diff f62dbea..b34550f log_abundance_char_zero_v2.txt
diff --git a/log_abundance_char_zero_v2.txt b/log_abundance_char_zero_v2.txt
index 15a410f..d1e22cd 100644
--- a/log_abundance_char_zero_v2.txt
+++ b/log_abundance_char_zero_v2.txt
@@ -1,63 +1,63 @@
-\documentclass[11pt,letterpaper]{article}
-\usepackage[T1]{fontenc}
-\usepackage[utf8]{inputenc}
-\usepackage{lmodern}
-\usepackage[margin=1in]{geometry}
-\usepackage{amsmath,amssymb,amsthm,mathtools}
-\usepackage{enumitem,booktabs,array}
-\usepackage{tikz-cd}
-\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue,allcolors=blue]{hyperref}
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\hypersetup{
pdftitle={Log abundance in characteristic zero (v2)},
pdfauthor={OpenAI},
pdflang={en-US}
}
-\newcommand{\Q}{\mathbb{Q}}
-\newcommand{\R}{\mathbb{R}}
-\newcommand{\C}{\mathbb{C}}
-\newcommand{\Z}{\mathbb{Z}}
-\newcommand{\PP}{\mathbb{P}}
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-\newtheorem{theorem}{Theorem}[section]
-\newtheorem{proposition}[theorem]{Proposition}
-\newtheorem{lemma}[theorem]{Lemma}
-\newtheorem{corollary}[theorem]{Corollary}
-\theoremstyle{definition}
-\newtheorem{definition}[theorem]{Definition}
-\newtheorem{assumption}[theorem]{Assumption}
-\theoremstyle{remark}
-\newtheorem{remark}[theorem]{Remark}
-\setlist{itemsep=3pt,topsep=5pt,leftmargin=*}
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\allowdisplaybreaks[1]
-\setlength{\emergencystretch}{2em}
+
\setcounter{tocdepth}{2}
-\title{Log abundance in characteristic zero\\[2pt]
- \large a rewritten, tightened edition}
+# Log abundance in characteristic zero\\[2pt]
+ \large a rewritten, tightened edition
+
-\begin{document}
\maketitle
-\begin{abstract}
+Abstract:
We prove the rational-boundary log abundance conjecture in every
dimension over algebraically closed fields of characteristic zero:
every nef $\Q$-Cartier log canonical divisor on a projective log
@@ -67,64 +67,64 @@ every dimension. The proof is a dimension induction; the hard input
is nonvanishing, closed by a reduction to positive characteristic and
a determinant argument. It is independent of logarithmic Iitaka
subadditivity except for a single use in the Albanese reduction.
-\end{abstract}
+
\tableofcontents
\newpage
%======================================================================
-\section*{A guide for the reader}
+# A guide for the reader
\addcontentsline{toc}{section}{A guide for the reader}
-\label{part:guide}
+
%======================================================================
-\subsection*{The statement we prove}
+## The statement we prove
-The \emph{log abundance conjecture} says this: if $K_X+B$ meets every
+The log abundance conjecture says this: if $K_X+B$ meets every
curve of $X$ nonnegatively, then some positive multiple of $K_X+B$ is
a genuine line bundle generated by its global sections. Nefness is a
numerical condition on curve degrees. Generation by sections is an
algebraic condition on actual functions. Log abundance says the first
forces the second.
-\begin{theorem}[Log abundance, main theorem]\label{thm:main}
+[Log abundance, main theorem]
Let $(X,B)$ be a normal projective log canonical pair over an
algebraically closed field of characteristic zero, with $B$ an
effective rational divisor and $K_X+B$ $\Q$-Cartier. If $K_X+B$ is
nef, then some positive Cartier multiple of $K_X+B$ is globally
generated.
-\end{theorem}
+
The body of the paper works over $\C$. The passage to an arbitrary
algebraically closed field of characteristic zero is bookkeeping and
-lives in Section~\ref{sec:completion}.
+lives in Section~sec:completion.
-\subsection*{The shape of the argument}
+## The shape of the argument
The proof is a dimension induction. In dimension $n$ it aims to
produce two outputs, in this order.
-\begin{enumerate}[label=(\roman*)]
-\item \textbf{Smooth canonical nonvanishing} (NV$_n$). A smooth
+[label=(\roman*)]
+\item Smooth canonical nonvanishing (NV$_n$). A smooth
projective complex $n$-fold $X$ with pseudo-effective $K_X$ has
$H^0(X,mK_X)\neq0$ for some $m>0$.
-\item \textbf{Good log minimal models} in dimension $n$ for all
+\item Good log minimal models in dimension $n$ for all
projective lc pairs with effective real boundary and pseudo-effective
adjoint. On a good model the adjoint is semiample. This is the
-inductive assertion; it yields Theorem~\ref{thm:main} in dimension
+inductive assertion; it yields Theorem~thm:main in dimension
$n$ by comparing the pair with its good model.
-\end{enumerate}
+
The two outputs are entangled. (ii) in dimension $n-1$ (the induction
-hypothesis) is used to \emph{prove} (i) in dimension $n$. And (i) in
+hypothesis) is used to prove (i) in dimension $n$. And (i) in
dimension $n$, together with the signed-representative theorem
-(Sections~\ref{sg:section}--\ref{sg:descent}), is used to \emph{prove}
+(Sections~sg:section--sg:descent), is used to prove
(ii) in dimension $n$. Together, (i)+(ii) in dimension $n$ supply the
induction hypothesis for the next dimension. The base case is a point.
The engine that turns nonvanishing into abundance is the
-\emph{signed-representative theorem} (Theorem~\ref{thm:signed}). Its
+signed-representative theorem (Theorem~thm:signed). Its
statement is subtle: it handles a nef $L=K_X+D$ on a $\Q$-factorial
dlt pair, with $D$ reduced and with $L$ rational-linearly equivalent
to a $\Q$-divisor with either positive or negative coefficients on
@@ -133,16 +133,16 @@ the components of $D$, provided
(ii) $L|_D$ is semiample. Under those hypotheses it produces abundance:
$\kappa(L)=\nu(L)$. The signed-representative theorem is what turns
a numerically-nonnegative adjoint into a genuinely semiample one.
-Its proof (Sections~\ref{sg:section}--\ref{sg:descent}) uses roots,
+Its proof (Sections~sg:section--sg:descent) uses roots,
Hodge modules, and a formal deformation argument; it is the technical
heart of the paper.
-\subsection*{Where the difficulty lives}
+## Where the difficulty lives
Smooth canonical nonvanishing is the bottleneck. Everything else in
the induction is formal once NV$_n$ is in hand:
-Section~\ref{sec:induction} proves
-Proposition~\ref{prop:inductive-reduction}, which makes good models of
+Section~sec:induction proves
+Proposition~prop:inductive-reduction, which makes good models of
$(X,\Delta)$ for real boundary once NV$_n$ holds. The signed-
representative theorem fills the zero-Iitaka case; the case of
positive Kodaira dimension is Fujino--Gongyo. Everything rests on
@@ -154,7 +154,7 @@ paper, and the earlier part of our reading guide has no more to say
about it beyond a signpost to the relevant sections.
\medskip\noindent
-\textbf{Stage 1: geometric exclusions} (Section~\ref{sec:exclusions}).
+Stage 1: geometric exclusions (Section~sec:exclusions).
The counterexample $X$ must:
(a) have irregularity zero, so numerical equivalence equals
$\Q$-linear equivalence on its rational divisors;
@@ -172,18 +172,18 @@ dominate the product. This last is Hanamura's birational-group
theorem applied to a positive canonical current.
\medskip\noindent
-\textbf{Stage 2: two opposing jet estimates} (Sections~\ref{sec:common-tools}
-and~\ref{sec:jets}). At a very general point of $X$, every section of
+Stage 2: two opposing jet estimates (Sections~sec:common-tools
+and~sec:jets). At a very general point of $X$, every section of
every Cartier multiple of a suitable normalized nef divisor $P$ has
-bounded vanishing order (the \emph{scalar jet bound}). But on the
+bounded vanishing order (the scalar jet bound). But on the
projective bundle over $X\times X$ associated to two copies of the
same line bundle, the very general Seshadri constant exceeds
-$2n+2$, so jets vanish to very high order (the \emph{two-slot
-separation}). The first bound consumes the curve exclusion (d); the
+$2n+2$, so jets vanish to very high order (the two-slot
+separation). The first bound consumes the curve exclusion (d); the
second is what produces the determinant incompatibility of Stage 3.
\medskip\noindent
-\textbf{Stage 3: the Frobenius determinant} (Section~\ref{fr:section}).
+Stage 3: the Frobenius determinant (Section~fr:section).
After reduction modulo a large prime, the two-slot jets give an
evaluation map of full generic rank on the product of two
Frobenius-twisted copies of the model. A canonical filtration of its
@@ -194,163 +194,162 @@ jet bound bounds its order at a diagonal point from above, while the
rank deficit bounds it from below. Contradiction for large primes.
\medskip\noindent
-\textbf{Stage 4: contradiction} (Section~\ref{fr:section}, end).
+Stage 4: contradiction (Section~fr:section, end).
The counterexample was impossible. NV$_n$ holds.
-\subsection*{Reading paths}
+## Reading paths
-\textbf{If you want only the statement and the skeleton.}
-Section~\ref{sec:introduction} (main theorem, conventions), then this
-guide, then the statement of Theorem~\ref{thm:signed} in
-Section~\ref{sg:section}, then the statement of
-Theorem~\ref{thm:smooth-nonvanishing} in Section~\ref{fr:section},
-then the proof of Theorem~\ref{thm:main} in Section~\ref{sec:completion}.
+If you want only the statement and the skeleton.
+Section~sec:introduction (main theorem, conventions), then this
+guide, then the statement of Theorem~thm:signed in
+Section~sg:section, then the statement of
+Theorem~thm:smooth-nonvanishing in Section~fr:section,
+then the proof of Theorem~thm:main in Section~sec:completion.
Everything else is either a lemma filling a gap in those two theorems
or a consequence downstream of the main theorem.
-\textbf{If you want to understand the engine.}
-Sections~\ref{sg:section}--\ref{sg:descent}: the root-and-adjunction
+If you want to understand the engine.
+Sections~sg:section--sg:descent: the root-and-adjunction
construction, the Hodge-module lifting, the deformation off the
boundary, the final descent. This is a self-contained subproof of
-Theorem~\ref{thm:signed} and does not require other parts of the
+Theorem~thm:signed and does not require other parts of the
paper.
-\textbf{If you want to understand the hard input.}
-Sections~\ref{sec:exclusions}--\ref{fr:section} in that order. The
+If you want to understand the hard input.
+Sections~sec:exclusions--fr:section in that order. The
section on jets depends on the section on exclusions; the section on
Frobenius depends on both and closes the loop.
-\textbf{The full proof, end to end.} Read linearly.
+The full proof, end to end. Read linearly.
-\subsection*{What is used from outside}
+## What is used from outside
The proof uses three external inputs:
-\begin{enumerate}[label=(\alph*)]
+[label=(\alph*)]
\item Logarithmic Iitaka subadditivity, stated below as
-Theorem~\ref{asm:iitaka}. It is used \emph{once}, in
-Lemma~\ref{ex:irregularity}, and nowhere else. The zero-boundary
+Theorem~asm:iitaka. It is used once, in
+Lemma~ex:irregularity, and nowhere else. The zero-boundary
case used there also follows from Hacon--Popa--Schnell and from
Cao--P\u{a}un.
\item Good minimal models of big klt adjoints, from Birkar--Cascini--
-Hacon--McKernan \cite{BCHM2010}. These are used in the signed-
+Hacon--McKernan [cite]. These are used in the signed-
representative engine and in the reduction at positive Kodaira
dimension.
\item Ordinary negative-ample Kodaira--Saito vanishing and the
-Du~Bois machinery of Saito \cite{Saito1990,Saito2000}, used in the
+Du~Bois machinery of Saito [cite], used in the
Hodge step of the engine.
-\end{enumerate}
+
No logarithmic Iitaka subadditivity is used in the engine, the
exclusions, the jet construction, or the Frobenius comparison. The
-companion preprint \cite{OpenAIMinimalModels2026} enters only in the
+companion preprint [cite] enters only in the
finite-generation corollary, after abundance has been proved.
-\subsection*{Notation}
+## Notation
$(X,B)$ always denotes a log canonical pair with $B$ effective,
rational boundary unless stated otherwise. ``Nef'' means nonnegative
on every curve. ``Semiample'' means a positive multiple is generated
by global sections. ``Abundant'' means $\kappa(L)=\nu(L)$, i.e.~the
Iitaka dimension equals the numerical dimension. Real-boundary
-statements occur only in the induction (Section~\ref{sec:induction});
+statements occur only in the induction (Section~sec:induction);
the main theorem has rational boundary. All other conventions are
-collected in Section~\ref{sec:introduction}, final subsection.
+collected in Section~sec:introduction, final subsection.
\newpage
%======================================================================
-\section{Introduction}\label{sec:introduction}
+# Introduction
%======================================================================
Let $(X,B)$ be a projective log canonical pair with rational boundary.
Its log canonical divisor $K_X+B$ packages the canonical class and
-the boundary. A rational Cartier divisor is \emph{nef} if its degree
-on every curve is nonnegative; it is \emph{semiample} if a positive
+the boundary. A rational Cartier divisor is nef if its degree
+on every curve is nonnegative; it is semiample if a positive
Cartier multiple is generated by global sections. Such a multiple
defines a morphism, so semiampleness turns numerical positivity into a
fibration described by sections. Log abundance asserts that semiampleness
-holds whenever nefness does (Theorem~\ref{thm:main}).
+holds whenever nefness does (Theorem~thm:main).
The proof runs by induction on $\dim X$. Its inductive hypothesis is
stronger than the main theorem: every projective lc pair with effective
-\emph{real} boundary and pseudo-effective \emph{real} Cartier adjoint
-has a \emph{good} log minimal model, on which the adjoint is semiample.
+real boundary and pseudo-effective real Cartier adjoint
+has a good log minimal model, on which the adjoint is semiample.
Smooth canonical nonvanishing --- the statement that a pseudo-effective
canonical class on a smooth projective variety has a nonzero
-pluricanonical section --- is proved at each dimension \emph{before}
+pluricanonical section --- is proved at each dimension before
the good-model output of that dimension is used. At the very end, descent
of the evaluation map of an actual Cartier line bundle transfers the
result over $\C$ to every algebraically closed field of characteristic
-zero (Section~\ref{sec:completion}).
+zero (Section~sec:completion).
Throughout, $\kappa(L)$ is the Iitaka dimension of a rational divisor
-$L$, with $\kappa(L)=-\infty$ if every positive integral system is
+$L$, with $\kappa(L)=-∞$ if every positive integral system is
empty; $\nu(L)$ is the numerical dimension. For nef $L$ we always have
$\kappa(L)\le\nu(L)$; abundance is the reverse inequality.
-\subsection*{Consequences covered by this paper}
+## Consequences covered by this paper
The consequences of the main theorem fall into four groups, each with
its own section, each recording the exact hypotheses of the cited
external theorems it consumes.
-\begin{itemize}
-\item \textbf{Good models, linear triviality, slc, and dlt extension}
-(Section~\ref{sec:consequences}). Good models over $\C$ for real
+
+\item Good models, linear triviality, slc, and dlt extension
+(Section~sec:consequences). Good models over $\C$ for real
boundaries; the smooth-canonical good model; rational-linear
triviality when $\kappa=0$; abundance for semi-log-canonical pairs
after normalization; supported extension of pluricanonical sections
across the reduced dlt boundary.
-\item \textbf{Finite generation} of rounded log canonical rings
-(Section~\ref{sec:canonical-rings}). Both absolute (projective over
+\item Finite generation of rounded log canonical rings
+(Section~sec:canonical-rings). Both absolute (projective over
a field of characteristic zero) and relative (proper over an
algebraic base over $\C$).
-\item \textbf{Classification-flavoured consequences}
-(Section~\ref{sec:classification}): rational connectedness via
+\item Classification-flavoured consequences
+(Section~sec:classification): rational connectedness via
Mumford's criterion; base of the MRC fibration via cotangent tensors;
finiteness of $\pi_1$ when $\kappa=0$ and $\chi\ne0$; quasi-projective
universal covers in the sense of Claudon--H\"oring--Koll\'ar.
-\item \textbf{Relative analytic abundance} (Section~\ref{sec:analytic-abundance}):
+\item Relative analytic abundance (Section~sec:analytic-abundance):
projective surjective morphisms of normal complex analytic varieties
with rational lc adjoint nef over the whole base, obtaining a relatively
generated Cartier multiple in a neighborhood of each compact base
subset.
-\end{itemize}
+
Each section states the theorem it applies precisely and separates the
review literature from the contribution of this paper.
-\subsection*{Historical development}
+## Historical development
The threefold case illustrates why nonvanishing and boundary geometry
-are central. Miyaoka \cite{Miyaoka1988} proved the numerical-dimension
+are central. Miyaoka [cite] proved the numerical-dimension
one case by studying an effective pluricanonical divisor and its
infinitesimal neighborhoods after suitable finite covers. Kawamata
-\cite{Kawamata1992} settled the remaining numerical-dimension two case
+[cite] settled the remaining numerical-dimension two case
for minimal threefolds; his Section~4 gives an alternative proof of
Miyaoka's case via finite covers and Hodge theory. Keel--Matsuki--
-McKernan \cite{KeelMatsukiMcKernan1994,KeelMatsukiMcKernan2004Correction,
-Matsuki2003Correction} proved the logarithmic theorem in dimension three.
-Fujino \cite{Fujino2000} extended the statement to semi-log-canonical
+McKernan [cite] proved the logarithmic theorem in dimension three.
+Fujino [cite] extended the statement to semi-log-canonical
threefolds, where sections must agree along the conductor before they
descend. This normalisation-and-gluing issue recurs in every dimension
whenever the coefficient-one boundary is reducible.
In higher dimensions, Birkar--Cascini--Hacon--McKernan
-\cite{BCHM2010} established minimal models for klt pairs in the
+[cite] established minimal models for klt pairs in the
big-boundary or log-general-type range, and finite generation of the
projective rational klt canonical ring without a bigness hypothesis
\cite[Corollary~1.1.2]{BCHM2010}. Two distinctions remain sharp on the
boundary of the positive cone.
-\begin{enumerate}[label=(\alph*)]
+[label=(\alph*)]
\item Nonvanishing produces sections but not semiampleness.
Hashizume \cite[Theorem~1.4]{Hashizume2018} reduces real-boundary lc
nonvanishing and ordinary minimal-model existence in dimension $\le n$
to smooth canonical nonvanishing in dimension $n$. The induction must
-then \emph{make} these models good. In positive Kodaira dimension the
+then make these models good. In positive Kodaira dimension the
required step is Fujino--Gongyo
\cite[Lemma~3.5]{FujinoGongyoRings2014}.
\item Kodaira dimension zero and numerical dimension zero are distinct
@@ -358,10 +357,10 @@ hypotheses. Gongyo \cite[Theorem~1.2]{Gongyo2013} proves that a
numerically trivial rational lc adjoint is $\Q$-linearly trivial. In
the zero-Iitaka argument below, numerical triviality forces an already
available effective representative to vanish.
-\end{enumerate}
-\begin{sloppypar}
-Liu and Xu \cite{LiuXu2025} handle numerical dimension at most one:
+
+
+Liu and Xu [cite] handle numerical dimension at most one:
good models for lc pairs of dimension at most five with nonnegative
invariant Kodaira dimension and numerical dimension at most one
\cite[Theorem~5.1]{LiuXu2025}, and a reduction to smooth nonvanishing
@@ -370,40 +369,40 @@ here has no numerical-dimension restriction. The ordinary-pair
specialisation of \cite[Corollary~1.2]{OpenAIMinimalModels2026}
concerns existence of a nef model and is not an input to the
good-model induction; we use it, after abundance has been proved, in
-the finite-generation part of Section~\ref{sec:canonical-rings}.
-\end{sloppypar}
+the finite-generation part of Section~sec:canonical-rings.
+
-\subsection*{The two tasks in the induction}
+## The two tasks in the induction
Assume good models exist in all dimensions below $n$. We must prove
(i)~NV$_n$ and (ii)~good models in dimension $n$. Both jobs use the
-same criterion for a \emph{signed representative} on a reduced boundary,
-which we state here once and then prove in Sections~\ref{sg:section}--%
-\ref{sg:descent}.
+same criterion for a signed representative on a reduced boundary,
+which we state here once and then prove in Sections~sg:section--%
+sg:descent.
Let $L$ be a nef divisor and $\nu(L)
=\max\{j\ :\ L^jH^{n-j}>0\}$ for $H$ ample and $n=\dim X$; this is the
numerical dimension. The signed-representative criterion is
\[
-\begin{gathered}
-L=K_X+D\text{ nef},\qquad L-cD\text{ pseudo-effective for some }c>0,\\
-L\sim_{\Q}\textstyle\sum_i a_iD_i,\qquad L|_D\text{ semiample as a
- rational line bundle},\\
-\Longrightarrow\quad \kappa(L)=\nu(L).
-\end{gathered}
+
+L=K_X+D nef,\qquad L-cD pseudo-effective for some c>0,\\
+L\sim_{\Q}\textstyle\sum_i a_iD_i,\qquad L|_D semiample as a
+ rational line bundle,\\
+⇒\quad \kappa(L)=\nu(L).
+
\]
Here $(X,D)$ is a projective $\Q$-factorial dlt pair, and $D=\sum_iD_i$
is the decomposition of the reduced boundary into prime components.
-The rational coefficients $a_i$ are allowed to be \emph{positive},
-\emph{negative}, or \emph{zero}. The hypothesis is semiampleness of
-$L|_D$ on the \emph{entire reduced scheme} $D$, so normalisation and
+The rational coefficients $a_i$ are allowed to be positive,
+negative, or zero. The hypothesis is semiampleness of
+$L|_D$ on the entire reduced scheme $D$, so normalisation and
conductor gluing are built in (they enter via
\cite[Theorem~1.5]{FujinoGongyo2014}). For the good-model task, positive
Kodaira dimension is Fujino--Gongyo \cite[Lemma~3.5]{FujinoGongyoRings2014};
the signed criterion handles the remaining zero-Iitaka case
-(Proposition~\ref{prop:inductive-reduction}).
+(Proposition~prop:inductive-reduction).
The signed criterion is proved geometrically.
A generated multiple of $L|_D$ gives a morphism from $D$ to projective
@@ -430,7 +429,7 @@ metric refinement of Ohsawa--Takegoshi extension under the hypothesis
that the boundary is purely log terminal and the effective representative
contains the coefficient-one divisor; Chan--Choi
\cite[Theorem~1.6.1]{ChanChoi2021} drop the containment condition. In
-the present paper the representative may \emph{have both signs}, and
+the present paper the representative may have both signs, and
compatibility is required across the whole reducible boundary. We
therefore construct compatible formal lifts on the whole reduced
boundary. Semiampleness on the boundary is Fujino--Gongyo
@@ -439,28 +438,28 @@ boundary. Semiampleness on the boundary is Fujino--Gongyo
The ingredients are familiar. Normalised cyclic covers and their
eigenspace decomposition follow Esnault--Viehweg
\cite[Section~3]{EsnaultViehweg1992}. Deligne's logarithmic Hodge theory
-\cite{Deligne1971} is the ambient framework. On the singular boundary
+[cite] is the ambient framework. On the singular boundary
we use Saito's projective strictness and graded de Rham vanishing with
his comparison to the Du~Bois complex
\cite[Theorem~2.14,Proposition~2.33]{Saito1990}
\cite[Theorem~0.2,Corollary~0.3]{Saito2000}. The paper's contribution
-here is to \emph{identify} the obstruction to lifting and to show that
+here is to identify the obstruction to lifting and to show that
these ordinary-variety statements apply to the covering construction;
-Section~\ref{sec:hodge} carries out exactly this identification.
+Section~sec:hodge carries out exactly this identification.
-\subsection*{Smooth nonvanishing}
+## Smooth nonvanishing
The proof of NV$_n$ assumes a counterexample
$X$ (smooth projective $n$-fold, $K_X$ pseudo-effective, $\kappa(X,K_X)
-=-\infty$) and proceeds through four stages.
+=-∞$) and proceeds through four stages.
-\paragraph{Exclusions}
-(Section~\ref{sec:exclusions}). The zero-boundary case of logarithmic
-Iitaka subadditivity (Theorem~\ref{asm:iitaka}) rules out positive
+Exclusions:
+(Section~sec:exclusions). The zero-boundary case of logarithmic
+Iitaka subadditivity (Theorem~asm:iitaka) rules out positive
irregularity via the Albanese map. This is its only use. Geometric
preparations then exclude three ways the counterexample could be
covered by lower-dimensional subvarieties:
-\begin{itemize}
+
\item families carrying a positive current pulled back from a
smaller variety;
\item families of curves of uniformly bounded genus and uniformly
@@ -468,17 +467,17 @@ bounded degree, after normalising the nef divisors to a fixed small
volume;
\item families of birational correspondences between fixed finite
covers whose graphs dominate the product.
-\end{itemize}
+
The first two use positive currents, algebraic webs, and an ascending
chain condition for Lelong numbers at valuations of bounded discrepancy.
The third uses Hanamura's birational-group theorem. The signed-
representative theorem plays a second role here: it forces the full
reduced support of any signed canonical representative to have a big
-logarithmic adjoint (Proposition~\ref{prop:signed-alternative}). This
+logarithmic adjoint (Proposition~prop:signed-alternative). This
controls the multisections of the two-slot construction below.
-\paragraph{Jets}
-(Sections~\ref{sec:common-tools} and~\ref{sec:jets}). The exclusions
+Jets:
+(Sections~sec:common-tools and~sec:jets). The exclusions
produce two incompatible estimates for vanishing at a moving point.
At a very general point, every nonzero section of every Cartier
multiple $kP$ of a suitable normalised nef divisor $P$ has vanishing
@@ -496,13 +495,13 @@ factor, and a fixed branch complement with bounded degree leaves only
finitely many covers. After fixing the covers, the correspondences
form a family of birational maps whose graphs dominate the product.
Hanamura then makes a fixed cover birational to an abelian variety;
-the ramification-and-current argument of Section~\ref{sec:exclusions}
+the ramification-and-current argument of Section~sec:exclusions
returns a pluricanonical section on the counterexample, a
contradiction. This closes the two-slot estimate and therefore the jet
separation.
-\paragraph{Frobenius}
-(Section~\ref{fr:section}). Once the projective bundle, its divisors,
+Frobenius:
+(Section~fr:section). Once the projective bundle, its divisors,
and the required finite collection of jet sections are fixed, reduction
modulo a large prime gives an evaluation map on the product of two
copies of the Frobenius-twisted model. That map has full generic rank.
@@ -516,14 +515,14 @@ bounds the order of the determinant at the selected diagonal point.
The rank deficit bounds the same order from below by $3/4$ of the rank.
Contradiction for large primes.
-Theorem~\ref{common:finite-data-frobenius} states this incompatibility
-for a \emph{fixed} smooth variety, \emph{fixed} jet sections, and a
-\emph{fixed} movable curve bound. Its proof opens
-Section~\ref{fr:section} and uses neither abundance nor subadditivity.
+Theorem~common:finite-data-frobenius states this incompatibility
+for a fixed smooth variety, fixed jet sections, and a
+fixed movable curve bound. Its proof opens
+Section~fr:section and uses neither abundance nor subadditivity.
The geometric application follows in the same section, using the fixed
inputs from the exclusions and from the jet constructions. The
numerical sub-adjunction and moving-kernel results needed for those
-constructions are proved in Section~\ref{sec:common-tools}.
+constructions are proved in Section~sec:common-tools.
The numerical part combines two established methods. The argument
following a moving base component uses parameter differentiation as in
@@ -541,40 +540,40 @@ Kitadai--Sumihiro \cite[Definition~3.1,Remark~3.2]{KitadaiSumihiro2008}.
Langer's instability estimate \cite[Corollary~2.5]{Langer2004}
controls the slopes of the graded bundles.
-\subsection*{The subadditivity input}
+## The subadditivity input
The all-dimensional argument uses the following companion result. Its
-application is confined to Lemma~\ref{ex:irregularity}.
+application is confined to Lemma~ex:irregularity.
-\begin{theorem}[Logarithmic Iitaka subadditivity,
-\cite[Corollary~6.2]{OpenAIIitaka2026}]\label{asm:iitaka}
+[Logarithmic Iitaka subadditivity,
+\cite[Corollary~6.2]{OpenAIIitaka2026}]
Let $f:X\to Y$ be a surjective morphism with connected fibres between
smooth connected projective complex varieties. Let $D_X,D_Y$ be
reduced effective SNC divisors, allowing zero boundaries, with
-$\Supp(f^*D_Y)\subseteq\Supp(D_X)$. For a very general smooth fibre
+$\Supp(f^*D_Y)⊆\Supp(D_X)$. For a very general smooth fibre
$F$, put $D_F=D_X|_F$. Then
\[
\kappa(X,K_X+D_X)\ge\kappa(F,K_F+D_F)+\kappa(Y,K_Y+D_Y).
\]
Here $D_F$ is reduced with simple normal crossings and
$K_F+D_F\sim(K_X+D_X)|_F$. The convention
-$(-\infty)+b=-\infty$ also holds when $b=-\infty$, and the zero
+$(-∞)+b=-∞$ also holds when $b=-∞$, and the zero
divisor on a point has Iitaka dimension zero.
-\end{theorem}
+
The support inclusion allows extra components in $D_X$; the morphism
need not be smooth away from $D_X$. The theorem imposes no nefness,
abundance, bigness, or good-model condition. Its zero-boundary case is
-used in the Albanese reduction of Section~\ref{sec:exclusions}; this
-is the \emph{only} subadditivity input in the proof. The companion's
+used in the Albanese reduction of Section~sec:exclusions; this
+is the only subadditivity input in the proof. The companion's
characteristic-zero specialisation
\cite[Corollary~6.3]{OpenAIIitaka2026} uses a geometric generic fibre;
the present paper keeps the complex, very-general-fibre statement
distinct. No fractional-boundary, orbifold, non-projective,
-Hodge-conjectural, or arithmetic extension of Theorem~\ref{asm:iitaka}
+Hodge-conjectural, or arithmetic extension of Theorem~asm:iitaka
is assumed here.
-The zero-boundary application in Lemma~\ref{ex:irregularity} also
+The zero-boundary application in Lemma~ex:irregularity also
follows from the external maximal-Albanese-dimension theorem of
Hacon--Popa--Schnell \cite[Theorem~1.1]{HaconPopaSchnell2017}: the
smooth base there maps generically finitely onto a subvariety of an
@@ -582,34 +581,34 @@ abelian variety, and Cao--P\u{a}un \cite[Theorem~1.1]{CaoPaunAbelian2016}
prove the underlying abelian-base case. Those cover the application,
not the full logarithmic statement.
-\subsection*{Reading order}
+## Reading order
-Section~\ref{sec:induction} sets out the dimension induction and proves
-the passage from nonvanishing to good models. Sections~\ref{sg:section}--%
-\ref{sg:descent} prove the signed-boundary criterion: first the root and
+Section~sec:induction sets out the dimension induction and proves
+the passage from nonvanishing to good models. Sections~sg:section--%
+sg:descent prove the signed-boundary criterion: first the root and
adjunction construction, then the infinitesimal lifting, then the final
-descent. Section~\ref{sec:exclusions} derives the geometric exclusions.
-After the local tools of Section~\ref{sec:common-tools},
-Section~\ref{sec:jets} constructs the scalar and two-slot jets.
-Section~\ref{fr:section} proves the independent Frobenius comparison and
+descent. Section~sec:exclusions derives the geometric exclusions.
+After the local tools of Section~sec:common-tools,
+Section~sec:jets constructs the scalar and two-slot jets.
+Section~fr:section proves the independent Frobenius comparison and
then uses it to establish smooth nonvanishing.
-Sections~\ref{sec:completion} and~\ref{sec:consequences} finish the
+Sections~sec:completion and~sec:consequences finish the
induction, descend to characteristic zero, and record the boundary
-consequences. Section~\ref{sec:canonical-rings} proves finite generation.
-Section~\ref{sec:classification} records the cotangent, fundamental-group,
-and quasi-projective-cover consequences. Section~\ref{sec:analytic-abundance}
+consequences. Section~sec:canonical-rings proves finite generation.
+Section~sec:classification records the cotangent, fundamental-group,
+and quasi-projective-cover consequences. Section~sec:analytic-abundance
records the relative analytic statement. A reader using only the
-finite-data comparison can start with Theorem~\ref{common:finite-data-frobenius}
-and its proof in Section~\ref{fr:section}; neither assumes the
+finite-data comparison can start with Theorem~common:finite-data-frobenius
+and its proof in Section~fr:section; neither assumes the
counterexample geometry.
-\subsection{Conventions}\label{par:conventions}
+## Conventions
-A \emph{variety} is integral. Canonical divisors are chosen compatibly
+A variety is integral. Canonical divisors are chosen compatibly
on birational models. For a divisor $E$ on a smooth model $p:W\to X$,
the log discrepancy is
\[
- a(E;X,B)=1+\operatorname{ord}_E\bigl(K_W-p^*(K_X+B)\bigr).
+ a(E;X,B)=1+ord_E(K_W-p^*(K_X+B)).
\]
Log canonical means $a(E;X,B)\ge0$ for every divisor over $X$; klt
means $a(E;X,B)>0$. We use the standard dlt, plt, and slc conventions.
@@ -625,13 +624,13 @@ associated rational maps. We write $\equiv$ for numerical equivalence
and $\sim_{\Q}$ for rational-linear equivalence.
Unless a different base field is specified, constructions are over $\C$.
-A \emph{very general} point of a variety avoids a countable union of
+A very general point of a variety avoids a countable union of
proper closed subvarieties. The transfer to arbitrary algebraically
-closed characteristic-zero fields is made in Section~\ref{sec:completion}
+closed characteristic-zero fields is made in Section~sec:completion
only after the complex argument has been completed.
%======================================================================
-\section{The inductive framework}\label{sec:induction}
+# The inductive framework
%======================================================================
The dimension induction has two outputs in each dimension: smooth
@@ -641,10 +640,10 @@ adjoint. The present section proves the passage from the first output
to the second, given good models strictly below the current dimension.
All varieties here are projective over $\C$.
-\subsection{Definitions and the induction hypothesis}
+## Definitions and the induction hypothesis
-For an effective real boundary $\Delta$, a \emph{good log minimal
-model} of $(X,\Delta)$ is a log minimal model on which the adjoint
+For an effective real boundary $\Delta$, a good log minimal
+model of $(X,\Delta)$ is a log minimal model on which the adjoint
divisor is semiample. For a real boundary, semiampleness means
real-linear equivalence to the pullback of an ample real divisor by a
contraction. We use the standard definition of log minimal model in
@@ -656,61 +655,61 @@ intersection-theoretic numerical dimension $\nu(X,A)$.
Fix an integer $n\ge1$.
-\begin{assumption}[Lower-dimensional good models]\label{mmp:lower-models}
+[Lower-dimensional good models]
Every projective log canonical pair of dimension $<n$, with effective
real boundary and pseudo-effective adjoint divisor, has a good log
minimal model.
-\end{assumption}
+
The base case (dimension zero) is a point. Assume
-Assumption~\ref{mmp:lower-models}. The induction runs as follows.
+Assumption~mmp:lower-models. The induction runs as follows.
-\begin{center}\small
-\begin{tabular}{@{}p{0.13\textwidth}p{0.40\textwidth}p{0.38\textwidth}@{}}
+\small
+{@{}p{0.13\textwidth}p{0.40\textwidth}p{0.38\textwidth}@{}}
\toprule
Stage & Hypothesis & Output and location \\
\midrule
-Base & Dimension $0$ & $\mathrm{GLM}_{\le0}$: a point \\
-Smooth step & Assumption~\ref{mmp:lower-models} and
- Theorem~\ref{asm:iitaka} &
- NV$_n$, Theorem~\ref{thm:smooth-nonvanishing} \\
-Good-model step & Assumption~\ref{mmp:lower-models} and NV$_n$ &
- $\mathrm{GLM}_{\le n}$,
- Proposition~\ref{prop:inductive-reduction} \\
-Nef conclusion & $\mathrm{GLM}_{\le n}$ &
- Semiampleness in dimension $n$, Lemma~\ref{mmp:comparison} \\
+Base & Dimension $0$ & $GLM_{\le0}$: a point \\
+Smooth step & Assumption~mmp:lower-models and
+ Theorem~asm:iitaka &
+ NV$_n$, Theorem~thm:smooth-nonvanishing \\
+Good-model step & Assumption~mmp:lower-models and NV$_n$ &
+ $GLM_{\le n}$,
+ Proposition~prop:inductive-reduction \\
+Nef conclusion & $GLM_{\le n}$ &
+ Semiampleness in dimension $n$, Lemma~mmp:comparison \\
\bottomrule
-\end{tabular}
-\end{center}
-Here $\mathrm{GLM}_{\le d}$ denotes good-model existence for all
+
+
+Here $GLM_{\le d}$ denotes good-model existence for all
projective complex lc pairs with effective real boundary and
pseudo-effective adjoint in dimensions at most $d$; NV$_n$ and
Semiampleness are the two other named outputs. The signed-
representative theorem used in both rows is proved in
-Sections~\ref{sg:section}--\ref{sg:descent}; its hypothesis of
+Sections~sg:section--sg:descent; its hypothesis of
semiampleness on the whole reduced boundary is supplied here by
-Proposition~\ref{prop:boundary-restriction}.
+Proposition~prop:boundary-restriction.
-\subsection{Comparison with a nef model}
+## Comparison with a nef model
-\begin{lemma}[Comparison with a nef model]\label{mmp:comparison}
+[Comparison with a nef model]
Let $(X,\Delta)$ be log canonical, and let $(X',\Delta')$ be a log
minimal model. On a common resolution
\[
X\xleftarrow{u}W\xrightarrow{v}X',
\]
there is an effective $v$-exceptional divisor $F$ with
-\begin{equation}\label{mmp:comparison-equation}
+
u^*(K_X+\Delta)=v^*(K_{X'}+\Delta')+F.
-\end{equation}
+
If $K_X+\Delta$ is nef, then $F=0$. Consequently, a nef rational
adjoint is semiample whenever it has a good log minimal model. If
$(X,\Delta)$ is klt, then so is its log minimal model, and the model
extracts no divisors.
-\end{lemma}
-\begin{proof}
+
+Proof.
Put $F=u^*(K_X+\Delta)-v^*(K_{X'}+\Delta')$. Discrepancy improvement
on divisors of $X$ gives $u_*F\ge0$, while $-F$ is $u$-nef because
$K_{X'}+\Delta'$ is nef. The negativity lemma gives $F\ge0$. At a
@@ -731,16 +730,16 @@ upstairs thus implies generation downstairs.
For a klt source, the coefficient at an extracted prime would be
strictly negative, an impossibility; the remaining discrepancy
inequalities make the model klt.
-\end{proof}
-\begin{proposition}[Semiampleness on the reduced boundary]
-\label{prop:boundary-restriction}
-Assume Assumption~\ref{mmp:lower-models}. Let $(V,D)$ be a projective
+
+[Semiampleness on the reduced boundary]
+
+Assume Assumption~mmp:lower-models. Let $(V,D)$ be a projective
$\Q$-factorial dlt $n$-fold with $D$ reduced and $M=K_V+D$ nef. Then
$M|_D$ is semiample as a rational line bundle on the reduced scheme $D$.
-\end{proposition}
-\begin{proof}
+
+Proof.
The ambient $V$ is klt. The reduced floor of a $\Q$-factorial dlt pair
satisfies Serre's condition $S_2$ and has ordinary double crossings in
codimension one; its irreducible components are normal.
@@ -752,7 +751,7 @@ finite direct image, hence Cohen--Macaulay. The divisor sequence then
shows that $D$ itself is Cohen--Macaulay.
Divisorial adjunction on the normalisation $\coprod D_i\to D$ gives lc
-pairs $(D_i,\operatorname{Diff}_{D_i}(D-D_i))$, where the different
+pairs $(D_i,Diff_{D_i}(D-D_i))$, where the different
includes the conductor with coefficient one. Removing that conductor
contribution defines an effective boundary on $D$; together with $D$ it
is an slc pair whose adjoint line bundle is $M|_D$. The divisible
@@ -760,46 +759,46 @@ residue identifications hold in codimension one, with even powers
eliminating residue signs at double crossings, and extend by $S_2$.
Each normalised adjoint is nef and semiample by
-Assumption~\ref{mmp:lower-models} and Lemma~\ref{mmp:comparison}.
+Assumption~mmp:lower-models and Lemma~mmp:comparison.
Semiampleness descends from the normalisation by the slc gluing
theorem \cite[Theorem~1.5, equivalently Theorem~4.3]{FujinoGongyo2014}.
This gives the required statement on the whole reduced scheme, not
merely on the individual components.
-\end{proof}
-\subsection{A uniform trivial-perturbation argument}
+
+## A uniform trivial-perturbation argument
We will also perturb a nef adjoint into a klt program. A single Cartier
index must survive every step: otherwise the smallness required of the
perturbation could change along the program. The following estimate
and line-bundle descent give that uniformity.
-\begin{lemma}\label{mmp:trivial-perturbation}
+
Let $(Z,\Delta)$ be a projective $\Q$-factorial dlt rational pair of
dimension $n$ with $L=K_Z+\Delta$ nef. Put $P=\Delta$ if the pair is
klt, and $P=\lfloor\Delta\rfloor$ otherwise. Fix $m>0$ with $mL$
Cartier. For every rational
\[
- 0<\epsilon<\frac{1}{4nm+2},
+ 0<ε<(1)/(4nm+2),
\]
-every step of an LMMP for $L-\epsilon P$ is $L$-trivial. On every
+every step of an LMMP for $L-ε P$ is $L$-trivial. On every
model of the program, the transform of $L$ is nef, and its multiple by
the same integer $m$ is Cartier.
-\end{lemma}
-\begin{proof}
-Both $\Delta-\epsilon P$ and $\Delta-\tfrac12P$ are effective klt
-boundaries. If $R$ is a ray negative for $L-\epsilon P$, nefness of $L$
+
+Proof.
+Both $\Delta-ε P$ and $\Delta-\tfrac12P$ are effective klt
+boundaries. If $R$ is a ray negative for $L-ε P$, nefness of $L$
makes $R$ negative for $L-\tfrac12P$. Choose a rational curve $C$
spanning $R$ and satisfying the length bound
\[
- -\bigl(L-\tfrac12P\bigr)\cdot C\le2n.
+ -(L-\tfrac12P)\cdot C\le2n.
\]
Writing $a=L\cdot C\ge0$ and $p=P\cdot C$, we have
\[
- p\le2a+4n,\qquad a<\epsilon p,\qquad (1-2\epsilon)a<4n\epsilon.
+ p\le2a+4n,\qquad a<ε p,\qquad (1-2ε)a<4nε.
\]
-The choice of $\epsilon$ gives $a<1/m$. Since $ma$ is a nonnegative
+The choice of $ε$ gives $a<1/m$. Since $ma$ is a nonnegative
integer, $a=0$.
For the driving klt contraction $c:Z\to B$, the line bundle $\OO_Z(mL)$
@@ -818,9 +817,9 @@ The driving boundary remains klt throughout its LMMP. The transformed
$\Delta-\tfrac12P$ remains effective and is smaller than the driving
boundary, so it too is klt. The same estimate and the same integer $m$
apply inductively at every subsequent step.
-\end{proof}
-\subsection{The good-model implication}
+
+## The good-model implication
We now assume smooth nonvanishing in the current dimension and prove
the good-model implication. The argument first reduces real nef
@@ -828,17 +827,17 @@ boundaries to rational ones. Positive Kodaira dimension is then
handled by lower-dimensional good models; what remains is the
zero-Iitaka case and its klt perturbations.
-\begin{proposition}[Inductive reduction to smooth nonvanishing]
-\label{prop:inductive-reduction}
-Assume Assumption~\ref{mmp:lower-models}. Assume also that every smooth
+[Inductive reduction to smooth nonvanishing]
+
+Assume Assumption~mmp:lower-models. Assume also that every smooth
projective $n$-fold with pseudo-effective canonical divisor has
nonnegative Kodaira dimension. Then every projective lc $n$-fold with
real boundary and pseudo-effective adjoint has a good log minimal
model. In particular, every nef rational lc adjoint in dimension $n$
is semiample.
-\end{proposition}
-\begin{proof}
+
+Proof.
Hashizume's reduction \cite[Theorem~1.4]{Hashizume2018} gives
nonvanishing and log minimal models for projective lc pairs with real
boundary in dimensions at most $n$. Pass to a $\Q$-factorial dlt log
@@ -867,22 +866,22 @@ many divisors and principal divisors and solving the resulting rational
linear system with nonnegative rational coefficients produces a
rational one. If $\kappa(Z,L)\ge1$, lower-dimensional good models give
a good minimal model by \cite[Lemma~3.5]{FujinoGongyoRings2014}, and
-Lemma~\ref{mmp:comparison} transfers semiampleness back to $Z$. It
+Lemma~mmp:comparison transfers semiampleness back to $Z$. It
remains to treat $\kappa(Z,L)=0$.
-\paragraph{The non-pseudo-effective perturbation.}
+The non-pseudo-effective perturbation.:
Put $P=\Delta$ in the klt case and $P=\lfloor\Delta\rfloor$ otherwise.
-Suppose $L-\epsilon P$ is not pseudo-effective for every sufficiently
-small $\epsilon>0$. Choose rational $\epsilon$ as in
-Lemma~\ref{mmp:trivial-perturbation}, and run the klt LMMP with
+Suppose $L-ε P$ is not pseudo-effective for every sufficiently
+small $ε>0$. Choose rational $ε$ as in
+Lemma~mmp:trivial-perturbation, and run the klt LMMP with
scaling to a Mori fibre space
\[
- (Z,\Delta-\epsilon P)\dashrightarrow(Z',\Delta'-\epsilon P')
+ (Z,\Delta-ε P)\dashrightarrow(Z',\Delta'-ε P')
\xrightarrow{f}T.
\]
Existence and termination in the non-pseudo-effective case are the
-established klt results of \cite{BCHM2010}. The program is crepant for
-$L$. The line-bundle descent in Lemma~\ref{mmp:trivial-perturbation},
+established klt results of [cite]. The program is crepant for
+$L$. The line-bundle descent in Lemma~mmp:trivial-perturbation,
applied also to the final contraction, gives a nef rational divisor
$A$ on $T$ with
\[
@@ -900,11 +899,11 @@ $(T,\Delta_T)$ is klt and
All the hypotheses hold: the total pair is globally klt and effective,
$f$ is a projective contraction, and the adjoint is rationally
linearly pulled back. In particular this use requires no conjecture on
-semiampleness of a moduli divisor. Assumption~\ref{mmp:lower-models}
-and Lemma~\ref{mmp:comparison} make $A$ semiample.
+semiampleness of a moduli divisor. Assumption~mmp:lower-models
+and Lemma~mmp:comparison make $A$ semiample.
Otherwise $P'$ is effective and relatively ample, since
-$K_{Z'}+\Delta'-\epsilon P'$ is relatively antiample. Some component
+$K_{Z'}+\Delta'-ε P'$ is relatively antiample. Some component
of the floor therefore dominates $T$. Take a crepant dlt blowup
$b:(\widetilde Z,\widetilde\Delta)\to(Z',\Delta')$ and choose the
strict transform $S$ of such a component. The restriction
@@ -922,11 +921,11 @@ not cover a complex vector space. Its norm is nonzero at the
downstairs point. Hence $A$ is semiample in this case as well, and
crepancy transfers the conclusion back to $L$.
-\paragraph{The klt pseudo-effective perturbation.}
-We next settle the klt case where $L-\epsilon\Delta$ is pseudo-effective
-for some small rational $\epsilon>0$. Nonvanishing gives
+The klt pseudo-effective perturbation.:
+We next settle the klt case where $L-ε\Delta$ is pseudo-effective
+for some small rational $ε>0$. Nonvanishing gives
\[
- L\sim_{\Q}G_0:=H_0+\epsilon\Delta,\qquad H_0\ge0.
+ L\sim_{\Q}G_0:=H_0+ε\Delta,\qquad H_0\ge0.
\]
Take a resolution $p:W\to Z$ with reduced SNC divisor $D$ containing
all exceptional divisors and the strict support of $H_0+\Delta$. Then
@@ -950,7 +949,7 @@ fix a common resolution
W\xleftarrow{q}U\xrightarrow{v}V.
\]
Its boundary is reduced. On $U$ the comparison divisor of
-Lemma~\ref{mmp:comparison} is exceptional over $V$; pushing $q^*G_W$
+Lemma~mmp:comparison is exceptional over $V$; pushing $q^*G_W$
forward by $v$ gives
\[
K_V+D_V\sim_{\Q}G_V=\sum_ib_iD_i,\qquad b_i>0,
@@ -964,14 +963,14 @@ comparison-divisor coefficient.) Effectivity and exceptionality in the
comparison preserve the adjoint section ring, so the nef adjoint on
$V$ still has Iitaka dimension zero.
-The small-perturbation hypothesis of Theorem~\ref{thm:signed} is now
+The small-perturbation hypothesis of Theorem~thm:signed is now
explicit. If $D_V\ne0$, choose rational $c$ with $0<c<\min_ib_i$; then
\[
K_V+(1-c)D_V\sim_{\Q}\sum_i(b_i-c)D_i\ge0.
\]
If $D_V=0$, pseudo-effectivity is immediate.
-Proposition~\ref{prop:boundary-restriction} supplies the required
-semiampleness on $D_V$. Theorem~\ref{thm:signed} then makes $K_V+D_V$
+Proposition~prop:boundary-restriction supplies the required
+semiampleness on $D_V$. Theorem~thm:signed then makes $K_V+D_V$
abundant. Its Iitaka dimension is zero, so its numerical dimension is
zero. For an ample divisor $H_V$ on $V$ we get
\[
@@ -991,15 +990,15 @@ good-model assertion in dimension $n$: for a non-nef pseudo-effective
klt adjoint, first take its klt log minimal model and apply what has
just been proved.
-\paragraph{The remaining non-klt case.}
+The remaining non-klt case.:
Finally suppose the original pair is not klt and a small floor
perturbation $L-\delta P$ is pseudo-effective. Choose rational
-$\epsilon>0$ with $2\epsilon\le\delta$, small enough to keep the
+$ε>0$ with $2ε\le\delta$, small enough to keep the
perturbed boundaries effective and klt. Convexity of the
pseudo-effective cone, applied between $L$ and $L-\delta P$, makes
both
\[
- L-\epsilon P,\qquad L-2\epsilon P,\qquad P=\lfloor\Delta\rfloor,
+ L-ε P,\qquad L-2ε P,\qquad P=\lfloor\Delta\rfloor,
\]
pseudo-effective klt adjoints. Their Iitaka dimensions are zero:
nonvanishing gives the lower bound, and adding the effective
@@ -1007,12 +1006,12 @@ perturbation bounds each by $\kappa(Z,L)=0$. The preceding klt case
gives good models for both of these perturbed adjoints, so their
Nakayama numerical dimensions are zero. Since
\[
- 2(L-\epsilon P)=L+(L-2\epsilon P),
+ 2(L-ε P)=L+(L-2ε P),
\]
and the last summand has an effective rational representative,
monotonicity and homogeneity of $\kappa_\sigma$ give
\[
- \kappa_\sigma(Z,L)\le\kappa_\sigma\bigl(Z,2(L-\epsilon P)\bigr)=0.
+ \kappa_\sigma(Z,L)\le\kappa_\sigma(Z,2(L-ε P))=0.
\]
The nef divisor $L$ is therefore numerically trivial. Nonvanishing is
now already available in this finishing case: write
@@ -1023,13 +1022,13 @@ abundance premise.
This settles the remaining zero-Iitaka case, so every rational nef
dlt adjoint considered above is semiample. Returning to the finite
rational simplex proves the real-boundary good-model assertion, and
-Lemma~\ref{mmp:comparison} descends semiampleness to every original
+Lemma~mmp:comparison descends semiampleness to every original
nef rational lc pair.
-\end{proof}
+
%======================================================================
-\section{The signed-representative engine}\label{sg:section}
+# The signed-representative engine
%======================================================================
The induction, at its key step, has a nef log canonical divisor
@@ -1046,7 +1045,7 @@ individual lemmas are for.
The semiample $L|_D$ gives a morphism
\[
- \phi:D\longrightarrow P:=\PP^s,\qquad
+ \phi:D→ P:=\PP^s,\qquad
N_0|_D\simeq\phi^*\OO_P(1)
\]
for a sufficiently divisible positive integer $m$, where $N_0:=\OO_X(mL)$.
@@ -1058,7 +1057,7 @@ component has image of dimension at most $r$, and that $G_+$ restricted
to $\Supp G_-\cup D_0$ has image of dimension less than $r$.
The root construction of
-Proposition~\ref{prop:signed-construction} produces, from this data,
+Proposition~prop:signed-construction produces, from this data,
a projective variety $\mathcal X$ with a reduced Cartier divisor
$S\subset\mathcal X$ and a finite representable morphism
$S\to D\times_P\mathcal P$ whose image covers the positive boundary,
@@ -1069,34 +1068,34 @@ Stein factorisation
\]
with $g$ finite representable and $p$ a projective bundle. The
factorisation is the geometric output that the Hodge step of
-Section~\ref{sec:hodge} consumes. That step lifts functions on the
+Section~sec:hodge consumes. That step lifts functions on the
positive boundary to every infinitesimal thickening, deformes them off
$D$, and produces a fixed moveable family of subvarieties on which $L$
is trivial. Descent along the resulting nef reduction
-(Section~\ref{sg:descent}) then proves abundance.
+(Section~sg:descent) then proves abundance.
The full theorem is:
-\begin{theorem}[Signed-boundary abundance criterion]\label{thm:signed}
+[Signed-boundary abundance criterion]
Let $X$ be a projective $\Q$-factorial dlt variety of dimension $n$,
let $D=\sum_iD_i$ be a reduced divisor whose support is the
coefficient-one part of the boundary, and let
$L=K_X+D$. Suppose:
-\begin{enumerate}[label=(\roman*)]
+[label=(\roman*)]
\item $L$ is nef;
\item $L-cD$ is pseudo-effective for some rational $c>0$;
\item $L|_D$ is semiample as a rational line bundle on the reduced
scheme $D$;
\item the support $D$ is nonempty and $0<\nu(L)<n$.
-\end{enumerate}
+
Then $\kappa(L)=\nu(L)$.
If instead $\nu(L)=n$, then $L$ is big (the numerical criterion for
a nef divisor). If $\nu(L)=0$, the case is either $D=0$, in which
case $L\sim_{\Q}0$ and there is nothing to prove, or the argument
below applies after shrinking $D$ to
-Proposition~\ref{prop:inductive-reduction}.
-\end{theorem}
+Proposition~prop:inductive-reduction.
+
The proof occupies the remainder of this section and the next; it does
not use the induction hypothesis, only established results of the
@@ -1105,18 +1104,18 @@ signed representative, allowing $a_i\in\Q$ of any sign, and decompose
$G=G_+-G_-$ into effective parts with disjoint prime supports;
$D_0=\sum_{a_i=0}D_i$ collects the coefficient-zero components.
-\begin{lemma}[Numerical location of the boundary]\label{sg:numerical-boundary}
-In the setting of Theorem~\ref{thm:signed}, assume $0<\nu(L)<n$.
+[Numerical location of the boundary]
+In the setting of Theorem~thm:signed, assume $0<\nu(L)<n$.
Put $r=\nu(L)-1$. Then:
-\begin{enumerate}[label=(\roman*)]
+[label=(\roman*)]
\item every component of $D$ has $\phi$-image of dimension at most $r$;
\item some component of $G_+$ has $\phi$-image of dimension exactly $r$;
\item the locus $\Supp G_+\cap(\Supp G_-\cup D_0)$ has $\phi$-image of
dimension less than $r$ (empty when $r=0$).
-\end{enumerate}
-\end{lemma}
-\begin{proof}
+
+
+Proof.
Part (i). Intersect the pseudo-effective class $L-cD$ with the product
of nef classes $L^vH^{n-v-1}$, $v=\nu(L)$, and $H$ ample. Every term
is nonnegative: $L^vH^{n-v-1}$ is nef intersection of $L$, and $(L-cD)$
@@ -1163,14 +1162,14 @@ have image dimension strictly less than $r$ wherever their images have
$r$-dimensional support, forcing the locus
$\Supp G_+ \cap(\Supp G_-\cup D_0)$ to have $\phi$-image of dimension
less than $r$.
-\end{proof}
+
We will also place the Stein image in a smooth target: this allows
differential operators and projective Hodge-module direct images to be
used on ordinary scheme charts. The infinitesimal neighborhoods will
later be cut down to a projective fiber before they are algebraized.
-\begin{proposition}\label{prop:signed-construction}
+
Choose $\ell$ divisible by $m$ and every nonzero integer $|ma_i|$,
and set $a=\ell/m$. There is a normal tame Deligne--Mumford stack
$\mathcal X$, considered on a neighborhood of a reduced Cartier
@@ -1178,13 +1177,13 @@ divisor $S$ of pure dimension $N=n-1$, together with a morphism
to $X$ and a reduced boundary
$T$ having no component in common with $S$, with the following
properties.
-\begin{enumerate}[label=\textup{(\roman*)}]
+[label=(\roman*)]
\item
The pair $(\mathcal X,S+T)$ is log canonical, and a fixed
isomorphism of reflexive sheaves is given by
\[
\omega_{\mathcal X}(S+T)\simeq\OO_{\mathcal X}(aS).
- \tag{\(\ast_{\mathrm{adj}}\)}\label{sg:ambient-adjunction}
+ \tag{\(\ast_{adj}\)}
\]
The support of $T$ is locally set-theoretically principal.
The ambient space and $S$ are Cohen--Macaulay on scheme charts.
@@ -1192,7 +1191,7 @@ Off $T$, the ambient space is Gorenstein and the displayed
isomorphism is ordinary Cartier adjunction data.
\item
-Put $J=(S\cap T)_{\mathrm{red}}$. Both $S$ and $J$ are Du Bois
+Put $J=(S\cap T)_{red}$. Both $S$ and $J$ are Du Bois
on scheme charts, the ideal $\mathcal I_{J/S}$ is maximal
Cohen--Macaulay, and
\[
@@ -1201,12 +1200,12 @@ Cohen--Macaulay, and
\simeq\OO_S(aS).
\]
The identification agrees off $J$ with the residue convention
-specified by \eqref{sg:ambient-adjunction}.
+specified by (sg:ambient-adjunction).
\item
There is a finite representable morphism
\[
- S\longrightarrow D\times_P\mathcal P
+ S→ D\times_P\mathcal P
\]
whose image covers $\Supp G_+$. The line $\OO_S(S)$ is the pullback
of $C$. The image of $J$ in $P$ has dimension less than $r$,
@@ -1219,7 +1218,7 @@ This is a morphism from the reduced divisor $S$; no extension
to a neighborhood of $S$ in $\mathcal X$ is asserted here.
Writing $h=p\circ g$ and
$\mathcal T=\Spec_{\mathcal P}h_*\OO_S$, the morphism $g$ factors
-through a closed embedding $\mathcal T\hookrightarrow\mathcal Y$,
+through a closed embedding $\mathcal T↪\mathcal Y$,
and
\[
g_*\OO_S=\OO_{\mathcal T},\qquad
@@ -1236,18 +1235,18 @@ Only this etale chart is extended; an extension of $g$ to $qS$
is not needed to define the chart.
The resulting $qS_U$ are quasi-projective schemes, separated
and quasi-finite over $X$.
-\end{enumerate}
+
All the adjunction and duality identifications are compatible
with changes of etale chart.
-\end{proposition}
-\begin{proof}
+
+Proof.
We construct the divisor and its adjunction data first, then place
its Stein image in a smooth projective bundle. The final step
verifies that the infinitesimal neighborhoods have the scheme
charts required for formal lifting.
-\smallskip\noindent\emph{Roots and a fixed adjunction isomorphism.}
+\smallskip\noindentRoots and a fixed adjunction isomorphism.
Let $\rho:\mathcal R\to X$ be the normalized simultaneous root stack of the Cartier
divisors $mG_+$, $mG_-$, and $mD_0$, taking an $\ell$-th root of
each divisor together with its section; the construction allows
@@ -1272,14 +1271,14 @@ slightly removes all zero-discrepancy places. The difference
between the two sides is a torsion integral Weil class, represented by
\[
\mathcal F=
- \bigl(\omega_{\mathcal R}(B)
- \otimes\OO_{\mathcal R}(-a(S_+-S_-))\bigr)^{**}.
+ (\omega_{\mathcal R}(B)
+ ⊗ \OO_{\mathcal R}(-a(S_+-S_-)))^{**}.
\]
Choose a periodicity isomorphism $\mathcal F^{[q]}\simeq\OO_{\mathcal R}$,
where square brackets denote reflexive powers. It defines multiplication
in the finite algebra
\[
- \mathcal A=\bigoplus_{i=0}^{q-1}\mathcal F^{[i]}.
+ \mathcal A=⊕_{i=0}^{q-1}\mathcal F^{[i]}.
\]
Let $\tau:\mathcal R^\dagger\to\mathcal R$ be the normalization
of $\Spec_{\mathcal R}\mathcal A$. In codimension one this is
@@ -1295,12 +1294,12 @@ on the cover stack itself, the adjoint isomorphism is equivariant
on every atlas. This equivariance will be needed for the
quotient below.
-\smallskip\noindent\emph{Separating the positive and negative parts.}
-Let $p_1:\widetilde{\mathcal R}\to\mathcal R^\dagger$ be the
+\smallskip\noindentSeparating the positive and negative parts.
+Let $p_1:\mathcal Rtilde\to\mathcal R^\dagger$ be the
normalized blowup of the ideal of $S_+\cap S_-$. The spaces
constructed so far are related by
\[
- \widetilde{\mathcal R}\xrightarrow{p_1}\mathcal R^\dagger
+ \mathcal Rtilde\xrightarrow{p_1}\mathcal R^\dagger
\xrightarrow{\tau}\mathcal R\xrightarrow{\rho}X.
\]
The blown-up ideal is
@@ -1318,7 +1317,7 @@ Cartier, and the divisors
\[
S'_+=p_1^*S_+-E,\qquad S'_-=p_1^*S_--E
\]
-are reduced Cartier and disjoint on $\widetilde{\mathcal R}$.
+are reduced Cartier and disjoint on $\mathcal Rtilde$.
Since no codimension-two two-branch stratum is contained
in $S_0$, its pullback $p_1^*S_0$ is reduced Cartier and has no
exceptional component. The total boundary
@@ -1337,7 +1336,7 @@ is effective Cartier, $\ord_F(B')>0$, whence
\]
Positive pair discrepancies also remain positive after the
boundary is dropped. Thus $V'$ is klt even at its
-higher-codimension singular loci. On $\widetilde{\mathcal R}$,
+higher-codimension singular loci. On $\mathcal Rtilde$,
put $S=S'_+$ and $T=S'_-+E+p_1^*S_0$. Near $S$, the section of the disjoint
divisor $S'_-$ is a unit, and the fixed linear equivalence reads
\[
@@ -1359,14 +1358,14 @@ description shows that $T|_S$ is generically reduced. Being
Cartier on the Cohen--Macaulay scheme $S$, it is reduced
everywhere.
-\smallskip\noindent\emph{Retaining the required root line.}
+\smallskip\noindentRetaining the required root line.
The construction so far has supplied reduced Cartier divisors
and fixed adjunction data. We now remove the root characters
that are unnecessary, while preserving the line attached to $S$.
-On $\widetilde{\mathcal R}$ the line $\OO(S-S'_-)$ has $\ell$-th power equal to the pullback
+On $\mathcal Rtilde$ the line $\OO(S-S'_-)$ has $\ell$-th power equal to the pullback
of $N_0$ and hence defines a morphism to the gerbe of $\ell$-th
roots of $N_0$ on $X$. Denote this gerbe by $\mathcal G$, and let
-$\kappa:\widetilde{\mathcal R}\to\mathcal X$ be the relative
+$\kappa:\mathcal Rtilde\to\mathcal X$ be the relative
coarse-space morphism over $\mathcal G$. On a trivializing chart, we quotient
by the kernel of the finite-group character on the indicated
root line. The quotients are ordinary quasi-projective schemes:
@@ -1386,7 +1385,7 @@ Cartierness of the reduced image $T$ is not needed.
Divisorial ramification occurs only on the boundary. Log
ramification thus preserves log canonicity and identifies the
descended adjoint isomorphism with
-\eqref{sg:ambient-adjunction}. The ambient chart and $S$ are
+(sg:ambient-adjunction). The ambient chart and $S$ are
Cohen--Macaulay because their structure sheaves are invariant
direct summands of the finite CM covers.
@@ -1398,19 +1397,19 @@ identifies the invariant log dualizing sheaf. Off $T$, this
isomorphism makes the canonical sheaf invertible, giving the
asserted Gorenstein property.
-\smallskip\noindent\emph{Duality on the boundary.}
+\smallskip\noindentDuality on the boundary.
The reduced supports $S$ and $J$ are unions of log canonical
centers, since intersections of lc centers are again unions
of lc centers. Both are therefore Du Bois by
\cite[Theorems~1.4 and~1.7]{KollarKovacs2010}.
To identify the ideal and its dual, work on a quotient chart
-and denote the finite cover above by $f:S^{\mathrm{up}}\to S$.
-Reducedness of $T^{\mathrm{up}}|_{S^{\mathrm{up}}}$ gives
+and denote the finite cover above by $f:S^{up}\to S$.
+Reducedness of $T^{up}|_{S^{up}}$ gives
\[
\mathcal I_{J/S}
=
- \bigl(f_*\OO_{S^{\mathrm{up}}}
- (-T^{\mathrm{up}}|_{S^{\mathrm{up}}})\bigr)^{\mathrm{inv}}.
+ (f_*\OO_{S^{up}}
+ (-T^{up}|_{S^{up}}))^{inv}.
\]
An invariant function vanishes on the reduced quotient image
exactly when its pullback vanishes on this reduced preimage.
@@ -1423,8 +1422,8 @@ Apply finite-map duality with trace and then take invariants:
\[
\mathcal Hom_S(\mathcal I_{J/S},\omega_S)
=
- \bigl(f_*\omega_{S^{\mathrm{up}}}
- (T^{\mathrm{up}}|_{S^{\mathrm{up}}})\bigr)^{\mathrm{inv}}
+ (f_*\omega_{S^{up}}
+ (T^{up}|_{S^{up}}))^{inv}
\simeq\OO_S(aS).
\]
The last isomorphism comes from Cartier adjunction upstairs
@@ -1435,13 +1434,13 @@ ordinary residue fixed by the ambient isomorphism. Normalized
trace preserves this residue convention, so the identifications
agree on overlaps.
-\smallskip\noindent\emph{The target of the lifting argument.}
+\smallskip\noindentThe target of the lifting argument.
On $S$, the root gerbe of $N_0$ is $D\times_P\mathcal P$.
Finiteness of the strict divisor, proved above, and removal
of the relative inertia kernel give a finite representable
map $S\to D\times_P\mathcal P$. Its image covers the positive
boundary, and the root line is $\OO_S(S)=C$.
-Lemma~\ref{sg:numerical-boundary} therefore gives image dimension
+Lemma~sg:numerical-boundary therefore gives image dimension
$r$ for $S$ in $P$. To check the smaller image of $J$, recall
that before the quotient $S'_-$ is disjoint from $S$, the
exceptional divisor $E$ maps into $\Supp G_+\cap\Supp G_-$,
@@ -1487,13 +1486,13 @@ fiber dimensions. The morphism $qS_U\to X$
is therefore quasi-finite as well. Zariski's main theorem
embeds $qS_U$ in a scheme finite over $X$, making it a
quasi-projective scheme as required.
-\end{proof}
-\section{Hodge theory and formal lifting}\label{sec:hodge}
+
+# Hodge theory and formal lifting
Our goal is to lift functions and a transverse parameter through
every infinitesimal neighborhood of the divisor constructed in
-Proposition~\ref{prop:signed-construction}. We recall its data.
+Proposition~prop:signed-construction. We recall its data.
\(\mathcal P=\sqrt[\ell]{\OO_{\PP^s}(1)}\) is a root gerbe, \(C\) is
its tautological line bundle, and
\[
@@ -1505,15 +1504,15 @@ projective bundle. The Stein factor of \(h\) is finite over
\(\mathcal P\); it has been embedded as \(\mathcal T\subset\mathcal Y\)
so that \(g_*\OO_S=\OO_{\mathcal T}\). The reduced Cartier
divisor \(S\subset\mathcal X\) has dimension \(N=n-1\). If
-\(J=(S\cap T)_{\mathrm{red}}\), then \(S,J\) are Du Bois,
+\(J=(S\cap T)_{red}\), then \(S,J\) are Du Bois,
\(\mathcal I_{J/S}\) is maximal Cohen--Macaulay, and
-\begin{equation}\label{hg:geometric-lines}
+
\mathcal B:=
\mathcal Hom_S(\mathcal I_{J/S},\omega_S)
\simeq \OO_S(aS)=g^*C^a,
\qquad
\OO_S(S)=g^*C,
-\end{equation}
+
for an integer \(a>0\). Off \(T\), the fixed ambient isomorphism
\(\omega_{\mathcal X}(S)\simeq\OO_{\mathcal X}(aS)\) supplies these
identifications by adjunction.
@@ -1524,7 +1523,7 @@ scheme etale chart \(U\to\mathcal Y\) of finite type, write
for the induced projective morphism. Denote by \(qS_U\) the
unique extension of \(S_U\to S\) to the nilpotent thickening
\(qS\). These extensions are compatible in \(q\).
-Proposition~\ref{prop:signed-construction} makes them quasi-projective
+Proposition~prop:signed-construction makes them quasi-projective
schemes: they are separated and quasi-finite over \(X\), and hence
open in schemes finite over the projective variety \(X\).
Powers and quotients of \(I\) below are pulled back to the appropriate
@@ -1532,20 +1531,20 @@ thickenings. On these quotients, \(g_*\) denotes direct image for
the common underlying topological map; a ringed-space map from a
thickening to \(U\) has not yet been constructed.
-\begin{proposition}\label{prop:formal-lifting}
+
For every such chart \(U\), every \(j\geq0\), and every \(k\geq1\),
the transition
\[
g_*(I^j/I^{j+k+1})
- \longrightarrow g_*(I^j/I^{j+k})
+ → g_*(I^j/I^{j+k})
\]
is surjective as a map of sheaves of abelian groups. Its kernel is
-\(\OO_{\mathcal T_U}\otimes C^{-j-k}\), where
+\(\OO_{\mathcal T_U} ⊗ C^{-j-k}\), where
\(\mathcal T_U=\mathcal T\times_{\mathcal Y}U\).
The statements are compatible with further etale changes of
charts. On an affine chart, these transitions are also
surjective on global sections.
-\end{proposition}
+
In their isolated-component case with numerical dimension one,
Liu--Xu use finite covers and Du Bois cohomology to obtain infinitesimal
@@ -1554,10 +1553,10 @@ may have positive dimension, so its local functions and the transverse
parameter must be lifted together.
For \(j\ge0\) and \(k\ge1\), the kernel of the transition from length \(k+1\) to length \(k\)
-in \(I^j/I^{j+k+1}\) is \(\OO_S\otimes C^{-j-k}\).
+in \(I^j/I^{j+k+1}\) is \(\OO_S ⊗ C^{-j-k}\).
The corresponding connecting homomorphism therefore takes values in
\[
- R^1g_*\OO_S\otimes C^{-j-k}.
+ R^1g_*\OO_S ⊗ C^{-j-k}.
\]
We will identify this sheaf with a negative twist of the lowest
filtered piece \(F_0M\) of a Hodge-module direct image. Lifting local
@@ -1567,7 +1566,7 @@ of the differential-operator symbol map. Negative-twist vanishing
will make that kernel have no global sections. The same graph
calculation then kills the error in lifting the transverse parameter.
-\subsection{The lowest Hodge piece}
+## The lowest Hodge piece
We work with graded-polarizable mixed Hodge modules on ordinary
complex algebraic varieties. The inputs are projective strictness
@@ -1586,7 +1585,7 @@ itself occupies degree zero. Accordingly, on a smooth chart \(V\),
the term in
degree \(-i\) of \(\Gr^F_k\DR_V(M)\) is
\[
- \Gr^F_{k-i}M\otimes_{\OO_V}\bigwedge^iT_V.
+ \Gr^F_{k-i}M ⊗ _{\OO_V}∧^iT_V.
\]
When \(F_{<0}M=0\), the lowest graded de Rham complex therefore
consists of the sheaf \(F_0M\) in degree zero.
@@ -1594,10 +1593,10 @@ consists of the sheaf \(F_0M\) in degree zero.
We now identify a Hodge module whose lowest piece contains the
coherent lifting obstructions. On each chart \(U\), write
\(J_U=J\times_{\mathcal Y}U\).
-For \(j:S_U\setminus J_U\hookrightarrow S_U\), put
+For \(j:S_U∖ J_U↪ S_U\), put
\[
- A^\bullet_U=\mathbf D\!\left(
- j_!\Q^H_{S_U\setminus J_U}[N]\right),
+ A^\bullet_U=\mathbf D\!(
+ j_!\Q^H_{S_U∖ J_U}[N]),
\qquad
A_{0,U}={}^{p}\!H^0A^\bullet_U,
\qquad
@@ -1610,7 +1609,7 @@ denote the corresponding direct image of differential modules.
These constructions commute with etale restriction. Thus the
\(M_U\) form a system \(M\) supported on \(\mathcal T\).
-\begin{lemma}\label{hg:lowest-piece}
+
The following identifications hold compatibly on the charts:
\[
F_{<0}A_{0,U}=0,\qquad F_0A_{0,U}=\mathcal B|_{S_U},
@@ -1618,14 +1617,14 @@ The following identifications hold compatibly on the charts:
F_{<0}M_U=0,\qquad
F_0M_U=R^1g_*(\mathcal B|_{S_U}).
\]
-In a smooth ambient embedding \(S_U\hookrightarrow V\) of
+In a smooth ambient embedding \(S_U↪ V\) of
codimension \(c\), the underlying module of \(A_{0,U}\) is
\(\mathcal H^c_{S_U}(\omega_V)\), localized off \(J_U\).
The lowest-piece inclusion is the adjunction, or Ext-to-support,
inclusion off \(J_U\), extended by meromorphic localization.
-\end{lemma}
-\begin{proof}
+
+Proof.
The proof must identify not just a coherent sheaf but its actual
inclusion into the differential module: the graph calculation
will differentiate classes in that inclusion. We first compute
@@ -1634,11 +1633,11 @@ comparison applied to the difference triangle for the constant
objects of \(S_U\) and \(J_U\) identifies its degree-zero
graded de Rham complex with \(\mathcal I_{J_U/S_U}\).
Coherent duality and the maximal-Cohen--Macaulay property give
-\begin{equation}\label{hg:derived-lowest}
+
\Gr^F_k\DR(A^\bullet_U)=0\quad(k<0),
\qquad
\Gr^F_0\DR(A^\bullet_U)=\mathcal B|_{S_U}[0].
-\end{equation}
+
Here \([N]\) cancels the dimension shift of the dualizing
complex. In the smooth case, our convention gives
\(\mathbf D(\Q^H[N])=\Q^H[N](N)\); its right differential module
@@ -1656,7 +1655,7 @@ row at that index, and therefore gives
\mathcal H^i\Gr^F_q\DR(A^\bullet_U)
=F_q\,{}^p\!H^iA^\bullet_U.
\]
-If \(q<0\), this contradicts Equation~\eqref{hg:derived-lowest}.
+If \(q<0\), this contradicts Equation~(hg:derived-lowest).
It follows that all these objects have \(F_{<0}=0\). The same
argument at index zero yields
\(F_0A_{0,U}=\mathcal B|_{S_U}\), while the lowest pieces of
@@ -1678,12 +1677,12 @@ of \(A_{0,U}\) in the statement.
The two descriptions must agree as inclusions, with the fixed
residue normalization. On the smooth locus of
-\(S_U\setminus J_U\), filtered duality and smooth graph
+\(S_U∖ J_U\), filtered duality and smooth graph
pushforward give the ordinary dualizing sheaf and its residue
inclusion. This is precisely Ext-to-support with the
normalization specified by ambient adjunction. Naturality in
smooth etale coordinates gives the same scalar on each chart.
-The agreement extends throughout \(S_U\setminus J_U\), because
+The agreement extends throughout \(S_U∖ J_U\), because
the first support-cohomology module has no sections supported
on a smaller-dimensional subset. One can see this from the
Cousin resolution of the smooth dualizing sheaf: its first term
@@ -1705,35 +1704,35 @@ Spencer complex survives: it is the direct image of
and identifies this sheaf as a coherent subsheaf of the unfiltered
degree-one direct image. All constructions in this identification
are canonical under etale changes of charts.
-\end{proof}
+
The obstruction sheaf for the transition indexed by \(j,k\) is thus
-\(F_0M\otimes C^{-(a+j+k)}\), by
-Equation~\eqref{hg:geometric-lines} and projection formula.
+\(F_0M ⊗ C^{-(a+j+k)}\), by
+Equation~(hg:geometric-lines) and projection formula.
We do not need all its global sections to vanish. The lifting proof
will show that the coordinate obstruction lies in the kernel of a
symbol map, and will apply the following vanishing to that kernel.
-\subsection{Negative-twist vanishing}
+## Negative-twist vanishing
The support of the system \(M\) just constructed is finite over
\(\mathcal P\). We prove the required vanishing for any system
with this support property.
-\begin{lemma}\label{hg:negative-vanishing}
+
Let \(M\) be a compatible system of mixed Hodge modules in perverse
degree zero on the scheme etale charts of \(\mathcal Y\). Suppose
its support is finite over \(\mathcal P\). Then, for every integer
\(k\), every positive integer \(b\), and every \(j<0\),
\[
- \mathbb H^j\!\left(
- \mathcal Y,\Gr^F_k\DR_{\mathcal Y}(M)\otimes p^*C^{-b}
- \right)=0.
+ \mathbb H^j\!(
+ \mathcal Y,\Gr^F_k\DR_{\mathcal Y}(M) ⊗ p^*C^{-b}
+ )=0.
\]
Here the graded de Rham complex is the descended coherent complex.
-\end{lemma}
-\begin{proof}
+
+Proof.
The line \(C\) lives on the root gerbe, so we first push the
complex to \(\mathcal P\). We will then pull back to projective
space, where \(C\) becomes \(\OO(1)\) and ordinary
@@ -1742,10 +1741,10 @@ support of \(M\), its direct image is concentrated in perverse
degree zero; call the resulting system on \(\mathcal P\) \(M'\).
On ordinary scheme
charts, projective strictness gives
-\begin{equation}\label{hg:graded-direct-image}
+
Rp_*\Gr^F_k\DR_{\mathcal Y}(M)
\simeq \Gr^F_k\DR_{\mathcal P}(M').
-\end{equation}
+
For the desired global hypercohomology vanishing, this identity must
hold for the descended complexes. We verify that compatibility
before passing to a finite cover.
@@ -1754,7 +1753,7 @@ For right differential modules, use the transfer bimodule
\[
\mathcal D_{\mathcal Y\to\mathcal P}
=
- \OO_{\mathcal Y}\otimes_{p^{-1}\OO_{\mathcal P}}
+ \OO_{\mathcal Y} ⊗ _{p^{-1}\OO_{\mathcal P}}
p^{-1}\mathcal D_{\mathcal P},
\]
with its order filtration. Derived tensor over
@@ -1775,17 +1774,17 @@ gradeds. The ordinary projective direct-image theorem on the base
charts supplies strictness and concentration in perverse degree
zero. Passing to the etale site does not change coherent
cohomology on those charts. This proves
-Equation~\eqref{hg:graded-direct-image} globally, as an identity
+Equation~(hg:graded-direct-image) globally, as an identity
computing coherent hypercohomology. The same verification will
apply to the finite representable map below.
To pass from the gerbe to a scheme, use the representable finite
surjective morphism
\[
- v:\PP^s_z\longrightarrow\mathcal P
+ v:\PP^s_z→\mathcal P
\]
defined by the power map
-\([z_0:\cdots:z_s]\mapsto[z_0^\ell:\cdots:z_s^\ell]\)
+\([z_0:...:z_s]\mapsto[z_0^\ell:...:z_s^\ell]\)
and the root \(\OO_{\PP^s_z}(1)\); thus
\(v^*C=\OO_{\PP^s_z}(1)\). Trivial roots on the standard affine
opens of \(\PP^s\) give scheme etale charts
@@ -1818,7 +1817,7 @@ infinity. The weight extensions give the required mixed object.
Vanishing for \(H\) will imply vanishing for \(M'\) once we
realize \(M'\) as a retract of \(v_+H\). On a chart one
-must use the \emph{whole} base-changed finite cover. Over
+must use the whole base-changed finite cover. Over
\(\check U_i\), this is a disjoint union of copies of the
corresponding affine coordinate chart of \(\PP^s_z\). The unit
on unshifted constants and its dual trace, formed using smooth
@@ -1837,7 +1836,7 @@ formula, and take perverse cohomology in degree zero. Finite
direct image is perverse exact, so it commutes with this
degree-zero cohomology. We obtain maps
\[
- M'\longrightarrow v_+H\longrightarrow M'
+ M'→ v_+H→ M'
\]
whose composite is multiplication by the degree of the whole
base-changed cover. Dividing the second map by that degree
@@ -1846,27 +1845,27 @@ retraction consequently descends on filtered differential modules
as well.
Applying the finite version of
-Equation~\eqref{hg:graded-direct-image} and projection formula
+Equation~(hg:graded-direct-image) and projection formula
now exhibits the hypercohomology in the statement as a direct
summand of
\[
- \mathbb H^j\!\left(
- \PP^s_z,\Gr^F_k\DR(H)\otimes\OO_{\PP^s_z}(-b)
- \right).
+ \mathbb H^j\!(
+ \PP^s_z,\Gr^F_k\DR(H) ⊗ \OO_{\PP^s_z}(-b)
+ ).
\]
Ordinary negative-ample Kodaira--Saito vanishing makes this group
zero for \(j<0\). For a mixed object, apply the vanishing to
the pure weight gradeds and then to the weight-filtration exact
sequences, whose Hodge filtrations are strict.
-\end{proof}
-\subsection{Lifting through the infinitesimal neighborhoods}
+
+## Lifting through the infinitesimal neighborhoods
We now combine the lowest-piece identification with negative-twist
vanishing. The remaining step is to identify the relation satisfied
by the coordinate and conormal obstructions.
-\begin{proof}[Proof of Proposition~\ref{prop:formal-lifting}]
+Proof. [Proof of Proposition~prop:formal-lifting]
We induct on \(k\), simultaneously for all powers \(I^j\).
The connecting homomorphisms will factor through the first
graded layers and together form a derivation. We must show
@@ -1883,16 +1882,16 @@ Define
\]
The first graded layer is
\[
- E_{j,1}=\OO_{\mathcal T_U}\otimes C^{-j}.
+ E_{j,1}=\OO_{\mathcal T_U} ⊗ C^{-j}.
\]
For adjacent lengths, the exact sequence has kernel
-\(g_*\OO_{S_U}\otimes C^{-j-k}\), and its connecting homomorphism
+\(g_*\OO_{S_U} ⊗ C^{-j-k}\), and its connecting homomorphism
takes values in
\[
- R^1g_*\OO_{S_U}\otimes C^{-j-k}.
+ R^1g_*\OO_{S_U} ⊗ C^{-j-k}.
\]
-\smallskip\noindent\emph{The obstruction derivation.}
+\smallskip\noindentThe obstruction derivation.
Assume inductively that all shorter transitions are surjective.
A section of
\(E_{j,k}\) with zero length-one term comes from \(E_{j+1,k-1}\)
@@ -1903,11 +1902,11 @@ factors through the length-one layer; for \(k=1\) this
factorization is immediate. More precisely, these factorizations
give additive maps
\[
- D_{k,j}:\OO_{\mathcal T_U}\otimes C^{-j}
- \longrightarrow R^1g_*\OO_{S_U}\otimes C^{-j-k}.
+ D_{k,j}:\OO_{\mathcal T_U} ⊗ C^{-j}
+ → R^1g_*\OO_{S_U} ⊗ C^{-j-k}.
\]
Together they define \(D_k\) on the graded algebra
-\(\bigoplus_{j\geq0}\OO_{\mathcal T_U}\otimes C^{-j}\), with
+\(⊕_{j\geq0}\OO_{\mathcal T_U} ⊗ C^{-j}\), with
graded shift \(k\); no \(\OO_U\)-linearity is asserted.
Its values are computed by choosing local
lifts and taking Cech differences. For a product, this
@@ -1915,33 +1914,33 @@ difference is the sum of the two first-order differences:
the product of two errors vanishes in the layer under
consideration. Thus \(D_k\) is a \(\C\)-derivation, with
values in the graded module whose degree-\(j\) term is
-\(R^1g_*\OO_{S_U}\otimes C^{-j}\).
+\(R^1g_*\OO_{S_U} ⊗ C^{-j}\).
Restrict its degree-zero part along
\(\OO_U\to\OO_{\mathcal T_U}\). This derivation factors
-through \(\Omega_U^1\), defining a section of
-\(T_U\otimes R^1g_*\OO_{S_U}\otimes C^{-k}\). As the chart
+through \(Ω_U^1\), defining a section of
+\(T_U ⊗ R^1g_*\OO_{S_U} ⊗ C^{-k}\). As the chart
varies, these sections descend to
-\begin{equation}\label{hg:obstruction-section}
- e\in H^0\!\left(
- \mathcal Y,T_{\mathcal Y}\otimes R^1g_*\OO_S\otimes C^{-k}
- \right)
+
+ e\in H^0\!(
+ \mathcal Y,T_{\mathcal Y} ⊗ R^1g_*\OO_S ⊗ C^{-k}
+ )
=
- H^0\!\left(
- \mathcal Y,T_{\mathcal Y}\otimes F_0M\otimes C^{-(a+k)}
- \right).
-\end{equation}
-The equality follows from Equation~\eqref{hg:geometric-lines},
-Lemma~\ref{hg:lowest-piece}, and projection formula. For the
+ H^0\!(
+ \mathcal Y,T_{\mathcal Y} ⊗ F_0M ⊗ C^{-(a+k)}
+ ).
+
+The equality follows from Equation~(hg:geometric-lines),
+Lemma~hg:lowest-piece, and projection formula. For the
descent assertion, function lifts pull back along the unique
etale thickenings. Ordinary flat base change on the reductions
pulls back their Cech classes in the coherent obstruction
sheaves. Differentials also pull back and generate under an
etale map, giving the required compatibility.
-\smallskip\noindent\emph{Coordinate errors and adjunction.}
+\smallskip\noindentCoordinate errors and adjunction.
We now compute this derivation in local coordinates. After shrinking \(U\), choose etale coordinates
-\(t_1,\ldots,t_d\), with \(d=\dim U\), and a frame of
+\(t_1,...,t_d\), with \(d=\dim U\), and a frame of
\(C^{-1}\). Write \(y\) for its pullback, a conormal frame.
Induction lifts the coordinates along \(g\) modulo \(I^k\)
and lifts \(y\) to \(I/I^{k+1}\). Lift them one step further
@@ -1953,12 +1952,12 @@ order farther than the coordinates: its next error lies in
\(I^k/I^{k+1}\). By formal etaleness of the coordinate map,
each \(h_i\) defines a local map from \((k+1)S_U\) to \(U\).
Write the overlap errors as
-\begin{equation}\label{hg:coordinate-errors}
+
h_{j,\lambda}-h_{i,\lambda}
=e_{ij,\lambda}y_i^k,
\qquad
y_j-y_i=\eta_{ij}y_i^{k+1}.
-\end{equation}
+
Thus \(D_k(t_\lambda)\) is represented by
\((e_{ij,\lambda}y^k)\), and \(D_k(y)\) by
\((\eta_{ij}y^{k+1})\). The powers of \(y\) record the
@@ -1979,11 +1978,11 @@ notation denotes a frame of the line bundle
it does not invert the nilpotent function \(y_i\) in its
structure sheaf. Realize the dualizing sheaves in support cohomology. Under
the trace inclusion
-\(\omega_{S_U}\hookrightarrow\omega_{(k+1)S_U}\), we have
-\begin{equation}\label{hg:dualizing-errors}
+\(\omega_{S_U}↪\omega_{(k+1)S_U}\), we have
+
b_j-b_i=-(a+k)\eta_{ij}\sigma,\qquad
y_i^kb_i=\sigma,\qquad y_i^{k+1}b_i=0.
-\end{equation}
+
The first identity is the binomial expansion of
\((1+\eta_{ij}y_i^k)^{-(a+k)}\): all quadratic terms vanish
because \(2k\geq k+1\). In the last two identities,
@@ -1991,12 +1990,12 @@ because \(2k\geq k+1\). In the last two identities,
through the trace inclusion. They express Cartier trace and
the annihilator of the thickening.
-\smallskip\noindent\emph{The graph calculation.}
+\smallskip\noindentThe graph calculation.
We next put both errors in the same differential module. Embed
\((k+1)S_U\) as a closed subscheme of a smooth open
\(Z\subset\overline Z=\PP^m\). The graph of the reduced map
\(g\) is closed in \(\overline Z\times U\), since it is proper
-over \(U\). Let \(\iota:S_U\hookrightarrow\overline Z\times U\)
+over \(U\). Let \(\iota:S_U↪\overline Z\times U\)
be this reduced graph embedding, and let \(\mathcal N_\Gamma\)
be the underlying unfiltered differential module of
\(\iota_+A_{0,U}\). This module is distinct from the direct-image
@@ -2008,16 +2007,16 @@ the graph of \(g\), so these sections lie in one support-cohomology
module rather than in different modules for the different lifts.
These sections need not lie in its lowest filtered piece.
We will prove the identity
-\begin{equation}\label{hg:graph-identity}
+
\delta_j(b_j)-\delta_i(b_i)
=
-(a+k)\eta_{ij}\sigma
+\sum_{\lambda=1}^d
- (e_{ij,\lambda}\sigma)\cdot\partial_{t_\lambda}.
-\end{equation}
+ (e_{ij,\lambda}\sigma)\cdot∂_{t_\lambda}.
+
On the right, each reduced dualizing section \(\sigma\) is
included in \(\mathcal N_\Gamma\) by the lowest-piece map of
-Lemma~\ref{hg:lowest-piece}. The right differential-operator
+Lemma~hg:lowest-piece. The right differential-operator
action is taken in this graph module.
The calculation also applies when \(S_U\) is singular. Put
@@ -2035,7 +2034,7 @@ Lift the \(h_{i,\lambda}\) locally to functions on \(Z\).
For a dualizing section \(b\) of the thickening, its graph
inclusion in \(Z\times U\) is the generalized fraction
\[
- \frac{b\,dt_1\wedge\cdots\wedge dt_d}
+ \frac{b\,dt_1\wedge...\wedge dt_d}
{\prod_{\lambda=1}^d(t_\lambda-h_{i,\lambda})}.
\]
It represents successive support cohomology in the additional
@@ -2050,7 +2049,7 @@ the two translations differ nilpotently on every section.
The change from \(h_i\) to \(h_j\) thus has a finite Taylor
expansion on \(b_j\). Again using \(2k\geq k+1\), products
-of two differences from Equation~\eqref{hg:coordinate-errors}
+of two differences from Equation~(hg:coordinate-errors)
annihilate \(b_j\). The linear terms are determined by
\[
(h_{j,\lambda}-h_{i,\lambda})b_j
@@ -2058,19 +2057,19 @@ annihilate \(b_j\). The linear terms are determined by
\]
The right action on a top differential form satisfies
\[
- \left(\frac{dt_\lambda}{t_\lambda-h}\right)
- \cdot\partial_{t_\lambda}
- =\frac{dt_\lambda}{(t_\lambda-h)^2}.
+ ((dt_\lambda)/(t_\lambda-h))
+ \cdot∂_{t_\lambda}
+ =(dt_\lambda)/((t_\lambda-h)^2).
\]
The doubled pole therefore contributes the positive sign in
-Equation~\eqref{hg:graph-identity}. Combining these terms
-with Equation~\eqref{hg:dualizing-errors} proves the identity.
+Equation~(hg:graph-identity). Combining these terms
+with Equation~(hg:dualizing-errors) proves the identity.
So far the calculation is off \(J_U\). Both its sections
and its identities extend meromorphically across \(J_U\) in
-the \emph{unfiltered} localized module, where arbitrary finite
+the unfiltered localized module, where arbitrary finite
pole orders are allowed.
-\smallskip\noindent\emph{Vanishing of the obstruction.}
+\smallskip\noindentVanishing of the obstruction.
Push the graph identity along the projection
\(\pi:\overline Z\times U\to U\), using the relative Spencer
complex. Closed direct image is perverse exact, and
@@ -2084,16 +2083,16 @@ differential. Its Cech difference is a boundary in total
degree one. A sufficiently refined ambient cover around
the closed graph computes this boundary; all the sheaves
in question are supported there. By
-Lemma~\ref{hg:lowest-piece}, the reduced cocycles in
-Equation~\eqref{hg:graph-identity} represent their
+Lemma~hg:lowest-piece, the reduced cocycles in
+Equation~(hg:graph-identity) represent their
\(R^1g_*\mathcal B\) classes in \(F_0M\). Right
differentiation in the \(U\)-coordinates acts on the
pushforward complex. Consequently the Cech classes satisfy
\[
-(a+k)[\eta_{ij}\sigma]
+\sum_{\lambda=1}^d
- [e_{ij,\lambda}\sigma]\cdot\partial_{t_\lambda}=0
- \qquad\text{in }M_U.
+ [e_{ij,\lambda}\sigma]\cdot∂_{t_\lambda}=0
+ \qquadin M_U.
\]
Here each bracketed class belongs to
\(R^1g_*\mathcal B=F_0M_U\); after differentiation, its
@@ -2104,11 +2103,11 @@ module. The inclusion \(F_1M\subset M\)
is an inclusion of actual sheaves, so vanishing in \(M\)
implies vanishing in \(F_1M\). Taking the order-one symbol
shows that the global section \(e\) from
-Equation~\eqref{hg:obstruction-section} is killed by
-\begin{equation}\label{hg:symbol-map}
- T_{\mathcal Y}\otimes F_0M
- \longrightarrow \Gr^F_1M,
-\end{equation}
+Equation~(hg:obstruction-section) is killed by
+
+ T_{\mathcal Y} ⊗ F_0M
+ → \Gr^F_1M,
+
after twisting by \(C^{-(a+k)}\).
Only the relation is used in this filtered step; no bound
on the Hodge filtration of the auxiliary cochain \((\delta_i(b_i))\)
@@ -2118,15 +2117,15 @@ Because \(F_{<0}M=0\), the entire complex at filtration
index one is
\[
\Gr^F_1\DR(M)=
- \bigl[T_{\mathcal Y}\otimes F_0M
- \longrightarrow\Gr^F_1M\bigr],
+ [T_{\mathcal Y} ⊗ F_0M
+ →\Gr^F_1M],
\]
in degrees \(-1\) and \(0\), with differential the symbol
-map in Equation~\eqref{hg:symbol-map}. After the indicated
+map in Equation~(hg:symbol-map). After the indicated
twist, its degree-\(-1\) hypercohomology is therefore exactly
the space of global sections of the kernel of that map.
The twist is negative,
-since \(a+k>0\). Lemma~\ref{hg:negative-vanishing}
+since \(a+k>0\). Lemma~hg:negative-vanishing
annihilates this space, and hence \(e=0\).
It remains to kill the conormal part of the derivation.
@@ -2151,21 +2150,21 @@ every degree, completing the induction and proving
surjectivity of all the sheaf transitions.
The kernels are the stated coherent sheaves
-\(\OO_{\mathcal T_U}\otimes C^{-j-k}\). If \(U\) is
+\(\OO_{\mathcal T_U} ⊗ C^{-j-k}\). If \(U\) is
affine, their first cohomology vanishes, so the exact
sequences of sheaves of abelian groups give surjectivity
on global sections as well. Successive choices of lifts
then give compatible formal lifts of any chosen functions
on \(\mathcal T_U\) and, after trivializing \(C\), of the
transverse conormal frame.
-\end{proof}
-\section{Compact null families and descent}
-\label{sg:descent}
-We now complete Theorem~\ref{thm:signed}. Retain $0<v<n$,
+# Compact null families and descent
+
+
+We now complete Theorem~thm:signed. Retain $0<v<n$,
$r=v-1$, $d=n-1-r=n-v$, and the construction of
-Proposition~\ref{prop:signed-construction}. The formal lifts
+Proposition~prop:signed-construction. The formal lifts
first produce compact subvarieties outside the entire boundary,
of dimension $n-v$, on which $L$ is numerically trivial.
Their existence bounds the dimension of the nef reduction;
@@ -2181,9 +2180,9 @@ the base has not been assumed. The construction below obtains a deformation
directly from the lifted functions, without requiring one ambient scheme
neighborhood for all orders.
-\begin{proposition}\label{prop:null-family}
+
Assume the transition surjectivity of
-Proposition~\ref{prop:formal-lifting}. There is a dominating
+Proposition~prop:formal-lifting. There is a dominating
algebraic family of integral projective subvarieties of $X$
of dimension
\[
@@ -2191,14 +2190,14 @@ of dimension
\]
whose general members avoid $D$ and on which $L$ is
numerically trivial.
-\end{proposition}
-\begin{proof}
+
+Proof.
We construct a formal deformation of a general boundary fiber,
algebraize it over $\C[[s]]$, and show that the resulting
subvarieties belong to a family dominating $X$.
-\smallskip\noindent\emph{Deforming a boundary fiber.}
+\smallskip\noindentDeforming a boundary fiber.
Choose a separated affine etale chart $U\to\mathcal Y$
near a general point of a top-dimensional component of its
Stein image $\mathcal T_U$. After shrinking, $C$ is trivial,
@@ -2207,7 +2206,7 @@ is flat over it. Since the image of $J$ in $P$ has dimension
less than $r$, we may also arrange that $S_U$ is disjoint
from $J_U$. Choose a closed point $t$ in this open set.
After a further shrinking, regular parameters
-$t_1,\ldots,t_r$ on $\mathcal T_U$ cut out only $t$.
+$t_1,...,t_r$ on $\mathcal T_U$ cut out only $t$.
The fiber
\[
F=S_{U,t}
@@ -2215,7 +2214,7 @@ The fiber
is projective and has pure dimension $d$. Along $F$, the
pullbacks of the parameters form a regular sequence. The ideal
sheaf $\mathcal I_{J/S}$ is maximal Cohen--Macaulay by
-Proposition~\ref{prop:signed-construction}(ii). Since $S_U$ is
+Proposition~prop:signed-construction(ii). Since $S_U$ is
disjoint from $J_U$, the etale pullback of this ideal sheaf is
$\mathcal I_{J_U/S_U}=\OO_{S_U}$; hence $S_U$ is
Cohen--Macaulay. The quotient $F$ by the regular sequence is
@@ -2223,10 +2222,10 @@ Cohen--Macaulay as well. We will use this property to control
the dimensions of the components after algebraization.
Write $I=\OO_{\mathcal X}(-S)$. By
-Proposition~\ref{prop:formal-lifting}, the transitions between
+Proposition~prop:formal-lifting, the transitions between
the sheaves $g_*(I^j/I^{j+k})$ are surjective as the length
$k$ increases. Their kernels are the coherent sheaves
-$g_*\OO_{S_U}\otimes C^{-j-k}$. Affineness of $U$ also gives
+$g_*\OO_{S_U} ⊗ C^{-j-k}$. Affineness of $U$ also gives
surjectivity on global sections. We can therefore lift the
parameters compatibly through all thickenings, and lift the
chosen conormal frame to a compatible element $\widetilde y$
@@ -2235,15 +2234,15 @@ that formally generates $I$.
Let $Z_q$ be the closed subscheme of $qS_U$ cut out by the
$r$ lifted parameters. Regard it as a scheme over
\[
- R_q=\C[s]/(s^q),\qquad s\longmapsto\widetilde y.
+ R_q=\C[s]/(s^q),\qquad s⟼\widetilde y.
\]
The construction has the diagram
\[
-\begin{tikzcd}[column sep=large]
+[column sep=large]
F \arrow[r,hook] \arrow[d] & Z_q \arrow[r] \arrow[d] &
X\times\Spec R_q \arrow[d] \\
\Spec\C \arrow[r,hook] & \Spec R_q \arrow[r,equal] & \Spec R_q.
-\end{tikzcd}
+
\]
The middle vertical map is defined by the lifted normal parameter.
The lifted base parameters cut out $Z_q$; they are not provided by a
@@ -2257,7 +2256,7 @@ proves flatness before imposing the parameters. The local
flatness criterion then preserves flatness when we quotient
by lifts of the regular sequence on the closed fiber.
-\smallskip\noindent\emph{Algebraization and boundary avoidance.}
+\smallskip\noindentAlgebraization and boundary avoidance.
The map $Z_q\to X\times\Spec R_q$ is quasi-finite. It is
also proper: its reduction is the map from the projective
scheme $F$, and nilpotent thickenings do not change
@@ -2270,7 +2269,7 @@ their multiplication and unit
\cite[Lemmas~30.24.1 and~30.24.3]{StacksExistence}.
The relative spectrum gives a finite morphism
\[
- Z\longrightarrow X_R
+ Z→ X_R
\]
with exactly the prescribed reductions. In particular,
$Z$ is projective over $R$. Formal flatness gives
@@ -2283,7 +2282,7 @@ proper over $R$, and must therefore be empty: any nonempty
closed subset of a proper $R$-scheme specializes to the
closed fiber. Recall also the exceptional divisor $E$ of the
normalized blowup separating $S_+$ and $S_-$ in
-Proposition~\ref{prop:signed-construction}. Its image in $X$
+Proposition~prop:signed-construction. Its image in $X$
lies in $\Supp G_+\cap\Supp G_-$, so it is already excluded
by avoidance of the pole part.
@@ -2320,7 +2319,7 @@ has pure dimension $d$. After a suitable finite extension of
$\C((s))$, choose a geometrically integral component of the
reduced base change of this image.
-\smallskip\noindent\emph{A family dominating $X$.}
+\smallskip\noindentA family dominating $X$.
We have constructed compact subvarieties outside $D$.
To obtain a dominating family, choose a positive component
$D_i$ with image dimension $r$, together with a component
@@ -2370,16 +2369,16 @@ dominates $X$. On each member, avoidance of $D$ makes
the rational section $s_0$ regular and nowhere zero.
It trivializes $N_0$, proving that $L$ is numerically
trivial on that member.
-\end{proof}
-\subsection{Descent along the nef reduction}
+
+## Descent along the nef reduction
The compact null subvarieties will bound the dimension of the
nef-reduction base. To descend the signed representative to
that base, we first prove a statement for vertical divisors
using relative nefness and a fiber-dimension bound.
-\begin{lemma}\label{sg:vertical-descent}
+
Let $f:Y\to B$ be a surjective projective morphism from a
normal integral variety to a smooth projective variety,
with geometrically connected integral generic fiber.
@@ -2390,9 +2389,9 @@ divisor that is relatively nef. Then
G_Y=f^*G_B
\]
for a $\Q$-divisor $G_B$ on $B$.
-\end{lemma}
-\begin{proof}
+
+Proof.
When $B$ is a point, every vertical divisor is zero.
When the relative dimension is zero, the fiber bound
makes $f$ finite. Its geometrically integral connected
@@ -2445,13 +2444,13 @@ part over the chosen point is thus a multiple of the
full fiber. Its missing component forces that multiple
to be zero, contradicting the retained component of
its support.
-\end{proof}
-\begin{proof}[Completion of the proof of Theorem~\ref{thm:signed}]
+
+Proof. [Completion of the proof of Theorem~thm:signed]
The cases $v=0$ and $v=n$ are covered by
-Lemma~\ref{sg:numerical-boundary}. For $0<v<n$,
-Propositions~\ref{prop:signed-construction},
-\ref{prop:formal-lifting}, and~\ref{prop:null-family}
+Lemma~sg:numerical-boundary. For $0<v<n$,
+Propositions~prop:signed-construction,
+prop:formal-lifting, and~prop:null-family
give a dominating family of compact $L$-trivial
subvarieties of dimension $d=n-v$.
@@ -2474,10 +2473,10 @@ the main component over a modification of the base,
resolve the base, and normalize the main transform.
This gives projective morphisms
\[
- \begin{tikzcd}
+
Y \arrow[r,"\pi"] \arrow[d,"f"'] & X\\
B &
- \end{tikzcd}
+
\]
with $\pi$ birational, $B$ smooth, $Y$ normal, and
geometrically connected integral generic fiber of $f$.
@@ -2510,7 +2509,7 @@ $b=0$, which would force $D=0$ and contradict $v>0$.
It follows that $G_Y=\pi^*G$ is vertical. It is also
relatively nef because $G_Y\sim_{\Q}\pi^*L$.
-Lemma~\ref{sg:vertical-descent} now gives
+Lemma~sg:vertical-descent now gives
\[
\pi^*L\sim_{\Q}f^*G_B
\]
@@ -2528,19 +2527,19 @@ that $G_B^b>0$, so $G_B$ is big. Pullback of
sections then gives $\kappa(X,L)\ge b=v$. The
inequality $\kappa(X,L)\le\nu(X,L)$ for nef divisors
provides the reverse bound and proves abundance.
-\end{proof}
-\section{Geometric exclusions for a nonvanishing counterexample}
-\label{sec:exclusions}
+
+# Geometric exclusions for a nonvanishing counterexample
+
We now prepare the smooth nonvanishing step. Assume only the
lower-dimensional good-model hypothesis of
-Assumption~\ref{mmp:lower-models}, and suppose that a smooth
+Assumption~mmp:lower-models, and suppose that a smooth
projective \(n\)-fold \(X\), with \(n>0\), satisfies
-\begin{equation}\label{ex:counterexample}
- K_X\ \text{is pseudo-effective},
- \qquad \kappa(X,K_X)=-\infty.
-\end{equation}
+
+ K_X\ is pseudo-effective,
+ \qquad \kappa(X,K_X)=-∞.
+
We first restrict its moving proper subvarieties and positive canonical
currents, and then exclude covering curves of bounded genus and normalized
degree. Two further conclusions will handle the remaining jet
@@ -2551,9 +2550,9 @@ place over \(\C\), and the two displayed properties persist on smooth
projective birational models. Numerical dimension for a
pseudo-effective divisor means Nakayama's numerical dimension
\(\kappa_\sigma\); for a nef divisor it agrees with the intersection
-definition used in Theorem~\ref{thm:signed}.
+definition used in Theorem~thm:signed.
-\subsection{The Albanese reduction and algebraic webs}
+## The Albanese reduction and algebraic webs
Subadditivity first shows that the Albanese map is constant. This
will let us pass from an effective numerical representative of a
@@ -2561,13 +2560,13 @@ rational divisor to an effective rational-linear representative.
We then construct the rational quotient generated by a moving
family of subvarieties.
-\begin{lemma}\label{ex:irregularity}
+
Every smooth projective birational model of \(X\) has irregularity zero.
Consequently, on such a model, or on a projective \(\Q\)-factorial
terminal birational model, numerical equivalence of rational divisors
implies their \(\Q\)-linear equivalence.
-\end{lemma}
-\begin{proof}
+
+Proof.
If \(q(X)>0\), resolve the Stein factorization of the Albanese map to
obtain a morphism \(f:W\to B\) with connected fibers, with \(W,B\)
smooth projective and \(\dim B>0\). The map from \(B\) to a subvariety
@@ -2583,18 +2582,18 @@ effective approximations with arbitrarily small ample error.
Adjunction identifies this restriction with \(K_F\). If \(F\) has
positive dimension, the induction hypothesis gives
\(\kappa(F,K_F)\geq0\); for a point the same assertion holds by
-convention. Applying Theorem~\ref{asm:iitaka} with both boundaries
+convention. Applying Theorem~asm:iitaka with both boundaries
empty gives \(\kappa(W,K_W)\geq0\), contradicting
-\eqref{ex:counterexample}.
+(ex:counterexample).
-This is the only use of Theorem~\ref{asm:iitaka} in the proof.
+This is the only use of Theorem~asm:iitaka in the proof.
Irregularity is a smooth birational invariant. Finally, when
\(\Pic^0=0\), a numerically trivial line bundle is torsion, since the
group of numerically trivial line bundles modulo \(\Pic^0\) is
finite. Clear denominators for rational divisors. On a terminal
\(\Q\)-factorial model, pull back to a smooth resolution and then
descend the rational linear equivalence.
-\end{proof}
+
To handle subvarieties through very general points, parameterize
them in countably many Hilbert families, then resolve and stratify.
@@ -2613,7 +2612,7 @@ dimension but is not of general type, the smooth general fibers
of this map have positive dimension. They form the new family
used below.
-\begin{lemma}[Algebraic web quotient]\label{ex:web}
+[Algebraic web quotient]
Let a smooth projective variety be dominated generically finitely by
the total space of a family with smooth integral projective general
fibers. On a dense regular open, form the distribution generated by
@@ -2621,8 +2620,8 @@ the tangent spaces of the fiber images, taking spans, saturation,
and Lie brackets. Its general leaves are dense opens of algebraic
subvarieties. Their closures are the general fibers of a rational
map, which has connected general fiber after Stein factorization.
-\end{lemma}
-\begin{proof}
+
+Proof.
Shrink to an open \(U\) where the evaluation has finitely many etale
sheets, the parameter map is smooth, and the generated involutive
distribution \(\mathcal F\) has constant rank. Its construction is
@@ -2673,7 +2672,7 @@ then everywhere by closure. A leaf closure of dimension \(\rk\mathcal F\) is
therefore unique through a general point. Assigning that closure
gives the desired rational map. Resolve its graph and take the
Stein factorization.
-\end{proof}
+
For positive closed \((1,1)\)-currents we shall use this quotient in
the following way. If a current vanishes on almost every parameter
@@ -2692,19 +2691,19 @@ brackets by the identities
Pullbacks by dominant morphisms and restrictions to almost every
smooth parameter fiber are defined by the same local potentials.
-\subsection{The general-type exclusion}
+## The general-type exclusion
The next proposition turns the lower-dimensional good-model hypothesis
into a restriction on every moving proper subvariety. Its proof passes
from Iitaka fibers to the algebraic quotient of their tangent
directions and then back to nonvanishing.
-\begin{proposition}\label{prop:general-type}
+
Every positive-dimensional proper subvariety through a very general
point of \(X\) is of general type on resolution. The assertion holds
also on every projective birational model.
-\end{proposition}
-\begin{proof}
+
+Proof.
Otherwise take a covering family of nongeneral-type subvarieties,
with smooth domains \(V\), and slice its parameters as above.
Restricting the ramification formula of its generically finite
@@ -2752,17 +2751,17 @@ It follows that \(T\) annihilates the web distribution of the
is supported on a proper algebraic set. The support theorem for
positive closed \((1,1)\)-currents expresses it as a finite
nonnegative real combination of prime divisors
-\cite{Siu1974,Demailly2012}. Its rational cohomology class then has
+[cite]. Its rational cohomology class then has
a nonnegative rational representative: solve the finite rational
linear system for the coefficients on the same face of the
-nonnegative orthant. Lemma~\ref{ex:irregularity} converts numerical
+nonnegative orthant. Lemma~ex:irregularity converts numerical
effectivity to \(\Q\)-linear effectivity, a contradiction.
-Otherwise Lemma~\ref{ex:web} gives a rational quotient with general
+Otherwise Lemma~ex:web gives a rational quotient with general
smooth fiber \(F\) on a smooth resolved graph, where
\(0<\dim F<n\). Its canonical divisor is pseudo-effective:
restrict the pseudo-effective canonical class of the smooth graph
-to a very general fiber, as in Lemma~\ref{ex:irregularity}.
+to a very general fiber, as in Lemma~ex:irregularity.
The moving \(G\)'s still generate its tangent space at general
points. Indeed the quotient is constant on every general web
member, hence induces a rational map on their parameter space;
@@ -2795,23 +2794,23 @@ corrected convention discussed in
\cite[Section~3 and Theorem~3.2]{Fujino2019Numerical},
causes no issue. The induction hypothesis supplies the base good
model, hence nonvanishing upstairs, contradicting
-\eqref{ex:counterexample}.
-\end{proof}
+(ex:counterexample).
+
+
-\begin{corollary}\label{ex:supporting}
Let \(Y\) be a projective \(\Q\)-factorial terminal birational model
of \(X\). If a rational divisor \(P\) has \(\kappa(Y,P)\geq1\),
then \(K_Y+cP\) is big for every rational \(c>0\).
Consequently, if \(\alpha\neq0\) is a supporting functional of the
pseudo-effective cone with \(\alpha(K_Y)=0\), then
\(\alpha(P)>0\).
-\end{corollary}
-\begin{proof}
+
+Proof.
Resolve a moving subsystem of a multiple of \(P\). If its map is
generically finite, \(P\) is big and the assertion follows from
pseudoeffectivity of \(K_Y\). Otherwise its general fiber is a
proper positive-dimensional subvariety through very general
-points and is of general type by Proposition~\ref{prop:general-type}.
+points and is of general type by Proposition~prop:general-type.
The canonical divisor of the smooth resolution is relatively big,
so adding a sufficiently large ample pullback from the image makes
it big. Since that pullback is bounded by a multiple of the
@@ -2821,9 +2820,9 @@ addition of the pseudo-effective \(P\), gives the assertion for
every \(c>0\). A nonzero nonnegative functional on a closed convex
cone is strictly positive on its interior. Applying it to
\(K_Y+cP\) proves the last assertion.
-\end{proof}
-\subsection{Valuations and positive currents}
+
+## Valuations and positive currents
The next two exclusions involve currents whose numerical intersection
gaps tend to zero. A limiting inequality alone is insufficient: we
@@ -2836,29 +2835,29 @@ its pullback at a divisorial valuation \(E\). The normalization is
such that a reduced smooth divisor has generic number one. We
write \(A_V(E)=a(E;V,0)\) for log discrepancy.
-\begin{lemma}\label{ex:valuation-acc}
+
For a fixed \(T\) and a fixed \(M\), the set
\[
\{\nu_E(T): A_V(E)\leq M\}
\]
satisfies the ascending chain condition.
-\end{lemma}
-\begin{proof}
+
+Proof.
We induct on the integer part of \(M\); discrepancies over a
smooth variety are positive integers, so the assertion is empty
when \(M<1\). Suppose that a strictly
increasing sequence exists, and discard initial terms so its
members exceed a fixed positive number \(c\).
Skoda integrability and change of variables give
-\begin{equation}\label{ex:skoda-valuation}
+
\nu_E(T)\leq A_V(E)\nu_x(T)
-\end{equation}
+
at a general point \(x\) of the center. Indeed every exponent
smaller than \(1/\nu_x(T)\) is locally integrable, whereas
integrability after pullback imposes the corresponding
discrepancy bound. Hence all centers lie in the fixed proper Siu
locus \(\{\nu_x(T)\geq c/M\}\), which is algebraic by Siu
-analyticity and projectivity \cite{Siu1974}. Passing to a
+analyticity and projectivity [cite]. Passing to a
subsequence, all centers lie in one irreducible component of
this locus.
@@ -2875,14 +2874,14 @@ Siu locus. Write \(T=c_D[D]+T'\) with \(T'\) positive and with
zero generic Lelong number along \(D\). The integers
\(\ord_E(D)\) are bounded: the fixed effective divisor \(D\) has
positive log canonical threshold, and
-\(\operatorname{lct}(D)\ord_E(D)\leq A_V(E)\).
+\(lct(D)\ord_E(D)\leq A_V(E)\).
Pass to a constant order. The residual numbers \(\nu_E(T')\)
are still strictly increasing, and after discarding a term have
-a positive lower bound. Equation~\eqref{ex:skoda-valuation}
+a positive lower bound. Equation~(ex:skoda-valuation)
places their centers in a fixed Siu locus of \(T'\), which does
not contain \(D\). Their intersection with \(D\) has codimension
at least two, so the preceding case applies.
-\end{proof}
+
To compare those Lelong numbers with movable algebraic systems, use
the following multiplier-ideal approximation
@@ -2890,28 +2889,28 @@ the following multiplier-ideal approximation
is a real algebraic class, one can choose integral line bundles
\(L_k\) such that \(c_1(L_k)-k\{T\}\) stays in a bounded,
uniformly sufficiently ample set and
-\(L_k\otimes\mathcal J(kT)\) is globally generated. Round the
+\(L_k ⊗ \mathcal J(kT)\) is globally generated. Round the
coefficients of \(k\{T\}\) in a fixed integral basis and add a
fixed sufficiently positive integral class. Nadel vanishing and
Castelnuovo--Mumford regularity give generation. For every
divisorial valuation,
-\begin{equation}\label{ex:multiplier-order}
+
\ord_E\mathcal J(kT)\leq k\nu_E(T).
-\end{equation}
+
The local Bergman weight is bounded below by the original weight
up to a constant; this inequality persists after pullback and
-gives \eqref{ex:multiplier-order}. These are statements for a
+gives (ex:multiplier-order). These are statements for a
fixed current, not uniform assertions over all currents.
-\begin{proposition}\label{prop:no-pullback}
+
Let \(Y\) be a fixed projective \(\Q\)-factorial terminal
birational model of \(X\), with \(K_Y\) nonbig. A positive current
in the pullback class of \(K_Y\) on a smooth resolution cannot,
on a dense regular Zariski open, annihilate the tangent spaces
of the general fibers of a rational fibration of positive
relative dimension.
-\end{proposition}
-\begin{proof}
+
+Proof.
Fix a current \(T\) as in the statement and suppose such a fibration
exists. Consider all dominant rational maps from \(Y\) with
connected general fiber of positive dimension for which the
@@ -2923,14 +2922,14 @@ vanishes. Their quotient will give a member of the same collection
with smaller base dimension.
If \(b=0\), the current vanishes on
a dense open; the divisorial-current argument in
-Proposition~\ref{prop:general-type} contradicts
-\eqref{ex:counterexample}. Thus \(0<b<n\).
+Proposition~prop:general-type contradicts
+(ex:counterexample). Thus \(0<b<n\).
Resolve the graph and the space carrying the current:
\[
- \begin{tikzcd}
+
W \arrow[r,"p"] \arrow[d,"h"'] & Y \\
B &
- \end{tikzcd}
+
\]
Here \(W,B\) are smooth projective and \(h\) has connected
general fiber. All further modifications replace this graph and
@@ -2968,7 +2967,7 @@ have center of codimension at least two on \(B\), contrary
to the property just established.
\smallskip
-\noindent\emph{Descent of the current.}
+\noindentDescent of the current.
Let \(T\) also denote the pulled-back current on \(W\).
Shrink to a smooth open \(B^0\) so that \(h\) is smooth
proper, the removed set contains no vertical divisor
@@ -3002,13 +3001,13 @@ Its class is real algebraic, since \(\{T'\}\) is the
algebraic class \(p^*K_Y\) minus real divisor classes and
algebraic intersection and pushforward preserve such
classes. Remove its divisorial parts along
-\(B\setminus B^0\), and call the result \(S\).
+\(B∖ B^0\), and call the result \(S\).
On a dense open \(T=h^*S\). Their difference is again closed
of order zero, so the same support theorem leaves only a
signed divisor sum. Over \(B^0\), this difference consists
only of the horizontal components removed above. Its coefficient
at a prime dominating a base prime that meets \(B^0\) is therefore
-zero. Along a base prime contained in \(B\setminus B^0\),
+zero. Along a base prime contained in \(B∖ B^0\),
the current \(S\) has zero generic Lelong number by construction.
At a prime upstairs dominating it, the generic local map is a
transverse power map times a submersion. Local target balls of
@@ -3024,14 +3023,14 @@ real algebraic class on \(B\), with \(T-h^*S\) a divisor current
whose only possible negative coefficients are \(p\)-exceptional.
Push this difference forward by \(p\). The exceptional terms
disappear and the remaining terms are effective, so
-\begin{equation}\label{ex:descent-domination}
- K_Y-T_B\{S\}\quad\text{is pseudo-effective}.
-\end{equation}
+
+ K_Y-T_B\{S\}\quadis pseudo-effective.
+
The horizontal components removed during descent contribute
effective divisors to the same difference.
\smallskip
-\noindent\emph{A negative canonical direction on the base.}
+\noindentA negative canonical direction on the base.
Choose a nonzero supporting functional \(\alpha\) of the
pseudo-effective cone at \(K_Y\), and put
\(\eta=T_B^*\alpha\). The corresponding functionals \(\eta'\)
@@ -3039,7 +3038,7 @@ on higher smooth base models are movable classes by
pseudo-effective/movable duality \cite[Theorem~0.2]{BDPP2013}.
They annihilate base-exceptional divisors. Since \(S\) is positive,
\(\eta\cdot\{S\}\geq0\). Applying \(\alpha\) to
-\eqref{ex:descent-domination} gives the reverse inequality,
+(ex:descent-domination) gives the reverse inequality,
because \(\alpha(K_Y)=0\). Hence \(\eta\cdot\{S\}=0\).
If \(c_D[D]\) is a positive Siu component of \(S\), both
\(S-c_D[D]\) and \([D]\) are positive. It follows that
@@ -3062,18 +3061,18 @@ both positive and negative coefficients: a nonzero effective
sum has positive intersection with an ample \((n-1)\)-fold
product on \(Y\), so cannot be numerically zero. The two sides
are thus nonzero effective rational divisors with no common
-prime component. Lemma~\ref{ex:irregularity} makes them
+prime component. Lemma~ex:irregularity makes them
\(\Q\)-linearly equivalent. Clearing denominators gives two
distinct linearly equivalent effective integral divisors and
hence a pencil, whose class is killed by \(\alpha\). This
-contradicts Corollary~\ref{ex:supporting}.
+contradicts Corollary~ex:supporting.
We claim that
-\begin{equation}\label{ex:negative-base}
+
K_B\cdot\eta<0.
-\end{equation}
+
The smooth general fiber \(F\) of \(h\) is of general type by
-Proposition~\ref{prop:general-type}. The relative-positivity
+Proposition~prop:general-type. The relative-positivity
theorem of Kov\'acs--Patakfalvi
\cite[Theorem~9.9]{KovacsPatakfalvi2017} applies to a log
canonical fiber space with smooth base, SNC total pair, and
@@ -3083,30 +3082,30 @@ subadditivity with the term \(h^*M\). Here both spaces are
smooth projective, the boundary is zero, and we take \(M=0\).
Invariance of plurigenera on the smooth locus identifies the
generic-fiber Kodaira dimension with that of \(F\). Thus
-\begin{equation}\label{ex:relative-positive}
+
\kappa(W,K_W-h^*K_B)\geq \kappa(F,K_F)=n-b>0.
-\end{equation}
+
This is an established general-type-fiber theorem, not another
-use or strengthening of Theorem~\ref{asm:iitaka}.
+use or strengthening of Theorem~asm:iitaka.
Choose compatible canonical divisors and set
\(P=K_Y-T_B(K_B)=p_*(K_W-h^*K_B)\).
For every sufficiently divisible \(m\), pushing forward an
effective divisor
-\(\operatorname{div}_W(\varphi)+m(K_W-h^*K_B)\)
+\(div_W(\varphi)+m(K_W-h^*K_B)\)
gives
-\(\operatorname{div}_Y(\varphi)+mP\geq0\).
+\(div_Y(\varphi)+mP\geq0\).
The resulting injection of sections preserves their ratios,
-so \(\kappa(Y,P)\geq1\). Corollary~\ref{ex:supporting} now gives
+so \(\kappa(Y,P)\geq1\). Corollary~ex:supporting now gives
\(0<\alpha(P)=-K_B\cdot\eta\), proving
-\eqref{ex:negative-base}.
+(ex:negative-base).
\smallskip
-\noindent\emph{Rational curves and an exact zero gap.}
+\noindentRational curves and an exact zero gap.
Apply the multiplier approximation to \(S\) on \(B\), and
principalize \(\mathcal J(kS)\) by \(r_k:B_k\to B\).
If \(F_k\) is its divisor, the class
\[
- P_k=\frac{r_k^*L_k-F_k}{k}
+ P_k=(r_k^*L_k-F_k)/(k)
\]
is nef. Put \(\eta_k=T_{B_k}^*\alpha\). The class \(\eta_k\)
is movable and \(F_k\) is effective, so
@@ -3121,9 +3120,9 @@ uses the fact that \(\eta_k\) kills exceptional divisors:
Fix an ample divisor \(H\) on \(B\), and write
\(\delta=-K_B\cdot\eta>0\). On \(B_k\), choose an ample
divisor \(H_k\) and a sufficiently small rational
-\(\epsilon_k>0\) so that the ample rational class
+\(ε_k>0\) so that the ample rational class
\[
- Q_k=P_k+k^{-1}r_k^*H+\epsilon_k H_k
+ Q_k=P_k+k^{-1}r_k^*H+ε_k H_k
\]
satisfies \(Q_k\cdot\eta_k=O(k^{-1})\). This is possible
because \(\eta_k\cdot r_k^*H=\eta\cdot H\) is fixed.
@@ -3143,7 +3142,7 @@ average of the ratios therefore gives one covering curve
\(\Gamma_k\) with
\[
\frac{Q_k\cdot\Gamma_k}{-K_{B_k}\cdot\Gamma_k}
- \leq\frac{2C}{\delta k}.
+ \leq(2C)/(\delta k).
\]
The quantitative bend-and-break estimate
\cite[Theorem~5]{MiyaokaMori1986} gives a rational curve
@@ -3152,10 +3151,10 @@ at most \(2\dim B\) times this ratio. Parameterization and
uncountability give a covering family of such rational curves,
denoted \(R_k\). Since \(Q_k-P_k-k^{-1}r_k^*H\) is ample
and both \(P_k\) and \(r_k^*H\) are nef, we obtain
-\begin{equation}\label{ex:small-curves}
+
P_k\cdot R_k=O(k^{-1}),\qquad
H\cdot r_{k*}R_k=O(1).
-\end{equation}
+
Shrink each parameter space so that the curve class and its
contacts with the exceptional divisors and the strict transforms
of the Siu divisors of \(S\) are constant. The general parametrized
@@ -3163,10 +3162,10 @@ member is free in characteristic zero. Consequently
\(K_{B_k}\cdot R_k\leq-2\). Put \(\bar R_k=r_{k*}R_k\).
Its bounded \(H\)-degree bounds \(K_B\cdot\bar R_k\), and
effectivity of the relative canonical divisor gives
-\begin{equation}\label{ex:contact-bound}
+
0\leq K_{B_k/B}\cdot R_k
=K_{B_k}\cdot R_k-K_B\cdot r_{k*}R_k=O(1).
-\end{equation}
+
In particular, the upper bound is independent of \(k\).
If an exceptional prime \(E\) has coefficient
\(a_E=A_B(E)-1\) in \(K_{B_k/B}\), a positive contact with
@@ -3180,23 +3179,23 @@ fixed degrees against \(\bar R_k\). Subtracting all these
divisorial parts from \(r_k^*S\) leaves a positive current.
Its restriction to almost every member of the family is
defined and positive, by the local-potential and Fubini
-argument following Lemma~\ref{ex:web}. Its degree is the
+argument following Lemma~ex:web. Its degree is the
first difference below; this proves that difference is
-nonnegative. Using \eqref{ex:multiplier-order} and
-\eqref{ex:small-curves} gives
-\begin{equation}\label{ex:zero-gap}
- \begin{split}
+nonnegative. Using (ex:multiplier-order) and
+(ex:small-curves) gives
+
+
0&\leq \{S\}\cdot\bar R_k
-\sum_E\nu_E(S)\,E\cdot R_k\\
&\leq \{S\}\cdot\bar R_k-(F_k/k)\cdot R_k
\leq O(k^{-1}).
- \end{split}
-\end{equation}
+
+
The sum includes exceptional primes and those strict
divisorial components; terms with zero contact are irrelevant.
For the last bound, the expression involving \(F_k\) equals
\(P_k\cdot R_k+(\{S\}-L_k/k)\cdot\bar R_k\).
-The first term is \(O(k^{-1})\) by \eqref{ex:small-curves};
+The first term is \(O(k^{-1})\) by (ex:small-curves);
the second has the same bound because \(L_k-k\{S\}\) stays
bounded and the \(H\)-degrees of \(\bar R_k\) are bounded.
@@ -3204,14 +3203,14 @@ Bounded ample degree gives only finitely many numerical
classes of the integral cycles \(\bar R_k\). Pass to one
class, so \(\{S\}\cdot\bar R_k\) is now one fixed real
number. The current \(S\) on \(B\) also remains fixed. By
-Lemma~\ref{ex:valuation-acc}, the sums in
-\eqref{ex:zero-gap} belong to an ACC set: there are a bounded
+Lemma~ex:valuation-acc, the sums in
+(ex:zero-gap) belong to an ACC set: there are a bounded
number of terms, their nonnegative integral multiplicities
are bounded, and all relevant discrepancies are bounded.
Finite sums of this kind preserve ACC, by successively
taking nonincreasing subsequences of their entries.
We can now replace the limiting gap by an exact equality: the
-nonnegative gaps in \eqref{ex:zero-gap} must equal zero for some
+nonnegative gaps in (ex:zero-gap) must equal zero for some
covering families. Otherwise a sequence
tending to zero has a strictly decreasing positive
subsequence, giving a strictly increasing sequence of the
@@ -3223,7 +3222,7 @@ complete curve is zero. A positive current on a curve is a
positive measure, so zero integral forces it to vanish.
Off the removed divisors the residual current agrees with
\(r_k^*S\).
-The discussion following Lemma~\ref{ex:web} and that lemma
+The discussion following Lemma~ex:web and that lemma
itself yield a positive-relative-dimensional rational
quotient of \(B\) whose vertical directions are annihilated
by \(S\) on an open. Composing with \(h\) and taking the
@@ -3231,9 +3230,9 @@ Stein factorization gives a quotient of smaller base dimension
whose vertical directions are annihilated by \(T\). It belongs
to the collection minimized at the start, contradicting the
choice of \(b\).
-\end{proof}
-\subsection{Excluding normalized bounded curves}
+
+## Excluding normalized bounded curves
We apply the same exact-zero mechanism to curves on the small
modifications of a fixed late model. The normalization by volume is
@@ -3244,22 +3243,22 @@ Run a canonical LMMP on \(X\) with scaling of a fixed very ample
rational divisor \(A\). For rational \(t>0\), its positive-threshold
stages give terminal \(\Q\)-factorial models \(Y_t\) on which
\(K_t+tA_t\) is nef, big, and semiample
-\cite{BCHM2010}. There are only finitely many divisorial
+[cite]. There are only finitely many divisorial
contractions, since each lowers Picard number. Fix a late model
\(Y\) after the last such contraction. All subsequent maps
\(Y\dashrightarrow Y_t\) are small, and are canonical
nonpositive birational maps. If the program terminates, use
the last model repeatedly. Define
-\begin{equation}\label{ex:volume-scale}
+
\mu_t=\vol(K_X+tA)^{1/n}.
-\end{equation}
+
Since \(K_X\) is not big, \(\mu_t\to0\). In particular \(K_Y\)
is nonbig. The transform \(A_t\) is effective up to rational
linear equivalence; all intersections below use general
covering curves, which are not contained in a chosen such
representative.
-\begin{lemma}[Exceptional current comparison]\label{ex:current-comparison}
+[Exceptional current comparison]
Fix a positive current \(T\) on a smooth resolution in the
pullback class of \(K_Y\). On a common smooth resolution of
\(Y\dashrightarrow Y_t\), write
@@ -3267,16 +3266,16 @@ pullback class of \(K_Y\). On a common smooth resolution of
p^*K_Y=p_t^*K_t+J_t,\qquad J_t\geq0,
\]
where \(J_t\) is exceptional over \(Y_t\).
-Then \(\nu_E(T)\geq\operatorname{coeff}_E J_t\) for every
+Then \(\nu_E(T)\geqcoeff_E J_t\) for every
prime on that resolution.
-\end{lemma}
-\begin{proof}
+
+Proof.
Pass to a common smooth model \(V\) also dominating the
fixed resolution carrying \(T\). Choose on the fixed
resolution a sufficiently positive line bundle \(H\).
For sufficiently divisible \(k\), multiplier approximation
gives a globally generated
-\(\OO(kp^*K_Y+H)\otimes\mathcal J(kT)\).
+\(\OO(kp^*K_Y+H) ⊗ \mathcal J(kT)\).
Pull this system to \(V\). For the finite collection of primes
on the common resolution whose coefficients we are comparing,
choose a general member \(D_k\) of the underlying effective
@@ -3301,20 +3300,20 @@ numerically trivial and exceptional, so the negativity lemma
applied with both signs makes it zero. Since
\(p_t^*D'_k\) is effective,
\[
- \operatorname{coeff}_E D_k
- \geq k\operatorname{coeff}_E J_t
- +\operatorname{coeff}_E(H-p_t^*H_t).
+ coeff_E D_k
+ \geq kcoeff_E J_t
+ +coeff_E(H-p_t^*H_t).
\]
The last coefficient is fixed for this model and \(t\).
-Combining with \eqref{ex:multiplier-order}, dividing by
-\(k\), and letting \(k\to\infty\) proves the assertion.
+Combining with (ex:multiplier-order), dividing by
+\(k\), and letting \(k\to∞\) proves the assertion.
The same proof can be made on each common model, so the
comparison applies to all divisorial valuations needed
later. No uniformity in \(t\) of the fixed error is required.
-\end{proof}
-\begin{proposition}\label{prop:bounded-curves}
-There are no sequences \(t_j\downarrow0\) and covering
+
+
+There are no sequences \(t_j↓0\) and covering
families of curves on \(Y_{t_j}\), with smooth projective
domains \(C_j\), for which both
\[
@@ -3324,8 +3323,8 @@ domains \(C_j\), for which both
are bounded by a fixed constant. Degrees mean degrees on
the domains, or equivalently intersection with their
pushforward cycles.
-\end{proposition}
-\begin{proof}
+
+Proof.
Suppose such families exist. Fix once and for all a smooth
resolution \(p:W\to Y\) and a positive current \(T\) in
the class \(p^*K_Y\); pseudoeffectivity supplies this current.
@@ -3354,10 +3353,10 @@ For a prime \(Z\) of \(U_t\) exceptional over \(Y\), the
restricted valuation of its function field is
\(r_Z\ord_E\) for a divisorial valuation \(E\) over \(Y\).
The coefficient of \(Z\) in the ramification difference is
-\begin{equation}\label{ex:ramification-cost}
- \operatorname{coeff}_Z(K_{U_t}-q_t^*K_Y)
+
+ coeff_Z(K_{U_t}-q_t^*K_Y)
=r_Za(E;Y,0)-1.
-\end{equation}
+
This follows by factoring through a model extracting \(E\)
and calculating the tame ramification of DVRs. It is
positive and bounded away from zero: \(Y\) is terminal,
@@ -3378,7 +3377,7 @@ the covering property. Thus
Each exceptional contact contributes
\((r_Za(E;Y,0)-1)(Z\cdot C_t)\) to this bounded sum.
The first factor is at least \(1/m\), and the second is
-a positive integer. Formula~\eqref{ex:ramification-cost}
+a positive integer. Formula~(ex:ramification-cost)
therefore bounds the number of contacted exceptional primes,
their log discrepancies, their ramification indices, and
their contact degrees. In particular the integers
@@ -3394,7 +3393,7 @@ The generic Lelong number of \(\rho_t^*T\) at \(Z\) is
\(r_Z\nu_E(T)\). Extract \(E\), remove its generic
divisorial part, and use the transverse-power-map
comparison for the zero-generic-number remainder.
-Lemma~\ref{ex:current-comparison} shows that these
+Lemma~ex:current-comparison shows that these
coefficients dominate the coefficients of the pulled-back
canonical difference \(J_t\). Its support is exceptional
over \(Y\) as well as \(Y_t\), since the map between these
@@ -3402,18 +3401,18 @@ models is small. Subtracting all exceptional divisorial
parts of \(\rho_t^*T\) leaves a positive current. Shrink the
parameter space so that the relevant divisor contacts are
constant, and restrict this residual current to almost every
-fiber. As in \eqref{ex:zero-gap}, its degree is the first
+fiber. As in (ex:zero-gap), its degree is the first
difference below and is nonnegative. Consequently
-\begin{equation}\label{ex:bounded-gap}
- \begin{split}
+
+
0&\leq q_t^*K_Y\cdot C_t
-\sum_{Z\ {\rm exceptional}}
r_Z\nu_E(T)\,Z\cdot C_t\\
&\leq K_t\cdot C_t
\leq (K_t+tA_t)\cdot C_t
=O(\mu_t).
- \end{split}
-\end{equation}
+
+
All divisor contacts used here are nonnegative, since the
curves cover the total space.
@@ -3423,31 +3422,31 @@ On the fixed smooth resolution \(p:W\to Y\), one has
-\ord_E(K_W-p^*K_Y)\leq a(E;Y,0),
\]
because the relative canonical divisor is effective.
-Thus Lemma~\ref{ex:valuation-acc} applies to the one fixed
+Thus Lemma~ex:valuation-acc applies to the one fixed
current \(T\) on \(W\), with a fixed discrepancy bound.
-The sums in \eqref{ex:bounded-gap} form an ACC set: their
+The sums in (ex:bounded-gap) form an ACC set: their
Lelong numbers belong to this ACC set, and both the number
of terms and their integral multiplicities
\(r_Z(Z\cdot C_t)\) are bounded. Pass to a fixed value of
\(q_t^*K_Y\cdot C_t\). Since \(\mu_t\to0\), the
-exact-zero argument of \eqref{ex:zero-gap} gives a family
+exact-zero argument of (ex:zero-gap) gives a family
for which the left gap is zero.
For that family, the restriction of the residual positive
current to almost every parameter fiber is a positive
measure of total mass zero, so it vanishes. On the dense
open away from the removed exceptional divisors it agrees
-with \(\rho_t^*T\). Apply Lemma~\ref{ex:web} to the family
+with \(\rho_t^*T\). Apply Lemma~ex:web to the family
evaluated by \(\rho_t:U_t\to W\). Its curve tangents generate
a rational fibration of positive relative dimension on the
smooth projective variety \(W\), and \(T\) annihilates the
fiber tangents on a dense open. Transport this rational
fibration through \(p:W\to Y\) on their common isomorphism
open.
-This contradicts Proposition~\ref{prop:no-pullback}.
-\end{proof}
+This contradicts Proposition~prop:no-pullback.
-\subsection{Special termination for signed representatives}
+
+## Special termination for signed representatives
To turn a signed canonical representative into a big logarithmic
adjoint, we will run a program whose negative rays meet the reduced
@@ -3455,9 +3454,9 @@ boundary. The following special-termination statement uses only the
lower-dimensional hypothesis; its proof keeps the programs on the
boundary separate from the ambient one.
-\begin{proposition}[Special termination over a point]
-\label{prop:special-termination}
-Assume Assumption~\ref{mmp:lower-models}.
+[Special termination over a point]
+
+Assume Assumption~mmp:lower-models.
Let \((V,\Delta)\) be a projective \(\Q\)-factorial dlt
\(n\)-fold with rational boundary and pseudo-effective adjoint.
Choose an effective ample rational divisor \(A\) such that
@@ -3469,9 +3468,9 @@ are eventually disjoint from the round-down of the boundary.
In particular, if at every stage the adjoint is rationally linearly
equivalent to a signed divisor supported on that round-down, the
program terminates at a log minimal model.
-\end{proposition}
-\begin{proof}
+
+Proof.
In an infinite ample-scaling program the scaling limit is zero:
a positive limit is covered by the termination theorem for a
dlt pair with an ample summand in the scaling divisor
@@ -3564,7 +3563,7 @@ of the nef pullback is pseudo-effective, as is the effective
comparison term, so $D_1+\lambda_j C_1$ is pseudo-effective.
Closedness of the pseudo-effective cone and $\lambda_j\to0$
show that $D_1$ is pseudo-effective.
-Assumption~\ref{mmp:lower-models} therefore gives an absolute
+Assumption~mmp:lower-models therefore gives an absolute
log minimal model for that initial pair. The concatenation
terminates by \cite[Theorem~1.9(iii)]{Birkar2012Special}, since
its zero limit is never attained. The floor-component argument
@@ -3580,26 +3579,26 @@ Special termination rules out all sufficiently late flips.
There are only finitely many divisorial contractions, since
each lowers the Picard number, and pseudo-effectivity excludes
a Mori fiber space as the endpoint. Thus the endpoint is nef.
-\end{proof}
-\subsection{The signed alternative}
-The jet construction in Section~\ref{sec:jets} will produce signed
+## The signed alternative
+
+The jet construction in Section~sec:jets will produce signed
rational representatives of \(K_t\). The following result turns
such a representative into a big logarithmic adjoint on a resolution.
-\begin{proposition}\label{prop:signed-alternative}
+
On any model \(Y_t\), let a signed rational divisor
represent \(K_t\) up to \(\Q\)-linear equivalence. On a
log resolution \(p:W\to Y_t\), let \(D\) be the reduced
SNC divisor consisting of its strict support and all
exceptional divisors. Then \(K_W+D\) is big.
-\end{proposition}
-\begin{proof}
+
+Proof.
Put \(P=K_W+D\), and suppose it is not big. It cannot
have Iitaka dimension at least one: by
-Corollary~\ref{ex:supporting}, \(K_W+cP\) would be big,
+Corollary~ex:supporting, \(K_W+cP\) would be big,
and
\((1+c)P=(K_W+cP)+D\) would then be big as well.
Hence \(\kappa(W,P)\leq0\).
@@ -3611,7 +3610,7 @@ that property. Every negative flip must meet the floor:
on its complement the representing rational section is
nowhere vanishing, so the adjoint has zero degree on
every complete curve there. By
-Proposition~\ref{prop:special-termination}, an infinite
+Proposition~prop:special-termination, an infinite
sequence would eventually have all flips disjoint from
the floor, a contradiction. Divisorial contractions
are finite in number. Thus the program terminates at
@@ -3622,8 +3621,8 @@ The boundary is reduced, \(K_Z\) is pseudo-effective
as the birational pushforward of \(K_W\), and the nef
adjoint has a signed rational representative supported
on \(D_Z\). Its restriction to \(D_Z\) is semiample
-by Proposition~\ref{prop:boundary-restriction}.
-Apply Theorem~\ref{thm:signed} with
+by Proposition~prop:boundary-restriction.
+Apply Theorem~thm:signed with
\(L=K_Z+D_Z\) and \(c=1\), so that the required
pseudo-effective class \(L-cD_Z\) is \(K_Z\).
It gives \(\kappa(Z,L)=\nu(Z,L)\geq0\).
@@ -3649,10 +3648,10 @@ Independence of exceptional divisor classes, by negativity,
identifies its coefficients with those of \(E\); hence
\(E\geq0\). Pushing forward by \(a\) shows that
\(K_W\) is \(\Q\)-linearly effective,
-contradicting \eqref{ex:counterexample}.
-\end{proof}
+contradicting (ex:counterexample).
+
-\subsection{Birational families between fixed covers}
+## Birational families between fixed covers
The last obstruction concerns fixed finite covers of a birational model
of the counterexample. If birational maps between two such covers form
@@ -3660,9 +3659,9 @@ an algebraic family whose graphs dominate their product, then one fixed
cover is birational to an abelian variety. The positive canonical current
rules this out.
-\begin{proposition}[Fixed-cover obstruction]\label{ex:fixed-cover}
+[Fixed-cover obstruction]
Let \(Y\) be a projective \(\Q\)-factorial terminal birational model
-of the counterexample \(X\) in Equation~\eqref{ex:counterexample}, and let
+of the counterexample \(X\) in Equation~(ex:counterexample), and let
\(T_i\to Y\), \(i=1,2\), be fixed finite surjective morphisms from normal
integral projective varieties.
There is no integral parameter variety \(A_0\) and closed subvariety
@@ -3672,9 +3671,9 @@ There is no integral parameter variety \(A_0\) and closed subvariety
flat over \(A_0\), with each fiber the integral graph of a birational
map \(T_1\dashrightarrow T_2\), for which the projection
\(\Gamma\to T_1\times T_2\) is dominant.
-\end{proposition}
-\begin{proof}
+
+Proof.
Suppose such a family exists, and denote its birational maps by
\(\phi_a:T_1\dashrightarrow T_2\).
Choose one map \(\phi_{a_0}\) in this family and use it to identify
@@ -3736,12 +3735,12 @@ divisorial current whose class is \(\deg(f)c_1(K_X)\) by the
projection formula. Rationality of the numerical class supplies
a rational effective representative, and irregularity zero turns
numerical effectivity into nonvanishing, as in
-Lemma~\ref{ex:irregularity}. This contradicts
-\(\kappa(X,K_X)=-\infty\).
-\end{proof}
+Lemma~ex:irregularity. This contradicts
+\(\kappa(X,K_X)=-∞\).
+
+
+# Tools for moving base components
-\section{Tools for moving base components}
-\label{sec:common-tools}
This section follows the base components of a complete kernel of jets
as the marked point moves. Their multiplicities give degree and
@@ -3749,7 +3748,7 @@ canonical-intersection bounds. A separate finiteness lemma treats the
covers that arise when a component projects generically finitely onto
a fixed base. These arguments use neither nonvanishing, abundance,
minimal models, nor logarithmic subadditivity.
-Section~\ref{sec:jets} applies them to moving components produced by
+Section~sec:jets applies them to moving components produced by
failure of the scalar or two-slot jet estimates.
All varieties in the moving-center arguments are over $\C$.
@@ -3765,7 +3764,7 @@ evaluation map after tensoring by the inverse line bundle.
is primary for the maximal ideal of $\mathcal O_{Z,\eta_V}$.
All powers of ideals below are ordinary powers.
-\subsection{Numerical subadjunction on the center}
+## Numerical subadjunction on the center
The first result converts a generic log canonical center into a
canonical-intersection inequality. Its testing class is allowed
@@ -3773,38 +3772,38 @@ to live on the center itself. The proof uses dlt adjunction and
the klt-trivial canonical bundle formula; it does not require the
auxiliary pair to be log canonical away from the generic center.
-\begin{lemma}[Numerical generic subadjunction]
-\label{common:numerical-subadjunction}
+[Numerical generic subadjunction]
+
Let $Z$ be a projective $\Q$-factorial klt variety over $\C$, let
-$\Theta\geq0$ be a rational divisor, and let $V\subset Z$ be an
-integral positive-dimensional subvariety. Suppose that $(Z,\Theta)$
+$Θ\geq0$ be a rational divisor, and let $V\subset Z$ be an
+integral positive-dimensional subvariety. Suppose that $(Z,Θ)$
is lc at the generic point of $V$ and that a divisor of log
discrepancy zero has center $V$. Let $f=\dim V$, let $P$ be a nef
rational Cartier divisor on $V$, and let $\rho:V^*\to V$ be a smooth
projective resolution, factoring through the normalization. Then
-\begin{equation}
-\label{common:subadjunction-inequality}
+
+
K_{V^*}\cdot(\rho^*P)^{f-1}
- \leq (K_Z+\Theta)|_V\cdot P^{f-1}.
-\end{equation}
+ \leq (K_Z+Θ)|_V\cdot P^{f-1}.
+
The right side is the intersection on $V$ of the restricted
rational Cartier class with $P^{f-1}$. No $\Q$-Gorenstein hypothesis
on the normalization of $V$ is required.
-\end{lemma}
-\begin{proof}
+
+Proof.
Apply the dlt blowup theorem for an effective rational divisor
\cite[Theorem~2.10]{FujinoHashizume2023}. It gives a projective
birational morphism $p:Q\to Z$ with $Q$ $\Q$-factorial and
-\begin{equation}
-\label{common:subadjunction-surplus}
- p^*(K_Z+\Theta)=K_Q+B_Q+E_Q,
-\end{equation}
+
+
+ p^*(K_Z+Θ)=K_Q+B_Q+E_Q,
+
where $(Q,B_Q)$ is dlt and $E_Q\geq0$ is the surplus over the
non-lc locus. In this construction the coefficients of the strict
boundary are truncated at one and the relevant extracted divisors
are reduced; the coefficients left over form $E_Q$. The surplus
-is absent over the open where $(Z,\Theta)$ is lc. In particular it
+is absent over the open where $(Z,Θ)$ is lc. In particular it
is absent over a neighborhood of the generic point of $V$.
There are lc strata of $(Q,B_Q)$ mapping onto $V$. Indeed over that
@@ -3814,14 +3813,14 @@ Choose a stratum $S$ minimal among those mapping onto $V$.
Iterated dlt adjunction makes $S$ normal and gives an effective
rational dlt different on $S$; its lc centers are precisely the
smaller ambient strata in $S$
-\cite{Kollar2013}. The divisor $E_Q$ does not contain $S$. Since
+[cite]. The divisor $E_Q$ does not contain $S$. Since
$Q$ is $\Q$-factorial, a Cartier multiple of $E_Q$ restricts to an
effective Cartier divisor on $S$. Adding that restriction to the
different gives an effective rational divisor $B_S$ such that
-\begin{equation}
-\label{common:subadjunction-stratum}
- K_S+B_S\sim_{\Q}(p|_S)^*\bigl((K_Z+\Theta)|_V\bigr).
-\end{equation}
+
+
+ K_S+B_S\sim_{\Q}(p|_S)^*((K_Z+Θ)|_V).
+
Over the generic point of $V$ this pair is klt: the surplus is
absent there, and a non-klt center of the different dominating
$V$ would be a smaller stratum, contradicting the choice of $S$.
@@ -3832,7 +3831,7 @@ factorization:
S\xrightarrow{q}V'\xrightarrow{\eta}V^\nu\xrightarrow{\nu}V.
\]
Here $q$ has connected fibers and $\eta$ is finite. Put
-$D_V=\nu^*((K_Z+\Theta)|_V)$ and
+$D_V=\nu^*((K_Z+Θ)|_V)$ and
$D'=\eta^*D_V$. We apply the canonical bundle formula on the
connected-fiber base $V'$, then push the resulting intersection
inequality through the finite map $\eta$. Its ramification will
@@ -3841,7 +3840,7 @@ canonical cycle on $V^\nu$ and the canonical divisor on $V^*$.
We verify the klt-trivial-fibration hypotheses for
$q:(S,B_S)\to V'$. The generic pair is sub-klt, and
-Equation~\eqref{common:subadjunction-stratum} is the required rational-linear
+Equation~(common:subadjunction-stratum) is the required rational-linear
pullback expression. For the rank condition, take a log resolution
over the generic point of $V'$. Since the boundary on $S$ is
effective and the generic pair is klt, the round-up of its
@@ -3857,10 +3856,10 @@ alone.
The canonical bundle formula, with Ambro's moduli positivity in
the form recalled by Fujino--Gongyo, now gives
-\begin{equation}
-\label{common:subadjunction-cbf}
+
+
D'\sim_{\Q}K_{V'}+B_{V'}+M_{V'},
-\end{equation}
+
where $M$ is a b-nef moduli divisor
\cite[Definition~3.1 and Theorem~3.5]{FujinoGongyoModuli2014}.
The definition requires sub-klt over the generic point; it does
@@ -3874,7 +3873,7 @@ image on $S$ has multiplicity $m_F\geq1$ and nonnegative boundary
coefficient $b_F$. Its discrepancy bounds the generic lc threshold
$t_F$ from above by
\[
- t_F\leq\frac{1-b_F}{m_F}\leq1.
+ t_F\leq(1-b_F)/(m_F)\leq1.
\]
Thus the discriminant coefficient $1-t_F$ is nonnegative. If the
fiber is already non-lc, $t_F$ can be negative; this only increases
@@ -3905,7 +3904,7 @@ by pushforward, is
\]
This identity of Weil cycles uses the discrete valuation rings at
generic prime divisors and does not require $K_{V^\nu}$ to be
-$\Q$-Cartier. Intersect Equation~\eqref{common:subadjunction-cbf} with
+$\Q$-Cartier. Intersect Equation~(common:subadjunction-cbf) with
$(P')^{f-1}$ and discard the nonnegative discriminant and moduli
intersections. Pushforward through $\eta$ and the nonnegative
ramification intersection then give
@@ -3917,19 +3916,19 @@ Finally the pushforward of $K_{V^*}$ to $V^\nu$ is its compatible
canonical cycle. All additional divisors on the resolution have
image of codimension at least two, so their intersection with
$f-1$ pulled-back Cartier powers is zero. Dividing by
-$\deg\eta$ proves Equation~\eqref{common:subadjunction-inequality}. When $f=1$
+$\deg\eta$ proves Equation~(common:subadjunction-inequality). When $f=1$
there are no birational exceptional divisors on the normal curve,
and the same calculation applies directly.
-\end{proof}
-\subsection{Degree and canonical bounds for a base component}
+
+## Degree and canonical bounds for a base component
We next apply numerical subadjunction to a boundary built from an
actual linear system. Its local multiplicity controls both the
degree of the center and the coefficient of that boundary.
-\begin{lemma}[A base component of high multiplicity]
-\label{common:base-component}
+[A base component of high multiplicity]
+
Let $Z$ be a projective $\Q$-factorial klt variety of dimension $d$,
let $P$ be a nef rational Cartier divisor, and let $k>0$ be an
integer for which $kP$ is Cartier. Let
@@ -3944,24 +3943,24 @@ $\mathfrak m=\mathfrak m_{\cO_{Z,\eta_V}}$. If
\qquad(h\in\Z_{>0}),
\]
then
-\begin{equation}
-\label{common:base-component-degree}
+
+
P^f\cdot V\leq (k/h)^aP^d.
-\end{equation}
+
There are a rational number $c$ with $0<c\leq ak/h$ and an
-effective rational divisor $\Theta\sim_{\Q}cP$ such that
-$(Z,\Theta)$ is lc at $\eta_V$ and has an lc place with center
+effective rational divisor $Θ\sim_{\Q}cP$ such that
+$(Z,Θ)$ is lc at $\eta_V$ and has an lc place with center
$V$. If $f>0$, on a smooth resolution $V^*$ one consequently has
-\begin{equation}
-\label{common:base-component-canonical}
+
+
K_{V^*}\cdot P^{f-1}
\leq (K_Z+cP)|_V\cdot P^{f-1}.
-\end{equation}
+
The degree estimate includes $f=0$; the canonical estimate is
asserted only for positive-dimensional $V$.
-\end{lemma}
-\begin{proof}
+
+Proof.
The local ring $R=\cO_{Z,\eta_V}$ is regular of dimension $a$.
Isolation means that $\mathfrak bR$ is $\mathfrak m$-primary.
Choose $a$ general sections of $\mathcal H$. Their local equations
@@ -3970,7 +3969,7 @@ ideal is contained in no smaller prime. Let $\mathfrak q$ be the
parameter ideal they generate. It is contained in $\mathfrak m^h$.
Regularity and the multiplicity comparison give
\[
- \operatorname{length}(R/\mathfrak q)
+ length(R/\mathfrak q)
=e(\mathfrak q,R)\geq e(\mathfrak m^h,R)=h^a.
\]
This is the generic intersection multiplicity along $V$.
@@ -3986,9 +3985,9 @@ the remaining components properly. Nefness of $P$ shows that
discarding effective components can only decrease their total
$P$-degree. After $a$ cuts the surviving effective cycle has
$P$-degree at most $k^aP^d$, and contains $V$ with multiplicity at
-least $h^a$. This proves Equation~\eqref{common:base-component-degree}.
+least $h^a$. This proves Equation~(common:base-component-degree).
-Let $\lambda=\operatorname{lct}_{\eta_V}(\mathfrak b)$ be the generic ideal
+Let $\lambda=lct_{\eta_V}(\mathfrak b)$ be the generic ideal
threshold. It is positive and rational. The valuation from
blowing up the smooth generic center has log discrepancy $a$
and ideal order at least $h$, so
@@ -4001,23 +4000,23 @@ divisor rather than apply subadjunction to an ideal formally.
Take a log resolution of $Z$ and $\mathfrak b$, so that the
pulled-back system has fixed divisor $F$ and a basepoint-free
moving part. Choose sufficiently many general members
-$D_1,\ldots,D_N$ of $\mathcal H$ and put
+$D_1,...,D_N$ of $\mathcal H$ and put
\[
- \Theta=\frac{\lambda}{N}(D_1+\cdots+D_N)
+ Θ=(\lambda)/(N)(D_1+...+D_N)
\sim_{\Q}cP,\qquad c=k\lambda.
\]
On the resolution the fixed contribution is $\lambda F$.
The moving contributions have coefficients $\lambda/N<1$ and,
by general choice, meet the fixed resolution divisor with simple
-normal crossings on the open in question. Thus $(Z,\Theta)$ is
+normal crossings on the open in question. Thus $(Z,Θ)$ is
lc at the generic point of $V$ and has the same lc place there
as the ideal threshold. Global log canonicity is unnecessary.
-For $f>0$, Lemma~\ref{common:numerical-subadjunction} gives
-Equation~\eqref{common:base-component-canonical}, and
+For $f>0$, Lemma~common:numerical-subadjunction gives
+Equation~(common:base-component-canonical), and
$c=k\lambda\leq ak/h$.
-\end{proof}
-\subsection{Differentiating the full moving jet kernel}
+
+## Differentiating the full moving jet kernel
Fixing a high order at a varying point produces a family of
linear subspaces of one fixed section space. A component common
@@ -4029,13 +4028,13 @@ is related to the method of Ein--K\"uchle--Lazarsfeld
\cite[Proposition~2.3]{EinKuchleLazarsfeld1995}; the ordinary-power
and primary-ideal details needed here are included.
-\begin{lemma}[Moving multiplicity]
-\label{common:moving-kernel}
+[Moving multiplicity]
+
Let $Z$ be an integral projective variety of dimension $d$, and
let $P$ be a nef, big, semiample rational Cartier divisor. Fix an integer $k$
with $kP$ Cartier and positive real numbers
\[
- \tau_0<\tau_1<\cdots<\tau_d,
+ \tau_0<\tau_1<...<\tau_d,
\qquad\tau_{i+1}-\tau_i=\delta>0.
\]
For a smooth point $x$, let $\mathcal H_i(x)$ be the entire
@@ -4055,33 +4054,33 @@ dominates $Z$. Each general $V_t$ has dimension in $[1,d-1]$,
is an isolated component of the higher base locus at its generic
point, and its entire higher-series base ideal is contained
there in
-\begin{equation}
-\label{common:moving-multiplicity-order}
+
+
\mathcal I_{V_t}^{h},\qquad h=\lceil k\delta/2\rceil,
\qquad k/h\leq2/\delta.
-\end{equation}
+
These are ordinary local ideal powers in the smooth ambient open.
The incidence base ideal lies in $\mathcal I_\Gamma^h$ on a dense
open of $\Gamma$. If, in addition, $Z$ is $\Q$-factorial and klt,
then for a general member, with $f=\dim V_t$, one has
-\begin{equation}\label{common:tracking-bounds}
- \begin{split}
+
+
P^f\cdot V_t&\le(2/\delta)^{d-f}P^d,\\
K_{V_t^*}P^{f-1}&\le(K_Z+cP)|_{V_t}P^{f-1},
\qquad 0<c\le2(d-f)/\delta .
- \end{split}
-\end{equation}
+
+
In particular the weaker bound $c\le2d/\delta$ also holds.
Positivity of $P^f\cdot V_t$ follows if $V_t$ meets the open
where the big semiample contraction is an isomorphism. No such
positivity is asserted for every exceptional subvariety.
-\end{lemma}
-\begin{proof}
+
+Proof.
On the smooth marked-point open, evaluation into the finite jet
bundle is a morphism from the constant bundle
-$H^0(Z,kP)\otimes\cO$ to a locally free sheaf. Remove the rank-jump
+$H^0(Z,kP) ⊗ \cO$ to a locally free sheaf. Remove the rank-jump
loci for the finitely many levels. Its kernels are then vector
subbundles whose fibers are exactly the full spaces
$\mathcal H_i(x)$. Their base loci are nested in increasing order
@@ -4175,23 +4174,23 @@ The specialized frame is a basis of the entire higher kernel,
so the containment concerns its whole base ideal.
If $Z$ is $\Q$-factorial and klt, apply
-Lemma~\ref{common:base-component} to the higher kernel.
+Lemma~common:base-component to the higher kernel.
Its generic base ideal is primary because $V_t$ is a base-locus
component, and its generic point is smooth by incidence dominance.
The bounds follow from $k/h\le2/\delta$. On the isomorphism open
of the big semiample contraction the image of $V_t$ has dimension
$f$, so the pullback of an ample class has positive top intersection
on $V_t$. This proves the stated positivity qualification.
-\end{proof}
-\subsection{Restriction to a suitable fiber}
+
+## Restriction to a suitable fiber
The restriction step is local and must be separated from the
global positivity and singularity assumptions of subadjunction.
-\begin{lemma}[Restriction of an isolated base ideal]
-\label{common:fiber-restriction}
-In the setting of Lemma~\ref{common:moving-kernel}, let $g:Z\to B$
+[Restriction of an isolated base ideal]
+
+In the setting of Lemma~common:moving-kernel, let $g:Z\to B$
be a morphism. There is a dense open of the incidence on which
the following holds. Choose $(t,z)$ in that open, let $b=g(z)$,
and suppose that $V_t$ is smooth over the smooth open of its
@@ -4202,13 +4201,13 @@ contained in $\mathcal I_F^h$.
If $F$ is proper in an integral ambient fiber component containing
it, the restricted series is nonzero there and $F$ is an isolated
base component at its generic point. Application of
-Lemma~\ref{common:base-component} on that ambient component
+Lemma~common:base-component on that ambient component
requires all its hypotheses, including projectivity, $\Q$-factoriality,
klt singularities, and smoothness at the generic point of $F$,
separately; they are not conclusions of this restriction lemma.
-\end{lemma}
-\begin{proof}
+
+Proof.
Choose the incidence point after removing the other base components,
the locus where the ideal containment is unavailable, and the
failures of generic smoothness for the incidence over its parameter
@@ -4234,20 +4233,20 @@ proper there, and the restricted series cannot vanish identically.
At its generic point the resulting ideal is primary. This is a
statement about this reduced general fiber and does not assert
transversality or nonzero restriction for arbitrary special fibers.
-\end{proof}
-\subsection{A fixed branch complement in a finite-type family}
+
+## A fixed branch complement in a finite-type family
For later applications, the local moving-center estimates are
followed by a global finiteness step. A degree bound alone does
not make a list of branched covers finite. The following statement
records exactly the additional family information that is needed.
-In the correspondence case of Proposition~\ref{prop:large-seshadri},
+In the correspondence case of Proposition~prop:large-seshadri,
it will turn bounded degree and branch divisors that do not sweep
the base into finitely many normal covers.
-\begin{lemma}[Finite covers after excluding sweeping branch divisors]
-\label{common:finite-covers}
+[Finite covers after excluding sweeping branch divisors]
+
Let $Y$ be a normal projective complex variety. Let
$\mathcal V\to T$ be a smooth projective family with integral
fibers, over an integral parameter variety, and let
@@ -4257,13 +4256,13 @@ at most a fixed integer $e$.
After a finite parameter extension and shrinking, spread the
geometric generic components of the zero divisor of the relative
Jacobian over $Y_{\rm sm}$ to divisors
-$\mathcal R_1,\ldots,\mathcal R_N$ in $\mathcal V$.
+$\mathcal R_1,...,\mathcal R_N$ in $\mathcal V$.
Retain the notation $\mathcal V,T,g$ after this base change.
Let $I$ be the set of indices $j$ for which the image closure
-$\overline{g_t((\mathcal R_j)_t)}$ is a divisor in $Y$ for general $t$.
+$g_t((\mathcal R_j)_t)bar$ is a divisor in $Y$ for general $t$.
Assume
\[
- \overline{g(\mathcal R_j)}\ne Y\qquad(j\in I).
+ g(\mathcal R_j)bar\ne Y\qquad(j\in I).
\]
Then, after shrinking again, one smooth nonempty open
$Y^\circ\subset Y$ makes every general finite normal Stein cover
@@ -4272,9 +4271,9 @@ many such covers of $Y$, up to isomorphism over $Y$.
The same conclusion holds simultaneously for finitely many
projections. The open and finite lists may depend on all fixed
parameters of the family.
-\end{lemma}
-\begin{proof}
+
+Proof.
The relative determinant is nonzero on a dense open because
the general maps are generically finite in characteristic zero.
Its zero divisor has finitely many irreducible components.
@@ -4293,11 +4292,11 @@ parameters disappear upon shrinking.
Put
\[
- B=Y_{\rm sing}\cup\bigcup_{j\in I}\overline{g(\mathcal R_j)}.
+ B=Y_{\rm sing}\cup∪_{j\in I}g(\mathcal R_j)bar.
\]
Each closed set in this finite union is proper by hypothesis,
so $B\ne Y$. The finite normal covers are unramified in
-codimension one over the smooth open $Y^\circ=Y\setminus B$.
+codimension one over the smooth open $Y^\circ=Y∖ B$.
Purity of the branch locus makes them finite \'etale there
\cite[Expos\'e~X, Theorem~3.1]{SGA1}.
The topological fundamental group of $Y^\circ$ is finitely
@@ -4318,9 +4317,9 @@ normal cover of $Y$ is the normalization in the function field
of its restriction to $Y^\circ$, and is determined by that
restriction. Taking the union of the proper image closures for
finitely many projections proves the simultaneous assertion.
-\end{proof}
-\section{Very-general jet estimates}\label{sec:jets}
+
+# Very-general jet estimates
This section derives two contrasting consequences of the assumed
counterexample: an upper bound for the vanishing of scalar sections,
@@ -4328,18 +4327,18 @@ and strong jet separation on a projective bundle over the product.
Both arguments use the exclusion of covering curves of bounded genus
and normalized degree.
-We retain the counterexample in Equation~\eqref{ex:counterexample}
-and the induction hypothesis of Assumption~\ref{mmp:lower-models}.
+We retain the counterexample in Equation~(ex:counterexample)
+and the induction hypothesis of Assumption~mmp:lower-models.
Thus \(X\) is smooth projective,
-\(K_X\) is pseudo-effective, and \(\kappa(X,K_X)=-\infty\), whereas
+\(K_X\) is pseudo-effective, and \(\kappa(X,K_X)=-∞\), whereas
good minimal models exist in smaller dimensions. We use the geometric
-exclusions of Section~\ref{sec:exclusions}. Dimension one is impossible, since a
+exclusions of Section~sec:exclusions. Dimension one is impossible, since a
smooth curve with pseudo-effective canonical divisor has genus at least
one. Hence \(n=\dim X\ge2\).
Throughout the section, fix the late terminal \(\Q\)-factorial model
\(Y\) from
-Proposition~\ref{prop:bounded-curves}. Write \(K=K_Y\) and \(A=A_Y\).
+Proposition~prop:bounded-curves. Write \(K=K_Y\) and \(A=A_Y\).
The subsequent scaling models \(Y_t\) are small modifications of \(Y\);
the transforms \(A_t\) are effective up to \(\Q\)-linear equivalence,
and \(K_t+tA_t\) is nef, big, and semiample. Put
@@ -4349,48 +4348,48 @@ and \(K_t+tA_t\) is nef, big, and semiample. Put
Here, in the volume on \(X\), \(A\) denotes the original ample divisor.
We have \(\mu_t\to0\). Choose an integer \(m>0\) with \(mK\) Cartier.
-\subsection{Normalization and two local tools}
+## Normalization and two local tools
We first normalize the small adjoint divisors so that their volumes
-stay bounded away from zero. Fix a small rational number \(\epsilon>0\).
-An integer \(q>0\) will be chosen after \(\epsilon\) and before \(t\).
-As rational \(t\downarrow0\),
+stay bounded away from zero. Fix a small rational number \(ε>0\).
+An integer \(q>0\) will be chosen after \(ε\) and before \(t\).
+As rational \(t↓0\),
choose positive integers \(r=r(t)\) with
-\begin{equation}\label{jet:normalization}
- r\mu_t\longrightarrow\epsilon,\qquad
+
+ r\mu_t→ε,\qquad
L=r(K+tA),\qquad L_b=L+bmK\quad(0\le b\le q).
-\end{equation}
+
Only rational weights \(b\) are used for model constructions;
volume functions in integrals are extended continuously to real weights.
The positive part of \(L_b\) is
the nef semiample class
\[
N_b=(r+bm)(K_s+sA_s)
- \quad\hbox{on }Y_s,\qquad s=\frac{rt}{r+bm}.
+ \quad\hbox{on }Y_s,\qquad s=(rt)/(r+bm).
\]
When emphasizing the weight, denote this axis model by \(Y(b)=Y_s\)
and write \(A_b\) for its transform of \(A\).
All references to positive parts below mean these classes on their
scaling models, or their pullbacks to common resolutions. Monotonicity
of volume in pseudo-effective order gives
-\begin{equation}\label{jet:volume-comparison}
+
\vol(L)\le \vol(L_b)\le
- \left(1+\frac{qm}{r}\right)^n\vol(L).
-\end{equation}
+ (1+(qm)/(r))^n\vol(L).
+
Indeed \(L_b-L=bmK\) is pseudo-effective, and
\((1+bm/r)L-L_b=bmtA\) is effective up to equivalence.
Consequently \(N_b^n\) is bounded above and bounded away from zero,
-uniformly for \(b\in[0,q]\), and tends uniformly to \(\epsilon^n\).
+uniformly for \(b\in[0,q]\), and tends uniformly to \(ε^n\).
More explicitly, for every fixed \(q\), once \(t\) is sufficiently
small depending on \(q\),
-\begin{equation}\label{jet:uniform-normalized-volume}
- \frac{\epsilon^n}{2}\le N_b^n\le2\epsilon^n
+
+ (ε^n)/(2)\le N_b^n\le2ε^n
\qquad(0\le b\le q).
-\end{equation}
+
The constants in this display are independent of \(q\); only the
threshold for \(t\) depends on the fixed choice of \(q\).
All unspecified constants in this section may depend on
-\(n,\epsilon,q,Y,A,m\), but not on sufficiently small \(t\) or on \(b\).
+\(n,ε,q,Y,A,m\), but not on sufficiently small \(t\) or on \(b\).
This normalization gives a convenient form of the curve exclusion.
A family of curves with uniformly bounded geometric genus and
@@ -4401,14 +4400,14 @@ small \(t\). In fact \(s/t\to1\) uniformly in \(b\), and
\]
is bounded above and away from zero. Thus such curves would have
uniformly bounded \((K_s+sA_s)\)-degree divided by \(\mu_s\), contrary
-to Proposition~\ref{prop:bounded-curves}. We call this consequence the
-\emph{normalized curve exclusion}.
+to Proposition~prop:bounded-curves. We call this consequence the
+normalized curve exclusion.
-We will apply the local estimates of Section~\ref{sec:common-tools}
+We will apply the local estimates of Section~sec:common-tools
on several different models. In each application, let \(Z\) denote the
ambient model of dimension \(d\), let \(P\) be its nef, big, semiample
class, and let \(V\) be a moving base component of dimension \(f\).
-Lemma~\ref{common:moving-kernel} applies to the \emph{entire} kernels
+Lemma~common:moving-kernel applies to the entire kernels
of jets in one Cartier degree \(kP\), at levels separated by a gap
\(\delta\). It gives vanishing along \(V\) in the ordinary ideal power
of order
@@ -4416,20 +4415,20 @@ of order
h=\lceil k\delta/2\rceil,\qquad k\delta\ge4,
\]
and, when \(Z\) is \(\Q\)-factorial and klt, the estimates
-\begin{equation}\label{jet:tracking-estimates}
+
P^f\cdot V\le (2/\delta)^{d-f}P^d,
\qquad K_{V^*}P^{f-1}\le(K_Z+cP)|_V P^{f-1},
\quad 0<c\le 2(d-f)/\delta\le2d/\delta.
-\end{equation}
+
Here \(V^*\) is a smooth resolution of \(V\). The canonical estimate
-uses Lemma~\ref{common:numerical-subadjunction}, which allows the nef
+uses Lemma~common:numerical-subadjunction, which allows the nef
class on the center itself and requires log canonicity only near its
generic point. The last, weaker coefficient is sufficient here. At every
application we choose the incidence point in the isomorphism open of the
big semiample contraction. Dominance of the incidence permits this choice;
it ensures that $P|_V$ is big and hence $P^f\cdot V>0$.
Exceptional components not meeting that open are not assigned this
-positivity. The separate fiber-restriction Lemma~\ref{common:fiber-restriction}
+positivity. The separate fiber-restriction Lemma~common:fiber-restriction
will be used only at a general incidence point where the scheme fiber is
reduced and the selected base component is isolated.
@@ -4437,10 +4436,10 @@ To apply normalized curve exclusion, we need curves with bounded
genus as well as bounded degree. The preceding intersection estimates
provide them on components of general type. Suppose a moving
\(f\)-fold \(V\) is of general type, \(P|_V\) is nef and big, and
-\begin{equation}\label{jet:curve-input}
+
0<P^f.V\le C_0,\qquad
K_{V^*}P^{f-1}\le C_1P^f.V.
-\end{equation}
+
Effective birationality gives a uniform pluricanonical degree whose
moving part, on a further resolution, is a big basepoint-free
Cartier divisor \(H\) defining a birational morphism
@@ -4470,7 +4469,7 @@ nonconstantly. The components sweep, and their chosen incidence points
can lie outside any prescribed countable union of proper closed
subsets. Hilbert schemes and spaces of maps have only countably many
components, so these curves can be parameterized in covering algebraic
-families. Thus, whenever Equation~\eqref{jet:curve-input} is
+families. Thus, whenever Equation~(jet:curve-input) is
established below, it remains to check general type and the stated
covering and projection conditions to invoke normalized curve exclusion.
@@ -4478,7 +4477,7 @@ We first apply the two local tools to scalar sections. Excessive
vanishing produces a proper moving component, whose general type
turns the numerical estimates into the forbidden curves.
-\begin{proposition}[Scalar jet bound]\label{prop:scalar-jets}
+[Scalar jet bound]
For sufficiently small \(t\), let \(P\) be the positive part of
\(2r(K+2tA)\), and put \(\rho=(P^n)^{1/n}\).
At a very general point, every nonzero section of every Cartier
@@ -4489,10 +4488,10 @@ spaces. Moreover
\[
2r\mu_t\le\rho\le4r\mu_t,
\]
-so in particular \(\rho\le8\epsilon\) for small \(t\).
-\end{proposition}
+so in particular \(\rho\le8ε\) for small \(t\).
-\begin{proof}
+
+Proof.
First, pseudo-effectivity of \(K\) gives the volume bounds:
\[
\vol(K+tA)\le\vol(K+2tA)\le2^n\vol(K+tA).
@@ -4508,47 +4507,47 @@ The full jet kernels at these levels are nonzero. Write \(\tau_0\)
for the lowest level. If the marked point were isolated in its base
locus, local Bezout for \(n\) general kernel sections would contribute
at least \((k\tau_0)^n>k^nP^n\), exceeding the total intersection.
-Lemma~\ref{common:moving-kernel} therefore supplies a moving proper
+Lemma~common:moving-kernel therefore supplies a moving proper
positive-dimensional component \(V\).
We can now apply the curve construction. The component is of general
type by
-Proposition~\ref{prop:general-type}. Here let \(Z\) be the scaling
+Proposition~prop:general-type. Here let \(Z\) be the scaling
model for \(K+2tA\), so that \(P=2r(K_Z+2tA_Z)\). The effective
transform \(A_Z\) does not contain a general such component, and
\(K_Z|_V\le(2r)^{-1}P|_V\) in effective order.
-Equations~\eqref{jet:tracking-estimates} consequently imply
-Equation~\eqref{jet:curve-input} with constants uniform in \(t\);
+Equations~(jet:tracking-estimates) consequently imply
+Equation~(jet:curve-input) with constants uniform in \(t\);
the gap is bounded above and away from zero because
-\(\rho\) is comparable to the fixed \(\epsilon\).
+\(\rho\) is comparable to the fixed \(ε\).
The resulting bounded-genus, bounded-normalized-degree covering
-curves contradict Proposition~\ref{prop:bounded-curves}.
+curves contradict Proposition~prop:bounded-curves.
The argument applies to every Cartier multiple, not only the divisible
degrees used for tracking: a section violating the bound in any Cartier
degree could be raised to a power in an arbitrarily divisible degree.
Its order and its degree increase by the same factor.
Finally, exceptional additions preserve the section spaces and the
orders at general points. This proves the assertion on resolutions.
-\end{proof}
-\subsection{The two-slot bundle}
+
+## The two-slot bundle
We next seek large jet separation by placing two copies of the
normalized adjoint divisor on a projective bundle. The weight
decomposition of its sections will let us compare moving components
with either copy of the base. On \(Y\times Y\), let
-\begin{equation}\label{jet:two-slot}
- Z_0=\PP(mK_1\oplus mK_2),\qquad
+
+ Z_0=\PP(mK_1 ⊕ mK_2),\qquad
\xi=\OO_{Z_0}(1),\qquad M=L_1+L_2+q\xi.
-\end{equation}
+
We use the convention that sections of \(k\xi\) are symmetric powers
of the indicated direct sum. Fixing one base slot gives the
one-slot bundle
-\(\PP(\OO_Y\oplus mK)\), with class \(L+q\xi\), up to a constant
-line from the fixed slot. We call it a \emph{slice}.
+\(\PP(\OO_Y ⊕ mK)\), with class \(L+q\xi\), up to a constant
+line from the fixed slot. We call it a slice.
The torus open in either bundle is the complement of its two axes.
-\begin{lemma}[Models, volume, and weights]\label{jet:bundle-models}
+[Models, volume, and weights]
Both the total class \(M\) and the slice class have big semiample
positive parts \(P\) on \(\Q\)-factorial klt models \(Z\), with
\[
@@ -4556,30 +4555,30 @@ positive parts \(P\) on \(\Q\)-factorial klt models \(Z\), with
\]
where \(c_t\) is bounded independently of small \(t\).
Their volumes satisfy, respectively,
-\begin{align}
- \vol(M)&=\frac{(2n+1)!}{(n!)^2}
- \int_0^q\vol(L_b)\vol(L_{q-b})\,db,\label{jet:total-volume}\\
- \vol(L+q\xi)&=(n+1)\int_0^q\vol(L_b)\,db.\label{jet:slice-volume}
-\end{align}
+
+ \vol(M)&=((2n+1)!)/((n!)^2)
+ \int_0^q\vol(L_b)\vol(L_{q-b})\,db,\\
+ \vol(L+q\xi)&=(n+1)\int_0^q\vol(L_b)\,db.
+
In particular both volumes are bounded above and below by positive
constants times \(q\). For each rational weight, on common
resolutions,
-\begin{equation}\label{jet:weight-domination}
+
P\sim_{\Q}N_{b,1}+N_{q-b,2}+E_b,\qquad E_b\ge0,
-\end{equation}
+
with the analogous one-term formula on slices.
At a generic point of a base divisor of a slice, and on a general
torus fiber, the full system has no fixed subtraction.
-\end{lemma}
-\begin{proof}
+
+Proof.
We construct the semiample models as models of big klt adjoints.
The weight decomposition then gives both the volume formulas and
the comparison with the axis models.
The product \(Y\times Y\) is \(\Q\)-factorial. To see this, take a
smooth resolution \(\widetilde Y\to Y\). Its irregularity is zero by
-Lemma~\ref{ex:irregularity}, so every line bundle on
+Lemma~ex:irregularity, so every line bundle on
\(\widetilde Y\times\widetilde Y\) is a tensor product of line bundles
from the two factors. The strict transform of a Weil divisor on
\(Y\times Y\) is therefore linearly equivalent to a sum of two
@@ -4599,7 +4598,7 @@ For \(0<\sigma\le1\), put
\[
\gamma=2\sigma+\frac{q(1+\sigma m)}r,\qquad
\Xi_t\sim_{\Q}\xi+
- \frac{(1+\sigma m)t}{\gamma}A_{\rm sum}.
+ ((1+\sigma m)t)/(\gamma)A_{\rm sum}.
\]
Both endpoint weight systems are big, so choose an effective rational
representative of \(\Xi_t\) containing neither axis, and denote the
@@ -4609,7 +4608,7 @@ bound independent of \(\sigma\).
Restrict to \(0<t\le\sigma/(1+m)\). Since \(\gamma\ge2\sigma\),
the coefficient
\[
- a=\frac{(1+\sigma m)t}{\gamma}
+ a=((1+\sigma m)t)/(\gamma)
\]
lies in \((0,1]\). Thus the numerical classes
\(\xi+aA_{\rm sum}\), and their restrictions to either fixed axis,
@@ -4643,50 +4642,50 @@ Lowering their coefficients to \(1-\sigma\) consequently shows that
\[
(Z_0,(1-\sigma)D+\gamma\Xi_t)
\]
-is klt. This choice respects the order: first \(\epsilon\), then
+is klt. This choice respects the order: first \(ε\), then
\(q\), then sufficiently small \(t\) with \(r\) as in
-Equation~\eqref{jet:normalization}.
+Equation~(jet:normalization).
A direct calculation gives its adjoint class
\[
K_{Z_0}+(1-\sigma)D+\gamma\Xi_t
- \sim_{\Q}\frac{1+\sigma m}{r}M.
+ \sim_{\Q}(1+\sigma m)/(r)M.
\]
The big klt adjoint has a good minimal model by
-\cite{BCHM2010}. Pushing the boundary to this model gives the stated
+[cite]. Pushing the boundary to this model gives the stated
\(\Delta\) and \(c_t=(1+\sigma m)/r\).
We turn to the volume formulas. The weight-\(l\) summand in degree
\(k\) has base classes \(kL_{l/k}\) and \(kL_{q-l/k}\); on a slice there is one
such factor. The section decomposition and asymptotic Riemann--Roch
-give Equations~\eqref{jet:total-volume}--\eqref{jet:slice-volume}
+give Equations~(jet:total-volume)--(jet:slice-volume)
as Riemann sums. One can justify passage to the integral without
assuming uniform asymptotic Riemann--Roch: partition \([0,q]\) into
rational bins, compare weight classes in each bin by adding and
subtracting the bin width times a fixed very ample divisor
dominating both \(mK\) and \(-mK\), take divisible-degree limits,
and then shrink the bins. Volume continuity gives the formulas.
-Equation~\eqref{jet:volume-comparison} supplies their uniform bounds
+Equation~(jet:volume-comparison) supplies their uniform bounds
and, in particular, proves bigness used above.
-In the range of Equation~\eqref{jet:uniform-normalized-volume},
+In the range of Equation~(jet:uniform-normalized-volume),
the bounds needed when choosing \(q\) are explicitly
\[
- \vol(M)\ge \frac{(2n+1)!}{4(n!)^2}\,q\epsilon^{2n},
+ \vol(M)\ge ((2n+1)!)/(4(n!)^2)\,qε^{2n},
\qquad
- \vol(L+q\xi)\le2(n+1)q\epsilon^n.
+ \vol(L+q\xi)\le2(n+1)qε^n.
\]
Their coefficients are independent of \(q\); the smallness threshold
for \(t\) is allowed to depend on \(q\).
It remains to establish the effective weight comparisons. On a common
-resolution and in compatible divisible degrees, let \(F_{\mathrm{full}}\)
+resolution and in compatible divisible degrees, let \(F_{full}\)
and \(F_b\) be the fixed divisors of the full system and its
weight-\(b\) subsystem, normalized by the degree. Since the second is a
-subsystem, \(F_b\ge F_{\mathrm{full}}\). Its fixed divisor consists of
+subsystem, \(F_b\ge F_{full}\). Its fixed divisor consists of
the toric monomial and the base-model fixed differences, and its moving
class is the sum of the indicated pullbacks of \(N_b\). Hence the full
moving class is that sum plus
-\(E_b=F_b-F_{\mathrm{full}}\ge0\). This proves
-Equation~\eqref{jet:weight-domination}. The same comparison in
+\(E_b=F_b-F_{full}\ge0\). This proves
+Equation~(jet:weight-domination). The same comparison in
individual sufficiently divisible degrees shows that every section
at that weight, after the full fixed subtraction, contains the
corresponding multiple of \(E_b\).
@@ -4696,11 +4695,11 @@ isomorphisms. The two endpoint systems are free there in divisible
degree, so together they have no common zero on the projective-line
fiber. The full system therefore has no fixed subtraction there.
The same holds on a general fiber. Finally, when restricting
-Equation~\eqref{jet:weight-domination} to a component sweeping the
+Equation~(jet:weight-domination) to a component sweeping the
torus open, choose a general component not contained in
\(\Supp E_b\). There are only countably many rational weights, so
these effective restrictions can be required simultaneously.
-\end{proof}
+
The bundle models are now available, and their volumes can be made
large by choosing \(q\). We use this growth to prove a lower bound
@@ -4708,18 +4707,18 @@ for local positivity. Recall that the Seshadri constant of a nef
\(\Q\)-Cartier class \(P\) on a projective variety \(Z\) at a smooth
point \(x\in Z\) is
\[
- \varepsilon(P;x)=
- \inf_{\substack{C\subset Z\ \mathrm{integral\ curve}\\x\in C}}
- \frac{P\cdot C}{\mult_x C}.
+ ε(P;x)=
+ \inf_{\substack{C\subset Z\ integral\ curve\\x\in C}}
+ (P\cdot C)/(\mult_x C).
\]
-\begin{proposition}[Large two-slot Seshadri constant]\label{prop:large-seshadri}
-For each fixed sufficiently small \(\epsilon>0\), there is an integer
+[Large two-slot Seshadri constant]
+For each fixed sufficiently small \(ε>0\), there is an integer
\(q>0\) such that, for all sufficiently small rational \(t>0\) and
-\(r\) as in Equation~\eqref{jet:normalization}, the positive part
-\(P\) of Equation~\eqref{jet:two-slot} has
+\(r\) as in Equation~(jet:normalization), the positive part
+\(P\) of Equation~(jet:two-slot) has
\[
- \varepsilon(P;x)>2n+2
+ ε(P;x)>2n+2
\]
at a very general smooth point of its torus open.
In particular, for some sufficiently divisible integer \(k>0\),
@@ -4727,21 +4726,21 @@ the complete system \(|kP|\) separates jets of order strictly
greater than \((2n+2)k\) at such a point.
The section spaces and these general-point jets agree on the
original bundle and on common birational resolutions.
-\end{proposition}
-\begin{proof}
+
+Proof.
The tracking lemma reduces failure of the assertion to a moving base
component. We will first analyze its fibers over the two base factors,
and then treat components that are generically finite over both.
Write \(d=2n+1\). Choose a gap \(\delta>0\) large enough that
\[
- 2(n+1)\epsilon^n(2/\delta)^n<1.
+ 2(n+1)ε^n(2/\delta)^n<1.
\]
This inequality will exclude a component that is a whole projective-line
fiber of a slice. Now choose \(q\) sufficiently large that
\[
- \left(\frac{(2n+1)!}{4(n!)^2}q\epsilon^{2n}\right)^{1/d}
+ (((2n+1)!)/(4(n!)^2)qε^{2n})^{1/d}
>2n+3+(d+1)\delta.
\]
For all sufficiently small \(t\), the \(d+1\) levels
@@ -4757,37 +4756,37 @@ contained in that system's base locus. Such a curve exists from
the assumed Seshadri bound and the choice \(\tau_0>2n+3\), whether or
not the infimum defining the constant is attained. Thus the lowest
kernel has a positive-dimensional base component, and
-Lemma~\ref{common:moving-kernel} supplies a
+Lemma~common:moving-kernel supplies a
moving proper component \(V\) and both estimates
-\eqref{jet:tracking-estimates}.
+(jet:tracking-estimates).
The numerical estimates already rule out components of general type.
For all moving components under consideration, the effective
-boundary \(\Delta\) of Lemma~\ref{jet:bundle-models} does not contain
+boundary \(\Delta\) of Lemma~jet:bundle-models does not contain
the component. Thus \(K_Z|_V\le c_tP|_V\) in effective order.
Whenever \(V\) is of general type, the curve construction following
-Equation~\eqref{jet:curve-input} and weight domination give a
+Equation~(jet:curve-input) and weight domination give a
contradiction. To treat the remaining components, we begin with
their fibers over a base factor.
-\paragraph{Restriction to a slice.}
+Restriction to a slice.:
Resolve the rational projection from the total model to the first copy
of \(Y\), and work on the open where this resolution and the original
bundle agree. Fixing the first coordinate there gives the opposite
-slice. Write \(Z_{\mathrm{sl}}\) for its model from
-Lemma~\ref{jet:bundle-models} and \(P_{\mathrm{sl}}\) for its positive
+slice. Write \(Z_{sl}\) for its model from
+Lemma~jet:bundle-models and \(P_{sl}\) for its positive
part; their dimension and volume are \(n+1\) and
-Equation~\eqref{jet:slice-volume}, respectively. We use these slice
+Equation~(jet:slice-volume), respectively. We use these slice
objects until returning to the total model below. Choose one sufficiently
divisible Cartier degree for the total model, the slice model, and
their comparison on a common resolution.
Choose a general incidence point on the common isomorphic torus open,
away from the other base components and where the ordinary-power
-containment of Lemma~\ref{common:moving-kernel} holds. Generic smoothness
+containment of Lemma~common:moving-kernel holds. Generic smoothness
allows us also to require that \(V\) be smooth over its actual image
in the first slot there. Let \(F\) be the reduced component of the
scheme fiber through that point, and suppose \(0<\dim F<n+1\).
-Lemma~\ref{common:fiber-restriction} now shows that, near that point, the
+Lemma~common:fiber-restriction now shows that, near that point, the
restricted higher-series base ideal has support exactly \(F\) and lies in the
ordinary power \(\mathcal I_F^h\), with \(h=\lceil k\delta/2\rceil\).
Because \(F\) is proper in the slice, the restricted series is nonzero
@@ -4796,26 +4795,26 @@ and its generic base ideal is primary.
The fixed differences of the full and slice models are units on a
further common open. After removing these fixed divisors and trivializing
the fixed-slot lines, all restricted higher-kernel sections form a
-subspace of \(H^0(Z_{\mathrm{sl}},kP_{\mathrm{sl}})\), with the same
+subspace of \(H^0(Z_{sl},kP_{sl})\), with the same
local base ideal. This subseries need not be the full jet kernel on
the slice: the numerical estimates come from
-Lemma~\ref{common:base-component}, applied to the actual restricted
+Lemma~common:base-component, applied to the actual restricted
subseries.
-The model \(Z_{\mathrm{sl}}\) is projective,
-\(\Q\)-factorial and klt, and \(P_{\mathrm{sl}}\) is nef, big and
+The model \(Z_{sl}\) is projective,
+\(\Q\)-factorial and klt, and \(P_{sl}\) is nef, big and
semiample. Choose the incidence point also in its smooth open.
The component \(F\) is proper, its generic base ideal is primary, and
the preceding restriction gives ordinary multiplicity at least
\(h\ge k\delta/2\). Writing \(f=\dim F\), the lemma therefore gives
\[
- \begin{aligned}
- P_{\mathrm{sl}}^f\cdot F
- &\le(2/\delta)^{n+1-f}P_{\mathrm{sl}}^{n+1},\\
- K_{F^*}P_{\mathrm{sl}}^{f-1}
- &\le(K_{Z_{\mathrm{sl}}}+cP_{\mathrm{sl}})|_F
- P_{\mathrm{sl}}^{f-1},
+
+ P_{sl}^f\cdot F
+ &\le(2/\delta)^{n+1-f}P_{sl}^{n+1},\\
+ K_{F^*}P_{sl}^{f-1}
+ &\le(K_{Z_{sl}}+cP_{sl})|_F
+ P_{sl}^{f-1},
\qquad 0<c\le2(n+1-f)/\delta.
- \end{aligned}
+
\]
The restricted components must also move sufficiently to apply curve
@@ -4834,10 +4833,10 @@ The restriction of its positive part to \(F\) is then nef and big,
so its top intersection on \(F\) is positive, as required for the
curve construction.
-\paragraph{Proper base images on a slice.}
+Proper base images on a slice.:
If \(F\) is generically finite over a proper positive-dimensional
image \(W\) in the remaining base, then \(W\), and hence \(F\), is
-of general type by Proposition~\ref{prop:general-type}.
+of general type by Proposition~prop:general-type.
The slice estimates and subadjunction give the required bounds on
this component, while weight domination bounds the degrees of its
projected curves. We therefore obtain forbidden bounded curves.
@@ -4846,17 +4845,17 @@ Over a proper base image, the other possibility is that \(F\) is
saturated over its base image \(W\): on the original bundle it is
the full projective-line bundle over \(W\).
Put \(w=\dim W=\dim F-1\). The nef weight comparison gives
-\begin{equation}\label{jet:saturated-degree}
- qN_b^w.W\le P_{\mathrm{sl}}^{w+1}.F.
-\end{equation}
-Indeed \(P_{\mathrm{sl}}-\pi^*N_b\) is effective on the resolved graph
+
+ qN_b^w.W\le P_{sl}^{w+1}.F.
+
+Indeed \(P_{sl}-\pi^*N_b\) is effective on the resolved graph
of the slice projection \(\pi\) to the base, so
successive mixed nef intersections give
-\(P_{\mathrm{sl}}^{w+1}.F\ge
-P_{\mathrm{sl}}(\pi^*N_b)^w.F\). The latter is
+\(P_{sl}^{w+1}.F\ge
+P_{sl}(\pi^*N_b)^w.F\). The latter is
\(qN_b^w.W\), since endpoint freeness gives fiber degree \(q\).
If \(W\) is a point, the lower bound \(q\) contradicts the slice
-Bezout upper bound \(2(n+1)q\epsilon^n(2/\delta)^n\), because our
+Bezout upper bound \(2(n+1)qε^n(2/\delta)^n\), because our
choice of \(\delta\) makes the latter strictly less than \(q\).
Suppose \(w>0\). The ruled component \(F\) itself need not be of
@@ -4876,7 +4875,7 @@ vanishes to this order. Their sum is
times the generic torus fiber, would contradict isolation of \(F\).
Our aim is to construct an effective boundary
-\(\Theta\sim_{\Q}c'L_b\), for some rational weight \(b\), such that
+\(Θ\sim_{\Q}c'L_b\), for some rational weight \(b\), such that
\(W\) is an lc center near its generic point and
\(c'\le2(n-w)/\delta\). Numerical subadjunction will then give a
canonical-intersection bound on \(W\), even though \(F\) is ruled.
@@ -4904,7 +4903,7 @@ To apply the intersection property for lc centers, we first realize
these ideal lc places by an actual divisor. Choose an integer
\(M_0>\max_l u_l\) and \(M_0\) general coefficient divisors
\(D_{l,j}\) from each system. Set
-\(\Theta=\sum_l(u_l/M_0)\sum_{j=1}^{M_0}D_{l,j}\).
+\(Θ=\sum_l(u_l/M_0)\sum_{j=1}^{M_0}D_{l,j}\).
On the simultaneous resolution its fixed crepant coefficients are
at most one, and its general moving divisors meet transversely
with coefficients below one. Thus the pair is lc near the generic
@@ -4918,8 +4917,8 @@ coordinate still has center in \(V(I_l)\).
Set
\[
- b=\frac{\sum_lu_l(l/k)}{\sum_lu_l},
- \qquad c'=k\sum_lu_l\le\frac{kc}{h}.
+ b=(\sum_lu_l(l/k))/(\sum_lu_l),
+ \qquad c'=k\sum_lu_l\le(kc)/(h).
\]
Since a weight-\(l\) coefficient divisor has class \(kL_{l/k}\),
the constructed boundary has class \(c'L_b\). Also
@@ -4930,30 +4929,30 @@ model the boundary has class \(c'N_b\). Numerical subadjunction
and \(N_b=(r+bm)K_{Y(b)}+rtA_b\) give
\[
K_{W^*}N_b^{w-1}
- \le\left(\frac1{r+bm}+c'\right)N_b^w.W.
+ \le(\frac1{r+bm}+c')N_b^w.W.
\]
Indeed a general \(W\) is not contained in a fixed effective
representative of \(A_b\), so its contribution to this intersection
is nonnegative. The slice degree estimate and
-Equation~\eqref{jet:saturated-degree} give, in turn,
+Equation~(jet:saturated-degree) give, in turn,
\[
0<N_b^w.W
- \le\frac{(2/\delta)^{n-w}}q P_{\mathrm{sl}}^{n+1}
- \le2(n+1)\epsilon^n(2/\delta)^{n-w}.
+ \le\frac{(2/\delta)^{n-w}}q P_{sl}^{n+1}
+ \le2(n+1)ε^n(2/\delta)^{n-w}.
\]
These are precisely the two bounds required in
-Equation~\eqref{jet:curve-input}, with constants independent of \(t\).
+Equation~(jet:curve-input), with constants independent of \(t\).
The variety \(W\) is of general type by
-Proposition~\ref{prop:general-type}; normalized curve exclusion
+Proposition~prop:general-type; normalized curve exclusion
again gives a contradiction.
-\paragraph{A slice multisection.}
+A slice multisection.:
Proper base images have now been excluded. The remaining proper
positive-dimensional slice case is
\(\dim F=n\), with \(F\) generically finite of degree \(e\) over
the whole base. Weight comparison and the slice degree bound give
\[
- eN_b^n\le P_{\mathrm{sl}}^n.F,
+ eN_b^n\le P_{sl}^n.F,
\]
so \(e\) is bounded uniformly in \(t\).
Here the two axes provide the boundary needed to use log general
@@ -4964,28 +4963,28 @@ effective integral cycle to \(Y\); call the resulting Weil divisors
\(D_1,D_2\). The two axis classes differ by the pullback of \(mK\),
and \(F\) has generic degree \(e\) over \(Y\). The projection formula
therefore gives
-\begin{equation}\label{jet:axis-signed}
+
D_1-D_2\sim_{\Q} \pm emK.
-\end{equation}
+
At the appropriate endpoint weight \(b\in\{0,q\}\), the difference
\(E_b|_F\) contains the corresponding toric-axis intersection with
coefficient \(q\). There is no full fixed subtraction above
-generic base-divisor points by Lemma~\ref{jet:bundle-models}.
+generic base-divisor points by Lemma~jet:bundle-models.
Intersecting with \(N_b^{n-1}\) and using mixed nef comparison yields
\[
- qN_b^{n-1}.D_i\le P_{\mathrm{sl}}^n.F.
+ qN_b^{n-1}.D_i\le P_{sl}^n.F.
\]
The other divisor has bounded degree for the same \(N_b\), because
-of Equation~\eqref{jet:axis-signed} and
+of Equation~(jet:axis-signed) and
\[
- 0\le K_{Y(b)}N_b^{n-1}\le\frac{N_b^n}{r+bm}.
+ 0\le K_{Y(b)}N_b^{n-1}\le(N_b^n)/(r+bm).
\]
We now apply the logarithmic curve construction on the base \(Y(b)\),
not on the multisection \(F\). The signed divisor
\(\pm(D_1-D_2)/(em)\) represents \(K_{Y(b)}\) by
-Equation~\eqref{jet:axis-signed}. On a log resolution of this base, add
+Equation~(jet:axis-signed). On a log resolution of this base, add
the reduced strict support of \(D_1+D_2\) and all exceptional divisors
-to the canonical divisor. Proposition~\ref{prop:signed-alternative}
+to the canonical divisor. Proposition~prop:signed-alternative
gives a big adjoint for the reduced strict support of \(D_1-D_2\) and
all exceptional divisors. Adding the remaining effective reduced boundary
shows that the resulting log canonical divisor is big as well.
@@ -4995,7 +4994,7 @@ degree no larger than \(D_1+D_2\).
Log effective birationality and the curve construction therefore
contradict normalized curve exclusion on the base itself.
-\paragraph{Reduction to a correspondence.}
+Reduction to a correspondence.:
We now return to the total model \(Z\) and its positive part \(P\).
The slice analysis has excluded fiber dimensions strictly between
zero and \(n+1\) over either slot. A full \((n+1)\)-dimensional fiber over
@@ -5011,7 +5010,7 @@ Thus only the case \(\dim V=n\), dominant with bounded degree over both
copies of \(Y\), remains. We next show that its branch divisors
cannot move; this will reduce the family to finitely many covers.
-\paragraph{Moving branch divisors.}
+Moving branch divisors.:
Consider the dominating algebraic family of these correspondences,
resolving maps after shrinking the parameter space. Suppose a
divisorial branch component of one projection moves. On a
@@ -5032,10 +5031,10 @@ The ramification formula over the terminal target bounds the latter
by \(K_{V^*}P^{n-1}\): its other ramification terms are nonnegative,
and the pullback of the target canonical class is pseudo-effective.
The total-space subadjunction estimate now gives
-\begin{equation}\label{jet:branch-degree}
+
N_b^{n-1}.J\le P^{n-1}.R
\le K_{V^*}P^{n-1}\le C_q.
-\end{equation}
+
The ramification coefficient of \(R\) is a positive integer, so is
at least one; the other ramification and exceptional terms are
nonnegative. The last bound is the total-space subadjunction
@@ -5054,27 +5053,27 @@ in codimension one. On resolving \(J^\nu\), exceptional divisors
map to codimension at least two and pair trivially with the
pulled-back \(N_b^{n-2}\). Consequently, with
\(d_J=N_b^{n-1}.J\),
-\begin{align}
+
K_{J^*}N_b^{n-2}
&\le (K_{Y(b)}+J)J N_b^{n-2}\notag\\
- &\le \frac{d_J}{r+bm}+\frac{d_J^2}{N_b^n}
- \le C'_q d_J.\label{jet:branch-adjunction}
-\end{align}
+ &\le (d_J)/(r+bm)+(d_J^2)/(N_b^n)
+ \le C'_q d_J.
+
For the second line, a general moving \(J\) is not a component of
a fixed effective representative of \(A_b\), giving the first
term; mixed Hodge index gives the second. The lower bound for
-\(N_b^n\) and Equation~\eqref{jet:branch-degree} give the final
+\(N_b^n\) and Equation~(jet:branch-degree) give the final
uniform constant.
A moving \(J\) sweeps very general base points and is of general
-type by Proposition~\ref{prop:general-type}.
-Equations~\eqref{jet:branch-degree}--\eqref{jet:branch-adjunction}
+type by Proposition~prop:general-type.
+Equations~(jet:branch-degree)--(jet:branch-adjunction)
therefore give forbidden bounded curves. Branch components can
be named after a finite parameter extension and shrinking.
It follows that, for all sufficiently small \(t\), all divisorial
branch components in the chosen family are fixed.
-\paragraph{Fixed covers.}
+Fixed covers.:
We now fix $t$. The branch complement is chosen from the one finite-type
family of correspondences under consideration, before any cover is
classified. After a finite parameter extension and shrinking, resolve
@@ -5083,11 +5082,11 @@ and follow the finitely many
irreducible relative ramification divisors. The argument just given
excludes each component whose divisorial images sweep the base: such a
component would supply the bounded-genus covering curves of
-Proposition~\ref{prop:bounded-curves}. For every remaining component the
+Proposition~prop:bounded-curves. For every remaining component the
closure of its divisorial evaluation image is a fixed proper closed
subset of $Y$, and the intersection estimates already bound the degrees
of both projections. These are precisely the hypotheses of
-Lemma~\ref{common:finite-covers} for the smooth projective family of
+Lemma~common:finite-covers for the smooth projective family of
resolved correspondences. It gives a smooth dense open $Y^0$ and only
finitely many finite normal Stein covers in either slot, all etale over
$Y^0$. Bad parameter fibers are removed by shrinking the parameter space,
@@ -5112,7 +5111,7 @@ cover choices, whereas the original incidence dominates the base
product. Its graph incidence has image of dimension \(2n\) in
\(T_1\times T_2\), because this product is finite over \(Y\times Y\).
The product is integral of dimension \(2n\), so the graph incidence
-dominates it. Proposition~\ref{ex:fixed-cover} excludes exactly this
+dominates it. Proposition~ex:fixed-cover excludes exactly this
family of birational maps between the two fixed covers.
Every possible moving component \(V\) has now led to a contradiction,
@@ -5124,10 +5123,10 @@ class downstairs gives the stated divisible jet-separating
multiple. Complete systems agree under the model comparisons,
so this conclusion transfers to the original torus open and to
common resolutions.
-\end{proof}
-\section{Frobenius comparison and smooth nonvanishing}
-\label{fr:section}
+
+# Frobenius comparison and smooth nonvanishing
+
The scalar and two-slot estimates lead to a contradiction through a
comparison in positive characteristic. We first prove that certain
@@ -5137,11 +5136,11 @@ minimal models, and logarithmic subadditivity. We then construct its
inputs from the counterexample geometry and obtain smooth canonical
nonvanishing.
-\subsection{The finite-data Frobenius theorem}
-\label{common:frobenius-tools}
+## The finite-data Frobenius theorem
+
We now work with arbitrary fixed data, independent of the scaling
-models and divisors of Section~\ref{sec:jets}. The comparison concerns
+models and divisors of Section~sec:jets. The comparison concerns
two orders of vanishing of an actual determinant. Fixed ordinary jets
will give a nonzero determinant after reduction. A small polarization bounds the
rank on the diagonal and therefore forces a large order there.
@@ -5155,30 +5154,30 @@ For a line bundle on a smooth variety, to generate jets through
order $a$ at a point means surjectivity to its stalk modulo
the $(a+1)$-st power of the maximal ideal.
-\begin{theorem}[Finite-data Frobenius incompatibility]
-\label{common:finite-data-frobenius}
+[Finite-data Frobenius incompatibility]
+
Let $W$ be a smooth connected projective complex variety of
dimension $n\ge2$. Let $L,H$ be rational Cartier divisors, with
$H$ ample, let $S$ be an integral Cartier divisor, and fix an
-integer $q>0$ and real numbers $r>1$, $\epsilon>0$. Suppose
-\begin{equation}\label{common:intersection-bounds}
- \begin{gathered}
- H^n\le(2\epsilon)^n,\qquad
+integer $q>0$ and real numbers $r>1$, $ε>0$. Suppose
+
+
+ H^n\le(2ε)^n,\qquad
K_WH^{n-1}\le2H^n/r,\qquad
(2L+qS)H^{n-1}\le4H^n,\\
- (14\epsilon)^n<\tfrac14,\qquad 80\epsilon<\tfrac34.
- \end{gathered}
-\end{equation}
+ (14ε)^n<\tfrac14,\qquad 80ε<\tfrac34.
+
+
Assume that the following fixed data exist.
-\begin{enumerate}[label=\textup{(\roman*)}]
+[label=(\roman*)]
\item A smooth integral complete-intersection flag from $W$ to
a curve $C$, whose successive divisor classes are $h_i=l_iH$
for fixed positive integers $l_i$, with
-$\mu_{\min}(\Omega_W^1|_C)\ge0$.
+$\mu_{\min}(Ω_W^1|_C)\ge0$.
\item On the projective bundle
\[
- Z=\mathbf P\bigl(\mathcal O_W(S)_1\oplus
- \mathcal O_W(S)_2\bigr)\longrightarrow W\times W,
+ Z=\mathbf P(\mathcal O_W(S)_1 ⊕
+ \mathcal O_W(S)_2)→ W\times W,
\qquad \xi=c_1(\mathcal O_Z(1)),\qquad M=L_1+L_2+q\xi,
\]
use the convention that the direct image of $\mathcal O_Z(a\xi)$
@@ -5190,19 +5189,19 @@ generate jets through order $(2n+2)k_0$ at $z_*$.
$\pi_x:\widehat W\to W$ with exceptional divisor $J$, and a
curve class
\[
- \gamma=f_*(A_1\cdots A_{n-1}),
+ \gamma=f_*(A_1... A_{n-1}),
\]
where $f:V\to\widehat W$ is one fixed birational morphism,
$V$ is smooth integral projective, and the $A_i$ are ample
integral Cartier divisors, such that $J\cdot\gamma>0$ and
-\begin{equation}\label{common:movable-input}
+
\pi_x^*(L+K_W/2+\lambda S)\cdot\gamma
- \le40\epsilon J\cdot\gamma\qquad(0\le\lambda\le q).
-\end{equation}
+ \le40ε J\cdot\gamma\qquad(0\le\lambda\le q).
+
It suffices to check the two endpoints in this affine inequality.
-\end{enumerate}
+
These data cannot coexist.
-\end{theorem}
+
There is no relation required between $x$ and the two base
coordinates of $z_*$. The divisors $L,S,K_W$ are not assumed
@@ -5212,13 +5211,13 @@ with geometric conditions from which one might construct them.
All data in its statement are fixed before the residual
characteristic tends to infinity.
-\subsubsection{Spreading the fixed witnesses}
+### Spreading the fixed witnesses
We prove the theorem in the next five steps. Put
-$\Lambda=\prod_{i=1}^{n-1}l_i$. The flag satisfies
-\begin{equation}\label{common:flag-degrees}
- [C]=\Lambda H^{n-1},\qquad h_i\cdot C=l_i\Lambda H^n.
-\end{equation}
+$Λ=\prod_{i=1}^{n-1}l_i$. The flag satisfies
+
+ [C]=Λ H^{n-1},\qquad h_i\cdot C=l_iΛ H^n.
+
Choose one sufficiently ample integral Cartier divisor $T_0$ so
that every $T_0+aS$, $0\le a\le(k_0-1)q$, has a section
nonvanishing at each of the two base coordinates of $z_*$.
@@ -5233,7 +5232,7 @@ surjection and nonvanishings persist. Birationality is preserved
by spreading an isomorphism between dense opens. All displayed
intersection numbers are constant.
-We also preserve $\mu_{\min}(\Omega_W^1|_C)\ge0$.
+We also preserve $\mu_{\min}(Ω_W^1|_C)\ge0$.
One way is to spread its characteristic-zero Harder--Narasimhan
filtration, make all its factors locally free, and use openness
of their semistability on curves; their degrees and ranks are
@@ -5256,7 +5255,7 @@ birational morphism and intersect with its fixed ample divisors.
This tests effective divisors on the reduction itself. No
specialization of a characteristic-zero effective cone is used.
-\begin{lemma}\label{common:positive-characteristic-jets}
+
For a sufficiently large residual characteristic $p$, set
\[
N_p=\lfloor p/k_0\rfloor,\qquad B_p=k_0N_pL+T_0.
@@ -5264,9 +5263,9 @@ For a sufficiently large residual characteristic $p$, set
The sections of $pq\xi+B_{p,1}+B_{p,2}$ generate jets through order
$(2n+2)k_0N_p$ at $z_*$. In particular, they surject onto the quotient
of its local ring by the $p$-th powers of all regular parameters.
-\end{lemma}
-\begin{proof}
+
+Proof.
Multiply $N_p$ copies of the fixed jet system. Every monomial of
total degree at most $(2n+2)k_0N_p$ is a product of $N_p$ monomials
of degree at most $(2n+2)k_0$. Taking sections with those leading
@@ -5298,9 +5297,9 @@ quotient by the $p$-th powers of regular parameters has top total
degree $(2n+1)(p-1)$. For large $p$ this is at most
$(2n+2)k_0N_p$. The underlying ordinary/Frobenius ideal comparison is
also recorded in \cite[proof of Proposition~2.12, Equation~(2.8)]{MustataSchwede2014}.
-\end{proof}
-\subsubsection{Full rank off the diagonal}
+
+### Full rank off the diagonal
Let $F:W\to W'$ be relative Frobenius to the base-field twist.
It is finite flat because $W$ is smooth. Write $S'$ for the twist
@@ -5310,64 +5309,64 @@ of $S$, so $F^*S'=pS$, and define
\qquad U_a=H^0(W,\cO_W(B_p+paS)).
\]
Projection formula gives an evaluation map on $W'\times W'$:
-\begin{equation}\label{common:two-slot-evaluation}
- \bigoplus_{\substack{a+b=q\\a,b\ge0}}
- U_a\otimes U_b\otimes
+
+ ⊕_{\substack{a+b=q\\a,b\ge0}}
+ U_a ⊗ U_b ⊗
\cO(-aS'_1-bS'_2)
- \longrightarrow \mathcal E\boxtimes\mathcal E.
-\end{equation}
+ → \mathcal E\boxtimes\mathcal E.
+
+
-\begin{lemma}\label{common:generic-rank}
-The map in Equation~\eqref{common:two-slot-evaluation} has generic rank $R$.
-\end{lemma}
+The map in Equation~(common:two-slot-evaluation) has generic rank $R$.
-\begin{proof}
+
+Proof.
Trivialize the two summands defining $Z$ near the selected base
points. Let $z$ be the torus coordinate, whose value at $z_*$ is
$z_0\ne0$. By the projective-bundle formula, the sections in
-Lemma~\ref{common:positive-characteristic-jets} are weight polynomials in $z$
+Lemma~common:positive-characteristic-jets are weight polynomials in $z$
with weight-$j$ coefficients in
\[
- H^0(W,B_p+jS)\otimes
+ H^0(W,B_p+jS) ⊗
H^0(W,B_p+(pq-j)S),\qquad 0\le j\le pq.
\]
Let $A_*$ be the tensor product of the local algebras of the two
base Frobenius fibers. The local quotient in
-Lemma~\ref{common:positive-characteristic-jets} is
+Lemma~common:positive-characteristic-jets is
\[
A_*[z]/(z^p-z_0^p),
\]
-free over $A_*$ with basis $1,z,\ldots,z^{p-1}$. Project to the
+free over $A_*$ with basis $1,z,...,z^{p-1}$. Project to the
coefficient of $1$ in this basis. Exactly the weights $j=pa$
survive, with multipliers $z_0^{pa}$. Their coefficients are the
values of the corresponding summands of
-Equation~\eqref{common:two-slot-evaluation}, in the chosen frames and the induced
+Equation~(common:two-slot-evaluation), in the chosen frames and the induced
twisted frames. As the full jet map is surjective, these coefficients
span $A_*$, of dimension $p^{2n}=R$. Thus the evaluation has full
rank at this pair of base points, and hence generically. The
calculation uses a basis over the possibly nonreduced algebra $A_*$,
so reducedness of the Frobenius fibers is unnecessary.
-\end{proof}
+
The jet calculation at $z_*$ proves generic full rank; the diagonal
estimate below will be tested at $(x',x')$, obtained from the
separately fixed point $x$.
-\subsubsection{The diagonal filtration}
+### The diagonal filtration
The Frobenius fiber product has the cartesian square
-\begin{equation}\label{common:frobenius-square}
-\begin{tikzcd}
- \Delta_F=W\times_{W'}W \arrow[r,"\mathrm{pr}_2"] \arrow[d,"\mathrm{pr}_1"'] & W \arrow[d,"F"]\\
+
+
+ \Delta_F=W\times_{W'}W \arrow[r,"pr_2"] \arrow[d,"pr_1"'] & W \arrow[d,"F"]\\
W \arrow[r,"F"'] & W'.
-\end{tikzcd}
-\end{equation}
+
+
Its reduced subscheme is the diagonal $W$. Put
\[
\mathcal L_p=
\cO_{\Delta_F}(B_{p,1}+B_{p,2}+pqS_1).
\]
-Write $h=F\circ\mathrm{pr}_1=F\circ\mathrm{pr}_2:
+Write $h=F\circpr_1=F\circpr_2:
\Delta_F\to W'$. On this fiber product,
\[
\cO(pS_1)\simeq h^*\cO_{W'}(S')\simeq\cO(pS_2).
@@ -5378,210 +5377,210 @@ Consequently every pair of weights $a+b=q$ gives the same line bundle:
\simeq\mathcal L_p.
\]
On the diagonal of $W'\times W'$, every source twist of
-Equation~\eqref{common:two-slot-evaluation} becomes $-qS'$.
+Equation~(common:two-slot-evaluation) becomes $-qS'$.
Finite base change and projection formula identify the twisted target as
\[
- (\mathcal E\otimes\mathcal E)(qS')\simeq h_*\mathcal L_p.
+ (\mathcal E ⊗ \mathcal E)(qS')\simeq h_*\mathcal L_p.
\]
The product sections defining the columns restrict to global sections
of this one bundle $\mathcal L_p$. Their values at a diagonal point
therefore lie in the image of $H^0(\Delta_F,\mathcal L_p)$ in the
corresponding fiber of $h_*\mathcal L_p$. This proves that the rank
at every diagonal point is at most
-\begin{equation}\label{common:diagonal-rank-upper}
+
h^0(\Delta_F,\mathcal L_p).
-\end{equation}
+
For a rank-$n$ bundle $V$ in characteristic $p$, denote by $A_j(V)$
the degree-$j$ part of its symmetric algebra modulo the $p$-th powers
of local generators. This construction is independent of frame:
the $p$-th power of a linear combination is the sum of the $p$-th
powers in characteristic $p$. Each $A_j(V)$ is locally free and a
-quotient of $V^{\otimes j}$. Set
+quotient of $V^{ ⊗ j}$. Set
\[
u=n(p-1),\qquad e_j=\rk A_j(V),\qquad
\sum_{j=0}^u e_j=p^n.
\]
Filtering $\mathcal L_p$ by powers of the ideal of the reduced
diagonal in $\Delta_F$ gives the graded bundles
-\begin{equation}\label{common:diagonal-graded}
- \mathcal G_j=\cO_W(2B_p+pqS)\otimes A_j(\Omega_W^1),
+
+ \mathcal G_j=\cO_W(2B_p+pqS) ⊗ A_j(Ω_W^1),
\qquad 0\le j\le u.
-\end{equation}
+
Indeed, in smooth local coordinates the algebra is generated by
parameter differences with their $p$-th powers zero, and its
-degree-one conormal is $\Omega_W^1$. This coordinate calculation
+degree-one conormal is $Ω_W^1$. This coordinate calculation
can be made étale locally or in completions and identifies the
global associated graded. This is the diagonal-ideal form of the canonical Frobenius filtration
\cite[Corollary~3.5]{KitadaiSumihiro2008}. The same calculation, using
only the bundle from the second factor, filters $F^*\mathcal E$ with pieces
\cite[Theorem~3.7]{Sun2008}
-\begin{equation}\label{common:Frob-pullback-graded}
- \cO_W(B_p)\otimes A_j(\Omega_W^1).
-\end{equation}
+
+ \cO_W(B_p) ⊗ A_j(Ω_W^1).
+
Complementary multiplication is a perfect pairing, giving
-\begin{equation}\label{common:socle-pairing}
- A_{u-j}(V)\simeq A_j(V)^\vee\otimes(\det V)^{p-1}.
-\end{equation}
+
+ A_{u-j}(V)\simeq A_j(V)^\vee ⊗ (\det V)^{p-1}.
+
On monomials, each exponent vector pairs with its complement to
-$(p-1,\ldots,p-1)$. The top line transforms by the character
+$(p-1,...,p-1)$. The top line transforms by the character
$(\det V)^{p-1}$: this is immediate on diagonal matrices and hence
- identifies the character of $\mathrm{GL}_n$ on that line.
+ identifies the character of $GL_n$ on that line.
-\subsubsection{One slope bound and one flag polynomial}
+### One slope bound and one flag polynomial
We bound the sections of every graded bundle in
-Equation~\eqref{common:diagonal-graded} by its rank times the same scalar
+Equation~(common:diagonal-graded) by its rank times the same scalar
polynomial in $p$. This uniformity lets us sum the ranks of the
graded pieces without an additional factor for their number.
We begin with a slope bound on the fixed curve; here slope means
ordinary degree divided by rank.
-\begin{lemma}\label{common:truncated-slope}
+
There is a constant $c\ge0$, independent of the sufficiently large
residual characteristic $p$ and of $j$, such that for
-$V=\Omega_W^1|_C$ one has
+$V=Ω_W^1|_C$ one has
\[
\mu_{\max}(A_j(V))\le(p-1)K_W\cdot C+nc.
\]
-\end{lemma}
-\begin{proof}
+
+Proof.
Write $F_C$ for absolute Frobenius of $C$. Langer's instability
estimate \cite[Corollary~2.5]{Langer2004} gives
-\begin{equation}\label{common:Langer-min}
+
L_{\min}(V):=
- \lim_{e\to\infty}p^{-e}\mu_{\min}(F_C^{e*}V)
- \ge-\frac{c}{p-1}.
-\end{equation}
+ \lim_{e\to∞}p^{-e}\mu_{\min}(F_C^{e*}V)
+ \ge-(c)/(p-1).
+
To check the uniform constant, the discrepancy from
$\mu_{\min}(V)\ge0$ is at most
$(\rk V-1)\deg A_C/(p-1)$, where $A_C$ is a nef line bundle such
-that $T_C\otimes A_C$ is globally generated. Its degree can be
+that $T_C ⊗ A_C$ is globally generated. Its degree can be
bounded in terms of the fixed genus, using a sufficiently positive
line bundle. Here the rank is $n$, irrespective of the fact that the base
is a curve.
We also need, for every integer $i\ge0$,
-\begin{equation}\label{common:tensor-min}
- \mu_{\min}(V^{\otimes i})\ge iL_{\min}(V).
-\end{equation}
+
+ \mu_{\min}(V^{ ⊗ i})\ge iL_{\min}(V).
+
Fix $p$ and $i$. Choose a line bundle $A_e$ of degree
$-\mu_{\min}(F_C^{e*}V)+O(1)$, rounded up with a fixed
genus-dependent additive constant, so that
-$F_C^{e*}V\otimes A_e$ is globally generated. This follows from
+$F_C^{e*}V ⊗ A_e$ is globally generated. This follows from
Serre duality: after subtracting any point, its minimum slope can
be made greater than $2g(C)-2$, which annihilates $H^1$ and makes
evaluation at that point surjective. For every vector-bundle
-quotient $Q$ of $V^{\otimes i}$, the bundle
-$F_C^{e*}Q\otimes A_e^{\otimes i}$ is then globally generated.
+quotient $Q$ of $V^{ ⊗ i}$, the bundle
+$F_C^{e*}Q ⊗ A_e^{ ⊗ i}$ is then globally generated.
Consequently
\[
p^e\mu(Q)+i\deg A_e\ge0.
\]
-Divide by $p^e$ and let $e\to\infty$ to prove
-Equation~\eqref{common:tensor-min}. This limit is taken for each fixed $p,i$;
+Divide by $p^e$ and let $e\to∞$ to prove
+Equation~(common:tensor-min). This limit is taken for each fixed $p,i$;
there is no interchange of Frobenius-iteration and characteristic
limits.
-Since $A_{u-j}(V)$ is a quotient of $V^{\otimes(u-j)}$, the perfect
-pairing Equation~\eqref{common:socle-pairing} yields
-\begin{align*}
+Since $A_{u-j}(V)$ is a quotient of $V^{ ⊗ (u-j)}$, the perfect
+pairing Equation~(common:socle-pairing) yields
+
\mu_{\max}(A_j(V))
&= (p-1)\deg\det V-\mu_{\min}(A_{u-j}(V))\\
- &\le(p-1)K_W\cdot C+\frac{(u-j)c}{p-1}\\
+ &\le(p-1)K_W\cdot C+((u-j)c)/(p-1)\\
&\le(p-1)K_W\cdot C+nc.
-\end{align*}
-\end{proof}
+
+
The divisors $B_p-pL=T_0-(p-k_0N_p)L$ range over a fixed finite
-list. Combining Equation~\eqref{common:intersection-bounds}, Equation~\eqref{common:flag-degrees}, Lemma~\ref{common:truncated-slope}
+list. Combining Equation~(common:intersection-bounds), Equation~(common:flag-degrees), Lemma~common:truncated-slope
therefore gives, uniformly for $0\le j\le u$,
-\begin{align}
+
\mu_{\max}(\mathcal G_j|_C)
- &\le p(4+2/r)\Lambda H^n+O(1)\notag\\
- &\le M_p:=7p\Lambda H^n+c_0,\label{common:common-slope-bound}
-\end{align}
+ &\le p(4+2/r)Λ H^n+O(1)\notag\\
+ &\le M_p:=7pΛ H^n+c_0,
+
where $c_0\ge0$ is fixed. The coefficient $7$ provides harmless
room in this common bound.
-\begin{lemma}\label{common:diagonal-sections}
+
For all sufficiently large $p$,
\[
h^0(\Delta_F,\mathcal L_p)
- \le R\bigl(7^nH^n+O(1/p)\bigr)<R/4.
+ \le R(7^nH^n+O(1/p))<R/4.
\]
The error constant is independent of the filtration index $j$.
-\end{lemma}
-\begin{proof}
+
+Proof.
Write the fixed flag as
\[
- W=W_n\supset W_{n-1}\supset\cdots\supset W_1=C,
+ W=W_n\supset W_{n-1}\supset...\supset W_1=C,
\qquad W_{n-k}\in|h_k|_{W_{n-k+1}}
\quad(1\le k\le n-1).
\]
At step $k$, fix the preceding nonnegative integers
-$b_1,\ldots,b_{k-1}$ and put
+$b_1,...,b_{k-1}$ and put
\[
\mathcal F_k=
\mathcal G_j|_{W_{n-k+1}}
- \Bigl(-\sum_{i<k}b_i h_i\Bigr).
+ (-\sum_{i<k}b_i h_i).
\]
For each $b_k\ge0$, the divisor restriction sequence is
\[
- 0\longrightarrow\mathcal F_k(-(b_k+1)h_k)
- \longrightarrow\mathcal F_k(-b_kh_k)
- \longrightarrow
+ 0→\mathcal F_k(-(b_k+1)h_k)
+ →\mathcal F_k(-b_kh_k)
+ →
\mathcal F_k(-b_kh_k)|_{W_{n-k}}
- \longrightarrow0.
+ →0.
\]
-Iterating it for $b_k=0,1,\ldots$ and then proceeding to the next
+Iterating it for $b_k=0,1,...$ and then proceeding to the next
member of the flag bounds the section space by
-\begin{equation}\label{common:flag-lattice-sum}
+
h^0(W,\mathcal G_j)\le
- \sum_{b_1,\ldots,b_{n-1}\ge0}
- h^0\left(C,\mathcal G_j|_C\Bigl(-\sum_i b_i h_i|_C\Bigr)\right).
-\end{equation}
+ \sum_{b_1,...,b_{n-1}\ge0}
+ h^0(C,\mathcal G_j|_C(-\sum_i b_i h_i|_C)).
+
At each stage the remainder is zero after sufficiently negative
ample twisting. Its stopping index need not be uniform in $j,p$
or in the preceding twists: enlarging the nonnegative finite sums
to the displayed lattice sum preserves the inequality.
-A summand in Equation~\eqref{common:flag-lattice-sum} vanishes when
+A summand in Equation~(common:flag-lattice-sum) vanishes when
$\sum_i b_i(h_i\cdot C)>M_p$, because then its maximum slope is
negative. Each nonzero summand has at most $e_j(M_p+1)$ sections.
Indeed, evaluation at $\lfloor M_p\rfloor+1$ distinct points is injective;
the kernel has negative maximum slope after subtracting those
-points. By Equation~\eqref{common:flag-degrees} it follows that
-\begin{align}
+points. By Equation~(common:flag-degrees) it follows that
+
h^0(W,\mathcal G_j)
&\le e_j(M_p+1)
\prod_{i=1}^{n-1}
- \left(1+\frac{M_p}{l_i\Lambda H^n}\right)\notag\\
- &=e_jp^n\bigl(7^nH^n+O(1/p)\bigr).
- \label{common:common-flag-polynomial}
-\end{align}
+ (1+(M_p)/(l_iΛ H^n))\notag\\
+ &=e_jp^n(7^nH^n+O(1/p)).
+
+
The leading coefficient follows from
-$\Lambda=\prod_i l_i$. More importantly, the whole scalar
+$Λ=\prod_i l_i$. More importantly, the whole scalar
polynomial on the first line is the same for every $j$.
-Sum over the filtration in Equation~\eqref{common:diagonal-graded}. The ranks
-satisfy $\sum_j e_j=p^n$, so Equation~\eqref{common:common-flag-polynomial} gives
+Sum over the filtration in Equation~(common:diagonal-graded). The ranks
+satisfy $\sum_j e_j=p^n$, so Equation~(common:common-flag-polynomial) gives
the asserted bound with $R=p^{2n}$. There is no extra factor for
the number of graded pieces. Finally
\[
- 7^nH^n\le(14\epsilon)^n<1/4
+ 7^nH^n\le(14ε)^n<1/4
\]
-by Equation~\eqref{common:intersection-bounds}, proving the strict inequality
+by Equation~(common:intersection-bounds), proving the strict inequality
for all large $p$.
-\end{proof}
-\subsubsection{The determinant has incompatible orders}
+
+### The determinant has incompatible orders
Choose $R$ generically independent columns of
-Equation~\eqref{common:two-slot-evaluation} from bases of its source vector spaces.
+Equation~(common:two-slot-evaluation) from bases of its source vector spaces.
Their determinant is a nonzero section
\[
0\ne\sigma\in H^0(W'\times W',\mathcal Q_1\boxtimes\mathcal Q_2)
@@ -5593,42 +5592,42 @@ $(\det\mathcal E)^s\boxtimes(\det\mathcal E)^s$, and the
chosen source determinant is
$\cO(-m_1S')\boxtimes\cO(-m_2S')$. Thus
\[
- \mathcal Q_i=(\det\mathcal E)^{\otimes s}
- \otimes\cO_{W'}(m_iS'),\qquad
+ \mathcal Q_i=(\det\mathcal E)^{ ⊗ s}
+ ⊗ \cO_{W'}(m_iS'),\qquad
\lambda_i=m_i/R.
\]
Every column weight lies between $0$ and $q$, so
$0\le\lambda_i\le q$ independently of the columns chosen.
Under the numerical identification with the base-field twist,
-\begin{equation}\label{common:slot-determinant-class}
+
\frac{c_1(\mathcal Q_i)}R
=\frac{c_1(\mathcal E)}s+\lambda_iS
- =\frac{B_p}p+\frac{p-1}{2p}K_W+\lambda_iS,
+ =\frac{B_p}p+(p-1)/(2p)K_W+\lambda_iS,
\qquad 0\le\lambda_i\le q.
-\end{equation}
+
The Frobenius first-Chern-class identity is also
\cite[Lemma~4.2]{Sun2008}; it is an identity of rational divisor classes. For the second equality, use
-Equation~\eqref{common:Frob-pullback-graded}. Complementary degrees in
-Equation~\eqref{common:socle-pairing} have total first Chern class
+Equation~(common:Frob-pullback-graded). Complementary degrees in
+Equation~(common:socle-pairing) have total first Chern class
$e_j(p-1)K_W$. Summing gives
\[
- \sum_jc_1(A_j(\Omega_W^1))=\frac{s(p-1)}2K_W.
+ \sum_jc_1(A_j(Ω_W^1))=\frac{s(p-1)}2K_W.
\]
Thus $c_1(F^*\mathcal E)=sB_p+s(p-1)K_W/2$, and Frobenius pullback of
twisted divisor classes multiplies by $p$, as required.
For $0\le\lambda\le q$, the fixed movable inequality gives
\[
- (L+K_W/2+\lambda S)\cdot\gamma\le40\epsilon J\cdot\gamma,
+ (L+K_W/2+\lambda S)\cdot\gamma\le40ε J\cdot\gamma,
\]
where pullback by $\pi_x$ is understood. This concerns only
intersections of fixed divisors and the fixed complete-intersection
witness, so it remains true on the reduction. The class in
-Equation~\eqref{common:slot-determinant-class} differs from its
+Equation~(common:slot-determinant-class) differs from its
left-hand divisor at $\lambda=\lambda_i$ by
\[
- \frac1p\bigl(B_p-pL-K_W/2\bigr).
+ \frac1p(B_p-pL-K_W/2).
\]
Its numerator belongs to the fixed finite list
$T_0-aL-K_W/2$, $0\le a<k_0$. The resulting error after
@@ -5639,52 +5638,52 @@ Let $x'$ be the twisted point and let $0\ne\tau_i\in
H^0(W',\mathcal Q_i)$. The strict transform of its effective divisor on
the blowup at $x'$ pairs nonnegatively with the twist of $\gamma$.
Since $J\cdot\gamma>0$, it follows that
-\begin{equation}\label{common:slot-order}
- \frac{\ord_{x'}(\tau_i)}R\le40\epsilon+O(1/p).
-\end{equation}
+
+ \frac{\ord_{x'}(\tau_i)}R\le40ε+O(1/p).
+
The constant is uniform in the columns and in $\tau_i$. This
argument tests sections on the reduction itself; it does not
require them to lift to characteristic zero.
K\"unneth identifies the space containing $\sigma$ with
-$H^0(W',\mathcal Q_1)\otimes H^0(W',\mathcal Q_2)$. Choose bases adapted to the
+$H^0(W',\mathcal Q_1) ⊗ H^0(W',\mathcal Q_2)$. Choose bases adapted to the
orders of vanishing at $x'$ in each factor. In each degree their
leading terms are linearly independent. Tensor products of these
leading terms remain independent in each bidegree of the two
disjoint sets of parameters; terms of different bidegrees cannot
cancel. Every nonzero tensor consequently has order at most the
sum of the two maximum basis orders. Applying
-Equation~\eqref{common:slot-order}, we obtain
-\begin{equation}\label{common:determinant-order-upper}
- \ord_{(x',x')}\sigma\le R\bigl(80\epsilon+O(1/p)\bigr).
-\end{equation}
+Equation~(common:slot-order), we obtain
+
+ \ord_{(x',x')}\sigma\le R(80ε+O(1/p)).
+
-On the other hand, Equation~\eqref{common:diagonal-rank-upper}, Lemma~\ref{common:diagonal-sections}
+On the other hand, Equation~(common:diagonal-rank-upper), Lemma~common:diagonal-sections
bound the matrix rank at $(x',x')$ by a number strictly smaller
than $R/4$. Trivialize its line bundles near that point. Invertible
constant row operations make more than $3R/4$ rows vanish at the
point. Each entry of those rows belongs to the maximal ideal, so
every term of the determinant has order at least $3R/4$. Therefore
-\begin{equation}\label{common:determinant-order-lower}
+
\ord_{(x',x')}\sigma\ge3R/4.
-\end{equation}
-Equations \eqref{common:determinant-order-upper} and
-\eqref{common:determinant-order-lower} contradict
-$80\epsilon<3/4$ for sufficiently large $p$.
+
+Equations (common:determinant-order-upper) and
+(common:determinant-order-lower) contradict
+$80ε<3/4$ for sufficiently large $p$.
-This contradiction proves Theorem~\ref{common:finite-data-frobenius}.
+This contradiction proves Theorem~common:finite-data-frobenius.
The full-rank calculation used $z_*$, while the diagonal estimate
holds at every point and was tested at the separately chosen $x$.
The errors were controlled by one fixed finite list of divisor
classes and one fixed flag polynomial. Thus their constants do
not depend on $p$, the filtration index, or the chosen columns.
-\subsection{The geometric witnesses and smooth nonvanishing}
-\label{fr:geometric-witnesses}
+## The geometric witnesses and smooth nonvanishing
+
We return to the notation and the hypothetical counterexample of
-Section~\ref{sec:jets}. Theorem~\ref{common:finite-data-frobenius}
+Section~sec:jets. Theorem~common:finite-data-frobenius
asks for fixed divisors, a small polarization and flag, finite two-slot
jets, and one actual strongly movable curve. We construct all these
inputs in characteristic zero. The positive parts on the scaling
@@ -5692,13 +5691,13 @@ models, rather than the original non-nef classes, provide the small
polarization. No minimal-model statement or pseudo-effective cone is
specialized to positive characteristic.
-\begin{theorem}[The smooth nonvanishing step]
-\label{thm:smooth-nonvanishing}
+[The smooth nonvanishing step]
+
Assume the lower-dimensional real-boundary good-model hypothesis of
-Assumption~\ref{mmp:lower-models}.
+Assumption~mmp:lower-models.
If \(X\) is a smooth projective complex variety of dimension \(n\) and
\(K_X\) is pseudo-effective, then \(\kappa(X,K_X)\geq0\).
-\end{theorem}
+
In dimension zero the assertion is immediate. On a smooth curve,
pseudo-effectivity of \(K_X\) gives \(g(X)\geq1\), hence
@@ -5711,24 +5710,24 @@ section. Fix \(m>0\) with \(mK\) Cartier, and write
L=r(K+tA).
\]
As before, \(r\) is a positive integer chosen so that
-\(r\mu_t\longrightarrow\epsilon\) as \(t\downarrow0\).
-Choose the rational number \(\epsilon>0\) sufficiently small for
-Propositions~\ref{prop:scalar-jets} and~\ref{prop:large-seshadri}, and also
+\(r\mu_t→ε\) as \(t↓0\).
+Choose the rational number \(ε>0\) sufficiently small for
+Propositions~prop:scalar-jets and~prop:large-seshadri, and also
so that
-\begin{equation}
-\label{fr:epsilon}
- (14\epsilon)^n<\frac14,\qquad
- 80\epsilon<\frac34.
-\end{equation}
+
+
+ (14ε)^n<\frac14,\qquad
+ 80ε<\frac34.
+
The strict inequalities leave room for the errors that tend to zero
with the characteristic. We next fix a sufficiently large integer
-\(q\) as in Proposition~\ref{prop:large-seshadri}, and then choose
+\(q\) as in Proposition~prop:large-seshadri, and then choose
a positive rational \(t\) small enough for the estimates below and for
\(r>qm+1\). Once those estimates hold, \(t\) and \(r\) remain fixed
through all the subsequent characteristic-zero constructions and
reductions.
-\subsubsection{A small polarization and a cotangent flag}
+### A small polarization and a cotangent flag
A small ample polarization and a complete-intersection flag will
control the number of sections on the Frobenius thickening of the
@@ -5743,12 +5742,12 @@ for \(K+tA\). In this application \(K,A,L\) also denote their pullbacks to
Let \(H_0\) be the pullback of the nef semiample positive part \(N_0\) of
\(L\) on its scaling model. The comparison of canonical pullbacks and
moving parts gives
-\begin{equation}
-\label{fr:limit-intersections}
- H_0^n\longrightarrow\epsilon^n,\qquad
+
+
+ H_0^n→ε^n,\qquad
LH_0^{n-1}=H_0^n,\qquad
- K_WH_0^{n-1}=KH_0^{n-1}\leq\frac{H_0^n}{r}.
-\end{equation}
+ K_WH_0^{n-1}=KH_0^{n-1}\leq(H_0^n)/(r).
+
To justify these intersections, recall that the maps
\(Y\dashrightarrow Y_t\) are small by the choice of the late model.
Every divisor exceptional over \(Y\) is therefore also exceptional
@@ -5761,18 +5760,18 @@ In particular, since \(qm/r<1\),
(2L+qmK)H_0^{n-1}
\le(2+qm/r)H_0^n<3H_0^n.
\]
-Choose \(t\) small enough that also \(H_0^n<(2\epsilon)^n\), and
+Choose \(t\) small enough that also \(H_0^n<(2ε)^n\), and
now fix \(t,r\) and the resolution \(W\). For a fixed ample divisor
-\(H_{\mathrm{amp}}\), put \(H=H_0+\eta H_{\mathrm{amp}}\), where
+\(H_{amp}\), put \(H=H_0+\eta H_{amp}\), where
\(\eta>0\) is rational. The strict margins just obtained, together
with \(K_WH_0^{n-1}\le H_0^n/r\), allow us to choose \(\eta\)
small enough that continuity gives
-\begin{equation}
-\label{fr:intersection-bounds}
- H^n\leq(2\epsilon)^n,\qquad
- K_WH^{n-1}\leq\frac{2H^n}{r},\qquad
+
+
+ H^n\leq(2ε)^n,\qquad
+ K_WH^{n-1}\leq(2H^n)/(r),\qquad
(2L+qmK)H^{n-1}\leq4H^n.
-\end{equation}
+
We next choose the flag for the section estimates. The canonical
divisor of \(W\) is pseudo-effective. Generic
@@ -5784,19 +5783,19 @@ Apply restriction to the Harder--Narasimhan filtration and its factors
large divisible integers $l_i$ such that $h_i=l_iH$ are integral very ample
classes, and choose a smooth integral complete-intersection flag
\[
- W=W_n\supset W_{n-1}\supset\cdots\supset W_1=C,
+ W=W_n\supset W_{n-1}\supset...\supset W_1=C,
\qquad W_{i-1}\in |h_{n-i+1}|_{W_i},
\]
for which
-\begin{equation}
-\label{fr:generic-nef}
- \mu_{\min}(\Omega_W^1|_C)\geq0.
-\end{equation}
+
+
+ \mu_{\min}(Ω_W^1|_C)\geq0.
+
Here and below slopes on \(C\) mean degree divided by rank. One may
obtain the flag by applying restriction to the finitely many
-Harder--Narasimhan quotients of \(\Omega_W^1\).
+Harder--Narasimhan quotients of \(Ω_W^1\).
-\subsubsection{A movable witness for all determinant weights}
+### A movable witness for all determinant weights
For the vanishing-order estimate, choose a very general point
\(x\in W\), away from the exceptional loci, and let
@@ -5806,28 +5805,28 @@ divisor \(J\). Set
D^+=2r(K+2tA)+2E.
\]
This big divisor has the scalar sections controlled by
-Proposition~\ref{prop:scalar-jets}: pushing to \(Y\) identifies the
+Proposition~prop:scalar-jets: pushing to \(Y\) identifies the
section spaces, since the added exceptional part does not change them.
If \(\rho=\vol(2r(K+2tA))^{1/n}\), then
\[
- \rho\leq4r\mu_t\leq8\epsilon
+ \rho\leq4r\mu_t\leq8ε
\]
for our sufficiently small \(t\). Thus every section of a sufficiently
divisible multiple of \(D^+\) has normalized order at \(x\) at most
-\(4\rho\leq32\epsilon\).
+\(4\rho\leq32ε\).
We convert this order bound into a curve inequality.
-It follows that \(\pi_x^*D^+-40\epsilon J\) is not pseudo-effective.
+It follows that \(\pi_x^*D^+-40ε J\) is not pseudo-effective.
Otherwise its convex combination with the big class \(\pi_x^*D^+\)
would make \(\pi_x^*D^+-cJ\) big for some rational
-\(32\epsilon<c<40\epsilon\), giving a section of excessive order.
+\(32ε<c<40ε\), giving a section of excessive order.
Movable-curve duality \cite[Theorem~2.2]{BDPP2013} therefore supplies an
actual covering curve class \(\gamma\) on \(\widehat W\) such that
-\begin{equation}
-\label{fr:curve-test}
+
+
(\pi_x^*D^+)\cdot\gamma
- <40\epsilon\,J\cdot\gamma,\qquad J\cdot\gamma>0.
-\end{equation}
+ <40ε\,J\cdot\gamma,\qquad J\cdot\gamma>0.
+
The strictness lets us fix an algebraic family that will survive
reduction. More precisely, the negative pairing can first be detected
by a strongly movable curve: take the pushforward of a general complete
@@ -5839,22 +5838,22 @@ rather than a limiting real movable class. Positivity of
pseudo-effectivity of \(\pi_x^*D^+\).
To turn this curve into the witness required by
-Theorem~\ref{common:finite-data-frobenius}, set $S=mK$ on $W$;
+Theorem~common:finite-data-frobenius, set $S=mK$ on $W$;
it is an integral Cartier divisor. We verify all its weight inequalities
before proceeding to the jet construction. Put
$Q_\lambda=L+K_W/2+\lambda S$ for $0\le\lambda\le q$.
The equality $K_W=K+E$ gives
-\begin{equation}\label{fr:slot-domination}
+
D^+-Q_\lambda
=(r-\tfrac12-\lambda m)K+3rtA+\tfrac32E.
-\end{equation}
+
This class is pseudo-effective in characteristic zero: $r>qm+1$,
$K$ is pseudo-effective, and $A,E$ are effective up to the stated
equivalences. Pairing with the fixed strongly movable class therefore gives
\[
\pi_x^*Q_\lambda\cdot\gamma
\le \pi_x^*D^+\cdot\gamma
- <40\epsilon J\cdot\gamma,
+ <40ε J\cdot\gamma,
\qquad J\cdot\gamma>0.
\]
These are the required movable inequalities. Only the two endpoint
@@ -5863,7 +5862,7 @@ affine in $\lambda$. The common theorem spreads these fixed numerical
inequalities and the actual birational complete-intersection witness;
it does not spread the pseudo-effective assertion used to obtain them.
-\subsubsection{A finite two-slot jet system}
+### A finite two-slot jet system
We must similarly realize the two-slot jet estimate by finitely many
sections before reducing. Their products will then supply the jet
@@ -5871,12 +5870,12 @@ orders needed for arbitrarily large characteristics.
On \(W\times W\), let
\[
- Z=\PP(mK_1\oplus mK_2),\qquad
+ Z=\PP(mK_1 ⊕ mK_2),\qquad
\xi=\OO_Z(1),\qquad
M=L_1+L_2+q\xi,
\]
with the symmetric-power convention for the projective bundle.
-Proposition~\ref{prop:large-seshadri} gives a semiample big model whose
+Proposition~prop:large-seshadri gives a semiample big model whose
Seshadri constant exceeds \(2n+2\) at a very general torus point.
On the open set where its birational contraction is an isomorphism,
the jet interpretation of the Seshadri constant consequently gives
@@ -5893,13 +5892,13 @@ Enlarge \(k_0\) to clear every denominator, in particular that of \(L\),
and fix finitely many sections realizing this jet surjection.
-\subsubsection{Applying the comparison}
+### Applying the comparison
All the preceding choices are now fixed. With $S=mK$, the dimension,
intersection and flag hypotheses of
-Theorem~\ref{common:finite-data-frobenius} are
-Equations~\eqref{fr:epsilon}, \eqref{fr:intersection-bounds},
-and~\eqref{fr:generic-nef}. The curve construction gives its movable
+Theorem~common:finite-data-frobenius are
+Equations~(fr:epsilon), (fr:intersection-bounds),
+and~(fr:generic-nef). The curve construction gives its movable
witness and both endpoint inequalities, and the finite jet system
gives the required surjection at $z_*$. This point need not lie over
the separately chosen testing point $x$. The proof of the common
@@ -5916,15 +5915,15 @@ minor vanishes to order at least \(3R/4\) at \((x',x')\), where \(x'\)
is the Frobenius image of the testing point \(x\). Its line bundle
is an actual external product. Applying the fixed curve inequality
to each factor bounds the same order above by
-\(R(80\epsilon+O(p^{-1}))\), a contradiction for large \(p\).
+\(R(80ε+O(p^{-1}))\), a contradiction for large \(p\).
The theorem proves these determinant estimates for the fixed witnesses
we have now supplied; it requires no additional geometric input here.
All constants are fixed before the residual prime varies.
The contradiction excludes the assumed counterexample and proves
-Theorem~\ref{thm:smooth-nonvanishing}.
+Theorem~thm:smooth-nonvanishing.
-\section{Completion and change of ground field}\label{sec:completion}
+# Completion and change of ground field
Smooth nonvanishing is now available in the order required by the
induction. We first finish the complex proof, including descent from
@@ -5932,21 +5931,21 @@ the good model to the original normal variety. We then descend the
globally generated Cartier multiple to an arbitrary algebraically
closed field of characteristic zero.
-\begin{proof}[Proof of Theorem~\ref{thm:main} over \(\C\)]
+Proof. [Proof of Theorem~thm:main over \(\C\)]
Use dimension induction in the real-boundary category of
-Proposition~\ref{prop:inductive-reduction}. In dimension zero the
+Proposition~prop:inductive-reduction. In dimension zero the
variety is a point. Suppose the good-model assertion holds in all
smaller dimensions. The signed-representative argument
-(Theorem~\ref{thm:signed}), the exclusions, and the jet estimates are
+(Theorem~thm:signed), the exclusions, and the jet estimates are
then available with exactly that lower-dimensional hypothesis.
-Theorem~\ref{thm:smooth-nonvanishing} establishes smooth
-nonvanishing in the present dimension. Proposition~\ref{prop:inductive-reduction}
+Theorem~thm:smooth-nonvanishing establishes smooth
+nonvanishing in the present dimension. Proposition~prop:inductive-reduction
therefore proves the real-boundary good-model assertion in the
present dimension. This is precisely the hypothesis needed to
continue to the next dimension.
Apply this to the rational lc pair \((X,B)\) of
-Theorem~\ref{thm:main}. Let \(f:(X_d,B_d)\to(X,B)\) be a crepant dlt
+Theorem~thm:main. Let \(f:(X_d,B_d)\to(X,B)\) be a crepant dlt
model and let \((V,B_V)\) be its good log minimal model. Take a common
resolution \(a:W\to X_d\), \(q:W\to V\), and put \(p=f\circ a\).
Then
@@ -5954,7 +5953,7 @@ Then
p^*(K_X+B)=a^*(K_{X_d}+B_d)=q^*(K_V+B_V).
\]
The first equality is crepancy. The second is the nef comparison in
-Lemma~\ref{mmp:comparison}, since \(K_{X_d}+B_d=f^*(K_X+B)\) is nef.
+Lemma~mmp:comparison, since \(K_{X_d}+B_d=f^*(K_X+B)\) is nef.
The right side is semiample because \((V,B_V)\) is a good model, so
the pullback of \(K_X+B\) is semiample. Normality gives
\(p_*\OO_W=\OO_X\). Choose a sufficiently divisible Cartier multiple
@@ -5963,21 +5962,21 @@ formula identifies its sections with those of its pullback.
Surjectivity of evaluation upstairs implies surjectivity downstairs:
otherwise a base point downstairs would make every pulled-back
section vanish on its nonempty fiber. Hence \(K_X+B\) is semiample.
-\end{proof}
-\begin{proposition}[Change of algebraically closed field]\label{prop:field-descent}
-The conclusion of Theorem~\ref{thm:main} over \(\C\) implies its
+
+[Change of algebraically closed field]
+The conclusion of Theorem~thm:main over \(\C\) implies its
conclusion over every algebraically closed field of characteristic zero.
-\end{proposition}
-\begin{proof}
+
+Proof.
Let \((X,B)\) be defined over such a field \(k\). Choose a finitely
generated subfield \(k_1\subset k\) over which the projective variety,
the rational boundary, a Cartier multiple of its adjoint, and a log
resolution are all defined. After enlarging \(k_1\) if necessary,
these data base change to the given data. Let \(k_0\) be its algebraic
closure inside \(k\), and denote the resulting pair by \((X_0,B_0)\).
-Choose an embedding \(k_0\hookrightarrow\C\), and write
+Choose an embedding \(k_0↪\C\), and write
\((X_{\C},B_{\C})\) for the complex base change.
The chosen log resolution tests log canonicity after either
@@ -5997,7 +5996,7 @@ nefness over that field. Conversely, curves over the smaller field
remain available after extension. Apply the converse to
\(k_0\subset k\): nefness of \(K_X+B\) gives nefness of
\(K_{X_0}+B_0\). Apply the forward direction to
-\(k_0\hookrightarrow\C\): its complex base change is nef as well.
+\(k_0↪\C\): its complex base change is nef as well.
The complex case supplies an integer \(m>0\), enlarged to a multiple
of the Cartier index fixed over \(k_0\), for which the Cartier
@@ -6005,7 +6004,7 @@ line bundle \(M=\OO_{X_0}(m(K_{X_0}+B_0))\) becomes
globally generated over \(\C\). Proper flat base change for sections
identifies
\[
- H^0(X_0,M)\otimes_{k_0}\C
+ H^0(X_0,M) ⊗ _{k_0}\C
= H^0(X_{\C},M_{\C}).
\]
Tensoring the evaluation map for \(M\) with \(\C\) therefore gives
@@ -6013,39 +6012,39 @@ the surjective complex evaluation map. Its coherent cokernel is
zero by faithful flatness. Extending from \(k_0\) to \(k\) preserves
this surjectivity, so the same integer \(m\) gives a globally
generated line bundle on the original \(X\).
-\end{proof}
-Together with Proposition~\ref{prop:field-descent}, the complex proof
-establishes Theorem~\ref{thm:main} in its stated generality.
-\section{Good models, linear triviality, and boundary consequences}
-\label{sec:consequences}
+Together with Proposition~prop:field-descent, the complex proof
+establishes Theorem~thm:main in its stated generality.
+
+# Good models, linear triviality, and boundary consequences
+
The real-boundary induction and the final line-bundle descent have
different natural scopes. We first record good-model existence over
\(\C\), including the precise smooth canonical comparison used by
fiber-space applications. We then deduce actual rational-linear
triviality in Iitaka dimension zero over every field covered by
-Theorem~\ref{thm:main}.
+Theorem~thm:main.
Finally, over \(\C\), normalization gluing and a section-extension
theorem give two consequences for reducible boundaries and nonnormal pairs.
-\begin{theorem}[Real-boundary good models over \(\C\)]
-\label{thm:good-models-complex}
+[Real-boundary good models over \(\C\)]
+
Every projective log canonical pair \((X,B)\) over \(\C\), with
effective real boundary and pseudo-effective real Cartier adjoint
\(K_X+B\), has a good log minimal model. Real semiampleness has the
-meaning fixed in Section~\ref{sec:induction}.
-\end{theorem}
-\begin{proof}
-The simultaneous induction in Section~\ref{sec:completion} starts with
+meaning fixed in Section~sec:induction.
+
+Proof.
+The simultaneous induction in Section~sec:completion starts with
dimension zero and proves the full real-boundary conclusion of
-Proposition~\ref{prop:inductive-reduction} at each dimension. In
+Proposition~prop:inductive-reduction at each dimension. In
particular, its conclusion applies to the given pair before any
rational-boundary or nefness specialization.
-\end{proof}
-\begin{corollary}[Smooth canonical good models]\label{cor:smooth-good-model}
+
+[Smooth canonical good models]
Let \(X\) be a smooth projective complex variety with pseudo-effective
\(K_X\). There is a normal projective \(\Q\)-factorial klt variety
\(V\) and a \(K_X\)-negative birational contraction \(X\dashrightarrow V\)
@@ -6054,15 +6053,15 @@ resolution \(p:W\to X\), \(q:W\to V\), compatible canonical divisors
satisfy
\[
p^*K_X=q^*K_V+E,
- \qquad E\geq0\quad\text{and }E\text{ is }q\text{-exceptional}.
+ \qquad E\geq0\quadand E is q-exceptional.
\]
In particular \(\kappa(X,K_X)=\kappa(V,K_V)\).
-\end{corollary}
-\begin{proof}
-Apply Theorem~\ref{thm:good-models-complex} to the klt pair \((X,0)\).
+
+Proof.
+Apply Theorem~thm:good-models-complex to the klt pair \((X,0)\).
Choose its \(\Q\)-factorial log minimal model, as in the construction
-of Proposition~\ref{prop:inductive-reduction}. The klt clause of
-Lemma~\ref{mmp:comparison} shows that it extracts no divisor, remains
+of Proposition~prop:inductive-reduction. The klt clause of
+Lemma~mmp:comparison shows that it extracts no divisor, remains
klt, and has the displayed effective exceptional comparison.
Its boundary is zero, being the strict transform of the zero boundary
with no extracted divisors.
@@ -6073,108 +6072,108 @@ divisor \(mE\) satisfies \(q_*\OO_W(mE)=\OO_V\), since \(V\) is
normal. Projection formula and the displayed comparison identify the
pluricanonical section spaces. Their ratios agree in the common
function field, proving the assertion about Iitaka dimension.
-\end{proof}
-\begin{lemma}\label{lem:semiample-zero}
+
+
Let \(Z\) be a nonempty integral projective variety over an algebraically
closed field, and let \(A\) be a semiample \(\Q\)-Cartier divisor.
If \(\kappa(Z,A)=0\), then \(A\sim_{\Q}0\).
-\end{lemma}
-\begin{proof}
+
+Proof.
Choose a positive integer \(m\) such that the line bundle
\(L=\OO_Z(mA)\) is globally generated. Its complete system gives
\[
- \phi:Z\longrightarrow\PP(H^0(Z,L)^*),\qquad
+ \phi:Z→\PP(H^0(Z,L)^*),\qquad
L\simeq\phi^*\OO(1).
\]
The image has dimension at most \(\kappa(Z,A)=0\), so it is a point.
A linear form nonzero at that point pulls back to a nowhere-vanishing
section of \(L\). Thus \(L\simeq\OO_Z\), an actual line-bundle
isomorphism, and \(A\sim_{\Q}0\).
-\end{proof}
-\begin{corollary}\label{cor:qlinear-zero}
-Let \((X,B)\) satisfy the hypotheses of Theorem~\ref{thm:main}.
+
+
+Let \((X,B)\) satisfy the hypotheses of Theorem~thm:main.
If \(\kappa(X,K_X+B)=0\), then \(K_X+B\sim_{\Q}0\).
In particular, every nef rational adjoint of Iitaka dimension zero
on a projective klt fivefold over an algebraically closed field of
characteristic zero is \(\Q\)-linearly trivial.
-\end{corollary}
-\begin{proof}
-Theorem~\ref{thm:main} gives semiampleness on the original normal
-variety. Lemma~\ref{lem:semiample-zero} gives a positive Cartier
+
+Proof.
+Theorem~thm:main gives semiampleness on the original normal
+variety. Lemma~lem:semiample-zero gives a positive Cartier
multiple with trivial associated line bundle. Klt pairs are lc, so
the fivefold assertion is a special case.
-\end{proof}
+
Thus the nef adjoint also has numerical dimension zero. This is a
consequence of the trivializing Cartier multiple, not an additional
hypothesis. The multiple may depend on the pair; no uniform index is
asserted.
-\subsection{Semi-log-canonical abundance and dlt extension}
+## Semi-log-canonical abundance and dlt extension
We use the standard semi-log-canonical convention, in which the underlying
projective scheme may be reducible or nonnormal. It is reduced, satisfies
\(S_2\), is pure-dimensional, and has ordinary double crossings in
codimension one. No component of the effective boundary is contained in the
codimension-one singular locus. If \(\nu:X^\nu\to X\) is the normalization,
-the boundary \(\Theta\)
+the boundary \(Θ\)
defined by
\[
- K_{X^\nu}+\Theta=\nu^*(K_X+\Delta)
+ K_{X^\nu}+Θ=\nu^*(K_X+\Delta)
\]
-includes the conductor boundary, and \((X^\nu,\Theta)\) is componentwise
+includes the conductor boundary, and \((X^\nu,Θ)\) is componentwise
log canonical.
These are the conventions of \cite[Definition~2.5]{FujinoGongyo2014}.
-\begin{corollary}[Semi-log-canonical abundance over \(\C\)]
-\label{cor:slc-abundance}
+[Semi-log-canonical abundance over \(\C\)]
+
Let \((X,\Delta)\) be a projective semi-log-canonical pair over \(\C\),
where \(\Delta\) is an effective rational divisor and \(K_X+\Delta\) is
\(\Q\)-Cartier. If \(K_X+\Delta\) is nef, then it is semiample.
-\end{corollary}
-\begin{proof}
+
+Proof.
Write \(X^\nu=\coprod_i X_i^\nu\) for the normalization and put
-\(\Theta_i=\Theta|_{X_i^\nu}\). Each \((X_i^\nu,\Theta_i)\) is a normal
+\(Θ_i=Θ|_{X_i^\nu}\). Each \((X_i^\nu,Θ_i)\) is a normal
projective log canonical pair with effective rational boundary. The adjoint
-\(K_{X_i^\nu}+\Theta_i\) is the pullback of \(K_X+\Delta\), so it is nef.
-Theorem~\ref{thm:main} makes each of these adjoints semiample. There are only
+\(K_{X_i^\nu}+Θ_i\) is the pullback of \(K_X+\Delta\), so it is nef.
+Theorem~thm:main makes each of these adjoints semiample. There are only
finitely many components, hence one common positive Cartier multiple is
globally generated on their disjoint union. Thus \(\nu^*(K_X+\Delta)\) is
semiample. The normalization gluing theorem
\cite[Theorem~1.5]{FujinoGongyo2014} now gives semiampleness on \(X\).
-\end{proof}
-\begin{corollary}[Supported extension for dlt pairs over \(\C\)]
-\label{cor:dlt-extension}
+
+[Supported extension for dlt pairs over \(\C\)]
+
Let \((X,\Delta)\) be a projective dlt pair over \(\C\), where \(\Delta\)
is an effective rational divisor, and put \(S=\lfloor\Delta\rfloor\).
Assume that \(K_X+\Delta\) is nef and that, for an effective rational
divisor \(D\),
\[
K_X+\Delta\sim_{\Q}D\geq0,
- \qquad S\subseteq\Supp D.
+ \qquad S⊆\Supp D.
\]
Then, for every integer \(m\geq2\) for which \(m(K_X+\Delta)\) is Cartier,
the restriction map
\[
- H^0\bigl(X,\OO_X(m(K_X+\Delta))\bigr)
- \longrightarrow
- H^0\bigl(S,\OO_X(m(K_X+\Delta))|_S\bigr)
+ H^0(X,\OO_X(m(K_X+\Delta)))
+ →
+ H^0(S,\OO_X(m(K_X+\Delta))|_S)
\]
is surjective.
-\end{corollary}
-\begin{proof}
+
+Proof.
Set \(B=\Delta-S\). The pair \((X,S+B)\) is dlt, \(B\) is an effective
rational divisor, and \(\lfloor S+B\rfloor=S\). Since a dlt pair is log
-canonical, Theorem~\ref{thm:main} makes its nef adjoint semiample.
+canonical, Theorem~thm:main makes its nef adjoint semiample.
The pair and the displayed effective representative therefore satisfy the
hypotheses of \cite[Proposition~5.12]{FujinoGongyo2014}, which gives the
stated restriction surjectivity for every Cartier multiple with \(m\geq2\).
This is the supported extension formulation in
\cite[Conjecture~5.8]{FujinoGongyo2014}.
-\end{proof}
+
The first corollary settles the rational-boundary form of the semi-log-canonical
abundance conjecture \cite[Conjecture~1.3]{FujinoGongyo2014} in every
@@ -6182,11 +6181,11 @@ dimension over \(\C\). The second settles the supported dlt extension
conjecture \cite[Conjectures~1.10 and~5.8]{FujinoGongyo2014} with the precise
Cartier-multiple range supplied by Proposition~5.12. Both are applications
of the cited implications after normal abundance has been established here.
-The support condition \(S\subseteq\Supp D\) is retained; no arbitrary
+The support condition \(S⊆\Supp D\) is retained; no arbitrary
restriction-surjectivity statement is asserted.
-\section{Finite generation of rational log canonical rings}
-\label{sec:canonical-rings}
+# Finite generation of rational log canonical rings
+
The preceding results also settle finite generation for rational lc pairs.
The finite-generation conjecture asks whether finitely many rounded
@@ -6207,18 +6206,18 @@ on \(X\) and nonnegative integers \(m,n\), the inclusions
\(\lfloor mD\rfloor+\lfloor nD\rfloor\leq\lfloor(m+n)D\rfloor\)
give the usual multiplication on the section algebras below.
-\begin{samepage}
-\begin{corollary}[Finite generation for rational lc pairs]
-\label{cor:lc-finite-generation}
+
+[Finite generation for rational lc pairs]
+
\leavevmode
-\begin{enumerate}[label=\textup{(\roman*)}]
+[label=(\roman*)]
\item Let \(k\) be an algebraically closed field of characteristic zero,
let \((X,\Delta)\) be a normal projective log canonical pair over \(k\), and assume
that \(\Delta\geq0\) is rational and \(D=K_X+\Delta\) is \(\Q\)-Cartier.
Then
\[
- R(X,D)=\bigoplus_{m\geq0}
- H^0\bigl(X,\OO_X(\lfloor mD\rfloor)\bigr)
+ R(X,D)=⊕_{m\geq0}
+ H^0(X,\OO_X(\lfloor mD\rfloor))
\]
is a finitely generated \(k\)-algebra.
@@ -6228,25 +6227,25 @@ proper morphism \(f:X\to S\) onto an algebraic variety \(S\), the relative
log canonical algebra
\[
\mathcal R(X/S,K_X+\Delta)
- =\bigoplus_{m\geq0}f_*\OO_X\bigl(\lfloor m(K_X+\Delta)\rfloor\bigr)
+ =⊕_{m\geq0}f_*\OO_X(\lfloor m(K_X+\Delta)\rfloor)
\]
is a finitely generated \(\OO_S\)-algebra.
-\end{enumerate}
-\end{corollary}
-\end{samepage}
-\begin{proof}
-In dimension zero, \(X\) is a point, and in \textup{(ii)} so is \(S\).
+
+
+
+Proof.
+In dimension zero, \(X\) is a point, and in (ii) so is \(S\).
The boundary and canonical divisor are zero, and the two algebras are
\(k[t]\) and \(\OO_S[t]\), respectively, with \(\deg t=1\).
We therefore assume \(\dim X\geq1\).
-We first prove \textup{(i)}. If \(D\) is not pseudo-effective, a nonzero
+We first prove (i). If \(D\) is not pseudo-effective, a nonzero
section \(s\) in degree \(m>0\) would give
\[
- \operatorname{div}(s)+mD
- =\bigl(\operatorname{div}(s)+\lfloor mD\rfloor\bigr)
- +\bigl(mD-\lfloor mD\rfloor\bigr)\geq0.
+ div(s)+mD
+ =(div(s)+\lfloor mD\rfloor)
+ +(mD-\lfloor mD\rfloor)\geq0.
\]
This would make \(D\) pseudo-effective. Thus all positive graded pieces
vanish in this case. Since \(H^0(X,\OO_X)=k\), we have \(R(X,D)=k\).
@@ -6257,13 +6256,13 @@ of the companion minimal-model theorem
pair \((Y,\Delta_Y)\) and a birational map \(\phi:X\dashrightarrow Y\)
extracting no divisors, with \(\Delta_Y=\phi_*\Delta\) and
\(D_Y=K_Y+\Delta_Y\) \(\Q\)-Cartier and nef. Log discrepancies do not decrease.
-Theorem~\ref{thm:main} makes the rational adjoint \(D_Y\) semiample.
+Theorem~thm:main makes the rational adjoint \(D_Y\) semiample.
On a common smooth resolution \(p:W\to X\), \(q:W\to Y\), choose
compatible canonical divisors. Then
-\begin{equation}\label{eq:ring-comparison}
- p^*D=q^*D_Y+E,\qquad E\geq0,\qquad E\text{ is }q\text{-exceptional}.
-\end{equation}
+
+ p^*D=q^*D_Y+E,\qquad E\geq0,\qquad E is q-exceptional.
+
Indeed, the coefficient of \(E\) at a prime divisor \(F\) is
\(a(F;Y,\Delta_Y)-a(F;X,\Delta)\), which is nonnegative by the discrepancy
comparison. Suppose \(F\) is not \(q\)-exceptional. Since \(\phi\) extracts no
@@ -6275,18 +6274,18 @@ For any normal variety \(Z\), rational Cartier divisor \(G\), and
\(m\geq0\), the nonzero global sections of \(\OO_Z(\lfloor mG\rfloor)\) are
exactly the rational functions \(s\) satisfying
\[
- \operatorname{div}_Z(s)+mG\geq0.
+ div_Z(s)+mG\geq0.
\]
This is equivalent to the usual inequality with \(\lfloor mG\rfloor\)
-because the coefficients of \(\operatorname{div}_Z(s)\) are integers.
+because the coefficients of \(div_Z(s)\) are integers.
If the inequality holds on \(X\), pull back its effective \(\Q\)-Cartier
-left side to \(W\) and push forward by \(q\). Equation~\eqref{eq:ring-comparison}
+left side to \(W\) and push forward by \(q\). Equation~(eq:ring-comparison)
and \(q_*E=0\) give the inequality on \(Y\). Conversely, pull back the
inequality on \(Y\), add \(mE\), and push forward by \(p\). Thus, inside
the common function field,
\[
- H^0\bigl(X,\OO_X(\lfloor mD\rfloor)\bigr)
- =H^0\bigl(Y,\OO_Y(\lfloor mD_Y\rfloor)\bigr)
+ H^0(X,\OO_X(\lfloor mD\rfloor))
+ =H^0(Y,\OO_Y(\lfloor mD_Y\rfloor))
\qquad(m\geq0).
\]
These identifications preserve multiplication, so they identify the full
@@ -6296,7 +6295,7 @@ It remains to use semiampleness on \(Y\). Choose \(r>0\) such that
\(rD_Y\) is Cartier and globally generated. Its complete linear system
defines \(\psi:Y\to\PP^N_k\) with
\(\OO_Y(rD_Y)\simeq\psi^*\OO_{\PP^N}(1)\). Let
-\(A=k[T_0,\ldots,T_N]\) act on \(R(Y,D_Y)\) by pulling back the coordinate
+\(A=k[T_0,...,T_N]\) act on \(R(Y,D_Y)\) by pulling back the coordinate
sections, each in degree \(r\). For \(0\leq i<r\), put
\(\mathcal F_i=\psi_*\OO_Y(\lfloor iD_Y\rfloor)\), which is coherent on
\(\PP^N_k\) because \(\psi\) is proper. Since
@@ -6307,30 +6306,30 @@ sections, each in degree \(r\). For \(0\leq i<r\), put
projection formula identifies the part in degrees congruent to \(i\)
modulo \(r\), as an \(A\)-module, with
\[
- \bigoplus_{\ell\geq0}H^0\bigl(\PP^N_k,\mathcal F_i(\ell)\bigr).
+ ⊕_{\ell\geq0}H^0(\PP^N_k,\mathcal F_i(\ell)).
\]
This is a finite \(A\)-module by the standard finite-generation theorem
for section modules \cite[Lemma~30.14.1(5)]{StacksSectionModules}.
There are only \(r\) residue classes, so \(R(Y,D_Y)\) is finite as an
\(A\)-module and hence finitely generated as a \(k\)-algebra. This proves
-\textup{(i)}.
+(i).
-For \textup{(ii)}, set \(n=\dim X\). In the notation of Fujino--Gongyo,
+For (ii), set \(n=\dim X\). In the notation of Fujino--Gongyo,
Conjecture \(C_{\leq n-1}\) asks for good minimal models of projective
\(\Q\)-factorial dlt pairs with effective real boundary and pseudo-effective
-adjoint in dimensions at most \(n-1\). Theorem~\ref{thm:good-models-complex}
+adjoint in dimensions at most \(n-1\). Theorem~thm:good-models-complex
supplies this statement. Its good-model convention is the one fixed in
-Section~\ref{sec:induction}, with a \(\Q\)-factorial dlt model as in
-Proposition~\ref{prop:inductive-reduction}. Theorem~1.1 of
-Fujino--Gongyo \cite{FujinoGongyoRings2014} therefore gives their
+Section~sec:induction, with a \(\Q\)-factorial dlt model as in
+Proposition~prop:inductive-reduction. Theorem~1.1 of
+Fujino--Gongyo [cite] therefore gives their
Conjecture \(B_n\): finite generation of the log canonical ring for
projective rational plt pairs of dimension \(n\) whose adjoint is big
and whose boundary has irreducible round-down. Their relative reduction,
Corollary~1.4, applies to the given rational lc pair and proper morphism
-onto \(S\), and gives exactly \textup{(ii)} without requiring \(X\) to
+onto \(S\), and gives exactly (ii) without requiring \(X\) to
be projective over \(\C\).
-Their Theorem~1.1 also gives \textup{(i)} directly when \(k=\C\).
-\end{proof}
+Their Theorem~1.1 also gives (i) directly when \(k=\C\).
+
The good-model input allows real boundaries, but both finite-generation
statements retain rational boundaries. In the absolute proof, rationality
@@ -6339,41 +6338,41 @@ assertion is finite generation over the algebraic base \(S\) in the complex
setting; it is not a claim about rounded rings for real boundaries or about
analytic bases. No uniform degree bound for generators is asserted.
-\section{Rational curves, cotangent positivity, and covers}
-\label{sec:classification}
+# Rational curves, cotangent positivity, and covers
+
We first combine smooth canonical nonvanishing with classical results on
rational curves and cotangent tensors. We then use smooth abundance to
remove the abundance premise from the quasi-projective-cover results of
Claudon, H\"oring and Koll\'ar. Throughout this section,
\(\kappa(X)=\kappa(X,K_X)\). We retain the convention
-\(\kappa(\mathrm{pt})=0\), and regard a point as rationally connected
+\(\kappa(pt)=0\), and regard a point as rationally connected
but not uniruled.
-\begin{corollary}[Uniruledness and Mumford's criterion]
-\label{cor:uniruled-mumford}
+[Uniruledness and Mumford's criterion]
+
Let \(X\) be a smooth connected projective complex variety.
-\begin{enumerate}[label=\textup{(\roman*)}]
-\item \(X\) is uniruled if and only if \(\kappa(X)=-\infty\).
+[label=(\roman*)]
+\item \(X\) is uniruled if and only if \(\kappa(X)=-∞\).
\item \(X\) is rationally connected if and only if
\[
- H^0\bigl(X,(\Omega_X^1)^{\otimes m}\bigr)=0
- \quad\text{for every integer }m>0.
+ H^0(X,(Ω_X^1)^{ ⊗ m})=0
+ \quadfor every integer m>0.
\]
-\end{enumerate}
-\end{corollary}
-\begin{proof}
+
+
+Proof.
For positive-dimensional \(X\), the theorem of
Boucksom--Demailly--P{\u a}un--Peternell
\cite[Corollary~0.3 and Theorem~2.6]{BDPP2013} says that \(X\) is
non-uniruled exactly when \(K_X\) is pseudo-effective.
-Corollary~\ref{cor:smooth-good-model} then gives \(\kappa(X)\geq0\):
+Corollary~cor:smooth-good-model then gives \(\kappa(X)\geq0\):
the semiample canonical divisor on the good model has a nonzero section
in a positive multiple, and the pluricanonical section spaces agree.
Conversely, a nonzero pluricanonical section makes \(K_X\) pseudo-effective. This proves
-\textup{(i)}.
+(i).
-For \textup{(ii)}, rational connectedness forces all positive covariant
+For (ii), rational connectedness forces all positive covariant
tensor differentials to vanish, as recalled by Lazi{\'c} and Peternell.
For the converse, their Proposition~2.4 reduces Mumford's criterion in
dimension \(n\) to tensor nonvanishing for smooth projective varieties
@@ -6382,81 +6381,81 @@ pseudo-effective \cite[Proposition~2.4]{LazicPeternell2017}.
If \(Y\) is such a variety of dimension \(d\), the preceding good-model
argument supplies a nonzero section of \(mK_Y\) for some integer \(m>0\).
Antisymmetrization embeds
-\(\Omega_Y^d\) into \((\Omega_Y^1)^{\otimes d}\) over \(\C\);
+\(Ω_Y^d\) into \((Ω_Y^1)^{ ⊗ d}\) over \(\C\);
tensoring this injection \(m\) times gives the required nonzero tensor
differential. This is the observation in
\cite[Remark~2.5]{LazicPeternell2017}, so their reduction applies in
every dimension. Both assertions for a point follow from our conventions.
-\end{proof}
+
For a smooth connected projective complex variety \(X\), let \(R(X)\)
be a smooth projective model of the base of its maximal rationally
connected (MRC) fibration. This base is determined up to birational
equivalence and is non-uniruled unless it is a point. We use
-\(\Omega_X^0=\OO_X\).
+\(Ω_X^0=\OO_X\).
+
+[Pluriforms and the rational quotient]
-\begin{corollary}[Pluriforms and the rational quotient]
-\label{cor:pluriforms}
-\begin{enumerate}[label=\textup{(\roman*)}]
+[label=(\roman*)]
\item For every smooth connected projective complex variety \(X\),
\[
\dim R(X)=
\max_{\substack{p\in\Z_{\geq0},\,k\in\Z_{>0}\\
- H^0(X,\operatorname{Sym}^k(\Omega_X^p))\ne0}} p.
+ H^0(X,Sym^k(Ω_X^p))\ne0}} p.
\]
\item A compact connected K\"ahler manifold \(X\) is rationally
connected if and only if
\[
- H^0\bigl(X,\operatorname{Sym}^k(\Omega_X^p)\bigr)=0
- \quad\text{for every pair of integers }p,k>0.
+ H^0(X,Sym^k(Ω_X^p))=0
+ \quadfor every pair of integers p,k>0.
\]
-\end{enumerate}
-\end{corollary}
-\begin{proof}
-Write \(r=\dim R(X)\) in \textup{(i)}. Resolve the MRC map to a morphism
+
+
+Proof.
+Write \(r=\dim R(X)\) in (i). Resolve the MRC map to a morphism
\(f:X'\to R(X)\), with \(X'\) smooth and projective. For a smooth
general fiber \(F\), the cotangent sequence is
\[
- 0\longrightarrow\OO_F^{\oplus r}
- \longrightarrow\Omega_{X'}^1|_F
- \longrightarrow\Omega_F^1\longrightarrow0.
+ 0→\OO_F^{ ⊕ r}
+ →Ω_{X'}^1|_F
+ →Ω_F^1→0.
\]
If \(p>r\), every graded piece of the induced exterior-power filtration
-on \(\Omega_{X'}^p|_F\) contains a positive exterior power of
-\(\Omega_F^1\): at most \(r\) factors can come from
-\(\OO_F^{\oplus r}\). Consequently, for \(k>0\), the graded pieces
+on \(Ω_{X'}^p|_F\) contains a positive exterior power of
+\(Ω_F^1\): at most \(r\) factors can come from
+\(\OO_F^{ ⊕ r}\). Consequently, for \(k>0\), the graded pieces
of the induced filtration on
-\(\operatorname{Sym}^k(\Omega_{X'}^p)|_F\) are direct summands of
-sums of positive tensor powers of \(\Omega_F^1\). Here we use
+\(Sym^k(Ω_{X'}^p)|_F\) are direct summands of
+sums of positive tensor powers of \(Ω_F^1\). Here we use
antisymmetrization and symmetrization over \(\C\). Rational
connectedness of \(F\) makes all their sections vanish, by
-Corollary~\ref{cor:uniruled-mumford}. A global section therefore
+Corollary~cor:uniruled-mumford. A global section therefore
vanishes on every sufficiently general fiber, hence on \(X'\).
Birational invariance of global covariant tensor differentials on smooth
projective varieties gives the same vanishing on \(X\). This is the
upper-bound argument of \cite[Section~3]{BrunebarbeCampana2015}.
If \(r>0\), the smooth non-uniruled base has \(\kappa(R(X))\geq0\) by
-Corollary~\ref{cor:uniruled-mumford}. A nonzero section of \(kK_{R(X)}\)
+Corollary~cor:uniruled-mumford. A nonzero section of \(kK_{R(X)}\)
for some \(k>0\) pulls back to a nonzero section of
-\(\operatorname{Sym}^k(\Omega_X^r)\). The pullback is taken on \(X'\)
+\(Sym^k(Ω_X^r)\). The pullback is taken on \(X'\)
and descends to \(X\) by the same birational invariance. If \(r=0\),
the term \(p=0\) gives a nonzero section and the same equality follows.
This is the equality \(r=r^-\) predicted in the remark ending
\cite[Section~3]{BrunebarbeCampana2015}.
-For \textup{(ii)}, the forward vanishing is the rational-curve argument
+For (ii), the forward vanishing is the rational-curve argument
recalled by Brunebarbe and Campana in the same section. Conversely, the
stated vanishing gives
-\(H^{2,0}(X)=H^0(X,\Omega_X^2)=0\). Kodaira's projectivity criterion
+\(H^{2,0}(X)=H^0(X,Ω_X^2)=0\). Kodaira's projectivity criterion
then makes \(X\) projective: the real degree-two cohomology is of type
\((1,1)\), so a rational class sufficiently close to a K\"ahler class
is still K\"ahler, and clearing denominators gives an integral K\"ahler
class. This is the projectivity reduction in
-\cite[Section~3]{BrunebarbeCampana2015}. Part \textup{(i)} now applies.
+\cite[Section~3]{BrunebarbeCampana2015}. Part (i) now applies.
Its maximum is zero because every term with \(p>0\) vanishes, so the
MRC base is a point and \(X\) is rationally connected.
-\end{proof}
+
We next use ordinary cotangent invariants, with no boundary. If
\(\mathcal L\) is a coherent rank-one subsheaf of a vector bundle on a
@@ -6465,37 +6464,37 @@ smooth projective variety, its double dual is a line bundle; put
\[
\omega(X)=
\max_{\substack{m>0\\
- \mathcal L\subset(\Omega_X^1)^{\otimes m}\\
- \operatorname{rank}\mathcal L=1}}
+ \mathcal L\subset(Ω_X^1)^{ ⊗ m}\\
+ rank\mathcal L=1}}
\kappa(\mathcal L),
\]
-where \(m\) is an integer and the value is \(-\infty\) if no such sheaf
+where \(m\) is an integer and the value is \(-∞\) if no such sheaf
has nonnegative Iitaka dimension. The inclusion of \(\mathcal L\) extends
to its double dual: away from codimension two these sheaves agree, and a
map into a locally free sheaf extends across that subset. Taking tensor
-powers then realizes the systems \(H^0(X,(\mathcal L^{**})^{\otimes a})\),
+powers then realizes the systems \(H^0(X,(\mathcal L^{**})^{ ⊗ a})\),
for integers \(a>0\), inside higher cotangent tensor powers. Thus allowing all \(m\) makes
this equivalent to Campana's definition using the dimensions of the
rational maps defined by \(H^0(X,\mathcal L)\). For positive-dimensional \(X\),
also put
\[
\kappa^+(X)=
- \max_{\substack{p>0\\0\ne\mathcal F\subset\Omega_X^p}}
+ \max_{\substack{p>0\\0\ne\mathcal F\subsetΩ_X^p}}
\kappa(\det\mathcal F).
\]
Here \(p\) is an integer, \(\mathcal F\) is coherent, and
-\(\det\mathcal F=(\bigwedge^{\rk\mathcal F}\mathcal F)^{**}\). These are the
-invariants used in Campana's Appendix~A to Taji \cite{Taji2014}.
+\(\det\mathcal F=(∧^{\rk\mathcal F}\mathcal F)^{**}\). These are the
+invariants used in Campana's Appendix~A to Taji [cite].
+
+[Cotangent invariants and the MRC base]
-\begin{corollary}[Cotangent invariants and the MRC base]
-\label{cor:cotangent-mrc}
Let \(X\) be a smooth connected projective complex variety. Then
\[
\omega(X)=\omega(R(X))=
- \begin{cases}
- -\infty,&\text{if }R(X)\text{ is a point},\\
- \kappa(R(X)),&\text{if }\dim R(X)>0.
- \end{cases}
+
+ -∞,&if R(X) is a point,\\
+ \kappa(R(X)),&if \dim R(X)>0.
+
\]
In particular, if \(X\) is positive-dimensional and non-uniruled, then
\(\omega(X)=\kappa(X)\). If \(X\) is positive-dimensional and
@@ -6503,27 +6502,27 @@ In particular, if \(X\) is positive-dimensional and non-uniruled, then
\[
\kappa^+(X)=\kappa(X).
\]
-\end{corollary}
-\begin{proof}
+
+Proof.
Campana's birationally invariant fibration properties
\cite[Appendix~A, property~4]{Taji2014} give
\(\omega(X)=\omega(R(X))\): the general MRC fiber is rationally
-connected and has \(\omega=-\infty\). Indeed any rank-one subsheaf with
+connected and has \(\omega=-∞\). Indeed any rank-one subsheaf with
nonnegative Iitaka dimension would give a section of a positive tensor
-power, contrary to Corollary~\ref{cor:uniruled-mumford}; a point has
-\(\omega=-\infty\) by the empty maximum. Birational invariance permits
+power, contrary to Corollary~cor:uniruled-mumford; a point has
+\(\omega=-∞\) by the empty maximum. Birational invariance permits
resolving the MRC map. If \(R(X)\) is a point, the displayed formula
-therefore follows without changing our convention \(\kappa(\mathrm{pt})=0\).
+therefore follows without changing our convention \(\kappa(pt)=0\).
Suppose that \(R=R(X)\) has positive dimension. It is non-uniruled, so
-\(\kappa(R)\geq0\) by Corollary~\ref{cor:uniruled-mumford}.
+\(\kappa(R)\geq0\) by Corollary~cor:uniruled-mumford.
To apply Campana's abundance deduction
\cite[Conjecture~A.2 and Remark~A.3]{Taji2014}, take a resolved Iitaka
fibration \(R'\to Z\), with \(R'\) and \(Z\) smooth and projective,
\(R'\) birational to \(R\), and \(\dim Z=\kappa(R)\). Its smooth
general fiber \(F\) has \(\kappa(F)=0\). If \(F\) is
-positive-dimensional, Corollary~\ref{cor:smooth-good-model} and
-Lemma~\ref{lem:semiample-zero} give a normal birational good model
+positive-dimensional, Corollary~cor:smooth-good-model and
+Lemma~lem:semiample-zero give a normal birational good model
\(V_F\) with \(K_{V_F}\sim_{\Q}0\). In particular its canonical
divisor is numerically trivial on its smooth locus. Campana's criterion
\cite[Appendix~A, Examples~A.1(2)]{Taji2014} says that the existence of
@@ -6541,36 +6540,36 @@ first stated specialization.
Campana also records
\(\omega(X)\geq\kappa^+(X)\geq\kappa(X)\)
\cite[Appendix~A, property~2]{Taji2014}. When \(\kappa(X)\geq0\),
-Corollary~\ref{cor:uniruled-mumford} makes \(X\) non-uniruled, and the
+Corollary~cor:uniruled-mumford makes \(X\) non-uniruled, and the
two outer terms are equal by the formula just proved. Thus
\(\kappa^+(X)=\kappa(X)\). This is the abundance consequence recalled
by Taji \cite[Theorem~1.3]{Taji2014}, who attributes it to Campana
\cite[Proposition~3.10]{Campana1995}.
-\end{proof}
-\begin{corollary}[Finite fundamental group in Kodaira dimension zero]
-\label{cor:finite-pi-one}
+
+[Finite fundamental group in Kodaira dimension zero]
+
Let \(X\) be a positive-dimensional smooth connected projective complex
variety. If \(\kappa(X)=0\) and \(\chi(X,\OO_X)\ne0\), then its ordinary
topological fundamental group \(\pi_1(X)\) is finite.
-\end{corollary}
-\begin{proof}
-Corollary~\ref{cor:cotangent-mrc} gives \(\kappa^+(X)=0\).
+
+Proof.
+Corollary~cor:cotangent-mrc gives \(\kappa^+(X)=0\).
Campana's criterion \cite[Corollary~5.3]{Campana1995}, in the form
recalled by Taji \cite[Theorem~1.6]{Taji2014}, gives the conclusion.
-\end{proof}
-\subsection{Quasi-projective covers}
+
+## Quasi-projective covers
Here a quasi-projective cover means a complex analytic covering space
biholomorphic to a quasi-projective variety; its deck transformations
need not be algebraic. The following application describes such covers
in terms of bundles over abelian varieties.
-\begin{corollary}[Quasi-projective covering spaces]
-\label{cor:quasi-projective-covers}
+[Quasi-projective covering spaces]
+
Let \(X\) be a connected normal projective complex variety.
-\begin{enumerate}[label=\textup{(\roman*)}]
+[label=(\roman*)]
\item Its universal cover is biholomorphic to a quasi-projective variety
if and only if there is a finite \'etale Galois cover \(X'\to X\)
and a locally trivial holomorphic fiber bundle \(X'\to A\), where
@@ -6585,25 +6584,25 @@ compact analytic subvarieties, such that
\widetilde X\simeq X'\times_A\widetilde A
\]
as covering spaces over \(X\).
-\end{enumerate}
-\end{corollary}
-\begin{proof}
-Theorem~\ref{thm:main} with zero boundary on a smooth projective complex
+
+
+Proof.
+Theorem~thm:main with zero boundary on a smooth projective complex
variety gives precisely Conjecture~1.2 of
-Claudon--H\"oring--Koll\'ar \cite{ClaudonHoeringKollar2013}.
-Their Theorem~1.1 and Corollary~1.5 therefore give \textup{(i)} and
-\textup{(ii)}, respectively.
-\end{proof}
+Claudon--H\"oring--Koll\'ar [cite].
+Their Theorem~1.1 and Corollary~1.5 therefore give (i) and
+(ii), respectively.
-The simply connected fiber condition belongs to \textup{(i)}, not to
-the general-cover statement \textup{(ii)}. Local triviality is holomorphic;
+
+The simply connected fiber condition belongs to (i), not to
+the general-cover statement (ii). Local triviality is holomorphic;
no global product after a finite cover or compact K\"ahler extension is
asserted.
-\section{Relative abundance for projective analytic morphisms}
-\label{sec:analytic-abundance}
+# Relative abundance for projective analytic morphisms
+
-The complex rational case of Theorem~\ref{thm:main} supplies the algebraic
+The complex rational case of Theorem~thm:main supplies the algebraic
fiber input in Fujino's relative analytic reduction
\cite[Theorem~1.10 and its proof]{FujinoRelativeAnalytic2025}.
We record only its normal rational case. Throughout this section, complex
@@ -6619,8 +6618,8 @@ projective integral curve contracted by \(\pi\), over every point of \(Y\).
These are the conventions of
\cite[Definitions~2.28, 2.32 and~2.48]{FujinoAnalyticMMP2022}.
-\begin{corollary}[Relative analytic abundance]
-\label{cor:analytic-abundance}
+[Relative analytic abundance]
+
Let \(\pi:X\to Y\) be a projective surjective morphism of normal complex
analytic varieties. Let \((X,\Delta)\) be a log canonical pair with
\(\Delta\) an effective rational divisor and \(D=K_X+\Delta\)
@@ -6634,15 +6633,15 @@ neighborhood \(U\supset W\) and a positive integer \(m\) such that
is generated relative to \(\pi_U:X_U\to U\). Equivalently, the evaluation
map
\[
- \pi_U^*(\pi_U)_*L\longrightarrow L
+ \pi_U^*(\pi_U)_*L→ L
\]
is surjective.
-\end{corollary}
-\begin{proof}
+
+Proof.
Since \(\pi\) is proper, \(\pi^{-1}(W)\) is compact. Local rational
Cartier indices therefore have a common multiple \(m_0\) on an open
neighborhood \(O\) of \(\pi^{-1}(W)\). The image of the closed set
-\(X\setminus O\) is closed and disjoint from \(W\), so there is an
+\(X∖ O\) is closed and disjoint from \(W\), so there is an
open neighborhood \(V\supset W\) with \(\pi^{-1}(V)\subset O\).
Thus \(m_0D\) is Cartier over \(V\).
@@ -6662,7 +6661,7 @@ On each normal projective component \(F\) of an analytically sufficiently
general fiber of \(S\to\pi'(S)\), restriction gives a log canonical
pair \((F,\Delta_F)\) with effective rational boundary and nef
adjoint \(K_F+\Delta_F=(K_S+\Delta_S)|_F\).
-Theorem~\ref{thm:main} makes this adjoint semiample, so its Iitaka
+Theorem~thm:main makes this adjoint semiample, so its Iitaka
and numerical dimensions agree. Thus \(D'|_S\) is abundant over
\(\pi'(S)\) for every stratum: this is precisely
\(\pi'\)-log abundance in
@@ -6680,13 +6679,13 @@ Generation upstairs therefore implies generation downstairs: a base point
downstairs would make every pulled-back section vanish on its nonempty
fiber.
-Choose finitely many of these neighborhoods \(U_{P_1},\ldots,U_{P_s}\)
+Choose finitely many of these neighborhoods \(U_{P_1},...,U_{P_s}\)
covering \(W\), and let \(m\) be a common multiple of
-\(m_0,m_{P_1},\ldots,m_{P_s}\). Put \(U=\bigcup_i U_{P_i}\).
+\(m_0,m_{P_1},...,m_{P_s}\). Put \(U=∪_i U_{P_i}\).
The divisor \(mD\) is Cartier over \(U\), and its line bundle is
relatively generated on each \(U_{P_i}\). Surjectivity of evaluation
is local on the base, so it holds over \(U\), as asserted.
-\end{proof}
+
The base \(Y\) need not be algebraic, compact, or Stein, and \(W\) need
not be a Stein compact subset. The integer \(m\) may depend on the pair,
@@ -6698,9 +6697,9 @@ requires \(X\) to be projective. The statement does not include real
boundaries or semi-log-canonical analytic pairs.
-\bibliographystyle{plain}
-\bibliography{references}
-\end{document}
+
+
+
%======================================================================
% references.bib (moved to the end of this file so a reader can start
@@ -6708,7 +6707,7 @@ boundaries or semi-log-canonical analytic pairs.
% below to the preamble, immediately before the document environment.)
%======================================================================
-% \begin{filecontents*}[overwrite]{references.bib}
+% [overwrite]{references.bib}
% @article{EinKuchleLazarsfeld1995,
% author = {Ein, Lawrence and K{\"u}chle, Oliver and Lazarsfeld, Robert},
@@ -6724,7 +6723,7 @@ boundaries or semi-log-canonical analytic pairs.
% author = {{OpenAI}},
% title = {{Orbifold and logarithmic Iitaka subadditivity}},
% howpublished = {OpenAI Math Release preprint
-% \href{https://github.com/openai/math/blob/main/preprints/Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026/paper.pdf}{OAI:Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026}},
+% OAI:Orbifold-and-logarithmic-Iitaka-subadditivity-September-26-2026},
% year = {2026},
% note = {Corollaries 6.2 and 6.3}
% }
@@ -7077,7 +7076,7 @@ boundaries or semi-log-canonical analytic pairs.
% author = {{The Stacks Project Authors}},
% title = {The {Stacks} Project: {Grothendieck}'s existence theorem, {I}},
% year = {2026},
-% howpublished = {\url{https://stacks.math.columbia.edu/tag/087V}},
+% howpublished = {https://stacks.math.columbia.edu/tag/087V},
% note = {Tag 087V, Lemmas 30.24.1 and 30.24.3; accessed September 26, 2026}
% }
@@ -7242,9 +7241,9 @@ boundaries or semi-log-canonical analytic pairs.
% @misc{OpenAIMinimalModels2026,
% author = {{OpenAI}},
-% title = {{Minimal models and Mori fibre spaces for generalized log canonical $\mathbb{Q}$-pairs}},
+% title = {{Minimal models and Mori fibre spaces for generalized log canonical $ℚ$-pairs}},
% howpublished = {OpenAI Math Release preprint
-% \href{https://github.com/openai/math/blob/main/preprints/Minimal-models-and-Mori-fibre-spaces-for-generalized-log-canonical-Q-pairs-September-24-2026/paper.pdf}{OAI:Minimal-models-and-Mori-fibre-spaces-for-generalized-log-canonical-Q-pairs-September-24-2026}},
+% OAI:Minimal-models-and-Mori-fibre-spaces-for-generalized-log-canonical-Q-pairs-September-24-2026},
% year = {2026},
% note = {Corollary 1.2}
% }
@@ -7318,7 +7317,7 @@ boundaries or semi-log-canonical analytic pairs.
% author = {{The Stacks Project Authors}},
% title = {The {Stacks} Project: Coherent sheaves on {Proj}, {I}},
% year = {2026},
-% howpublished = {\url{https://stacks.math.columbia.edu/tag/01YS}},
+% howpublished = {https://stacks.math.columbia.edu/tag/01YS},
% note = {Tag 01YS, Lemma 30.14.1(5); accessed September 27, 2026}
% }
@@ -7359,4 +7358,4 @@ boundaries or semi-log-canonical analytic pairs.
% url = {https://arxiv.org/abs/1611.08768v2},
% note = {Version 2; Theorem 1.1}
% }
-% \end{filecontents*}
+%