64c3814
==== TABLE OF CONTENTS ====
Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/paper.tex
Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/references.tex
Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/sections/01-introduction.tex
Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/sections/02-foundations.tex
Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/sections/03-comparison.tex
Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/sections/04-low-places.tex
Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/sections/05-witnesses.tex
Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/sections/06-cycles.tex
Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/sections/07-termination.tex
===========================
==== BEGIN Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/paper.tex ====
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\hypersetup{pdftitle={Termination of generalized-canonical flips on compact Kahler fourfolds},pdfauthor={OpenAI}}
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\title{Termination of generalized-canonical flips\\on compact K\"ahler fourfolds}
\author{OpenAI}
\date{October 6, 2026}
\begin{document}
\maketitle
\input{sections/01-introduction}
\input{sections/02-foundations}
\input{sections/03-comparison}
\input{sections/04-low-places}
\input{sections/05-witnesses}
\input{sections/06-cycles}
\input{sections/07-termination}
\input{references}
\end{document}
==== END Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/paper.tex ====
==== BEGIN Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/references.tex ====
\begin{thebibliography}{99}
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==== END Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/references.tex ====
==== BEGIN Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/sections/01-introduction.tex ====
\begin{abstract}
Every sequence of generalized-canonical flips on compact K\"ahler
fourfolds with rational boundary and fixed rational analytically nef
b-divisor terminates when the small morphisms are projective. We prove
this for normal globally Weil $\Q$-factorial models with boundary
coefficients less than one, allowing exceptional log discrepancies equal
to one. No scaling rule or pseudo-effectivity assumption is required.
The theorem applies to existing projective small diagrams with the
specified opposite ample signs.
\end{abstract}
\section{Introduction and the main theorem}\label{sec:introduction}
A flip replaces a small contraction on which an adjoint divisor is
negative by a small contraction on which its transform is positive.
Termination asks whether this local improvement can continue
indefinitely. Generalized pairs enlarge the adjoint from $K_X+B$ to
$K_X+B+M_X$, where $M_X$ is the trace of nef data on a higher
birational model. The nef contribution can change discrepancies even
where the ordinary boundary is absent. On a compact K\"ahler space,
one must also distinguish analytic nefness from nonnegative degrees
on curves.
We prove termination in the generalized-canonical case in dimension
four, for rational nef data fixed throughout the sequence. The proof
does not choose the contractions. It applies to every sequence of
projective small diagrams with the required opposite ample signs.
Exceptional log discrepancies may equal one. The resulting canonical
singularities need not be smooth in codimension two, so the argument
must establish smoothness precisely along the flipped surfaces that
it counts. We first specify the global analytic category of the theorem.
\subsection{The data and the statement}
All spaces are complex, and a fourfold is irreducible and
four-dimensional. A K\"ahler form on a normal space is
given by smooth strictly plurisubharmonic local potentials in analytic
embeddings. A class in the real Bott--Chern space
$H^{1,1}_{\mathrm{BC}}(X,\R)$ is \emph{analytically nef} if it lies
in the closure of the K\"ahler cone. For a rational line bundle or
a $\Q$-Cartier divisor, this means that its first Chern class is nef
in this sense.
We call a normal compact space \emph{globally Weil $\Q$-factorial} if
every prime Weil divisor defined on the whole space is $\Q$-Cartier
and some positive reflexive power of its canonical sheaf is a line
bundle. This condition does not assert factoriality of the local
analytic rings, or $\Q$-Cartierness of every rank-one reflexive sheaf.
Compatible actual canonical divisor representatives on the birational
models are part of the supplied data. We do not infer their existence
from the canonical rational line-bundle condition alone.
Fix a projective birational morphism
\[
\pi:X'\longrightarrow X
\]
of normal compact K\"ahler spaces and an analytically nef
$\Q$-Cartier divisor $M'$ on $X'$. These data define a rational
b-divisor $\bM$: on a model dominating $X'$ its trace is the pullback
of $M'$, and on another model its trace is the pushforward from a
common higher model. We write $M_T$ for the trace on a model $T$.
The divisor $M'$ need not be effective. Replacing $X'$ by a higher
model and $M'$ by its pullback leaves $\bM$ unchanged.
Let $B\geq0$ be a rational divisor of finite support with coefficients
less than one, and assume $D=K_X+B+M_X$ is $\Q$-Cartier. On a smooth
model $p:W\to X$ dominating $X'$, define the actual rational divisor
$\Delta_W$ by
\begin{equation}\label{eq:structure}
K_W+\Delta_W+M_W=p^*(K_X+B+M_X).
\end{equation}
For a prime divisor $E$ on $W$, its \emph{generalized log discrepancy}
is
\[
a(E;X,B+\bM)=1-\operatorname{coeff}_E\Delta_W.
\]
Here a \emph{prime over $X$}, or a \emph{divisorial place}, is a prime
divisor on a proper bimeromorphic model, identified with its strict
transforms on common higher models. Its centre is its reduced
irreducible image, and it is \emph{exceptional} if this image has
codimension at least two. Equation~\eqref{eq:structure} on higher
models defines its discrepancy independently of the carrying model.
The pair is \emph{generalized klt} if all these discrepancies are
positive, and \emph{generalized canonical} if the discrepancies of
exceptional primes are at least one. Replacing the latter inequality
by strict inequality gives \emph{generalized terminal}. Throughout
the theorem we impose positivity for every prime as well as the
canonical bound for exceptional primes. Accordingly, exceptional
coefficients of $\Delta_W$ may be zero or negative.
We use \emph{projective} for a proper morphism admitting a relatively ample
holomorphic line bundle. A $\Q$-Cartier divisor is relatively ample
if a positive Cartier multiple is so. A proper bimeromorphic
morphism is \emph{small} if its exceptional locus contains no prime
divisor. A small birational map identifies the prime divisors on its
two models, and hence defines strict transforms of boundaries and
canonical divisors.
\begin{theorem}\label{thm:termination}
Let $X_0$ be a normal irreducible globally Weil $\Q$-factorial compact
K\"ahler fourfold, and let $B_0\geq0$ be a rational divisor with
coefficients in $[0,1)$. Fix compatible actual canonical divisors on
the birational models. Let $\bM$ be the rational b-divisor represented
by an analytically nef $\Q$-Cartier divisor $M'$ on a projective
birational morphism $X'\to X_0$ of normal compact K\"ahler spaces.
Assume
\[
\begin{aligned}
a(E;X_0,B_0+\bM)&>0&&\text{for every prime over $X_0$},\\
a(E;X_0,B_0+\bM)&\geq1&&\text{for every exceptional prime}.
\end{aligned}
\]
Consider a sequence of normal globally Weil $\Q$-factorial compact
K\"ahler fourfolds
\[
X_0\dashrightarrow X_1\dashrightarrow X_2\dashrightarrow\cdots .
\]
On $X_i$, let $B_i$ be the strict transform of $B_0$, let $M_i$ be
the trace of this same $\bM$, and suppose
$D_i=K_{X_i}+B_i+M_i$ is $\Q$-Cartier. Suppose that every step is
given by a diagram
\begin{equation}\label{eq:step-diagram}
X_i\xrightarrow{\ f_i\ }Z_i
\xleftarrow{\ f_i^+\ }X_{i+1}
\end{equation}
of projective small bimeromorphic morphisms with connected fibres,
where $Z_i$ is normal compact K\"ahler, both morphisms are
nonisomorphisms, $-D_i$ is $f_i$-ample, and $D_{i+1}$ is
$f_i^+$-ample. Then the sequence is finite.
\end{theorem}
The canonical discrepancy bound and positivity are assumed only on
the initial pair; both persist by the comparison proved below.
The theorem imposes no pseudo-effectivity or scaling condition and
no relative Picard-number restriction. It concerns finiteness of
given diagrams; existence of a desired next step is a separate question.
\subsection{Preceding results and the present argument}
In the relative projective algebraic setting, Kawamata, Matsuda,
and Matsuki proved termination for ordinary terminal
$\Q$-factorial fourfolds with empty boundary. Their fourfold argument
combines Shokurov's discrepancy difficulty with a descent of surface
classes \cite[Theorem~5-1-15 and its proof]{KMM1987}.
Fujino extended the discrepancy and surface-count method to ordinary
canonical projective fourfold pairs with effective rational boundary
\cite{Fujino2004}. The corrected statement with
$\lfloor B\rfloor=0$ is \cite[Theorem~5.1]{Fujino2005Addendum}; its
flipping contractions need not have relative Picard number one.
Two parts of this work directly underlie our proof. First, the target
of a canonical flip is terminal near general points of its
codimension-two flipped locus, which permits the blowup discrepancy
calculation \cite[Lemma~2.2]{Fujino2004}. Second, the corrected
low-discrepancy count makes its exclusions by centre, so that they
apply to repeated blowups above a boundary surface and not only to
its first blowup
\cite[Proposition~3.1 and Lemma~3.2]{Fujino2005Addendum}.
Generalized pairs retain nef information on a higher birational
model; see the generalized polarized pairs of
Birkar and Zhang \cite[Definition~1.4 and Section~4]{BirkarZhang2016}.
Chen and Tsakanikas proved termination of every sequence of flips for
pseudo-effective four-dimensional NQC log canonical generalized pairs
in the relative projective algebraic setting
\cite[Theorem~1.1]{ChenTsakanikas2023}; see also
\cite[Theorem~4.4]{Moraga2025}.
Here NQC means that the nef data are finite nonnegative real
combinations of nef $\Q$-Cartier data.
The rational nef datum used here is NQC in the algebraic setting.
Their result allows log canonical singularities and real NQC data.
Theorem~\ref{thm:termination} instead concerns canonical singularities
and fixed rational data, permits nonprojective compact K\"ahler spaces,
and applies without pseudo-effectivity. The smooth-centre discrepancy
formula for generalized pairs also appears in
\cite[Lemma~2.9]{ChenTsakanikas2023}.
In the K\"ahler setting, Das, Hacon, and P\u{a}un prove termination
for ordinary dlt compact K\"ahler fourfold pairs whose adjoint is rationally linearly
equivalent to an effective divisor, assuming the successive models
remain K\"ahler \cite[Theorem~6.6]{DHP2024}.
Hacon and Xie prove termination of a minimal model program with
scaling for strongly $\Q$-factorial compact K\"ahler generalized klt
pairs whose boundary plus nef part is big, in arbitrary dimension
\cite[Theorem~1.5]{HX2026}. The present theorem treats every given
sequence in its canonical rational setting, without either of these
positivity or scaling conditions.
The ordinary terminal K\"ahler fourfold argument in
\cite[Section~4]{OrdinaryKahler} uses spans of analytic cycle classes
in place of algebraic cycle classes. We retain this organization and
establish the generalized-canonical version. Its essential points are
local ordinary terminality along flipped surfaces, a finite count of
places below log discrepancy two allowing zero exceptional boundary
coefficients, and a fixed denominator for witness discrepancies at
smooth general points. The cycle comparison is proved for compact
reduced analytic spaces, so it applies directly to boundary components
even when they are nonnormal. The proof reproduces these arguments
in full and invokes no termination theorem. Its analytic negativity
input is Wang's lemma \cite[Lemma~1.3 and Appendices~A--B]{Wang2021}.
The references to \cite{OrdinaryKahler} concern only this negativity
statement, terminal smoothness in codimension two, and the cycle
comparison. No canonical bundle formula or absolute supporting-contraction
theorem from that paper is used: the small projective diagrams are part
of the hypotheses of Theorem~\ref{thm:termination}.
\subsection{Proof strategy}
Dimension four links the two parts of the proof. Smallness bounds both
exceptional loci by dimension two, and the opposite ample signs force
their dimensions to sum to at least three. Once the target locus
contains no surface, the source locus must therefore contain one.
A surface also has codimension two: its blowup at a smooth general
point has ordinary log discrepancy two. This connects the discrepancy
threshold to the surface classes that will be counted.
Surfaces contained in the source and target exceptional loci are
called \emph{flipping} and \emph{flipped surfaces}, respectively.
First we prove strict discrepancy increase for every place centred
in either exceptional locus. Thus a place exceptional over the target
and centred in a flipped surface has log discrepancy strictly greater
than one, even when its source discrepancy equals one. Effectivity of
the boundary and of the difference between the pulled-back nef trace
and the nef datum then give ordinary terminality near a general point
of that surface. Terminal smoothness in codimension two makes the
target smooth there. This is the precise smoothness needed in the proof.
Blowing up a flipped surface at such points supplies a global prime
divisor, called its \emph{witness}. Choose one integer $m>0$ such that
$mM'$ is Cartier and $mB_0$ has integral coefficients. The integral
Weil divisor $mM_T$ is Cartier near the smooth general point of every
flipped surface on a target $T$. The witness calculation consequently
puts its target log discrepancy in
\[
(1,2]\cap m^{-1}\Z.
\]
Its discrepancy strictly increases at each use, so each fixed place
can serve as a witness at most $m$ times. This count uses no uniform
bound on the Cartier indices of the nef traces elsewhere on the models.
We combine this finite set of values with a finite set of possible
places, treating the positive boundary coefficients in decreasing
order. If a flipped surface lies in a component of coefficient $b$,
its witness has source discrepancy strictly below $2-b$.
On the first model of a fixed tail, monotonicity keeps its discrepancy
below $2-b$. The low-place lemma puts these divisors in a finite set,
except possibly those centred on surfaces in a positive boundary
component of coefficient $d$, with discrepancy at least $2-d$. Thus $d>b$.
The induction has already
excluded flipping surfaces in those higher-coefficient components,
so these centres persist through the common isomorphic opens and
cannot become flipped surfaces. At each coefficient stage the
possible divisorial witnesses thus belong to a fixed finite set.
Once these witnesses are exhausted, the target exceptional locus
contains no surface in the boundary components of the current
coefficient. On each such component,
Borel--Moore localization compares
its analytic surface classes with those of the source component.
Every removed source surface gives a nonzero kernel class by
K\"ahler positivity. The rank consequently drops at each remaining
step affecting such a surface. After the boundary components have
been treated, the witness argument with threshold two excludes every
remaining flipped surface. The same cycle comparison on the fourfolds
then gives strict rank loss at every step.
Section~\ref{sec:foundations} fixes the analytic foundations and
common models. Section~\ref{sec:comparison} proves the strict
one-step comparison. Section~\ref{sec:low-places} establishes the
low-place count. Section~\ref{sec:witnesses} proves generic
terminality along flipped surfaces and constructs witnesses with
finitely many possible discrepancy values. Section~\ref{sec:cycles} proves the
cycle-rank comparison, and Section~\ref{sec:termination} combines
these ingredients.
==== END Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/sections/01-introduction.tex ====
==== BEGIN Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/sections/02-foundations.tex ====
\section{Analytic foundations and common models}\label{sec:foundations}
We will compare actual divisors on common models and later prove
smoothness at general points of flipped surfaces. This section
records the analytic facts needed for these two purposes. All
discrepancy comparisons use the fixed b-divisor $\bM$ from
Theorem~\ref{thm:termination}.
\subsection{Places and projective resolutions}\label{subsec:places}
A \emph{place} is a global prime divisor on a normal proper
bimeromorphic model, identified with its strict transforms on common
higher models. Its centre is its reduced irreducible image. A place
is exceptional over a model when its centre has codimension at least
two. This description does not identify places solely through fields
of global meromorphic functions. Properness makes the centres compact
analytic subspaces. When checking a local singularity condition, we
also use divisors on proper modifications of analytic neighbourhoods;
we refer to these explicitly as \emph{local places}.
Every divisor in a pullback comparison is an actual divisor, with
the compatible canonical representatives prescribed in
Theorem~\ref{thm:termination}. Ordinary relative canonical
coefficients can be computed from local pluricanonical equations
after clearing a Cartier multiple; over a smooth space they are
the orders of the Jacobian. These computations localize. Indeed,
a ratio of representatives that is a unit off a codimension-two
subset of a normal base extends as a unit across that subset.
Thus agreement on the isomorphic opens gives the same local
discrepancy coefficients.
For a proper bimeromorphic morphism $r:V\to T$ of normal spaces,
the set of points with positive-dimensional fibres is closed
analytic. Off this subset the morphism is finite and
bimeromorphic, hence an isomorphism by normality. The subset has
codimension at least two: its inverse image is a proper analytic
subset of $V$, and the fibre-dimension theorem excludes a
codimension-one image with positive-dimensional fibres. Fibres
are connected by Stein factorization. A point where the morphism
is a local isomorphism is isolated in its fibre, so it cannot
belong to a connected positive-dimensional fibre. The exceptional
locus is therefore exactly the inverse image of this subset.
We use normalization, proper mapping, fibre dimension, Stein
factorization, and resolution and principalization in the complex
analytic category. The last two permit simultaneous resolution
of finitely many divisors and coherent ideals by projective
modifications; see
\cite[Theorems~1.6, 1.10, and~13.2(1)]{BierstoneMilman1997}.
Projective morphisms are preserved under base change and
restriction to closed analytic subspaces; finite morphisms are
projective. The projective morphisms used here can also be
composed: a sufficiently large twist of one relative ample
line bundle by the pullback of the next is relatively ample
for the composite, with compactness giving a uniform choice.
\begin{proposition}[Common models]\label{prop:common-models}
Any finite portion of the sequence in
Theorem~\ref{thm:termination} has a smooth common model dominating
its members and $X'$, projective over each member. It may also
dominate any prescribed projective modification of one of those
members. For a chosen member $T$, it can be arranged that the
exceptional locus over $T$ is divisorial and that its union with
the strict transform of $\Supp B_T$ has simple normal crossings.
The model carries $\bM$ as the pullback of $M'$.
\end{proposition}
\begin{proof}
Enlarge the finite portion to an initial segment starting at $X_0$.
For one step, take the reduced component of
$X_i\times_{Z_i}X_{i+1}$ containing the graph over the common
isomorphic open. It is projective over both models, and its
normalization remains so. Combine these graphs and $X'\to X_0$
by successive dominating components of fibre products, then
normalize and resolve. To include a prescribed projective
modification of a member, take one further fibre product over
that member before resolving. Base change, restriction, finite
normalization, and composition show that all resulting morphisms
to the members of the segment are projective.
Fix $T$. Choose a closed analytic subset of codimension at
least two outside which the map is an isomorphism, $T$ is
smooth, and the boundary components are smooth and disjoint;
include their singular loci and pairwise intersections in this
subset. Principalize the pullback of its ideal and resolve
the total divisor without changing this open set. The inverse
image of the chosen subset is now a divisor whose components
are exceptional over $T$; it contains the entire exceptional
locus. Resolving together with the strict boundary gives the
claimed normal-crossing condition. Pulling back $M'$ preserves
the fixed data.
\end{proof}
A further common resolution with a model carrying any prescribed
place allows comparison of its discrepancy. That carrying model
need not itself be projective, and the further resolution is only
required to be proper. Projectivity of the models in
Proposition~\ref{prop:common-models} will be used when applying
negativity and when constructing divisorial witnesses.
\subsection{Negativity for actual divisors}
For a projective morphism, relative nefness of a rational divisor
in the following lemma is the degree condition on all contracted
curves. This is distinct from absolute analytic nefness.
\begin{lemma}[Analytic negativity]\label{lem:negativity}
Let $r:V\to T$ be a projective bimeromorphic morphism of normal
complex spaces and let $F$ be a $\Q$-Cartier divisor on $V$.
If $-F$ has nonnegative degree on every contracted curve and
$r_*F$ is effective, then $F$ is effective.
\end{lemma}
This is \cite[Lemma~1.3 and Appendix~A]{Wang2021}, after
clearing a Cartier multiple. The relative degree criterion in
the projective case is explicit in \cite[Appendix~B]{Wang2021}.
Effectivity is that of the actual divisor, not merely its class.
The corresponding statement is
\cite[Lemma~2.3(i)]{OrdinaryKahler}. We apply it only when $r_*F=0$.
An effective $\Q$-Cartier divisor on a normal space pulls back
effectively: after multiplying, its local meromorphic equation
has no divisorial poles, hence is holomorphic by normality.
It also has nonnegative degree on every compact curve not
contained in its support, by restriction to the normalization
of that curve. We will use both facts.
\subsection{Ordinary terminal singularities in codimension two}
The following local statement will be applied on an open set
meeting a flipped surface. We include its surface proof, as in
\cite[Lemma~4.4]{OrdinaryKahler}. Ordinary terminality here means
that every local exceptional place has ordinary log discrepancy
strictly greater than one.
\begin{lemma}\label{lem:terminal-smooth}
A normal complex fourfold with a canonical rational line bundle
and ordinary terminal singularities is smooth in codimension two.
\end{lemma}
\begin{proof}
Suppose the singular locus has a codimension-two component.
Near its general smooth point, choose a holomorphic projection
to $\C^2$ submersive on that component. Take a log resolution
which is an isomorphism over the regular locus. Choose a general
fibre $S$ of the projection: its inverse image $\widetilde S$
is smooth, misses exceptional strata whose images have dimension
less than two, and meets all remaining strata transversely. These
choices follow by generic smoothness on the resolution and its
finitely many strata.
Write $\rho:\widetilde S\to S$ for the induced resolution. Terminal
singularities are klt, so the ambient space has rational singularities
by \cite[Theorem~3.12]{FujinoAnalytic2022}, and is Cohen--Macaulay
\cite[Remark~2.1]{Greb2011}. The fibre has the expected codimension
two. Its two defining parameters therefore form a regular sequence,
so $S$ is Cohen--Macaulay. It is regular away from isolated points
by the general choice of fibre. Serre's criterion makes $S$ normal.
For the regular sequence, the Koszul calculation of the dualizing
sheaf gives $\omega_S\simeq\omega_T\otimes\mathcal O_S$, where
$T$ denotes the ambient neighbourhood; the two slice equations
trivialize the normal determinant. This uses the local analytic
Ext description of the dualizing sheaf
\cite[pp.~3--4]{RSW2017}. Away from the isolated intersections with
the ambient singular locus, a local Cartier pluricanonical power
restricts accordingly, and the identification extends reflexively
over the normal surface. Applying adjunction also on the smooth
resolution therefore restricts its relative canonical equation to
\[
A:=K_{\widetilde S}-\rho^*K_S=\sum_j a_jE_j,
\qquad a_j>0.
\]
Here each exceptional curve is a transverse restriction of an
exceptional divisor on the resolution; terminality gives its
positive coefficient.
A normal surface germ with this property is smooth. Indeed, if
exceptional curves remain, negative definiteness
\cite[Section~4.8(e)]{Grauert1962} gives $A^2<0$,
so $A\cdot E_j<0$ for some $j$. The pulled-back canonical divisor
has degree zero on $E_j$, and therefore
$K_{\widetilde S}\cdot E_j<0$. Since $E_j^2<0$, adjunction gives
\[
0\leq p_a(E_j)
=1+\tfrac12(K_{\widetilde S}\cdot E_j+E_j^2)\leq0.
\]
Both negative intersection numbers are integers, so each equals
$-1$. Thus $E_j$ is a smooth rational $(-1)$-curve. Contract it
to a smooth point by \cite[Section~4.8(a)]{Grauert1962}. The map to
the normal surface germ factors
through this contraction, and pushforward of the relative
canonical equation leaves positive coefficients on all remaining
exceptional curves. Repetition ends with a finite bimeromorphic
map to the normal surface germ, which is an isomorphism.
But the transverse slice at the chosen singular fourfold
point has embedding dimension at least that of the fourfold
germ minus two, hence greater than two. It is singular, a
contradiction.
\end{proof}
In Proposition~\ref{prop:ordinary-terminal} we will verify ordinary
terminality on an open set meeting each flipped surface before
applying the lemma there.
==== END Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/sections/02-foundations.tex ====
==== BEGIN Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/sections/03-comparison.tex ====
\section{Discrepancies across one flip}\label{sec:comparison}
We first establish the properties of an individual step that will be
used throughout the termination argument. The two relative ample
signs produce an effective divisor on the normalized graph. Its
support contains the entire inverse image of the exceptional loci;
this is what gives strict increase for every place centred in either
locus.
\begin{proposition}[Comparison across a flip]\label{prop:comparison}
Let
\[
T\overset{f}{\longrightarrow}Z
\overset{g}{\longleftarrow}T^+
\]
be a step satisfying the hypotheses of Theorem~\ref{thm:termination} between normal compact K\"ahler fourfolds. Write
\[
D_T=K_T+B_T+M_T,\qquad
D_{T^+}=K_{T^+}+B_{T^+}+M_{T^+},
\]
where the nef traces come from the same fixed b-divisor $\bM$.
Then the following assertions hold.
\begin{enumerate}
\item The exceptional images coincide:
\[
A:=f(\Exc f)=g(\Exc g).
\]
With reduced structures, put
\[
L=f^{-1}(A)=\Exc f,\qquad
L^+=g^{-1}(A)=\Exc g.
\]
These are closed analytic subsets, and
\[
T\setminus L\simeq T^+\setminus L^+,
\qquad \dim L,\dim L^+\leq2,\qquad \dim A\leq1.
\]
\item For every divisorial place $E$ over the two models,
\begin{equation}\label{eq:discrepancy-monotonicity}
a(E;T^+,B_{T^+}+\bM)
\geq a(E;T,B_T+\bM).
\end{equation}
The inequality is strict if the centre of $E$ on $T$ is contained in
$L$, or if its centre on $T^+$ is contained in $L^+$.
A place is exceptional over $T$ if and only if it is exceptional over
$T^+$. In particular, the generalized canonical condition of
Theorem~\ref{thm:termination} is preserved. If it holds on $T$, then
every exceptional place with centre contained in $L^+$ satisfies
\begin{equation}\label{eq:target-strict-discrepancy}
a(E;T^+,B_{T^+}+\bM)>1.
\end{equation}
\item The exceptional loci satisfy
\begin{equation}\label{eq:exceptional-dimensions}
\dim L+\dim L^+\geq3.
\end{equation}
Consequently, if $L^+$ contains no surface, then $L$ contains a surface.
\end{enumerate}
\end{proposition}
\begin{proof}
\emph{The common exceptional image.}
For each of the two morphisms, the exceptional locus is precisely the
inverse image of the locus of positive-dimensional fibres, by the
discussion in Section~\ref{subsec:places}. Off the union of these
two images, the morphisms initially give isomorphic opens whose
complements upstairs have codimension at least two. Thus prime
divisors already correspond, before equality of the images is known.
Let $U\subset Z$ be the open set where $f$ is an isomorphism. Transport
$D_T|_{f^{-1}(U)}$ to an actual $\Q$-Cartier divisor $D_U$ on $U$.
The two adjoints agree under the identification of prime divisors
induced by the small map: this holds for the canonical and boundary
terms by their prescribed transforms, and for the nef traces by the
pushforward rule for the fixed b-divisor. Hence on $g^{-1}(U)$ the
divisors $D_{T^+}$ and $g^*D_U$ agree away from the exceptional locus
of $g$. That locus contains no divisor, so the two Weil divisors
agree everywhere on $g^{-1}(U)$. Clearing Cartier multiples and using
normality gives equality as $\Q$-Cartier divisors as well.
If $g$ had a positive-dimensional fibre over a point of $U$, its
projectivity would provide a curve $C$ in that fibre. The equality
just proved would give $D_{T^+}\cdot C=0$, contrary to $g$-ampleness.
Thus $g$ is also an isomorphism over $U$. Interchanging $f$ and $g$
and using $f$-ampleness of $-D_T$ proves the reverse inclusion of their
isomorphism loci. This establishes the asserted common image $A$
and the identification of the complements of $L$ and $L^+$.
Smallness gives $\dim L,\dim L^+\leq2$. Every fibre over $A$ is
positive dimensional, so the fibre-dimension theorem gives
$\dim A\leq1$.
\emph{An effective divisor on the graph.}
Let $G$ be the normalization of the component of $T\times_ZT^+$
dominating $Z$. Its projections fit into the commutative diagram
\begin{equation}\label{eq:normalized-graph}
\begin{array}{ccc}
G&\xrightarrow{\ q\ }&T^+\\[3pt]
{\scriptstyle p}\big\downarrow&&
\big\downarrow{\scriptstyle g}\\[3pt]
T&\xrightarrow{\ f\ }&Z.
\end{array}
\qquad h:=f\circ p=g\circ q.
\end{equation}
The morphisms $p,q,h$ are projective and bimeromorphic, as in
Proposition~\ref{prop:common-models}. Define the actual $\Q$-Cartier
divisor
\begin{equation}\label{eq:graph-divisor}
F:=p^*D_T-q^*D_{T^+}.
\end{equation}
It vanishes off $h^{-1}(A)$. Since both $L$ and $L^+$ have codimension
at least two, $F$ is exceptional over both models; in particular,
$p_*F=0$.
If $C$ is a curve contracted by $p$, then $q_*C$ is either zero or a
curve contracted by $g$. The projection formula therefore gives
\[
(-F)\cdot C=D_{T^+}\cdot q_*C\geq0.
\]
Thus $-F$ is $p$-nef in the relative curve-degree sense. Applying
Lemma~\ref{lem:negativity} to a positive Cartier multiple of $F$
proves that $F\geq0$.
There is a strict inequality on every curve contracted by $h$.
Indeed, for such an irreducible curve $C$, the relative ample signs
give
\[
D_T\cdot p_*C\leq0,\qquad
D_{T^+}\cdot q_*C\geq0,
\]
with strict inequality whenever the corresponding image is a curve.
The two images cannot both be points: the map
$G\to T\times_ZT^+$ is finite onto the graph component, so it cannot
contract $C$. Hence
\begin{equation}\label{eq:graph-curve-negativity}
F\cdot C=D_T\cdot p_*C-D_{T^+}\cdot q_*C<0
\qquad\text{for every $h$-contracted irreducible curve $C$.}
\end{equation}
We next show that this strict inequality determines the whole support:
\begin{equation}\label{eq:graph-support}
\Supp F=h^{-1}(A).
\end{equation}
Fix $z\in A$. The fibre $h^{-1}(z)$ is projective and connected;
connectedness also follows directly from the proper bimeromorphic
morphism $h$ and normality of $Z$. It is positive dimensional because
it surjects onto $f^{-1}(z)$. Its reduction has finitely many
irreducible components. A zero-dimensional irreducible component
would be a point disjoint from every other component, contradicting
connectedness. Therefore every point belongs to a positive-dimensional
projective component. Intersecting such a component with hyperplanes
through the point produces an irreducible curve through that point.
An effective $\Q$-Cartier divisor has nonnegative degree on every
compact irreducible curve not contained in its support: a positive
multiple is an effective Cartier divisor, whose canonical section
restricts to a nonzero holomorphic section on the normalization of
that curve. By \eqref{eq:graph-curve-negativity}, every curve in
$h^{-1}(z)$ is therefore contained in $\Supp F$. The preceding
paragraph puts every point of the fibre in $\Supp F$. This proves
$h^{-1}(A)\subset\Supp F$; the reverse inclusion was already noted.
\emph{Discrepancies of all places.}
Choose a sufficiently high smooth common model carrying the fixed
nef data and dominating $G$, as supplied by
Proposition~\ref{prop:common-models}. To compare a specified place
$E$, further dominate a model representing it, so that $E$ is a prime
divisor on a smooth model $W$ with morphism $r:W\to G$. The same
canonical divisor $K_W$ and the same nef trace $M_W$ occur in both
discrepancy equations:
\[
\begin{aligned}
K_W+\Delta_T+M_W&=(p\circ r)^*D_T,\\
K_W+\Delta_{T^+}+M_W&=(q\circ r)^*D_{T^+}.
\end{aligned}
\]
Their difference is the equality of actual divisors
\[
r^*F=\Delta_T-\Delta_{T^+}.
\]
Taking the coefficient at $E$ yields
\begin{equation}\label{eq:graph-discrepancy-identity}
a(E;T^+,B_{T^+}+\bM)-a(E;T,B_T+\bM)
=\operatorname{coeff}_E(r^*F).
\end{equation}
Pullback of the effective $\Q$-Cartier divisor $F$ is effective, proving
\eqref{eq:discrepancy-monotonicity}.
If the centre of $E$ on $T$ lies in $L$, its centre on $G$ lies in
$p^{-1}(L)=h^{-1}(A)$; the same conclusion holds if its centre on
$T^+$ lies in $L^+$. By \eqref{eq:graph-support}, this centre is
contained in $\Supp F$. A local equation of a positive effective
Cartier multiple of $F$ vanishes on that centre. Its pullback has
positive order along $E$: it is a nonzero holomorphic function on
the inverse image of the coordinate neighbourhood and vanishes on
the part of $E$ lying there. Consequently the
right-hand side of \eqref{eq:graph-discrepancy-identity} is strictly
positive. This proves strictness for an arbitrary place with either
specified centre condition.
A place whose centre is a prime divisor on a normal model is
represented by that divisor, since a proper bimeromorphic morphism
to a normal space is an isomorphism at general points of each prime
divisor on the target. Every prime divisor
on $T$ meets $T\setminus L$, and its corresponding prime divisor on
$T^+$ meets $T^+\setminus L^+$. The common isomorphism identifies
their orders. Thus a place is nonexceptional on one side exactly
when it is nonexceptional on the other. Together with
\eqref{eq:discrepancy-monotonicity}, this preserves positivity for
all places and the lower bound $1$ for exceptional places. For a
place centred in $L^+$, strictness improves that lower bound to
\eqref{eq:target-strict-discrepancy}.
\emph{The dimension inequality.}
The step is nontrivial, so $A\neq\varnothing$. Equation
\eqref{eq:graph-support} implies $F\neq0$. Every irreducible component
$P$ of $\Supp F$ is a divisor on the four-dimensional normal space
$G$, hence has dimension three. Its finite image under
$G\to T\times_ZT^+$ is contained in $L\times_ZL^+$. Therefore
\[
3=\dim P\leq\dim(L\times_ZL^+)\leq\dim L+\dim L^+,
\]
which proves \eqref{eq:exceptional-dimensions}. If $L^+$ contains no
surface, then $\dim L^+\leq1$, while smallness gives $\dim L\leq2$.
The inequality forces $\dim L=2$, so a two-dimensional irreducible
component of $L$ is the required surface.
\end{proof}
We will call an irreducible compact analytic surface contained in $L$
a \emph{flipping surface}, and one contained in $L^+$ a
\emph{flipped surface}. Applying Proposition~\ref{prop:comparison}
successively shows that every pair in a sequence as in
Theorem~\ref{thm:termination} remains generalized canonical. The
strict bound \eqref{eq:target-strict-discrepancy} concerns places
centred in the flipped locus and will supply the local terminal
condition used to study its surfaces.
==== END Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/sections/03-comparison.tex ====
==== BEGIN Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/sections/04-low-places.tex ====
\section{Places of small discrepancy}
\label{sec:low-places}
Fix a model $T$ in the sequence of Theorem~\ref{thm:termination}.
Proposition~\ref{prop:comparison} gives log discrepancy at least one
for every exceptional place over $T$. We will show that, apart from a
finite set of places, an exceptional place of discrepancy below two is
centred on a surface in a positive boundary component. Its discrepancy
is then bounded below by two minus that component's coefficient.
This description will restrict the possible divisors used to count
flipped surfaces.
\subsection{The nef trace and local estimates}
Write $B_T$ and $M_T$ for the boundary and the trace of $\bM$ on $T$.
They are $\Q$-Cartier: each is a finite rational combination of globally
defined prime Weil divisors, to which global Weil $\Q$-factoriality applies.
Choose a smooth model
\[
r\colon W\longrightarrow T
\]
that is projective over $T$ and carries the fixed nef data, as supplied by
Proposition~\ref{prop:common-models}. Define the actual divisors
\begin{equation}
\Delta_W=r^*(K_T+B_T+M_T)-K_W-M_W,
\qquad J_W=r^*M_T-M_W.
\label{eq:boundary-and-defect}
\end{equation}
Thus the coefficient of a prime $E$ in $\Delta_W$ is
$1-a(E;T,B_T+\bM)$.
\begin{lemma}[Effectivity of the nef-trace defect]
\label{lem:nef-defect}
For every smooth model $r\colon W\to T$ that is projective over $T$ and
carries the fixed nef data, the divisor $J_W$ in
\eqref{eq:boundary-and-defect} is an effective
$r$-exceptional $\Q$-Cartier divisor.
\end{lemma}
\begin{proof}
The pushforward of both $r^*M_T$ and $M_W$ is $M_T$, so $J_W$ is
$r$-exceptional. For every curve $C$ contracted by $r$,
\[
(-J_W)\cdot C=M_W\cdot C\geq 0.
\]
To see the last inequality, let $s\colon W\to X'$ be the morphism to the
fixed carrier. Since $M_W=s^*M'$, its degree on $C$ is zero if $s(C)$ is a
point, and otherwise is the degree of $M'$ on $s(C)$ multiplied by the
degree of the induced map of normalized curves. Analytic nefness of $M'$
implies that these curve degrees are nonnegative: restrict a sequence of
K\"ahler classes converging to $c_1(M')$ to the normalized curve and pass to
the limit. We use this consequence of analytic nefness only for the
relative degree calculation above. Lemma~\ref{lem:negativity}, applied to
$J_W$ and $r_*J_W=0$, now gives $J_W\geq0$.
\end{proof}
We will use the following local estimate both to check ordinary terminality
and to locate places of small generalized discrepancy. It applies to local
analytic places as well as to places represented by divisors on compact
models.
\begin{lemma}[A local differential estimate]
\label{lem:jacobian}
Let $W$ be a smooth complex space of dimension $n$, and let $E$ be a prime
divisor on a smooth proper birational model of an open subset of $W$.
Suppose its centre $C$ has codimension $c$. At a general smooth point of
$C$, choose coordinates $x_1,\ldots,x_n$ with
$C=(x_1=\cdots=x_c=0)$. Writing $v=\operatorname{ord}_E$, one has
\begin{equation}
a(E;W,0)\geq\sum_{j=1}^{c}v(x_j)\geq c.
\label{eq:jacobian-bound}
\end{equation}
If $C$ is contained in exactly one positive component $H$ of a rational
divisor $\Delta$, and $H$ is smooth at the general point of $C$ with
coefficient $d\in(0,1)$, choose the coordinates so that $H=(x_1=0)$.
Then
\begin{equation}
a(E;W,\Delta)
\geq (1-d)v(x_1)+\sum_{j=2}^{c}v(x_j)
\geq c-d.
\label{eq:one-positive-bound}
\end{equation}
If $C$ is contained in no positive component of $\Delta$, then
$a(E;W,\Delta)\geq c$.
\end{lemma}
\begin{proof}
Compute at a general smooth point of $E$ above the chosen coordinate
neighbourhood. If $t$ is a local equation of $E$, write
$x_j=t^{v(x_j)}u_j$ for $1\leq j\leq c$, where $u_j$ is a unit at the
general point of $E$. Each $v(x_j)$ is a positive integer. In expanding
the pullback of $dx_1\wedge\cdots\wedge dx_n$, at most one factor in a
nonzero term can contribute a differential $dt$. Thus its order along
$E$ is at least $\sum_{j=1}^c v(x_j)-1$. The remaining coordinate
differentials are holomorphic and do not lower this order. Adding one to
the coefficient of the relative canonical divisor proves
\eqref{eq:jacobian-bound}.
For the boundary statement, each component with nonpositive coefficient
contributes a nonnegative amount to the log discrepancy. Components not
containing $C$ have order zero along $E$. Subtracting just
$d\,v(x_1)$ from \eqref{eq:jacobian-bound} therefore gives
\eqref{eq:one-positive-bound}. If there is no positive component through
$C$, nothing has to be subtracted.
\end{proof}
\subsection{A finite exceptional set of places}
We next describe the places with generalized log discrepancy below two.
The finite exceptional set is attached to one fixed model; its size need
not be uniform along the sequence. The distinction according to the
centre of a place follows the corrected discrepancy argument of
Fujino~\cite[Proposition~3.1 and Lemma~3.2]{Fujino2005Addendum}. We give
the full argument, including the repeated blowups that are needed when a
boundary coefficient exceeds $1/2$.
\begin{lemma}[Places of log discrepancy below two]
\label{lem:low-places}
Let $(T,B_T+\bM)$ be any model of a sequence as in Theorem~\ref{thm:termination}. There is a
finite set $\mathcal F_T$ of exceptional places over $T$ with the following
property. If $E$ is exceptional over $T$,
\[
a(E;T,B_T+\bM)<2,
\qquad E\notin\mathcal F_T,
\]
then its centre $S$ on $T$ is a surface contained in a unique positive
component $H$ of $B_T$. The space $T$ and the reduced boundary support are
smooth at the general point of $S$. If the coefficient of $H$ is $d$,
then
\begin{equation}
a(E;T,B_T+\bM)\geq2-d.
\label{eq:low-place-threshold}
\end{equation}
If $B_T=0$, there are only finitely many exceptional places with log
discrepancy below two.
\end{lemma}
\begin{proof}
Choose a smooth modification $r\colon W\to T$, projective over $T$ and
carrying the nef data, such that its exceptional locus is divisorial and, together with
the strict transform of $\Supp B_T$, has simple normal crossings.
Starting with this simple normal crossing divisor, blow up the
intersections of pairs of positive strict transforms until they are
mutually disjoint. Each intersection is a finite disjoint union of smooth
codimension-two subspaces. Blowing it up separates the chosen pair,
preserves simple normal crossings, and cannot make a previously separated
pair meet. Thus finitely many such blowups suffice.
Keep the notation $W$ for the resulting model and $\Delta_W$ for its
boundary in \eqref{eq:boundary-and-defect}. Its positive components are
exactly the strict transforms of the positive components of $B_T$, with
the same coefficients. They are smooth and pairwise disjoint. Every
component exceptional over $T$ has nonpositive coefficient by
generalized canonicity; zero coefficients are allowed. On a higher
smooth model $u\colon V\to W$,
the nef terms cancel because $M_V=u^*M_W$, giving
\[
K_V+\Delta_V=u^*(K_W+\Delta_W).
\]
Hence generalized discrepancies over $T$ can be computed as ordinary
discrepancies for $(W,\Delta_W)$, including its negative coefficients.
Put all $r$-exceptional prime divisors on $W$ into $\mathcal F_T$.
There are finitely many, since $W$ is compact. A further exceptional
place $E$ has centre $C$ on $W$ of codimension at least two: a place with
divisorial centre on a normal model is represented by that divisor.
Lemma~\ref{lem:jacobian} shows that
$a(E;W,\Delta_W)<2$ forces $C$ to have codimension two and to lie in
one of the positive components, say $H$ with coefficient $d$. The same
lemma gives
\begin{equation}
a(E;W,\Delta_W)\geq2-d.
\label{eq:resolution-low-bound}
\end{equation}
If $C$ is not contained in $\Exc(r)$, then $r$ is an isomorphism at its
general point. Its image is the surface $S$ required in the statement;
smoothness and uniqueness of the positive component descend from $W$.
It remains to treat the centres contained in $\Exc(r)$. Such a centre
$C$, being codimension two and contained in $H$, must be an irreducible
component of $H\cap F$ for some exceptional divisor $F$ on $W$.
There are only finitely many such components, and each is smooth by
simple normal crossings. We show that a fixed one contributes only
finitely many places of log discrepancy below two.
Near $C$, the only positive component of $\Delta_W$ is $H$.
Deleting the nonpositive components can only lower discrepancies, so
\[
a(E;W,dH)\leq a(E;W,\Delta_W)<2
\]
for every place under consideration with centre $C$. We can therefore
work with the smooth pair $(W,dH)$ and the fixed smooth codimension-two
centre $C\subset H$.
Set $W_0=W$ and $H_0=H$. Blow up $C$ to obtain $W_1$, with exceptional
divisor $F_1$ and strict transform $H_1$. If $E$ is not $F_1$, its
centre on $W_1$ is contained in $F_1$ and has codimension at least two.
The crepant boundary on $W_1$ has sole positive component $dH_1$; the
coefficient of $F_1$ is $-(1-d)$. Lemma~\ref{lem:jacobian} therefore
forces the centre of $E$ to have codimension two and to lie in $H_1$.
It is consequently the entire intersection
\[
C_1=H_1\cap F_1.
\]
This intersection is a smooth irreducible section of the exceptional
$\mathbb P^1$-bundle over $C$, hence isomorphic to $C$.
Repeat this construction: after $W_j$ has been obtained, blow up
$C_j=H_j\cap F_j$ to obtain $W_{j+1}$, with new exceptional divisor
$F_{j+1}$. Writing $C_0=C$, the maps and
their centres are
\begin{equation}\label{eq:forced-centres}
W_j=\operatorname{Bl}_{C_{j-1}}W_{j-1},\qquad
C_j=H_j\cap F_j\simeq C_{j-1}\qquad(j\geq1).
\end{equation}
The new strict transform
$H_{j+1}$ meets $F_{j+1}$ in a smooth section and is disjoint from the
strict transform of $F_j$. Inductively, it misses every older exceptional
divisor. The codimension-two blowup formula gives the coefficient
\begin{equation}
\operatorname{coeff}_{F_j}(\Delta_j)=-j(1-d),
\qquad a(F_j;W,dH)=1+j(1-d),
\label{eq:forced-chain-coefficients}
\end{equation}
where $K_{W_j}+\Delta_j$ is the pullback of $K_W+dH$.
Indeed the new coefficient is the sum of the coefficients of $H_j$ and
$F_j$, minus one. This is $-(j+1)(1-d)$.
Thus the only way a place of discrepancy below two can remain
unextracted is for its centre to follow these successive intersections.
At every stage, if $E$ has not yet appeared, the argument just given
forces its centre to be $C_j$. Choose local transverse coordinates
$x,y$ with $H_j=(x=0)$ and $F_j=(y=0)$, and put
$v=\operatorname{ord}_E$. Retaining the negative coefficient of $F_j$
in the differential estimate gives
\begin{equation}
\begin{split}
a(E;W,dH)&\geq (1-d)v(x)+\bigl(1+j(1-d)\bigr)v(y)\\
&\geq (1-d)+\bigl(1+j(1-d)\bigr)
=2-d+j(1-d).
\end{split}
\label{eq:forced-chain-cutoff}
\end{equation}
For
\[
J=\left\lceil\frac{d}{1-d}\right\rceil,
\]
the right side is at least two. Every place with discrepancy below two
and centre $C$ must therefore be one of $F_1,\ldots,F_J$. Equivalently,
the possible indices in \eqref{eq:forced-chain-coefficients} satisfy
$j<1/(1-d)$; the strict endpoint is included in the cutoff calculation.
All these blowups have global smooth compact analytic centres. Thus the
divisors just enumerated are global places. Conversely, an arbitrary
place can be followed on common proper resolutions even when its
original carrying model was not projective; its centres map onto the
preceding centres. The argument forcing $C_j$ therefore covers every
place over $C$, not only those chosen in advance from this blowup chain.
Add the finitely many divisors obtained for all the finitely many
intersections $H\cap F$ to $\mathcal F_T$. Every place outside this set
has the centre described earlier, and
\eqref{eq:resolution-low-bound} gives
\eqref{eq:low-place-threshold}. If the boundary has no positive
component, Lemma~\ref{lem:jacobian} already excludes every further place
of discrepancy below two, proving the last assertion.
The proof used only nonpositivity of the exceptional coefficients on
$W$, so exceptional places of log discrepancy exactly one cause no
additional case.
\end{proof}
\begin{remark}
The second and subsequent blowups in the preceding proof are essential.
For example, when $d=2/3$, the first two divisors have log discrepancies
$4/3$ and $5/3$, and the third has discrepancy two. This is the phenomenon
in Fujino's correction \cite[Example~2.1]{Fujino2005Addendum}. The finite
exceptional set in Lemma~\ref{lem:low-places} is defined by centres and
contains the whole finite chain over each exceptional intersection.
\end{remark}
==== END Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/sections/04-low-places.tex ====
==== BEGIN Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/sections/05-witnesses.tex ====
\section{Divisorial witnesses for flipped surfaces}
\label{sec:witnesses}
A surface in the flipped locus supplies a divisor whose discrepancy has
just increased. The fixed rational nef data constrain the new discrepancy
to a finite set. The construction requires smoothness at the general
point of the flipped surface. Generalized canonicity alone does not
supply that smoothness; strict increase on the target exceptional locus
does. We first establish this local consequence and then construct a
global divisor by blowing up the surface.
For ordinary canonical pairs, this use of terminality near a flipped
surface appears in Fujino \cite[Lemma~2.2]{Fujino2004}.
The proof here includes the nef-trace defect and checks the local
singularity condition from the hypotheses on global places.
\begin{proposition}[Terminality near a flipped surface]
\label{prop:ordinary-terminal}
Let $T\dashrightarrow T^+$ be a step in the sequence of
Theorem~\ref{thm:termination}, and let $S\subset L^+$ be an irreducible
flipped surface. There is an analytic open subset $U\subset T^+$
meeting $S$ on which the underlying space has ordinary terminal
singularities. Consequently $T^+$ is smooth at a general point of $S$.
\end{proposition}
\begin{proof}
Choose a smooth model $r\colon W\to T^+$, projective over $T^+$ and
carrying the fixed nef data, with divisorial exceptional locus as in
Proposition~\ref{prop:common-models}. Using the divisors of
\eqref{eq:boundary-and-defect}, write the ordinary relative canonical
equation as
\begin{equation}
\Theta_W:=r^*K_{T^+}-K_W
=\Delta_W-r^*B_{T^+}-J_W.
\label{eq:ordinary-boundary}
\end{equation}
The pullback $r^*B_{T^+}$ is effective: after clearing a Cartier
multiple, a local equation for the effective boundary is holomorphic
by normality, and remains holomorphic after pullback.
Lemma~\ref{lem:nef-defect} gives $J_W\geq0$. Generalized canonicity
therefore makes every exceptional coefficient of $\Theta_W$
nonpositive. Its nonexceptional coefficients are zero, since it is
an ordinary relative canonical divisor.
An exceptional prime $P$ on $W$ whose image contains $S$ has image
exactly $S$: its irreducible image has dimension at most two.
Proposition~\ref{prop:comparison} gives
\[
a(P;T^+,B_{T^+}+\bM)>a(P;T,B_T+\bM)\geq1,
\]
because its target centre lies in $L^+$. Thus its coefficient in
$\Delta_W$, and hence in $\Theta_W$, is strictly negative.
There are only finitely many exceptional prime divisors on $W$.
Delete from $T^+$ the images of those with coefficient zero in
$\Theta_W$, and call the resulting open set $U$. Proper mapping
makes the deleted set closed analytic. None of these images contains
$S$, so their finite union cannot contain the irreducible surface $S$;
in particular, $U\cap S$ is a dense open subset of $S$.
We check ordinary terminality on $U$ using local analytic places.
Restrict the resolution and \eqref{eq:ordinary-boundary} over $U$.
Every local component of its exceptional divisor now has strictly
negative coefficient in $\Theta_W$. If a local exceptional place
$E$ over $U$ has divisorial centre on this resolution, it is the place
of that divisor. The divisor must be exceptional over $U$, since
otherwise $E$ would not be exceptional. Its negative coefficient
therefore gives ordinary log discrepancy greater than one.
For any remaining local exceptional place, take a common local
resolution. Its centre on the smooth space $r^{-1}(U)$ has
codimension at least two. Since $\Theta_W\leq0$,
Lemma~\ref{lem:jacobian} gives
\[
a(E;U,0)=a(E;r^{-1}(U),\Theta_W|_{r^{-1}(U)})
\geq a(E;r^{-1}(U),0)\geq2.
\]
This checks all local exceptional places and proves ordinary
terminality. The canonical rational line bundle restricts to $U$,
so Lemma~\ref{lem:terminal-smooth} applies there. Its singular locus
has codimension at least three and cannot contain the surface
$S\cap U$. Thus $T^+$ is smooth at a general point of $S$.
\end{proof}
\subsection{The blowup divisor and its discrepancy}
Fix an integer $m>0$ such that $mM'$ is a Cartier divisor and $mb$ is an
integer for every positive coefficient $b$ of $B_0$. The same integer
clears all boundary coefficients on every model, since the boundaries are
strict transforms. Moreover
\begin{equation}
mM_T\text{ is an integral Weil divisor on every model }T.
\label{eq:integral-nef-traces}
\end{equation}
Indeed, on a common model carrying the data, $mM_W$ is the pullback of the
integral Cartier divisor $mM'$, and pushforward preserves integral Weil
coefficients. Although \eqref{eq:integral-nef-traces} does not assert
global Cartierness of $mM_T$, it implies Cartierness on the smooth locus
of $T$.
\begin{proposition}[A witness for every flipped surface]
\label{prop:witnesses}
Consider the $i$-th step of a sequence as in Theorem~\ref{thm:termination}, with target
$T^+=X_{i+1}$. Let $S$ be an irreducible surface contained in its flipped
locus. There is a global prime place $E_S$, with centre $S$ on $T^+$,
obtained by blowing up $S$ at its general smooth point. It is exceptional
over every model of the sequence. Write
\[
B_{i+1}=\sum_{\ell} b_{\ell}H_{\ell},
\qquad b_{\ell}>0,
\]
and let $r\colon W\to T^+$ be a smooth model that is projective over
$T^+$ and carries both $E_S$ and the fixed nef data. With
$J_W=r^*M_{i+1}-M_W$, one has
\begin{equation}
a(E_S;X_{i+1},B_{i+1}+\bM)
=2-\sum_{\ell}b_{\ell}\operatorname{mult}_S(H_{\ell})
-\operatorname{coeff}_{E_S}(J_W).
\label{eq:witness-discrepancy}
\end{equation}
In particular,
\begin{equation}
a(E_S;X_{i+1},B_{i+1}+\bM)
\in\Lambda_m:=(1,2]\cap\tfrac1m\Z
=\left\{1+\tfrac1m,1+\tfrac2m,\ldots,2\right\}.
\label{eq:witness-grid}
\end{equation}
Its discrepancy on the source is strictly below two. If $S$ is contained
in a component of $B_{i+1}$ of coefficient $b>0$, then more precisely
\begin{equation}
a(E_S;X_i,B_i+\bM)
< a(E_S;X_{i+1},B_{i+1}+\bM)\leq2-b.
\label{eq:witness-threshold}
\end{equation}
\end{proposition}
\begin{proof}
By Proposition~\ref{prop:ordinary-terminal}, a general point of the
surface $S$ is a smooth point of both $T^+$ and $S$.
Blow up the global coherent
ideal of the reduced analytic subspace $S$. Over the open subset where
$T^+$ and $S$ are smooth, this is the blowup of a smooth codimension-two
centre, with one exceptional prime dominating that open subset of $S$.
After normalization the preimage of $S$ is a compact analytic subset
with finitely many irreducible components. One of these contains this
exceptional divisor and dominates $S$, by proper mapping; the smooth
calculation gives uniqueness. It is a global prime divisor, and we
denote its place by $E_S$. Resolve further and use
Proposition~\ref{prop:common-models} to obtain a smooth model $W$,
projective over $T^+$, carrying it and the nef data. The centre on $T^+$ is $S$, so it is
exceptional there. Smallness of every step preserves exceptionalness,
as in Proposition~\ref{prop:comparison}, and therefore it is exceptional
over every $X_j$.
The ordinary log discrepancy of the blowup at a smooth codimension-two
centre is two. Pulling back a boundary component $H_{\ell}$ subtracts
$b_{\ell}\operatorname{mult}_S(H_{\ell})$, where the generic
multiplicity is a nonnegative integer and is zero unless $S\subset
H_{\ell}$. The identity
\[
\Delta_W=r^*(K_{T^+}+B_{i+1})-K_W+J_W
\]
then proves \eqref{eq:witness-discrepancy}. This computation only uses
the smooth general point of the centre. The boundary components may be
singular or intersect one another along $S$.
Lemma~\ref{lem:nef-defect} gives
$\operatorname{coeff}_{E_S}(J_W)\geq0$. It also lies in
$\tfrac1m\Z$. To verify this denominator claim, restrict to a smooth
open subset of $T^+$ meeting a general point of $S$.
By \eqref{eq:integral-nef-traces}, $mM_{i+1}$ is an integral Cartier
divisor there, so its pullback has integral order along $E_S$.
The divisor $mM_W$ is integral Cartier as the pullback of $mM'$.
Consequently the coefficient of
\[
mJ_W=r^*(mM_{i+1})-mM_W
\]
along $E_S$ is an integer. This is a local calculation near the general
point of $S$, and does not require $mM_{i+1}$ to be Cartier elsewhere.
Every term subtracted from two in \eqref{eq:witness-discrepancy} is
nonnegative and belongs to $\tfrac1m\Z$. Since the target centre is
contained in $L_i^+$, strict increase in
Proposition~\ref{prop:comparison} and generalized canonicity on the
source give
\[
a(E_S;X_{i+1},B_{i+1}+\bM)
>a(E_S;X_i,B_i+\bM)\geq1.
\]
These facts prove \eqref{eq:witness-grid}. If $S$ is contained in a component
of coefficient $b$, its generic multiplicity is at least one, so
\eqref{eq:witness-discrepancy} bounds the target discrepancy by $2-b$.
This proves \eqref{eq:witness-threshold}; without that containment the
same argument gives the strict source bound below two.
\end{proof}
Call a place chosen by Proposition~\ref{prop:witnesses} a \emph{witness}
for the corresponding flipped surface. The possible target discrepancies
of witnesses form the fixed finite set $\Lambda_m$. A place that is used
again must reach a strictly larger member of this set.
\begin{lemma}[Finite use of each witness]
\label{lem:finite-use}
Let $I$ be any set of step indices in a sequence as in Theorem~\ref{thm:termination}. For every
$i\in I$, choose a flipped surface of the $i$-th step and a witness $E_i$
as in Proposition~\ref{prop:witnesses}. Then each fixed place $E$
occurs among the $E_i$ at most $m$ times. Consequently, if all the
$E_i$ belong to a fixed finite collection $\mathcal F$, then
$\#I\leq m\,\#\mathcal F$.
\end{lemma}
\begin{proof}
Suppose $i<j$ are two indices for which $E_i=E_j=E$. Monotonicity and
the strict increase when $E$ is used at step $j$ give
\[
a(E;X_{i+1},B_{i+1}+\bM)
\leq a(E;X_j,B_j+\bM)
<a(E;X_{j+1},B_{j+1}+\bM).
\]
Thus the target discrepancies at all uses of this fixed place are
strictly increasing elements of $\Lambda_m$. This set has $m$ elements,
so there can be at most $m$ uses. Summing this bound over
$\mathcal F$ proves the last assertion. No discreteness of the
discrepancy at intervening steps is required.
\end{proof}
==== END Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/sections/05-witnesses.tex ====
==== BEGIN Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/sections/06-cycles.tex ====
\section{Loss of surface classes}\label{sec:cycles}
Once a target exceptional locus contains no surface, each surface in the
source exceptional locus forces a decrease in the rank of analytic
surface classes. We prove this first for compact reduced analytic
spaces, so that it applies both to the fourfolds and to their possibly
nonnormal boundary components. Restriction to the common open carries
source surface classes onto all target surface classes. A removed
source surface lies in the kernel, and K\"ahler positivity makes its
class nonzero.
The use of surface-cycle ranks follows the algebraic argument of
Fujino \cite[Lemma~2.3]{Fujino2004}; we give the compact analytic form,
as in \cite[Section~4.1]{OrdinaryKahler}, with the additional generality
needed for these boundary components.
For a compact reduced complex analytic space $V$, define
\begin{equation}\label{eq:cycle-rank}
\begin{split}
\mathcal C_2(V)
&:=\operatorname{span}_{\R}\{[S]\in H_4(V,\R):
S\subset V\text{ is an irreducible compact analytic surface}\},\\
c_2(V)&:=\dim_{\R}\mathcal C_2(V).
\end{split}
\end{equation}
Here $[S]$ is the image in $H_4(V,\R)$ of the fundamental class of $S$,
with the complex orientation on its regular locus. All homology groups
in this section have real coefficients.
We recall the topology that makes this definition and its use below
valid for singular analytic spaces. A compact reduced analytic space
admits a finite triangulation compatible with any prescribed finite
collection of closed analytic subsets. A subset of complex dimension
$d$ is triangulated in real dimension at most $2d$. One can obtain this
statement for an abstract analytic space as follows. Choose a compatible
Whitney stratification
\cite[Chapter~III, Propositions~1.5 and~2.2.2]{Teissier1982}, finitely
many analytic charts $z_j:U_j\hookrightarrow\mathbb C^{N_j}$, and nonnegative
smooth functions $\rho_j$ supported compactly in $U_j$, with positivity
sets covering the space. Extending each block by zero gives an embedding
\[
x\longmapsto\bigl(\rho_j(x),\rho_j(x)z_j(x)\bigr)_j
\]
into a Euclidean space. Indeed, a block with $\rho_j(x)>0$ recovers its
chart coordinate as $w/t$. Locally all blocks extend smoothly to an
ambient chart, and this left inverse shows that the embedding is locally
an ambient smooth embedding. It therefore 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} give a
triangulation compatible with the strata and smooth on them. Compactness
makes it finite; smoothness on the strata gives the stated dimension
bound. In particular, $H_4(V,\R)$ is finite-dimensional and $c_2(V)$
is a finite nonnegative integer.
We will also use Borel--Moore homology $H_*^{\mathrm{BM}}$, defined by
locally finite chains. It agrees with ordinary homology on compact
spaces, and it allows a fundamental class to be restricted to an open
subset even when that subset is noncompact. For a closed analytic
subset $A\subset V$ of a compact analytic space, compatible
triangulations give the localization exact sequence
\begin{equation}\label{eq:bm-localization}
\cdots\longrightarrow H_j(A,\R)\longrightarrow H_j(V,\R)
\xrightarrow{\ j^*\ } H_j^{\mathrm{BM}}(V\setminus A,\R)
\longrightarrow H_{j-1}(A,\R)\longrightarrow\cdots.
\end{equation}
This is the homological sequence for the closed subset and its open
complement. In a compatible triangulation, the Borel--Moore group of
the complement is the homology of the relative chain complex for
$(V,A)$, so the sequence follows from the long exact sequence of that
pair; see
\cite[Sections~1.2 and~1.6]{BorelHaefliger1961}.
If $S$ is an irreducible surface, its singular
locus has real dimension at most two. Applying this sequence to that
locus extends the oriented fundamental class of $S_{\mathrm{reg}}$
uniquely to a class in $H_4(S,\R)$. This is the analytic fundamental
class used in \eqref{eq:cycle-rank}. Its restriction to any open subset
is the corresponding Borel--Moore fundamental class; see also
\cite[Section~3.1 and Theorem~3.2]{BorelHaefliger1961}.
K\"ahler positivity supplies the nonvanishing needed for strict rank
loss. If $V$ is a closed reduced analytic subspace of a compact
K\"ahler space $Y$, and $S\subset V$ is an irreducible surface, then
the local potentials of a K\"ahler form $\omega$ on $Y$ define a class
$[\omega]\in H^2(Y,\R)$. Here is the construction on the possibly
singular space. Let $\mathcal{PH}_Y$ be the sheaf of functions locally
equal to the real part of a holomorphic function. The differences of
the potentials form a cocycle in this sheaf. The connecting
homomorphism for the exact sequence
\[
0\longrightarrow\R_Y\xrightarrow{\,\sqrt{-1}\,}\mathcal O_Y
\xrightarrow{\,\operatorname{Re}\,}\mathcal{PH}_Y
\longrightarrow0
\]
gives $[\omega]$, with the normalization agreeing with the de~Rham
class on smooth spaces. Constant-sheaf cohomology agrees with
singular cohomology because analytic spaces are locally contractible,
as also follows from the compatible triangulations above. The
construction commutes with holomorphic pullback; hence its pullback
to a resolution is the de~Rham class of the pulled-back form. We obtain
\begin{equation}\label{eq:positive-surface-class}
\bigl\langle[\omega]^2,[S]\bigr\rangle
=\int_{S_{\mathrm{reg}}}\omega^2>0.
\end{equation}
The pairing here is taken in $Y$. To compute it, choose a resolution
$r:\widetilde S\to S$ that is an isomorphism over $S_{\mathrm{reg}}$.
Since $r_*[\widetilde S]=[S]$, the pairing equals
$\int_{\widetilde S}(r^*\omega)^2$. The pulled-back form is
semipositive everywhere and positive on the open set mapping to
$S_{\mathrm{reg}}$, which proves the strict inequality. Consequently
the class of $S$ is nonzero in $H_4(Y,\R)$ and hence also in
$H_4(V,\R)$.
\begin{lemma}[Cycle loss]\label{lem:cycle-loss}
Let $P$ and $Q$ be compact reduced complex analytic spaces, and let
$L_P\subset P$ and $L_Q\subset Q$ be closed analytic subsets with
dense complements. Suppose an isomorphism
\[
U=P\setminus L_P\simeq Q\setminus L_Q
\]
is represented by a proper bimeromorphic graph: there is a compact
reduced analytic space $G$ with proper projections $p:G\to P$ and
$q:G\to Q$ such that the common open
\begin{equation}\label{eq:cycle-graph-open}
p^{-1}(P\setminus L_P)=q^{-1}(Q\setminus L_Q)
\end{equation}
is dense in $G$ and maps isomorphically to both displayed complements.
If $\dim L_Q\leq1$, there is a surjection
\[
\mathcal C_2(P)\longrightarrow\mathcal C_2(Q),
\qquad\text{and thus}\qquad c_2(P)\geq c_2(Q).
\]
If $L_P$ contains an irreducible surface whose class in $H_4(P,\R)$ is nonzero,
then $c_2(P)>c_2(Q)$. In particular, strict inequality holds whenever
$L_P$ contains a surface and $P$ is a closed analytic subspace of a
compact K\"ahler space.
\end{lemma}
\begin{proof}
The restriction from $Q$ will identify its fourth homology with the
fourth Borel--Moore homology of the common open. We then compare the
surface classes by their restrictions there.
Since $L_Q$ has real dimension at most two,
$H_4(L_Q,\R)=H_3(L_Q,\R)=0$. The degree-four part of
\eqref{eq:bm-localization} is therefore
\[
0\longrightarrow H_4(Q,\R)
\xrightarrow{\ j_Q^*\ }H_4^{\mathrm{BM}}(U,\R)
\longrightarrow0.
\]
Define the linear map
\begin{equation}\label{eq:cycle-comparison-map}
\Phi=(j_Q^*)^{-1}\circ j_P^*:
H_4(P,\R)\longrightarrow H_4(Q,\R).
\end{equation}
To determine $\Phi$ on surface classes, we first check that strict
transforms are analytic, without a normality hypothesis. Let
$S\subset P$ be an irreducible surface not contained in $L_P$.
Removing a proper analytic subset from an irreducible analytic space
leaves an irreducible space: its regular locus remains connected after
removing an analytic subset of positive complex codimension
\cite[Section~3.3]{BorelHaefliger1961}. Thus
$S\cap U$ is irreducible. The analytic inverse image $p^{-1}(S)$ has
a unique irreducible component $\widetilde S$ meeting the open where
$p$ and $q$ are isomorphisms; there it agrees with $S\cap U$.
This component is a compact irreducible surface. Remmert's proper
mapping theorem shows that
\[
S^+:=q(\widetilde S)\subset Q
\]
is an irreducible analytic surface, agreeing with $S\cap U$ on $U$.
It is the strict transform of $S$. The same construction with $p$ and
$q$ interchanged gives strict transforms in the other direction.
The restrictions of $[S]$ and $[S^+]$ to $U$ are the same oriented
fundamental class, so \eqref{eq:cycle-comparison-map} gives
$\Phi([S])=[S^+]$. If instead $S\subset L_P$, its restriction
vanishes and $\Phi([S])=0$. It follows that $\Phi$ maps
$\mathcal C_2(P)$ into $\mathcal C_2(Q)$. Every irreducible surface
in $Q$ meets $U$, since $\dim L_Q\leq1$; its strict transform in $P$
therefore maps back to its class. The induced map of cycle spans is
surjective. Finally, a nonzero surface class supported in $L_P$ lies
in its kernel. Finite-dimensionality gives the strict rank inequality,
and \eqref{eq:positive-surface-class} gives the stated K\"ahler case.
\end{proof}
Here are the two applications to a flip
$T\to Z\leftarrow T^+$ with exceptional loci $L,L^+$.
Let $A\subset Z$ be the common exceptional image furnished by
Proposition~\ref{prop:comparison}. On the normalized graph of the
flip, with projections $p,q$ and common composite $h$ to $Z$, one has
\[
p^{-1}(T\setminus L)=h^{-1}(Z\setminus A)
=q^{-1}(T^+\setminus L^+).
\]
Both projections are isomorphisms there. Thus this graph gives exactly
the diagram required by Lemma~\ref{lem:cycle-loss} for $P=T$, $Q=T^+$.
For a boundary component $H\subset T$ and its transform
$H^+\subset T^+$, give both spaces their reduced structures and put
\[
L_H=H\cap L,\qquad L_{H^+}=H^+\cap L^+.
\]
Smallness ensures that the complementary opens are dense and
identified. In the compact analytic fibre product $H\times_Z H^+$,
take the reduced irreducible component $G_H$ containing their common
open graph, with projections $p_H,q_H$ and common composite $h_H$ to
$Z$. This component exists because the fibre product over
$Z\setminus A$ is precisely that graph. Its projections are proper,
and the common open is dense in it. Moreover
\[
p_H^{-1}(H\setminus L_H)=h_H^{-1}(Z\setminus A)
=q_H^{-1}(H^+\setminus L_{H^+}),
\]
with both projections isomorphisms on this open. This verifies
\eqref{eq:cycle-graph-open}, including equality of the inverse images;
normality of $H$ and $H^+$ is unnecessary.
In particular, when $H^+$ contains no flipped surface,
$\dim L_{H^+}\leq1$, so $c_2(H)\geq c_2(H^+)$. The inequality is
strict if $H$ contains a flipping surface, because $H$ is a closed
analytic subspace of the K\"ahler fourfold $T$. The surjectivity used
throughout is the surjectivity on the spans of surface classes proved
above; no surjectivity of \eqref{eq:cycle-comparison-map} on all homology
is needed.
==== END Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/sections/06-cycles.tex ====
==== BEGIN Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/sections/07-termination.tex ====
\section{Termination by descending boundary coefficients}
\label{sec:termination}
We now combine the two counts. At each positive boundary coefficient,
the witness count first excludes flipped surfaces, and the cycle count
then excludes flipping surfaces. The same witness count at threshold
two removes all remaining flipped surfaces, after which the ranks of
surface classes on the fourfolds decrease at every step.
Write $L_i$ and $L_i^+$ for the exceptional loci of step $i$, so that
flipping surfaces lie in $L_i$ and flipped surfaces lie in $L_i^+$.
Proposition~\ref{prop:comparison} gives
\begin{equation}\label{eq:final-dimension}
\dim L_i\leq2,\qquad \dim L_i^+\leq2,\qquad
\dim L_i+\dim L_i^+\geq3.
\end{equation}
Smallness identifies the finitely many positive boundary
components and preserves their coefficients.
\begin{lemma}[Persistence of a surface centre]\label{lem:persistence}
Let $T\dashrightarrow T^+$ be a step of a sequence as in
Theorem~\ref{thm:termination}, and let $E$ be an exceptional place
whose centre on $T$ is a surface $S$. If $S$ is not contained in the
source exceptional locus, its target centre is the strict transform
of $S$, a surface meeting the common isomorphic open. If $S$
lies in a boundary component, its transform lies in the
corresponding component.
\end{lemma}
\begin{proof}
Represent the place on a common higher model. Its image onto $S$ is
surjective by the definition of the centre and properness. The inverse
image of the dense open portion of $S$ on which the step is an
isomorphism is nonempty and dense in $E$. There the two centre maps agree.
The closure of this portion's target image is both the strict
transform of $S$ and the whole target centre. The assertions
follow.
\end{proof}
The following count isolates the argument used at each coefficient,
including the final value $b=0$. Write
$a_i(E)=a(E;X_i,B_i+\bM)$ and fix the integer $m$ of
Proposition~\ref{prop:witnesses}.
\begin{lemma}[Counting flipped surfaces]
\label{lem:coefficient-witness-count}
Consider a tail beginning with $X_N$ of a sequence as in
Theorem~\ref{thm:termination}. Let $b$ be either zero or a positive
boundary coefficient. Suppose no flipping surface at any step of the
tail lies in a boundary component of coefficient greater than $b$.
If $b>0$, only finitely many steps of the tail have a flipped surface
contained in a coefficient-$b$ boundary component. If $b=0$, only
finitely many steps of the tail have any flipped surface.
\end{lemma}
\begin{proof}
At every step $i$ under consideration, choose one such flipped surface
and its witness $E_i$ from Proposition~\ref{prop:witnesses}. The place
is exceptional over $X_N$, and discrepancy monotonicity gives
\begin{equation}\label{eq:tail-witness-bound}
a_N(E_i)\leq a_i(E_i)<a_{i+1}(E_i)\leq2-b.
\end{equation}
For $b=0$ this is the witness bound for an arbitrary flipped surface.
Apply Lemma~\ref{lem:low-places} on the fixed model $X_N$. If
$E_i\notin\mathcal F_{X_N}$, its centre there is a surface contained
in a positive boundary component of coefficient $d$, and
\[
2-d\leq a_N(E_i)<2-b.
\]
Thus $d>b$. By the hypothesis on flipping surfaces, this surface
centre is not contained in $L_N$. Lemma~\ref{lem:persistence} carries
it to a surface in the corresponding coefficient-$d$ component on
$X_{N+1}$. At each successive step up to and including step $i$, the
same hypothesis and lemma apply. Consequently the centre on $X_{i+1}$
meets $X_{i+1}\setminus L_i^+$. This contradicts its definition as
the chosen flipped surface, which is contained in $L_i^+$.
Every chosen witness therefore belongs to the fixed finite set
$\mathcal F_{X_N}$. Lemma~\ref{lem:finite-use} permits each place to
be used at most $m$ times, since its discrepancies on witness targets
are strictly increasing members of $(1,2]\cap m^{-1}\Z$. There are
at most $m\,\#\mathcal F_{X_N}$ steps under consideration.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:termination}]
Suppose the sequence is infinite. Proposition~\ref{prop:comparison}
preserves generalized canonicality: every exceptional place has
log discrepancy at least one on every model. The strict bound greater
than one used in the witness count holds on the target of its witness
step by strict discrepancy increase. No terminality of the whole
model is used.
Order the distinct positive boundary coefficients as
$b_1>\cdots>b_s>0$. Inductively pass to tails on which no flipping
or flipped surface lies in a component of any coefficient already
treated. Start with the full sequence. For the next coefficient
$b=b_j$, the induction hypothesis satisfies the assumption of
Lemma~\ref{lem:coefficient-witness-count}. Passing beyond the finitely
many steps it counts leaves a tail with no flipped surface in any
coefficient-$b$ component.
Index those components on the initial model by a finite set $I_b$,
and write $H_{i,\alpha}$ for their strict transforms on $X_i$,
$\alpha\in I_b$. Smallness preserves this indexing. Each target
component now satisfies
\[
\dim(H_{i+1,\alpha}\cap L_i^+)\leq1.
\]
Otherwise this intersection would contain a flipped surface.
The boundary-component application of Lemma~\ref{lem:cycle-loss}
therefore gives
\[
c_2(H_{i,\alpha})\geq c_2(H_{i+1,\alpha}),
\]
with strict inequality whenever $H_{i,\alpha}$ contains a flipping
surface. The hypotheses were checked in Section~\ref{sec:cycles}
for the reduced components, without normalizing them. In particular,
K\"ahler positivity in $X_i$ makes every removed surface class nonzero
in $H_4(H_{i,\alpha},\R)$.
It follows that the finite nonnegative integer
\[
r_b(i):=\sum_{\alpha\in I_b}c_2(H_{i,\alpha})
\]
does not increase and decreases strictly at each step with a flipping
surface in a coefficient-$b$ component. Only finitely many such
steps occur. Pass beyond them to finish this stage.
After all positive coefficients have been treated, the hypothesis of
Lemma~\ref{lem:coefficient-witness-count} holds with $b=0$. If
$B_0=0$, the positive-coefficient induction is empty and this hypothesis
already holds on the original sequence. Pass beyond the finitely many
steps having a flipped surface. On the resulting tail,
$\dim L_i^+\leq1$, whereas \eqref{eq:final-dimension} forces $L_i$ to
contain a surface. Lemma~\ref{lem:cycle-loss}, applied to the
fourfolds themselves, gives
\[
c_2(X_i)>c_2(X_{i+1})
\]
at every remaining step. These are finite nonnegative integers, so
they cannot decrease strictly along an infinite tail. This proves
termination.
\end{proof}
==== END Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds//build/sections/07-termination.tex ====