*\OO_P\simeq\OO_J.
\]
For an automorphism \(\sigma\in G\) covering \(h\in\operatorname{Aut}(J)\),
the identities \(v\sigma=hv\) and \(\sigma^*\omega_P\simeq\omega_P\)
therefore imply \(h^*\OO_J(D_J)\simeq\OO_J(D_J)\).
Put \(d=\deg D_J\) and write \(J=\C/\Lambda\). If translation \(t_x\)
stabilizes \(\OO_J(D_J)\), then \(t_x^*D_J-D_J\) is principal. Its
point sum in \(J\) is \(-dx\), whereas a principal divisor has point
sum zero. To recall the latter fact, lift its meromorphic function to
an elliptic function \(F\), and choose a fundamental parallelogram
\(\mathcal P\) with boundary avoiding its zeros and poles. For a lattice
basis \(\omega_1,\omega_2\), the residue theorem and pairing opposite
edges give
\[
\sum_{z\in\mathcal P}\ord_z(F)z
=\frac{1}{2\pi i}\int_{\partial\mathcal P}z\frac{F'(z)}{F(z)}\,dz
=\omega_1 n_2-\omega_2 n_1\in\Lambda,
\]
where \(n_i=(2\pi i)^{-1}\int_{z_0}^{z_0+\omega_i}F'(z)/F(z)\,dz\in\Z\), since
the endpoint values of \(F\) agree. It follows that \(dx=0\) in \(J\).
If \(d>0\), all stabilizing translations lie in the finite group
\(J[d]=(\tfrac1d\Lambda)/\Lambda\). The infinitely many translations
above therefore force \(d=0\). Effectivity gives \(D_J=0\), so \(\eta\)
is nowhere vanishing.
The ratio of any holomorphic two-form to \(\eta\) is holomorphic on the
compact connected \(P\), hence constant. Thus \(\eta\) spans
\(H^0(P,\omega_P)\), and every \(\sigma\in G\) satisfies
\(\sigma^*\eta=c_\sigma\eta\) for a nonzero constant \(c_\sigma\).
Every element of \(H\) is \(\eta^{\otimes qk}\)
times a meromorphic function from \(J\). The corresponding function
space is therefore preserved by the base action. The union of the pole sets of
a basis is finite and intrinsic to this vector space, hence invariant
under the base group. Infinitely many translations preserve no nonempty
finite subset. Thus these functions have no poles and are constant.
The fiber class belongs to \(\NS(P)_\R\), has square zero, and is nonzero
by its positive K\"ahler area. Nonpositivity of \(\NS(P)_\R\) makes it
a radical vector. The degenerate case of
Lemma~\ref{lem:lattice-scalar} gives the same finite scalar image on \(H\).
This completes the algebraic-dimension-one case and the proof of
Proposition~\ref{prop:finite-isotropy}.
For later use, finiteness of isotropy also controls all transports in a
fixed degree between two objects of the same orbit. Choose one comparison
from an orbit representative to each member. Every other transport to
that member is a self-transport of the representative, whose image is
finite, followed by this chosen transport. Thus only finitely many linear
transport actions occur in that degree, even if the groupoid has infinitely
many bimeromorphic arrows.
==== END Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//scalars.tex ====
==== BEGIN Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//references.bib ====
@article{Miyaoka1988,
author = {Miyaoka, Yoichi},
title = {Abundance conjecture for 3-folds: case {$\nu=1$}},
journal = {Compositio Mathematica},
volume = {68}, number = {2}, pages = {203--220}, year = {1988},
url = {https://www.numdam.org/article/CM_1988__68_2_203_0.pdf}
}
@article{Kawamata1992,
author = {Kawamata, Yujiro},
title = {Abundance theorem for minimal threefolds},
journal = {Inventiones Mathematicae}, volume = {108}, pages = {229--246}, year = {1992},
doi = {10.1007/BF02100604}
}
@article{Fujino2000,
author = {Fujino, Osamu},
title = {Abundance theorem for semi log canonical threefolds},
journal = {Duke Mathematical Journal}, volume = {102}, number = {3}, pages = {513--532}, year = {2000},
url = {https://www.math.kyoto-u.ac.jp/~fujino/Abundance.pdf}
}
@misc{Kollar2013Sources,
author = {Koll{\'a}r, J{\'a}nos}, title = {Sources of log canonical centers},
year = {2012}, note = {arXiv:1107.2863v3, November 14, 2012},
url = {https://arxiv.org/abs/1107.2863v3}
}
@article{HoeringPeternell2016,
author = {H{\"o}ring, Andreas and Peternell, Thomas},
title = {Minimal models for {K}{\"a}hler threefolds},
journal = {Inventiones Mathematicae}, volume = {203}, pages = {217--264}, year = {2016},
doi = {10.1007/s00222-015-0592-x},
note = {Locators refer to the author manuscript dated January 16, 2015},
url = {https://math.univ-cotedazur.fr/~hoering/articles/a21-kaehler-mmp.pdf}
}
@misc{DasHacon2024,
author = {Das, Omprokash and Hacon, Christopher D.},
title = {The log minimal model program for {K}{\"a}hler 3-folds},
year = {2024}, note = {arXiv:2009.05924v4, April 9, 2024},
url = {https://arxiv.org/abs/2009.05924v4}
}
@misc{DasHacon2024OnMMP,
author = {Das, Omprokash and Hacon, Christopher},
title = {On the Minimal Model Program for {K}{\"a}hler 3-folds},
year = {2024},
note = {arXiv:2306.11708v2, June 26, 2024},
url = {https://arxiv.org/abs/2306.11708v2}
}
@misc{DasOu2025,
author = {Das, Omprokash and Ou, Wenhao},
title = {On the log abundance for compact {K}{\"a}hler threefolds {II}},
year = {2025}, note = {arXiv:2306.00671v4, June 12, 2025},
url = {https://arxiv.org/abs/2306.00671v4}
}
@misc{DasHaconPaun2024,
author = {Das, Omprokash and Hacon, Christopher D. and P{\u a}un, Mihai},
title = {On the 4-dimensional minimal model program for {K}{\"a}hler varieties},
year = {2024}, note = {arXiv:2205.12205v3, April 9, 2024},
url = {https://arxiv.org/abs/2205.12205v3}
}
@misc{HoeringLazicLehn2025,
author = {H{\"o}ring, Andreas and Lazi{\'c}, Vladimir and Lehn, Christian},
title = {Nonvanishing results for {K}{\"a}hler varieties},
year = {2025}, note = {arXiv:2508.14634v2, October 21, 2025},
url = {https://arxiv.org/abs/2508.14634v2}
}
@misc{Das2026,
author = {Das, Swapnajit},
title = {Abundance for semi log canonical compact {K}{\"a}hler threefolds},
year = {2026}, note = {arXiv:2604.28085v2, September 13, 2026},
url = {https://arxiv.org/abs/2604.28085v2}
}
@article{McMullen2002,
author = {McMullen, Curtis T.}, title = {Dynamics on {K3} surfaces: {Salem} numbers and {Siegel} disks},
journal = {Journal f{\"u}r die reine und angewandte Mathematik}, volume = {545}, pages = {201--233}, year = {2002},
doi = {10.1515/crll.2002.036},
note = {Cited author manuscript dated January 19, 2001},
url = {https://people.math.harvard.edu/~ctm/papers/home/text/papers/k3/k3.pdf}
}
@article{Wang2021,
author = {Wang, Juanyong},
title = {On the {Iitaka} conjecture {$C_{n,m}$} for {K}{\"a}hler fibre spaces},
journal = {Annales de la Facult{\'e} des Sciences de Toulouse. Math{\'e}matiques},
volume = {30}, number = {4}, pages = {813--897}, year = {2021},
doi = {10.5802/afst.1690}
}
@misc{Temkin2017,
author = {Temkin, Michael},
title = {Functorial desingularization over {$\mathbb{Q}$}: boundaries and the embedded case},
year = {2017}, note = {arXiv:0912.2570v3, February 21, 2017},
url = {https://arxiv.org/abs/0912.2570v3}
}
@misc{Fujino2026Fujiki,
author = {Fujino, Osamu},
title = {On vanishing theorems for non-compact analytic spaces},
year = {2026}, note = {Author version 0.25, April 29, 2026},
url = {https://www.math.kyoto-u.ac.jp/~fujino/fujiki-vanishing3.pdf}
}
@misc{Fujino2023AnalyticVanishing,
author = {Fujino, Osamu},
title = {Vanishing theorems for projective morphisms between complex analytic spaces},
year = {2023}, note = {arXiv:2205.14801v7, October 14, 2023},
url = {https://arxiv.org/abs/2205.14801v7}
}
@article{Houzel1961,
author = {Houzel, Christian},
title = {G{\'e}om{\'e}trie analytique locale, {IV}},
journal = {S{\'e}minaire Henri Cartan}, volume = {13}, number = {2},
pages = {1--15}, year = {1960--1961}, note = {Expos{\'e} 21},
url = {https://www.numdam.org/item/SHC_1960-1961__13_2_A8_0.pdf}
}
@misc{Namikawa2002,
author = {Namikawa, Yoshinori},
title = {Projectivity criterion of {Moishezon} spaces and density of projective symplectic varieties},
year = {2001}, note = {arXiv:math/0101019v6, November 12, 2001},
url = {https://arxiv.org/abs/math/0101019v6}
}
@article{Sard1942,
author = {Sard, Arthur},
title = {The measure of the critical values of differentiable maps},
journal = {Bulletin of the American Mathematical Society},
volume = {48}, pages = {883--890}, year = {1942},
url = {https://webhomes.maths.ed.ac.uk/~v1ranick/papers/sard.pdf}
}
@article{FujinoGongyo2014,
author = {Fujino, Osamu and Gongyo, Yoshinori},
title = {Log pluricanonical representations and the abundance conjecture},
journal = {Compositio Mathematica}, volume = {150}, number = {4},
pages = {593--620}, year = {2014}, doi = {10.1112/S0010437X13007495},
note = {Cited author manuscript},
url = {https://www.math.kyoto-u.ac.jp/~fujino/fg-comp-final.pdf}
}
@article{Buchdahl1999,
author = {Buchdahl, Nicholas P.}, title = {On compact {K}{\"a}hler surfaces},
journal = {Annales de l'Institut Fourier}, volume = {49}, number = {1},
pages = {287--302}, year = {1999}, doi = {10.5802/aif.1674},
url = {https://www.numdam.org/article/AIF_1999__49_1_287_0.pdf}
}
@misc{ProkhorovShramov2017,
author = {Prokhorov, Yuri and Shramov, Constantin},
title = {Automorphism groups of compact complex surfaces},
year = {2019}, note = {arXiv:1708.03566v5, July 3, 2019},
url = {https://arxiv.org/abs/1708.03566v5}
}
@incollection{KodairaCompactAnalytic,
author = {Kodaira, Kunihiko}, title = {On compact analytic surfaces},
booktitle = {Analytic Functions}, series = {Princeton Mathematical Series},
number = {24}, publisher = {Princeton University Press}, pages = {121--135}, year = {1960},
note = {Theorems 2 and 4, original pp. 121--122, reproduced in Collected Works, vol. III (1975)},
url = {https://api.pageplace.de/preview/DT0400.9781400869879_A26113086/preview-9781400869879_A26113086.pdf}
}
@article{ClaudonHoeringLin2019,
author = {Claudon, Beno{\^i}t and H{\"o}ring, Andreas and Lin, Hsueh-Yung},
title = {The fundamental group of compact {K}{\"a}hler threefolds},
journal = {Geometry \& Topology}, volume = {23}, number = {7}, pages = {3233--3271}, year = {2019},
doi = {10.2140/gt.2019.23.3233},
url = {https://msp.org/gt/2019/23-7/gt-v23-n7-p01-s.pdf}
}
@misc{StacksProject,
author = {{The Stacks Project Authors}}, title = {{The Stacks Project}},
note = {Tags 09NU, 02FQ, and 08A8}, url = {https://stacks.math.columbia.edu}
}
@misc{SGA4Continuity,
author = {{SGA 4}},
title = {Th{\'e}orie des topos et cohomologie {\'e}tale des sch{\'e}mas, Tome 2},
note = {Expos{\'e} Vbis by B. Saint-Donat. Cited retypeset version 71766d9 (2024-07-30), Lemma 4.1.3, PDF p. 95},
url = {https://www.normalesup.org/~forgogozo/SGA4/tomes/tome2.pdf}
}
@misc{TemkinNonembedded,
author = {Temkin, Michael},
title = {Functorial desingularization of quasi-excellent schemes in characteristic zero: the non-embedded case},
year = {2011}, note = {arXiv:0904.1592v2, December 11, 2011},
url = {https://arxiv.org/abs/0904.1592v2}
}
@article{FujinoFujisawaSaito2014,
author = {Fujino, Osamu and Fujisawa, Taro and Saito, Morihiko},
title = {Some Remarks on the Semipositivity Theorems},
journal = {Publications of the Research Institute for Mathematical Sciences},
volume = {50}, number = {1}, pages = {85--112}, year = {2014},
doi = {10.4171/PRIMS/125},
note = {Preprint locators refer to arXiv:1302.6180v2, June 24, 2013},
url = {https://arxiv.org/abs/1302.6180v2}
}
@article{Saito1990Kahler,
author = {Saito, Morihiko},
title = {Decomposition theorem for proper {K}{\"a}hler morphisms},
journal = {Tohoku Mathematical Journal, Second Series},
volume = {42}, number = {2}, pages = {127--148}, year = {1990},
doi = {10.2748/tmj/1178227650},
url = {https://www.jstage.jst.go.jp/article/tmj1949/42/2/42_2_127/_pdf/-char/en}
}
@misc{Saito2022Kahler,
author = {Saito, Morihiko},
title = {Some remarks on decomposition theorem for proper {K}{\"a}hler morphisms},
year = {2022}, note = {arXiv:2204.09026v5, May 26, 2022},
url = {https://arxiv.org/abs/2204.09026v5}
}
@misc{SabbahSchnell2026,
author = {Sabbah, Claude and Schnell, Christian},
title = {Pure complex {H}odge modules},
year = {2026}, note = {MHM Project, version 2},
url = {https://perso.pages.math.cnrs.fr/users/claude.sabbah/MHMProject/mhm.pdf}
}
@article{AcquistapaceBrogliaTognoli1979,
author = {Acquistapace, F. and Broglia, F. and Tognoli, A.},
title = {An embedding theorem for real analytic spaces},
journal = {Annali della Scuola Normale Superiore di Pisa, Classe di Scienze},
series = {4}, volume = {6}, number = {3}, pages = {415--426}, year = {1979},
note = {Series 4},
url = {https://www.numdam.org/article/ASNSP_1979_4_6_3_415_0.pdf}
}
@article{Lojasiewicz1964,
author = {{\L}ojasiewicz, S.}, title = {Triangulation of semi-analytic sets},
journal = {Annali della Scuola Normale Superiore di Pisa, Classe di Scienze},
series = {3}, volume = {18}, number = {4}, pages = {449--474}, year = {1964},
note = {Series 3},
url = {https://www.numdam.org/article/ASNSP_1964_3_18_4_449_0.pdf}
}
@article{Serre1956GAGA,
author = {Serre, Jean-Pierre},
title = {G{\'e}om{\'e}trie alg{\'e}brique et g{\'e}om{\'e}trie analytique},
journal = {Annales de l'Institut Fourier}, volume = {6}, pages = {1--42}, year = {1956},
doi = {10.5802/aif.59},
url = {https://www.numdam.org/article/AIF_1956__6__1_0.pdf}
}
@article{Grauert1960ProperMapping,
author = {Grauert, Hans},
title = {Ein {Theorem} der analytischen {Garbentheorie} und die {Modulr{\"a}ume} komplexer {Strukturen}},
journal = {Publications Math{\'e}matiques de l'IH{\'E}S},
volume = {5}, pages = {5--64}, year = {1960},
doi = {10.1007/BF02684746},
note = {Section 7, no. 1, Satz 1, p. 60 (PDF p. 59)},
url = {https://www.numdam.org/item/PMIHES_1960__5__5_0.pdf}
}
@article{Grothendieck1961TechniquesII,
author = {Grothendieck, Alexander},
title = {Techniques de construction en g{\'e}om{\'e}trie analytique. {II}.
{G{\'e}n{\'e}ralit{\'e}s} sur les espaces annel{\'e}s et les espaces analytiques},
journal = {S{\'e}minaire Henri Cartan},
year = {1960-1961}, volume = {13}, number = {1}, pages = {1--14},
note = {Expos{\'e} no. 9},
url = {https://www.numdam.org/item/SHC_1960-1961__13_1_A5_0/}
}
@article{Grothendieck1961TechniquesIII,
author = {Grothendieck, Alexander},
title = {Techniques de construction en g{\'e}om{\'e}trie analytique. {III}.
{Produits} fibr{\'e}s d'espaces analytiques},
journal = {S{\'e}minaire Henri Cartan},
year = {1960-1961}, volume = {13}, number = {1}, pages = {1--11},
note = {Expos{\'e} no. 10},
url = {https://www.numdam.org/item/SHC_1960-1961__13_1_A6_0/}
}
@article{Grothendieck1961TechniquesVIII,
author = {Grothendieck, Alexander},
title = {Techniques de construction en g{\'e}om{\'e}trie analytique. {VIII}.
{Rapport} sur les th{\'e}or{\`e}mes de finitude de {Grauert} et {Remmert}},
journal = {S{\'e}minaire Henri Cartan},
year = {1960-1961}, volume = {13}, number = {2}, pages = {1--10},
note = {Expos{\'e} no. 15},
url = {https://www.numdam.org/item/SHC_1960-1961__13_2_A2_0/}
}
@article{Fujiki1975,
author = {Fujiki, Akira},
title = {On the Blowing Down of Analytic Spaces},
journal = {Publications of the Research Institute for Mathematical Sciences},
year = {1975}, volume = {10}, number = {2}, pages = {473--507},
doi = {10.2977/prims/1195192006},
url = {https://www.jstage.jst.go.jp/article/kyotoms1969/10/2/10_2_473/_article}
}
@article{CollinsTosatti2015,
author = {Collins, Tristan C. and Tosatti, Valentino},
title = {{K}{\"a}hler currents and null loci},
journal = {Inventiones Mathematicae},
volume = {202}, number = {3}, pages = {1167--1198}, year = {2015},
doi = {10.1007/s00222-015-0585-9},
note = {Cited arXiv:1304.5216v5, February 20, 2015; Theorem 1.1, p. 2},
url = {https://arxiv.org/abs/1304.5216v5}
}
@article{Boucksom2004,
author = {Boucksom, S{\'e}bastien},
title = {Divisorial {Zariski} decompositions on compact complex manifolds},
journal = {Annales scientifiques de l'{\'E}cole Normale Sup{\'e}rieure},
year = {2004}, volume = {Ser. 4, 37}, number = {1}, pages = {45--76},
doi = {10.1016/j.ansens.2003.04.002},
url = {https://www.numdam.org/articles/10.1016/j.ansens.2003.04.002/}
}
@incollection{BoucksomGuedj2013,
author = {Boucksom, S{\'e}bastien and Guedj, Vincent},
title = {Regularizing Properties of the {K{\"a}hler--Ricci} Flow},
booktitle = {An Introduction to the {K{\"a}hler--Ricci} Flow},
editor = {Boucksom, S{\'e}bastien and Eyssidieux, Philippe and Guedj, Vincent},
series = {Lecture Notes in Mathematics}, volume = {2086},
pages = {189--237}, year = {2013}, publisher = {Springer}, address = {Cham},
doi = {10.1007/978-3-319-00819-6_4},
url = {https://link.springer.com/chapter/10.1007/978-3-319-00819-6_4}
}
@article{Demailly1985,
author = {Demailly, Jean-Pierre},
title = {Mesures de {Monge-Amp{\`e}re} et caract{\'e}risation g{\'e}om{\'e}trique des vari{\'e}t{\'e}s alg{\'e}briques affines},
journal = {M{\'e}moires de la Soci{\'e}t{\'e} Math{\'e}matique de France},
volume = {2e s{\'e}rie, 19}, pages = {1--124}, year = {1985},
doi = {10.24033/msmf.320},
url = {https://www.numdam.org/item/10.24033/msmf.320.pdf}
}
@article{DemaillyPaun2004,
author = {Demailly, Jean-Pierre and P{\u a}un, Mihai},
title = {Numerical characterization of the {K}{\"a}hler cone of a compact {K}{\"a}hler manifold},
journal = {Annals of Mathematics},
volume = {159}, number = {3}, pages = {1247--1274}, year = {2004},
doi = {10.4007/annals.2004.159.1247},
url = {https://annals.math.princeton.edu/2004/159-3/p05}
}
@article{Ancona1982,
author = {Ancona, Vincenzo},
title = {Faisceaux amples sur les espaces analytiques},
journal = {Transactions of the American Mathematical Society},
volume = {274}, number = {1}, pages = {89--100}, year = {1982},
note = {Definitions 2.1--2.2, p. 91; Proposition 2.5, p. 92}
}
@article{Grauert1962,
author = {Grauert, Hans},
title = {{{\"U}ber} {Modifikationen} und exzeptionelle analytische {Mengen}},
journal = {Mathematische Annalen},
volume = {146}, pages = {331--368}, year = {1962},
doi = {10.1007/BF01441136},
note = {Section 3, no. 7, Satz 8, p. 353; Section 2, no. 4, Definition 3 and Satz 5, pp. 339--340},
url = {https://link.springer.com/article/10.1007/BF01441136}
}
@misc{Fujino2009Fundamental,
author = {Fujino, Osamu},
title = {Fundamental theorems for the log minimal model program},
year = {2009},
note = {Author version 4.02, September 24, 2009; Section 4.8 and Theorem 4.10, pp. 11--12},
url = {https://www.math.kyoto-u.ac.jp/~fujino/fundamental6.pdf}
}
@article{Serre1955FAC,
author = {Serre, Jean-Pierre}, title = {Faisceaux alg{\'e}briques coh{\'e}rents},
journal = {Annals of Mathematics}, volume = {61}, number = {2}, pages = {197--278}, year = {1955},
doi = {10.2307/1969915},
url = {https://www.college-de-france.fr/media/jean-pierre-serre/UPL5435398796951750634_Serre_FAC.pdf}
}
@article{CampanaHoeringPeternell2016,
author = {Campana, Fr{\'e}d{\'e}ric and H{\"o}ring, Andreas and Peternell, Thomas},
title = {Abundance for {K}{\"a}hler threefolds},
journal = {Annales scientifiques de l'{\'E}cole Normale Sup{\'e}rieure},
year = {2016}, volume = {Ser. 4, 49}, number = {4}, pages = {971--1025},
doi = {10.24033/asens.2301},
url = {https://numdam.org/articles/10.24033/asens.2301/}
}
@article{HoeringPeternell2015,
author = {H{\"o}ring, Andreas and Peternell, Thomas},
title = {{Mori} Fibre Spaces for {K}{\"a}hler Threefolds},
journal = {Journal of Mathematical Sciences, the University of Tokyo},
year = {2015}, volume = {22}, number = {1}, pages = {219--246},
url = {https://www.ms.u-tokyo.ac.jp/journal/pdf/jms220107.pdf}
}
@article{GuenanciaPaun2025,
author = {Guenancia, Henri and P{\u a}un, Mihai},
title = {{Bogomolov--Gieseker} inequality for log terminal {K}{\"a}hler threefolds},
journal = {Communications on Pure and Applied Mathematics},
year = {2025}, volume = {78}, number = {11}, pages = {2206--2244},
doi = {10.1002/cpa.70000},
note = {With an appendix by Fr{\'e}d{\'e}ric Campana, Andreas H{\"o}ring,
and Thomas Peternell. Appendix locators refer to arXiv:2405.10003v4,
March 2, 2025},
url = {https://arxiv.org/abs/2405.10003v4}
}
@misc{DasHaconYanez2026,
author = {Das, Omprokash and Hacon, Christopher and Y{\'a}{\~n}ez, Jos{\'e} Ignacio},
title = {{MMP} for Generalized Pairs on {K}{\"a}hler 3-folds},
year = {2026}, note = {arXiv:2305.00524v3, June 29, 2026},
url = {https://arxiv.org/abs/2305.00524v3}
}
@misc{DasOu2022,
author = {Das, Omprokash and Ou, Wenhao},
title = {On the Log Abundance for Compact {K}{\"a}hler {$3$}-folds},
year = {2023}, note = {arXiv:2201.01202v3, February 21, 2023},
url = {https://arxiv.org/abs/2201.01202v3}
}
@misc{OpenAILogAbundance2026,
author = {{OpenAI}},
title = {{Log abundance in characteristic zero}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Log-abundance-in-characteristic-zero-September-24-2026/paper.pdf}{OAI:Log-abundance-in-characteristic-zero-September-24-2026}},
year = {2026},
note = {Corollaries 11.5--11.6}
}
@misc{OpenAILifting2026,
author = {{OpenAI}},
title = {{Lifting sections from the reduced support of an adjoint}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Lifting-sections-from-the-reduced-support-of-an-adjoint-September-27-2026/paper.pdf}{OAI:Lifting-sections-from-the-reduced-support-of-an-adjoint-September-27-2026}},
year = {2026}
}
==== END Abundance-after-nonvanishing-for-compact-Kahler-fourfolds//references.bib ====