==== TABLE OF CONTENTS ====
Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//01-introduction.tex
Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//02-analytic.tex
Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//03-steps.tex
Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//04-terminal.tex
Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//05-scaling.tex
Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//06-perturbation.tex
Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//07-endpoints.tex
Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//08-relative.tex
Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//08-resolutions.tex
Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//09-threefold.tex
Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//10-special.tex
Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//11-threshold.tex
Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//12-supply.tex
Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//13-local-convention.tex
Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//dependencies.tex
Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//main.tex
Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//references.bib
===========================
==== BEGIN Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//01-introduction.tex ====
\section{Introduction}\label{sec:introduction}
The minimal model program seeks a birational model whose canonical
class, or more generally whose adjoint $K_X+\Delta$, is nef. When no
such model is compatible with pseudo-effectivity, its expected outcome
is a Mori fibre space: a fibration along which the adjoint is negative.
Each birational step contracts an extremal ray or replaces a small
contraction by its flip. Constructing the individual steps and choosing
a sequence that reaches an endpoint are separate problems.
For compact Kähler fourfolds, we prove that a suitable choice of ordinary
negative steps gives a finite program. It starts on the given rational
klt pair in the global Weil-divisor $\Q$-factorial category and reaches
the endpoint dictated by pseudo-effectivity. The absence of a global
ample polarization makes both the contractions and the finiteness
argument analytic. A second issue is categorical: a divisor or reflexive
sheaf defined on the entire compact space carries information different
from a divisor germ on an arbitrary analytic neighbourhood. The proof
keeps this distinction, and keeps the actual canonical rational line
bundle, throughout the program.
\subsection{Category and the finite-program theorem}
All varieties are reduced irreducible second-countable complex analytic
spaces. A Kähler form on a normal space has smooth strictly
plurisubharmonic local potentials in local embeddings. Projectivity of a
morphism means the existence of a relatively ample holomorphic line
bundle.
\begin{definition}\label{def:global-factoriality}\label{def:strong-factoriality}
A normal compact space $X$ is \emph{globally Weil $\Q$-factorial} if
every prime Weil divisor defined on the whole of $X$ is $\Q$-Cartier and
some positive reflexive power of its canonical sheaf $\omega_X$ is a line
bundle. It is \emph{globally strongly $\Q$-factorial} if every coherent
rank-one reflexive sheaf $\cF$ on $X$ has an invertible positive reflexive
power $\cF^{[m]}=(\cF^{\otimes m})^{**}$.
\end{definition}
The Weil convention is that of \cite[Definition~2.1]{DHY2023};
the strong convention is used in \cite[Section~2]{HX2026}.
The strong condition implies the Weil
condition. Neither definition imposes $\Q$-factoriality on every analytic
local ring. The distinction matters even for projective varieties; an
example is given in Appendix~\ref{sec:local-convention}.
A rational line bundle is an element of $\Pic(X)\otimes_{\Z}\Q$.
An equality of rational line bundles means an isomorphism of holomorphic
line bundles after a common positive integral multiple.
We write $K_X$ additively for the canonical sheaf. When a canonical Weil
divisor is given, we keep that representative and transport it through
the program. We will also prove the intrinsic variant in which only the
canonical rational line bundle is specified. In that variant, identities
of adjoints mean isomorphisms of holomorphic line bundles after a common
positive multiple. Relative canonical divisors are defined using
compatible local canonical generators. In particular, a numerical
identity alone never specifies such a divisor.
For an effective rational boundary $\Delta$, put $D=K_X+\Delta$.
We use \emph{log discrepancies} throughout:
\[
a(E;X,\Delta)
=1+\operatorname{coeff}_E\bigl(K_W-p^*(K_X+\Delta)\bigr)
\]
on a smooth model $p:W\to X$ carrying $E$. A pair is klt when all these
numbers are positive. It is terminal when it is klt and they are greater
than one for every exceptional divisor over $X$. The same convention
will apply to the auxiliary real and generalized pairs.
Let $\BC(X)$ denote real Bott--Chern cohomology of closed $(1,1)$-forms
with local potentials, and let $\NA(X)$ be the closed cone of positive
currents of bidimension $(1,1)$ modulo their pairings with these classes.
A class is \emph{nef} if it belongs to the closure of the Kähler cone,
and \emph{pseudo-effective} if it contains a closed positive
$(1,1)$-current with local potentials. These terms for a rational line
bundle refer to its first Chern class. A class is \emph{big} if it
contains a Kähler current, meaning a closed current that dominates
a positive multiple of a Kähler form.
For a projective contraction
$f:X\to Z$, write $\rho(X/Z)$ for the dimension of the space of global
real line-bundle classes modulo zero degree on every contracted curve,
and put
\[
\rho_{\mathrm{BC}}(X/Z)
=\dim_{\R}\bigl(\BC(X)/f^*\BC(Z)\bigr).
\]
An \emph{ordinary $D$-negative step} is a projective divisorial
contraction, or the flip of a projective small contraction, of a
$D$-negative extremal ray of $\NA(X)$. The contraction has connected
fibres and a normal compact Kähler target; its relative rank is one and
$-D$ is relatively ample. For a flip the transformed adjoint is ample
over the same target. The boundary is pushed forward in a divisorial
step and strictly transformed in a flip. A \emph{Mori fibre space} has
the same contraction properties and a base of strictly smaller
dimension.
\begin{theorem}[Finite ordinary programs]\label{thm:finite}
Let $X$ be a normal globally Weil $\Q$-factorial compact Kähler
fourfold, with a canonical Weil divisor $K_X$, and let $\Delta\geq0$
be a rational divisor such that $(X,\Delta)$ is klt. Then there is a
finite sequence of ordinary negative steps
\[
(X,\Delta)=(X_0,\Delta_0)\dashrightarrow\cdots
\dashrightarrow(X_n,\Delta_n).
\]
Every $X_i$ is normal, compact Kähler and globally Weil
$\Q$-factorial; every $(X_i,\Delta_i)$ is klt, and
$D_i=K_{X_i}+\Delta_i$ is $\Q$-Cartier. The canonical divisors are
the transforms of the specified datum. The following alternatives hold.
\begin{enumerate}[label=\textup{(\roman*)}]
\item If $D=K_X+\Delta$ is pseudo-effective, then $D_n$ is nef.
The composite $\phi:X\dashrightarrow X_n$ extracts no prime divisor,
and
\[
a(E;X,\Delta)\leq a(E;X_n,\Delta_n)
\]
for every prime divisor over the two models, with strict inequality
for every prime on $X$ contracted by $\phi$.
\item If $D$ is not pseudo-effective, there is a projective surjective
morphism $g:X_n\to S$ with connected fibres, where $S$ is normal
compact Kähler, $\dim S<4$, $\rho(X_n/S)=1$, and $-D_n$ is
$g$-ample.
\end{enumerate}
For each negative contraction, including $g$, the relative
Bott--Chern dimension is one. If $X$ is globally strongly
$\Q$-factorial, so is every $X_i$.
The same conclusions hold in the intrinsic formulation with the
canonical rational line bundle in place of a chosen canonical Weil
divisor. In either formulation the first model is exactly $X$.
\end{theorem}
Theorem~\ref{thm:finite} gives an affirmative solution of the
finite-program form of the rational klt fourfold minimal model
conjecture in this global compact Kähler category. The quantifier is
existence of one finite ordinary program; its proof uses auxiliary
models without replacing the first model $X$. In the pseudo-effective
case, the endpoint assertion is nefness, with the discrepancy properties
displayed above.
\subsection{History and the additional argument}
The projective minimal model program supplies the geometric framework:
negative extremal contractions, flips and the comparison of discrepancies.
Kawamata--Matsuda--Matsuki developed the four-dimensional terminal
termination argument using discrepancies and the loss of algebraic
surface classes~\cite[Theorem~5-1-15 and
Lemmas~5-1-16--5-1-17]{KMM1987}; Kollár--Mori give a systematic account
of the birational tools~\cite{KM1998}. Fujino extended the
fourfold termination method to canonical pairs with rational boundary
\cite{Fujino2004,Fujino2005Addendum}. His addendum identifies a
necessary correction to the low-discrepancy count: all valuations over
certain smooth boundary surfaces must be treated together, rather than
only the divisor obtained by the first blowup.
In the compact Kähler setting, Höring--Peternell constructed minimal
models for non-uniruled threefolds~\cite{HP2016}.
Das--Hacon--Păun proved the fourfold minimal model theorem for dlt
pairs whose adjoint is $\Q$-linearly equivalent to an effective divisor
\cite[Theorem~1.1]{DHP2024}. Their work also develops analytic
finite-generation and birational tools. Fujino's relative analytic
MMP supplies ordinary cone, base-point-free and flip theorems for
projective morphisms between complex analytic spaces~\cite{Fujino2022}.
These are relative projective constructions; neither the source nor
the final model is required to be projective over a point.
Generalized pairs provide a way to carry a nef class on a higher model
through the program. Das--Hacon--Yáñez developed their compact Kähler
threefold theory and the analytic language used here~\cite{DHY2023}.
Hacon--Xie's cone and transcendental base-point-free theorems supply
contractions in the compact Kähler category
\cite[Theorems~1.3--1.4]{HX2026}. Their generalized MMP with scaling
terminates when the boundary plus nef part is big
\cite[Theorem~1.5]{HX2026}; their Theorem~1.1 gives Mori fibre
models in the non-pseudo-effective case and good models in its stated
big cases. These results furnish the steps, the auxiliary programs,
and the finiteness of marked models away from scaling parameter zero
that we use below.
Matsumura--Zhong prove termination with Kähler scaling for strongly
$\Q$-factorial compact Kähler klt pairs whose pseudo-effective adjoint
has numerical dimension zero. They also obtain minimal models after
a small crepant strong $\Q$-factorial modification in that case
\cite[Theorems~6.1 and~6.3]{MatsumuraZhong2026}.
The remaining issue for Theorem~\ref{thm:finite} is the
pseudo-effective adjoint without a bigness or effectivity assumption.
We follow the limiting-model strategy of Birkar
\cite[Theorem~1.2 and Section~4]{Birkar2009}. In that strategy,
termination for a limiting terminal pair is combined with rational
decomposition of a nef real adjoint and a sufficiently small
perturbation. Birkar already works with real boundaries in the
projective category. Here the required replacements concern compact
Kähler models, potentially transcendental scaling data, and varying
projective contraction bases.
Three points make this transfer possible. First, the trace of the one
initial Kähler form is identified as a positive current with local
potentials on every relevant model. Second, we prove ordinary terminal
fourfold termination with effective real boundary by combining the
corrected low-discrepancy argument with compact analytic cycle classes.
Third, actual line bundles numerically trivial over the relevant
birational contractions descend without an additional tensor power.
The latter fact preserves the Cartier indices of finitely many rational
nef adjoints and therefore the positive intersection gap used in the
perturbation argument. None of these auxiliary constructions changes
the initial model of the ordinary program.
\subsection{Proof strategy and organization}
Sections~\ref{sec:analytic}--\ref{sec:counts} establish the common
analytic, birational and termination tools. The proof of
Theorem~\ref{thm:finite} then has four stages.
\begin{enumerate}[label=\textup{\arabic*.}]
\item Fix a Kähler form on the original space and run ordinary steps
with its transformed generalized nef data as a scaling direction.
If there were infinitely many steps, marked-model finiteness would
force their scaling parameters to tend to zero.
\item Extract the stabilized finite set of low-discrepancy places.
On a fixed auxiliary model the boundaries decrease to an effective
ordinary terminal boundary.
\item Run a finite program for that limiting terminal pair and
express its nef adjoint as a positive combination of rational nef
adjoints. Their fixed Cartier indices give a uniform positive lower
bound for nonzero degrees on curves.
\item A sufficiently small generalized perturbation has a finite
program on which all these rational adjoints are crepant. At its
endpoint, descent to an earlier flip base makes the originally
negative adjoint trivial on its contracted ray, a contradiction.
\end{enumerate}
Section~\ref{sec:endpoints} identifies the endpoint and concludes the
finite-program proof. All extractions occur on auxiliary models;
the ordinary sequence itself begins on $X$.
Section~\ref{sec:analytic} constructs the trace and comparison tools,
and Section~\ref{sec:absolute} supplies ordinary steps and exact bundle
descent. Section~\ref{sec:counts} proves the terminal theorem before
it is used. Sections~\ref{sec:scaling} and~\ref{sec:perturbation}
carry out the four stages above, and Section~\ref{sec:endpoints}
identifies the endpoints. Appendix~\ref{sec:local-convention} gives
a smooth projective example showing why analytic local
$\Q$-factoriality cannot replace the global convention in the theorem.
Figure~\ref{fig:dependencies} records the dependencies of this argument.
\input{figures/dependencies}
==== END Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//01-introduction.tex ====
==== BEGIN Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//02-analytic.tex ====
\section{Classes, currents, and exceptional divisors}\label{sec:analytic}
The scaling argument transports one K\"ahler form from the original
space through several birational models. Its trace need not be smooth,
or even have local potentials without further argument. We establish
the precise trace relation, its positivity, and the comparison rules
used below. Equalities between Bott--Chern classes will be distinguished
from equalities between divisors or currents.
\subsection{Pullback of classes}
We use the following cohomological descent theorem. A space is in
\emph{Fujiki's class $\mathcal C$} if it is bimeromorphic to a compact
K\"ahler manifold.
\begin{lemma}[Pullback and descent]\label{lem:pullback-image}
Let $f:T\to Z$ be a proper surjective morphism with connected fibres between normal
compact analytic spaces with rational singularities. Suppose that $T$
is in class $\mathcal C$ and that either the general fibre is rationally
connected or $R^jf_*\cO_T=0$ for every $j>0$. Then
\[
f^*: \BC(Z)\longrightarrow\BC(T)
\]
is injective, and its image consists exactly of the classes having
degree zero on every $f$-vertical curve. If $T$ and $Z$ are K\"ahler,
a class on $Z$ is nef if and only if its pullback is nef.
\end{lemma}
\begin{proof}
The pullback assertion is \cite[Lemma~2.39]{HX2026}; nef descent is
\cite[Lemma~2.38]{DHP2024}. In particular, the first assertion applies
to any proper bimeromorphic morphism between compact spaces with
rational singularities whose source is in class $\mathcal C$: its
general fibre is a point. For a projective klt log Fano contraction,
relative vanishing supplies the higher-direct-image hypothesis once
rationality of the base has been established.
\end{proof}
\subsection{Positive currents on a normal model}
An $f$-exceptional real divisor is a finite real linear combination of
prime divisors whose images have codimension at least two.
We next give the descent statement needed when an exceptional
correction has either sign. Pseudo-effectivity requires a positive
current with local plurisubharmonic potentials; we do not assume
that an arbitrary positive current on a singular space has them.
\begin{lemma}[Exceptional descent]\label{lem:positive-descent}
Let $p:U\to T$ be a resolution of a normal compact K\"ahler space,
with $U$ compact K\"ahler. Let $\alpha\in\BC(T)$ and let $E$ be a
$p$-exceptional real divisor. If $p^*\alpha+[E]$ is pseudo-effective,
then $\alpha$ is pseudo-effective. No sign condition on $E$ is needed.
Pseudo-effective classes pull back to resolutions, and nef classes
on $T$ are pseudo-effective.
\end{lemma}
\begin{proof}
Choose a smooth representative $\theta$ of $\alpha$ with local smooth
potentials. On the smooth compact K\"ahler space $U$, a positive
representative of the given class has the form
\begin{equation}\label{eq:positive-upstairs}
S=p^*\theta+[E]+\ddc v
\end{equation}
for a global distribution $v$. Remove from $T$ its singular locus
and the image of the exceptional locus, obtaining a smooth open set
$T^\circ$ whose complement has codimension at least two. The map
$p$ is an isomorphism there. Equation~\eqref{eq:positive-upstairs}
gives a global $\theta$-plurisubharmonic function $v^\circ$ on
$T^\circ$. We do not assert that $v$ is quasi-plurisubharmonic along
$E$.
On a coordinate neighbourhood where $\theta=\ddc\rho$, extend
$\rho+v^\circ$ first across the removed codimension-two subset of
the regular locus, and then across the singular locus. The first
extension is the plurisubharmonic Hartogs theorem; the second is its
normal-space version of Grauert--Remmert, recalled in
\cite[Section~2.1, pp.~927--928]{CMM2017}. In the latter statement,
every plurisubharmonic function on the regular locus of a normal space
extends uniquely to the whole space; local upper boundedness at the
singular points is not a further hypothesis to be deduced from $v$.
We apply it only after obtaining a plurisubharmonic function on the
entire regular locus. These extensions are unique.
On overlaps their differences are the original smooth
pluriharmonic differences of the functions $\rho$. Subtracting
$\rho$ therefore gives a global potential $v_T$, and
\[
\theta+\ddc v_T\geq0
\]
is a current with local plurisubharmonic potentials representing
exactly $\alpha$. This proves descent, including for signed $E$.
For pullback, compose local plurisubharmonic potentials with the
resolution. They cannot become identically $-\infty$ on a nonempty
open subset, since a resolution is an isomorphism on a dense open
set. Thus they define the pulled-back positive current; see
\cite[Section~2.1, p.~928]{CMM2017}.
Finally, if $\alpha$ is nef, then $p^*\alpha$ is nef on $U$.
For a K\"ahler form $\omega_U$, choose positive representatives of
$p^*\alpha+\varepsilon[\omega_U]$. Their masses are bounded as
$\varepsilon\downarrow0$, so a weak limit is a positive current in
$p^*\alpha$. On the smooth K\"ahler manifold $U$ it has local
plurisubharmonic potentials. The first part, with $E=0$, descends
this current to the required class on $T$.
\end{proof}
\subsection{Generalized pairs and negativity}
We recall the generalized-pair language in the form needed here
\cite[Section~2.1 and Definition~2.7]{DHY2023}. A closed \emph{b-$(1,1)$
current} is a compatible collection of closed $(1,1)$-currents on
proper birational models: pushforward along a morphism between models
recovers the current on the lower model. It \emph{descends} to a
model $U$ if its trace there and all higher traces have local
potentials, and the classes of the higher traces are pullbacks of
its class on $U$.
It is \emph{b-nef} if that class is nef on some such model.
We also use the relative version for real combinations of line-bundle
data on a carrier projective over a specified base. Here nefness
means nonnegative degree on curves contracted to that base. The
structure equation and discrepancy definition below are unchanged.
A generalized pair on $T$ consists of a boundary $B\geq0$, a b-nef
datum $\mathbf M$, and a log resolution $\nu:U\to T$ on which the
datum descends, together with an actual real divisor $B_U$ such that
$\nu_*B_U=B$ and
\begin{equation}\label{eq:generalized-structure}
[K_U+B_U]+[\mathbf M_U]=\nu^*L,
\qquad L\in\BC(T).
\end{equation}
The support of $B_U$ may be taken to have simple normal crossings.
Negativity makes this structure boundary unique; passing to higher
resolutions gives the other structure boundaries. The generalized
log discrepancy of a prime $P$ on such a resolution is
\[
a(P,T,B+\mathbf M)=1-\operatorname{coeff}_P B_U.
\]
The pair is generalized klt, or \emph{gklt}, if all these discrepancies
are positive, and generalized log canonical, or \emph{glc}, if they
are all nonnegative. The \emph{gklt locus} is the open subset where
all discrepancies of primes centred there are positive; its
complement is the generalized non-klt locus. We use the word
\emph{terminal} for an ordinary or
generalized klt pair whose discrepancies at all exceptional primes
are strictly greater than one. Negative coefficients are permitted
in $B_U$, although the boundary $B$ on the model is effective.
Generalized klt spaces have rational singularities
\cite[Theorem~2.19]{DHY2023}.
The negativity lemma concerns actual divisors, not arbitrary
Bott--Chern classes. We record also the strict form needed for
fourfold termination.
\begin{lemma}[Negativity and strict comparison]\label{lem:negativity}
\leavevmode
\begin{enumerate}[label=\textup{(\roman*)}]
\item Let $f:T\to Z$ be a proper bimeromorphic morphism of normal
complex spaces and let $E$ be a real Cartier divisor such that $-E$
is $f$-nef. Then $E\geq0$ if and only if $f_*E\geq0$.
Consequently an $f$-exceptional, $f$-numerically trivial real Cartier
divisor is zero.
\item If, in addition, $f$ has connected projective fibres and
$E\geq0$ is $f$-anti-nef, then every fibre is either contained in
$\Supp E$ or disjoint from $\Supp E$.
\item Consider a nontrivial small birational diagram
\[
T\xrightarrow{\ f\ }Z\xleftarrow{\ f^+\ }T^+
\]
of normal compact spaces, with projective contractions on both
sides. Let $B\geq0$ be a real boundary, let $B^+$ be its strict
transform, and suppose the ordinary adjoints
$D=K_T+B$ and $D^+=K_{T^+}+B^+$ are real Cartier, with $-D$ relatively
ample on the left and $D^+$ relatively ample on the right.
The two exceptional loci have the same image $A\subset Z$. On a
common resolution $p:V\to T$, $q:V\to T^+$,
\begin{equation}\label{eq:ordinary-flip-comparison}
p^*D-q^*D^+=F,\qquad F\geq0,
\end{equation}
where $F$ is an actual divisor exceptional over both models.
For every prime divisor $P$ over these spaces,
\[
a(P,T^+,B^+)\geq a(P,T,B),
\]
and the inequality is strict if the centre of $P$ on either side
is contained in that side's exceptional locus.
\item For the diagram in \textup{(iii)}, if $\dim T=4$, then $\dim A\leq1$ and
\[
\dim\Exc(f)+\dim\Exc(f^+)\geq3.
\]
\item The discrepancy conclusions in \textup{(iii)} also hold for
generalized pairs with the same fixed b-nef datum and transformed
boundaries in a projective small diagram, provided the source log
class has negative degree on every curve contracted on that side,
and the target log class has positive degree on every curve
contracted on the other side. They hold likewise for a projective divisorial
contraction $f:T\to T'$ when the target boundary is the pushforward,
the b-datum is unchanged, and the source log class has negative
degree on every $f$-vertical curve. In this divisorial case strictness holds at every prime
whose source centre lies in $\Exc(f)$.
This part also applies when the fixed b-datum is nef only over a
specified base.
\end{enumerate}
\end{lemma}
\begin{proof}
Part~(i) is the analytic negativity lemma
\cite[Lemma~2.5]{HX2026}; apply it to both signs for the last assertion.
For (ii), points of a connected projective fibre can be joined by a
chain of irreducible curves. If the fibre met both $\Supp E$ and
its complement, some curve in such a chain would meet the support
without being contained in it. Its intersection with the effective
real Cartier divisor would be strictly positive, contrary to
$f$-anti-nefness. The assertion for a real Cartier divisor follows
locally from positive combinations of effective rational Cartier
divisors, as explained below.
For (iii), let $G$ be the normalization of the graph in
$T\times_ZT^+$, with projections $p_G,q_G$ and map $h:G\to Z$.
Compatible canonical data give the actual real Cartier divisor
\[
F_G=p_G^*D-q_G^*D^+.
\]
This difference is defined locally by compatible meromorphic canonical
generators and therefore glues, even when the canonical sheaves were
specified as $\Q$-line bundles. Smallness makes it exceptional over
both sides. It is effective by (i), applied over $T$: on a
$p_G$-vertical curve the degree of $-F_G$ is the degree of
$q_G^*D^+$, which is nonnegative.
For every $h$-vertical curve $C$ one has
\begin{equation}\label{eq:graph-strict-degree}
F_G\cdot C<0.
\end{equation}
Indeed, normalization is finite onto the graph, so the two projections
cannot both contract $C$. The nonzero degrees contributed by $p_G^*D$
and $-q_G^*D^+$ are both negative. An effective real Cartier divisor
has nonnegative degree on a curve outside its support. For completeness,
this fact and its strict pullback analogue hold even if its individual
prime components are not $\Q$-Cartier: locally express it in the real
span of finitely many Cartier divisors. Vanishing of coefficients
outside its support and nonnegativity of the remaining coefficients
are rational linear conditions. The resulting rational polyhedral
cone expresses it as a positive real combination of effective
$\Q$-Cartier divisors. The usual effective Cartier assertions apply
to each term; summing the nonnegative local intersection numbers
gives the asserted global degree inequality. In particular, the pullback has positive coefficient
at every prime whose centre is contained in the support.
The morphism $p_G$ has connected fibres because $T$ is normal;
hence $h=f p_G$ has connected fibres. Its fibres are projective
because $G$ is finite over the graph in the fibre product. Each
point of a positive-dimensional fibre lies on a curve in that fibre. Equation
\eqref{eq:graph-strict-degree} therefore places the entire fibre in
$\Supp F_G$. To identify the relevant fibres, suppose $f$ is an
isomorphism over an open subset of $Z$. There the log divisor on
$T^+$ is pulled back from $Z$, by smallness and agreement in codimension
one. Relative ampleness excludes an $f^+$-vertical curve there;
projectivity, connectedness and normality then make $f^+$ an
isomorphism there as well. Interchanging the sides proves that their
exceptional images coincide. Thus $h^{-1}(A)\subset\Supp F_G$.
Pull $F_G$ back to a resolution on which $P$ appears. Its coefficient
at $P$ is exactly
$a(P,T^+,B^+)-a(P,T,B)$, with our log-discrepancy convention.
Effectivity gives monotonicity, and the support assertion gives the
claimed strictness. This proves (iii).
For (iv), a small exceptional locus in a fourfold has dimension at
most two and has positive-dimensional fibres, so its image has
dimension at most one. Moreover $F_G\ne0$ by
\eqref{eq:graph-strict-degree}. A component of its support has
dimension three and maps finitely into
$\Exc(f)\times_Z\Exc(f^+)$. The dimension of this product is at most
the sum of the dimensions of its two factors. This proves (iv).
For (v), choose a common smooth resolution $p:V\to T$,
$q:V\to T'$ carrying the fixed b-datum and projective over the
common contraction base. Such a resolution is obtained by taking
projective modifications of the graph that resolve the maps to a
carrier of the datum. Subtract the actual
structure boundaries to obtain a real Cartier divisor
$F=B_{T,V}-B_{T',V}$. Its coefficients are the differences of
generalized log discrepancies, and cancellation of the fixed
b-part gives
\[
[F]=p^*L_T-q^*L_{T'}.
\]
Let $h:V\to Z$ be the morphism to the common contraction base;
in the divisorial case $Z=T'$ and $q=h$. It is projective by this
choice and has connected fibres because it is bimeromorphic onto
the normal space $Z$. The boundary-pushforward
condition makes $F$ exceptional over $Z$. The relative signs of
the log classes give $F\cdot C\leq0$ on every $h$-vertical curve,
with strict inequality whenever $p(C)$ or, in the small case,
$q(C)$ is a curve. Thus (i) gives $F\geq0$.
If $y$ lies in an exceptional image, a vertical curve downstairs
can be lifted to a curve on $V$ dominating it: use a component of
its projective inverse image and cut by relatively ample
hyperplanes. Such a curve has negative $F$-degree and hence lies
in $\Supp F$. Part~(ii) now puts the entire fibre $h^{-1}(y)$ in
the support. Pullback to a higher resolution has positive
coefficient at every prime centred in this support. This proves
all the strict discrepancy inequalities, including for a centre
on the flipped side. No trace current on the normalized graph
is assumed to be Cartier; the actual divisor is constructed from
the structure boundaries on the smooth resolution.
\end{proof}
\subsection{A fixed nef b-part and its traces}
For the finite-program proof, fix a K\"ahler form $h$ on the original
space $X$, and set $H=[h]$. It defines a positive b-current
$\mathbf H$: for a marked birational model $T$, choose a common
resolution $p:V\to T$, $q:V\to X$ and put
\begin{equation}\label{eq:actual-H-trace}
\mathbf H_T=p_*q^*h.
\end{equation}
Compatibility on higher resolutions makes this independent of the
choice of resolution \cite[Claim~2.5]{DHY2023}. Throughout the proof
this is one fixed current datum; we do not replace it by arbitrary
cohomologous currents. The class of the trace, when it exists, will
be denoted by $H_T$.
\begin{lemma}[The exceptional trace relation]\label{lem:class-trace}
Let $T$ be a normal compact K\"ahler space with rational singularities,
marked birationally to $X$. Suppose that on a common smooth K\"ahler
resolution $p:V\to T$, $q:V\to X$ there are a class $H_T\in\BC(T)$
and an actual $p$-exceptional real divisor $J_T$ satisfying
\begin{equation}\label{eq:class-trace}
p^*H_T=q^*H+[J_T].
\end{equation}
Then $J_T\geq0$. The class $H_T$ and the divisor $J_T$ are unique,
and the correction on a higher resolution is the pullback of $J_T$.
For an effective boundary $B_T$ with real Cartier ordinary adjoint,
the generalized structure boundary for the nef datum $t\mathbf H$ is
\begin{equation}\label{eq:trace-boundary}
B_{T,V}(t)=p^*(K_T+B_T)-K_V+tJ_T,\qquad t\geq0.
\end{equation}
In particular, decreasing $t$ increases every generalized log
discrepancy. If the generalized pair is gklt, its ordinary pair is
klt; if all its exceptional log discrepancies exceed one, the
ordinary pair is terminal.
Conversely, suppose $t>0$ and a generalized pair with boundary $B_T$
and this fixed datum $t\mathbf H$ is given by
\eqref{eq:generalized-structure}, with log class $L$ and real Cartier
ordinary adjoint. Then \eqref{eq:class-trace} holds with
\[
H_T=\frac{L-[K_T+B_T]}{t}.
\]
\end{lemma}
\begin{proof}
On every $p$-vertical curve, $-J_T$ has the same degree as the nef
class $q^*H$. Lemma~\ref{lem:negativity}(i) gives $J_T\geq0$.
Subtract two possible relations. The difference of the correction
divisors is $p$-exceptional and numerically trivial over $T$, hence
zero. Pullback injectivity from Lemma~\ref{lem:pullback-image} then
gives uniqueness of the class as well.
Pulling back \eqref{eq:class-trace} to a higher resolution gives the
claimed correction there by uniqueness. Conversely, if the relation
is initially known on a higher resolution $r:V'\to V$, push its
correction down to $V$. The difference between the upstairs
correction and the pullback of this pushforward is $r$-exceptional
and numerically trivial over $V$, so is zero. Injectivity of $r^*$
gives the relation on $V$.
Equation~\eqref{eq:trace-boundary} follows by adding $K_V$ and the
fixed nef class $tq^*H$ to its two sides. Its coefficients give
\[
a(P,T,B_T+t\mathbf H)
=a(P,T,B_T)-t\operatorname{coeff}_P J_T
\]
on every sufficiently high resolution. This proves the discrepancy
assertions.
For the converse, let $B_V$ be the given actual generalized structure
boundary and put $B_V^{\rm ord}=p^*(K_T+B_T)-K_V$. Both push forward
to $B_T$, so
\[
J_T=\frac{B_V-B_V^{\rm ord}}{t}
\]
is an actual $p$-exceptional divisor. Subtract the ordinary log class
from \eqref{eq:generalized-structure} to obtain
\eqref{eq:class-trace}. This constructs the divisor before applying
negativity; it does not infer a divisor from an arbitrary class
difference.
\end{proof}
The converse is particularly useful for an extraction carrying a
pulled-back generalized log class. It shows that the divided
difference defining its trace is the same class whenever the marked
model is the same, even if its boundary was obtained at a different
scaling parameter.
\begin{corollary}[Positivity of the trace]\label{cor:big-trace}
Under the hypotheses of Lemma~\ref{lem:class-trace}, the actual trace
$\mathbf H_T$ has local plurisubharmonic potentials and represents
$H_T$. The class $H_T$ is big.
\end{corollary}
\begin{proof}
The positive current $q^*h+[J_T]$ represents $p^*H_T$.
The proof of Lemma~\ref{lem:positive-descent} constructs a positive
current $S_T$ in $H_T$ with local potentials. On the big open where
$p$ is an isomorphism and the correction is absent, it agrees with
$p_*q^*h$. Their difference is a closed current of order zero
supported in codimension at least two. Such a current vanishes:
after a local embedding it has bidimension
$(\dim T-1,\dim T-1)$, whereas its support has smaller dimension
\cite[Chapter~III, Corollary~2.11]{DemaillyCADG2012}.
Thus $S_T$ is precisely the actual trace in
\eqref{eq:actual-H-trace}. This step is needed because a trace of
a positive b-current need not have local potentials a priori
\cite[Remark~2.6(ii)]{DHY2023}.
The class $q^*H$ is nef and has positive top self-intersection:
$q^*h$ is semipositive and is positive on the dense open where $q$
is an isomorphism. By \cite[Theorem~0.5]{DemaillyPaun2004}, it contains
a K\"ahler current $R\geq\delta\omega_V$ for some $\delta>0$ and
some K\"ahler form $\omega_V$. Fix a K\"ahler form $\omega_T$ and
choose $C>0$ with $p^*\omega_T\leq C\omega_V$. For
$0<\varepsilon<\delta/C$,
\[
R+[J_T]-\varepsilon p^*\omega_T\geq0.
\]
Its class is $p^*(H_T-\varepsilon[\omega_T])$.
Lemma~\ref{lem:positive-descent} shows that
$H_T-\varepsilon[\omega_T]$ is pseudo-effective. Hence $H_T$
contains a K\"ahler current, as required.
\end{proof}
\subsection{Comparison with the same nef datum}
The preceding results supply the positivity needed for scaling. We
finish with the comparison that later excludes divisorial steps in
the limiting program and identifies the two nef pullbacks in the
final contradiction.
\begin{lemma}[Small-model comparison]\label{lem:small-comparison}
Let $(T,B_T+\mathbf M)$ and $(T',B_{T'}+\mathbf M)$ be generalized
pairs with the same b-nef datum, whose log classes are $L_T$ and
$L_{T'}$. Suppose the marked birational map $T\dashrightarrow T'$
extracts no divisors and $B_{T'}$ is the pushforward of $B_T$.
On a common sufficiently high resolution $p:V\to T$, $q:V\to T'$, put
\[
F=B_{T,V}-B_{T',V}.
\]
Then $F$ is an actual divisor exceptional over $T'$, and
\begin{equation}\label{eq:small-comparison}
p^*L_T-q^*L_{T'}=[F],\qquad
\operatorname{coeff}_P F
=a(P,T',B_{T'}+\mathbf M)-a(P,T,B_T+\mathbf M).
\end{equation}
If the map is small, $F$ is exceptional over both models. In this
case nefness of $L_{T'}$ implies $F\geq0$, and nefness of both log
classes implies $F=0$. These conclusions also hold relatively over
a common base. Even without smallness, equality of the pullback classes
in \eqref{eq:small-comparison} implies equality of the structure boundaries.
\end{lemma}
\begin{proof}
Subtract the two structure equations on $V$. The trace of the fixed
b-datum cancels, giving \eqref{eq:small-comparison}. At a prime
dominating a divisor on $T'$, the corresponding boundary coefficient
on $T$ is unchanged, so its coefficient in $F$ is zero. This proves
exceptionality over $T'$. Smallness gives the same assertion over
$T$.
If $L_{T'}$ is nef, then $-F$ is $p$-nef, because its degree on a
$p$-vertical curve is the degree of $q^*L_{T'}$. Negativity gives
$F\geq0$. If $L_T$ is also nef, then $F$ is $q$-nef, so applying
negativity with the opposite sign gives $F\leq0$. The same proof
uses only relative nefness when the models lie over a common base.
For equal pullback classes, the already constructed exceptional
divisor is relatively numerically trivial, and is zero by
Lemma~\ref{lem:negativity}(i).
\end{proof}
All equalities of pulled-back log classes in this lemma are
cohomological. Equality of actual currents was used only in the
separate trace-identification argument. These distinctions allow
the ordinary steps and the generalized auxiliary constructions to
share one fixed nef datum.
==== END Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//02-analytic.tex ====
==== BEGIN Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//03-steps.tex ====
\section{Ordinary steps on the given space}\label{sec:absolute}
We construct the contraction and flip of every negative analytic ray in
arbitrary dimension. The input is an ordinary rational klt pair in the
\emph{global Weil-divisor} category. Generalized pairs enter only through
the cone and base-point-free theorems; the steps themselves are ordinary.
The main point is to preserve actual line bundles, including their fixed
Cartier indices, rather than only numerical classes.
\subsection{Integral descent}
Bundles are written additively in numerical expressions. For example,
$L-(K_T+C)$ denotes the rational line bundle obtained from $L$ and the
rational adjoint. All degrees in the following lemma are degrees on
compact curves contracted by the indicated morphism.
\begin{lemma}[Integral descent]\label{lem:descent}
Let $f:T\to Z$ be a projective bimeromorphic contraction of normal complex
spaces, and let $L$ be a line bundle numerically trivial over $Z$.
Suppose that every point of $Z$ has a neighbourhood $U$ on which there is
an ordinary rational klt pair $(f^{-1}U,C_U)$ with rational adjoint and
\[
L-(K_{f^{-1}U}+C_U)\quad\text{$f$-nef over $U$}.
\]
Then $f_*L$ is a line bundle and the evaluation map is an isomorphism
\begin{equation}\label{eq:integral-descent}
f^*f_*L\simeq L.
\end{equation}
In particular the conclusion applies if the local ordinary adjoints are
relatively antiample. It also applies to a global ordinary rational klt
pair $(T,C)$ whenever $L-(K_T+C)$ is $f$-nef. Clearing one denominator
therefore descends a rational line bundle as an actual rational line
bundle.
\end{lemma}
\begin{proof}
Fix $z\in Z$ and shrink to a connected Stein neighbourhood $U$ on which
the stated pair is defined. A line bundle $M$ on $f^{-1}U$ has a
Cartier-divisor representative there: the coherent direct image $f_*M$
has rank one over the nonempty open where $f$ is an isomorphism, so
Cartan's Theorem~A supplies a nonzero section. Its pullback is a section
of $M$ that is not identically zero, and its zero divisor is Cartier.
This argument takes place over $U$, not on the whole compact space.
Every rational line bundle $N$ on $f^{-1}U$ is also relatively big.
Indeed, choose $q>0$ with $qN$ a line bundle and an $f$-ample line bundle
$A$. The same direct-image argument gives a nonzero section of
$qN-A$, hence
\[
N\simq q^{-1}A+q^{-1}E\qquad(E\geq0).
\]
This is relative bigness in the analytic sense
\cite[Definition~2.46]{Fujino2022}. Apply it to
$N=L-(K_T+C_U)$. The klt base-point-free theorem gives, after shrinking
around $z$, relative generation of $L^k$ for \emph{every} sufficiently
large integer $k$ \cite[Theorem~6.2 and Remark~6.3]{Fujino2022}.
Choose finitely many such generating sections near the compact fibre
$F=f^{-1}(z)$. The morphism they define on the projective reduced fibre
$F_{\mathrm{red}}$ is constant on each irreducible component: a
positive-dimensional image would yield a curve with positive degree
against the pullback of the hyperplane bundle, contrary to the
numerical triviality of $L^k$. Such a curve is obtained by successive
general hyperplane sections of a component. Connectedness of $F$ makes
the images of all its components the same point. A linear combination
of the generating sections is consequently nonzero at every point of
$F_{\mathrm{red}}$.
This section is a generator of $L^k$ in the local ring at every point of
$F$: its residue is nonzero, and hence its representing function is a
unit. This also treats a nonreduced fibre. Properness allows us to
shrink $U$ away from the image of the section's zero set. Thus $L^k$ is
trivial on $f^{-1}U$. Apply the argument to consecutive integers $k$ and
$k+1$, and use one common neighbourhood. Their quotient trivializes
$L$ itself. Since $f_*\cO_T=\cO_Z$, on this neighbourhood $f_*L\simeq
\cO_U$, and evaluation is an isomorphism. These local conclusions give
\eqref{eq:integral-descent} globally. No additional tensor power enters
the integral conclusion.
\end{proof}
The nef variant is important even when the adjoint has degree zero over
$f$. For example, if $P=K_T+C$ is a rational klt adjoint, $\ell P$ is
Cartier, and $P$ is numerically trivial over $f$, apply
Lemma~\ref{lem:descent} to $L=\ell P$. Its difference from the adjoint
is $(\ell-1)P$, so the \emph{same} index $\ell$ descends.
\subsection{Supporting classes and projectivity}
A class is \emph{modified big} if it is the birational pushforward of a big class
\cite[Definition~2.8]{HX2026}. In particular, a big class is modified big.
We use the following consequences of the cone and transcendental
base-point-free theorems \cite[Theorems~1.3 and~1.4]{HX2026}. For a
compact K\"ahler gklt pair with a descending nef part $\beta$, the cone
$\NA(T)$ is the sum of its adjoint-nonnegative part and rays of rational
curves. Only finitely many rays are needed when the boundary plus
$\beta$ is big. If its adjoint class $\alpha$ is nef and the boundary
plus $\beta$ is modified big, there is a Moishezon contraction
$f:T\to Z$ to a normal compact K\"ahler space with $\alpha=f^*\gamma$
for a K\"ahler class $\gamma$ on $Z$. Neither assertion requires strong
factoriality. Projectivity in our weaker category will be proved from
the ordinary rational adjoint.
\begin{lemma}[Supports with a K\"ahler margin]\label{lem:absolute-support}
Let $(T,B)$ be an ordinary rational klt pair on a normal compact K\"ahler
space. Assume that $B$ is $\Q$-Cartier and that $D=K_T+B$ is a rational
line bundle. If $D$ is not nef, $\NA(T)$ has a $D$-negative extremal ray.
Every such ray $R$ is generated by a rational curve and has a nonempty
relatively open set $\mathcal U_R\subset R^\perp$ of classes satisfying
\begin{equation}\label{eq:absolute-support}
\alpha\text{ is nef},\qquad
\NA(T)\cap\alpha^\perp=R,\qquad
\alpha-c_1(D)\text{ is K\"ahler}.
\end{equation}
\end{lemma}
\begin{proof}
Fix a K\"ahler class $\omega$ and write $d=c_1(D)$. Cone duality applies
because klt singularities are rational. The slice
\[
S=\{v\in\NA(T):\omega\cdot v=1\}
\]
is compact. Indeed, fix smooth representatives of a basis of $\BC(T)$.
Each is bounded above and below by a multiple of a K\"ahler form, so all
its pairings on $S$ are bounded; the slice is closed. If $D$ is not nef,
the minimum of $d$ on $S$ is negative. An extreme point of its minimum
face gives a negative extremal ray.
Fix any such ray and let $r\in S$ be its normalized point. Choose
$\varepsilon>0$ with $(d+\varepsilon\omega)\cdot r<0$. The descending
nef part $\varepsilon\omega$ changes no discrepancy, and
$B+\varepsilon\omega$ is big since $B$ is effective and $\Q$-Cartier.
The finite form of the cone theorem gives
\begin{equation}\label{eq:absolute-finite-cone}
\NA(T)=\NA(T)_{d+\varepsilon\omega\geq0}
+\sum_{j=1}^{N}\R_{\geq0}[C_j],
\end{equation}
where the $C_j$ are rational curves. Normalize their classes in $S$
and discard repetitions. By extremality, $r$ is one of these points,
say $r_1$. Let $K$ be the convex hull of the others and of
$S\cap\{d+\varepsilon\omega\geq0\}$. This is compact and excludes
$r$; otherwise extremality of $r$ would fail.
If $K$ is nonempty, strict separation gives a linear functional $h$
with $h(r)=0$ and $h>0$ on $K$, after subtracting a multiple of $\omega$.
Regard it as a Bott--Chern class by duality. For all sufficiently small
$\delta>0$, $h-\delta d$ is strictly positive on $K$ and at $r$.
Since $S=\operatorname{conv}(\{r\}\cup K)$, it is strictly positive on
$S$, hence K\"ahler. Thus $\alpha=h/\delta$ has the required properties.
Its positive minimum on $K$ persists under small perturbations in
$R^\perp$; openness of the K\"ahler cone also preserves
$\alpha-d$. These perturbations give $\mathcal U_R$. If $K$ is empty,
then $S=\{r\}$ and $-d$ is K\"ahler; a small neighbourhood of $0$ in
$R^\perp$ has the required properties. This includes the case
$R^\perp=\{0\}$.
\end{proof}
\begin{lemma}[Relative positivity]\label{lem:relative-positivity}
Let $f:T\to Z$ be a proper morphism of compact complex spaces, with $T$
and $Z$ K\"ahler. Suppose that $L$ is a rational line bundle and
\[
c_1(L)=\kappa-f^*\gamma,
\]
where $\kappa$ is K\"ahler on $T$ and $\gamma$ is a smooth real
Bott--Chern class on $Z$. Then $L$ is $f$-ample and $f$ is projective.
Conversely, a normal space projective over a compact K\"ahler space is
compact K\"ahler if it has a global relatively ample line bundle.
\end{lemma}
\begin{proof}
Clear the denominator of $L$ and choose any smooth Hermitian metric
on that line bundle. The Bott--Chern identity says that the difference
between its curvature and the indicated difference of forms is
$\ddc$ of a global smooth function. Adjusting the metric by the
exponential of that function gives the required curvature, with the
same denominator restored at the end. Smooth functions and metrics
here have their normal-space meaning in local embeddings.
On every fibre its curvature is positive. Positivity of a line bundle
on a compact complex space implies ampleness, and for a proper analytic
map ampleness on every fibre is equivalent to relative ampleness
\cite[Corollary~1.12 and Definition~3.1--Remark~3.2]{FujinoVanishing2026}.
This criterion does not presuppose projectivity and applies to
nonreduced fibres as well. Existence of the relatively ample line bundle
makes $f$ projective.
For the converse, local relative embeddings give metrics on a relatively
ample bundle with positive curvature in fibre directions. Glue their
weights by a partition of unity on the base. The resulting metric
retains that positivity. Add a sufficiently large multiple of a base
K\"ahler form to its curvature. The sum is strictly positive; compactness
allows one multiple to work everywhere.
\end{proof}
\subsection{The ordinary contraction and flip}
We can now perform each step and preserve the category. The next
proposition records both the numerical ranks of the negative contraction
and the actual canonical algebra of a flip. This description identifies the transformed adjoint as an actual
rational line bundle.
\begin{proposition}[Ordinary negative steps]\label{prop:ordinary-step}
Let $(T,B)$ be an ordinary rational klt pair on a normal compact K\"ahler
space, globally $\Q$-factorial in the Weil-divisor sense, with canonical
sheaf a rational line bundle. Put $D=K_T+B$. Every $D$-negative
extremal ray $R\subset\NA(T)$ has a projective contraction $f:T\to Z$
with connected fibres and normal compact K\"ahler target, such that
\begin{equation}\label{eq:ordinary-negative-bc}
f^*\BC(Z)=R^\perp,\qquad
\rho(T/Z)=1,\qquad -D\text{ is }f\text{-ample}.
\end{equation}
Here $\rho$ is the rank of global line-bundle degrees on contracted
curves. The target has rational singularities and
$R^j f_*\cO_T=0$ for $j>0$. If $\dim Z<\dim T$, this is a Mori fibre
space. Otherwise $f$ is bimeromorphic, and precisely one of the
following occurs.
\begin{enumerate}[label=\textup{(\roman*)}]
\item The exceptional locus is one prime divisor $E$. Put $T'=Z$,
$B'=f_*B$ and $D'=K_{T'}+B'$. Then
\begin{equation}\label{eq:ordinary-divisorial-difference}
D=f^*D'+eE\qquad(e>0).
\end{equation}
\item The contraction is small. For an adjoint Cartier index $r>0$,
the intrinsic algebra
\begin{equation}\label{eq:ordinary-flip-algebra}
\cR_r=\bigoplus_{m\geq0}f_*\cO_T(mrD)
\end{equation}
is locally finitely generated, and $T'=\Proj_Z\cR_r$ is the ordinary
flip. It has a small projective morphism $f^+:T'\to Z$ with connected
fibres. Put $B'=\phi_*B$ and $D'=K_{T'}+B'$, where
$\phi:T\dashrightarrow T'$ is the induced small map. For a sufficiently
divisible choice of $r$ there is an actual isomorphism
\begin{equation}\label{eq:ordinary-tautological}
\cO_{\Proj_Z\cR_r}(1)\simeq\cO_{T'}(rD').
\end{equation}
In particular $D'$ is a rational line bundle and is $f^+$-ample.
Passing to a sufficiently divisible Veronese leaves $T'$ unchanged.
\end{enumerate}
In both birational cases, $T'$ is compact K\"ahler and globally
Weil-divisor $\Q$-factorial, its canonical sheaf is a rational line
bundle, and $(T',B')$ is klt. Strong global $\Q$-factoriality is
preserved when imposed on $T$. A supplied global canonical Weil divisor
is transported to a canonical Weil divisor on $T'$. No prime divisor is
extracted. All log discrepancies weakly increase; the increase is strict
for every original prime contracted by the step, and, for a flip, for
every place centred in either exceptional locus.
\end{proposition}
\begin{proof}
Choose $\alpha\in\mathcal U_R$ from Lemma~\ref{lem:absolute-support}
and put $\kappa=\alpha-c_1(D)$. Insert the descending nef part
$\kappa$ into the ordinary pair. It changes no discrepancy, while
$B+\kappa$ is big. Transcendental base-point-freeness gives
$f:T\to Z$ and $\gamma$ K\"ahler with $\alpha=f^*\gamma$.
Thus $c_1(-D)=\kappa-f^*\gamma$, and
Lemma~\ref{lem:relative-positivity} proves projectivity and relative
ampleness. This establishes projectivity without strong factoriality.
For every curve $C\subset T$,
\begin{equation}\label{eq:ordinary-contracted-curves}
f(C)\text{ is a point}\quad\Longleftrightarrow\quad
\alpha\cdot C=0\quad\Longleftrightarrow\quad[C]\in R.
\end{equation}
The rational curve generating $R$ is contracted, so $f$ is nontrivial.
Relative vanishing gives $R^j f_*\cO_T=0$ for $j>0$
\cite[Theorem~5.2]{Fujino2022}; the log Fano base has rational
singularities \cite[Lemma~8.8]{DHP2024}. Lemma~\ref{lem:pullback-image}
now identifies $f^*\BC(Z)$ with the classes of degree zero on all
vertical curves, which is exactly $R^\perp$. The global line-bundle
rank is also one, since all those curves are proportional and the
rational line bundle $D$ detects their ray. This proves
\eqref{eq:ordinary-negative-bc}. The lower-dimensional case has all
the stated Mori fibre-space properties. In equal dimension,
connectedness gives generic degree one, so $f$ is bimeromorphic.
\smallskip\noindent
\emph{Divisorial contractions.}
Suppose $E$ is an exceptional prime. It is $\Q$-Cartier. If
$E\cdot R\geq0$, it would be $f$-nef, contradicting negativity for a
nonzero effective exceptional divisor. Thus $E\cdot R<0$. Every
vertical curve lies in $E$, since an effective $\Q$-Cartier divisor
has nonnegative degree on a curve outside its support. Every
positive-dimensional projective fibre is covered by curves. A
zero-dimensional local component of a connected fibre would be isolated,
and the birational form of Zariski's main theorem would make $f$ an
isomorphism there. Consequently $\Exc(f)=E$.
For a prime Weil divisor $A_Z$ on $Z$, let $A_T$ be its strict transform.
Choose $t\in\Q$ so $(A_T+tE)\cdot R=0$. The coefficient is rational
by taking the ratio of the two rational degrees on one integral curve.
A Cartier multiple of $A_T+tE$ descends by Lemma~\ref{lem:descent}:
its difference from $D$ is relatively ample. The descended line bundle
agrees with the corresponding reflexive power of $\cO_Z(A_Z)$ off
$f(E)$, a subset of codimension at least two. Reflexivity identifies
them everywhere. Hence every global prime on $Z$ is $\Q$-Cartier.
If a global canonical Weil divisor is given, push it forward. It agrees
with the canonical sheaf on the big open where $f$ is an isomorphism, so
reflexivity shows that it remains a canonical Weil representative.
Alternatively, start with an invertible reflexive power of $\omega_T$,
adjust it by a rational multiple of $E$ to degree zero, and descend after
clearing denominators. On the same big open the result is a reflexive
power of $\omega_Z$; hence that power is invertible everywhere. This
proves the canonical assertion without assuming a global meromorphic
canonical section.
Now $D'$ is a rational line bundle and its canonical identification away
from $E$ gives \eqref{eq:ordinary-divisorial-difference}. Intersection
with $R$ gives $e>0$. If $b_E$ is the coefficient of $E$ in $B$, then
\[
a(E;Z,B')=1-b_E+e>1-b_E=a(E;T,B)>0.
\]
Pullback of the effective divisor $eE$ gives weak increase for all other
log discrepancies. Thus the target pair is klt.
\smallskip\noindent
\emph{The small canonical model.}
Suppose $f$ is small. Finite generation of the ordinary relative log
canonical ring \cite[Theorem~1.18]{Fujino2022} gives local finite
generation of \eqref{eq:ordinary-flip-algebra}. When canonical data are
written as sheaves, this application can be made over each Stein open
in $Z$: nonzero sections of rank-one proper direct images provide the
meromorphic representatives there. Changing representatives induces
compatible graded isomorphisms of the intrinsic algebra.
The ordinary analytic flip theorem \cite[Theorem~1.14]{Fujino2022}
constructs precisely its relative canonical model. It has no local
factoriality hypothesis. Its proof glues the unique local log canonical
models, giving a normal small projective $f^+:T'\to Z$ globally. The
fibres are connected because $Z$ is normal. A finite cover of the
compact base permits one common sufficiently divisible Veronese
generated in degree one. Replace $r$ by that multiple. The resulting
tautological bundle is relatively ample and agrees with
$\cO_T(rD)$ on the common big open. Its reflexive extension is
$\cO_{T'}(rD')$, proving \eqref{eq:ordinary-tautological} as an actual
bundle isomorphism. This also proves that $D'$ is a rational line
bundle. The relative Proj is invariant under this Veronese operation.
By Lemma~\ref{lem:relative-positivity}, $T'$ is compact K\"ahler.
For a global prime $A'$ on $T'$, transform it to $A$ on $T$ and choose
$t\in\Q$ so $(A+tD)\cdot R=0$. Descend a Cartier multiple by
Lemma~\ref{lem:descent}. On $T'$ its pullback agrees on the common big
open with the same multiple of $A'+tD'$. Reflexivity extends the
identification; since $D'$ is already a rational line bundle, $A'$ is
$\Q$-Cartier. This proves global Weil-divisor factoriality. A given
canonical Weil divisor is transported on that common open and then
reflexively. In the sheaf formulation, repeat the descent argument with
an invertible power of $\omega_T$ adjusted by $tD$; undoing the twist
by $tD'$ proves the rational-line-bundle property of $\omega_{T'}$.
On a common smooth resolution $p:V\to T$, $q:V\to T'$,
Lemma~\ref{lem:negativity} gives the actual discrepancy divisor
\begin{equation}\label{eq:ordinary-small-difference}
p^*D=q^*D'+F,\qquad F\geq0,
\end{equation}
exceptional over both sides, with the asserted strictness over the
flipping and flipped loci. This also proves klt preservation.
\smallskip\noindent
\emph{Optional strong factoriality.}
In the divisorial case, take any coherent rank-one reflexive sheaf
$\cF$ on $Z$. Its reflexive pullback to $T$ is coherent of rank one.
Strong factoriality supplies an invertible reflexive power. Adjust it
by a rational multiple of $E$, descend as above, and compare off $f(E)$.
A reflexive power of $\cF$ is therefore a line bundle.
In the small case start with such a sheaf $\cF'$ on $T'$. On a common
smooth resolution put
\[
\cM=(q^*\cF'/\text{torsion})^{**},\qquad
\cF=(p_*\cM)^{**}.
\]
Here $\cM$ is a line bundle and $\cF$ is coherent reflexive of rank one;
both give the required transform on the common big open. An invertible
power of $\cF$, adjusted by a rational multiple of $D$, descends.
Pull back the descended bundle and undo the twist by $D'$. After
clearing denominators, reflexivity identifies the resulting line bundle
with a reflexive power of $\cF'$. Thus the strong condition is
preserved. Only sheaves defined on the whole compact spaces have been
used.
\end{proof}
\subsection{Transport of Bott--Chern classes}
For later scaling we need to transport one fixed nef datum. The following
construction uses the rank of the negative contraction only. It makes
no assertion that all Bott--Chern classes on a flipped space come from
the preceding space.
\begin{lemma}[Class transport]\label{lem:class-transport}
For a birational step of Proposition~\ref{prop:ordinary-step}, let
$f':T'\to Z$ mean $f^+$ in the small case and $\id_Z$ in the divisorial
case. Every $\theta\in\BC(T)$ has a unique expression
$\theta=f^*\eta+s\,c_1(D)$, with $\eta\in\BC(Z)$ and $s\in\R$.
Define
\begin{equation}\label{eq:class-transport}
\theta'= (f')^*\eta+s\,c_1(D').
\end{equation}
On a common resolution, $p^*\theta-q^*\theta'$ is the class of an
actual divisor exceptional over $T'$. This rule sends the class of a
global Weil divisor to that of its transform or pushforward. Moreover,
it propagates the fixed-nef-part trace relation of
Lemma~\ref{lem:class-trace}, with an actual effective exceptional
correction on the new model. The same conclusions hold for ordinary
real-boundary flips on strongly globally $\Q$-factorial compact K\"ahler
spaces constructed by \cite[Proposition~4.2]{HX2026}.
\end{lemma}
\begin{proof}
Since $D\cdot R\ne0$, subtracting a unique multiple of $c_1(D)$ makes
$\theta$ vanish on $R$. Equation~\eqref{eq:ordinary-negative-bc} and
pullback injectivity give the unique $\eta$. On a common resolution
over $Z$ the two pullbacks of $\eta$ agree. Therefore
\[
p^*\theta-q^*\theta'
=s\,[p^*D-q^*D'].
\]
The expression in brackets is the actual exceptional discrepancy
divisor already constructed in the proof of the proposition.
If $\theta=c_1(A)$ for a global Weil divisor $A$, the difference
$p^*A-q^*A'$ is likewise exceptional over $T'$, where $A'$ is its
transform or pushforward. Subtracting the two identities represents
$q^*(\theta'-c_1(A'))$ by a $q$-exceptional divisor. That divisor is
numerically trivial over $q$, so negativity in both signs makes it
zero. Pullback injectivity gives $\theta'=c_1(A')$.
Finally suppose the old trace $H_T$ of a fixed nef class on an earlier
model has, on a common high resolution, the relation
\[
p^*H_T=p_0^*H+[J_T].
\]
If $H_T=f^*\eta+s\,c_1(D)$, the new correction is the actual divisor
\[
J_{T'}=J_T-s(p^*D-q^*D'),\qquad
q^*H_{T'}=p_0^*H+[J_{T'}].
\]
Every old exceptional divisor remains exceptional
over $T'$: neither kind of step extracts divisors. The resulting
correction $J_{T'}$ is therefore an actual $q$-exceptional divisor.
Its negative is $q$-nef because $p_0^*H$ is nef. Negativity gives
$J_{T'}\geq0$, and uniqueness and compatibility are those of
Lemma~\ref{lem:class-trace}.
For the stated real-boundary extension, the cited contraction theorem
supplies the negative-side Bott--Chern rank and the ordinary real flip.
The difference $p^*D-q^*D'$ is still an actual exceptional real
discrepancy divisor, by Lemma~\ref{lem:negativity}. The decomposition,
transport formula, and correction argument above therefore apply
unchanged. Rationality of $D$ was not used in these arguments.
\end{proof}
\begin{remark}\label{rem:ordinary-continuation}
Proposition~\ref{prop:ordinary-step} starts on the given pair and can be
reapplied after every finite prefix of ordinary steps. It allows every
negative ray. Its construction is independent of any termination
claim, and makes no assertion about the positive-side quotient
$\BC(T')/(f^+)^*\BC(Z)$ or about equality of the full Bott--Chern ranks
across a flip.
\end{remark}
==== END Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//03-steps.tex ====
==== BEGIN Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//04-terminal.tex ====
\section{Termination for ordinary terminal fourfold pairs}\label{sec:counts}
We prove termination of ordinary flips for compact K\"ahler fourfold
pairs with effective real boundary and log discrepancy $a(E)>1$
for every exceptional place. Two finite counts
replace projectivity of the ambient fourfold: analytic cycle classes
control contractions, and low-discrepancy places control flipped surfaces.
The bases of successive projective contractions may vary.
We first establish these counts, including their generalized-pair form
needed for extraction later in the proof.
\subsection{Loss of analytic cycle classes}
For a compact complex analytic space $V$ and an integer $k\geq 0$, put
\[
\begin{split}
\mathcal C_k(V)
&:=\operatorname{span}_{\R}\{[S]\in H_{2k}(V,\R):
S\subset V\text{ irreducible compact analytic},\ \dim S=k\},\\
c_k(V)&:=\dim_{\R}\mathcal C_k(V).
\end{split}
\]
Fundamental classes here use the complex orientation on the regular loci.
Compact analytic spaces admit finite triangulations compatible with any
specified finite collection of closed analytic subsets, with simplicial
dimension at most twice the complex dimension. Thus $c_k(V)$ is finite.
We justify this topological input after Lemma~\ref{lem:cycle-loss}.
For such a triangulation, Borel--Moore localization is the relative
simplicial-chain sequence for a closed analytic subset and its complement.
If $V$ is K\"ahler and $S$ is such a subspace, then
\begin{equation}\label{eq:positive-cycle}
\langle[\omega]^k,[S]\rangle=\int_{S_{\mathrm{reg}}}\omega^k>0
\end{equation}
for a K\"ahler form $\omega$ on $V$. Thus $[S]\neq 0$, including when
$V$ or $S$ is singular.
\begin{lemma}[Cycle loss]\label{lem:cycle-loss}
Let $X\dashrightarrow Y$ be a bimeromorphic transformation of compact
K\"ahler spaces, represented by a proper bimeromorphic diagram.
Suppose it identifies
\[
U=X\setminus A\simeq Y\setminus B,
\]
where $A$ and $B$ are closed analytic subspaces and $\dim B<k$.
Then there is a surjection
\[
\mathcal C_k(X)\longrightarrow\mathcal C_k(Y).
\]
If $A$ contains an irreducible compact analytic $k$-dimensional subspace,
then $c_k(X)>c_k(Y)$.
Consequently, in a sequence of bimeromorphic transformations of
$n$-dimensional compact K\"ahler spaces which extract no prime divisors,
only finitely many transformations contract a prime divisor. A small
transformation preserves $c_{n-1}$.
\end{lemma}
\begin{proof}
The Borel--Moore localization sequence for $B\subset Y$ contains
\[
H_{2k}(B,\R)\longrightarrow H_{2k}(Y,\R)
\xrightarrow{\ j_Y^*\ }H_{2k}^{\mathrm{BM}}(U,\R)
\longrightarrow H_{2k-1}(B,\R).
\]
The outside groups vanish: the real dimension of $B$ is at most $2k-2$.
Thus $j_Y^*$ is an isomorphism. Compose restriction from $X$ with its
inverse to obtain a linear map
\[
H_{2k}(X,\R)\xrightarrow{\ j_X^*\ }H_{2k}^{\mathrm{BM}}(U,\R)
\xrightarrow{\ (j_Y^*)^{-1}\ }H_{2k}(Y,\R).
\]
For an irreducible $k$-cycle $S\subset X$ not contained in $A$, its
restriction is the fundamental class of $S\cap U$. Closing its image in
$Y$ gives its strict transform: analyticity follows by taking the proper
image of the corresponding strict transform in the given diagram.
The displayed map therefore sends $[S]$ to this transformed cycle class.
It sends $[S]$ to zero when $S\subset A$.
Conversely, no $k$-dimensional subspace of $Y$ is contained in $B$.
Its strict transform in $X$ consequently maps to its fundamental class.
This proves the asserted surjection of cycle spans. If $S\subset A$
has dimension $k$, its nonzero class, by \eqref{eq:positive-cycle}, is
in the kernel. This proves the strict inequality.
For a transformation extracting no prime divisor, choose the common
isomorphic open so that the complement in the target has codimension
at least two. Every contracted source prime lies in the source
complement. Apply the result with $k=n-1$ at every step. The
nonnegative integer $c_{n-1}$ cannot decrease infinitely often. For a
small transformation both complements have codimension at least two,
so applying the result in both directions gives equality.
\end{proof}
This argument does not assert surjectivity on full homology, and does
not compare the Bott--Chern spaces of the two models.
\begin{proof}[Topological justification]
We give the abstract-space reduction for the finite compatible
triangulation used above. Teissier's compatible Whitney stratification
\cite[Chapter~III, Propositions~1.5 and~2.2.2]{Teissier1982} has finitely
many strata on a compact space. Choose finitely many analytic charts
$z_j:U_j\hookrightarrow\C^{N_j}$ and nonnegative smooth cutoffs
$\rho_j$ compactly supported in these charts whose positivity sets
cover the space. Extend $\rho_j$ and $\rho_j z_j$ by zero. The map
\[
x\longmapsto\bigl(\rho_j(x),\rho_j(x)z_j(x)\bigr)_j
\]
is a topological embedding into a Euclidean space. Near a point where
$\rho_j>0$, its components extend smoothly to the ambient chart and the
$j$-th block has the smooth left inverse $(t,w)\mapsto w/t$.
It is therefore locally an ambient smooth embedding, which preserves
the Whitney conditions. Mather's control data
\cite[Section~7, Proposition~7.1, and Section~8]{Mather1970} and
Goresky's triangulation theorem \cite[Section~5]{Goresky1978} now give
a compatible triangulation, smooth on each stratum. Compactness makes
it finite, and smoothness on strata gives the dimension bound.
Borel--Moore localization is then the relative simplicial-chain
sequence for a closed analytic subset and its complement.
\end{proof}
\subsection{Low places on compact log smooth data}
We use generalized discrepancies in the sense of
Section~\ref{sec:analytic}. A place is called exceptional over $X$ when
its center on $X$ has codimension at least two.
The generalized klt statement below supplies the finite extraction set in
Section~\ref{sec:scaling}. Its coordinate estimate and finite centre-blowup
argument also enter the terminal surface analysis in this section.
\begin{lemma}[Low places and their projective realization]\label{lem:low-places}
Let $(X,B+\bM)$ be a generalized pair on a compact normal complex
space, with effective real boundary of finite support and nef
b-$(1,1)$ data descending to a model projective over $X$. Fix a projective log resolution $p:W\to X$ carrying those data.
\begin{enumerate}[label=\textup{(\roman*)}]
\item If the pair is generalized klt, there is $\epsilon>0$ such that
every prime divisor over $X$ has log discrepancy at least $\epsilon$,
and only finitely many exceptional places have
\[
a(E;X,B+\bM)<1+\epsilon.
\]
\item If the pair is generalized terminal and
$b=\max(\{0\}\cup\{\operatorname{coeff}_D B\})$, there are only
finitely many exceptional places with
\[
a(E;X,B+\bM)<2-b.
\]
\end{enumerate}
In each assertion the finite set of exceptional places can be represented
simultaneously by prime divisors on a smooth model projective over $X$.
In particular, any prescribed finite set of exceptional places with
$a(E;X,B+\bM)\leq 1$ for a generalized klt pair has such a realization.
This conclusion does not use an extraction or termination theorem.
\end{lemma}
\begin{proof}
Let $L\in\BC(X)$ be the generalized log class and let
$M_W=[\bM_W]$ be the nef trace class on $W$. Write the crepant formula as
\begin{equation}\label{eq:carrier-boundary-count}
[K_W+\Delta]+M_W=p^*L.
\end{equation}
This is the crepant boundary identity defining generalized
discrepancies; it is an equality of Bott--Chern classes when the nef
data are transcendental. The boundary $\Delta$ is an actual real
divisor with simple normal crossing support. Since the nef data
descend to $W$, every further generalized discrepancy is the ordinary
discrepancy for $(W,\Delta)$. The coefficients of $\Delta$ need not be
nonnegative. We may enlarge its listed simple normal crossing divisor
by components of coefficient zero.
Here a closed stratum means an irreducible component of an intersection
of listed components; it is smooth, and is the closure of its open
stratum.
We give the local estimate and the realization argument explicitly;
they are the log smooth calculations underlying
\cite[Proposition~2.36]{KM1998} and
\cite[Proposition~2.8]{CT2023}.
Write $\Delta=\sum_j d_jD_j$ and put $w_j=1-d_j$.
Suppose first that every $w_j$ is positive, and set
\[
\delta=\min\bigl(\{1\}\cup\{w_j\}_j\bigr)>0.
\]
Let $E$ be exceptional over a smooth model with this log smooth
boundary. At a general point of its center $C$, let $c=\codim C$,
and choose local parameters $x_1,\ldots,x_c$ for $C$ so that the
boundary components containing $C$ are $x_1=0,\ldots,x_r=0$.
For $v=\operatorname{ord}_E$, the Jacobian calculation gives
\begin{equation}\label{eq:coordinate-discrepancy-count}
a(E;W,\Delta)\geq
\sum_{j=1}^{r}(1-d_j)v(x_j)+\sum_{j=r+1}^{c}v(x_j).
\end{equation}
Indeed, on a smooth model containing $E$, with local parameter $t$
for $E$, write $x_j=t^{v(x_j)}u_j$ at its general point. In a top
exterior differential at most one factor can use the term involving
$dt$ that lowers the order by one. Thus the relative Jacobian has
order at least $\sum_{j=1}^c v(x_j)-1$. Adding one and subtracting
the boundary orders proves \eqref{eq:coordinate-discrepancy-count}.
The remaining coordinates along $C$ are holomorphic and contribute
no negative orders.
In particular every discrepancy is at least $\delta$: for a divisor
already on the smooth model it is its listed weight, or one if it is
not listed; for an exceptional divisor use
\eqref{eq:coordinate-discrepancy-count}. If $C$ is not a stratum of
the listed simple normal crossing divisor, then $c>r$. When $r>0$,
the right hand side is at least $\delta+1$; when $r=0$, it is at
least $c\geq 2\geq\delta+1$. Hence
\begin{equation}\label{eq:nonstratum-cutoff}
C\text{ not a stratum}\quad\Longrightarrow\quad
a(E;W,\Delta)\geq 1+\delta.
\end{equation}
Now fix a place $E$ with discrepancy strictly below $1+\delta$.
If it is not already a divisor on $W$, its center is a stratum by
\eqref{eq:nonstratum-cutoff}. Blow up that closed stratum.
The center is smooth, the blowup is projective, and the new listed
boundary is again simple normal crossing. If the center is the
intersection of the components indexed by $J$, then the new exceptional
component has weight
\begin{equation}\label{eq:stratum-weight-sum}
w_{\mathrm{new}}=\sum_{j\in J}w_j.
\end{equation}
Thus all weights remain at least $\delta$, so the same argument
applies on the next model. It excludes a non-stratum center at
every stage until $E$ becomes a divisor.
This process is finite. While $E$ is exceptional, a blowup of its
smooth center of codimension $c\geq2$ changes its ordinary
empty-boundary discrepancy by
\[
a(E;W_{\mathrm{new}},0)
=a(E;W_{\mathrm{old}},0)-(c-1)v(\mathcal I_C).
\]
The subtracted quantity is a positive integer, whereas an ordinary
log discrepancy over a smooth space is a positive integer.
Consequently only finitely many such blowups are possible.
Thus $E$ is obtained by successive stratum blowups, which is the
meaning of a \emph{toroidal place} here. This also realizes it on a projective model without first assuming
that the original model on which $E$ was given was projective over $X$.
For completeness, there are only finitely many such toroidal places.
A toroidal prime over a fixed irreducible stratum is determined by a
primitive nonzero vector $(m_j)_{j\in J}$ of nonnegative integers
in the normal boundary directions. Its discrepancy is
\begin{equation}\label{eq:monomial-discrepancy-count}
\sum_{j\in J}m_jw_j.
\end{equation}
These statements follow also directly from the successive stratum
blowups: a blowup inserts the sum of the relevant coordinate rays,
and the discrepancy transforms by \eqref{eq:stratum-weight-sum}.
The vector determines the monomial order and is unchanged when a
boundary equation is multiplied by a unit. There are finitely many
irreducible strata on the compact model. Since each $w_j\geq\delta$,
an upper bound $T$ on \eqref{eq:monomial-discrepancy-count} bounds
every $m_j$ by $T/\delta$. This leaves finitely many vectors and
hence finitely many places. Add the finitely many prime divisors
already on $W$ which are exceptional over $X$.
For a generalized klt pair, all weights in
\eqref{eq:carrier-boundary-count} are positive. The preceding
argument with $\epsilon=\delta$ proves \textup{(i)} and its projective
realization assertion. In particular $a\leq1$ lies strictly below
$1+\delta$.
No rationality of the weights was used: their fixed positive minimum
bounds the nonnegative integer vectors. Thus the conclusion applies
to real boundaries and real nef data.
For a generalized terminal pair, the coefficient of every
$p$-exceptional component in $\Delta$ is negative, since its
log discrepancy is greater than one. The other coefficients are
the original boundary coefficients and are at most $b<1$.
All the weights on $W$ are therefore at least $1-b$.
Repeat the argument with $\delta=1-b$ and $T=2-b$ to prove
\textup{(ii)}. Subsequent stratum blowups preserve this lower bound
by \eqref{eq:stratum-weight-sum}.
Finally, to represent the entire finite set at once, take the
dominating component of the fiber product of the finitely many
projective realizations over $W$, normalize, and resolve projectively.
The strict transforms of the designated prime divisors remain
divisors on this common model. Its composite map to $X$ is projective.
\end{proof}
\begin{remark}\label{rem:strict-low-cutoff}
The strict inequalities in Lemma~\ref{lem:low-places} are necessary.
For example, in $\mathbf P^n$, $n\geq3$, let $H$ be a hyperplane
with coefficient $b\in[0,1)$. Blowing up any codimension-two linear
subspace contained in $H$ gives log discrepancy $2-b$ for
$(\mathbf P^n,bH)$. There are infinitely many such places, although
this pair is terminal. With empty boundary their discrepancy is two.
Below we count places strictly below a threshold and use a step at
which a place reaches that threshold to obtain a strict decrease.
\end{remark}
\subsection{Surface centres of low-discrepancy places}
The terminal termination argument must allow many places of discrepancy
below two above a moving boundary surface. After recalling smoothness in
codimension two, Lemma~\ref{lem:terminal-low-centres} identifies the infinite
families that will be harmless. Excluding only
the first blowup above each surface would be incorrect; the correction
in \cite[Proposition~3.1 and Lemma~3.2]{Fujino2005Addendum} is essential.
\begin{lemma}\label{lem:terminal-codim-two}
Let $(T,\Phi)$ be an ordinary klt pair on a normal complex fourfold,
with effective real boundary, $K_T$ a $\Q$-line bundle, and
$a(E;T,\Phi)>1$ for every exceptional prime divisor $E$ over $T$.
Then $T$ is smooth in codimension two.
\end{lemma}
\begin{proof}
Dropping the effective boundary shows that $T$ has terminal
singularities. Suppose its singular locus has a codimension-two
component. Near a general smooth point of this component choose a
holomorphic projection to $\C^2$ submersive along it. Take a general
fibre transverse both to the regular locus and, on a fixed log
resolution, to the exceptional divisor and its strata. These
transversality conditions hold after shrinking and choosing a general
value. The resulting surface is a complete intersection in the
Cohen--Macaulay space $T$, since klt singularities are rational, and is
regular outside isolated points. It is therefore normal.
Adjunction of a local Cartier pluricanonical power, first on the regular
locus and then by reflexivity, restricts the relative canonical equality
to the surface resolution. All its exceptional discrepancy coefficients
are positive. A normal surface germ with this property is smooth.
Indeed, negative definiteness of the exceptional intersection matrix
shows that a positive exceptional discrepancy divisor has negative
intersection with some exceptional curve. That curve has negative
canonical degree; adjunction makes it a smooth rational $(-1)$-curve.
Contract it smoothly and repeat. The remaining discrepancy coefficients
stay positive by pushforward. Eventually no exceptional curve remains,
and the proper bimeromorphic map to the normal surface is an isomorphism.
On the other hand, a slice of a singular fourfold point by two equations
has embedding dimension at least the ambient embedding dimension minus
two, and hence greater than two. The chosen surface is singular there,
a contradiction.
\end{proof}
\begin{lemma}\label{lem:terminal-low-centres}
Let $(T,\Phi)$ be an ordinary klt pair on a compact normal complex
fourfold, where $\Phi\geq0$ is real and
$a(E;T,\Phi)>1$ for every exceptional prime divisor $E$ over $T$.
Apart from finitely many exceptional places, every exceptional place with
$a(E;T,\Phi)<2$ has centre a surface $S\subset T$ with the following
properties: $T$ and the reduced support of $\Phi$ are generically smooth
along $S$, exactly one positive boundary component contains $S$, and,
if its coefficient is $b$, then
\[
a(E;T,\Phi)\geq 2-b.
\]
All places above such a surface are allowed in this conclusion, not just
the exceptional divisor of its first blowup.
\end{lemma}
\begin{proof}
Choose a log resolution on which the positive strict boundary components
are smooth and mutually disjoint. Starting with simple normal crossing
support, blow up intersections of the positive components to separate
them. Every exceptional crepant coefficient is negative by terminality.
Thus the only positive components on this resolution are those disjoint
strict transforms, with their original coefficients.
Consider a place not already on the resolution. At its general centre,
apply \eqref{eq:coordinate-discrepancy-count}, dropping negative boundary
coefficients. A centre of codimension at least three has log discrepancy
at least $3-b>2$, if it lies in a positive component of coefficient $b$,
and at least three otherwise. A codimension-two centre outside the
positive support has log discrepancy at least two. Thus a place with
$a<2$ has codimension-two centre in one positive component, and its log
discrepancy is at least $2-b$.
If this centre lies in the exceptional locus, it is an irreducible
component of the intersection of that positive component and an
exceptional divisor. There are finitely many such centres on the compact
resolution. We check that each supports only finitely many places with
$a<2$. It suffices to retain only the smooth component of coefficient
$b$, since deleting the negative coefficients lowers discrepancies.
Make this reduction once on the fixed resolution; on subsequent blowups
retain all exceptional coefficients in the crepant boundary.
Blow up the fixed centre at its general point. Unless the given place
has appeared, the same estimate forces its next centre to be the
intersection of the positive strict transform and the new exceptional
divisor. Indeed, the strict transform of the positive divisor maps isomorphically
to that divisor near the generic point of the original centre, because
the centre is Cartier within it. A codimension-two centre over that
generic point must therefore be this unique intersection. A proper
subset has codimension at least three and discrepancy at least $3-b$;
a centre outside the positive divisor has discrepancy at least two.
Thus no other branch is possible. The process reaches the given place
after finitely many blowups, by the decreasing positive integer
empty-boundary discrepancy used in Lemma~\ref{lem:low-places}.
For this reduced pair, the $k$th exceptional divisor on the branch has
crepant coefficient $-k(1-b)$ and log discrepancy
\[
1+k(1-b),\qquad k=1,2,\ldots.
\]
Indeed, the first coefficient is $b-1$, and blowing up the intersection
of the newest exceptional divisor with the coefficient-$b$ strict
transform changes $-k(1-b)$ to $b-k(1-b)-1=-(k+1)(1-b)$.
Thus only
$k<1/(1-b)$ can occur before the cutoff two is reached. This proves
finiteness over every fixed exceptional centre. These generic blowups
are realized globally by blowing up the closures of the centres.
Add the finitely many exceptional primes on the chosen resolution.
All other centres under consideration lie outside its exceptional locus
at their general points, where the resolution is an isomorphism. Their
images are precisely the surfaces described in the statement.
\end{proof}
We now have the two counts needed for termination. A flipped surface
will give a place whose discrepancy strictly increases into a fixed
finite set. Once such surfaces disappear, a compact surface-class rank
will decrease at every remaining step.
\subsection{Real terminal pairs}
\begin{proposition}\label{prop:terminal-real-four}
Let $(T_0,\Phi_0)$ be an ordinary klt pair on a normal globally
$\Q$-factorial compact K\"ahler fourfold, with $K_{T_0}$ a
$\Q$-line bundle and effective real boundary. Suppose
$a(E;T_0,\Phi_0)>1$ for every exceptional prime divisor $E$ over $T_0$.
Every sequence of ordinary $(K_{T_i}+\Phi_i)$-flips starting at this
pair terminates, provided all models are normal globally $\Q$-factorial
compact K\"ahler fourfolds, each $K_{T_i}$ is a $\Q$-line bundle, and
each flip is given by projective small contractions
\[
T_i\xrightarrow{\ f_i\ }Z_i
\xleftarrow{\ f_i^+\ }T_{i+1}
\]
with $-(K_{T_i}+\Phi_i)$ relatively ample for $f_i$ and
$K_{T_{i+1}}+\Phi_{i+1}$ relatively ample for $f_i^+$.
Here $\Phi_{i+1}$ is the strict transform of $\Phi_i$.
No pseudo-effectivity or bigness assumption is required.
\end{proposition}
\begin{proof}
By Lemma~\ref{lem:negativity}, discrepancies do not decrease, and they
increase strictly for a place centred in either exceptional locus.
Since the maps are small, the exceptional places are the same on all
models. Thus every pair remains terminal in the stated sense, and its
ambient fourfold is smooth in codimension two by
Lemma~\ref{lem:terminal-codim-two}.
Write the distinct positive coefficients of the boundary in decreasing
order. We successively remove, by discarding finitely many initial
steps, every flipping and flipped surface contained in a component of
each coefficient. After the positive coefficients we treat the value
zero, at which stage every flipped surface is considered.
\smallskip\noindent
\emph{Eliminating flipped surfaces.}
Fix the current coefficient $b$, and suppose that no flipping or flipped
surface is contained in a component of coefficient greater than $b$.
For $b=0$ assume that all positive coefficients have already been treated.
If $S$ is a flipped surface contained in a coefficient-$b$ component,
or any flipped surface when $b=0$, blow it up at its general point.
The exceptional prime $E$ has new log discrepancy
\begin{equation}\label{eq:surface-witness-values}
a(E;T_{i+1},\Phi_{i+1})
=2-\sum_{l=1}^{s}b_l\operatorname{mult}_{S}\Phi_{l,i+1},
\qquad 1<a(E;T_{i+1},\Phi_{i+1})\leq2-b,
\end{equation}
where $\Phi_{l,i+1}$ are the positive prime boundary components and
$b_l$ their fixed coefficients. The blowup place is defined globally
by blowing up the closed surface and resolving; its generic point
suffices for this computation.
The values in \eqref{eq:surface-witness-values} belong to a finite set,
even for irrational coefficients. Indeed, terminality gives
\[
\sum_l b_l\operatorname{mult}_{S}\Phi_{l,i+1}<1,
\qquad
0\leq\operatorname{mult}_{S}\Phi_{l,i+1}<1/b_l.
\]
Each multiplicity is an integer, and there are finitely many positive
components. We use no discreteness assertion for all discrepancies.
Strictness in Lemma~\ref{lem:negativity} gives
$a(E;T_i,\Phi_i)<2-b$. Its discrepancy on the first model of the
current tail is therefore also strictly below $2-b$. By
Lemma~\ref{lem:terminal-low-centres}, outside a finite set such a place
would initially have a surface centre in one boundary component of
coefficient $c$, with discrepancy at least $2-c$. Necessarily $c>b$.
That centre persists as a surface in this component at every later
step: at each step the map is an isomorphism at its general point,
since otherwise the centre itself would be a flipping surface in a
higher-coefficient component. It cannot become a flipped surface
either, by the same higher-coefficient exclusion. Thus it cannot be
the place just constructed. All these witness places belong to a
fixed finite set.
Each member of that finite set is used only finitely often. Between
uses its discrepancy does not decrease, and at each use it strictly
increases to one of the finitely many values
\eqref{eq:surface-witness-values}. Consequently only finitely many
steps have a flipped surface of the current kind. Pass beyond them.
\smallskip\noindent
\emph{Eliminating flipping surfaces inside the boundary.}
Suppose $b>0$. Fix a coefficient-$b$ component and normalize its
transforms on the two sides of a flip. Denote the normalizations by
$D_i^\nu,D_{i+1}^\nu$, and normalize their common image in $Z_i$ to
obtain $B_i$. The induced maps to $B_i$ are proper and bimeromorphic.
They are isomorphisms away from the inverse image of the common
exceptional image in $Z_i$, whose dimension is at most one by
Lemma~\ref{lem:negativity}. Finite normalization preserves this bound.
On $D_{i+1}^\nu$ its inverse image also has dimension at most one,
because no flipped surface is contained in the boundary component.
Every compact surface in $B_i$ has a strict transform on $D_i^\nu$,
so pushforward gives a surjection
\[
\mathcal C_2(D_i^\nu)\longrightarrow\mathcal C_2(B_i).
\]
On the other side, Borel--Moore localization along the exceptional
subset of dimension at most one makes restriction in degree four
injective. Compatibility with proper pushforward and the common
isomorphic open therefore gives an injection
\[
\mathcal C_2(D_{i+1}^\nu)\lhook\joinrel\longrightarrow
\mathcal C_2(B_i).
\]
These are assertions about surface spans; no surjectivity on the full
fourth homology of the negative side is needed. Only compact analytic
topology is used on the normalized components and their common image;
the positivity needed next comes from the ambient Kähler fourfold.
If a flipping surface lies in this boundary component, a surface above
it on $D_i^\nu$ maps to a set of dimension at most one in $B_i$.
Its class is nonzero: its image in $T_i$ is a surface, and the square
of a K\"ahler class on $T_i$ has strictly positive integral over it.
One may compute this integral on a resolution of the surface.
The finite normalization has positive generic degree on this surface,
so the same nonvanishing holds upstairs. Its class is a nonzero
kernel vector in the displayed surjection. Hence
\[
c_2(D_i^\nu)\geq c_2(D_{i+1}^\nu),
\]
with strict inequality whenever the component contains a flipping
surface. Summing over the finitely many coefficient-$b$ components
shows that only finitely many such steps occur. This completes the
induction at $b$.
\smallskip\noindent
\emph{The remaining flips.}
After the positive coefficients have been treated, perform the
flipped-surface argument with $b=0$. No flipped surface remains.
For every remaining flip, Lemma~\ref{lem:negativity} gives
\[
\dim\Exc(f_i)+\dim\Exc(f_i^+)\geq3.
\]
Smallness bounds both dimensions by two, so $\Exc(f_i)$ contains a
surface and $\dim\Exc(f_i^+)\leq1$. Apply
Lemma~\ref{lem:cycle-loss} with $k=2$ to the common isomorphic open.
It gives $c_2(T_i)>c_2(T_{i+1})$ at every remaining step, impossible
for an infinite sequence of nonnegative integers.
\end{proof}
The algebraic discrepancy and cycle-count strategy originates in
\cite[Theorem~5-1-15 and Lemmas~5-1-16--5-1-17]{KMM1987}
and \cite{Fujino2004,Fujino2005Addendum}.
The proof above supplies the real-boundary argument and uses compact
analytic cycle classes with K\"ahler positivity. It applies those
geometric tools one step at a time, without a common projective base.
\begin{corollary}\label{prop:terminal-four}
Let $X_0$ be a normal globally strongly $\Q$-factorial compact
K\"ahler fourfold with terminal singularities.
Every sequence starting at $X_0$ of ordinary empty-boundary negative
divisorial contractions and flips, with the projectivity and compact
K\"ahler conclusions of Proposition~\ref{prop:ordinary-step}, terminates.
No pseudo-effectivity hypothesis is required for this assertion about
birational sequences.
\end{corollary}
\begin{proof}
For flips, terminality persists by Lemma~\ref{lem:negativity}.
For a divisorial contraction, \eqref{eq:ordinary-divisorial-difference}
gives $K_{X_i}=f^*K_{X_{i+1}}+eE$ with $e>0$.
The lost prime has target log discrepancy $1+e>1$; every other
exceptional place has target discrepancy at least its source
discrepancy, which was greater than one. Thus terminality persists
throughout the sequence.
Lemma~\ref{lem:cycle-loss} with $k=3$ bounds the number of divisorial
steps. Any infinite sequence would therefore have a tail consisting
only of flips, contrary to Proposition~\ref{prop:terminal-real-four}
with zero boundary.
\end{proof}
==== END Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//04-terminal.tex ====
==== BEGIN Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//05-scaling.tex ====
\section{Scaling and a limiting terminal pair}\label{sec:scaling}
We now construct a particular ordinary program on the pair of
Theorem~\ref{thm:finite}. If that program were infinite, we would obtain
strongly \(\Q\)-factorial auxiliary models carrying nef generalized
adjoints and a decreasing sequence of effective boundaries. Their limit
will be an ordinary terminal pair. These auxiliary extractions are used
only to compare the original program with a terminating one; the
original program starts on \(X\).
Fix a smooth Kähler form \(h\) on \(X\), with class \(H\), large enough
that \(K_X+\Delta+H\) is nef. Throughout this section
\(\mathbf H\) denotes the fixed b-nef datum obtained by pulling back
this very form. On subsequent models its trace class is denoted by
\(H_T\), as in Lemma~\ref{lem:class-trace}. We do not replace the
fixed datum by arbitrary cohomologous current representatives.
\subsection{Constructing the scaling program}
\begin{proposition}\label{prop:scaling-construction}
Let \((X,\Delta)\) satisfy the hypotheses of Theorem~\ref{thm:finite}.
There is an ordinary negative-ray program, finite or infinite,
\[
(X,\Delta)=(X_0,\Delta_0)\dashrightarrow
(X_1,\Delta_1)\dashrightarrow\cdots,
\]
in the same global Weil-divisor \(\Q\)-factorial compact Kähler
category. It stops when the ordinary adjoint is nef or a negative ray
has a Mori fibre contraction. Put
\[
D_i=K_{X_i}+\Delta_i,\qquad
L_i(t)=D_i+tH_i,\qquad H_i=H_{X_i},\qquad \lambda_{-1}=1.
\]
At every stage before stopping there is a parameter
\(0<\lambda_i\leq\lambda_{i-1}\) and a contracted extremal ray
\(R_i\) such that
\[
D_i\cdot R_i<0,\qquad L_i(\lambda_i)\cdot R_i=0,
\qquad L_i(\lambda_i)\text{ is nef}.
\]
The generalized pairs \((X_i,\Delta_i+t\mathbf H)\) are gklt for
\(0\leq t\leq\lambda_{i-1}\). A birational step is crepant for the
data at \(t=\lambda_i\), and discrepancies do not decrease across
that step for any \(0\leq t\leq\lambda_i\).
\end{proposition}
\begin{proof}
On \(X_0\), the nef datum descends, so its addition changes no
discrepancies. Thus all the initial generalized pairs are gklt.
Suppose the assertions have been established through \(X_i\).
If \(D_i\) is nef, stop. Otherwise define
\[
\lambda_i=\min\{t\in[0,\lambda_{i-1}]:L_i(t)\text{ is nef}\}.
\]
The set is nonempty by induction and is closed and convex. Its minimum
is positive because \(D_i\) is not nef.
We first find a ray at this threshold. Choose
\(t_k\uparrow\lambda_i\) with \(\lambda_i/2<t_k<\lambda_i\).
The generalized cone theorem gives an extremal ray negative for
\(L_i(t_k)\). Since \(L_i(\lambda_i)\) is nef, that ray is also
negative for \(L_i(\lambda_i/2)\). The trace \(H_i\) is big by
Corollary~\ref{cor:big-trace}. Hence
\(\Delta_i+(\lambda_i/2)H_i\) is big, and the cone theorem gives
only finitely many negative rays for this latter generalized pair
\cite[Theorem~1.3]{HX2026}. One ray \(R_i\) therefore occurs for
infinitely many \(k\). On this fixed ray, continuity gives
\(L_i(\lambda_i)\cdot R_i=0\). Moreover
\(H_i\cdot R_i>0\), so \(D_i\cdot R_i<0\).
Apply Proposition~\ref{prop:ordinary-step} to this ordinary negative
ray, and denote its contraction by \(f_i:X_i\to Z_i\). A fibre
contraction gives the stopping alternative. Otherwise make the
ordinary divisorial step or flip. Lemma~\ref{lem:class-transport}
transports the trace \(H_i\) to \(H_{i+1}\) while preserving its
relation to the fixed b-datum. By Lemma~\ref{lem:pullback-image}, the
class \(L_i(\lambda_i)\) descends to \(Z_i\). Its descended class
is nef, and its pullback on the new model is
\(L_{i+1}(\lambda_i)\). Thus the two threshold classes have equal
pullbacks on a common resolution. Their difference is represented by
the actual exceptional discrepancy divisor, which vanishes by
negativity as in Lemma~\ref{lem:small-comparison}. The step is
consequently generalized crepant at the threshold.
For completeness, this also proves the induction on singularities.
On a common resolution, the log discrepancy at each prime \(E\) is
affine in \(t\). Its difference across the step is zero at
\(t=\lambda_i\) and is the ordinary discrepancy increase at zero.
Consequently
\begin{align}
&a(E,X_{i+1},\Delta_{i+1}+t\mathbf H)
-a(E,X_i,\Delta_i+t\mathbf H)\notag\\
&\qquad=\left(1-\frac{t}{\lambda_i}\right)
\bigl(a(E,X_{i+1},\Delta_{i+1})-a(E,X_i,\Delta_i)\bigr)
\geq0\label{eq:scaling-interpolation}
\end{align}
for \(0\leq t\leq\lambda_i\). The generalized pairs on this
interval remain gklt, and the new upper-threshold class is nef. The
ordinary steps preserve all the stated properties of the original
category by Proposition~\ref{prop:ordinary-step}.
\end{proof}
\subsection{An infinite program has vanishing thresholds}
We record first why an infinite program has a tail consisting only of
flips. Use the cycle-space notation of Lemma~\ref{lem:cycle-loss}:
\(c_3(T)\) is the dimension of the span of compact analytic
three-dimensional cycle classes in \(H_6(T,\R)\). It is finite.
A small modification of fourfolds is an isomorphism away from sets of
dimension at most two, so that lemma, applied in both directions,
preserves \(c_3\). A divisorial contraction strictly lowers it: an
exceptional divisor has a nonzero cycle class, detected by integration
of the cube of a Kähler form, and maps to a set of dimension at most
two. Therefore only finitely many divisorial steps occur.
\begin{lemma}\label{lem:scaling-zero}
If the program of Proposition~\ref{prop:scaling-construction} is
infinite, then \(\lambda_i\to0\).
\end{lemma}
\begin{proof}
Fix \(0<\epsilon<1\) and a resolution \(\nu:U\to X\). Consider the
compact segment of generalized data
\[
\mathcal P_\epsilon
=\{(\Delta,t\nu^*H):\epsilon\leq t\leq1\}
\subset \operatorname{Div}_{\R}(X)\times\BC(U).
\]
The upstairs nef data here are the pullbacks of the fixed smooth form.
Every pair in this segment is gklt, its upstairs class is nef, and its
boundary-plus-trace is modified big, since \(tH\) is Kähler on \(X\).
This real segment is allowed in the finiteness theorem for weak log
canonical models \cite[Theorem~5.15]{HX2026}; neither rationality of
\(H\) nor strong \(\Q\)-factoriality of \(X\) is required there.
Let \(\phi_i:X\dashrightarrow X_i\) be the accumulated map, and
suppose \(\lambda_i\geq\epsilon\). It is a weak log canonical
model of the original generalized pair at parameter \(\lambda_i\).
Indeed, it extracts no divisors, its target adjoint is nef, and
discrepancies do not decrease: apply
\eqref{eq:scaling-interpolation} to every preceding step at the fixed
parameter \(\lambda_i\), which is no larger than any preceding
threshold. On a common resolution the original adjoint is the
pullback of the target nef class plus an effective exceptional
divisor. Pseudo-effectivity descends by
Lemma~\ref{lem:positive-descent}, so the original data belong to the
pseudo-effective portion of \(\mathcal P_\epsilon\).
The finiteness conclusion concerns the maps \(\phi_i\), up to
isomorphisms of their targets commuting with the maps from \(X\).
These marked maps are pairwise distinct in the flip tail. In fact,
given two stages, an exceptional prime has strictly increasing
ordinary discrepancy at the first intervening flip by
Lemma~\ref{lem:negativity}, and later steps
cannot decrease it. A target isomorphism compatible with the marking
would identify the transformed boundary and all its discrepancies,
a contradiction. This is also the distinction used in the proof of
\cite[Theorem~5.16]{HX2026}.
There are therefore only finitely many \(i\) with
\(\lambda_i\geq\epsilon\). Since \(\epsilon>0\) was arbitrary,
the thresholds tend to zero.
\end{proof}
An infinite program would thus provide nef generalized adjoints with
arbitrarily small nef parts. We next put their nonterminal places on
auxiliary models, so that the boundary can be allowed to converge on
one fixed space.
\subsection{Stabilizing the extractions}
\begin{proposition}\label{prop:terminal-limit}
Suppose the program of Proposition~\ref{prop:scaling-construction} is
infinite. After discarding finitely many stages, there are proper
bimeromorphic morphisms
\[
\mu_i:Y_i\longrightarrow X_i
\]
with \(Y_i\) strongly \(\Q\)-factorial and compact Kähler, and
effective real boundaries \(\Theta_i\), with the following properties.
\begin{enumerate}[label=\textup{(\roman*)}]
\item The natural maps between the \(Y_i\) are isomorphisms in
codimension one, and
\begin{equation}\label{eq:extracted-nef-adjoint}
Q_{i,Y_i}:=K_{Y_i}+\Theta_i+\lambda_iH_{Y_i}
=\mu_i^*L_i(\lambda_i)
\end{equation}
is nef. Each ordinary pair \((Y_i,\Theta_i)\) is klt and terminal.
\item On a fixed first model \(Y=Y_{i_0}\), write
\(\Theta_{i,Y}\) for the strict transform of \(\Theta_i\).
These boundaries have a fixed finite containing support and decrease
coefficientwise to an effective real boundary \(\Theta_Y\).
The ordinary pair \((Y,\Theta_Y)\) is klt and terminal.
\item The strict transform of \(\Theta_Y\) on each \(Y_i\)
pushes forward to \(\Delta_i\) under \(\mu_i\).
\end{enumerate}
Here terminal means that every exceptional prime has log discrepancy
strictly greater than one.
\end{proposition}
\begin{proof}
Discard the finitely many divisorial steps. All remaining maps
\(X_i\dashrightarrow X_{i+1}\) are small. For each such \(i\), put
\[
\mathcal E_i=\{E\text{ exceptional over }X_i:
a(E,X_i,\Delta_i+\lambda_i\mathbf H)\leq1\}.
\]
The fixed nef datum has a carrier projective over \(X_i\): take a
common resolution of the finitely many preceding projective ordinary
step diagrams. For a flip its graph is projective over either side,
so this construction and a further projective log resolution retain
projectivity over \(X_i\). On this resolution the datum is the
pullback of \(h\). Lemma~\ref{lem:low-places} therefore makes
\(\mathcal E_i\) finite. It allows real coefficients and counts
places of log discrepancy exactly one as well as those below one.
Smallness identifies the primes exceptional over successive
\(X_i\). Their log discrepancies at the current threshold agree by
crepancy. On the new model, decreasing the coefficient of the nef
part increases discrepancies: in the trace formula of
Lemma~\ref{lem:class-trace}, the correction divisor is effective.
Thus, for every such prime,
\begin{equation}\label{eq:decreasing-extraction-sets}
\begin{split}
a(E,X_{i+1},\Delta_{i+1}+\lambda_{i+1}\mathbf H)
&\geq a(E,X_{i+1},\Delta_{i+1}+\lambda_i\mathbf H)\\
&=a(E,X_i,\Delta_i+\lambda_i\mathbf H).
\end{split}
\end{equation}
It follows that \(\mathcal E_{i+1}\subseteq\mathcal E_i\).
After another truncation they equal one finite set \(\mathcal E\).
Only this set stabilizes; no discreteness of the real discrepancy
values is asserted.
Apply the exact extraction theorem
\cite[Corollary~2.29]{HX2026} to the generalized pair at
\(\lambda_i\) and the prescribed set \(\mathcal E\).
It gives a strongly \(\Q\)-factorial compact Kähler model
\(\mu_i:Y_i\to X_i\) extracting precisely those primes.
Its compact Kähler output is supplied by the relative MMP in its proof.
The input of this theorem need not be
strongly \(\Q\)-factorial, and its nef datum and crepant boundary
may be real.
Let \(\Theta_i\) be the crepant generalized boundary on \(Y_i\).
On the transform of a divisor of \(X_i\), its coefficient is the
coefficient in \(\Delta_i\); on an extracted prime \(E\), it is
\[
1-a(E,X_i,\Delta_i+\lambda_i\mathbf H)\in[0,1).
\]
Thus \(\Theta_i\) is effective. The generalized crepant relation
gives \eqref{eq:extracted-nef-adjoint}. More explicitly, subtract
the ordinary class \(K_{Y_i}+\Theta_i\) from the pulled-back
generalized adjoint and divide by \(\lambda_i>0\).
Lemma~\ref{lem:class-trace} identifies the result as the trace
\(H_{Y_i}\) of our fixed b-datum, with an effective exceptional
correction on a common resolution. Equation
\eqref{eq:extracted-nef-adjoint} is nef because its right-hand side
is a pullback of a nef class.
Every prime exceptional over \(Y_i\) is exceptional over \(X_i\)
and is absent from \(\mathcal E\), so its generalized log
discrepancy is greater than one. Dropping the nef part weakly
increases this discrepancy. Hence \((Y_i,\Theta_i)\) is ordinary
terminal. Together with the boundary coefficients below one, this
also proves that it is klt.
The flip tail identifies all prime divisors on the \(X_i\), and
each \(Y_i\) extracts exactly the same additional primes.
Consequently the \(Y_i\) are naturally isomorphic in codimension
one. Fix the first one, \(Y=Y_{i_0}\). On its primes inherited
from \(X_{i_0}\), the coefficients of \(\Theta_{i,Y}\) are
constant. On the extracted primes, they decrease by
\eqref{eq:decreasing-extraction-sets}. They are nonnegative and
have a fixed finite containing support, so their coefficientwise
limit \(\Theta_Y\) exists and
\[
0\leq\Theta_Y\leq\Theta_{i,Y}\leq\Theta_{i_0,Y}.
\]
Terminality of the limit follows by comparison with the first
boundary on this fixed model. Since \(Y\) is \(\Q\)-factorial,
the effective real divisor \(\Theta_{i_0,Y}-\Theta_Y\) has
nonnegative order at every divisorial valuation. Therefore
\[
a(E,Y,\Theta_Y)\geq a(E,Y,\Theta_{i_0,Y})>1
\]
for every exceptional prime \(E\). All boundary coefficients
remain below one, so the limit is klt as well. This argument uses
decrease of the boundary, rather than any assertion that terminality
is closed under limits. Finally, only coefficients of
\(\mu_i\)-exceptional primes vary. The transformed limiting
boundary thus pushes forward to \(\Delta_i\), proving (iii).
\end{proof}
The infinite-program assumption has now produced an ordinary terminal
pair on a fixed strongly \(\Q\)-factorial model, together with the
nef adjoints \eqref{eq:extracted-nef-adjoint}. In the next section we
apply terminal termination to that limiting pair and compare a small
perturbation with one of these nef adjoints.
==== END Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//05-scaling.tex ====
==== BEGIN Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//06-perturbation.tex ====
\section{The limiting model and the perturbation contradiction}
\label{sec:perturbation}
We now show that the ordinary program constructed in
Proposition~\ref{prop:scaling-construction} is finite. An infinite program
would give, by Proposition~\ref{prop:terminal-limit}, a decreasing family of
boundaries on small modifications of one terminal pair. We first make the
limiting ordinary adjoint nef. We then compare it with one sufficiently
nearby generalized adjoint. The decisive point is that this comparison can
be made without changing the Cartier indices of finitely many rational
nef adjoints. The rational decomposition and perturbation method follows
Birkar's argument \cite[Remark~3.1, Lemma~3.2, and Section~4]{Birkar2009}.
We give the details needed for the fixed Kähler b-part and for exact
line-bundle descent on the analytic contractions.
\subsection{A nef model of the limiting ordinary pair}
Recall the data supplied by Proposition~\ref{prop:terminal-limit} under the
assumption that the scaling program is infinite. After truncation, all
its steps are flips. There are strongly $\Q$-factorial compact Kähler
models $\mu_i:Y_i\to X_i$, isomorphic to one another in codimension one,
and effective boundaries $\Theta_i$ such that
\begin{equation}
Q_{i,Y_i}:=\mu_i^*L_i(\lambda_i)
=K_{Y_i}+\Theta_i+\lambda_i H_{Y_i}
\quad\text{is nef}.
\label{eq:extracted-nef}
\end{equation}
Here $L_i(t)=D_i+tH_{X_i}$, $D_i=K_{X_i}+\Delta_i$, and $\lambda_i\to0$.
The nef b-part $\mathbf H$ is the fixed datum determined by the initial
Kähler form. On a fixed first model $Y$, the transforms $\Theta_{i,Y}$
decrease coefficientwise, on a fixed finite support, to an effective real
boundary $\Theta_Y$. The ordinary pair $(Y,\Theta_Y)$ is terminal.
The transform of $\Theta_Y$ on each $Y_i$ pushes forward to $\Delta_i$.
\begin{proposition}
\label{prop:limiting-nef}
For these data there is a finite sequence of ordinary
$(K_Y+\Theta_Y)$-flips
\[
Y\dashrightarrow W
\]
such that $W$ is strongly $\Q$-factorial and compact Kähler, the transformed
ordinary pair $(W,\Theta_W)$ is terminal, and
$P:=K_W+\Theta_W$ is nef. In particular, $W$ is isomorphic in codimension
one to every $Y_i$.
\end{proposition}
\begin{proof}
Run the ordinary real-boundary program from $(Y,\Theta_Y)$ using
\cite[Proposition~4.2]{HX2026}, with zero nef part.
We show inductively that every step is a flip. Let $T$ be a current
model, so that $T$ is small to every $Y_i$, and write
\[
P_T=K_T+\Theta_T,\qquad
Q_{i,T}=K_T+\Theta_{i,T}+\lambda_i H_T .
\]
The trace $H_T$ exists by Lemma~\ref{lem:class-trace}: along these
strongly $\Q$-factorial real-boundary flips we use the canonical
Bott--Chern transport of \cite[Proposition~4.2]{HX2026}. Its pullback
comparison has an actual exceptional-divisor correction, so the same
trace relation propagates with the fixed b-data.
Together with the effective transformed boundary, the actual trace
relation supplies the generalized structure boundary for $Q_{i,T}$;
no generalized klt condition is required for the small-model comparison
below. The finite boundary support and
$\lambda_i\to0$ give $Q_{i,T}\to P_T$ in $\BC(T)$.
Fix a $P_T$-negative extremal ray and its contraction. For all sufficiently
large $i$ this same ray is $Q_{i,T}$-negative. Take a common smooth
Kähler resolution $p:U\to T$, $q:U\to Y_i$. The boundaries are transforms
of one another and the b-part is unchanged. Thus
Lemma~\ref{lem:small-comparison}, together with
\eqref{eq:extracted-nef}, gives an actual divisor $E_i$ such that
\begin{equation}
p^*Q_{i,T}=q^*Q_{i,Y_i}+[E_i],
\qquad E_i\ge0,
\qquad E_i\text{ is exceptional over both models}.
\label{eq:limit-comparison}
\end{equation}
Suppose that the contraction were divisorial. Choose a general point
of an exceptional prime outside the codimension-at-least-two image of
$\Exc(p)$. A positive-dimensional projective fibre through this point
contains an irreducible curve $C$. Its strict transform $\widetilde C$
on $U$ is not contained in $\Supp(E_i)$. Consequently
\[
Q_{i,T}\cdot C
=q^*Q_{i,Y_i}\cdot\widetilde C+E_i\cdot\widetilde C\ge0,
\]
contrary to the choice of the ray. For a fibre-type contraction the
same argument uses a general point of $T$ instead. It also rules out
ending the program with a Mori fibre space.
Therefore every step is an ordinary flip. Terminality persists by the
ordinary discrepancy comparison. These flips have the exact relative
properties needed for the auxiliary termination theorem. The negative
contraction is projective by \cite[Proposition~4.2]{HX2026}; the positive
one is proper birational between rational-singularity spaces, hence
strongly Moishezon by \cite[Lemma~2.39]{HX2026}, and projective by
\cite[Lemma~2.40]{HX2026}. For a step
$T\xrightarrow{f}Z\xleftarrow{f^+}T^+$, the signed ordinary adjoints
$-P_T$ and $P_{T^+}$ are relatively Kähler real line bundles.
To obtain relative ampleness in this intrinsic line-bundle setting,
write either signed adjoint $L$ in a finite real span of integral line
bundles, and choose a base class $\gamma$ such that
$c_1(L)+h^*\gamma$ is Kähler, where $h=f$ or $f^+$.
Openness of the Kähler cone gives a rational simplex around the
coefficient vector of $L$ whose rational vertices $L_\nu$ still have
$c_1(L_\nu)+h^*\gamma$ Kähler. Lemma~\ref{lem:relative-positivity}
makes every $L_\nu$ relatively ample, and their positive convex
combination $L$ is therefore relatively ample as a real line bundle. Thus
Proposition~\ref{prop:terminal-real-four} makes this sequence finite.
Its endpoint has nef ordinary adjoint, as asserted.
\end{proof}
\subsection{Rational nef boundaries and a positive intersection gap}
The following argument gives a rational description near a nef ordinary
adjoint, even though the limiting boundary has real coefficients.
Only the ordinary boundary coefficients vary in this argument; the
transcendental b-part plays no role.
\begin{lemma}
\label{lem:rational-nef-polytope}
Let $W$ be a globally $\Q$-factorial compact Kähler fourfold with $K_W$
a $\Q$-line bundle. Let $B\ge0$ be a real boundary such that $(W,B)$ is
klt and $K_W+B$ is nef. There are effective rational klt boundaries
$B^1,\ldots,B^r$ and positive real numbers $u_1,\ldots,u_r$ such that
\[
\sum_{j=1}^r u_j=1,\qquad
B=\sum_{j=1}^r u_jB^j,\qquad
K_W+B^j\ \text{is nef for every }j.
\]
The boundaries $B^j$ can be chosen on the positive support of $B$.
\end{lemma}
\begin{proof}
If $B=0$, take $r=1$ and $B^1=0$. Otherwise work in the finite-dimensional
coefficient space on $\Supp(B)$. A sufficiently small simplex with
rational vertices $C^0,\ldots,C^d$ contains $B$ in its interior and
consists of effective klt boundaries. Indeed, all coefficients of $B$
on this support are positive, and the klt inequalities are open on one
log resolution. Put $D_k=K_W+C^k$.
Choose a rational number $\delta>0$ smaller than all the barycentric
coordinates of $B$. Let $\mathcal P$ be the smaller rational polytope
in the simplex defined by requiring every barycentric coordinate to
be at least $\delta$. We claim that its nef locus
\[
\mathcal N=\{C\in\mathcal P:K_W+C\text{ is nef}\}
\]
is a rational polytope.
Consider an extremal ray $R$ that is negative for $K_W+C$ for some
$C\in\mathcal P$. At least one vertex adjoint, say $D_a$, is negative
on $R$. Among all irreducible rational curves whose classes span $R$,
choose $\Gamma$ minimizing $-D_a\cdot\Gamma$. Such curves exist by the
cone theorem \cite[Theorem~1.3]{HX2026}, and a minimum exists because
a Cartier multiple of $D_a$ puts their positive degrees in a discrete
subset of $\R_{>0}$.
This choice minimizes the numerical scale of the curve class on $R$.
For any other vertex $D_k$ negative on $R$, the cone theorem supplies
a rational curve $\Gamma_k$ on $R$ with
$0<-D_k\cdot\Gamma_k\le8$. Since $[\Gamma]$ is a positive multiple
of $[\Gamma_k]$ with multiplier at most one, it follows that
$-D_k\cdot\Gamma\le8$ as well. Vertices nonnegative on $R$ need no
estimate. Thus all the numbers $d_k=D_k\cdot\Gamma$ satisfy $d_k\ge-8$.
Write a member of $\mathcal P$ that is negative on $R$ as
$\sum t_kC^k$, where $t_k\ge\delta$ and $\sum t_k=1$. Then
$\sum t_kd_k<0$, so for each $k$,
\[
t_kd_k<8(1-t_k).
\]
In particular,
\begin{equation}
-8\le d_k<\frac8\delta\qquad(0\le k\le d).
\label{eq:finite-intersection-vectors}
\end{equation}
Each $D_k$ has a fixed global Cartier index on $W$. Hence its degrees
belong to a fixed discrete subgroup of $\R$, and
\eqref{eq:finite-intersection-vectors} permits only finitely many
vectors $(d_0,\ldots,d_d)$. Their nonnegativity inequalities are rational.
These finitely many inequalities define exactly $\mathcal N$ inside
$\mathcal P$. Indeed, a ray never negative on $\mathcal P$ imposes
no further condition. For any other ray the chosen vector tests its
nonnegativity. Finally, positivity on all extremal rays is positivity
on $\NA(W)$: intersect this cone with the hyperplane where a fixed
Kähler class has value one and use the resulting compact convex slice.
Nef-cone duality now proves the claim. Express $B$ as a convex
combination of rational vertices of $\mathcal N$, omitting vertices
with zero weight. These are the required $B^j$.
\end{proof}
Apply Lemma~\ref{lem:rational-nef-polytope} to $(W,\Theta_W)$ from
Proposition~\ref{prop:limiting-nef}. Fix a decomposition
\begin{equation}
\Theta_W=\sum_{j=1}^r u_jB^j,\qquad
P=\sum_{j=1}^r u_jP^j,\qquad P^j=K_W+B^j,
\label{eq:nef-decomposition}
\end{equation}
and positive integers $\ell_j$ such that $\ell_jP^j$ is an integral
line bundle, also clearing the coefficients of $B^j$.
Define
\begin{equation}
c=\min_{1\le j\le r}\frac{u_j}{\ell_j}>0,
\qquad
m\in\N,\quad (m-1)c>8.
\label{eq:gap-and-amplification}
\end{equation}
Whenever the transforms of the $P^j$ remain nef and the same $\ell_j$
remain Cartier indices, every curve $\Gamma$ satisfies
\begin{equation}
P\cdot\Gamma>0\quad\Longrightarrow\quad P\cdot\Gamma\ge c.
\label{eq:positive-gap}
\end{equation}
At least one nonnegative summand in \eqref{eq:nef-decomposition} is then
positive, and its degree is at least $1/\ell_j$.
The next argument preserves exactly these indices.
\subsection{A nearby program that is trivial for the limiting class}
\begin{proposition}
\label{prop:trivial-perturbation}
For the extracted data of Proposition~\ref{prop:terminal-limit} and the
model $W$ of Proposition~\ref{prop:limiting-nef}, take $i$ sufficiently
large. There is a finite generalized $Q_{i,W}$-program
$W\dashrightarrow W_i$ consisting only of flips, with the following
properties:
\begin{enumerate}
\item $Q_{i,W_i}$ is nef;
\item every step is trivial for the transformed class $P$ and for
each $P^j$ in \eqref{eq:nef-decomposition};
\item the transformed $P^j$ are nef adjoints of ordinary rational
klt pairs, and the fixed multiples $\ell_jP^j$ are line bundles;
\item every $Q_{i,W_i}$-trivial curve is $P_{W_i}$-trivial.
\end{enumerate}
\end{proposition}
\begin{proof}
On $W$, set $G_i=\Theta_{i,W}-\Theta_W\ge0$. Keep the integer $m$ in
\eqref{eq:gap-and-amplification} fixed and consider the generalized
adjoint
\begin{equation}
A_W=K_W+\Theta_W+mG_i+m\lambda_iH_W
=mQ_{i,W}-(m-1)P .
\label{eq:amplified-adjoint}
\end{equation}
For large $i$ its pair is gklt. To see this, choose one log resolution
carrying the finite boundary support and the fixed b-data. The
crepant boundary coefficients vary continuously with $G_i$ and
$\lambda_i$; at the limit they are those of the ordinary klt pair
$(W,\Theta_W)$. Thus they remain strictly less than one.
The boundary $\Theta_W+mG_i$ is effective. The pair for $Q_{i,W}$
is the convex interpolation with weight $1/m$ between this pair and
$(W,\Theta_W)$, so it is gklt too.
The sum $\Theta_{i,W}+\lambda_iH_W$ is big:
$\lambda_i>0$, the trace $H_W$ is big by Corollary~\ref{cor:big-trace},
and $\Theta_{i,W}$ is effective. The model $W$ is strongly
$\Q$-factorial. Choose a sufficiently large Kähler scaling class and
apply \cite[Theorem~1.5]{HX2026} to obtain a finite $Q_{i,W}$-program.
All its steps are flips and its endpoint is nef. Indeed, as long as
the current model $T$ is small to $Y_i$, the comparison
\[
p^*Q_{i,T}=q^*Q_{i,Y_i}+[E],\qquad E\ge0
\]
holds on a common resolution, with $E$ exceptional over both models.
The curve-through-a-general-point argument in
Proposition~\ref{prop:limiting-nef} excludes a negative divisorial
contraction or a fibre-type contraction. It applies with this fixed
$i$, without any limiting argument.
We prove the remaining assertions by induction along these flips.
Suppose that on the current model $T$ the ordinary pairs
$(T,B^j_T)$ are klt, the $P^j_T=K_T+B^j_T$ are nef with
$\ell_jP^j_T$ line bundles, and the amplified pair with class
\[
A_T=mQ_{i,T}-(m-1)P_T
\]
is gklt. These conditions hold initially. A $Q_{i,T}$-negative
extremal ray is $A_T$-negative because $P_T$ is nef. The cone theorem
for the amplified pair gives a rational curve $\Gamma$ spanning that
ray with $A_T\cdot\Gamma\ge-8$. If $P_T\cdot\Gamma>0$, then
\eqref{eq:positive-gap} implies
\[
mQ_{i,T}\cdot\Gamma
=A_T\cdot\Gamma+(m-1)P_T\cdot\Gamma
\ge-8+(m-1)c>0,
\]
a contradiction. Hence the ray is $P_T$-trivial. Since every $u_j$
is positive and every $P^j_T$ is nef, it is also $P^j_T$-trivial for
each $j$.
Let $f:T\to Z\leftarrow T^+:f^+$ be this flip. For the ordinary
rational klt pair $(T,B^j_T)$, the line bundle
$M_j=\ell_jP^j_T$ is numerically trivial over $Z$, and
\[
M_j-(K_T+B^j_T)=(\ell_j-1)P^j_T
\]
is relatively nef. Lemma~\ref{lem:descent} gives an actual line
bundle $M_{j,Z}$ with $M_j=f^*M_{j,Z}$.
The pullback $(f^+)^*M_{j,Z}$ agrees on the common big open set with
the reflexive sheaf of $\ell_j(K_{T^+}+B^j_{T^+})$. Normality and
reflexivity extend the agreement. Thus the \emph{same} integer
$\ell_j$ is a Cartier index on $T^+$; no further multiple is taken.
It follows that both adjoints $P^j_T$ and $P^j_{T^+}$ pull back from
the same class on $Z$. On a common smooth compact Kähler resolution
$p:V\to T$, $q:V\to T^+$, their pullbacks are therefore equal.
The class $p^*P^j_T$ is nef, and nef descent along $q$ in
Lemma~\ref{lem:pullback-image} shows that $P^j_{T^+}$ is nef: the target
$T^+$ is generalized klt and hence has rational singularities.
Their actual discrepancy-change divisor is zero by negativity, so
the pairs remain klt and the comparison is crepant.
In particular the same is true of their convex combination $P_T$.
Finally, on a common resolution the discrepancy-change divisor for
the amplified pair is now $m$ times that for the $Q_i$-pair, because
the contribution from $P_T$ is zero. The latter divisor is effective
along the chosen generalized flip. The amplified pair therefore
remains gklt. This completes the induction, including the fixed-index
assertion and the validity of \eqref{eq:positive-gap} at the endpoint.
It remains to prove assertion (4). Write $Q=Q_{i,W_i}$ and
$P'=P_{W_i}$. Both are nef. Suppose a $Q$-trivial curve has positive
$P'$-degree. The face
\[
\mathcal F=\{\gamma\in\NA(W_i):Q\cdot\gamma=0\}
\]
then has a compact Kähler slice on which $P'$ takes a positive value.
Maximize $P'$ on that slice and choose an extreme point of its
maximizing face. This point is extreme in the slice of $\mathcal F$.
Since $\mathcal F$ is a face of $\NA(W_i)$, it spans an extremal ray
of the full cone. On this ray $Q=0$ and $P'>0$, so the amplified
class $A=mQ-(m-1)P'$ is negative. Its cone theorem gives a rational
generator $\Gamma$ with
\[
-8\le A\cdot\Gamma
=-(m-1)P'\cdot\Gamma
\le-(m-1)c<-8,
\]
which is impossible.
\end{proof}
\subsection{Descent to the original flipping base}
The preceding construction has produced a nef limiting class that
vanishes on every curve killed by the nearby nef generalized class.
We now descend it to the base of one of the original flips. This
forces the original negative adjoint to have degree zero on its
chosen ray, giving the contradiction.
\begin{proposition}
\label{prop:scaling-finite}
Let $(X,\Delta)$ satisfy the hypotheses of
Proposition~\ref{prop:scaling-construction}. The ordinary program
constructed there, with scaling of the fixed initial Kähler b-part,
is finite.
\end{proposition}
\begin{proof}
Suppose it were infinite. Use Proposition~\ref{prop:terminal-limit}
and choose $W$, then a sufficiently large $i$ and $W_i$, as in
Propositions~\ref{prop:limiting-nef} and
\ref{prop:trivial-perturbation}. Let $f_i:X_i\to Z_i$ be the
contraction of the chosen $D_i$-negative ray in the original flip
tail. The threshold class $L_i(\lambda_i)$ descends to $Z_i$.
Take a common smooth compact Kähler resolution
$p:V\to W_i$ and $q:V\to Y_i$. Put
$s=\mu_iq$ and $r=f_is$. The relevant maps are shown in
Figure~\ref{fig:perturbation-descent}.
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[
>=Latex,
every node/.style={font=\small},
map/.style={->,semithick}
]
\node (V) at (0,2.5) {$V$};
\node (Wi) at (-2.4,1.25) {$W_i$};
\node (Yi) at (2.4,1.25) {$Y_i$};
\node (Xi) at (2.4,0) {$X_i$};
\node (Zi) at (0,-1.15) {$Z_i$};
\draw[map] (V)--node[above left] {$p$}(Wi);
\draw[map] (V)--node[above right] {$q$}(Yi);
\draw[map] (Yi)--node[right] {$\mu_i$}(Xi);
\draw[map] (Xi)--node[below right] {$f_i$}(Zi);
\draw[map] (V)--node[left,pos=.63] {$r=f_i\mu_iq$}(Zi);
\draw[dashed,semithick] (Wi)--node[above,pos=.28] {small}(Yi);
\end{tikzpicture}
\caption{The final comparison is made on $V$.
The class $p^*P_{W_i}$ descends along $r$; no morphism from $W_i$
to $X_i$ or to $Z_i$ is required.}
\label{fig:perturbation-descent}
\end{figure}
The models $W_i$ and $Y_i$ are small modifications with the same
transformed boundary and b-data. Their $Q_i$-classes are both nef.
Applying Lemma~\ref{lem:small-comparison} in both directions gives
\begin{equation}
p^*Q_{i,W_i}=q^*Q_{i,Y_i}
=s^*L_i(\lambda_i).
\label{eq:equal-nef-pullbacks}
\end{equation}
Consequently this class has zero degree on every $r$-vertical curve.
For such a curve $C$, if $p(C)$ is a point then
$p^*P_{W_i}\cdot C=0$ immediately. Otherwise the projection formula
and \eqref{eq:equal-nef-pullbacks} show that $p(C)$ is
$Q_{i,W_i}$-trivial. Proposition~\ref{prop:trivial-perturbation}(4)
then gives $p^*P_{W_i}\cdot C=0$ as well.
The map $r$ is proper and birational. Its target $Z_i$ is normal
compact Kähler with rational singularities by the ordinary-step
construction, and its source $V$ is smooth compact Kähler.
Its fibres are connected, and its general fibre is a point.
Thus all the hypotheses of Lemma~\ref{lem:pullback-image} apply to
$r$. There is a class $\xi\in\BC(Z_i)$ with
\begin{equation}
p^*P_{W_i}=r^*\xi.
\label{eq:limiting-descent}
\end{equation}
We next form the ordinary canonical-and-boundary comparison.
Using compatible canonical data, the difference
\begin{equation}
F=p^*(K_{W_i}+\Theta_{W_i})
-s^*(K_{X_i}+\Delta_i)
\label{eq:final-exceptional-divisor}
\end{equation}
is an actual real divisor on $V$, with the relative canonical terms
interpreted as in Section~\ref{sec:analytic}.
It is exceptional over $X_i$: on every prime that is not
$s$-exceptional the two boundary coefficients agree, since the
limiting boundary on $Y_i$ pushes to $\Delta_i$ and $W_i$ is small
to $Y_i$. This constructs the divisor before any appeal to its
cohomology class.
By \eqref{eq:limiting-descent},
\[
[F]=p^*P_{W_i}-s^*D_i
=s^*(f_i^*\xi-D_i).
\]
Hence $F$ is numerically trivial over $X_i$.
Lemma~\ref{lem:negativity}, applied to $F$ and to $-F$, gives $F=0$.
Pullback injectivity for $s$ now yields
\[
D_i=f_i^*\xi\quad\text{in }\BC(X_i).
\]
Its intersection with the ray contracted by $f_i$ is therefore zero,
contradicting the strict $D_i$-negativity with which that ray was chosen.
The scaling program is finite.
\end{proof}
==== END Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//06-perturbation.tex ====
==== BEGIN Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//07-endpoints.tex ====
\section{The two endpoints}\label{sec:endpoints}
The scaling program is finite by Proposition~\ref{prop:scaling-finite}.
We now identify its endpoint from the pseudo-effectivity of the original
adjoint. Only ordinary birational comparisons are needed for this last step.
\begin{lemma}\label{lem:pe-step}
Let $(T,B)\dashrightarrow(T',B')$ be an ordinary negative divisorial
contraction or flip between normal compact Kähler klt pairs, with rational
boundaries and rational line-bundle adjoints. Then $K_T+B$ is pseudo-effective if
and only if $K_{T'}+B'$ is pseudo-effective.
\end{lemma}
\begin{proof}
Set $D=K_T+B$ and $D'=K_{T'}+B'$. The ordinary comparison in
Proposition~\ref{prop:ordinary-step} and Lemma~\ref{lem:negativity} gives,
on a common smooth compact Kähler resolution $p:V\to T$, $q:V\to T'$,
\[
p^*D=q^*D'+F,\qquad F\geq0,
\]
where $F$ is an actual divisor exceptional over $T'$.
If $D'$ is pseudo-effective, its pullback plus $[F]$ is pseudo-effective;
descent along $p$ shows that $D$ is pseudo-effective.
Conversely, if $D$ is pseudo-effective, exceptional descent along $q$
applied to $q^*D'+[F]$ proves that $D'$ is pseudo-effective.
Both uses of pullback and descent are justified by
Lemma~\ref{lem:positive-descent}.
\end{proof}
\begin{lemma}\label{lem:mori-not-pe}
Let $g:Y\to S$ be a projective surjective morphism of normal compact Kähler
varieties with $\dim S<\dim Y$. If a rational line bundle $D$ satisfies
$-D$ $g$-ample, then $D$ is not pseudo-effective.
\end{lemma}
\begin{proof}
Suppose that $c_1(D)$ contains a closed positive current $T$ with local
plurisubharmonic potentials. Choose a point $y$ where such a potential is
finite and which lies on a positive-dimensional component of a fibre of
$g$. This is possible on the dense open locus of fibres of the expected
dimension: the pole set of a plurisubharmonic potential contains no open
subset. That fibre component is projective, so hyperplane sections through
$y$ give an integral curve $C$ through $y$ contained in the fibre.
Let $\nu:C^\nu\to Y$ denote its normalization followed by inclusion.
Local potentials pull back under $\nu$ to subharmonic functions or to the
constant $-\infty$. Finiteness at a preimage of $y$ excludes the latter
alternative on the connected curve. Thus they define a positive current
$\nu^*T$ in the class $\nu^*c_1(D)$, and
\[
D\cdot C=\int_{C^\nu}\nu^*T\geq0.
\]
This contradicts $D\cdot C<0$, which follows from relative ampleness.
The choice of $y$ is essential: no restriction of $T$ to a curve contained
in its pole set has been used.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:finite}]
Run the scaling construction on the given pair $(X,\Delta)$.
Proposition~\ref{prop:ordinary-step} supplies each ordinary negative step,
and Proposition~\ref{prop:scaling-finite} proves that the construction
stops after finitely many such steps. Write its endpoint as $(Y,\Gamma)$.
All intermediate varieties are normal compact Kähler fourfolds, all
boundaries are the rational transforms of $\Delta$, and all pairs remain
ordinary klt and globally Weil-divisor $\Q$-factorial with the stipulated
canonical datum. The optional strong global property is preserved by the
same Proposition. The auxiliary extractions used to prove finiteness do
not change this program, which starts on $X$ itself.
The stopping rule gives either a nef adjoint $K_Y+\Gamma$ or a Mori
contraction $g:Y\to S$. In the latter case
Proposition~\ref{prop:ordinary-step} supplies a normal compact Kähler base,
a projective surjective morphism with connected fibres, $\dim S<4$,
relative numerical rank $\rho(Y/S)=1$, and relative ampleness of
$-(K_Y+\Gamma)$. These are precisely the required Mori fibre space
conditions, with relative rank computed from global Cartier classes.
The same Proposition gives relative Bott--Chern dimension one for every
negative contraction, including $g$.
By Lemma~\ref{lem:pe-step}, the adjoint at every stage has the same
pseudo-effectivity status as $K_X+\Delta$. If the latter is
pseudo-effective, Lemma~\ref{lem:mori-not-pe} excludes the Mori alternative,
so the endpoint is nef. If it is not pseudo-effective, the endpoint
cannot be nef, since nef classes are pseudo-effective by
Lemma~\ref{lem:positive-descent}; hence the endpoint has the asserted Mori
fibre structure.
For completeness, the nef endpoint has the discrepancy properties of a
log minimal model. No step extracts a divisor. On a common resolution of
the finite sequence, with maps $p:V\to X$ and $q:V\to Y$, the ordinary
effective comparison divisors add to
\[
p^*(K_X+\Delta)=q^*(K_Y+\Gamma)+F,
\qquad F\geq0,
\]
with $F$ exceptional over $Y$. Consequently, for every prime divisor $E$
over these models,
\[
a(E,Y,\Gamma)\geq a(E,X,\Delta).
\]
If a prime divisor on $X$ is contracted, its discrepancy increases strictly
at the divisorial step that contracts its transform and never decreases
thereafter. Thus the displayed inequality is strict for every such
prime. Here $a$ denotes the log discrepancy, as throughout the paper.
If the initial adjoint is nef, the construction takes no steps; conversely,
a zero-step nef outcome requires the initial adjoint to be nef. In the
non-pseudo-effective case a zero-step program is also allowed when the
first selected contraction already gives a Mori fibre structure on $X$.
This completes both endpoint assertions.
\end{proof}
==== END Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//07-endpoints.tex ====
==== BEGIN Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//08-relative.tex ====
\section{Relative programs for rational nef line data}\label{sec:relative}
Rational line bundles on projective modifications provide the relative
constructions used in the contraction induction and the companion
flip-termination theorem. We establish ordinary representatives over Stein
neighborhoods, elementary relative steps, a finite auxiliary program,
and extraction of prescribed divisors. All these constructions take
place over a fixed compact base. Their numerical spaces use degrees
of global line bundles, rather than the full Bott--Chern space of the
ordinary contractions above.
The nef data in this section are rational line bundles, distinct from the
possibly transcendental nef classes used in the supporting-contraction proof.
\subsection{Rational bundles and generalized pairs}
\begin{convention}\label{conv:rational-bundles}
A rational line bundle is an element of $\Pic(T)\otimes_{\Z}\Q$.
An equality of rational line bundles means an isomorphism of holomorphic
line bundles after a positive common multiple. A rank-one reflexive
sheaf is rationally invertible if one of its positive reflexive powers
is invertible. We use additive notation for these objects, their
tensor products, and their reflexive transforms. In particular, $K_T$
denotes the reflexive canonical sheaf, without a choice of a global
meromorphic canonical form.
For a projective morphism $T\to V$, the words relatively nef and
relatively ample, applied to rational line bundles, refer respectively
to degrees on curves in the fibers and ampleness of a positive integral
power. Real combinations of rational line bundles have the corresponding
numerical meanings. Absolute nefness of a bundle on a compact K\"ahler
space implies relative nefness in this sense.
\end{convention}
\begin{definition}\label{def:generalized-data}
Rational generalized b-line data on a normal compact space $T$ consist
of an effective rational divisor $B$, a projective modification
$p\colon W\to T$, and a rational line bundle $M_W$ on $W$.
The bundle is pulled back on every higher model; this b-object is denoted
by $\bM$. We require $M_W$ to be nef, or to be nef over a specified
base in a relative argument. Its trace on $T$ is obtained by taking a
proper direct image of a cleared power and then its double dual. The
adjoint
\[
D_T=K_T+B+M_T
\]
is required to be rationally invertible. We may increase $W$ to a smooth
projective log resolution carrying the data. On such a model the
boundary $B_W$ is determined by the crepant formula
\begin{equation}\label{eq:generalized-crepant}
K_W+B_W+M_W=p^*D_T.
\end{equation}
The equality uses the natural meromorphic identifications over the
isomorphism locus. Log discrepancy is
$a(E;T,B+\bM)=1-\operatorname{coeff}_E B_W$, evaluated on a model
carrying $E$. Generalized klt and generalized lc, abbreviated gklt and
glc, mean that all these discrepancies are positive and nonnegative,
respectively. A generalized terminal pair is gklt and has
$a(E;T,B+\bM)>1$ for every exceptional prime divisor over $T$.
A generalized dlt pair, abbreviated gdlt, is a glc pair for which there
is a Zariski open set $T^\circ\subset T$ such that
$(T^\circ,B|_{T^\circ})$ is simple normal crossing, the b-data descend
there, and $T^\circ$ contains the general point of every
zero-discrepancy center. We choose the carrier and subsequent log
resolutions to be isomorphisms on this open set. Thus the centers in
$T^\circ$ are strata of the coefficient-one boundary.
\end{definition}
We also allow real combinations of a fixed finite set of rational data
for local cone arguments and real scaling. The extraction and termination
statements below use rational
data.
\begin{lemma}[Traces and exceptional comparisons]\label{lem:trace}
Let $p\colon W\to T$ be a projective modification of normal compact
spaces.
\begin{enumerate}[label=\textup{(\roman*)}]
\item For a line bundle $L$ on $W$, $(p_*L)^{**}$ is coherent and
rank one. If its appropriate reflexive power is invertible, the natural
comparison between that power pulled back to $W$ and the corresponding
power of $L$ is an exceptional Cartier divisor. This assertion uses
actual bundles and is compatible with further pullback.
\item Rank-one reflexive sheaves can be transported coherently across
a bimeromorphic map by pullback, proper direct image, and double dual on
a common projective resolution. Their transforms agree on the common
codimension-one open set and are independent of the chosen resolution
when the map is small.
If $L=L'+G$ is an identity of rational bundles on the carrier with
$G$ a rational divisor, their reflexive traces satisfy
$L_T=L'_T+p_*G$. The trace of $K_W$, or of its pullback to a higher
model, is $K_T$.
\item If $T$ is globally strongly $\Q$-factorial and $\bM$ is nef
over a base over which $p$ is projective, then
\begin{equation}\label{eq:nef-trace-defect}
F_M:=p^*M_T-M_W\geq0
\end{equation}
is exceptional. Consequently, forgetting the nef part of a gklt,
respectively gdlt, pair on $T$ gives an ordinary klt, respectively dlt,
pair. A gdlt pair becomes gklt after replacing $B$ by
$B-\epsilon\lfloor B\rfloor$, for every sufficiently small
$\epsilon>0$.
\end{enumerate}
\end{lemma}
\begin{proof}
Proper direct image preserves coherence. On the isomorphism locus its
rank is one, and double dual gives a rank-one reflexive sheaf on the
normal target. Locally on $T$, choose meromorphic generators of this
coherent sheaf and of a cleared invertible power. Pulling back their
comparison gives a meromorphic comparison with the bundle upstairs.
One can see that the comparison is meromorphic, rather than merely
defined on a punctured open set, by using a local finite presentation
of $p_*L$: its generators evaluate to holomorphic sections of $L$, and
their ratios supply local denominators. The comparisons agree on
overlaps because they agree on the isomorphism locus. Their divisor
is exceptional. This also proves compatibility with powers and
pullback. A global meromorphic frame was not used.
For a common resolution $W$ with maps to the two spaces, first pull
back the sheaf, remove torsion, and take its double dual; after an
additional projective principalization it is a line bundle. Proper
direct image to the other space followed by double dual is coherent.
On a common codimension-one open set it is the required transform.
Normality gives uniqueness of a reflexive extension across codimension
at least two, proving the assertion for small maps.
The additive trace identity is checked at every codimension-one point
of $T$, where the modification is an isomorphism, and then extended
reflexively. The same observation applies to canonical sheaves and
their pullbacks. It requires compatible local canonical forms only.
Global strong $\Q$-factoriality makes $M_T$ rationally invertible, so
the first assertion defines $F_M$. Its negative is $p$-nef and has
zero pushforward. Lemma~\ref{lem:negativity} gives $F_M\geq0$. The ordinary crepant boundary is $B_W-F_M$:
\eqref{eq:generalized-crepant} gives explicitly
\[
p^*(K_T+B)-K_W=B_W+M_W-p^*M_T=B_W-F_M.
\]
Thus ordinary log discrepancies are at least the generalized ones.
Every ordinary zero center is a generalized zero center and is
therefore generically in the specified simple normal crossing open
set. This proves the klt and dlt assertions.
The effective divisor $\lfloor B\rfloor$ is rational Cartier on $T$.
Decreasing it subtracts the effective divisor
$\epsilon p^*\lfloor B\rfloor$ from the crepant boundary. A former
positive discrepancy stays positive. Every former zero-discrepancy
place has center in the floor, since its center is generically a
coefficient-one stratum, and hence has strictly positive order on the
pullback of the floor. Its new discrepancy is positive as well. The
boundary on $T$ remains effective for $0<\epsilon<1$.
\end{proof}
\subsection{Ordinary representatives over Stein neighborhoods}
We use the following local analytic inputs with their stated domains.
A Stein compact has property \textup{(P)} if its intersections with
analytic sets defined near the compact have finitely many connected
components.
For a projective morphism over a Stein compact satisfying property
\textup{(P)}, the ordinary klt cone theorem has finitely many rays in
each region truncated by a relatively ample divisor
\cite[Theorem~7.2]{Fujino2022}. The base point free theorem generates
every sufficiently large integral power of a nef line bundle $L$ when
$aL-(K+\Delta)$ is relatively ample
\cite[Theorems~6.2 and~6.5]{Fujino2022}. Ordinary klt flips exist for
small projective morphisms of normal analytic spaces without a
$\Q$-factoriality assumption \cite[Theorem~1.14]{Fujino2022}. Finally, on
a smooth source the multigraded adjoint ring with simultaneous SNC
subunit rational boundaries and a common relatively ample rational
part is locally finitely generated
\cite[Theorem~3.1]{DHP2024}. We explain the passages from these local
statements to our data and compact base.
\begin{lemma}[Local ordinary representatives]\label{lem:ordinary-representative}
Let $\pi\colon T\to V$ be projective and bimeromorphic, let
$(T,B+\bM)$ be gklt rational data, and suppose the carrier is
projective over $T$ and $M_W$ is nef over $V$. Let $H$ be a
$\pi$-ample rational line bundle. Near any sufficiently small Stein
compact in $V$ there are an effective ordinary klt boundary $\Delta$,
an effective rational divisor $H_0\simq H$, and $\eta>0$ such that
\begin{equation}\label{eq:ordinary-replacement}
K_T+\Delta\simq D_T,\qquad \Delta\geq\eta H_0.
\end{equation}
These are actual rational line-bundle equivalences on that
neighborhood. For $0\leq\delta\leq\eta/2$,
$\Delta-\delta H_0$ is effective klt and represents $D_T-\delta H$.
For finitely many rational adjoints of this type on a common initial
space, the representatives can have a common positive ample part.
Their convex combinations have the same properties. The local cone
conclusions also hold for gklt real combinations of finitely many
rational data.
\end{lemma}
\begin{proof}
Write $p\colon W\to T$ for a smooth carrier, with $p$ projective, and put
$\rho=\pi p$. Work on a Stein neighborhood $U$ of the prescribed
compact. Choose a $\rho$-ample rational line bundle $A$. For an
integer $m$ clearing denominators, the sheaf
\[
\rho_*\cO_W\bigl(m(M_W-A)\bigr)
\]
is coherent and has rank one over the bimeromorphic isomorphism locus.
It is nonzero. Cartan's Theorem~A gives a nonzero section over $U$,
whose pullback defines an effective divisor. Hence, after division
by $m$,
\begin{equation}\label{eq:relative-kodaira-section}
M_W\simq A+E,\qquad E\geq0.
\end{equation}
This proves relative bigness directly, including when $M_W=0$.
The divisor $E$ need not be exceptional. The equivalence comes from
a section of the specified bundle, and introduces no unspecified
numerical or base twist.
Keep $W$ and the $\rho$-ample bundle $A$ fixed. Choose an auxiliary
log resolution of the finite fixed supports only to test the
singularities. On that auxiliary resolution there are finitely many
crepant coefficients over the compact preimage, all strictly below
one. Hence a sufficiently small rational $\epsilon>0$ makes
$(W,B_W+\epsilon E)$ sub-klt.
The rational bundle
\[
A_\epsilon=(1-\epsilon)M_W+\epsilon A
\]
is $\rho$-ample. Choose a rational $\eta>0$ such that
$A_\epsilon-\eta p^*H$ is still $\rho$-ample. On a smaller
neighborhood, sufficiently large powers of these relatively ample
bundles are relatively generated. Cartan's Theorem~A supplies finitely
many sections generating over the compact preimage; shrink once more
so that they generate throughout. Divided general members give
\[
H_0\geq0,\quad H_0\simq H,
\qquad G\geq0,\quad
G\simq A_\epsilon-\eta p^*H.
\]
Choose the members on $T$ and $W$, respectively, with large
denominators, and test generality on the auxiliary log resolution.
Their pulled-back systems remain free; we do not claim that a pulled-back
ample bundle stays ample. Analytic
Bertini gives
\[
C_W=B_W+\epsilon E+G+\eta p^*H_0
\]
sub-klt. Here generation is used for the ample systems only; the
nonzero section producing $E$ was not claimed to generate its bundle.
The analytic generation and Bertini statements used in this step are
\cite[Theorems~2.12 and~2.20]{DHP2024}.
Push down to obtain
\[
\Delta=B+\epsilon p_*E+p_*G+\eta H_0\geq\eta H_0.
\]
Only exceptional coefficients of $B_W$ could be negative, so
$\Delta$ is effective. The actual upstairs identity is
$K_W+C_W\simq p^*D_T$. Choose an integer clearing this identity,
push forward the resulting line-bundle isomorphism, and take double
duals. Agreement in codimension one gives
\begin{equation}\label{eq:replacement-cartier}
\cO_T\bigl(m(K_T+\Delta)\bigr)\simeq\cO_T(mD_T).
\end{equation}
In particular, the new ordinary adjoint is rational Cartier. We have
not inferred Cartierness of newly chosen local divisors from global
strong $\Q$-factoriality. The divisor
\[
K_W+C_W-p^*(K_T+\Delta)
\]
is exceptional and $p$-numerically trivial by the same line-bundle
identity. Lemma~\ref{lem:negativity} in both signs makes it zero.
Thus $C_W$ is the ordinary crepant boundary, and $(T,\Delta)$ is klt.
Subtracting $\delta H_0$ gives the asserted representation of
$D_T-\delta H$ and leaves the common ample part $(\eta/2)H_0$.
For finitely many vertices, use one carrier and one $H_0$, choose their
effective divisors in \eqref{eq:relative-kodaira-section}, and resolve
all fixed supports together. Take a common sufficiently small
$\epsilon$ and then a common $\eta$. Choose the finitely many free
members generally. On the resulting common resolution all the
crepant coefficients are uniformly below one. Convex combinations
are therefore klt and keep the same ample part. The parameter space
can be the simplex of vertex weights; no affine independence of the
original vertices is required.
For real data, first express the bundle in a finite real span of
rational bundles on the carrier. Every signed rational generator has
an effective representative over the Stein neighborhood by the same
Cartan argument. Thus a real nef bundle also has an expression
$M_W\sim_{\R}A+E$ with $A$ relatively ample and $E\geq0$ in a fixed
finite divisor span. The preceding small-$\epsilon$ construction
applies. A real relatively ample bundle can be written as a positive
combination of finitely many nearby rational relatively ample bundles;
divided general members of those bundles give the required real
boundary. The ordinary real klt cone theorem is consequently
applicable. Only the rational construction is used for line-bundle
descent and for the finite adjoint rings.
\end{proof}
Lemma~\ref{lem:ordinary-representative} also supplies the hypothesis
needed to apply integral descent to a relatively antiample generalized
klt adjoint. Indeed its ordinary representative is klt and has the
same relative line-bundle class. If an integral line bundle $L$ has
zero degree on every contracted curve, then
$aL-(K_T+\Delta)$ is relatively ample. Lemma~\ref{lem:descent}
therefore gives the actual identity $L=f^*f_*L$; the consecutive-power
argument in its proof introduces no extra integral multiple. Clearing
one denominator gives the corresponding assertion for rational line
bundles. The representatives may differ on the Stein neighborhoods:
the descended bundle and its evaluation map are intrinsic and glue.
\subsection{Global relative degrees and elementary steps}
For a projective morphism $\pi\colon T\to V$ of compact spaces, let
$N^1_{\mathrm{gl}}(T/V)_{\Q}$ be the space of global rational line
bundles modulo equality of degrees on all curves contracted by $\pi$.
Put $N^1_{\mathrm{gl}}(T/V)=N^1_{\mathrm{gl}}(T/V)_{\Q}\otimes\R$,
and let $N_{1,\mathrm{gl}}(T/V)$ be the dual space generated by relative
curve functionals. These spaces are finite-dimensional: pull back to
a compact resolution and use the first Chern class in its finite
dimensional cohomology. Write
\[
\overline{\mathrm{NE}}_{\mathrm{gl}}(T/V)
\subset N_{1,\mathrm{gl}}(T/V)
\]
for the closed cone generated by these functionals. A relatively ample
global line bundle gives a compact normalized slice of this cone.
Here are details about the compactness and the local cone statements
which will be used. Choose a finite cover of $V$ by interiors of
Stein compacts $W_\nu$ in Stein open sets $U_\nu$. We choose them
semianalytic in coordinate charts, so they satisfy property
\textup{(P)}. On each chart the ordinary
relative numerical cone has a compact slice normalized by the
restriction of a fixed global relatively ample bundle $H$. Restricting
global bundles defines a linear projection from the local curve space
to $N_{1,\mathrm{gl}}(T/V)$. The image of that slice is compact and
still lies in $H=1$. The convex hull of these finitely many compact
images is compact and is precisely the global normalized slice.
Indeed every relative curve lies over one of the compacts, and passage
to closure commutes with this finite compact convex hull.
Apply Lemma~\ref{lem:ordinary-representative} on the charts to a gklt
adjoint $J$. The local cone theorem and projection show that, for each
$c>0$, only finitely many generating curve rays are needed in the
global region $(J+cH)<0$. The remaining summand is in
$(J+cH)\geq0$. If $J$ is nef, apply the same argument to the
ordinary representative of $J-\delta H$, with a further ample
perturbation $(\delta/2)H$. In the slice $H=1$ this gives a finite
set of curve rays near $J=0$ and a remainder on which
$J\geq\delta/2$. In particular the face $J=0$ is generated by
finitely many curves. It also gives the following useful openness
statement: if a linear perturbation is positive on that zero face,
it is positive on the whole slice for all sufficiently small positive
parameters. This follows by checking the finite rays and using the
uniform $\delta/2$ margin on the remainder.
For a rational gklt adjoint, every negative global extremal ray has a
rational supporting line bundle. To see this, use the finite
decomposition in a truncated negative neighborhood of that ray.
The normalized point of an extremal ray is outside the compact convex
hull of the other generators and the remaining nonnegative part.
Separation gives a nef functional vanishing only on that ray, with a
strict margin on the other part. These functionals contain an open
subset of its perpendicular hyperplane. The ray is generated by a
curve, so the hyperplane is rational; choose a rational functional
there. It is a rational line-bundle class by the definition of
$N^1_{\mathrm{gl}}$. Subtracting a sufficiently small positive
multiple of the adjoint is strictly positive on the normalized slice.
After rescaling the support, its difference from the adjoint is
relatively ample. Strict positivity here implies ampleness on each
projective fiber by Kleiman's criterion, and relative ampleness follows
over a neighborhood of each fiber. This reasoning involves line
bundle degrees on projective fibers only.
\begin{lemma}[Sheaves and signs across a step]\label{lem:sheaf-step}
Let $T$ be globally strongly $\Q$-factorial. Suppose a projective
bimeromorphic contraction $f\colon T\to Z$ has all its contracted
curves in one nonzero ray for global rational line-bundle degrees.
Let $J$ be a rational line bundle with $-J$ $f$-ample, admitting the
local klt representatives required in Lemma~\ref{lem:descent}.
\begin{enumerate}[label=\textup{(\roman*)}]
\item If $f$ is divisorial, its exceptional locus is a single prime
$E$, with $E\cdot R<0$, and $Z$ is globally strongly $\Q$-factorial.
Every adjoint pushed down to $Z$ is rational Cartier.
\item If $f$ is small and its $J$-positive small projective model
$f^+\colon T^+\to Z$ exists, then $T^+$ is globally strongly
$\Q$-factorial. Every global rational line bundle $L$ on $T$ has a
rationally invertible transform $L^+$, and
\begin{equation}\label{eq:step-line-adjustment}
L=f^*A+cJ\quad\Longrightarrow\quad
L^+=(f^+)^*A+cJ^+
\end{equation}
for an actual rational line bundle $A$ on $Z$, where
$c=(L\cdot R)/(J\cdot R)\in\Q$. In particular, a negative, zero,
or positive degree for $L$ becomes respectively relative ampleness,
relative triviality, or relative antiampleness for $L^+$.
\end{enumerate}
\end{lemma}
\begin{proof}
For an exceptional prime $E$, global strong $\Q$-factoriality makes $E$
rational Cartier. If it were $f$-nef,
Lemma~\ref{lem:negativity}\textup{(i)} applied to $-E$ would give
$-E\geq0$. Thus $E\cdot R<0$. Every contracted curve
must lie in $E$, since an effective rational Cartier divisor has
nonnegative intersection with a curve not contained in its support.
Every point of a nontrivial projective fiber lies on a curve in that
fiber. Hence the entire exceptional locus is $E$, and no second
exceptional prime exists.
For a global rank-one reflexive sheaf $\cF$ on $Z$, form its reflexive
pullback to $T$. A positive power is a line bundle. Subtract a
rational multiple of $E$ so that its degree is zero on $R$ and apply
Lemma~\ref{lem:descent}. The descended rational bundle agrees with
$\cF$ to that power on the complement of a codimension-two set in
$Z$. Reflexivity extends the identity. This proves the global strong
condition and, in particular, applies to the pushed-forward canonical,
boundary, and nef-trace sheaves.
In the small case, $L-cJ$ has zero degree and descends by
Lemma~\ref{lem:descent}. The right-hand side of
\eqref{eq:step-line-adjustment} is a rational line bundle on $T^+$
and agrees with the transform of $L$ on the common open set. It is
therefore the reflexive transform. The sign conclusions follow from
relative ampleness of $J^+$ and the sign of $c$.
Conversely, transport an arbitrary global rank-one reflexive sheaf
$\cG$ on $T^+$ to $T$ as in Lemma~\ref{lem:trace}. A reflexive power
of its transform is invertible on $T$. The preceding argument makes
its transform a rational line bundle on $T^+$. It agrees with the
corresponding reflexive power of $\cG$ on the common codimension-one
open set, and hence everywhere. This is global strong $\Q$-factoriality
on $T^+$; no assertion about arbitrary open subsets is involved.
\end{proof}
\begin{definition}\label{def:relative-elementary}
An elementary relative step is a
projective divisorial contraction
or a small projective negative-to-positive adjoint surgery over a
normal compact K\"ahler base, with globally strongly $\Q$-factorial
source and output. All contracted curves on the negative side span
one nonzero ray when paired with global rational line bundles. The
boundary is pushed forward or strictly transformed, and the same
b-data are used on common higher models. The small positive model is
the relative Proj of the actual adjoint algebra. No condition on its
relative Bott--Chern dimension is included.
\end{definition}
\begin{proposition}[Relative continuation]\label{prop:relative-steps}
Let $\pi\colon T\to V$ be a projective bimeromorphic morphism of
normal compact K\"ahler spaces. Assume $T$ is globally strongly
$\Q$-factorial and $(T,B+\bM)$ is rational gdlt, with projective
carrier and nef data over $V$. If its adjoint is not curve-nef over
$V$, there is an elementary negative step over $V$. Its target is
projective over $V$, compact K\"ahler, globally strongly
$\Q$-factorial, and gdlt with the transformed data. The construction
is available after every finite prefix whose final adjoint is not
curve-nef over $V$. For gklt data one may choose
any negative extremal ray in the global degree cone.
\end{proposition}
\begin{proof}
First assume gklt. The global cone construction above gives a negative
ray and a nef rational supporting line bundle $L$ which vanishes
precisely on that ray and for which a multiple minus the adjoint is
relatively ample. Apply the ordinary base point free theorem on the
finite Stein cover, using Lemma~\ref{lem:ordinary-representative}.
It generates a common positive power of $L$ relative to $V$.
The relative section morphism and its Stein factorization give
$f\colon T\to Z$ with connected fibers and normal target, contracting
exactly the curves on the selected ray. The target is projective over
$V$: it is finite over the image in the projectivization of the
coherent sheaf of relative sections. Since $T\to V$ is bimeromorphic,
$Z$ has the same dimension and $f$ is bimeromorphic. The negative
adjoint is relatively ample, because on every $f$-fiber its class is
a positive multiple of a relatively ample class in the one-dimensional
global degree space.
If $f$ is divisorial, apply Lemma~\ref{lem:sheaf-step}. If it is
small, apply Lemma~\ref{lem:ordinary-representative} locally over
$Z$, followed by the ordinary analytic klt flip theorem
\cite[Theorem~1.14]{Fujino2022}. Equivalently, the actual relative
adjoint algebra is locally finitely generated by
\cite[Corollary~3.7]{DHP2024}; the flip theorem also supplies its
normality and smallness. The local models glue by the intrinsic
algebra. Compactness permits a common Veronese generated in degree
one. On the common codimension-one open set its tautological bundle
is the stated adjoint power. Both sheaves are reflexive on the normal
positive side, so this identity extends globally. Thus the positive
adjoint is rational Cartier and relatively ample as an actual bundle.
Lemma~\ref{lem:sheaf-step} gives global strong $\Q$-factoriality.
All spaces constructed are projective over a compact K\"ahler space.
A relatively ample line bundle has a metric with positive curvature
in the fiber directions; adding a sufficiently large pullback of a
K\"ahler form on the base gives a K\"ahler form. Thus the spaces are
compact K\"ahler. The common carrier remains projective over the
models: take the main component of the graph over $Z$ and resolve it.
The pulled-back nef data are still nef over $V$.
For gdlt data, replace $B$ by
$B-\epsilon\lfloor B\rfloor$ with small rational $\epsilon>0$.
By Lemma~\ref{lem:trace} this is gklt; denote its adjoint by $D_\epsilon$.
Normalize by a relatively ample bundle $H$, and choose a relative
curve class $z$ with $H\cdot z=1$ and $D\cdot z=-a<0$.
Choose $\epsilon$ so that
$|(D_\epsilon-D)\cdot u|<a/4$ on the whole compact normalized slice.
Choose an extreme point $u_0$ of the face where $D$ attains its
minimum on this slice. It is an extreme point of the entire slice,
so it spans a global extremal ray $R$. Since
$D\cdot u_0\leq D\cdot z=-a$, the uniform perturbation bound gives
$D_\epsilon\cdot u_0<-3a/4$. Thus $R$ is negative for both
adjoints. The gklt construction contracts this global extremal ray
for the perturbed pair.
The original adjoint has the same negative sign on the ray. In the
small case Lemma~\ref{lem:sheaf-step} identifies its positive transform
with a positive multiple of the perturbed positive adjoint modulo a
base pullback. In the divisorial case that lemma makes its pushdown
rational Cartier. These are therefore steps for the original data.
Finally Lemma~\ref{lem:negativity} proves the discrepancy comparisons.
For gklt, positivity is preserved. For gdlt, a new zero-discrepancy
center cannot have its general point in an exceptional locus, by
strictness. Its corresponding old zero center is generically in the
unchanged good open set. The transformed boundary is SNC and the
b-data descend there. This gives a gdlt good open set on the output.
The same hypotheses hold for the next relative morphism to $V$, so
the construction continues whenever relative nefness fails.
\end{proof}
\subsection{Finiteness on a compact relative base}
The next proposition constructs some finite relative program. It will
be used to replace a single step of an arbitrary sequence by a finite
program on a higher model. It does not assert termination of every
relative program. We first prove the finite-model statement needed
for this construction.
\begin{lemma}[Local finiteness of marked ample models]\label{lem:marked-models}
Let $T\to V$ be projective and bimeromorphic and let
$D_0,\ldots,D_s$ be rational gklt adjoints on $T$ with a common
projective carrier and nef data over $V$. On a smaller neighborhood
of a prescribed Stein compact, the rational adjoints in the simplex
of their convex combinations have only finitely many marked relative
ample models. Here a marking is the bimeromorphic map from $T$,
and an ample model is required to have an effective exceptional
pullback difference. No local $\Q$-factoriality of $T$ is assumed.
\end{lemma}
\begin{proof}
Lemma~\ref{lem:ordinary-representative} supplies ordinary klt
representatives with a common positive relatively ample part.
Choose a common projective log resolution $a\colon S\to T$ by
successive blowups and principalizations with centers exceptional over
$T$, resolving all fixed supports. This choice can be made with an
effective exceptional divisor $F$ such that $-F$ is $a$-ample:
for each blowup take its effective tautological exceptional divisor,
and inductively add it to a sufficiently large positive multiple of the
pullback of the divisor for the preceding composite. Relative
ampleness under composition proves the assertion; further such blowups
preserve it. For the identity resolution take $F=0$.
This is the blowup-resolution construction used in
\cite[Theorem~2.13, Remark~2.14, and the proof of Corollary~3.7]{DHP2024}.
For each vertex choose every exceptional coefficient strictly between
the maximum of zero and its crepant coefficient and one. We obtain
\[
K_S+\Gamma_j=a^*D_j+E_j,
\qquad E_j\geq0\text{ exceptional},\quad
0\leq\Gamma_j<1.
\]
All supports, including that of $F$, are SNC. The ample
part downstairs permits a common relatively ample part upstairs.
More explicitly, scale the chosen $F$ by a small positive rational number so that
$a^*H-F$ is ample over $V$, where $H$ is the fixed ample part
downstairs. For a sufficiently small $\tau>0$ put
$A=\tau(a^*H-F)$. The finitely many exceptional coefficients of the
$\Gamma_j$ have positive margins from both zero and one. Then
$\Gamma_j-A$ is effective with coefficients below one; their
supports, including that of $F$, are SNC. One can either use this
relatively ample rational divisor $A$ directly or replace it by a
divided general member and make the corresponding simultaneous
linear-equivalence changes in the boundaries.
The multigraded ring of the $K_S+\Gamma_j$ is locally finitely
generated by \cite[Theorem~3.1]{DHP2024}. For nonnegative integers
$n_j$, the correction $\sum n_jE_j$ is effective and exceptional
over $T$. Normality gives
$a_*\cO_S(\sum n_jE_j)=\cO_T$. The projection formula therefore
identifies every multigraded piece with the corresponding piece for
the $D_j$, compatibly with multiplication. Passing to one common
Veronese removes all rounding. Thus their actual multigraded ring
is locally finitely generated as well.
For completeness, finite generation gives the required finiteness as
follows. On a sufficiently small neighborhood choose homogeneous
generators with multidegrees $d_1,\ldots,d_N\in\N^{s+1}$. For a
rational weight $w$ in the simplex, a point of the relative spectrum
is semistable for the diagonal ring of weight $w$ precisely when
$w$ is in the cone generated by those $d_i$ whose generators are
nonzero at the point. To verify this equivalence, clear the rational
coefficients of such a cone expression and form the corresponding
monomial; conversely a nonvanishing homogeneous section contains a
nonvanishing monomial of that weight. There are only finitely many
subsets of the generators and hence finitely many resulting open
sets. The diagonal Proj is the quotient of its open set, obtained by
the degree-zero localizations of these monomials. Uniqueness of this
quotient shows that weights with the same open set give the same
ample model. All identifications commute with the map from $T$,
which is determined on the bimeromorphic isomorphism locus. A finite
cover of the Stein compact makes the number of possibilities finite
there. This is also the marked finite-model conclusion of
\cite[Theorem~E]{Fujino2022}, now with the ordinary representatives
and the smooth reduction justified.
\end{proof}
\begin{proposition}[A finite relative gklt run]\label{prop:relative-finite}
In the setting of Proposition~\ref{prop:relative-steps}, assume the
pair is gklt. There is a finite sequence of elementary steps over
$V$ whose final adjoint is curve-nef over $V$.
\end{proposition}
\begin{proof}
Let $D=K_T+B+M_T$. We may suppose $D$ is not already relatively
nef.
\smallskip
\noindent\emph{A scaling class valid on every later model.}
Choose global rational line bundles $L_1,\ldots,L_r$ whose
degrees form a basis of $N^1_{\mathrm{gl}}(T/V)_{\Q}$, and write
the class of $D$ as $d\in\Q^r$.
These bundles continue to span the degree space after every finite
sequence of elementary steps. Indeed every line bundle on a later
model can be transported to a rational line bundle on the preceding
one by Lemma~\ref{lem:trace} and global strong $\Q$-factoriality.
For a divisorial contraction its pullback is also such a bundle;
exceptional-divisor adjustments are available on the source.
Moreover a bundle numerically trivial over $V$ remains so after a
step. It first descends across that contraction by
Lemma~\ref{lem:descent}. Any curve on the new side over $V$ has a
projective lift to a common model and then a curve upstairs mapping
onto it with positive degree. Intersecting the pullbacks of the
descended bundle proves its zero degree. Thus every later relative
curve determines a rational functional $v\in\Q^r$, nonzero because
the later space has a relatively ample line bundle.
Choose a real vector $h\in\R^r$ defining a real linear combination
$H$ of the chosen rational line bundles, with $H$ and $D+H$
relatively ample. We choose it outside
the following countable family of proper rational hyperplanes:
\begin{align}
v(h)&=0 &&(0\ne v\in\Q^r),\label{eq:scaling-generic1}\\
v(d)w(h)-w(d)v(h)&=0
&&\left(\begin{array}{l}v,w\in\Q^r\text{ independent},\\
(v(d),w(d))\ne(0,0)\end{array}\right).
\label{eq:scaling-generic2}
\end{align}
The ample conditions define a nonempty open set, and a countable
union of proper hyperplanes cannot exhaust it. The rational
functionals here range over all possibilities, so this single choice
works at every later model; no enumeration of the models is required.
\smallskip
\noindent\emph{The scaling steps and their strictly decreasing thresholds.}
On a current model, let $D_i$ be the reflexive transform of $D$,
and let $H_i$ be the same real linear combination of the reflexive
transforms of the rational line bundles defining $H$; across a
divisorial step these transforms are the corresponding pushforwards.
Regard $H$
also as nef b-data over $V$, carried on the initial space. Its
pullback to a common carrier is relatively nef, so for every
$t\geq0$ the initial generalized pair with adjoint $D+tH$ is gklt.
At each preceding step of threshold $t_j$, its adjoint has negative
degree for $0\leq t<t_j$ and zero degree for $t=t_j$.
Lemma~\ref{lem:negativity} consequently shows that the transformed
pair is gklt throughout the parameter interval below the preceding
threshold.
Start with the relatively ample class $D+H$. On a current model
take the lower endpoint $t_i$ of the nef segment extending downward
from the preceding parameter. If $t_i=0$, then $D_i$ is nef and
the process stops. Otherwise $J_i=D_i+t_iH_i$ is nef and not ample.
The local cone argument preceding Lemma~\ref{lem:sheaf-step}, applied
to this real gklt adjoint, shows that its zero face is generated by
finitely many curves. A generating curve cannot have $D_i$-degree
zero, because then its $H_i$-degree would be zero, contrary to
\eqref{eq:scaling-generic1}. Two independent such rays would violate
\eqref{eq:scaling-generic2}. Hence the face is one ray in the global
degree space. Its $D_i$-degree is negative: if it were positive,
lowering $t$ would be positive on this face, and the uniform cone
margin would give nefness below $t_i$, a contradiction.
Apply Proposition~\ref{prop:relative-steps} to this $D_i$-negative
ray. The threshold class descends. Here descent for a real class
requires no real-bundle base point free theorem: the perpendicular
hyperplane of the rational ray is rational, so express the threshold
class as a real combination of rational line-bundle classes of degree
zero and descend each by Lemma~\ref{lem:descent}. The threshold
comparison is crepant and its transform is nef.
The thresholds strictly decrease. In a small step, the zero face on
the positive side is again generated by curves by the real gklt
local cone argument. Equations \eqref{eq:scaling-generic1}--
\eqref{eq:scaling-generic2} allow at most one ray. Flipped curves
supply that ray and have positive $D$-degree. Lowering the parameter
is therefore positive on it, and the uniform margin makes the class
ample for a nonempty interval below the old threshold. In a
divisorial step there are no zero curves at the old threshold:
lifting one to the old space would make it lie on the contracted
zero ray while still mapping onto a curve. The same margin again
gives an ample interval. At every stage, the open interval between
successive thresholds consists of relatively ample classes: an
interior zero curve would have zero degree for two parameters, hence
zero $H_i$-degree, contrary to \eqref{eq:scaling-generic1}.
\smallskip
\noindent\emph{Finitely many marked ample models.}
It remains to exclude infinitely many such intervals. Choose finitely
many rational relatively ample bundles $H^{(1)},\ldots,H^{(s)}$
near $H$, with $H$ in the interior of their convex hull and with
$D+H^{(j)}$ relatively ample. Consider the fixed rational simplex
of weights with vertices
\begin{equation}\label{eq:scaling-polytope}
D,\ D+H^{(1)},\ldots,D+H^{(s)}.
\end{equation}
At an interior parameter of any one of the alleged infinitely many
ample intervals, all preceding intersections for that parameter are
strictly negative and final ampleness is open. Choose a nearby
rational point of \eqref{eq:scaling-polytope} retaining these finitely
many strict conditions. The current model is its marked ample
model: on a common resolution its pullback difference from the
initial adjoint is effective and exceptional over the current model
by composition of the preceding comparisons. The corresponding
relative section algebras agree, since pushing an effective
exceptional correction to a normal target gives the structure sheaf.
Each vertex in \eqref{eq:scaling-polytope} has nef b-data on the
initial model and is gklt. Apply Lemma~\ref{lem:marked-models} on
each member of a finite Stein compact cover of $V$. It gives finitely
many local marked ample models, hence finitely many possible tuples
of such local models. Two global models giving the same tuple are
uniquely isomorphic over the interiors of these compacts: the
identification is fixed on the common bimeromorphic open set and
extends by the marked local isomorphism. The identifications agree
on overlaps by normality, so they glue. Thus only finitely many
global marked ample models occur.
A marked model cannot recur across a nontrivial intervening step.
For a fixed marking $T\dashrightarrow Y$, choose a common resolution.
The pullback difference for $D+tH$ is affine in $t$. Its coefficient
inequalities of effectivity are convex, and the relatively ample cone
of $Y$ is convex. The parameters for which $Y$ is its marked ample
model therefore form an interval. If it occurred in two separated
ample intervals of the run, it would also be the ample model at every
intermediate ample parameter. Uniqueness of the relative Proj would
identify each intermediate model with the same marking. But adjacent
models across a nontrivial elementary step have different markings,
as their adjoint has opposite relative signs. This is impossible.
Finitely many marked models therefore imply finitely many steps.
\end{proof}
\begin{remark}
The proof concerns some finite relative run over a bimeromorphic
compact base. The finite-cover argument for ordinary pairs is also
explicit in \cite[Theorems~2.1 and~2.4]{FujinoCompact2026}. We have
given the additional generalized-data, global-sheaf, and marked-model
arguments, because an ordinary theorem assuming $\Q$-factoriality near
a Stein compact does not by itself prove the statement above.
\end{remark}
\subsection{Extraction of prescribed low places}
\begin{proposition}[Exact extraction]\label{prop:extraction}
Let $(T,B+\bM)$ be a rational gklt pair on a globally strongly
$\Q$-factorial compact K\"ahler space, with projective nef carrier.
For any specified finite set $\mathcal E$ of exceptional prime
divisors over $T$ with
$a(E;T,B+\bM)\leq1$, there is a projective modification
$h\colon U\to T$ such that:
\begin{enumerate}[label=\textup{(\roman*)}]
\item $U$ is compact K\"ahler and globally strongly $\Q$-factorial;
\item the $h$-exceptional prime divisors are exactly $\mathcal E$;
\item the crepant generalized pair on $U$ is gklt with effective
boundary.
\end{enumerate}
\end{proposition}
\begin{proof}
The projective realization statement in Lemma~\ref{lem:low-places}
places the finitely many designated divisors on a projective log
resolution carrying the nef data. Its proof uses only log-smooth
discrepancy calculations and stratum blowups, independently of
extraction. Write this resolution as $p\colon W\to T$.
Keep every designated coefficient at its crepant value $1-a(E)$,
which lies in $[0,1)$. Keep the nonexceptional coefficients as well.
For every other exceptional prime choose a coefficient in $[0,1)$
strictly greater than its crepant coefficient. Since the pair is
gklt, there are only finitely many such choices and they are possible.
With this effective SNC boundary $\Gamma$ we have
\[
K_W+\Gamma+M_W=p^*D_T+G,
\qquad G\geq0,
\]
where the support of $G$ is exactly the set of undesired exceptional
primes. The new pair is gklt.
Apply Proposition~\ref{prop:relative-finite} over $T$. At every
stage the adjoint is the pullback of $D_T$ plus the effective
transform of $G$, and the latter remains exceptional over $T$.
A divisorial negative step cannot contract a prime outside its
support: a general covering contracted curve in that prime avoids
containment in the support and has nonnegative intersection with
the effective rational Cartier divisor $G$, whereas its adjoint
degree is negative. Small steps contract no prime divisor. Hence
every designated prime survives, and no nonexceptional prime of the
resolution is lost.
At the relative nef endpoint $h\colon U\to T$, the remaining
effective exceptional correction $G_U$ is $h$-nef. Negativity gives
$G_U\leq0$, hence $G_U=0$. Every undesired prime has therefore been
contracted. The equality is now crepant, and the retained boundary
is effective. The global strong condition, K\"ahlerness, and gklt
property follow at each step from Proposition~\ref{prop:relative-steps}.
The exceptional primes are exactly the specified ones, as claimed.
\end{proof}
==== END Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//08-relative.tex ====
==== BEGIN Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//08-resolutions.tex ====
\section{Effective resolutions and fiberwise nonvanishing}\label{sec:resolution-fibers}
Fujiki class~$\mathcal C$ consists of compact complex spaces
bimeromorphic to compact Kähler manifolds. For a rational adjoint,
$\kappa$ denotes its Kodaira dimension, defined by sections of divisible
positive multiples and equal to $-\infty$ when all such section spaces
vanish. A fibration is a surjective holomorphic map with connected
fibres.
We first make the boundary effective on a resolution while preserving
the divisible adjoint section spaces. We then establish fiberwise
nonvanishing with one common divisible degree.
\begin{lemma}\label{lem:supply-log-resolution}
Let $(X,B)$ be a rational klt pair with $D=K_X+B$ rational Cartier, and let
$p:Z\to X$ be a projective log resolution. There is an effective rational
SNC divisor $\Gamma$ with coefficients strictly less than one such that
\begin{equation}\label{eq:supply-log-resolution}
K_Z+\Gamma\simq p^*D+E,
\qquad E\geq0\quad\text{and}\quad E\text{ is }p\text{-exceptional}.
\end{equation}
These are actual rational line bundles with their birational
identification. If $D$ is analytically pseudo-effective, then so is
$K_Z+\Gamma$. For every sufficiently divisible positive integer $m$,
\begin{equation}\label{eq:supply-section-descent}
p_*\cO_Z\bigl(m(K_Z+\Gamma)\bigr)
\simeq \cO_X(mD).
\end{equation}
\end{lemma}
\begin{proof}
Define the crepant subboundary $\Gamma_0$ by
$K_Z+\Gamma_0=p^*D$, using the canonical identification over the locus
where $p$ is an isomorphism. Local canonical generators and their
Jacobians define this equality without choosing a global canonical
divisor. Klt singularities give
$\operatorname{coeff}_P\Gamma_0<1$ for every prime divisor $P$ on $Z$.
The nonexceptional coefficients are those of $B$ and are nonnegative.
Set
\[
\Gamma=\sum_P\max\{\operatorname{coeff}_P\Gamma_0,0\}P,
\qquad E=\Gamma-\Gamma_0.
\]
Both supports lie in the SNC divisor of the log resolution. This proves
\eqref{eq:supply-log-resolution}. A positive singular metric on a
Cartier multiple of $D$ pulls back under the dominant map $p$; multiplying
by the divisor metric of the effective correction proves
pseudo-effectivity upstairs.
Take $m$ clearing the Cartier indices and all coefficients in
\eqref{eq:supply-log-resolution}. The image of $\Supp E$ has codimension
at least two in the normal space $X$, and
$p_*\cO_Z(mE)=\cO_X$. Indeed, a local section of $\cO_Z(mE)$ is a
meromorphic function with poles allowed only along $E$. It gives a
holomorphic function off that codimension-two image, which extends by
normality; its pullback agrees with the original section on a dense open
set. The reverse inclusion is immediate. The actual line-bundle
identity
\[
\cO_Z\bigl(m(K_Z+\Gamma)\bigr)
\simeq p^*\cO_X(mD)\otimes\cO_Z(mE)
\]
and the projection formula give \eqref{eq:supply-section-descent}.
\end{proof}
The next lemma separates two quantifiers in restricting a positive
metric. Such a metric initially restricts on almost every smooth
fiber. Projective nonvanishing in dimension at most three and proper
holomorphic cohomology then supply a single nonzero adjoint multiple
on every fiber of the indicated smooth locus.
\begin{lemma}\label{lem:supply-fibers}
Let $g:Z\to T$ be a surjective holomorphic map with connected fibers
between smooth compact connected Kähler manifolds. Let $\Gamma$ be an
effective rational SNC divisor with coefficients strictly less than one.
Suppose that $K_Z+\Gamma$ is analytically pseudo-effective and that the
very general smooth fiber is projective of dimension at most three.
Then
\[
\kappa(F,K_F+\Gamma|_F)\geq0
\]
for a very general smooth fiber $F$. Moreover, one positive divisible
degree gives a nonzero section on every fiber over the smooth base locus
where the boundary strata restrict smoothly.
\end{lemma}
\begin{proof}
\emph{Restriction of the metric.}
Remove the critical values of $g$ and the proper images on which a
boundary stratum fails to restrict smoothly. The remaining nonempty
open set $T^\circ\subset T$ is connected. On it the fibers are smooth,
$\Gamma|_{F_t}$ is an effective SNC boundary with coefficients less than
one, and adjunction is an identity of actual rational line bundles:
\begin{equation}\label{eq:supply-fiber-adjunction}
(K_Z+\Gamma)|_{F_t}=K_{F_t}+\Gamma|_{F_t}.
\end{equation}
Here the constant one-dimensional factor coming from
$K_T|_t$ is immaterial to the line-bundle isomorphism on $F_t$.
Choose $r>0$ so that $L=\cO_Z(r(K_Z+\Gamma))$ is a line bundle. Give $L$
a singular Hermitian metric of nonnegative curvature. Its local weights
are plurisubharmonic. In product coordinates for the submersion over
$T^\circ$, their local integrability and Fubini's theorem show that the
weights restrict to plurisubharmonic functions, rather than identically
$-\infty$, on almost every fiber. A countable cover gives this assertion
simultaneously for the local weights; their transition identities are
preserved under restriction. Thus $L|_{F_t}$ is analytically
pseudo-effective for almost every $t\in T^\circ$.
\smallskip\noindent
\emph{Projective nonvanishing on the fibers.}
For almost every such parameter the fiber is also projective: the
exceptions to the very-general projectivity assumption have measure
zero. On a smooth projective manifold the analytic pseudo-effective
cone restricted to divisor classes is the closure of the effective
divisor cone, by \cite[Proposition~1.2]{BDPP2013}. Hence the projective
klt pair $(F_t,\Gamma|_{F_t})$ has pseudo-effective adjoint. The
projective log MMP in dimensions at most three produces a nef model.
For example, in dimension three the ordinary cone and flip theorems,
together with \cite[Theorem~1.2]{CT2023} with zero nef part, give
termination; pseudo-effectivity excludes a Mori fiber space. The
projective log abundance theorem
\cite[Theorem~1.1]{KMMAbundance1994}, together with
\cite[\S6.1]{MatsukiCorrection2003}, gives a nonzero section of a
multiple of the nef adjoint. Dimensions one and two use the classical
log abundance theorem. The effective pullback comparison on a common
resolution pulls this section back to a section of a multiple of the
original adjoint. It descends to $F_t$ by the projection formula for the
modification. Consequently
\begin{equation}\label{eq:supply-ae-nonvanishing}
H^0(F_t,L^{\otimes m_t}|_{F_t})\ne0
\quad\text{for some }m_t>0
\end{equation}
for almost every $t\in T^\circ$. If the fiber is a point this assertion
holds directly.
\smallskip\noindent
\emph{A common positive degree.}
To obtain the required quantifier, put
\[
A_m=\bigl\{t\in T^\circ:
h^0(F_t,L^{\otimes m}|_{F_t})\geq1\bigr\},
\qquad m=1,2,\ldots .
\]
These are closed analytic jumping loci: the restricted family is proper
and smooth, and $L^{\otimes m}$ is locally free and flat over its base,
so proper holomorphic cohomology and base change apply
\cite[\S8, Theorem~3, p.~62]{Douady1974}. By
\eqref{eq:supply-ae-nonvanishing}, their countable union contains almost
every point of $T^\circ$. If every $A_m$ were proper, that union would
have measure zero. Therefore some $A_m$ equals $T^\circ$, since this
base is connected. This proves both assertions. In particular,
almost-everywhere restriction of a metric has not been identified with
the very-general assertion by definition.
\end{proof}
==== END Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//08-resolutions.tex ====
==== BEGIN Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//09-threefold.tex ====
\section{Arbitrary termination in dimension three}\label{sec:threefold}
We establish the lower-dimensional termination needed for special
termination. Throughout this section we use the rational generalized
pairs and elementary steps of Section~\ref{sec:relative}.
The spaces are compact K\"ahler and globally
strongly $\Q$-factorial, and the elementary steps have the negative-side
one-ray condition for degrees of global rational line bundles. All higher
nef data are fixed as a rational b-line bundle. The projective bases of the
steps may change. No pseudo-effectivity assumption is made in this section.
We first treat generalized terminal pairs. We then bound Cartier indices
under uniform discrepancy bounds and use that bound to lift a general
sequence to generalized terminal models. This follows the threefold
strategy of \cite[Section~3]{CT2023}; the surface, index, and lifting
arguments below establish its analytic form with varying bases.
\subsection{Generalized terminal pairs}
\begin{proposition}\label{prop:terminal-three}
Let $(X_0,B_0+\bM)$ be a rational generalized terminal pair on a normal
globally strongly $\Q$-factorial compact K\"ahler threefold, with fixed
rational b-line data as in Definition~\ref{def:generalized-data}.
Every sequence of small elementary steps in the sense of
Section~\ref{sec:relative} starting from
this pair terminates. The contraction bases may vary. Here generalized
terminal means generalized klt with all exceptional log discrepancies
greater than one.
\end{proposition}
\begin{proof}
Write the sequence as
\[
(X_i,B_i+\bM)\dashrightarrow (X_{i+1},B_{i+1}+\bM),
\qquad X_i\xrightarrow{f_i}Z_i\xleftarrow{f_i^+}X_{i+1}.
\]
The boundary is transformed strictly, so its coefficients belong to one
finite set. Smallness and Lemma~\ref{lem:negativity} preserve generalized
terminality. Forgetting the effective boundary and the effective nef defect
increases discrepancies, so each $X_i$ is an ordinary terminal threefold.
In particular its singularities are isolated, and the general point of every
flipped curve is smooth.
Choose an integer $p>0$ for which $p\bM$ is represented by an integral line
bundle on a carrier, and an integer $N$ divisible by $p$ and by the
denominators of the boundary coefficients. These integers continue to work
on higher carriers throughout any finite part of the sequence. If $C$ is
a flipped curve on $X_{i+1}$, the exceptional valuation $E_C$ of its generic
blowup has log discrepancy
\begin{equation}\label{eq:curve-discrepancy-three}
a(E_C;X_{i+1},B_{i+1}+\bM)
=2-\sum_j m_jb_j-\frac{q}{p},
\qquad m_j,q\in\Z_{\geq0}.
\end{equation}
Indeed, the ordinary blowup contribution is two, the $m_j$ are the generic
multiplicities of the boundary components, and the remaining term is the
coefficient of the nef defect. At this smooth generic point the integral
trace of $p\bM$ is Cartier. The defect therefore has coefficients in
$p^{-1}\Z$ and is effective by Lemma~\ref{lem:trace}. This proves
\eqref{eq:curve-discrepancy-three}, including its signs.
Let $b$ be the largest boundary coefficient, with $b=0$ when the boundary
is empty. For $b>0$ test the flipped curves contained in a coefficient-$b$
component; for $b=0$ test every flipped curve. The value in
\eqref{eq:curve-discrepancy-three} belongs to the fixed finite set
\[
\mathcal A_b=N^{-1}\Z\cap(0,2-b].
\]
For each $t\in\mathcal A_b$, set
\[
d_t(i)=\#\{E\text{ exceptional over }X_i:
a(E;X_i,B_i+\bM)<t\}.
\]
Lemma~\ref{lem:low-places} makes these numbers finite, since the terminal
cutoff is the strict inequality $a<2-b$. Small transformations identify
the sets of exceptional valuations, and discrepancy monotonicity makes
every $d_t(i)$ nonincreasing. For the valuation of a tested curve,
Lemma~\ref{lem:negativity} gives
\[
a(E_C;X_i,B_i+\bM)
<a(E_C;X_{i+1},B_{i+1}+\bM)=t.
\]
Thus $d_t$ drops at that step. The nonnegative integer
$\sum_{t\in\mathcal A_b}d_t(i)$ can drop only finitely often. If $b=0$,
every nontrivial small step has a flipped curve, so the proof is complete.
Suppose $b>0$. On a tail, no flipped curve is contained in any of the
coefficient-$b$ components. Fix such a component $D_i$, and let
$D_i^\nu$ denote its normalization. The two images of $D_i$ and
$D_{i+1}$ in $Z_i$ agree, because the ambient transformation is an
isomorphism in codimension one. Denote the normalization of this common
image by $V_i$. There are projective birational morphisms
\[
D_i^\nu\longrightarrow V_i\longleftarrow D_{i+1}^\nu.
\]
The morphism on the right contracts no curve and is therefore finite:
every positive-dimensional projective fiber contains a curve. A finite
birational morphism to a normal space is an isomorphism. Hence there is a
projective birational morphism $D_i^\nu\to D_{i+1}^\nu$. It contracts
exactly the curves whose images in $X_i$ are flipping curves contained in
$D_i$. By Lemma~\ref{lem:cycle-loss}, with $k=1$, each such contraction
strictly decreases the dimension of the span of curve classes of the compact
normalized surface. There are finitely many coefficient-$b$ components,
so only finitely many further steps have a flipping curve contained in one
of them.
On the resulting tail put $D_i^{\max}=\sum_{b_j=b}D_{i,j}$ and replace
$B_i$ by $B_i-bD_i^{\max}$. Every contracted curve $C$ has
$D_i^{\max}\cdot C\geq0$, because it is not contained in this effective
$\Q$-Cartier divisor. Thus the new adjoint
\[
A_i'=K_{X_i}+B_i-bD_i^{\max}+M_{X_i}
\]
is negative on the old ray. To check its positive transform, write
$A_i=K_{X_i}+B_i+M_{X_i}$ and choose the rational number
$c=(A_i'\cdot C)/(A_i\cdot C)>0$. The global one-ray condition and
Lemma~\ref{lem:descent} give an actual identity of rational line bundles
\[
A_i'=cA_i+f_i^*L_i,\qquad L_i\in\Pic(Z_i)\otimes\Q.
\]
Its strict transform is
$A_{i+1}'=cA_{i+1}+(f_i^+)^*L_i$, which is $f_i^+$-ample.
Consequently the same tail is a sequence of elementary steps for the new
generalized pair. Decreasing the effective boundary preserves generalized
terminality. Induction on the number of distinct positive boundary
coefficients proves the proposition.
\end{proof}
\subsection{Uniform indices for the terminal lift}
To lift a generalized klt sequence to generalized terminal models, we
will need the discrepancies of the finitely many exceptional places with
log discrepancy at most one to stabilize. A uniform Cartier index will
put those discrepancies in a fixed rational grid. We first establish
the local index statement and the surface estimate used to obtain that
uniform bound.
\begin{lemma}\label{lem:terminal-local-index}
Let $(T,x)$ be a complex analytic terminal threefold germ of canonical
Cartier index $r$. If $D$ is an integral Weil divisor germ that is
$\Q$-Cartier, then $rD$ is Cartier. If $r>1$, there is an exceptional
valuation centered at $x$ with ordinary log discrepancy $1+1/r$.
\end{lemma}
\begin{proof}
For the first assertion, use the analytic index-one cover
$\pi:(T^\sharp,x^\sharp)\to(T,x)$, a cyclic cover of degree $r$ that is
\'etale away from the distinguished point. The terminal threefold
classification makes $T^\sharp$ smooth or an isolated cDV hypersurface
germ; see \cite{Mori1985}. A sufficiently small punctured representative
$U^\sharp$ is simply connected, by the connectivity theorem for links of
isolated hypersurfaces \cite[Theorems~2.10 and~5.2]{Milnor1968}. For a convergent
analytic defining equation, finite determinacy gives an analytically
equivalent sufficiently high Taylor polynomial
\cite[Theorem~6.2, p.~24]{DeJongEtAl1998}, so the same connectivity
statement applies; the smooth case has the same property. For every
$m>0$, the analytic Kummer sequence implies
that the $m$-torsion of $\Pic(U^\sharp)$ is an image of
$H^1(U^\sharp,\boldsymbol\mu_m)=0$. The local divisor class group
injects into this Picard group: a rank-one reflexive sheaf trivial on the
punctured normal germ is trivial on the germ by reflexive extension.
It follows that the local class group of $T^\sharp$ is torsion-free.
Since $D$ is $\Q$-Cartier, the integral divisor $\pi^*D$ has torsion class
and is therefore principal, say $\pi^*D=\operatorname{div}(u)$ for a
meromorphic germ $u$. The meromorphic norm gives
\[
\operatorname{div}\operatorname{Norm}_{\pi}(u)
=\pi_*\operatorname{div}(u)
=\pi_*\pi^*D=rD.
\]
Thus $rD$ is Cartier; compare the index divisibility statement in
\cite[Lemma~5.1]{Kawamata1988}. The argument above proves the analytic
assertion for divisor germs already known to be $\Q$-Cartier.
The second assertion is the terminal small-discrepancy theorem
\cite[Appendix, pp.~193--195]{KawamataAppendix1992}. Its proof uses the
analytic quotient classification and gives ordinary discrepancy $1/r$, or
log discrepancy $1+1/r$ in our convention.
\end{proof}
When the germ lies on a compact terminal space, the last valuation can be
seen on a global resolution. In fact, its ordinary crepant boundary has
only negative exceptional coefficients. A valuation still exceptional
over that smooth resolution has ordinary log discrepancy at least two.
Thus a local valuation of discrepancy $1+1/r<2$ is already represented by
an exceptional component of the restricted global resolution. Since its
center on $T$ is the isolated point $x$, that component gives a global
exceptional prime. This observation permits the use of
Lemma~\ref{lem:terminal-local-index} with global discrepancy bounds.
\begin{lemma}\label{lem:exceptional-surface-curve}
Let $h:U\to T$ be a projective bimeromorphic morphism of normal compact
complex threefolds. Suppose that $U$ is klt and that $P$ is an exceptional
prime divisor with $K_U+P$ rational Cartier. Given finitely many effective
$\Q$-Cartier divisors $D_j$ on $U$, none having $P$ as a component, there
is a curve $C\subset P$ contracted by $h$, contained in none of the $D_j$,
such that
\[
(K_U+P)\cdot C\geq-3.
\]
\end{lemma}
\begin{proof}
Let $\nu:P^\nu\to P$ be the normalization and let
$\mu:S\to P^\nu$ be its minimal resolution. Divisorial adjunction gives
\begin{equation}\label{eq:arbitrary-prime-different}
\nu^*((K_U+P)|_P)=K_{P^\nu}+\Delta_{P^\nu},
\qquad \Delta_{P^\nu}\geq0,
\end{equation}
as an identity of rational adjoint bundles. Neither normality of $P$ nor
log canonicity of $(U,P)$ is needed for effectivity of the different in this
formula. This is the ordinary different of a reduced prime, computed at
codimension-two points of $U$; the construction and effectivity are given in
\cite[Section~14.1(i)--(iii), pp.~40--41]{FujinoFundamental2010},
and analytic adjunction is included in
\cite[Theorem~1.1 and the following setup, p.~1]{FujinoInversion2023}.
Here is the local analytic effectivity argument. On a transverse klt
surface $T$, take a small smooth quotient $q:V\to T$ and write
$P_T$ for the reduced boundary curve and $C=(q^{-1}P_T)_{\mathrm{red}}$.
The quotient is unramified at the general points of these curves, so
$q^*(K_T+P_T)=K_V+C$. Let $\rho:C^\nu\to P_T^\nu$ be the induced
map of normalizations. Residue adjunction for the reduced plane curve
$C$ has an effective conductor divisor $A$, and the canonical
ramification formula for $\rho$ has an effective divisor $R$.
Compatibility of residues with pullback gives
\[
\rho^*\operatorname{Diff}_{P_T^\nu}(0)=A+R.
\]
At a branch of ramification degree $e$, its coefficient downstairs is
$(c+e-1)/e\geq0$, where $c$ is the conductor coefficient. This proves
the analytic effectivity in \eqref{eq:arbitrary-prime-different}
without an lc hypothesis on $(U,P)$ or normality of $P$.
The sum on the right of \eqref{eq:arbitrary-prime-different} is rational
Cartier; its individual canonical term need not be. If that term is used
separately, take its numerical pullback in the sense of Mumford, determined
by intersection zero with all exceptional curves. The negative definite
exceptional intersection matrix makes this pullback linear and makes the
pullback of an effective divisor effective. In particular the calculation
does not presume $\Q$-factoriality of $P^\nu$.
For completeness, the effective correction can also be checked directly
on $S$. For every irreducible $\mu$-exceptional curve $E$, adjunction on
the smooth surface gives
\[
K_S\cdot E=2p_a(E)-2-E^2\geq0.
\]
Indeed, an exceptional smooth rational $(-1)$-curve is excluded by
minimality over $P^\nu$; every other exceptional curve satisfies the
displayed inequality. The pullback of the adjoint in
\eqref{eq:arbitrary-prime-different} therefore has the form
\begin{equation}\label{eq:surface-effective-correction}
(\nu\mu)^*((K_U+P)|_P)=K_S+\Delta_S,
\qquad \Delta_S\geq0.
\end{equation}
To obtain the sign, the residue comparison defines $\Delta_S$ as a rational
divisor with pushforward $\Delta_{P^\nu}\geq0$. Since $K_S$ is
$\mu$-nef, the divisor $-\Delta_S$ is $\mu$-nef. Apply
Lemma~\ref{lem:negativity}\textup{(i)} to $E=\Delta_S$ to obtain
$\Delta_S\geq0$. This is equivalently the
numerical-pullback calculation above, and proves
\eqref{eq:surface-effective-correction} as an actual bundle comparison.
The image $h(P)$ has dimension zero or one. In the latter case, general
smooth fiber components of $S\to h(P)$ are moving curves $F$ with
$K_S\cdot F=2g(F)-2\geq-2$. One may use the Stein factorization to
describe these fibers over a smooth compact curve. Choose a general fiber
component not contained in $\Supp\Delta_S$, the conductor, or the
pullbacks of the $D_j$.
If $h(P)$ is a point, $P$ is a projective fiber subspace, and $S$ is a
smooth projective surface. Take a surface minimal model $\rho:S\to S_0$.
If $K_{S_0}$ is nef, use general ample curves on $S_0$. Otherwise the
classification of smooth projective surfaces gives ruling fibers with
canonical degree $-2$, or lines on $\mathbb P^2$ with canonical degree
$-3$. Their general strict transforms $F$ satisfy $K_S\cdot F\geq-3$,
because $K_S-\rho^*K_{S_0}$ is effective. In all cases the curves cover
$S$, so $F$ may be chosen not contained in the finitely many excluded
divisors. This is the surface argument also used in
\cite[Lemma~3.3]{CT2023}.
The curve $F$ maps birationally to a curve $C\subset P$: it is not
contained in the exceptional or conductor locus. By the projection
formula and \eqref{eq:surface-effective-correction},
\[
(K_U+P)\cdot C=(K_S+\Delta_S)\cdot F
\geq K_S\cdot F\geq-3.
\]
Its choice also ensures $C\not\subset\Supp D_j$ for every $j$.
\end{proof}
The surface estimate bounds the denominator introduced by extracting one
exceptional place of log discrepancy at most one. We now turn it into a
bound for every global rank-one reflexive sheaf.
\begin{lemma}\label{lem:index-bound}
For every integer $n\geq0$ and $\varepsilon>0$, there is an integer
$I(n,\varepsilon)>0$ with the following property. Let
$(T,B+\bM)$ be a rational generalized klt compact K\"ahler threefold pair,
with $T$ globally strongly $\Q$-factorial. Suppose that
\begin{enumerate}[label=\textup{(\roman*)}]
\item every log discrepancy is at least $\varepsilon$;
\item at most $n$ exceptional valuations have log discrepancy at most one;
\item every exceptional log discrepancy greater than one is at least
$1+\varepsilon$.
\end{enumerate}
Then $\cF^{[I(n,\varepsilon)]}$ is invertible for every global coherent
rank-one reflexive sheaf $\cF$ on $T$.
\end{lemma}
\begin{proof}
We induct on the number of exceptional places with log discrepancy at
most one. The terminal case is controlled by local canonical indices.
For the induction step, extract one place, bound its intersection degree
using the preceding surface estimate, and descend an integral line bundle.
Replace $\varepsilon$ by $\min\{\varepsilon,1\}$, so assume
$0<\varepsilon\leq1$. We construct integers divisible by those from the
preceding induction stage. When $n=0$, the pair is generalized terminal.
The underlying space is ordinary terminal and
\[
a(E;T,0)\geq a(E;T,B+\bM)\geq1+\varepsilon
\]
for every exceptional valuation. By
Lemma~\ref{lem:terminal-local-index} and the observation following that
lemma, every canonical point index $r>1$ satisfies
$r\leq1/\varepsilon$. Each local representative of $\cF$ is
$\Q$-Cartier by the global strong condition, and its Cartier index divides
$r$. Thus one may take
\[
I(0,\varepsilon)=\operatorname{lcm}
\{1,\ldots,\lfloor1/\varepsilon\rfloor\}.
\]
Local invertibility of this reflexive power at every point is precisely
global invertibility; no trivialization on all of $T$ is asserted.
Suppose $n>0$ and put $I'=I(n-1,\varepsilon)$. If there is no exceptional
valuation of discrepancy at most one, the preceding case applies. Otherwise
choose one such place, of discrepancy $a\in[\varepsilon,1]$, and apply
Proposition~\ref{prop:extraction} to obtain a projective extraction
$h:U\to T$ with that single exceptional prime $P$. Write its effective
crepant boundary as
\[
B_U=B_U^{\mathrm{str}}+(1-a)P.
\]
Crepancy preserves all discrepancies and lowers the number of exceptional
low places by one. The induction hypotheses therefore hold on $U$, so
$I'$ clears the index of every global rank-one reflexive sheaf on $U$.
First, $-P$ is $h$-ample. If $H$ is an $h$-ample line bundle, its reflexive
pushforward is rational Cartier on $T$. Lemma~\ref{lem:trace} gives
\[
H=h^*H_T+cP
\]
as a rational bundle comparison. The divisor $cP$ is $h$-ample and
$h$-exceptional, hence is nonpositive by negativity. Since $h$ has a
positive-dimensional fiber, $c\neq0$, and therefore $c<0$.
The same trace comparison for the nef data gives
\[
M_U=h^*M_T-sP,\qquad s\geq0.
\]
Apply Lemma~\ref{lem:exceptional-surface-curve} with the components of
$B_U^{\mathrm{str}}$ as the excluded divisors. For its contracted curve
$C\subset P$, the other boundary has nonnegative degree and
$M_U\cdot C=-sP\cdot C\geq0$. Crepancy yields
\[
-3\leq(K_U+P+B_U^{\mathrm{str}}+M_U)\cdot C
=aP\cdot C<0.
\]
Consequently
\begin{equation}\label{eq:exceptional-degree-bound}
0<-P\cdot C\leq\frac{3}{\varepsilon}.
\end{equation}
Now let $\cF$ be any global rank-one reflexive sheaf on $T$, and let
$\cF_U$ be its reflexive strict transform. In additive rational-bundle
notation, Lemma~\ref{lem:trace} gives
\begin{equation}\label{eq:sheaf-extraction-comparison}
h^*\cF=\cF_U+qP,\qquad q\in\Q.
\end{equation}
This comparison is defined by local meromorphic frames agreeing off the
exceptional locus, and requires no global meromorphic section of $\cF$.
The sheaves $\cF_U^{[I']}$ and $\cO_U(I'P)$ are line bundles. Intersecting
\eqref{eq:sheaf-extraction-comparison} with $C$ gives
\[
q=\frac{I'\cF_U\cdot C}{-I'P\cdot C},
\qquad k:=-I'P\cdot C\in\Z,
\quad 1\leq k\leq\frac{3I'}{\varepsilon}.
\]
The numerator is an integer. Define
\begin{equation}\label{eq:index-recurrence}
I(n,\varepsilon)
=I'\operatorname{lcm}
\{1,\ldots,\lceil3I'/\varepsilon\rceil\}.
\end{equation}
Then $I:=I(n,\varepsilon)$ is divisible by $I'$ and $Iq\in I'\Z$.
Thus $I\cF_U+(Iq)P$ represents an integral line bundle $L$ on $U$,
with its prescribed comparison to $h^*\cF^{[I]}$ off the exceptional
locus. Equation~\eqref{eq:sheaf-extraction-comparison} shows that $L$ has
degree zero on every $h$-contracted curve.
Increase the coefficient $1-a$ of $P$ by a sufficiently small positive
rational number $\delta$. On a fixed log resolution carrying the data,
the inequalities defining generalized klt remain strict for all sufficiently
small $\delta$, so the new pair is effective and generalized klt. Its
adjoint is
\[
h^*(K_T+B+M_T)+\delta P,
\]
which is $h$-antiample. Lemma~\ref{lem:descent} now descends $L$
\emph{integrally}. More explicitly, local ordinary klt replacement and
the analytic base-point-free theorem generate $L^m$ for every sufficiently
large integer $m$; descend two consecutive powers and take their quotient.
No further index is introduced. The descended line bundle agrees with
$\cF^{[I]}$ away from $h(P)$, a set of codimension at least two in $T$.
Reflexive extension identifies them on $T$. This proves the induction,
including the case $a=1$.
\end{proof}
\subsection{Nef endpoints and terminal lifting}
The uniform index bound will make the low discrepancies constant on a
tail of any small sequence. After extracting those places, we must compare
each lifted relative program with the prescribed positive model downstairs.
The next Lemma provides both the discrepancy comparison during the run
and the crepant morphism at its nef endpoint. It will also be used for
strata and threshold lifts.
\begin{lemma}\label{lem:nef-end-comparison}
Let $Y_0,Y,T_+$ be normal compact spaces projective and bimeromorphic over
a normal space $S$, and let $W$ be a common smooth modification
projective over each of the three spaces, with maps
$p_0:W\to Y_0$, $p:W\to Y$, and $q:W\to T_+$.
Let $A_0,A,A_+$ be rational line bundles on the respective spaces.
Suppose their prescribed meromorphic comparisons give
\begin{equation}\label{eq:two-target-comparison}
P_0=P+G=P_++F,
\qquad P_0=p_0^*A_0,\quad P=p^*A,\quad P_+=q^*A_+,
\end{equation}
where $G$ is effective and $p$-exceptional, and $F$ is effective and
$q$-exceptional. If $A_+$ is nef over $S$, then $F-G\geq0$.
If also $A$ is nef over $S$, then $P=P_+$. If, in addition, $A_+$ is
ample over $S$, the induced bimeromorphic map $Y\dashrightarrow T_+$
is a projective morphism $g$, and $A=g^*A_+$ as rational line bundles
with the given comparison.
\end{lemma}
\begin{proof}
Set $E=P-P_+=F-G$. Over $Y$, the divisor $-E=P_+-P$ is nef, while
\[
p_*(-E)=-p_*F\leq0,
\]
since $G$ is $p$-exceptional. Lemma~\ref{lem:negativity} gives
$-E\leq0$, proving the first assertion. If $A$ is nef over $S$, then
$E$ is nef over $T_+$ and
\[
q_*E=-q_*G\leq0,
\]
since $F$ is $q$-exceptional. Negativity gives $E\leq0$, and hence
$E=0$. The two exceptional supports need not coincide.
Assume $A_+$ is relatively ample. A fiber of $p$ maps by $q$ into a
projective fiber of $T_+\to S$. Equality $p^*A=q^*A_+$ gives degree zero
on every curve in that fiber. A nonconstant image would be detected by a
curve of positive degree against $A_+$, so $q$ is constant on each
connected fiber of $p$. Proper factorization through the normal space
$Y$ gives $q=g\circ p$. The map $g$ is projective because $Y\to S$ is
projective and $T_+\to S$ is separated. Finally, pullback by $p$ is
injective on rational line bundles, by $p_*\cO_W=\cO_Y$ and the
projection formula; therefore $A=g^*A_+$. This proves the actual bundle
identity, not merely numerical equality. Compare
\cite[Lemma~2.12]{LMT2022} for the algebraic form of this comparison.
\end{proof}
\begin{theorem}\label{thm:gklt-three}
Let $(X_0,B_0+\bM)$ be a rational generalized klt pair of dimension at
most three on a normal globally strongly $\Q$-factorial compact K\"ahler
space, with fixed rational b-line data as in
Definition~\ref{def:generalized-data}. Every sequence of elementary
birational steps in the sense of
Section~\ref{sec:relative} starting from this pair terminates. The
compact K\"ahler contraction bases may vary.
\end{theorem}
\begin{proof}
In dimension at most two there are no nontrivial small birational
contractions, and Lemma~\ref{lem:cycle-loss} gives the result. In dimension
three that lemma first removes a finite divisorial prefix. Relabel the
remaining small sequence as
\[
(X_i,B_i+\bM)\dashrightarrow(X_{i+1},B_{i+1}+\bM),\qquad i\geq1.
\]
All boundaries are now strict transforms of $B_1$.
By Lemma~\ref{lem:low-places}, there is $\varepsilon_0>0$ such that the
initial set of exceptional valuations with discrepancy below
$1+\varepsilon_0$ is finite. Since discrepancies increase and small maps
preserve exceptionality, the sets
\[
\mathcal L_i=\{E\text{ exceptional over }X_i:
a(E;X_i,B_i+\bM)\leq1\}
\]
form a decreasing sequence of finite sets. After discarding a finite
prefix, they equal one set $\mathcal L=\{E_1,\ldots,E_n\}$.
At the first model of this tail, the finitely many exceptional values in
$(1,1+\varepsilon_0)$ have a positive minimum distance from one; use
$\varepsilon_0$ itself if there are no such values. Choose
$\varepsilon>0$ no larger than this distance, than $\varepsilon_0$, than
one, and than the positive lower bound for all discrepancies of the initial
gklt pair furnished by Lemma~\ref{lem:low-places}. Monotonicity and the
stabilization of $\mathcal L$ show that, on every model of this tail,
all discrepancies are at least $\varepsilon$, and every exceptional
discrepancy greater than one is at least $1+\varepsilon$.
Apply Lemma~\ref{lem:index-bound} with these fixed $n$ and
$\varepsilon$. Let $r$ clear the coefficients of $B_1$ and let $p$ clear
the higher nef datum. For example
\[
J=I(n,\varepsilon)\operatorname{lcm}(r,p)
\]
makes each $J(K_{X_i}+B_i+M_{X_i})$ an integral line bundle, and makes
$J\bM$ integral on every higher carrier. Comparing the actual adjoint
bundles on a smooth carrier shows that every discrepancy belongs to
$J^{-1}\Z$: the canonical bundle there is integral and the coefficients
of the corresponding integral divisor difference are integers. The same
argument applies after any further resolution. Thus each nondecreasing
sequence
\[
a(E_\ell;X_i,B_i+\bM)\in J^{-1}\Z\cap[\varepsilon,1]
\]
is eventually constant. After another finite prefix, all these values
are constant simultaneously.
Use Proposition~\ref{prop:extraction} to extract exactly
$E_1,\ldots,E_n$ over the first model of this final tail:
\[
h_1:(Y_1,\Delta_1+\bM)\longrightarrow(X_1,B_1+\bM).
\]
The boundary is effective and crepant. Every valuation exceptional over
$Y_1$ is exceptional over $X_1$ and is not in $\mathcal L$, so its
discrepancy is greater than one. Hence this is a globally strongly
$\Q$-factorial generalized terminal model. If $n=0$, take $h_1=\id$.
Consider the first remaining flip over $Z_1$. Starting from $Y_1$, use
Proposition~\ref{prop:relative-steps} to take relative steps over $Z_1$
while the adjoint is not nef. We show that no divisorial contraction can
occur in this run. At any finite stage $Y_j$, a common smooth model gives
\[
P_0=P_j+G_j=P_++F,
\]
where $P_0$ is pulled back from $Y_1$, $P_j$ from the current lifted
adjoint, and $P_+$ from $K_{X_2}+B_2+M_{X_2}$. These are the prescribed
adjoint comparisons. The accumulated error $G_j$ is effective and
exceptional over $Y_j$; the error $F$ of the original flip is effective
and exceptional over $X_2$. The positive adjoint is ample over $Z_1$.
The first assertion of Lemma~\ref{lem:nef-end-comparison}, which does not
require nefness on $Y_j$, gives $G_j\leq F$.
If a divisor were lost, its strict transform from $Y_1$ would have a
positive coefficient in $G_j$, by the strict discrepancy comparison at
its contraction. But every divisor on $Y_1$ is either the transform of a
divisor on $X_1$, or one of the $E_\ell$. In the first case its
coefficient in $F$ is zero because the original map is small. In the
second case that coefficient is
\[
a(E_\ell;X_2,B_2+\bM)-a(E_\ell;X_1,B_1+\bM)=0
\]
by stabilization. Both cases contradict $G_j\leq F$. Thus the entire
lifted run is small. It stays generalized terminal, and
Proposition~\ref{prop:terminal-three} forces it to reach a nef endpoint
after finitely many steps.
At that endpoint, Lemma~\ref{lem:nef-end-comparison} gives equality of
the two adjoint pullbacks and a projective crepant morphism
\[
h_2:(Y_2,\Delta_2+\bM)\longrightarrow(X_2,B_2+\bM).
\]
Here $Y_2$ denotes the endpoint, rather than the first intermediate model
of the finite run. Since the lifted transformations were small, the
exceptional primes of $h_2$ are exactly the same $E_\ell$, with the same
boundary coefficients. We may therefore repeat this construction over
each successive base $Z_i$.
Each nontrivial flip below forces at least one step above. Indeed, take a
negative curve $C\subset X_i$ over $Z_i$. Projectivity of $h_i$ supplies
a curve $\Gamma\subset Y_i$ mapping onto $C$ with positive degree. The
crepant identity gives
\[
(K_{Y_i}+\Delta_i+M_{Y_i})\cdot\Gamma
=\deg(\Gamma/C)(K_{X_i}+B_i+M_{X_i})\cdot C<0.
\]
Thus the starting lifted adjoint is not nef over $Z_i$.
An infinite original sequence would consequently concatenate to an
infinite sequence of small generalized terminal elementary steps, with
fixed rational boundary coefficients and the same rational nef b-line
data. All spaces remain compact K\"ahler and globally strongly
$\Q$-factorial. The bases may change, exactly as allowed in
Proposition~\ref{prop:terminal-three}, which gives the contradiction.
\end{proof}
==== END Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//09-threefold.tex ====
==== BEGIN Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//10-special.tex ====
\section{Adjunction and special termination}\label{sec:special}
We use the generalized pairs and elementary steps of
Section~\ref{sec:relative}. In particular, an elementary step is
projective over its own normal compact K\"ahler contraction base, is
negative for the adjoint on the negative side, and has a relatively
ample transformed adjoint on the positive side. The models on which
elementary programs are run are globally strongly $\Q$-factorial.
The bases in a sequence need not be the same. All boundaries and
higher line-bundle data in this section are rational. The higher
data are nef and are carried by projective modifications.
Our goal is special termination: eventually both exceptional loci
avoid every generalized log canonical center. Restriction to a center
reduces its proof to termination in a smaller dimension. The center
need not satisfy the global strong factoriality condition, so we also
construct a crepant gdlt modification before running the smaller
program.
These assertions are proved in a fixed order. Special termination
in dimension $d$ uses gdlt modifications and gdlt termination only in
dimensions less than $d$. It then gives gdlt modifications in
dimension $d$. For $d\leq3$, removing the floor and applying
Theorem~\ref{thm:gklt-three} gives gdlt termination in dimension $d$.
The induction ends with special termination and crepant gdlt
modifications in dimension four; it uses full gdlt termination only
through dimension three.
We first establish adjunction and discrepancy comparison on the
normalized centers.
\subsection{Adjunction on normalized strata}
We write $\operatorname{Nklt}(V,A+\bM)$ for the union of the
generalized log canonical centers of a glc pair. A stratum below
means the closure of a coefficient-one stratum on the good snc open
set in the definition of gdlt. Its normalization is always taken
before adjunction is iterated.
\begin{lemma}[Adjunction with a fixed higher denominator]
\label{lem:stratum-adjunction}
Let $(V,A+\bM)$ be a rational gdlt pair on a globally strongly
$\Q$-factorial normal compact K\"ahler space. Suppose that $pM_W$
is a holomorphic line bundle on a projective carrier $W\to V$, for
an integer $p>0$. If $T$ is the normalization of a generalized log
canonical center, then iterated divisorial adjunction along a
coefficient-one chain gives a rational gdlt pair
$(T,A_T+\bM^T)$ with effective boundary and an identity of rational
holomorphic adjoint bundles
\begin{equation}\label{eq:stratum-adjunction}
K_T+A_T+M_T^T
=\nu^*(K_V+A+M_V),
\end{equation}
where $\nu:T\to V$ is the natural map. The b-line $\bM^T$ is
represented by restricting $\bM$ to the strict stratum on a higher
model; its denominator divides $p$. Its generalized log canonical
centers map to generalized log canonical centers of $V$ properly
contained in $\nu(T)$.
More precisely, for a nonnegative coefficient set $I\subset[0,1]$
define
\begin{equation}\label{eq:adjunction-coefficients}
\mathcal D_p(I)=
\left\{
1-\frac1r+\frac{\sum_{j=1}^{\ell}k_jd_j+k/p}{r}
\ \middle|\
\begin{array}{l}
r\in\N_{>0},\ \ell,k,k_j\in\Z_{\geq0},\\
d_j\in I,\ \sum_{j=1}^{\ell}k_jd_j+k/p\leq1
\end{array}
\right\}.
\end{equation}
If the coefficients of $A$ belong to $I$, those after $c$
successive adjunctions belong to $\mathcal D_p^{\circ c}(I)$,
where the superscript denotes the $c$-fold iterate.
If $I$ satisfies the descending chain condition (DCC), so does every
one of these sets. No global
strong $\Q$-factoriality, or uniform Cartier index for $A_T$, is
asserted for $T$.
\end{lemma}
\begin{proof}
Choose a projective log resolution $h:W\to V$ carrying the nef
data which is an isomorphism at the general points of the good
strata under consideration. We may replace $W$ further whenever
necessary, preserving these general points. Since $V$ is globally
strongly $\Q$-factorial, the trace $M_V$ is a rational line bundle.
Lemma~\ref{lem:trace} and Lemma~\ref{lem:negativity} give
\begin{equation}\label{eq:adjunction-nef-defect}
H=h^*M_V-M_W\geq0,
\qquad H\text{ is }h\text{-exceptional}.
\end{equation}
Use compatible local canonical representatives to write
\[
K_W+A_W+M_W=h^*(K_V+A+M_V),\qquad
K_W+A_W^{\rm o}=h^*(K_V+A).
\]
These are actual rational sheaf comparisons, and
$A_W=A_W^{\rm o}+H$.
We first justify the ordinary companion in this construction.
The inequality above says that ordinary log discrepancies of
$(V,A)$ are at least the generalized log discrepancies. In
particular $(V,A)$ is lc. An ordinary zero-discrepancy place is
also a generalized zero-discrepancy place. Its center therefore
meets the good open set. Conversely, every generalized center
meets that set, where $H=0$, the data descend, and the log formula
is the ordinary snc formula. Such a center is an ordinary
coefficient-one stratum. Thus $(V,A)$ is ordinary dlt and its lc
centers are exactly the generalized ones. This argument also
explains why forgetting the nef part does not introduce an
additional zero center outside the good open set; compare the
algebraic formulation in \cite[Remark~2.6(1)]{CT2023}.
Let $S\subset V$ be a prime component of $\lfloor A\rfloor$ and
let $S_W$ be its strict transform. It is smooth after the chosen
resolution. The coefficient of $S_W$ is one in both formulas.
The residue isomorphism restricts them to $S_W$ as
\begin{align*}
(K_W+A_W+M_W)|_{S_W}
&=K_{S_W}+(A_W-S_W)|_{S_W}+M_W|_{S_W},\\
(K_W+A_W^{\rm o})|_{S_W}
&=K_{S_W}+(A_W^{\rm o}-S_W)|_{S_W}.
\end{align*}
The normal space $S_W$ maps to the normalization $S^\nu$.
Push forward the indicated boundaries and the nef trace to
$S^\nu$. Over its smooth large open set, the residue
identification is the adjunction identification of the canonical
sheaf. It therefore defines, by reflexive extension, the
rational adjoint bundle on $S^\nu$ as the pullback of the
ambient adjoint. This construction gives an actual line-bundle
isomorphism after a common positive multiple. The prescribed
residue identification, and not equality of first Chern classes,
specifies the comparison.
The restricted log boundary on $S_W$ is snc with coefficients at
most one. Thus the resulting generalized pair is sub-lc before
effectivity is checked. Every zero stratum of that restricted
log formula is an intersection of $S_W$ with coefficient-one
components of the ambient log formula. Its image is an ambient
generalized center and hence meets the good open set. Over that
set the restriction is snc and its nef data descend. The same
reasoning applies to the ordinary companion. In particular,
after effectivity is established, both restricted pairs have the
appropriate dlt good-open property. Their zero centers agree:
on a common carrier their log boundaries differ by the effective
restriction of $H$, while every generalized zero center meets
the open set where that restriction vanishes.
Here is the local effectivity and coefficient calculation, with
the issue of nonfactorial intermediate strata made explicit.
Suppose at some stage of the chain that $U$ is the normal space
already obtained, $C$ is its effective generalized boundary, and
$S\subset\lfloor C\rfloor$ is the next prime. Carry along the
ordinary dlt companion just constructed. Its floor has the same
strata. Both adjoints are rational Cartier, although the
individual trace $M_U$ need not be rational Cartier on $U$.
Fix a prime divisor on $S^\nu$ whose coefficient is to be
computed, and work at an analytically general point of its image,
a codimension-two locus of $U$.
Take a general transverse surface slice there. This operation
can be performed by fixing general values of local parameters
along that locus on a simultaneous embedded resolution. We
choose the parameters transverse to the finitely many resolution
strata dominating the locus. At a general point in question
the normal ambient space is Cohen--Macaulay: a normal space is
$S_2$, and its non-Cohen--Macaulay locus has codimension at least
three. The sliced surface is $S_2$ and regular in codimension
one, hence normal. The transverse restrictions of the
resolution formulas are crepant surface formulas, by
complete-intersection adjunction. One can check this first
away from the distinguished point and then extend the
isomorphism of the rational Cartier adjoints. In particular,
the ordinary surface pair is dlt along the coefficient-one
branch. The boundary multiplicities computed on the slice are
the multiplicities at the original general point.
We use the following local surface fact. An ordinary dlt
surface germ along a coefficient-one branch is either snc for
the reduced boundary, or is a plt germ with a smooth pair
chart modulo a small cyclic group of some order $r$. The
branch in the chart is a coordinate curve. The different of
that branch alone is $1-1/r$, and the local Cartier index of
any integral divisorial sheaf divides $r$. This is the log
surface classification of
\cite[Theorem~4.15(1),(3), pp.~119--120, and Proposition~4.18]{KM1998}.
Its use here
is in the analytic-germ category: the numerical plt resolution
criterion gives a Hirzebruch--Jung string with the strict
boundary meeting an end; the corresponding normal analytic
surface germ is the cyclic quotient. It is a classification
of the surface germ, and does not assert local factoriality
of the higher-dimensional space $U$. In the snc case set
$r=1$.
Write the additional effective boundary on the slice as a sum
with coefficients $d_j$. Pulling to the cyclic chart and
restricting to its smooth coordinate branch gives nonnegative
integral intersection multiplicities $k_j$. Its contribution
to the different is $\sum_j k_jd_j/r$. For the generalized
part, restrict a common higher model and the line bundle
representing $p\bM$ to the higher surface. Define its trace on
the sliced surface by this proper pushforward; the restricted
higher log formula fixes the identification on the punctured
surface. This avoids interchanging an arbitrary reflexive
hull with specialization. The cyclic quotient fact makes
$rp$ times this trace Cartier. Consequently its pullback
minus the higher nef bundle is an exceptional divisor with
denominator dividing $rp$. It is effective by
Lemma~\ref{lem:negativity}, because the higher bundle is nef
on every curve of the projective surface resolution. It
does not contain the strict coefficient-one branch, and its
restriction to that branch has coefficient $k/(rp)$ for
some $k\in\Z_{\geq0}$. The resulting coefficient is therefore
\begin{equation}\label{eq:surface-generalized-different}
1-\frac1r+\frac{\sum_jk_jd_j+k/p}{r}.
\end{equation}
This proves effectivity of the generalized different.
Omitting the nef correction proves effectivity of the ordinary
different carried along in the construction. The coefficient
is at most one by the sub-lc log formula already established.
The local analytic adjunction and nef-defect calculation also
appears explicitly in the proof of
\cite[Lemma~3.1, pp.~5--6]{HXACC2023}; the local gdlt
modification in that proof is not used here as a global
modification theorem.
We can now repeat the construction. On a resolution the chosen
chain is a chain of snc strata, and its successive restrictions
have the two log formulas just considered. A zero stratum of
any restricted formula is a zero stratum of the preceding one;
its image is therefore an ambient center meeting the original
good open set. This proves the center and gdlt assertions at
every stage. Closures and normalizations introduce no finite
cover of the chosen center: each selected component is
generically isomorphic over it, so its normalization is its
unique normalization. The restricted higher model is
projective over that normalization and is compact K\"ahler.
The higher line bundle remains nef, and its denominator still
divides $p$, under restriction and subsequent pullback.
This proves \eqref{eq:stratum-adjunction} and the asserted
description of the b-data. A fixed chain on a common higher
model also shows that these b-data are compatible under any
birational comparison which preserves the general points of
the chain.
It remains to prove the DCC assertion rather than infer a
bound on all Cartier indices. If $I$ is DCC, its positive
elements, when present, have a positive minimum. The same is
true after adjoining $1/p$. A sum of such elements bounded
by one consequently has a bounded number of nonzero summands.
Finite sums of a nonnegative DCC set satisfy DCC: from any
sequence of tuples of a fixed length one can successively
choose a subsequence on which each coordinate is
nondecreasing. Thus the numerator sums in
\eqref{eq:adjunction-coefficients} satisfy DCC. For bounded
$r$, their images in that formula satisfy DCC. In a strictly
decreasing sequence of displayed coefficients with first
term $c_0<1$, the inequality
\[
1-\frac1r\leq c\leq c_0
\quad\Longrightarrow\quad r\leq\frac1{1-c_0}
\]
bounds $r$. A sequence starting at one reduces to this case
after its first strict decrease. Therefore
$\mathcal D_p(I)$ is DCC, and induction proves the same for
each finite iterate.
\end{proof}
\subsection{Comparison on a stratum}
Adjunction supplies an effective generalized boundary with coefficients
in a fixed DCC set. To use those coefficients along a sequence, we need
monotonicity of the restricted discrepancies. Strictness must also
detect an ambient exceptional intersection of arbitrary codimension
on the stratum.
\begin{lemma}[Restricted discrepancy comparison]
\label{lem:stratum-comparison}
Suppose that
\[
(V,A+\bM)\dashrightarrow (V^+,A^++\bM)
\]
is a small elementary gdlt step over $Z$. Let $T$ be a
normalized generalized log canonical center whose general
point and coefficient-one chain are unchanged by the step,
and let $T^+$ denote the corresponding normalization on the
positive side. Put
$D_T=K_T+A_T+M_T^T$ and $D_{T^+}=K_{T^+}+A_{T^+}+M_{T^+}^T$.
Both strata map projectively and bimeromorphically to the
normalization $S$ of their common image in $Z$. The adjoints
are respectively antiample and ample over $S$.
On a common smooth higher model, with projective maps $p$ and
$q$ to $T$ and $T^+$, respectively,
\begin{equation}\label{eq:restricted-positive-difference}
p^*D_T-q^*D_{T^+}=F_T\geq0.
\end{equation}
For every prime divisor $E$ on a proper bimeromorphic analytic model
of the stratum, the generalized discrepancies do not decrease.
They increase strictly if its center on either stratum is
contained in the corresponding ambient exceptional locus.
\end{lemma}
\begin{proof}
The restricted morphisms factor through $S$ by normality.
They are projective, since restriction and this finite
factorization preserve relative ampleness; they are
bimeromorphic because the ambient step is an isomorphism
at the general point of the stratum. Restriction of the
relatively ample adjoint, and then pullback by the
normalization, gives the asserted signs.
Take a common projective log resolution of the ambient step,
carrying its common nef data and preserving the general
points of the chosen snc chain. If $a,b$ are its projections,
Lemma~\ref{lem:negativity} gives the actual divisor comparison
\[
F=a^*(K_V+A+M_V)-b^*(K_{V^+}+A^++M_{V^+})\geq0.
\]
This difference is anti-nef over $Z$. Its support contains
the full fibers over the non-isomorphism locus in $Z$:
opposite relative ampleness makes the difference nontrivial
over each such point, and the fiber-support assertion of
Lemma~\ref{lem:negativity} applies. The strict transform of
the chosen stratum is not contained in $\Supp F$, since its
general point lies in the common isomorphism open set.
Restrict the two log formulas along the common snc chain.
At each restriction the residue canonical bundles are the
same, and the common higher nef bundles are the same.
They cancel. Each restricted difference is effective,
because the next strict stratum is not contained in its
support. At the last stage this gives
\eqref{eq:restricted-positive-difference} as an equality
with its prescribed meromorphic identification. The
coefficient of $E$ in this difference is the increase of its
generalized log discrepancy.
For strictness, first make $E$ divisorial on a smooth higher
stratum. This can be achieved by projective blowups of its
successive centers; perform the same blowups in the ambient
resolution and resolve further, preserving the generic
chain. If the center of $E$ is in the ambient exceptional
locus, its entire inverse image in the common ambient model
lies in $\Supp F$, by the full-fiber assertion. Therefore
the center on the higher stratum lies in the support of the
restricted effective Cartier divisor. Its defining local
equation has positive order at $E$. This proves strictness
on either side. In particular, the argument detects an
intersection in arbitrary codimension on the original
stratum; it does not require that intersection already to
be a divisor there. The algebraic counterparts are
\cite[Lemma~2.8]{LMT2022} and
\cite[Proposition~3.2]{Moraga2020}; the argument above uses
the analytic projective negativity statement directly.
\end{proof}
\subsection{The triangular induction}
We state the three assertions together to make their dependency order
explicit. The first stops steps near the generalized log canonical
centers; the second constructs crepant models; the third supplies the
lower-dimensional termination used in the next induction stage.
\begin{theorem}[Special termination]\label{thm:special}
Let $d\leq4$, and let
\[
(V_0,A_0+\bM)\dashrightarrow(V_1,A_1+\bM)
\dashrightarrow\cdots
\]
be a sequence of elementary steps for a rational gdlt pair
on globally strongly $\Q$-factorial normal compact K\"ahler
spaces of dimension $d$, with fixed higher nef b-line data.
There exists $i_0$ such that, for every $i\geq i_0$, the
exceptional loci on both sides of the step are disjoint
from all generalized log canonical centers on those sides.
No common contraction base for the sequence is assumed.
\end{theorem}
\begin{proposition}[Crepant gdlt modification]
\label{prop:gdlt-mod}
Let $(V,A+\bM)$ be a rational glc pair of dimension at most
four on a normal compact K\"ahler space. Its higher nef
line data are carried by a projective modification, and
$K_V+A+M_V$ is a rational holomorphic line bundle.
There is a projective bimeromorphic morphism $h:Y\to V$
such that $Y$ is normal compact K\"ahler and globally
strongly $\Q$-factorial, $(Y,A_Y+\bM)$ is gdlt with
$A_Y\geq0$, and
\[
K_Y+A_Y+M_Y=h^*(K_V+A+M_V).
\]
Every $h$-exceptional prime has coefficient one in $A_Y$,
or equivalently has generalized log discrepancy zero over
$(V,A+\bM)$. Global strong $\Q$-factoriality is not assumed
for $V$.
\end{proposition}
\begin{theorem}[Gdlt termination in dimensions at most three]
\label{thm:gdlt-three}
Every sequence as in Theorem~\ref{thm:special} in dimension
at most three is finite. The assertion concerns arbitrary
choices of the permitted elementary steps, and does not
assume pseudo-effectivity of the adjoint.
\end{theorem}
\begin{proof}[Joint proof of Theorem~\ref{thm:special},
Proposition~\ref{prop:gdlt-mod}, and Theorem~\ref{thm:gdlt-three}]
For each $d$ in increasing order we prove special
termination, then the modification assertion, and, if
$d\leq3$, gdlt termination. In dimension zero these
statements are immediate. Fix $d>0$, assume all the
appropriate assertions in smaller dimensions, and first
prove special termination in dimension $d$.
\smallskip
\noindent\emph{Stabilizing centers.}
There are only finitely many divisorial steps, by
Lemma~\ref{lem:cycle-loss}; hence we may discard a finite
prefix and consider only small steps. Generalized
discrepancies do not decrease. A zero-discrepancy place on
the positive side was consequently a zero-discrepancy
place before the step, and strictness in
Lemma~\ref{lem:negativity} excludes its center from either
exceptional locus. Thus every new center is the transform
of an old center with unchanged general point. If the
exceptional locus contains the general point of an old
center, that center disappears. There are finitely many
centers on the initial model, so after another finite
prefix their list is unchanged and all steps are
isomorphisms at their general points.
For each center choose a chain of coefficient-one strata
at its general point. Transform this choice along the
sequence. Lemma~\ref{lem:stratum-adjunction} gives the
corresponding normalized strata $T_i$ and their adjoints
$D_{T_i}$. The b-line data on these strata are the same
under their birational identifications, with one fixed
higher denominator. The ambient boundary coefficients
belong to a fixed finite set. The coefficients after
adjunction therefore belong to one fixed DCC set, depending
only on that set, the denominator, and the length of the
chosen chain.
We now induct on the dimension $e$ of a surviving center
to show eventual disjointness. A point center is already
disjoint after the stabilization just made. For a center
of positive dimension $e$, its proper zero substrata have
smaller dimension and there are finitely many of them.
Discarding a further finite prefix, we may assume that
both exceptional loci avoid every such substratum at
every remaining step. In particular, the induced maps
on $T_i$ are isomorphisms in neighborhoods of their
non-gklt loci.
\smallskip
\noindent\emph{Finitely many extractions on the stratum.}
Identify divisorial places over the $T_i$ through common proper
bimeromorphic analytic models. If a prime
divisor $E$ is extracted by $T_i\dashrightarrow T_{i+1}$,
then $E$ is a divisor on $T_{i+1}$ and its center there is
contained in the ambient exceptional locus. Otherwise
the ambient isomorphism near its general point would
provide its strict transform as a divisor on $T_i$.
Lemma~\ref{lem:stratum-comparison} gives
\[
a(E;T_i,A_{T_i}+\bM^T)
<a(E;T_{i+1},A_{T_{i+1}}+\bM^T)\leq1,
\]
where the last inequality uses effectivity of the boundary.
By monotonicity its discrepancy on $T_0$ is less than one.
Its center on $T_0$ cannot be contained in
$\operatorname{Nklt}(T_0,A_{T_0}+\bM^T)$: the maps are
isomorphisms near that locus and its transforms, so a
valuation centered there cannot acquire a divisorial
center through the transformations under consideration.
There are only finitely many such valuations by the
off-non-gklt assertion of Lemma~\ref{lem:low-places}.
In addition to its exceptional valuations, include the
finitely many divisors already on $T_0$ having discrepancy
less than one; these are components of its positive
boundary. For any one of this finite set, the coefficients
on the models where it is extracted form a strictly
decreasing sequence: discrepancy never decreases, and
each new extraction gives a strict increase. Those
coefficients are in the fixed DCC set. Thus each
valuation is extracted only finitely often. Altogether
there are finitely many extractions.
\smallskip
\noindent\emph{Removing divisorial changes and stabilizing coefficients.}
After these extractions, the induced bimeromorphic maps
extract no divisors. Lemma~\ref{lem:cycle-loss}, applied
to the span of prime $(e-1)$-cycles of the compact
K\"ahler strata, shows that only finitely many further
maps contract a divisor. In dimension one a proper
bimeromorphic map between normal curves is already an
isomorphism. We may therefore assume that every induced
map is an isomorphism in codimension one.
With primes matched on this tail, their boundary
coefficients do not increase, by
Lemma~\ref{lem:stratum-comparison}. No component with
coefficient zero can acquire a positive coefficient.
Only the finite boundary support at the start of the
tail needs to be considered, and DCC stabilizes each
coefficient. Consequently
\[
(T_i\dashrightarrow T_{i+1})_*A_{T_i}=A_{T_{i+1}}.
\]
Let $S_i$ be the normalization of their common image in
the contraction base. Neither $T_i\to S_i$ nor
$T_{i+1}\to S_i$ contracts a prime divisor. Indeed such
a prime is matched in codimension one on the other
stratum, but its center is in the corresponding ambient
exceptional locus. Strict discrepancy increase would
contradict coefficient stabilization.
We have thus obtained small diagrams over the varying
$S_i$, with antiample and ample adjoints, respectively,
and strict-transform boundary and b-data. The effective
difference of Lemma~\ref{lem:stratum-comparison} is now
exceptional over both stratum models: its coefficient
at a divisor on either model is zero by the matched data.
If one of the induced birational maps is an isomorphism,
the matched boundary and b-line identify its adjoints
as actual rational bundles with their canonical
meromorphic comparison. The restricted difference is
then zero on a common model. The strictness assertion
shows that this step has no ambient exceptional
intersection with the stratum. Thus only nontrivial
small stratum surgeries remain to be excluded.
\smallskip
\noindent\emph{Lifting the small surgeries in dimension $e<d$.}
The induction hypothesis in dimension $e$ gives a
projective crepant gdlt modification
$h_0:Y_0\to T_0$ with globally strongly
$\Q$-factorial source. Over $S_0$ run the relative
elementary steps of Proposition~\ref{prop:relative-steps}
for its crepant adjoint. By gdlt termination in
dimension $e$, this run is finite and ends at a model
$Y_1$ whose adjoint is nef over $S_0$.
On a common smooth model of the initial lift, its
endpoint, and $T_1$, write the actual comparisons as
\[
P_0=P_1+G=P_++F.
\]
Here $P_0$ pulls back the initial crepant adjoint,
$P_1$ pulls back the adjoint of $Y_1$, and $P_+$ pulls
back $D_{T_1}$. The relative steps give $G\geq0$
exceptional over $Y_1$. The restricted comparison
just established, pulled through the initial crepant
modification, gives $F\geq0$ exceptional over $T_1$.
Lemma~\ref{lem:nef-end-comparison} applies: $P_1$ is
nef over $S_0$ and $D_{T_1}$ is ample there. It gives
$P_1=P_+$ and a projective crepant morphism
$h_1:Y_1\to T_1$.
This endpoint is again a gdlt modification of the
required type. The elementary steps preserve gdlt
and effectivity. They extract no prime divisor, and
the stratum map below is small. Any prime on $Y_1$
exceptional over $T_1$ was therefore exceptional on
$Y_0$ over $T_0$. If it survives, its coefficient is
still one, since boundaries are pushed forward.
Crepancy identifies that coefficient with its
zero-discrepancy extraction over $T_1$.
Repeat over $S_1,S_2,\ldots$. Each nontrivial stratum
surgery forces at least one lifted elementary step.
Indeed its negative morphism has a contracted curve
$C$ with $D_{T_i}\cdot C<0$. Projectivity gives a
curve on the lift dominating $C$ with positive
degree, on which the pulled-back adjoint is negative.
The lifted model is therefore not already nef over
$S_i$. Infinitely many nontrivial stratum surgeries
would give an infinite concatenation of elementary
gdlt steps in dimension $e$. This contradicts the
lower-dimensional gdlt termination hypothesis, which
allows varying compact contraction bases. There
are only finitely many such surgeries. The preceding
isomorphism case now proves disjointness for the
chosen center.
This completes the induction on $e$. Taking the
maximum of the finitely many resulting indices gives
special termination in dimension $d$, on both sides
of every remaining step. This proves
Theorem~\ref{thm:special} in that dimension.
\smallskip
\noindent\emph{Constructing the modification in dimension $d$.}
Now let $(V,A+\bM)$ be an arbitrary rational glc pair
as in Proposition~\ref{prop:gdlt-mod} in dimension $d$.
Take a projective log resolution $h:W\to V$ with $W$ smooth,
carrying the higher nef bundle. Let
\[
\Gamma=h_*^{-1}A+\sum_{E\subset\Exc(h)}E.
\]
The pair $(W,\Gamma+\bM)$ is gdlt: its boundary is
snc and its data descend to $W$. Its adjoint has
the actual comparison
\begin{equation}\label{eq:gdlt-modification-excess}
K_W+\Gamma+M_W=h^*(K_V+A+M_V)+Q,
\qquad
Q=\sum_{E\subset\Exc(h)}
a(E;V,A+\bM)E\geq0.
\end{equation}
The support of $Q$ is contained in $\lfloor\Gamma\rfloor$.
Run relative elementary steps over $V$ using
Proposition~\ref{prop:relative-steps}. On every model
the pushforward of \eqref{eq:gdlt-modification-excess}
remains the identity of its adjoint with the pulled-back
original adjoint plus the effective exceptional
transform of $Q$. Every surviving component of that
transform is a floor component.
If this run were infinite, special termination in
dimension $d$, just proved, would make its tail
disjoint from the floor. A contracted curve on such
a tail lies outside the support of $Q$ and has
nonnegative degree against this effective rational
Cartier divisor. Its degree against the pulled-back
original adjoint is zero. This contradicts negativity
of the elementary step. Thus the run is finite;
continuation ensures it ends at relative nefness.
On the endpoint $Y$, the remaining transform $Q_Y$
is nef over $V$ and exceptional, so
Lemma~\ref{lem:negativity} gives $Q_Y\leq0$. It is
also effective, hence zero. The comparison is
crepant. The resulting model is projective over $V$,
compact K\"ahler, globally strongly $\Q$-factorial,
and gdlt, by the relative construction. Any prime
exceptional over $V$ is the transform of an
exceptional prime on $W$, and has coefficient one
in the pushed boundary. This proves
Proposition~\ref{prop:gdlt-mod} in dimension $d$.
\smallskip
\noindent\emph{Gdlt termination in dimension $d\leq3$.}
Finally suppose $d\leq3$ and consider any infinite
elementary gdlt sequence in this dimension. Discard
its finite divisorial prefix and apply special
termination. On the resulting tail all exceptional
loci avoid $\lfloor A_i\rfloor$. Replace its boundary
by $A_i-\lfloor A_i\rfloor$ on each model, retaining
the same b-line. The resulting pairs are effective
and gklt. To check the latter, discrepancies cannot
decrease on removing an effective rational Cartier
floor. Each old zero-discrepancy place has center
in the floor, and the order of its pullback is
positive, so its discrepancy becomes positive.
Equivalently this is the snc calculation on a fixed
carrier and the gdlt good-open condition.
The floor has degree zero on every contracted curve
on both sides, because it is disjoint from the
exceptional loci. Thus all steps of this tail are
still negative and positive, respectively, for the
new adjoints. Their boundaries are strict transforms,
their nef data are unchanged, and their one-ray
condition for global line-bundle degrees is
unchanged. They are consequently elementary gklt
steps in the category of
Theorem~\ref{thm:gklt-three}. That theorem excludes
an infinite tail. This proves
Theorem~\ref{thm:gdlt-three} in dimension $d$ and
completes the triangular induction through
dimension four.
\end{proof}
\begin{remark}
The proof constructs auxiliary strong-$\Q$-factorial
models only after adjunction, or over the space to
which a gdlt modification is requested. It never
assumes that a normalized stratum is locally
$\Q$-factorial. Its uses of curve-nefness occur over
individual projective bimeromorphic morphisms;
the termination assertion concerns the entire
sequence of compact models with varying bases.
No analytic special-termination theorem is deduced
solely by citing its projective counterpart.
\end{remark}
==== END Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//10-special.tex ====
==== BEGIN Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//11-threshold.tex ====
\section{Termination for elementary sequences with a nef b-line summand}\label{sec:threshold}
We prove termination for the elementary sequences of
Section~\ref{sec:relative} when the adjoint has an effective rational
summand and a nef rational b-line summand. Special termination
allows us to delete the coefficient-one boundary from a tail of any
purported infinite sequence. Repeating this operation will give strictly
increasing generalized log canonical thresholds, contrary to ACC.
All programs and line-bundle comparisons take place on the full compact
spaces. The algebraic floor-deletion argument of
\cite[Lemma~2.20 and Theorem~4.1]{CT2023} provides a useful comparison;
the relative lifts needed here will follow from the analytic results already
proved.
\begin{theorem}\label{thm:expression-termination}
Let $(X,B)$ be a rational klt pair on a globally strongly $\Q$-factorial
compact K\"ahler fourfold. Suppose there are an effective rational divisor
$N$ on $X$ and rational nef b-line data $\bM$, carried by a projective
modification of $X$, such that
\begin{equation}\label{eq:threshold-expression}
K_X+B\simq N+M_X
\end{equation}
as actual rational line bundles. Then every birational elementary
$(K_X+B)$-sequence in the category of Section~\ref{sec:relative}
is finite. In particular, this holds for every sequence constructed by
Proposition~\ref{prop:ordinary-step}, independently of its ray choices.
\end{theorem}
The nefness in this statement is nefness of the higher rational line bundle.
No global meromorphic frame for that bundle, or uniform Cartier index for
its traces on the successive spaces, is assumed.
\subsection{Thresholds and their lifts}
The first lemma identifies the threshold on a fixed resolution. The
negative curve ensures that the testing boundary on that resolution is
nonzero, which makes the threshold finite.
\begin{lemma}\label{lem:rational-threshold}
Let $V$ be a globally strongly $\Q$-factorial compact K\"ahler space,
let $B,N\geq0$ be rational divisors, and let $\bM$ be rational nef b-line
data on a projective higher model. Set
\[
T=N+M_V,\qquad A(t)=K_V+B+tN+tM_V.
\]
Suppose $(V,B+aN+a\bM)$ is generalized klt for some rational $a\geq0$,
and $T\cdot C<0$ for a compact curve $C\subset V$. Then
\[
s=\sup\{t\geq0:(V,B+tN+t\bM)\text{ is generalized lc}\}
\]
is a finite rational number strictly greater than $a$. The pair is
generalized lc at $s$ and generalized klt at every $0\leq t<s$.
\end{lemma}
\begin{proof}
Choose a projective log resolution $p:W\to V$ carrying $\bM$ and resolving
all the divisor comparisons below. Write
\[
K_W+C_W=p^*(K_V+B),\qquad
H=p^*(N+M_V)-M_W.
\]
These are the comparisons of actual rational sheaves from
Lemma~\ref{lem:trace}. Strong $\Q$-factoriality makes $N$ and $M_V$
rational Cartier, and
\begin{equation}\label{eq:threshold-defect}
H=p^*N+(p^*M_V-M_W)\geq0
\end{equation}
by Lemma~\ref{lem:negativity}. After increasing $W$, the support of
$C_W+H$ is simple normal crossing. The crepant higher boundary at
parameter $t$ is $C_W+tH$.
The divisor $H$ is nonzero. Otherwise $p^*T=M_W$ is nef. Since $p$ is
projective, there is a compact curve $\widetilde C\subset W$ mapping onto
$C$ with positive degree: take a component dominating $C$ in $p^{-1}(C)$
and intersect it with sufficiently many general relative hyperplanes.
The projection formula would give
\[
0\leq M_W\cdot\widetilde C
=\deg(\widetilde C/C)\,T\cdot C<0.
\]
Here only degrees on curves are used; no absolute curve criterion for
analytic nefness is needed.
For the finitely many prime components $E$ with
$h_E=\operatorname{coeff}_E H>0$, put
$c_E=\operatorname{coeff}_E C_W$. The log-smooth discrepancy criterion
gives
\begin{equation}\label{eq:rational-threshold-formula}
s=\min_{h_E>0}\frac{1-c_E}{h_E}.
\end{equation}
Indeed all coefficients of $C_W+aH$ are strictly below one, and components
with $h_E=0$ impose no new bound. Formula~\eqref{eq:rational-threshold-formula}
proves finiteness, rationality, attainment, and $s>a$, as well as the
claimed singularities at and below $s$.
\end{proof}
At the threshold we need a generalized dlt model to use special
termination. The next lemma lifts each step of an infinite sequence to
a finite, nonempty string on such models. It uses the existence of a
finite relative generalized klt run from
Proposition~\ref{prop:relative-finite}; all comparisons concern one flip
base at a time.
\begin{lemma}\label{lem:threshold-lift}
Consider an infinite sequence of small elementary transformations
\[
X_i\dashrightarrow X_{i+1},\qquad
X_i\xrightarrow{f_i}Z_i\xleftarrow{f_i^+}X_{i+1},
\]
between globally strongly $\Q$-factorial compact K\"ahler spaces of
dimension at most four. Let
$B_i,N_i\geq0$ be strict transforms of rational divisors, let $\bM$ be
fixed rational nef b-line data, and put
\[
T_i=N_i+M_{X_i},\qquad A_i(t)=K_{X_i}+B_i+tT_i.
\]
Suppose that for rational numbers $0\leq a<s$:
\begin{enumerate}[label=\textup{(\roman*)}]
\item $(X_i,B_i+aN_i+a\bM)$ is generalized klt and
$(X_i,B_i+sN_i+s\bM)$ is generalized lc for every $i$;
\item $T_i$ and $A_i(t)$ have negative degree on the contracted ray
and their transforms are relatively ample on the positive side,
for every rational $t\geq a$.
\end{enumerate}
Then the sequence lifts to a concatenation of nonempty finite strings of
elementary generalized dlt steps at parameter $s$. At the ends of the
strings there are projective crepant morphisms
\[
h_i:(Y_i,\Delta_i+s\bM)\longrightarrow
(X_i,B_i+sN_i+s\bM)
\]
with effective generalized dlt boundary, extracting only divisors of
coefficient one.
\end{lemma}
\begin{proof}
Take $h_1$ by Proposition~\ref{prop:gdlt-mod}. Suppose the current $h_i$
has been constructed. We choose a nearby klt parameter, run a finite
relative program, and identify its endpoint with a crepant model of
$X_{i+1}$.
\smallskip\noindent
\emph{Choosing a nearby klt parameter.}
For $t\in[a,s]$, define its crepant boundary by
\[
K_{Y_i}+\Delta_i(t)+tM_{Y_i}=h_i^*A_i(t).
\]
This boundary depends affinely on $t$. Its nonexceptional coefficients
are $b+tn$ with $b,n\geq0$. Its exceptional coefficients equal one at
$s$. There are only finitely many exceptional primes, so for a rational
$t_i<s$ sufficiently close to $s$, with $t_i\geq a$, the boundary
$\Delta_i(t_i)$ is effective. In particular, a nonexceptional coefficient
zero at the positive parameter $s$ has $b=n=0$ and stays zero. The pair
downstairs is generalized klt for $a\leq t<s$: its discrepancies are
affine combinations of the positive discrepancies at $a$ and the
nonnegative ones at $s$. Thus the crepant pair at $t_i$ upstairs is
effective and generalized klt.
Choose a curve in the ray of $f_i$. By the one-ray condition on global
rational line-bundle degrees, there is a positive rational number $q_i$
such that $A_i(t_i)-q_iA_i(s)$ has zero degree on all $f_i$-contracted
curves. Apply Lemma~\ref{lem:descent}, using the generalized klt
antiample adjoint at $t_i$, to obtain a rational line bundle $L_i$ on
$Z_i$ with
\begin{equation}\label{eq:threshold-proportionality}
A_i(t_i)\simq q_i A_i(s)+f_i^*L_i.
\end{equation}
After pullback by $h_i$, this is an actual identity modulo a rational
bundle from $Z_i$, not merely an equality of numerical classes.
\smallskip\noindent
\emph{Running the finite relative program.}
The composite $Y_i\to Z_i$ is projective and bimeromorphic. Apply
Proposition~\ref{prop:relative-finite} to the effective generalized klt
pair at $t_i$ to obtain some finite relative elementary run ending nef
over $Z_i$. Transforming~\eqref{eq:threshold-proportionality} through
each step shows that the same run is negative, and positive after each
flip, for the full adjoint at $s$. It is therefore a generalized dlt
run at $s$, by Proposition~\ref{prop:relative-steps}, and its endpoint
is nef for this adjoint as well.
\smallskip\noindent
\emph{Identifying the endpoint.}
To check the endpoint, on a common resolution write
\begin{equation}\label{eq:threshold-comparison}
P_0=P_j+G_j=P_++F.
\end{equation}
Here $P_0$ is the pullback of $h_i^*A_i(s)$, $P_j$ is the pullback of the
lifted endpoint adjoint, and $P_+$ is the pullback of $A_{i+1}(s)$.
Negativity for the lifted run gives $G_j\geq0$, exceptional over its
endpoint, and negativity for the original small step gives $F\geq0$,
exceptional over $X_{i+1}$. Both endpoint adjoints are nef over $Z_i$,
and $A_{i+1}(s)$ is ample there. Lemma~\ref{lem:nef-end-comparison}
applies. Explicitly, if $p$ and $q$ denote the maps to these two targets,
then $P_j-P_+=F-G_j$ is $q$-nef with pushforward $-q_*G_j\leq0$;
its negative is $p$-nef with pushforward $-p_*F\leq0$. The two
negativity inequalities give $P_j=P_+$. Relative ampleness and
projective connected fibers then give the crepant projective morphism
from the lifted endpoint to $X_{i+1}$.
Every prime exceptional for this morphism is the transform of an
$h_i$-exceptional prime: the original step is small and the lifted run
extracts none. If it survives, its coefficient one is preserved by the
boundary pushforward rule. This proves the required assertion for
$h_{i+1}$. The string is nonempty, since $h_i^*A_i(s)$ has negative
degree on a curve lifting an $f_i$-contracted curve and so is not nef
over $Z_i$. Induction proves the lemma. The choice of $t_i$ is made
separately at each finite stage.
\end{proof}
\subsection{Deleting the floor and raising the threshold}
\begin{proof}[Proof of Theorem~\ref{thm:expression-termination}]
From an infinite sequence we will construct successively larger
thresholds with fixed coefficient data. At each repetition,
special termination will let us remove the floor while retaining an
infinite sequence with the required signs.
\smallskip\noindent
\emph{Step 1: the infinite tail and its first threshold.}
Suppose there is an infinite sequence. By Lemma~\ref{lem:cycle-loss},
it has only finitely many divisorial steps. Truncate after the last of
them and relabel the resulting sequence by $X_1\dashrightarrow X_2
\dashrightarrow\cdots$. Taking actual reflexive traces transports
\eqref{eq:threshold-expression} through this finite prefix. The
effective divisor remains effective and rational, and a projective
common higher model carries the pullback of the original nef line
bundle. Thus the relabelled input still has the required expression.
We will repeat the following construction. Its input is an infinite
small elementary sequence with rational families
\begin{equation}\label{eq:threshold-family}
(X_i,B_i+tN_i+t\bM),\qquad
A_i(t)=K_{X_i}+B_i+tN_i+tM_{X_i},
\quad T_i=N_i+M_{X_i},
\end{equation}
and a rational $a\geq0$ such that the pairs at $a$ are generalized klt,
and both $T_i$ and $A_i(t)$ have the negative-to-positive signs of each
step for all rational $t\geq a$. The divisors $B_i,N_i$ are effective
strict transforms. Initially $a=0$, the nef part at zero is zero, and
\[
T_i\simq K_{X_i}+B_i,
\qquad A_i(t)\simq(1+t)(K_{X_i}+B_i),
\]
so these conditions hold.
Let $s$ be the threshold of the first pair in~\eqref{eq:threshold-family}.
Lemma~\ref{lem:rational-threshold} applies to a contracted curve and gives
$s\in\Q$, $a<s<\infty$. Since $A_i(s)$ is negative on the input side
and ample on the positive side at each step, discrepancy monotonicity
shows that every pair in the sequence at $s$ is generalized lc.
Lemma~\ref{lem:threshold-lift} gives an infinite concatenation of
generalized dlt elementary steps at $s$, with compatible crepant
endpoint morphisms to the $X_i$.
\smallskip\noindent
\emph{Step 2: deleting the floor after special termination.}
Apply Lemma~\ref{lem:cycle-loss} and Theorem~\ref{thm:special} to this
concatenation. It has a tail consisting of small steps whose exceptional
loci, on both sides, avoid the support of the coefficient-one boundary.
Every lifted string is finite and nonempty, so an endpoint occurs beyond
any prescribed finite prefix. Start the tail at such an endpoint
$h:Y\to X_i$ and relabel. Write the union of the supports of $B_i,N_i$
as $\bigcup P$, with coefficients $b_P,n_P\geq0$. If $\widehat P$
denotes the strict transform on $Y$, the crepant boundary at $s$ is
\[
\Delta(s)=\sum_P(b_P+sn_P)\widehat P+\sum_{E\text{ exceptional for }h}E.
\]
Set
\begin{equation}\label{eq:retained-threshold-coefficients}
\widetilde B=\sum_{b_P+sn_P<1}b_P\widehat P,
\qquad
\widetilde N=\sum_{b_P+sn_P<1}n_P\widehat P.
\end{equation}
Then
\[
\widetilde B+s\widetilde N
=\Delta(s)-\lfloor\Delta(s)\rfloor.
\]
Deleting the floor of a generalized dlt pair gives a generalized klt
pair with the same nef data. Indeed every place of discrepancy zero
has center in that floor, whose rational Cartier pullback has positive
order at the place; all other discrepancies were already positive.
Consequently $(Y,\widetilde B+s\widetilde N+s\bM)$ is generalized klt.
Both divisors in~\eqref{eq:retained-threshold-coefficients} are effective.
\smallskip\noindent
\emph{Step 3: retaining the signs and increasing the threshold.}
Let their strict transforms define the new family on every model of
the lifted tail. Its signs require checking for the testing direction,
as well as for its adjoint. Do this over each original flip base $Z_i$.
At the initial endpoint of its lifted string, the trace comparison gives
\begin{equation}\label{eq:floor-supported-test}
h_i^*T_i=\widetilde N_i+M_{Y_i}+F_i,
\qquad
\Supp F_i\subseteq\Supp\lfloor\Delta_i(s)\rfloor.
\end{equation}
To see the support assertion, the difference between pullback and strict
transform of $N_i$, and the difference between pullback and b-line trace
of $M_{X_i}$, are $h_i$-exceptional. Every such prime has coefficient
one in $\Delta_i(s)$. The remaining difference consists of the discarded
$n_P\widehat P$, also supported in the floor. This is a comparison of
actual rational sheaves; a sign for $F_i$ is unnecessary.
On $X_i$, both $A_i(s)$ and $T_i$ have negative degree on the contracted
ray, and their ratio is a positive rational number $c_i$.
Lemma~\ref{lem:descent}, applied using any nearby generalized
klt parameter below $s$, gives a rational line bundle $Q_i$ on $Z_i$
such that
\begin{equation}\label{eq:floor-test-proportionality}
h_i^*A_i(s)\simq
c_i(\widetilde N_i+M_{Y_i}+F_i)+(f_i h_i)^*Q_i.
\end{equation}
Transform this identity through the finite lifted string. The tail is
small, so all its boundary divisors, traces, and $F_i$ transform strictly.
The support of the transform of $F_i$ remains in the transformed floor.
All contracted and flipped curves are disjoint from that floor.
Equation~\eqref{eq:floor-test-proportionality} therefore shows that
\(\widetilde T=\widetilde N+M_{\mathrm{trace}}\) has negative degree on
every contracted ray and positive relatively ample transform on the
other side. For the ampleness assertion, the same identity, with the
floor bundles trivial near the exceptional fibers, identifies its
restriction there with a positive rational multiple of the full ample
adjoint; fiberwise ampleness is the relative ampleness criterion.
Likewise subtracting the floor changes neither sign of the adjoint at
$s$. On each side, for every rational $u\geq s$, the new adjoint is
\[
\widetilde A(u)=\widetilde A(s)+(u-s)\widetilde T,
\]
so it has the same negative-to-positive signs. At later endpoints the
retained coefficients still obey~\eqref{eq:retained-threshold-coefficients}:
the downstairs maps are small, and an exceptional divisor of an endpoint
morphism cannot become the transform of a downstairs prime. All
boundaries upstairs are pushforwards, hence strict transforms on this
tail. Thus the families constructed over consecutive bases agree.
We have obtained another infinite small elementary sequence satisfying
the same conditions, now with $a=s$. The generalized klt condition at
$s$ holds throughout the tail by discrepancy monotonicity. It also
implies the generalized klt condition at parameter zero, by
\eqref{eq:threshold-defect}. Lemma~\ref{lem:rational-threshold} shows
that its next threshold is finite, rational, and strictly larger than
$s$. The construction can therefore be repeated indefinitely if the
original sequence was infinite. It produces
\begin{equation}\label{eq:increasing-thresholds}
0<s_1<s_2<s_3<\cdots.
\end{equation}
\smallskip\noindent
\emph{Step 4: the fixed data for analytic ACC.}
Let $I$ be the finite set consisting of zero and the coefficients of the
first $B$, and let $J$ consist of zero, the coefficients of the first
$N$, and $1/p$, where $p>0$ makes $pM_{W_0}$ integral on the
original nef carrier $W_0$. At every repetition, the new boundary and testing
coefficients are obtained solely by omissions from the previous ones,
as in~\eqref{eq:retained-threshold-coefficients}. Newly extracted
coefficient-one primes are removed before the next threshold input.
The boundary coefficients therefore stay in $I$, the testing-boundary
coefficients stay in $J$, and the nef part at parameter zero remains
zero. On every required higher model the testing nef data are
$\frac1p$ times the pullback of the same integral nef line bundle.
The denominator $p$ is fixed. Common projective models are needed only
for finitely many maps at a time and do not change $p$.
We verify explicitly that each $s_j$ belongs to the analytic generalized
threshold set with these data. Choose a general point of a computing
center on its compact input, and a relatively compact Stein neighborhood
$U$ of that point. Restrict a projective log resolution carrying the
data to $U$, and denote it by $g:W\to U$. The computing place still
meets this restriction. Hence the local threshold equals $s_j$: the
pair is lc at $s_j$, whereas the same place has negative discrepancy
for every larger parameter.
Let $\cL$ represent the integral line bundle $pM_W$. The coherent sheaf
$g_*\cL$ has rank one on the isomorphism locus of $g$. Cartan's
Theorem~A supplies a section nonzero at a point of that locus, and thus
a nonzero section of $\cL$ on $W$. Its divisor $P'$ is Cartier and
represents $\cL$. In particular $P'$ is $g$-nef and $M_W$ is represented
by $\frac1pP'$. The induced meromorphic identification gives precisely
the original trace and pullback defect. Compatible local canonical
representatives may be chosen in the same way from the coherent
canonical sheaf. Neither choice changes the boundary or testing-boundary
coefficients, the discrepancies, or the factor $1/p$. The testing
boundary is rational Cartier because its components are restrictions
of global rational Cartier divisors on the strongly $\Q$-factorial input.
The analytic generalized threshold theorem
\cite[Theorem~1.1 and Theorem~3.3; see also Definition~2.3 and
Remark~2.5]{HXACC2023} consequently applies in dimension four with the
fixed DCC sets $I,J$. Its nef summands are actual relatively nef Cartier
divisors with the fixed weight $1/p$; it requires no common bound for
Cartier indices of their downstairs traces. It prohibits the strictly
increasing sequence~\eqref{eq:increasing-thresholds}. This contradiction
proves the theorem.
\end{proof}
==== END Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//11-threshold.tex ====
==== BEGIN Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//12-supply.tex ====
\section{Nef b-line decompositions}\label{sec:supply}
\begin{proposition}\label{prop:expression}
Let $(X,B)$ be a globally strongly $\Q$-factorial compact Kähler
fourfold pair, with $B$ effective and rational, $(X,B)$ klt, and
$D=K_X+B$ rational Cartier and analytically pseudo-effective. Suppose
there is a projective log resolution $p:W\to X$, with $W$ smooth and
compact Kähler, such that $W$ is non-uniruled or Moishezon. Then there
exist a smooth compact Kähler manifold $U$, a
projective modification $r:U\to X$, an analytically nef rational
holomorphic line bundle $M_U$, and an effective rational divisor $N$ on
$X$ such that, for the rational b-line bundle $\bM$ determined by $M_U$,
\begin{equation}\label{eq:supply-expression}
D\simq N+M_X.
\end{equation}
The equivalence is an identity of actual rational reflexive sheaves.
\end{proposition}
\begin{proof}
Choose a projective log resolution $p:W\to X$ as in the statement.
We first describe how to transfer a nef adjoint on a minimal model reached
by a finite program from $W$ to a rational b-line summand over $X$.
Suppose that a finite program on $W$ ends at a model
$Y$, and that $H_W$ and $H_Y$ are its initial and final actual rational
adjoints. Both morphisms defining each step are projective. The main
components of the successive fiber products of their graphs, followed
by normalization and resolution, therefore give a diagram
\[
\begin{array}{ccc}
&U&\\[-2pt]
a\swarrow&&\searrow q\\[-2pt]
W&&Y
\end{array}
\]
whose two arrows are projective modifications. The construction uses
only the finitely many steps of this program: projectivity follows
successively by base change and composition. In particular $r=pa$ is
projective over the original $X$, and $U$ is compact Kähler.
The stepwise comparisons in Lemma~\ref{lem:negativity}, pulled back to
this common model, give
\begin{equation}\label{eq:supply-common-comparison}
a^*H_W\simq q^*H_Y+G,\qquad G\geq0.
\end{equation}
Set $M_U=q^*H_Y$. When $H_Y$ is nef, so is $M_U$: pull back metrics with
arbitrarily small negative curvature and compare the pulled-back fixed
form with a Kähler form on $U$. In the projective case $U$ is itself
projective, and the pullback of a nef rational bundle is analytically
nef by the projective metric characterization. For a common multiple
$m$, the trace of $M_U$ is
represented by the rank-one reflexive sheaf
\[
\bigl(r_*\cO_U(mM_U)\bigr)^{**}.
\]
Coherence follows from properness, and global strong $\Q$-factoriality
on $X$ makes a further reflexive power invertible. By
Lemma~\ref{lem:trace}, tracing \eqref{eq:supply-common-comparison} gives
\begin{equation}\label{eq:supply-trace-comparison}
\operatorname{Tr}_r(a^*H_W)\simq M_X+r_*G.
\end{equation}
Indeed this is the usual tensor identity at each codimension-one point
of $X$, where $r$ is an isomorphism, and both sides extend reflexively.
It retains the actual identifications of the line bundles. In
particular, no global meromorphic frame of either canonical bundle or
of $M_U$ is chosen.
\medskip\noindent
\textit{Non-uniruled resolution.}
By \cite[Theorem~1.1]{Ou2025}, $K_W$ is analytically pseudo-effective.
Starting with the smooth empty-boundary pair $(W,0)$, apply
Lemma~\ref{lem:absolute-support} and Proposition~\ref{prop:ordinary-step}
whenever the current canonical bundle is not nef.
Lemma~\ref{lem:pe-step} preserves its
pseudo-effectivity, and Lemma~\ref{lem:mori-not-pe} excludes a
fiber-type contraction at every stage. Terminality persists by
Lemma~\ref{lem:negativity}. Corollary~\ref{prop:terminal-four} rules
out an infinite sequence. Continuation from every non-nef model thus
gives a finite program ending at a terminal compact Kähler $Y$ with
analytically nef actual rational canonical bundle $K_Y$.
Apply \eqref{eq:supply-common-comparison} with $H_W=K_W$ and $H_Y=K_Y$.
The reflexive trace of $a^*K_W$ on $X$ is $K_X$: over the common
codimension-one open set the canonical sheaves are canonically
identified, and normality determines their reflexive extensions.
Equation~\eqref{eq:supply-trace-comparison} therefore yields
\[
K_X+B\simq M_X+(r_*G+B).
\]
Take $N=r_*G+B$. This argument uses the trace of $K_W$; it does not
require the discrepancies comparing $K_W$ with $p^*D$ to be
nonnegative. The terminal fourfold result used here was established
independently of Theorem~\ref{thm:expression-termination}.
\medskip\noindent
\textit{Uniruled Moishezon resolution.}
The smooth Kähler manifold $W$ is then projective. By
Lemma~\ref{lem:supply-log-resolution}, choose an effective rational SNC
klt boundary $\Gamma$ satisfying
\begin{equation}\label{eq:supply-projective-start}
H_W:=K_W+\Gamma\simq p^*D+E_0,
\qquad E_0\geq0\quad\text{and }E_0\text{ is }p\text{-exceptional}.
\end{equation}
This adjoint is analytically pseudo-effective, hence numerically
pseudo-effective in the projective sense by
\cite[Proposition~1.2]{BDPP2013}.
Run the ordinary projective klt MMP on $(W,\Gamma)$. Divisorial steps
are finite, and every remaining flip sequence terminates by
\cite[Theorem~1.1]{CT2023}, applied to a pseudo-effective NQC lc
fourfold with zero nef part. These are projective varieties over a
point, as required by that theorem's conventions. Pseudo-effectivity
persists by Lemma~\ref{lem:pe-step}, so Lemma~\ref{lem:mori-not-pe}
excludes a fiber-type negative contraction. The resulting model $Y$ has nef
rational adjoint $H_Y=K_Y+\Gamma_Y$.
Form \eqref{eq:supply-common-comparison} for this finite program.
Since $E_0$ is exceptional over $X$, the reflexive trace of
$a^*H_W$ in \eqref{eq:supply-projective-start} is $D$. Thus
\eqref{eq:supply-trace-comparison} gives
\[
D\simq M_X+r_*G.
\]
Take $N=r_*G$. The projective fourfold theorem has supplied a nef
adjoint; fourfold abundance is not used.
\end{proof}
==== END Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//12-supply.tex ====
==== BEGIN Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//13-local-convention.tex ====
\section{Why analytic local factoriality is different}
\label{sec:local-convention}
Global Weil-divisor $\Q$-factoriality does not require every analytic local
ring to be $\Q$-factorial. The distinction already occurs for projective
threefolds with one ordinary double point; see
\cite[Definition~5.1, Remark~(b)]{Reid}. The following product example
explains why the stronger local requirement cannot be inserted into
Theorem~\ref{thm:finite}.
\begin{proposition}\label{prop:local-convention}
There is a smooth projective fourfold $X$ with pseudo-effective, non-nef
canonical divisor for which no ordinary $K_X$-negative program can reach a
nef model while keeping every model analytically locally
$\Q$-factorial. Its unique possible first negative birational contraction
is divisorial. Its target is nevertheless globally factorial and admits
a nef canonical divisor.
\end{proposition}
\begin{proof}
\emph{Construction.}
Let $b:B\to\Pbb^4$ be the blowup of a point, let
$H=b^*\cO_{\Pbb^4}(1)$, and let $F$ be the exceptional divisor.
The graph of projection from that point embeds $B$ in
$\Pbb^4\times\Pbb^3$, and the restriction of $\cO(1,1)$ is $2H-F$.
Thus $2H-F$ is very ample. Since $H$ is globally generated,
\[
6H-2F=2(2H-F)+2H
\]
is very ample as well. Take a general smooth member
$U\in|6H-2F|$.
Its image $U_0\subset\Pbb^4$ is smooth away from the blown-up point.
The quadratic jets of the defining sextics at that point are arbitrary,
so a general member has a nondegenerate quadratic initial term.
The holomorphic Morse lemma identifies its unique singularity with
\[
xy-zw=0.
\]
The restriction $\pi:U\to U_0$ resolves this node, with exceptional
quadric
\[
E=F|_U\simeq\Pbb^1\times\Pbb^1,
\qquad \cO_U(E)|_E=\cO_E(-1,-1).
\]
Adjunction gives
\begin{equation}\label{eq:local-example-canonical}
K_U=H|_U+E,
\end{equation}
since $K_B=-5H+3F$.
Integral weak Lefschetz for the smooth ample threefold $U\subset B$
\cite[Section~7, Corollary~7.3]{MilnorMorse1963} gives
\[
H^1(U,\Z)=0,
\qquad H^2(U,\Z)=\Z[H|_U]\oplus\Z[E].
\]
This is the blowup calculation underlying the classical example in
\cite[Definition~5.1, Remark~(b)]{Reid}.
Let $C$ be a smooth elliptic curve and set $X=U\times C$.
Its canonical divisor is the pullback of \eqref{eq:local-example-canonical},
so is effective. If $\ell$ is either ruling in $E\times\{c\}$, then
$K_X\cdot\ell=-1$; in particular $K_X$ is not nef.
\emph{All negative contractions have the same target.}
Put $D=E\times C$ and let $h=\{e\}\times C$ be a horizontal curve.
A curve not contained in $D$ has nonnegative degree against both the
effective divisor $D$ and the nef hyperplane pullback. Thus every
$K_X$-negative curve lies in $D$.
The two rulings of $E$ have the same class in $H_2(U,\R)$: both have degrees
$0,-1$ against the displayed basis of $H^2(U,\R)$.
Künneth and $H^1(U,\R)=0$ now show that every integral curve
$\gamma\subset D$ has class
\begin{equation}\label{eq:local-example-curve}
[\gamma]=a[\ell]+b[h],\qquad a,b\geq0.
\end{equation}
Here $a$ is the sum of the projection degrees to the two factors of
$E=\Pbb^1\times\Pbb^1$, and $b$ is the projection degree to $C$.
Moreover $K_X\cdot\gamma=-a$, so negativity means $a>0$.
Let $f:X\to Z$ be any nontrivial projective birational contraction with
connected fibres, normal compact Kähler target, and $-K_X$ relatively
ample. Choose a Kähler class $\omega_Z$ and set $\alpha=f^*\omega_Z$.
For a curve, vanishing of its degree against $\alpha$ is equivalent to
being contracted by $f$.
In particular $\alpha\cdot\ell\geq0$ and $\alpha\cdot h>0$: the
horizontal curve cannot be contracted because $K_X\cdot h=0$.
A contracted curve $\gamma$ has the form
\eqref{eq:local-example-curve} with $a>0$, and
\[
0=\alpha\cdot\gamma
=a(\alpha\cdot\ell)+b(\alpha\cdot h).
\]
It follows that $b=0$ and $\alpha\cdot\ell=0$.
Thus all rulings in every $E\times\{c\}$ are contracted, and $f$ is not
small.
Set $q=\pi\times\id_C:X\to U_0\times C$.
Its nontrivial fibres are the quadrics $E\times\{c\}$, which are connected
by their rulings, so $f$ is constant on every $q$-fibre.
Conversely, every curve in an $f$-fibre has the form
\eqref{eq:local-example-curve} with $b=0$, and is contracted by $q$.
A connected projective fibre is connected by chains of curves, so $q$ is
constant on every $f$-fibre. The two targets are isomorphic: the reduced
image of $(q,f)$ in their product projects properly, bijectively and
bimeromorphically to each normal target, hence finitely and isomorphically.
Therefore $q$ is the only possible first negative birational contraction.
It is divisorial, and its relative numerical rank is one because both
quadric rulings have the same nonzero numerical class.
\emph{Failure of analytic local $\Q$-factoriality.}
Near the singular locus of $U_0\times C$, consider the prime analytic
divisor
\[
P=\{x=z=0\}\times\text{(a disk)}
\ \subset\ \{xy-zw=0\}\times\text{(a disk)}.
\]
On the blowup of the singular axis, its strict transform meets each
exceptional quadric in a ruling line. If $nP$ were Cartier near an axis
point for some integer $n>0$, its pullback would have the form
$n\widetilde P+kE_{\mathrm{exc}}$.
The associated line bundle restricts trivially to the quadric over that
point, because it is pulled back from a line bundle at a point.
On the other hand its restriction is
\[
\cO_{\Pbb^1\times\Pbb^1}(n-k,-k),
\]
up to interchanging the factors. Triviality forces $k=0$ and $n=0$, a
contradiction. The target is not analytically locally $\Q$-factorial.
Since $K_X$ is not nef and there is no small negative contraction,
there is no permissible first step under that local convention.
\emph{Compatibility with the global convention.}
The same example has a valid one-step global program. Indeed,
$-E$ is $\pi$-ample, so $-K_X$ is $q$-ample, and adjunction on $U_0$ gives
\[
\cO_{U_0\times C}(K_{U_0\times C})
\simeq\operatorname{pr}_1^*\cO_{U_0}(1),
\]
which is nef. On the blowup of the node the exceptional divisor has log
discrepancy two; its smooth product with $C$ is a log resolution with the
same log discrepancy. Hence the target is terminal, in particular klt.
It remains to check that the local divisor $P$ is not a global
factoriality obstruction. The displayed integral cohomology of $U$ and
Hodge decomposition give $H^1(U,\cO_U)=H^2(U,\cO_U)=0$ and
$\Pic(U)=\Z[H|_U]\oplus\Z[E]$.
The exponential sequence and Künneth therefore give
\[
\Pic(X)=\Z[H]\oplus\Z[D]\oplus\operatorname{pr}_C^*\Pic(C).
\]
Strict transform and divisor pushforward identify
\[
\Cl(U_0\times C)=\Pic(X)/\Z[D].
\]
Every surviving class descends to a Cartier class, either from the
hyperplane bundle on $U_0$ or from a line bundle on $C$.
Chow's theorem makes every global analytic Weil divisor on this projective
target algebraic, so every such divisor is Cartier
\cite[Section~4, no.~19, Proposition~13]{SerreGAGA1956}.
GAGA identifies its global coherent
rank-one reflexive sheaves with algebraic divisorial sheaves, which are
therefore invertible as well
\cite[Section~3, no.~12, Theorems~2--3]{SerreGAGA1956}. The plane $P$ is defined only on an analytic
neighbourhood; it need not extend to a divisor on the whole compact space.
\end{proof}
==== END Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//13-local-convention.tex ====
==== BEGIN Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//dependencies.tex ====
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[
box/.style={draw=linkblue!65,rounded corners=2pt,align=center,
fill=linkblue!3,text width=5.1cm,inner sep=6pt,font=\small},
edge/.style={-{Stealth[length=2mm]},draw=linkblue!80,thick}]
\node[box] (steps) at (0,0)
{Positive traces and ordinary steps\\Exact line-bundle descent};
\node[box] (terminal) at (6.2,0)
{Discrepancies and analytic cycle ranks\\Terminal fourfold termination\\with real boundary};
\node[box] (scaling) at (0,-2)
{Ordinary scaling from $X$\\Vanishing thresholds\\Stabilized auxiliary extraction};
\node[box] (limit) at (6.2,-2)
{The limiting terminal pair\\A finite small program\\to a nef ordinary adjoint};
\node[box] (gap) at (0,-4.1)
{Rational nef decomposition\\Fixed Cartier indices\\and a positive intersection gap};
\node[box] (perturb) at (6.2,-4.1)
{A nearby generalized program\\Crepant rational adjoints\\Descent to the original flip base};
\node[box,text width=11.3cm] (finite) at (3.1,-6.1)
{The original negative ray becomes trivial: contradiction\\
A finite ordinary program and its nef or Mori endpoint
(Theorem~\ref{thm:finite})};
\draw[edge] (steps)--(scaling);
\draw[edge] (terminal)--(limit);
\draw[edge] (scaling)--(limit);
\draw[edge] (limit.south west)--(gap.north east);
\draw[edge] (gap)--(perturb);
\draw[edge] (limit)--(perturb);
\draw[edge] (perturb)--(finite);
\end{tikzpicture}
\caption{The finite-program argument. The ordinary sequence starts on
$X$. Extraction and the limiting and nearby programs are auxiliary
comparisons. Exact bundle descent preserves the indices used in the
positive intersection gap; the last descent occurs on a common
resolution, as in Figure~\ref{fig:perturbation-descent}.}
\label{fig:dependencies}
\end{figure}
==== END Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//dependencies.tex ====
==== BEGIN Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//main.tex ====
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\hypersetup{pdftitle={Finite ordinary minimal model programs on compact Kähler fourfolds},pdfauthor={OpenAI}}
\title[Minimal model programs on Kähler fourfolds]{Finite ordinary minimal model programs\\on compact Kähler fourfolds}
\author{OpenAI}
\date{September 24, 2026}
\begin{document}
\begin{abstract}
We prove that a globally Weil-$\Q$-factorial compact Kähler klt fourfold pair with effective rational boundary and canonical rational line bundle admits a finite ordinary minimal model program starting on the given pair. It ends at a nef model when the adjoint class is pseudo-effective and at a Mori fibre space otherwise.
\end{abstract}
\maketitle
\tableofcontents
\input{sections/01-introduction}
\input{sections/02-analytic}
\input{sections/03-steps}
\input{sections/04-terminal}
\input{sections/05-scaling}
\input{sections/06-perturbation}
\input{sections/07-endpoints}
\appendix
\input{sections/13-local-convention}
\enlargethispage{2\baselineskip}
\bibliographystyle{amsplain}
\begingroup
\renewcommand{\baselinestretch}{0.95}
\bibliography{references}
\endgroup
\end{document}
==== END Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//main.tex ====
==== BEGIN Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//references.bib ====
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}
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@misc{DHY2023,
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}
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==== END Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds//references.bib ====