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\title{Log abundance in characteristic zero\\[2pt]
       \large a rewritten, tightened edition}
\author{OpenAI}
\date{September 24, 2026 (v2)}



\begin{document}
\maketitle

\begin{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
canonical pair with effective rational boundary is semiample. Over
$\C$, the same proof establishes smooth canonical nonvanishing in
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}
\addcontentsline{toc}{section}{A guide for the reader}
\label{part:guide}
%======================================================================

\subsection*{The statement we prove}

The \emph{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}
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}.

\subsection*{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
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
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
$n$ 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
dimension $n$, together with the signed-representative theorem
(Sections~\ref{sg:section}--\ref{sg:descent}), is used to \emph{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
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
the components of $D$, provided
(i) $L-cD$ is pseudo-effective for some $c>0$, and
(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,
Hodge modules, and a formal deformation argument; it is the technical
heart of the paper.

\subsection*{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
$(X,\Delta)$ 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
NV$_n$.

The proof of NV$_n$ assumes a counterexample and derives a
contradiction in four stages. This is the pivotal movement of the
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}).
The counterexample $X$ must:
(a) have irregularity zero, so numerical equivalence equals
$\Q$-linear equivalence on its rational divisors;
(b) not be covered by proper subvarieties of non-general type;
(c) admit no positive canonical current annihilating the fibers of a
rational fibration (this rules out families of subvarieties pulled
back from a smaller ambient);
(d) admit no covering families of curves with uniformly bounded
genus and bounded normalized degree (this rules out the
``multi-section'' cases that would otherwise survive the two-slot
analysis);
(e) and, on any terminal birational model, admit no family of
birational correspondences between fixed finite covers whose graphs
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
every Cartier multiple of a suitable normalized nef divisor $P$ has
bounded vanishing order (the \emph{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
second is what produces the determinant incompatibility of Stage 3.

\medskip\noindent
\textbf{Stage 3: the Frobenius determinant} (Section~\ref{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
restriction to the diagonal of the Frobenius square cuts that rank
down by a factor of four. The determinant of any maximal minor is a
nonzero section of an external product of two line bundles; the scalar
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).
The counterexample was impossible. NV$_n$ holds.

\subsection*{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}.
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
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
paper.

\textbf{If you want to understand the hard input.}
Sections~\ref{sec:exclusions}--\ref{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.

\subsection*{What is used from outside}

The proof uses three external inputs:
\begin{enumerate}[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
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-
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
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
finite-generation corollary, after abundance has been proved.

\subsection*{How to read the 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});
the main theorem has rational boundary. All other conventions are
collected in Section~\ref{sec:introduction}, final subsection.

\newpage

%======================================================================
\section{Introduction}\label{sec: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
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}).

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.
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}
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}).

Throughout, $\kappa(L)$ is the Iitaka dimension of a rational divisor
$L$, with $\kappa(L)=-\infty$ 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}

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
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
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
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}):
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}

The threefold case illustrates why nonvanishing and boundary geometry
are central. Miyaoka \cite{Miyaoka1988} 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
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
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
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*)]
\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
required step is Fujino--Gongyo
\cite[Lemma~3.5]{FujinoGongyoRings2014}.
\item Kodaira dimension zero and numerical dimension zero are distinct
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:
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
in that numerical range \cite[Theorem~5.4]{LiuXu2025}. The induction
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}

\subsection*{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}.

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

\[
\begin{gathered}
L=K_X+D\text{ nef},\qquad L-cD\text{ pseudo-effective for some }c>0,\\
L\sim_{\Q}\textstyle\sum_i a_iD_i,\qquad L|_D\text{ semiample as a
  rational line bundle},\\
\Longrightarrow\quad \kappa(L)=\nu(L).
\end{gathered}
\]

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
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}).

The signed criterion is proved geometrically.
A generated multiple of $L|_D$ gives a morphism from $D$ to projective
space. Over a general point of the image of a positive-coefficient
component, a root construction separates that component from the
negative and coefficient-zero boundary. A filtered Hodge-module
calculation lifts local parameters on this image, together with a
parameter transverse to the lifted boundary, through all infinitesimal
neighborhoods. Fixing the lifted base parameters and varying the
transverse one deforms a boundary fiber off $D$ to a projective
subvariety. Its variations give a dominating family of subvarieties on
which $L$ is numerically trivial. The dimension of this family is
$n-\nu(L)$, which bounds the base dimension of the nef reduction of
$L$ from below by $\nu(L)$; descent along the reduction then
identifies $L$ as the pullback of a big divisor on the base, so
$\kappa(L)\ge\nu(L)$, hence abundance.

The formal-deformation shape has a direct precedent in Miyaoka's
construction, who credits Reid with the suggestion to analyse a
pluricanonical divisor \cite[p.~220]{Miyaoka1988}. Boundary extension
provides an alternative approach: Demailly--Hacon--P\u{a}un
\cite[Theorem~1.7,Corollary~1.8]{DemaillyHaconPaun2013} use a singular-
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
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
\cite[Theorem~1.5]{FujinoGongyo2014}.

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
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
these ordinary-variety statements apply to the covering construction;
Section~\ref{sec:hodge} carries out exactly this identification.

\subsection*{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.

\paragraph{Exclusions}
(Section~\ref{sec:exclusions}). The zero-boundary case of logarithmic
Iitaka subadditivity (Theorem~\ref{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
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
controls the multisections of the two-slot construction below.

\paragraph{Jets}
(Sections~\ref{sec:common-tools} and~\ref{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
order bounded by $k$ times a small normalised constant. On the other
hand, on the projective bundle over $X\times X$ associated to two
copies of a line bundle, the Seshadri constant at a very general point
exceeds $2n+2$, so jets of very high order exist. The two summands
let the argument move in either factor.

The first estimate fails only if the corresponding jet kernel has a
proper moving base component. Slicing and the curve exclusions leave
only components that project generically finitely in both factors,
hence give correspondences; their branch divisors cannot sweep either
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}
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,
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.
Its restriction to the ordinary diagonal is controlled by sections on
the Frobenius thickening of the diagonal, where the two relative
Frobenius maps agree; a canonical filtration of those sections cuts
the rank by a factor of four. The determinant of any nonzero maximal
minor is a section of an external product of two line bundles. The
first estimate --- expressed by a fixed movable curve on a blowup ---
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.
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}.

The numerical part combines two established methods. The argument
following a moving base component uses parameter differentiation as in
Ein--K\"uchle--Lazarsfeld
\cite[Proposition~2.3]{EinKuchleLazarsfeld1995}; the paper supplies
the tracking and adjunction estimates needed here. After reduction to
positive characteristic, ordinary jets are compared to Frobenius
jets using the local containment recorded by Musta\c{t}\u{a}--Schwede
\cite[Equation~(2.8)]{MustataSchwede2014}. The resulting evaluation
map is studied through Sun's canonical Frobenius filtration
\cite[Theorem~3.7]{Sun2008} (its curve case is treated in
Joshi--Ramanan--Xia--Yu \cite[Section~5.3]{JoshiRamananXiaYu2006}),
using the description by powers of the diagonal ideal due to
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 all-dimensional argument uses the following companion result. Its
application is confined to Lemma~\ref{ex:irregularity}.

\begin{theorem}[Logarithmic Iitaka subadditivity,
\cite[Corollary~6.2]{OpenAIIitaka2026}]\label{asm:iitaka}
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
$F$, 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
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
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}
is assumed here.

The zero-boundary application in Lemma~\ref{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
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}

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
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
then uses it to establish smooth nonvanishing.
Sections~\ref{sec:completion} and~\ref{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}
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
counterexample geometry.

\subsection{Conventions}\label{par:conventions}

A \emph{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
\[
 a(E;X,B)=1+\operatorname{ord}_E\bigl(K_W-p^*(K_X+B)\bigr).
\]
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.

For non-nef pseudo-effective divisors, statements about numerical
dimension use Nakayama's $\kappa_\sigma$ as specified in the relevant
reduction; these are not silently interchanged with the
intersection-theoretic numerical dimension $\nu$.

For a divisor with nonempty positive system, the Iitaka dimension is
the maximum dimension of the images of its positive systems under the
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
proper closed subvarieties. The transfer to arbitrary algebraically
closed characteristic-zero fields is made in Section~\ref{sec:completion}
only after the complex argument has been completed.

%======================================================================
\section{The inductive framework}\label{sec:induction}
%======================================================================

The dimension induction has two outputs in each dimension: smooth
canonical nonvanishing (NV$_n$) and good log minimal models for
projective lc pairs with effective real boundary and pseudo-effective
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}

For an effective real boundary $\Delta$, a \emph{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
which extracted prime divisors are included with coefficient one.

For a pseudo-effective divisor $A$ we write $\kappa_\sigma(X,A)$ for
Nakayama's numerical dimension. When $A$ is nef it agrees with the
intersection-theoretic numerical dimension $\nu(X,A)$.

Fix an integer $n\ge1$.

\begin{assumption}[Lower-dimensional good models]\label{mmp:lower-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.

\begin{center}\small
\begin{tabular}{@{}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} \\
\bottomrule
\end{tabular}
\end{center}

Here $\mathrm{GLM}_{\le d}$ 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
semiampleness on the whole reduced boundary is supplied here by
Proposition~\ref{prop:boundary-restriction}.

\subsection{Comparison with a nef model}

\begin{lemma}[Comparison with a nef model]\label{mmp:comparison}
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}
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
prime of $W$ not $v$-exceptional, the coefficient of $F$ is zero unless
that prime is extracted on $X'$. In the latter case it equals the
negative of the log discrepancy over $(X,\Delta)$; it is therefore
nonpositive and so zero. Hence $F$ is $v$-exceptional. If the source
adjoint is nef, $F$ is also $v$-nef, and negativity gives $F=0$.

The equality of pullbacks transfers semiampleness of a rational adjoint.
Indeed a proper birational morphism onto a normal variety has direct
image of its structure sheaf equal to that structure sheaf, so the
projection formula identifies global sections of each Cartier multiple
with sections of its pullback. If every pulled-back section vanished
over a point downstairs, it could not generate upstairs; generation
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
$\Q$-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}
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.

For the $S_2$ assertion, work locally with the cyclic class cover of
the $\Q$-Cartier divisor $D$. This cover is quasi-\'etale and klt, hence
Cohen--Macaulay. The eigensheaf $\OO_V(-D)$ is a direct summand of the
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
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
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}.
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}

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
\[
 0<\epsilon<\frac{1}{4nm+2},
\]
every step of an LMMP for $L-\epsilon P$ 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 $L$
makes $R$ negative for $L-\tfrac12P$. Choose a rational curve $C$
spanning $R$ and satisfying the length bound
\[
 -\bigl(L-\tfrac12P\bigr)\cdot C\le2n.
\]
Writing $a=L\cdot C\ge0$ and $p=P\cdot C$, we have
\[
 p\le2a+4n,\qquad a<\epsilon p,\qquad (1-2\epsilon)a<4n\epsilon.
\]
The choice of $\epsilon$ gives $a<1/m$. Since $ma$ is a nonnegative
integer, $a=0$.

For the driving klt contraction $c:Z\to B$, the line bundle $\OO_Z(mL)$
is numerically trivial over $B$. The line-bundle clause of the
contraction theorem gives its descent to a line bundle $\cA_B$ on $B$
with no change of $m$ \cite[Theorem~3.74(3)]{Fujino2017}. (One may also
see the unchanged index directly from relative basepoint freeness: two
consecutive sufficiently large powers descend, so their quotient
descends.) The line bundle $\cA_B$ is nef because its pullback
$\OO_Z(mL)$ is nef. For a flip $c^+:Z^+\to B$, its pullback
$(c^+)^*\cA_B$ is the line bundle of the transformed $mL$. This proves,
separately, that the transformed $L$ is nef and that $mL$ remains
Cartier with the same $m$.

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}

We now assume smooth nonvanishing in the current dimension and prove
the good-model implication. The argument first reduces real nef
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
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}
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
minimal model. Fix the support of its boundary, impose the rational
affine constraints that its coefficient-one components remain equal
to one, and take a sufficiently small rational polytope around the
given boundary within those constraints. On a fixed log resolution
witnessing dlt, the strict discrepancy inequalities remain strict on
this neighbourhood, so its boundaries remain dlt. The
rational-polytope theorem for nef adjoints
\cite[Remark~3.1, Proposition~3.2(3)]{Birkar2011II}, applied to all
extremal rays and intersected with this neighbourhood, expresses the
given real boundary as a finite rational simplex of dlt boundaries
with nef adjoints. It is enough to prove semiampleness for these
rational adjoints: their positive real combinations are semiample by
multiplication of the associated contractions. (One may also rationalise
semiampleness of a rational adjoint directly: the finite system of
linear equations expressing its divisor and principal divisors has
rational data, so a real solution with positive coefficients has a
rational one.)

We therefore work with a rational dlt pair $(Z,\Delta)$ and
$L=K_Z+\Delta$ nef. Real nonvanishing implies rational nonvanishing
here: writing a real effective representative in terms of finitely
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
remains to treat $\kappa(Z,L)=0$.

\paragraph{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
scaling to a Mori fibre space
\[
 (Z,\Delta-\epsilon P)\dashrightarrow(Z',\Delta'-\epsilon P')
 \xrightarrow{f}T.
\]
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},
applied also to the final contraction, gives a nef rational divisor
$A$ on $T$ with
\[
 K_{Z'}+\Delta'\sim_{\Q}f^*A.
\]
The base $T$ has dimension less than $n$.

If $(Z,\Delta)$ is klt, the transformed original pair $(Z',\Delta')$
is klt as well. Ambro's descent theorem \cite[Theorem~0.2]{Ambro2005}
gives an effective rational boundary $\Delta_T$ such that
$(T,\Delta_T)$ is klt and
\[
 A\sim_{\Q}K_T+\Delta_T.
\]
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.

Otherwise $P'$ is effective and relatively ample, since
$K_{Z'}+\Delta'-\epsilon P'$ 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
$f_S=(f\circ b)|_S:S\to T$ is surjective. Crepancy and adjunction
give an lc pair on $S$ whose nef adjoint is rationally linearly
equivalent to $f_S^*A$; it is semiample by the lower-dimensional
hypothesis. Semiampleness of a rational line bundle descends along a
proper surjection: use the equality of sections after Stein
factorisation for the connected fibre, and norms for the remaining
finite morphism. For the latter: choose a section of a basepoint-free
multiple nonzero simultaneously at all points of the chosen finite
fibre; such a section exists because each vanishing condition is a
proper linear subspace, and finitely many proper linear subspaces do
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
\[
 L\sim_{\Q}G_0:=H_0+\epsilon\Delta,\qquad H_0\ge0.
\]
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
\[
 K_W+D=p^*L+R,\qquad R\ge0.
\]
Every component of $D$ has positive coefficient in
$G_W:=p^*G_0+R\sim_{\Q}K_W+D$. Indeed, at strict boundary components
the difference between coefficient one and a klt boundary coefficient
is positive, and at exceptional components the coefficient contributed
by the adjoint comparison is the positive log discrepancy. The
pushforward $p_*G_W$ is bounded coefficientwise by a fixed multiple of
$G_0$, and sections inject under birational pushforward, so
\[
 0\le\kappa(W,K_W+D)\le\kappa(Z,G_0)=0.
\]

Take a $\Q$-factorial dlt log minimal model $(V,D_V)$ of $(W,D)$, and
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 $q^*G_W$
forward by $v$ gives
\[
 K_V+D_V\sim_{\Q}G_V=\sum_ib_iD_i,\qquad b_i>0,
 \qquad\Supp G_V=D_V.
\]
(For completeness: at a component extracted on $V$, positivity also
holds since its log discrepancy over $(W,D)$ is zero, its centre lies
in $D$, and the pullback of the full-support effective $G_W$ has
positive order there. At a prime retained on $V$, there is no
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
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 $K_V+D_V$
abundant. Its Iitaka dimension is zero, so its numerical dimension is
zero. For an ample divisor $H_V$ on $V$ we get
\[
 G_V\cdot H_V^{n-1}=(K_V+D_V)\cdot H_V^{n-1}=0.
\]
Effectivity forces $G_V=0$, and its full support then gives $D_V=0$.

Since $G_W=p^*G_0+R$ with $R\ge0$, on the same common resolution
\[
 0\le q^*p^*G_0\le q^*G_W,\qquad v_*q^*G_W=G_V=0.
\]
Thus $q^*p^*G_0$ is effective and $v$-exceptional. It is nef because
it is rationally linearly equivalent to $(p\circ q)^*L$. The
negativity lemma for $v$ forces it to vanish. Hence $G_0=0$ and
$L\sim_{\Q}0$. Together with the preceding cases this settles the klt
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.}
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
perturbed boundaries effective and klt. Convexity of the
pseudo-effective cone, applied between $L$ and $L-\delta P$, makes
both
\[
 L-\epsilon P,\qquad L-2\epsilon P,\qquad P=\lfloor\Delta\rfloor,
\]
pseudo-effective klt adjoints. Their Iitaka dimensions are zero:
nonvanishing gives the lower bound, and adding the effective
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
\[
 2(L-\epsilon P)=L+(L-2\epsilon P),
\]
and the last summand has an effective rational representative,
monotonicity and homogeneity of $\kappa_\sigma$ give
\[
 \kappa_\sigma(Z,L)\le\kappa_\sigma\bigl(Z,2(L-\epsilon P)\bigr)=0.
\]
The nef divisor $L$ is therefore numerically trivial. Nonvanishing is
now already available in this finishing case: write
$L\sim_{\Q}E\ge0$. For an ample divisor $H$, $E\cdot H^{n-1}=0$ forces
$E=0$. Hence $L\sim_{\Q}0$, without any further nonvanishing or
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
nef rational lc pair.
\end{proof}


%======================================================================
\section{The signed-representative engine}\label{sg:section}
%======================================================================

The induction, at its key step, has a nef log canonical divisor
$L=K_X+D$ on a $\Q$-factorial dlt pair $(X,D)$ with $D$ reduced. The
hypothesis of the signed criterion is that some rational-line
representative of $L$ --- not necessarily the boundary itself --- is
supported on $D$ with either signs. The difficulty is that $L|_D$ is
semiample but $L$ itself is only nef.

The engine of this section converts that hypothesis into abundance.
The outline, already given on page~\pageref{par:engineering}, is
repeated here in numerical form so the reader knows what the
individual lemmas are for.

The semiample $L|_D$ gives a morphism
\[
 \phi:D\longrightarrow P:=\PP^s,\qquad
 N_0|_D\simeq\phi^*\OO_P(1)
\]
for a sufficiently divisible positive integer $m$, where $N_0:=\OO_X(mL)$.
Write $s_0$ for the rational section of $N_0$ having divisor $mG$, where
$G=G_+-G_-$ comes from the signed representative and $G_\pm,E\ge0$. Let
$r+1$ be the dimension of the image of a positive-coefficient
component of $D$ in $P$; a lemma below shows that every other
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,
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,
such that $S$ is (generically) its own root line and carries a
Stein factorisation
\[
 S\xrightarrow{g}\mathcal Y\xrightarrow{p}\PP^s
\]
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
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.

The full theorem is:

\begin{theorem}[Signed-boundary abundance criterion]\label{thm:signed}
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*)]
\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}

The proof occupies the remainder of this section and the next; it does
not use the induction hypothesis, only established results of the
minimal model program. We write $L\sim_{\Q}\sum_ia_iD_i$ for the
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 $0<\nu(L)<n$.
Put $r=\nu(L)-1$. Then:
\begin{enumerate}[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}
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 $(L-cD)$
is pseudo-effective. The result is
\[
 0\le(L-cD)L^vH^{n-v-1}
   =-c\sum_iD_iL^vH^{n-v-1}.
\]
Each $D_iL^vH^{n-v-1}$ is nonnegative, so all vanish. If $v=0$,
ampleness forces $D=0$, contradicting the hypothesis $D\ne0$. For
$v>0$, semiampleness of $L|_D$ identifies the numerical dimension of
each $D_i$'s image in $P$ with the numerical dimension of $L|_{D_i}$,
which equals the intersection $D_iL^{\nu(L)-1}H^{n-\nu(L)}=\dim\phi(D_i)$.
This intersection is zero by the vanishing just proved, so the image
dimension of every $D_i$ is at most $r=\nu(L)-1$.

Part (ii). By the same vanishing,
\[
 0<L^{\nu(L)}H^{n-\nu(L)}=\sum_ia_iD_iL^rH^{n-v},
\]
so at least one $a_i>0$ and at least one term is strictly positive.
The corresponding component of $G_+$ attains image dimension exactly
$r$.

Part (iii). Consider the symmetric matrix $Q_{ij}=D_iD_jL^rH^{n-r-2}$.
Distinct effective $\Q$-Cartier divisors have effective intersection
cycles, so the off-diagonal entries are nonnegative. The mixed Hodge
index theorem gives the ambient intersection form at most one positive
direction: approximate by ample classes and apply the ordinary Hodge
index theorem on complete-intersection surfaces. The class $L$ is
isotropic and orthogonal to every $D_i$ (this is the vanishing from
part (i)), while its pairing with $H$ is strictly positive:
\[
 Q(L,L)=L^{r+2}H^{n-r-2}=0,\quad
 Q(L,D_i)=D_iL^{r+1}H^{n-r-2}=0,\quad
 Q(L,H)=L^{r+1}H^{n-r-1}>0.
\]
The intersection form on the orthogonal complement of $L$ is therefore
negative semidefinite, and this holds on the span of the $D_i$. Pulling
back to a resolution preserves the conclusion, so smoothness of $X$
is not needed. But any strictly positive off-diagonal entry would
contribute a positive direction to the span; the entries therefore
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
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*)}]
\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}
\]
The support of $T$ is locally set-theoretically principal.
The ambient space and $S$ are Cohen--Macaulay on scheme charts.
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
on scheme charts, the ideal $\mathcal I_{J/S}$ is maximal
Cohen--Macaulay, and
\[
 \mathcal B:=
 \mathcal Hom_S(\mathcal I_{J/S},\omega_S)
 \simeq\OO_S(aS).
\]
The identification agrees off $J$ with the residue convention
specified by \eqref{sg:ambient-adjunction}.

\item
There is a finite representable morphism
\[
 S\longrightarrow D\times_P\mathcal P
\]
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 $r$,
whereas the image of $S$ has dimension $r$.

\item
There is a smooth projective bundle $p:\mathcal Y\to\mathcal P$
and a representable projective morphism $g:S\to\mathcal Y$.
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 $\mathcal T\hookrightarrow\mathcal Y$,
and
\[
 g_*\OO_S=\OO_{\mathcal T},\qquad
 \OO_S(S)=g^*C,\qquad \mathcal B=g^*C^a.
\]

\item
For a separated quasi-projective scheme chart
$U\to\mathcal Y$ that is etale and of finite type, put
$S_U=S\times_{\mathcal Y}U$.
The etale morphism $S_U\to S$ extends compatibly to every
nilpotent thickening $qS$ in $\mathcal X$.
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}
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.}
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
empty divisors.  Denote the tautological root divisors by
$S_+,S_-,S_0$ on $\mathcal R$. Their reducedness can be checked
at a generic prime of downstairs multiplicity $b$. The chosen
divisibility conditions give $b\mid\ell$. A normalized chart of
$y^\ell=x^b$ has ramification index $\ell/b$, and $y$ has order
one.  Since $m$ divides $\ell$, this also applies to $D_0$.
The divisors are Cartier, so normality and generic reducedness
give their reducedness everywhere.

Put $B=S_++S_-+S_0$ on $\mathcal R$. With this full reduced
boundary, log ramification gives a log
canonical pair and the relation
\[
 K_{\mathcal R}+B\sim_{\Q}a(S_+-S_-).
\]
The ambient charts are klt.  They are klt off the boundary,
and, on a log resolution, decreasing every boundary coefficient
slightly removes all zero-discrepancy places. The difference
between the two sides is a torsion integral Weil class, represented by
\[
 \mathcal F=
 \bigl(\omega_{\mathcal R}(B)
       \otimes\OO_{\mathcal R}(-a(S_+-S_-))\bigr)^{**}.
\]
Choose a periodicity isomorphism $\mathcal F^{[q]}\simeq\OO_{\mathcal R}$,
where square brackets denote reflexive powers. It defines multiplication
in the finite algebra
\[
 \mathcal A=\bigoplus_{i=0}^{q-1}\mathcal F^{[i]}.
\]
Let $\tau:\mathcal R^\dagger\to\mathcal R$ be the normalization
of $\Spec_{\mathcal R}\mathcal A$. In codimension one this is
the ordinary cover of a torsion line bundle and is etale. The
cover preserves log canonicity, klt ambient singularities, and
the reduced Cartier boundary. From now on $S_+,S_-,S_0$ denote
their pullbacks to $\mathcal R^\dagger$.

Tautological evaluation on this cover trivializes the pullback
of $\mathcal F$ in codimension one and hence fixes the adjoint
isomorphism. Because this evaluation is a sheaf morphism
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
normalized blowup of the ideal of $S_+\cap S_-$. The spaces
constructed so far are related by
\[
 \widetilde{\mathcal R}\xrightarrow{p_1}\mathcal R^\dagger
 \xrightarrow{\tau}\mathcal R\xrightarrow{\rho}X.
\]
The blown-up ideal is
locally generated by two elements, so the ordinary blowup embeds
in a relative projective line.  Its fibers have dimension at
most one, as do the fibers after finite normalization.  Hence
each exceptional divisor lies over a codimension-two component
of the intersection.  Downstairs such a component is a dlt
stratum, generically SNC.  Separate normalized Kummer charts
and purity for the index cover at the smooth generic locus
give the same description upstairs.

The tautological exceptional divisor $E$ is consequently reduced
Cartier, and the divisors
\[
 S'_+=p_1^*S_+-E,\qquad S'_-=p_1^*S_--E
\]
are reduced Cartier and disjoint on $\widetilde{\mathcal R}$.
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
\[
 S'_++S'_-+E+p_1^*S_0
\]
is crepant and log canonical.  To check the ambient singularities
as well, denote this boundary on a chart $V'$ by $B'$.  The
crepant equality gives log canonicity for every valuation.
Outside $B'$, the blowup is an isomorphism to the old klt
boundary complement.  A divisor $F$ with
$a(F;V',B')=0$ therefore has center in $\Supp B'$.  Since $B'$
is effective Cartier, $\ord_F(B')>0$, whence
\[
 a(F;V',0)=a(F;V',B')+\ord_F(B')>0.
\]
Positive pair discrepancies also remain positive after the
boundary is dropped.  Thus $V'$ is klt even at its
higher-codimension singular loci. On $\widetilde{\mathcal R}$,
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
\[
 \omega(S+T)\simeq\OO(aS).
\]
Klt singularities make the ambient charts Cohen--Macaulay.
All the boundary divisors in this display are Cartier, so the
isomorphism also makes these charts Gorenstein.

We also need finiteness of $S\to S_+$ on $\mathcal R^\dagger$.
Locally write $S_+=(u=0)$ and $S_-=(v=0)$. Before normalization,
the strict transform of $S_+$ lies in the blowup chart with
ratio $u/v$, where its equation is $u/v=0$. Thus each fiber of
this strict transform over $S_+$ has at most one point.
Properness and finite normalization prove finiteness of $S\to S_+$.
A codimension-one point of $S\cap T$ therefore lies over a
codimension-two intersection downstairs.  The generic SNC
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.}
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
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
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:
the covers used above are finite, the blowup is projective,
and finite quotients preserve quasi-projectivity.  These
chartwise quotients patch.

Before taking the quotient, the section of the disjoint divisor
$S'_-$ is a unit near $S$. There the retained line
$\OO(S-S'_-)$ identifies with $\OO(S)$, so the line of $S$ and
its section both descend through the kernel quotient. Continue
to write $S,T$ for their reduced images under $\kappa$; they
now lie on $\mathcal X$. In particular $S$
remains Cartier.  The finite-group norm of a local equation
for $T$ upstairs makes its image set-theoretically principal;
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
Cohen--Macaulay because their structure sheaves are invariant
direct summands of the finite CM covers.

More explicitly, the kernel acts trivially on the retained root
line and hence on $\OO(aS)$.  Equivariance of the fixed
isomorphism gives the same kernel character on $\omega(S+T)$.
Taking invariants descends the isomorphism, and log ramification
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.}
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
\[
 \mathcal I_{J/S}
 =
 \bigl(f_*\OO_{S^{\mathrm{up}}}
       (-T^{\mathrm{up}}|_{S^{\mathrm{up}}})\bigr)^{\mathrm{inv}}.
\]
An invariant function vanishes on the reduced quotient image
exactly when its pullback vanishes on this reduced preimage.
The displayed invariant sheaf is maximal Cohen--Macaulay:
the upstairs sheaf is invertible on a CM scheme, finite
pushforward preserves its depth over the quotient, and in
characteristic zero invariants form a direct summand.

Apply finite-map duality with trace and then take invariants:
\[
 \mathcal Hom_S(\mathcal I_{J/S},\omega_S)
 =
 \bigl(f_*\omega_{S^{\mathrm{up}}}
       (T^{\mathrm{up}}|_{S^{\mathrm{up}}})\bigr)^{\mathrm{inv}}
 \simeq\OO_S(aS).
\]
The last isomorphism comes from Cartier adjunction upstairs
and descent of the line of $S$.  Finite-map duality applies
in the presence of ramification; it requires no invertibility
of $\omega_S$ itself.  Off $J$, the identification is the
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.}
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
$r$ 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 $\Supp G_+\cap\Supp G_-$,
and $p_1^*S_0$ maps into $D_0$. Taking the quotient changes none
of these images in $X$. The image of $J$ in $D$ is consequently
contained in
\[
 \Supp G_+\cap(\Supp G_-\cup D_0).
\]
The same lemma gives image dimension less than $r$ for this
locus in $P$. We have thus obtained a
representable projective morphism $h:S\to\mathcal P$; it remains
to embed its Stein image in the required smooth bundle.

The Stein algebra defines a finite stack
$\mathcal T=\Spec_{\mathcal P}h_*\OO_S$ over $\mathcal P$.
This coherent algebra is generated as a module by vector
bundles on $\mathcal P$.  Indeed, decompose it by the finitely
many inertia characters, twist each summand by a power of $C$
so that it descends to $P$, and use Serre generation there.
The resulting vector-bundle surjection embeds $\mathcal T$ in
a vector bundle over $\mathcal P$.  Since $\mathcal T$ is
proper over the base, it remains closed in the projective
completion. This completion is $\mathcal Y$ and gives the map
$g:S\to\mathcal Y$ from the reduced divisor.
The Stein construction yields $g_*\OO_S=\OO_{\mathcal T}$,
and the line identities follow from those already established.

Finally, start with the etale chart $S_U\to S$ obtained from $g$.
Invariance of the etale site under nilpotent thickening extends
this chart uniquely to $qS_U\to qS$, compatibly as $q$ varies.
This construction does not use a map $qS\to\mathcal Y$;
lifting the map from $S$ is the task of the next section.
The reduction $S_U$ is a scheme; its extension $qS_U$ is an
algebraic space with trivial inertia.  The chosen chart is
separated, the map to the root gerbe is finite, and that gerbe
has finite diagonal, so $qS_U$ is separated over $X$.
The finite representable map $S\to D\times_P\mathcal P$
and the etale map $S_U\to S$ show that the finite-type
geometric fibers of $S_U\to X$ are zero-dimensional.
Nilpotent thickening changes neither geometric points nor
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}

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.
\(\mathcal P=\sqrt[\ell]{\OO_{\PP^s}(1)}\) is a root gerbe, \(C\) is
its tautological line bundle, and
\[
 S\xrightarrow{g}\mathcal Y\xrightarrow{p}\mathcal P,
 \qquad h=p\circ g.
\]
Here \(g\) is representable and projective, and \(p\) is a smooth
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,
\(\mathcal I_{J/S}\) 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.

Put \(I=\OO_{\mathcal X}(-S)\). For a separated quasi-projective
scheme etale chart \(U\to\mathcal Y\) of finite type, write
\(S_U=S\times_{\mathcal Y}U\) and again \(g:S_U\to U\)
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
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
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
\[
 g_*(I^j/I^{j+k+1})
       \longrightarrow g_*(I^j/I^{j+k})
\]
is surjective as a map of sheaves of abelian groups.  Its kernel is
\(\OO_{\mathcal T_U}\otimes C^{-j-k}\), 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
motion \cite[Theorem~3.1]{LiuXu2025}. Here the semiample boundary image
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 \(\OO_S\otimes C^{-j-k}\).
The corresponding connecting homomorphism therefore takes values in
\[
 R^1g_*\OO_S\otimes C^{-j-k}.
\]
We will identify this sheaf with a negative twist of the lowest
filtered piece \(F_0M\) of a Hodge-module direct image.  Lifting local
functions gives Cech cocycles in this coherent sheaf.  A calculation
with their graphs shows that the coordinate cocycle lies in the kernel
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}

We work with graded-polarizable mixed Hodge modules on ordinary
complex algebraic varieties.  The inputs are projective strictness
and the usual functorial operations
\cite[Theorem~2.14 and Section~4]{Saito1990}, the comparison with the
Du Bois complex and its coherent dual
\cite[Theorem~0.2, Corollary~0.3 and Proposition~5.3]{Saito2000}, and
Kodaira--Saito vanishing \cite[Proposition~2.33]{Saito1990}.
On a smooth stack, we use compatible systems of these objects on
scheme etale charts and descend their filtered differential modules.
Thus the argument requires no Hodge-module theory on stacks.

All differential modules in this section are right modules with an
increasing filtration.  In the Spencer de Rham complex the module
itself occupies degree zero.  Accordingly, on a smooth chart \(V\),
the term in
degree \(-i\) of \(\Gr^F_k\DR_V(M)\) is
\[
 \Gr^F_{k-i}M\otimes_{\OO_V}\bigwedge^iT_V.
\]
When \(F_{<0}M=0\), the lowest graded de Rham complex therefore
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
\[
 A^\bullet_U=\mathbf D\!\left(
             j_!\Q^H_{S_U\setminus J_U}[N]\right),
 \qquad
 A_{0,U}={}^{p}\!H^0A^\bullet_U,
 \qquad
 M_U={}^{p}\!H^1g_*A_{0,U}.
\]
Here \(\mathbf D\) is Hodge-module duality, \({}^pH^i\) denotes
perverse cohomology, and \(g_*\) in the definition of \(M_U\)
is the derived Hodge-module direct image.  Later, \(g_+\) will
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:
\[
 F_{<0}A_{0,U}=0,\qquad F_0A_{0,U}=\mathcal B|_{S_U},
 \qquad
 F_{<0}M_U=0,\qquad
 F_0M_U=R^1g_*(\mathcal B|_{S_U}).
\]
In a smooth ambient embedding \(S_U\hookrightarrow V\) of
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}
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
the sheaf by duality.  Du Bois
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
has \(\omega\) as its lowest piece, at index zero.

To pass to perverse cohomology, work locally on a fixed chart
and let \(q\) be the smallest filtration index occurring in any
perverse cohomology object of \(A^\bullet_U\).  Boundedness of
the complex and goodness of the filtrations provide such a lower
bound.  At index \(q\), each of these objects has a graded
Spencer complex consisting only of its degree-zero term.  The
spectral sequence for perverse truncation has a single nonzero
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}.
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
the other perverse cohomology objects vanish.  This passage
does not require the shifted constant complex to be perverse.

We next identify the underlying unfiltered module.  Ambient duality in \(V\)
identifies the dual of the reduced constant complex with
\[
 R\Gamma_{[S_U]}(\omega_V)[c].
\]
The degree-zero module is the first nonzero support cohomology,
\(\mathcal H^c_{S_U}(\omega_V)\).  The support of \(J_U\) is locally
set-theoretically principal in \(S_U\) by the geometric
construction.  Lift a local defining function to \(V\) and
invert it.  This exact meromorphic localization realizes
\(j_*\) on the underlying module and gives the description
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
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 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
supported on \(S_U\) is a sum of injective modules at the
codimension-\(c\) generic points.  Localization extends the
agreement across \(J_U\).  We have therefore identified the
actual lowest-piece inclusion, including its residue convention.

We have now identified \(F_0A_{0,U}\) and its inclusion before
direct image.  To identify \(F_0M_U\), compute \(g_+\) by a
graph embedding followed by a smooth
projection.  At filtration zero, only the top term of the relative
Spencer complex survives: it is the direct image of
\(\mathcal B|_{S_U}\) on the graph.  Projective strictness yields
\[
 F_{<0}M_U=0,\qquad
 F_0M_U=R^1g_*(\mathcal B|_{S_U}),
\]
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.
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}

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 \(j<0\),
\[
 \mathbb H^j\!\left(
   \mathcal Y,\Gr^F_k\DR_{\mathcal Y}(M)\otimes p^*C^{-b}
 \right)=0.
\]
Here the graded de Rham complex is the descended coherent complex.
\end{lemma}

\begin{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
Kodaira--Saito vanishing applies.  Since \(p\) is finite on the
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.

For right differential modules, use the transfer bimodule
\[
 \mathcal D_{\mathcal Y\to\mathcal P}
 =
 \OO_{\mathcal Y}\otimes_{p^{-1}\OO_{\mathcal P}}
                         p^{-1}\mathcal D_{\mathcal P},
\]
with its order filtration.  Derived tensor over
\(\mathcal D_{\mathcal Y}\), followed by derived direct image,
forms the filtered direct image; equivalently, these operations
can be carried out on Rees modules.  De Rham on the base is a
further derived tensor with \(\OO_{\mathcal P}\), filtered from
degree zero.  The filtered Spencer resolution, associativity of
derived tensor, and projection formula identify this with direct
image of the de Rham complex upstairs.  Indeed, tensoring the
transfer module over the base differential operators with the
base structure sheaf gives \(\OO_{\mathcal Y}\).

These sheaf constructions are canonical on the etale site, and the
Spencer resolutions are locally free over differential operators.
They commute with chart changes already before taking associated
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
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
\[
 v:\PP^s_z\longrightarrow\mathcal P
\]
defined by the power map
\([z_0:\cdots:z_s]\mapsto[z_0^\ell:\cdots:z_s^\ell]\)
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
\(\check U_i\to\mathcal P\).  The restriction of \(v\) to
\(z_i\ne0\) factors through \(\check U_i\).  On each such
chart take the derived Hodge-module inverse image of \(M'\)
and then its degree-zero perverse cohomology.  Thus the object
being constructed chartwise is
\[
 H={}^pH^0(v^*M').
\]
The chosen root and functoriality identify these objects on
overlaps, producing a system on the Zariski opens of \(\PP^s_z\).

We check that \(H\) is a global graded-polarizable mixed Hodge
module, so that ordinary vanishing applies to it.  The perverse,
filtered, and weight data glue, as do the strict-support
decompositions of the pure weight gradeds, by uniqueness.  On a
smooth connected dense stratum of an irreducible support, a
polarization on a nonempty Zariski open extends to a flat pairing:
the fundamental group of that open surjects onto the fundamental
group of the stratum.  Hodge compatibility extends by continuity,
and the extended flat pairing remains nondegenerate and positive.
The local Hodge modules already give boundary quasi-unipotence.
Saito's extension theorem for polarizable variations
\cite[Section~3.b]{Saito1990} and uniqueness of strict-support
extension identify the global pure object with the glued object.
Projectivity of \(\PP^s_z\) leaves no additional boundary at
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
\(\check U_i\), 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
duality in equal dimensions, compose to multiplication by the
degree.  This can first be checked on the finite etale locus,
where the trace sums over the sheets.  On each connected smooth
base chart, the shifted constant Hodge module is the rank-one
intersection complex, with endomorphism ring \(\Q\), and its
endomorphisms are determined on a dense open.  The composition
therefore equals the degree globally, with no contribution
supported on the branch locus.  This argument permits
ramification.

Tensor the unit and trace with \(M'\), apply proper projection
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
\[
 M'\longrightarrow v_+H\longrightarrow M'
\]
whose composite is multiplication by the degree of the whole
base-changed cover.  Dividing the second map by that degree
gives the retraction.  Each map is canonical under etale base change; the
retraction consequently descends on filtered differential modules
as well.

Applying the finite version of
Equation~\eqref{hg:graded-direct-image} and projection formula
now exhibits the hypercohomology in the statement as a direct
summand of
\[
 \mathbb H^j\!\left(
   \PP^s_z,\Gr^F_k\DR(H)\otimes\OO_{\PP^s_z}(-b)
 \right).
\]
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}

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}]
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
that this derivation vanishes both on functions from \(U\)
and on a local generator of \(I/I^2\).  The graph calculation
relates these two errors in the differential module \(M_U\).
Its order-one symbol, together with negative-twist vanishing,
first kills the function error.  The original relation then
kills the conormal error.

Define
\[
 E_{j,k}=g_*(I^j/I^{j+k}),\qquad j\geq0,\quad k\geq1.
\]
The first graded layer is
\[
 E_{j,1}=\OO_{\mathcal T_U}\otimes C^{-j}.
\]
For adjacent lengths, the exact sequence has kernel
\(g_*\OO_{S_U}\otimes C^{-j-k}\), and its connecting homomorphism
takes values in
\[
 R^1g_*\OO_{S_U}\otimes C^{-j-k}.
\]

\smallskip\noindent\emph{The obstruction derivation.}
Assume inductively that all shorter transitions are surjective.
A section of
\(E_{j,k}\) with zero length-one term comes from \(E_{j+1,k-1}\)
when \(k>1\).  Induction lifts it locally from \(E_{j+1,k}\),
so its connecting class vanishes.  Induction also makes
\(E_{j,k}\to E_{j,1}\) surjective.  The obstruction thus
factors through the length-one layer; for \(k=1\) this
factorization is immediate.  More precisely, these factorizations
give additive maps
\[
 D_{k,j}:\OO_{\mathcal T_U}\otimes C^{-j}
       \longrightarrow R^1g_*\OO_{S_U}\otimes C^{-j-k}.
\]
Together they define \(D_k\) on the graded algebra
\(\bigoplus_{j\geq0}\OO_{\mathcal T_U}\otimes C^{-j}\), 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
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
\(R^1g_*\OO_{S_U}\otimes C^{-j}\).

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
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)
 =
 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
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.}
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
\(C^{-1}\).  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
on an open cover of \(S_U\), obtaining functions
\(h_{i,\lambda}\) modulo \(I^{k+1}\) and generators \(y_i\)
modulo \(I^{k+2}\).  The conormal generator must be lifted one
order farther than the coordinates: its next error lies in
\(I^{k+1}/I^{k+2}\), whereas a coordinate 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
target line bundles; \(e_{ij,\lambda}\) and \(\eta_{ij}\)
are their coefficients on the reduction.  Denote the induced
frame of
\(\mathcal B|_{S_U}\) by \(\sigma=y^{-a}\).

Off \(J_U\), the fixed adjunction isomorphism gives
\[
 \omega_{(k+1)S_U}
 \simeq\OO_{\mathcal X}((a+k)S)|_{(k+1)S_U}.
\]
Via this isomorphism, the local equation \(y_i\) defines a
dualizing generator, denoted \(b_i=y_i^{-(a+k)}\).  This
notation denotes a frame of the line bundle
\(\OO_{\mathcal X}((a+k)S)\) restricted to the thickening;
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}
 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,
\(\sigma\) is viewed in the dualizing sheaf of the thickening
through the trace inclusion.  They express Cartier trace and
the annihilator of the thickening.

\smallskip\noindent\emph{The 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 \(\iota:S_U\hookrightarrow\overline Z\times U\)
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
module \(M_U\) on \(U\).  For each local graph lift \(h_i\),
Ext-to-support cohomology sends the dualizing section \(b_i\)
to a section \(\delta_i(b_i)\) of \(\mathcal N_\Gamma\).
All the lifted graphs have the same reduced support, namely
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}
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
action is taken in this graph module.

The calculation also applies when \(S_U\) is singular.  Put
\(c_0=\dim Z-N\).  Ext-to-support identifies the dualizing
sheaf of the thickening with its annihilator submodule in
\(\mathcal H^{c_0}_{S_U}(\omega_Z)\).  To see this, use the
support-cohomology spectral sequence.  Support cohomology
vanishes below \(c_0\), so in total degree \(c_0\) the Ext
group consists of homomorphisms from the structure sheaf of
the thickening to the first support-cohomology module.  In
particular, no local complete-intersection hypothesis on
\(S_U\subset Z\) is needed.

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
\[
 \frac{b\,dt_1\wedge\cdots\wedge dt_d}
      {\prod_{\lambda=1}^d(t_\lambda-h_{i,\lambda})}.
\]
It represents successive support cohomology in the additional
coordinate equations, equivalently the Koszul residue for the
graph, and can be computed etale-locally near the graph branch.
After taking support cohomology from \(Z\), the translated
new coordinates form a regular sequence; their support
cohomology is consequently concentrated in the degree equal
to their number.  Either set of translated coordinates gives
the same localization of this support-torsion module, because
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}
annihilate \(b_j\).  The linear terms are determined by
\[
 (h_{j,\lambda}-h_{i,\lambda})b_j
       =e_{ij,\lambda}\sigma.
\]
The right action on a top differential form satisfies
\[
 \left(\frac{dt_\lambda}{t_\lambda-h}\right)
       \cdot\partial_{t_\lambda}
   =\frac{dt_\lambda}{(t_\lambda-h)^2}.
\]
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.
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
pole orders are allowed.

\smallskip\noindent\emph{Vanishing 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
\(\pi_+\iota_+A_{0,U}=g_+A_{0,U}\).  Thus degree-one holonomic
differential-module cohomology of this Spencer direct image is
the underlying module of
\(M_U={}^{p}\!H^1g_*A_{0,U}\).
The cochain \((\delta_i(b_i))\) belongs to the top, degree-zero
term, from which there is no outgoing relative Spencer
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
\(R^1g_*\mathcal B\) classes in \(F_0M\).  Right
differentiation in the \(U\)-coordinates acts on the
pushforward complex.  Consequently the Cech classes satisfy
\[
 -(a+k)[\eta_{ij}\sigma]
 +\sum_{\lambda=1}^d
       [e_{ij,\lambda}\sigma]\cdot\partial_{t_\lambda}=0
 \qquad\text{in }M_U.
\]
Here each bracketed class belongs to
\(R^1g_*\mathcal B=F_0M_U\); after differentiation, its
image belongs to \(F_1M_U\).  Thus every term of the relation
lies in \(F_1M\), and the \(\eta\)-term lies in \(F_0M\).
The equality itself was proved in the unfiltered differential
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}
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 \((\delta_i(b_i))\)
has been imposed.

Because \(F_{<0}M=0\), the entire complex at filtration
index one is
\[
 \Gr^F_1\DR(M)=
 \bigl[T_{\mathcal Y}\otimes F_0M
       \longrightarrow\Gr^F_1M\bigr],
\]
in degrees \(-1\) and \(0\), with differential the symbol
map in Equation~\eqref{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}
annihilates this space, and hence \(e=0\).

It remains to kill the conormal part of the derivation.
With coordinates and a line trivialization fixed locally,
\(e=0\) says that every class
\([e_{ij,\lambda}\sigma]\) vanishes in \(F_0M\), and thus
in \(M\).  Its actual differential-operator image vanishes
as well, before passing to symbols.  Returning to the
unfiltered relation gives
\[
 (a+k)[\eta_{ij}\sigma]=0.
\]
Injectivity of the lowest-piece inclusion and \(a+k>0\)
then give \(D_k(y)=0\).  The degree-zero derivation also
vanishes: its restriction to \(\OO_U\) is the coordinate
error \(e\), and \(\OO_U\to\OO_{\mathcal T_U}\) is
surjective because \(\mathcal T_U\) is a closed subscheme
of \(U\).  The coefficient
algebra and the conormal frame \(y\) generate the full
graded length-one algebra locally.  Thus \(D_k=0\) in
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
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 $0<v<n$,
$r=v-1$, $d=n-1-r=n-v$, and the construction of
Proposition~\ref{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;
descent along that reduction then proves abundance.

The immediate geometric problem is to move a projective fiber off the
boundary while keeping it projective. A normal coordinate supplies the
direction of motion; parameters on the image of the boundary family keep
the fiber dimension fixed. The global root and Hodge constructions have
already supplied these functions formally. Their lifts concern the common
underlying topological map of the thickenings; a map of ringed spaces to
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
algebraic family of integral projective subvarieties of $X$
of dimension
\[
 d=n-1-r=n-v
\]
whose general members avoid $D$ and on which $L$ is
numerically trivial.
\end{proposition}

\begin{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.}
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,
that component is smooth of pure dimension $r$, and $S_U$
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 $t$.
The fiber
\[
 F=S_{U,t}
\]
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
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
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
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
surjectivity on global sections.  We can therefore lift the
parameters compatibly through all thickenings, and lift the
chosen conormal frame to a compatible element $\widetilde y$
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
\[
 R_q=\C[s]/(s^q),\qquad s\longmapsto\widetilde y.
\]
The construction has the diagram
\[
\begin{tikzcd}[column sep=large]
F \arrow[r,hook] \arrow[d] & Z_q \arrow[r] \arrow[d] &
X\times\Spec R_q \arrow[d] \\
\Spec\C \arrow[r,hook] & \Spec R_q \arrow[r,equal] & \Spec R_q.
\end{tikzcd}
\]
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
preexisting morphism of the ambient thickening to the base.

The schemes $Z_q$ are compatible under reduction, have
closed fiber $F$, and are flat over $R_q$.  For flatness,
the powers of the Cartier generator first identify the
successive $s$-layers of $qS_U$ with $\OO_{S_U}$.  This
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.}
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
properness of a finite-type morphism.  Thus it is finite.
Put $R=\C[[s]]$.  Projective Grothendieck existence
algebraizes the compatible finite algebra sheaves on
$X\times\Spec R_q$ to a finite algebra on
$X_R=X\times\Spec R$.  Full faithfulness algebraizes
their multiplication and unit
\cite[Lemmas~30.24.1 and~30.24.3]{StacksExistence}.
The relative spectrum gives a finite morphism
\[
 Z\longrightarrow X_R
\]
with exactly the prescribed reductions.  In particular,
$Z$ is projective over $R$.  Formal flatness gives
flatness along the closed fiber; on the generic fiber
over $\C((s))$ it is automatic.  Hence $Z$ is flat over $R$.

The images of the pole and coefficient-zero boundary parts
miss $F$.  Their inverse images in $Z$ are closed and
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 $X$
lies in $\Supp G_+\cap\Supp G_-$, so it is already excluded
by avoidance of the pole part.

On the formal neighborhood from which the $Z_q$ were cut out,
the root construction therefore identifies the divisor of the
pulled-back section $s_0$ with $\ell S$.  Comparing its
root-line frame with the formal Cartier generator
$\widetilde y$ gives a frame in which this section is
$\widetilde y^\ell$.  Restricting to the $Z_q$ gives
compatible trivializations
\[
 N_0|_{\widehat Z}\simeq\OO_{\widehat Z},
 \qquad s_0|_{\widehat Z}=s^\ell.
\]
Full faithfulness algebraizes this line-bundle trivialization
and its section identity on $Z$; no algebraic map from $Z$
to the root stack is needed.  The generic fiber of $Z$ therefore misses
the positive part of $D$ as well, and hence avoids all of
$D$.

We next check the dimension assertion needed for the family.
At a closed point $z$ of $F$, flatness makes $s$ a
nonzerodivisor in $\OO_{Z,z}$, and its quotient is the
Cohen--Macaulay ring $\OO_{F,z}$ of dimension $d$.
Thus $\OO_{Z,z}$ is Cohen--Macaulay of dimension $d+1$.
These local rings are also equidimensional: a Cohen--Macaulay local ring
essentially of finite type over the excellent discrete
valuation ring $R$ has this property.
Every component of $Z$ meets $F$ by properness, and no
component is contained in $F$ by flatness.  The dimension
formula now gives dimension $d$ for every component of the
generic fiber.  Its finite image in $X_{\C((s))}$ therefore
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 $X$.}
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
of $S$ covering it.  As the general chart and $t$ vary,
the images of points of $F$ contain a dense open subset
of $D_i$.  We will show that each point in this open subset
lies in the closed evaluation image of an algebraic family
of $d$-dimensional subvarieties disjoint from $D$.

Now consider all Hilbert schemes of $d$-dimensional
subvarieties of $X$.  The geometrically integral loci
parametrizing members disjoint from $D$ have a countable
stratification by integral parameter spaces with
irreducible universal families.  Denote the irreducible
closures of their evaluation images in $X$ by $W_\alpha$.
Fix a point $x$ in the dense open subset of $D_i$, and a
point of a boundary fiber $F$ mapping to it.  Since $Z$ has
no vertical component, this point lies in the closure of a
generic component.  That component has dimension $d$ by
the preceding argument.  Pass to a finite extension of the
discrete valuation field over which the reduced generic
components are geometrically integral, and let $R'$ be the
extended discrete valuation ring.  Choose a generic component
after this base change whose closure contains a point over
the chosen point of $F$.  Flatness still excludes vertical
components, so such a choice is possible.  Its reduced
finite image is geometrically integral.  Take the schematic
closure of that image in $X_{R'}$.
The structure sheaf of the closure has no uniformizer
torsion, so the closure is flat over the discrete valuation
ring and defines a morphism to the Hilbert scheme.
Its generic member lies in one of the strata above.
The corresponding $W_\alpha$ is closed and contains the
generic image, hence also its special-fiber support and
the point $x$.  Thus every point of the chosen open subset
of $D_i$ belongs to some $W_\alpha$.  The generic component
and the index $\alpha$ may depend on $x$.

An irreducible variety over the uncountable field $\C$
cannot have a dense open covered by countably many proper
closed subsets.  Some $W_\alpha$ therefore contains $D_i$.
By definition it also contains points outside $D$.
Since $D_i$ is a prime divisor in the integral variety
$X$, the only proper irreducible closed subset containing
it is $D_i$ itself.  Hence $W_\alpha=X$, and the family
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}

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.
Suppose all fibers have dimension at most
$\dim Y-\dim B$.  Let $G_Y$ be a vertical $\Q$-Cartier
divisor that is relatively nef.  Then
\[
 G_Y=f^*G_B
\]
for a $\Q$-divisor $G_B$ on $B$.
\end{lemma}

\begin{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
generic fiber makes it birational, and normality of $B$
then makes it an isomorphism.  The assertion follows in
both cases.  We may therefore assume that the base
dimension and the relative dimension are positive.

The fiber bound forces every vertical prime divisor to
map onto a prime divisor of $B$.  For each such prime
$P$, write its full pullback as $f^*P=\sum_jm_jE_j$.
Let $b_j$ be the coefficient of $G_Y$ along $E_j$, with
zero coefficients included, and assign coefficient
$\min_j(b_j/m_j)$ to $P$ in $G_B$.  Only finitely many
of these coefficients are nonzero.  The residual divisor
\[
 R'=G_Y-f^*G_B
\]
is effective, vertical, $\Q$-Cartier, and relatively nef;
over each base prime, at least one component is absent
from its support.  It remains to prove that $R'=0$.

Suppose $R'\ne0$, and choose a base prime under its
support.  Intersect $B$ with general hyperplanes to
obtain a complete-intersection curve meeting that prime
at a general point; if $\dim B=1$, use $B$ itself.
Normal Bertini makes its inverse image in $Y$ normal.
This inverse image is integral: generic geometric
integrality supplies the dominating component, and
the fiber bound rules out further vertical components.
Intersect upstairs with $\dim Y-\dim B-1$ general
very ample hyperplanes.  The result is a normal integral
surface over the base curve, with geometrically connected
generic fiber.  Over the chosen general base point,
these cuts retain curves both in the residual support
and in a component missing from it.  The curves remain
distinct because the original components were distinct
at the generic point of the base prime.

Resolve this surface.  The pullback of the residual
divisor stays effective, vertical, and relatively nef.
It has zero intersection with the full fiber and
nonnegative intersection with every fiber component.
All those component intersections must therefore be
zero.  The fiber is connected, by generic connectedness
and Stein factorization over the normal base curve.
Its intersection matrix is negative semidefinite,
with kernel generated by the full fiber.  The residual
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}]
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}
give a dominating family of compact $L$-trivial
subvarieties of dimension $d=n-v$.

Apply the nef-reduction theorem to a Cartier multiple of
$L$ \cite[Theorem~2.1]{BauerEtAl2002}.  The resulting almost
holomorphic rational map has connected fibers.  The
divisor $L$ is numerically trivial on its compact general
fibers and has positive degree on every noncontracted
curve through a very general point of $X$.  Write $b$
for the base dimension.  A compact $L$-trivial $d$-fold
through such a point must be contracted: otherwise,
general ample slices through the point would produce a
noncontracted curve of $L$-degree zero.  Consequently
\[
 b\le n-d=v.
\]

To apply vertical descent, resolve the graph, flatten
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 $f$.
Every fiber has dimension at most $n-b$.  Indeed,
flatness and generic integrality keep the flat main
transform integral after the base modification, and
finite normalization preserves the bound on fiber
dimension.  This bound is the needed hypothesis;
normalization need not preserve flatness.

We claim that $\pi^*D$ has no horizontal component.
Restrict to a very general fiber of $f$.  There,
$\pi^*L$ is numerically trivial and $\pi^*(L-cD)$
restricts to a pseudo-effective class.  To justify the
second assertion, choose a sequence of effective approximants with
ample errors tending to zero. Shrink to a flat open of the base with
geometrically integral fibers. This is possible because the generic
fiber is geometrically integral. On this common open,
the locus over which a given support contains the whole fiber is
proper closed, by upper semicontinuity of fiber dimension.
Avoid the countably many such bad closed loci in the base.
Every approximant then restricts effectively to a very general fiber,
and passage to the limit gives pseudo-effectivity of the restriction.
The resulting class is $-c\pi^*D$.
A negative nonzero effective divisor cannot be
pseudo-effective on a projective variety, as its
intersection with an ample power shows.  The restriction
of $\pi^*D$ must therefore be zero.  This also excludes
$b=0$, 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
\[
 \pi^*L\sim_{\Q}f^*G_B
\]
for a $\Q$-divisor $G_B$ on the smooth base.  Every
curve in $B$ is dominated by a curve in $Y$, where
the pullback has nonnegative degree; hence $G_B$ is
nef.  Numerical dimension is unchanged by these pullbacks;
the intersection formula gives
\[
 v=\nu(X,L)=\nu(Y,\pi^*L)=\nu(B,G_B)\le b.
\]
Together with $b\le v$, this yields $b=v=\nu(B,G_B)$.
Since $G_B$ is nef on the $b$-dimensional base, it follows
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}

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
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}
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
configurations: the reduced-boundary adjoint of a signed canonical
representative is big, and fixed finite covers admit no family of
birational maps whose graphs dominate their product. The argument takes
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}.

\subsection{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
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}
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
of \(\Alb(X)\) is generically finite. Wedges of invariant one-forms
on the abelian variety therefore give a nonzero section of \(K_B\):
at the generic point, choose \(\dim B\) independent pulled-back
one-forms. Thus \(\kappa(B,K_B)\geq0\).

For a very general smooth fiber \(F\), the restriction of the
pseudo-effective class \(K_W\) is pseudo-effective. For example, restrict
a positive current representing it to almost every fiber, or use
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
empty gives \(\kappa(W,K_W)\geq0\), contradicting
\eqref{ex:counterexample}.

This is the only use of Theorem~\ref{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.
On a suitable open parameter space the domains form a smooth
projective family with integral fibers, and evaluation is dominant.
When each fiber maps generically finitely onto its image, general
ample cuts of the parameter space make total evaluation generically
finite while preserving dominance. We resolve and compactify this
total evaluation before using its ramification divisor.
Invariance of plurigenera for smooth projective families
\cite[Theorem~1]{Paun2007} permits the same construction for the
Iitaka fibers of general members: choose a divisible pluricanonical
system on the general member, spread it by base change, and resolve
its relative rational map. When the member has nonnegative Kodaira
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}
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
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}
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
algebraic: spans and brackets stabilize at the generic point after
finitely many operations, and descend from the etale sheets. The
nonempty open parts of the integral parameter fibers are connected.
A chain step moves between two points in the image of one such
open fiber. We use only steps lying in \(U\), so each step remains
in one analytic leaf of \(\mathcal F\).

For \(x\in U\), endpoints of chains of at most \(k\) steps form a
constructible set, by the algebraic fiber-product construction and
Chevalley's theorem. Every endpoint lies in the analytic leaf
through \(x\). A locally closed smooth variety contained in an
analytic leaf has dimension at most \(\rk\mathcal F\). Indeed, in a
Frobenius chart the leaf meets at most countably many plaques: use
finite plaque chains in a countable foliation atlas. The transverse
coordinate map on any connected smooth piece then has countable
image and is constant.

There is therefore a maximal dimension among the closures of all
such endpoint loci; it is attained for some finite \(k\). The loci
are nested because we allow at most \(k\) steps, including the
identity step. Choose an irreducible component \(V\) of maximal
dimension and a dense open \(O\subset V\) of reachable points.
On each etale sheet, the parameter map defines the algebraic
relation consisting of pairs in the same fiber. Restrict its
first coordinate to the inverse image of \(O\), take the component
through the diagonal, and map the second coordinate to \(U\).
The closure of this image contains \(O\), hence \(V\), while its
points are reachable in at most \(k+1\) steps. Maximality forces
that closure to equal \(V\). At a general
diagonal point, smoothness of the parameter map identifies its
vertical tangent with the corresponding web tangent. Every web
tangent is consequently tangent to \(V\), and so is every bracket. Hence
\(\dim V\geq\rk\mathcal F\), while the reverse inequality was proved
above. A plaque is thus open in \(V\). The equations of \(V\),
pulled back to the connected immersed leaf, vanish on a nonempty
open and hence on the entire leaf. Its closure is exactly \(V\).

These invariant subvarieties of dimension \(\rk\mathcal F\) have countably many
Hilbert or Chow parameter spaces. Tangency on the regular locus is
an algebraic condition after stratification, so one such family
dominates. An invariant irreducible variety meeting \(U\) contains
the leaf through any of its points in \(U\): the tangent vector
fields preserve its reduced ideal, first on its smooth locus and
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
fiber away from a fixed removed divisor, it annihilates the
corresponding fiber tangents on a dense open. In submersion
coordinates, choose locally integrable plurisubharmonic potentials.
Fubini's theorem makes the pure fiber Hessian zero as a
distribution; positivity then makes the mixed coefficients in
fiber directions zero as well. A generically etale evaluation
transfers this assertion to the target. Annihilation passes to
brackets by the identities
\[
 \mathcal L_v T=d(\iota_vT)+\iota_v(dT),\qquad
 \iota_{[v,w]}T=\mathcal L_v(\iota_wT)-\iota_w(\mathcal L_vT).
\]
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 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}
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
total evaluation shows that \(K_V\) dominates the restriction of
the pulled-back \(K_X\) by an effective divisor. Thus \(K_V\) is
pseudo-effective. The induction hypothesis gives a good minimal
model of \(V\), so \(\kappa(V)\geq0\). Since \(V\) is not of general
type, its general Iitaka fibers have positive dimension and
Kodaira dimension zero. Spreading these fibers gives another
covering family with smooth domains \(G\), of dimension less than
\(n\), and \(\kappa(G)=0\). Each \(G\) has a good minimal model and
\(\kappa_\sigma(K_G)=0\).

To exploit this zero-Iitaka-dimension family, fix a positive current
\(T\) representing \(K_X\). After slicing and
resolving the total evaluation of the \(G\)'s, its restriction to
almost every parameter fiber, plus the effective restricted
ramification divisor, is a positive current in \(K_G\).
Every positive current in that class is the current of the fixed
canonical divisor of \(G\). To see this, let \(R\) be such a
current and take a common resolution
\(u:H\to G\), \(v:H\to G_{\min}\) with the good minimal model.
The current
\[
 u^*R+[K_H-u^*K_G]
\]
is positive and represents \(K_H\). Since \(K_{G_{\min}}\) is
torsion, this class is represented by an effective
\(v\)-exceptional divisor. Its intersection with the
\((\dim G-1)\)-st power of an ample pullback from \(G_{\min}\)
is zero. That form is strictly positive off the exceptional
locus, so the current is supported there. The support theorem
makes it divisorial, and independence of exceptional divisor
classes, by negativity, determines its coefficients. Pushing
forward by \(u\) determines \(R\) itself.
After shrinking the parameter space, these fixed canonical
divisors are the fixed divisor of a sufficiently divisible
relative pluricanonical system. Remove this divisor and the
excluded loci of the construction from the total family. Their
images under generically finite evaluation are proper algebraic
subsets of \(X\).

It follows that \(T\) annihilates the web distribution of the
\(G\)'s on a dense open. If that distribution has full rank, \(T\)
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
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
effectivity to \(\Q\)-linear effectivity, a contradiction.

Otherwise Lemma~\ref{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}.
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;
restricting to a general quotient fiber preserves dominance of
evaluation. A pluricanonical rational map of \(F\) is constant
along each such \(G\). Slice the parameters inside \(F\) to make
the total evaluation generically finite: ramification injects the
restricted pluricanonical sections into sections of a multiple of
\(K_G\), and these span a space of dimension at most one.
Fix a nonzero denominator section and work where it does not
vanish. Its restriction is nonzero on a general member of the
covering family, so all ratios are constant on that member. Their
differentials vanish on the web and its brackets, so the
pluricanonical map is constant on \(F\). Lower-dimensional
nonvanishing therefore gives \(\kappa(F)=0\), and its good minimal
model gives \(\kappa_\sigma(K_F)=0\).

The quotient has now reached the exact setting of the
numerical-zero-fiber reduction of
Gongyo--Lehmann \cite[Theorem~1.3 and Corollary~4.5]{GongyoLehmann2013}.
For a projective \(\Q\)-factorial klt rational pair with a
connected-fiber morphism and numerical dimension zero on the
general fiber, that theorem produces a klt pair on a smooth
birational base whose good-minimal-model existence is equivalent.
Here the total space is the smooth graph, the boundary is zero,
and the base has dimension less than \(n\). The required numerical
dimension is \(\kappa_\sigma\), which is zero by the good model of
\(F\); thus the numerical-dimension qualification, with the
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}

\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}
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}.
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
resolved moving subsystem, pushforward gives \(K_Y+aP\) big for
some \(a>0\). Convexity with the pseudo-effective \(K_Y\), and then
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}

The next two exclusions involve currents whose numerical intersection
gaps tend to zero. A limiting inequality alone is insufficient: we
need one family on which the gap is exactly zero. The ACC statement
below supplies that step for a fixed current and bounded discrepancies.

For a positive closed \((1,1)\)-current \(T\) on a smooth projective
variety \(V\), let \(\nu_E(T)\) denote the generic Lelong number of
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}
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
subsequence, all centers lie in one irreducible component of
this locus.

If, after passage to a subsequence, the centers lie in a fixed
codimension-at-least-two subvariety, principalize its ideal.
All lifted centers lie in the exceptional locus, where the
relative canonical divisor has positive integral order.
The discrepancy bound for the new smooth ambient space has
therefore decreased by at least one. Pull back \(T\) and apply
induction.

Otherwise the centers lie in a fixed prime divisor \(D\) of the
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
\(\operatorname{lct}(D)\ord_E(D)\leq A_V(E)\).
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}
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
\cite[Theorems~5.11, 6.27, and 14.2]{Demailly2012}. On a smooth projective variety, if \(\{T\}\)
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
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
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}
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
pulled-back current annihilates the fiber tangents on a dense
regular open of a resolved graph. Choose one with smallest base
dimension \(b\). We will descend the current to this base and
find a covering family of curves on which the descended current
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 \(0<b<n\).
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
base by birational models; they do not change the chosen rational
map or the function field of its base.
Flatten the main component over a modification of the base,
resolve that base, normalize the main transform, and then resolve
upstairs. Every vertical prime on the flat
transform maps to a divisor of the base: a prime over base
codimension \(c\) would have generic fiber dimension at least
\(n-b+c-1\), whereas all fibers have dimension at most \(n-b\).
A prime of \(W\) either maps onto a prime of the flat transform,
where this dimension count applies, or is exceptional over that
transform. In the latter case it is also exceptional over \(Y\),
because the transform maps birationally to \(Y\). Thus every
vertical divisor on \(W\) over base codimension at least two is
exceptional over \(Y\). This remains true after later resolutions
and base modifications: a prime dominating a divisor of \(Y\)
already has divisorial center on the old base, and its center on
a higher base model is the strict transform of that divisor.

For divisors and numerical classes put
\[
 T_B=p_*h^*.
\]
This strict pullback is well defined on numerical classes because
\(Y\) is \(\Q\)-factorial: numerical classes upstairs decompose
into pulled-back classes from \(Y\) and exceptional classes.
It preserves pseudoeffectivity. For a higher base model
\(r:B'\to B\), its analogue satisfies
\(T_{B'}r^*=T_B\), by computing on a common graph.
The operator \(T_{B'}\) kills divisors exceptional over \(B\):
otherwise a prime in such a pullback that dominates a divisor of \(Y\) would
have center of codimension at least two on \(B\), contrary
to the property just established.

\smallskip
\noindent\emph{Descent 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
over \(B^0\), and the remaining open in every fiber is
nonempty and connected. Away from that fixed removed
algebraic set upstairs, \(T\) descends to a positive current
\(S^0\).
Indeed in submersion coordinates the horizontal coefficient
distributions are independent of fiber variables by closedness,
and the open parts of general fibers are connected. They
therefore glue to \(S^0\) on \(B^0\). The difference
\(T-h^*S^0\) is closed of order zero and supported on the
removed algebraic set. The support theorem expresses it as a
signed divisor sum; a closed order-zero \((1,1)\)-current
supported in codimension at least two is zero. Its horizontal coefficients are
nonnegative, since a pullback has zero generic Lelong number
along a horizontal divisor.

Subtract those finitely many global Siu components from
\(T\), obtaining a positive current \(T'\) that equals
\(h^*S^0\) on all of \(h^{-1}(B^0)\). Choose a K\"ahler form
\(\omega\) representing an ample algebraic class and put
\(d=n-b\). Fiber integration gives
\(h_*(\omega^d)=c>0\), constant on \(B^0\), by closedness.
The projection formula therefore gives
\[
 c^{-1}h_*(T'\wedge\omega^d)|_{B^0}=S^0.
\]
The left side defines a global positive closed extension.
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 \(S\).
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 \(B\setminus B^0\),
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
a radius equal to a fixed power of the source radius give the
Lelong comparison, so \(h^*S\) also has zero generic number
there. Positivity of \(T\) then makes the coefficient of
\(T-h^*S\) nonnegative. Thus all coefficients at primes
dominating base primes are nonnegative.
The only possible negative coefficients lie over base
codimension at least two and are exceptional over \(Y\).
We have therefore obtained a positive closed current \(S\) in a
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}
The horizontal components removed during descent contribute
effective divisors to the same difference.

\smallskip
\noindent\emph{A 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 \(\eta'\)
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,
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
\(\eta\cdot D=0\), or equivalently \(\alpha(T_BD)=0\).

Before applying multiplier approximation, we need to control the
divisorial part of this descended current. Only finitely many prime
divisors on \(B\) have positive
divisorial Lelong coefficient for \(S\). Otherwise their
nonzero strict pullbacks are effective divisors on \(Y\)
killed by \(\alpha\), with no common prime components.
Indeed, a nonexceptional prime of \(Y\) cannot map into the
intersection of two base primes.
Only finitely many base primes have zero strict pullback,
since their total pullbacks are supported in the finite
exceptional locus of \(p\). In the finite-dimensional rational
space of divisor classes, infinitely many remaining strict
pullbacks have a rational relation. This relation must have
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
\(\Q\)-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}.

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
theorem of Kov\'acs--Patakfalvi
\cite[Theorem~9.9]{KovacsPatakfalvi2017} applies to a log
canonical fiber space with smooth base, SNC total pair, and
log-general-type geometric generic fiber; for a rational base
divisor \(M\) with \(\kappa(M)\geq0\), it gives relative
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}.
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
\(\operatorname{div}_W(\varphi)+m(K_W-h^*K_B)\)
gives
\(\operatorname{div}_Y(\varphi)+mP\geq0\).
The resulting injection of sections preserves their ratios,
so \(\kappa(Y,P)\geq1\). Corollary~\ref{ex:supporting} now gives
\(0<\alpha(P)=-K_B\cdot\eta\), proving
\eqref{ex:negative-base}.

\smallskip
\noindent\emph{Rational 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
\[
 P_k=\frac{r_k^*L_k-F_k}{k}
\]
is nef. Put \(\eta_k=T_{B_k}^*\alpha\). The class \(\eta_k\)
is movable and \(F_k\) is effective, so
\(0\leq P_k\cdot\eta_k\leq (L_k\cdot\eta)/k\).
Since \(\eta\cdot\{S\}=0\) and \(L_k-k\{S\}\) stays in a
bounded set, this gives the first estimate below. The second
uses the fact that \(\eta_k\) kills exceptional divisors:
\[
 0\leq P_k\cdot\eta_k=O(k^{-1}),\qquad
 K_{B_k}\cdot\eta_k=K_B\cdot\eta<0.
\]
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
\[
 Q_k=P_k+k^{-1}r_k^*H+\epsilon_k H_k
\]
satisfies \(Q_k\cdot\eta_k=O(k^{-1})\). This is possible
because \(\eta_k\cdot r_k^*H=\eta\cdot H\) is fixed.
Approximate \(\eta_k\) by a positive sum
\(\gamma_k=\sum_{i=1}^{N_k} a_i[C_i]\), with \(a_i>0\),
of covering-curve classes,
closely enough that
\[
 Q_k\cdot\gamma_k\leq C/k,\qquad
 -K_{B_k}\cdot\gamma_k\geq\delta/2
\]
for a constant \(C\) independent of \(k\). Keep only terms
with \(K_{B_k}\cdot C_i<0\). Their total anticanonical
degree is at least \(\delta/2\), while their total
\(Q_k\)-degree is at most \(C/k\). Taking the weighted
average of the ratios therefore gives one covering curve
\(\Gamma_k\) with
\[
 \frac{Q_k\cdot\Gamma_k}{-K_{B_k}\cdot\Gamma_k}
 \leq\frac{2C}{\delta k}.
\]
The quantitative bend-and-break estimate
\cite[Theorem~5]{MiyaokaMori1986} gives a rational curve
through a general point of \(\Gamma_k\) with \(Q_k\)-degree
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
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
it contributes \(a_E(E\cdot R_k)\) to this bounded sum.
Both factors are positive integers. Thus the number of
exceptional primes met by \(R_k\), their log discrepancies
\(A_B(E)=a_E+1\), and their contact degrees are uniformly
bounded. Contacts with the finitely many strict transforms
of divisorial Lelong components of \(S\) are bounded by their
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
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}
 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 second has the same bound because \(L_k-k\{S\}\) stays
bounded and the \(H\)-degrees of \(\bar R_k\) are bounded.

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
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
covering families. Otherwise a sequence
tending to zero has a strictly decreasing positive
subsequence, giving a strictly increasing sequence of the
complementary sums.

For a family with zero gap, restrict the residual current to
almost every parameter fiber as above. Its integral on that
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
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
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}

We apply the same exact-zero mechanism to curves on the small
modifications of a fixed late model. The normalization by volume is
chosen so that its scale tends to zero, while the current and the
discrepancy lattice stay fixed.

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
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}
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
\[
 p^*K_Y=p_t^*K_t+J_t,\qquad J_t\geq0,
\]
where \(J_t\) is exceptional over \(Y_t\).
Then \(\nu_E(T)\geq\operatorname{coeff}_E J_t\) for every
prime on that resolution.
\end{lemma}
\begin{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
\(\OO(kp^*K_Y+H)\otimes\mathcal J(kT)\).
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
divisor system. At every such prime \(E\), it has multiplicity
\(\ord_E\mathcal J(kT)\); global generation after removal
of the ideal ensures no additional generic vanishing.
Here the notation \(H\) also denotes its pullback to \(V\).

Put \(D'_k=p_{t*}D_k\) and \(H_t=p_{t*}H\). The first is
effective, and both are \(\Q\)-Cartier because \(Y_t\) is
\(\Q\)-factorial. Pushing the rational linear relation gives
\[
 D'_k\sim_\Q kK_t+H_t.
\]
Thus the exceptional divisor
\(D_k-p_t^*D'_k\) is rationally equivalent to
\[
 kJ_t+H-p_t^*H_t.
\]
The two exceptional divisors are equal: their difference is
numerically trivial and exceptional, so the negativity lemma
applied with both signs makes it zero. Since
\(p_t^*D'_k\) is effective,
\[
 \operatorname{coeff}_E D_k
 \geq k\operatorname{coeff}_E J_t
       +\operatorname{coeff}_E(H-p_t^*H_t).
 \]
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.
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
families of curves on \(Y_{t_j}\), with smooth projective
domains \(C_j\), for which both
\[
 g(C_j),\qquad
 \frac{(K_{t_j}+t_jA_{t_j})\cdot C_j}{\mu_{t_j}}
\]
are bounded by a fixed constant. Degrees mean degrees on
the domains, or equivalently intersection with their
pushforward cycles.
\end{proposition}
\begin{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.
Neither \(W\) nor \(T\) will depend on \(t\). Let \(g_0\)
be a common upper bound for the genera, and choose an integer
\(m>0\) such that \(mK_Y\) is Cartier.

We first compare the curves on these fixed models without
changing their domains. Slice the parameter space so that the
smooth total family has dimension \(n\), the parameter space
has dimension \(n-1\), and evaluation is generically finite.
It has rational maps to \(Y\), to \(W\), and to the common
models comparing \(Y\) and \(Y_t\). Each target is proper,
so each map extends at codimension-one points by the valuative
criterion. Its indeterminacy locus has dimension at most
\(n-2\), and its image cannot dominate the parameter space.
Remove these finitely many images from that space, then resolve
and compactify away from the remaining complete fibers.
Denote the resulting maps by
\(\rho_t:U_t\to W\) and \(q_t=p\circ\rho_t:U_t\to Y\).
The smooth general parameter fiber is still the original
\(C_t\), with the same genus. The current we use on this
varying space is always \(\rho_t^*T\).

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)
       =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,
so \(a(E;Y,0)>1\), and the index \(m\) puts these log
discrepancies in \(m^{-1}\Z\). In particular
\(a(E;Y,0)\geq1+1/m\). The
remaining ramification coefficients are nonnegative.
On the general parameter fiber,
\[
 K_{U_t}\cdot C_t=2g(C_t)-2,\qquad
 q_t^*K_Y\cdot C_t\geq0.
 \]
The latter inequality follows from pseudoeffectivity and
the covering property. Thus
\[
 0\leq (K_{U_t}-q_t^*K_Y)\cdot C_t\leq 2g_0-2.
\]
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}
therefore bounds the number of contacted exceptional primes,
their log discrepancies, their ramification indices, and
their contact degrees. In particular the integers
\(r_Z(Z\cdot C_t)\) are uniformly bounded. Also
\[
 q_t^*K_Y\cdot C_t\in m^{-1}\Z\cap[0,2g_0-2],
\]
so these degrees take only finitely many values.

The ramification estimate has bounded the possible contacts. We next
compare their Lelong contributions with the canonical differences.
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
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
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
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.

On the fixed smooth resolution \(p:W\to Y\), one has
\[
 a(E;W,0)=a(E;Y,0)
       -\ord_E(K_W-p^*K_Y)\leq a(E;Y,0),
\]
because the relative canonical divisor is effective.
Thus Lemma~\ref{ex:valuation-acc} applies to the one fixed
current \(T\) on \(W\), with a fixed discrepancy bound.
The sums in \eqref{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
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
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}

\subsection{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
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}.
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
\((V,\Delta+A)\) is dlt and \(K_V+\Delta+A\) is nef.
The LMMP for \(K_V+\Delta\) with scaling of \(A\) has special
termination: if the program is infinite, its flipping loci
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}
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
\cite[Theorem~1.9(ii), equivalently Theorem~4.1(ii)]
{Birkar2012Special}. Discard the finitely many divisorial
contractions and write the flips as \(V_i\dashrightarrow V_{i+1}/Z_i\),
with scaling numbers \(\lambda_i>0\) tending to zero.
Fix a component \(S_1\) of \(\lfloor\Delta_1\rfloor\), write
\(S_i\) for its normal strict transform, and let \(T_i\) be the
normalization of its image in \(Z_i\). Since the ambient contraction
is small, \(S_i\to T_i\) is projective birational. The standard
discrepancy argument makes \(S_i\dashrightarrow S_{i+1}\)
an isomorphism in codimension one after finitely many steps
\cite[proof of Lemma~3.6]{Birkar2010}.

To identify the lower-dimensional input, consider the relative
program induced on this one component of the floor.
Take a small projective \(\Q\)-factorialization
\(p_i:S'_i\to S_i\), and put
\[
 D_i=K_{S'_i}+B_i=p_i^*((K_{V_i}+\Delta_i)|_{S_i}),
 \qquad C_i=p_i^*(A_i|_{S_i}).
\]
Adjunction gives a dlt pair \((S'_i,B_i)\), with \(B_i,C_i\geq0\),
and \((S'_i,B_i+\lambda_i C_i)\) is lc. The scaling divisor
has no floor component, so its restriction is effective.
The number \(\lambda_i\) is rational, being the ratio of
intersections of rational divisors on the contracted rational curve.
The globally nef divisor
\(L_i=K_{V_i}+\Delta_i+\lambda_i A_i\) descends rationally
linearly across \(V_i\to Z_i\). Indeed, for sufficiently small
rational \(\delta>0\), the pair
\((V_i,(1-\delta)\Delta_i)\) is klt and its negative adjoint
is ample over \(Z_i\). Relative basepoint freeness applies to a
Cartier multiple of \(L_i\); its numerical triviality and the
connected fibers give descent. Restricting and pulling back yields
\[
 D_i+\lambda_i C_i\sim_{\Q}0/T_i.
\]
The descended divisor on $T_i$ is nef: its pullback to $S'_i$
is the pullback of the restriction of the globally nef $L_i$.

By \cite[Theorem~1.1(3)]{Birkar2012Special}, a \(D_i\)-LMMP
over \(T_i\) with scaling of a fresh ample divisor terminates.
The theorem applies to the effective rational lc boundary
\(B_i+\lambda_i C_i\), with \(\lambda_i C_i\) rational Cartier
and the driving pair \(\Q\)-factorial dlt. Since the base map
is birational, the endpoint is a log minimal model, not a Mori
fiber space. Comparison with the relatively ample model
\(S_{i+1}\) identifies this endpoint with a small
\(\Q\)-factorialization \(S'_{i+1}\), as in
\cite[Remarks~2.9--2.10]{Birkar2012Special}.

The next point is that these finite relative programs concatenate
into an absolute program with scaling of the transforms of \(C_1\).
Throughout the \(i\)-th piece, \(D_i+\lambda_i C_i\) remains
the pullback of a nef divisor on \(T_i\), hence is globally nef.
Here $D_i$ and $C_i$ also denote their transforms along this
relative program. A contracted negative ray $R$ satisfies
\[
 (D_i+\lambda_i C_i)\cdot R=0,
 \qquad D_i\cdot R<0.
\]
It follows that $C_i\cdot R>0$. Every smaller coefficient of
$C_i$ leaves this ray negative, while $D_i+\lambda_i C_i$ is
globally nef. Thus the absolute scaling threshold at that step
is exactly $\lambda_i$.
The relative cone is a face of the absolute cone, cut out by
the pullback of an ample divisor on the projective base \(T_i\);
its extremal rays are therefore absolute extremal rays.
Thus the pieces concatenate as asserted in the cited remarks.
Rescaling the initial scaling divisor by \(\lambda_1\), if
necessary, makes its sum with the initial boundary lc and nef.

If the concatenation were infinite, its positive scaling numbers
would tend to zero. Fix a late scaling value $\lambda_j$.
On every earlier ray $R$ contracted at threshold $\lambda_i$,
the calculation above gives
\[
 (D_i+\lambda_j C_i)\cdot R
   =(\lambda_j-\lambda_i)C_i\cdot R\le0.
\]
Thus all preceding steps are nonpositive for the adjoint with
this fixed coefficient $\lambda_j$. On a common resolution of
the initial and the $j$-th models, the pullback of
$D_1+\lambda_j C_1$ equals the pullback of the late nef divisor
$D_j+\lambda_j C_j$ plus an effective comparison divisor.
Push this equality down to the initial model. The pushforward
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
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
of \cite[Lemma~3.6]{Birkar2010} now gives special termination.
The only good-model input in this argument concerns the absolute
pair on the lower-dimensional floor component. In particular it
does not require an effective representative in dimension \(n\).

For the final assertion, a curve disjoint from the round-down
has degree zero against any signed divisor supported there.
It therefore cannot generate an adjoint-negative flipping ray.
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
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}
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,
and
\((1+c)P=(K_W+cP)+D\) would then be big as well.
Hence \(\kappa(W,P)\leq0\).

Run the dlt LMMP for \(P\) with ample scaling.
The divisor \(P\) has a signed rational representative
supported on the floor \(D\), and its transforms retain
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
sequence would eventually have all flips disjoint from
the floor, a contradiction. Divisorial contractions
are finite in number. Thus the program terminates at
a \(\Q\)-factorial dlt log minimal model
\((Z,D_Z)\).

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
\(L=K_Z+D_Z\) and \(c=1\), so that the required
pseudo-effective class \(L-cD_Z\) is \(K_Z\).
It gives \(\kappa(Z,L)=\nu(Z,L)\geq0\).
The Iitaka dimension is at most zero, as proved above,
so both dimensions are zero. Intersecting
\(K_Z+D_Z\equiv0\) with an ample
\((n-1)\)-fold product, and using pseudoeffectivity of
\(K_Z\), forces \(D_Z=0\). The signed relation then
gives \(K_Z\sim_\Q0\).

The signed representative of \(K_W\) is supported on
\(D\): pull back the given representative of \(K_t\)
and add the relative canonical divisor. Since \(D_Z=0\),
every component of this support is contracted over \(Z\).
On a common resolution \(a:V\to W\), \(b:V\to Z\),
we therefore have \(a^*K_W\sim_\Q E\) for a signed
\(b\)-exceptional divisor \(E\). The class \(a^*K_W\)
is pseudo-effective. A positive current in this class has
zero intersection with the \((n-1)\)-st power of an ample
pullback from \(Z\), so is supported on the exceptional
locus. The support theorem makes it effective divisorial.
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}

\subsection{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
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}
Let \(Y\) be a projective \(\Q\)-factorial terminal birational model
of the counterexample \(X\) in Equation~\eqref{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
\[
 \Gamma\subset A_0\times T_1\times T_2
\]
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}
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
\(T_2\) birationally with \(T_1\). The maps
\(g_a=\phi_{a_0}^{-1}\circ\phi_a\) are then birational selfmaps of
\(T_1\). The evaluation \((a,u)\mapsto(u,g_a(u))\) is dominant onto
\(T_1\times T_1\); in particular, the images of a general point under
these selfmaps are dense.

We use the birational automorphism group to identify this fixed cover.
The cover is non-uniruled, being generically finite over the
non-uniruled \(Y\).  Hanamura's non-uniruled birational-group
theorem gives a smooth projective birational model for which the
reduced birational group is a group scheme, locally of finite type,
and its identity component is an abelian variety of regular automorphisms
\cite[Theorems~2.1--2.2]{Hanamura1988};
see also \cite[Proposition~3.7 and Theorems~3.8--3.9]{Blanc2017}.
After conjugating the maps to that model, shrink the irreducible
parameter variety anew so that their graph closures are flat
with integral fibers and both graph projections remain birational on
every fiber. This is a family in the flat graph functor of
\cite[Definition~3.4]{Blanc2017}, which is represented by the
birational scheme \cite[Proposition~3.7]{Blanc2017}. It therefore
defines a morphism to that scheme;
because the parameter variety is reduced, this morphism factors
through its reduction. Its connected image lies in a single component,
hence in a translate of the identity component. Translate the family
by one member so that it lies in that identity component; this
preserves dominance of evaluation. Thus the identity component, an
abelian variety, has a dense orbit. That orbit is the quotient by the
stabilizer of a general point. A quotient of an abelian variety is
again an abelian variety, so the fixed cover is birational to an
abelian variety.

It remains to see why this abelian cover contradicts the original
counterexample.  Resolve the generically finite rational map from
that abelian variety to \(X\):
on a smooth model \(U\), ramification gives
\[
 K_U=f^*K_X+R,\qquad R\ge0.
\]
The canonical class of \(U\) also has the effective representative
\(E_A\) exceptional over the abelian variety: choosing a nowhere-zero
top form on that variety gives \(K_U\sim E_A\). This second choice
of representative need not be the canonical divisor compatible with
the displayed ramification formula. Let \(T\) be a positive current in \(K_X\).
The current \(f^*T+[R]\) is positive and represents \(K_U\).
Let \(\alpha\) be an ample class on the abelian variety. The
intersection of \(K_U\) with the \((n-1)\)-st power of its pullback
is zero, since \(K_U\sim E_A\) and \(E_A\) is exceptional.
The positive current \(f^*T+[R]\) therefore has zero mass against
that power. On the open where the birational map to the abelian
variety is an isomorphism, the pulled-back ample form is strictly
positive, so the current vanishes there. Its support is consequently
exceptional over the abelian variety. The positive summand \(f^*T\)
is supported there as well, and the support theorem makes it divisorial.
Push forward this summand alone: \(f_*(f^*T)\) is a positive
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}

\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
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
failure of the scalar or two-slot jet estimates.

All varieties in the moving-center arguments are over $\C$.
A rational Cartier divisor is a rational divisor some positive
integral multiple of which is Cartier. A requirement that $kP$
be Cartier means that $kP$ is an actual integral Cartier divisor.
For a nef
class on an integral subvariety, intersections are computed after
pullback to its normalization and a resolution. For a linear
subseries of a line bundle, its base ideal is the image of its
evaluation map after tensoring by the inverse line bundle.
``Isolated at the generic point of $V$'' means that this ideal
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}

The first result converts a generic log canonical center into a
canonical-intersection inequality. Its testing class is allowed
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}
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 $(Z,\Theta)$
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}
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}
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}
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 a neighborhood of the generic point of $V$.

There are lc strata of $(Q,B_Q)$ mapping onto $V$. Indeed over that
lc neighborhood the construction is crepant, and the center of a
log-discrepancy-zero divisor on the dlt model is an lc stratum.
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
$Q$ 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}
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 $S$.

Factor the map through the normalization $V^\nu$ and its Stein
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
$D'=\eta^*D_V$. 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
contribute an effective term. The final comparison is between the
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
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
discrepancy divisor has coefficient zero along every strict
boundary component and has nonnegative coefficients only on
exceptional divisors. Over this dense klt open, its pushforward to the normal $S$ is
$\cO_S$: a rational function allowed poles only over exceptional
centers extends across codimension at least two. Connectedness
in the Stein factorization therefore gives generic rank one for
the required discrepancy round-up pushforward. This verifies the
rank condition, rather than deducing it from connected fibers
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
not require the pair to be globally sub-lc. Thus the possible
non-lc fibers caused by $E_Q|_S$ are permitted.

The discriminant $B_{V'}$ on this initial base is effective. To
check this at a prime divisor $F\subset V'$, work over its generic
point, where the base is regular. A component of its inverse
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
\[
 t_F\leq\frac{1-b_F}{m_F}\leq1.
\]
Thus the discriminant coefficient $1-t_F$ is nonnegative. If the
fiber is already non-lc, $t_F$ can be negative; this only increases
that coefficient.

Let $P'=\eta^*\nu^*P$. On a higher projective base
$\mu:\widehat V'\to V'$ where the moduli divisor is nef, its
trace satisfies $M_{V'}=\mu_*M_{\widehat V'}$. The projection
formula gives
\[
 M_{V'}\cdot(P')^{f-1}
 =M_{\widehat V'}\cdot(\mu^*P')^{f-1}\geq0.
\]
Only this intersection is asserted to be nonnegative; the trace
$M_{V'}$ need not itself be nef. Likewise
$B_{V'}\cdot(P')^{f-1}\geq0$ by effectivity and nefness.
All intersections with Weil divisors here mean their cycle
intersections with rational Cartier powers, so they are defined
even when the separate canonical or discriminant divisor is not
$\Q$-Cartier.

Choose compatible canonical Weil divisors. Since $\eta$ is finite
and separable, the codimension-one ramification formula, followed
by pushforward, is
\[
 \eta_*K_{V'}=(\deg\eta)K_{V^\nu}+\eta_*R_\eta,
 \qquad R_\eta\geq0.
\]
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
$(P')^{f-1}$ and discard the nonnegative discriminant and moduli
intersections. Pushforward through $\eta$ and the nonnegative
ramification intersection then give
\[
 (\deg\eta)D_V\cdot(\nu^*P)^{f-1}
 \geq(\deg\eta)K_{V^\nu}\cdot(\nu^*P)^{f-1}.
\]
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 $f=1$
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}

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}
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
$0\ne\mathcal H\subset H^0(Z,\cO_Z(kP))$ be a linear subspace
with proper base locus and base ideal $\mathfrak b$.
Suppose that $V$ is an integral base-locus component of dimension
$f<d$, isolated at its generic point $\eta_V$, and that
$\eta_V\in Z_{\rm sm}$. Write $a=d-f$ and
$\mathfrak m=\mathfrak m_{\cO_{Z,\eta_V}}$. If
\[
 \mathfrak b\cO_{Z,\eta_V}\subset\mathfrak m^h
 \qquad(h\in\Z_{>0}),
\]
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
$V$. 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}
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
form a system of parameters in $R$, since an $\mathfrak m$-primary
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
\[
 \operatorname{length}(R/\mathfrak q)
 =e(\mathfrak q,R)\geq e(\mathfrak m^h,R)=h^a.
\]
This is the generic intersection multiplicity along $V$.

The global intersection can be performed without assuming that
the series has no other base components. Before each cut, discard
the components of the current effective cycle that lie in the
base locus. Before the final cut their dimensions are greater
than $f$, so none can contain $V$: otherwise $V$ would not be a
base-locus component. Discarding them therefore leaves the
calculation at $\eta_V$ unchanged. A general next section meets
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}.

Let $\lambda=\operatorname{lct}_{\eta_V}(\mathfrak b)$ 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
\[
 0<\lambda\leq a/h.
\]
Because $\mathfrak bR$ is primary, a divisor computing this
threshold has center $V$. We realize the ideal threshold by a
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
\[
 \Theta=\frac{\lambda}{N}(D_1+\cdots+D_N)
 \sim_{\Q}cP,\qquad c=k\lambda.
\]
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
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
$c=k\lambda\leq ak/h$.
\end{proof}

\subsection{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
to two successive base loci acquires high multiplicity, because
parameter derivatives of the higher kernel still belong to the
lower kernel. We first state the parameter-family conclusion. Local restriction
to a fiber is treated separately below. The differentiation argument
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}
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
\[
 \tau_0<\tau_1<\cdots<\tau_d,
 \qquad\tau_{i+1}-\tau_i=\delta>0.
\]
For a smooth point $x$, let $\mathcal H_i(x)$ be the entire
subspace of sections of $kP$ vanishing at $x$ to order at least
$\lceil k\tau_i\rceil$. Suppose that for very general $x$ all these
subspaces are nonzero and the base locus of $\mathcal H_0(x)$
has a positive-dimensional component through $x$.
If $k\delta\geq4$, then, after an algebraic parameter extension
and restriction to nonempty opens, there are an irreducible
parameter space $T$, a fixed adjacent pair of levels, and a
family of integral subvarieties $V_t$ common to the two base
loci such that the incidence
\[
 \Gamma=\{(t,z):z\in V_t\}\subset T\times Z
\]
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}
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
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
as $i$ increases. Starting with the stipulated component through
the mark, follow a component containing it at each later level.
Every such component has dimension between $1$ and $d-1$: the
kernel is nonzero, so its base locus is proper. Among the $d+1$
nested components two successive ones have the same dimension
and hence are equal.

This tracking can be made algebraic for the fixed $k$. Follow the
finitely many components of the geometric generic base loci,
their inclusions, and their incidences with the mark. They are
defined after a finite extension of the generic parameter field.
Spread over a corresponding irreducible parameter variety and
shrink for the required flatness, geometric integrality, and
constant-rank conditions. There are only finitely many choices
of adjacent step; one of them gives a family dominant over the
marked-point open. It gives the asserted $T$ and $\Gamma$.
The evaluation $\Gamma\to Z$ is dominant because the moving mark
belongs to each $V_t$ and its map to $Z$ is dominant.

We now prove ordinary high-order vanishing. Work on a smooth
parameter open, and denote the higher-kernel bundle there by
$\mathcal K$. Evaluation on $T\times Z$ gives a map from the
pullback of $\mathcal K$ to the pullback of $\cO_Z(kP)$.
Let $\mathfrak b_T$ be its base ideal. After trivializing the
target line bundle, a local frame of $\mathcal K$ generates
$\mathfrak b_T$. We show that every frame section lies in
$\mathcal I_\Gamma^h$.
Each frame element has the form
\[
 s(t,z)=\sum_\alpha c_\alpha(t)s_\alpha(z),
\]
where the $s_\alpha$ are a fixed basis of the global space
$H^0(Z,kP)$. Parameter differentiation with $z$ fixed only
differentiates the coefficients. Every derivative, at each fixed
parameter, is consequently a global section of the same line
bundle $kP$.

Write $x(t)$ for the moving mark. In local coordinates $z-x(t)$,
the section starts in order $\lceil k\tau_{i+1}\rceil$. Each parameter
derivative lowers this order by at most one. A derivative of
order $r$ therefore belongs to the lower kernel if
\[
 r\leq\lceil k\tau_{i+1}\rceil-\lceil k\tau_i\rceil.
\]
The right side is at least $k\delta-1$. Since $k\delta\geq4$,
every derivative of order less than $h=\lceil k\delta/2\rceil$ is
allowed. It thus vanishes along $\Gamma$, which lies in the
lower base locus. A change of higher-kernel frame contributes
only parameter coefficients and derivatives of orders no larger
than $r$, so the conclusion holds for any local frame.

Here a first-order normal-space assertion must be upgraded to
all orders. At a general smooth point of $\Gamma$, its projection
to $Z$ is submersive. Thus
\[
 T_{(t,z)}(T\times Z)
 =T_{(t,z)}\Gamma+(T_tT\times\{0\}).
\]
In particular, parameter directions span the normal space to
$\Gamma$. Choose parameter coordinates
$t=(t',t'')$, where $t'$ has length
$a=\codim_{T\times Z}\Gamma$, such that the normal equations have
an invertible Jacobian minor in the $t'$ directions. The analytic
implicit-function theorem writes
\[
 \Gamma:\quad t'=F(t'',z).
\]
Set $u=t'-F(t'',z)$. With $t''$ and $z$ held fixed, differentiation
in $t'$ is exactly differentiation in $u$, to every order. Use a
frame of $kP$ depending only on $z$. The vanishing on $\Gamma$
of all parameter derivatives of order below $h$ says that every
normal Taylor coefficient of degree below $h$ vanishes. Therefore
$s\in\mathcal I_\Gamma^h$ in the ordinary analytic local ring.
Equivalently, in arbitrary relative normal coordinates a product
of $r$ normal vector fields expands into parameter derivatives
of orders at most $r$, so no unaccounted higher-coordinate terms
arise.

The sections and ideals are algebraic. Faithful flatness of the
analytic local ring over the algebraic local ring gives the same
ideal membership algebraically. Applying this to a finite frame
and clearing the finitely many denominators gives containment
on a dense algebraic open of the incidence. Generic smoothness
of $\Gamma\to T$ identifies its scheme fiber there with the
reduced general component $V_t$. Restriction therefore gives
$\mathcal I_\Gamma^h\cO_{\{t\}\times Z}=\mathcal I_{V_t}^h$ locally.
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.
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}

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 $g:Z\to B$
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
actual image at $z$. Let $F$ be the reduced component through
$z$ of its scheme fiber. Locally on $Z_b$ at $z$, restriction of
the higher kernel's base ideal has support exactly $F$ and is
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
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}
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
space and for the component over its actual image. These are proper
closed subsets after shrinking. A prescribed very general condition
on the evaluation can also be imposed because the incidence dominates
$Z$. The component is then chosen through this point, rather than
by selecting a possibly special fiber in advance.

Locally the original base support is exactly $V_t$. Smoothness
over the actual image makes its scheme fiber reduced at the chosen
point. Thus in the ambient fiber
\[
 \mathcal I_{V_t}\mathcal O_{Z_b}=\mathcal I_F,\qquad
 \mathcal I_{V_t}^{h}\mathcal O_{Z_b}=\mathcal I_F^{h}
\]
near that point. Restricting the actual sections extends their
base ideal, which proves both asserted local properties.
No other base component contains the chosen point, so none can
contain the component $F$ through it. If this component is proper
in the chosen integral ambient component, the base support is
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}

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},
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}
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
$g:\mathcal V\to Y$ be a morphism whose general fiber maps
$g_t:V_t\to Y$ are dominant and generically finite of degree
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 $\mathcal V$.
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 $t$.
Assume
\[
 \overline{g(\mathcal R_j)}\ne Y\qquad(j\in I).
\]
Then, after shrinking again, one smooth nonempty open
$Y^\circ\subset Y$ makes every general finite normal Stein cover
of $g_t$ finite \'etale over $Y^\circ$. There are only finitely
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}
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.
Shrink away vertical components and make a finite parameter extension
to define the geometric generic components individually. Their
closures give the divisors $\mathcal R_j$ in the statement.
Shrink further to keep their fiber-image dimensions constant.
These components account for every branch prime of the general
finite Stein cover: a ramification prime there has a strict
transform on $V_t$, because a proper birational morphism to a
normal variety is an isomorphism at each generic divisorial
point. A divisor exceptional over the Stein cover has image of
codimension at least two on that cover and hence on $Y$, so it
cannot hide a branch prime. Components confined to special
parameters disappear upon shrinking.

Put
\[
 B=Y_{\rm sing}\cup\bigcup_{j\in I}\overline{g(\mathcal R_j)}.
\]
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 $Y^\circ=Y\setminus B$.
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
generated. Indeed, triangulate the compact semialgebraic space
$Y$ compatibly with its closed subset $B$
\cite[Theorem~1.10 in the author's notes]{Coste2007}.
After barycentric subdivision the subcomplex for $B$ is full.
On its complement, normalize the sum of the barycentric
coordinates at vertices outside that subcomplex; decreasing
the other coordinates to zero gives a deformation retraction
onto a finite subcomplex. Its fundamental group is finitely
generated. A finitely generated group has only
finitely many permutation representations in each symmetric
group of degree at most $e$. Riemann existence
\cite[Expos\'e~XII, Theorem~5.1]{SGA1} therefore gives
finitely many finite \'etale covers of these degrees. A finite
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}

This section derives two contrasting consequences of the assumed
counterexample: an upper bound for the vanishing of scalar sections,
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}.
Thus \(X\) is smooth projective,
\(K_X\) is pseudo-effective, and \(\kappa(X,K_X)=-\infty\), whereas
good minimal models exist in smaller dimensions. We use the geometric
exclusions of Section~\ref{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 \(A=A_Y\).
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
\[
 \mu_t=\vol(X,K_X+tA)^{1/n}.
\]
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}

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 \(t\downarrow0\),
choose positive integers \(r=r(t)\) with
\begin{equation}\label{jet:normalization}
 r\mu_t\longrightarrow\epsilon,\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
\[
 N_b=(r+bm)(K_s+sA_s)
 \quad\hbox{on }Y_s,\qquad s=\frac{rt}{r+bm}.
\]
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}
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 \(\epsilon^n\).
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
 \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 \(b\).

This normalization gives a convenient form of the curve exclusion.
A family of curves with uniformly bounded geometric genus and
\(N_b\)-degree cannot cover the corresponding \(Y_s\) for arbitrarily
small \(t\).  In fact \(s/t\to1\) uniformly in \(b\), and
\[
 (r+bm)\mu_s=\vol(L_b)^{1/n}
\]
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}.

We will apply the local estimates of Section~\ref{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
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
\[
 h=\lceil k\delta/2\rceil,\qquad k\delta\ge4,
\]
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
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}
will be used only at a general incidence point where the scheme fiber is
reduced and the selected base component is isolated.

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
\cite[Theorem~4.0.1(3)]{HMX2018}.  Thus
\(HP^{f-1}\le C_2P^f.V\).  Mixed Hodge index gives
\[
 H^jP^{f-j}\le C_2^jP^f.V\quad(0\le j\le f);
\]
no lower bound on \(P^f.V\) is needed here.  For \(f>1\), cut by
\(f-1\) general members of \(|H|\). Their \(P\)-degree is
\(H^{f-1}P\le C_2^{f-1}C_0\), and their images under the birational
morphism defined by \(H\) have degree \(H^f\le C_2^fC_0\).
A general plane projection is birational on such an image curve;
the arithmetic genus of the plane image therefore bounds its geometric
genus in terms of this degree. For \(f=1\), the
canonical-degree bound already bounds the genus.
We will also use this construction for a log smooth pair of log
general type with reduced boundary, using coefficients in the fixed set
\(\{1\}\) in effective birationality.

If such components sweep an ambient space mapping to \(Y\), and
their curves project nonconstantly with \(N_b\)-degree bounded by
their \(P\)-degree, we obtain forbidden covering curves on \(Y_s\).
When the component has positive-dimensional image in the base, general
complete intersections from the big birational system project
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
established below, it remains to check general type and the stated
covering and projection conditions to invoke normalized curve exclusion.

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}
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
multiple \(kP\) has order at most \(4k\rho\).
The same statement holds on a common resolution after adding an
effective exceptional divisor that does not change the section
spaces.  Moreover
\[
 2r\mu_t\le\rho\le4r\mu_t,
\]
so in particular \(\rho\le8\epsilon\) for small \(t\).
\end{proposition}

\begin{proof}
First, pseudo-effectivity of \(K\) gives the volume bounds:
\[
 \vol(K+tA)\le\vol(K+2tA)\le2^n\vol(K+tA).
\]
Suppose the asserted order bound fails for arbitrarily small \(t\).
There are countably many Cartier degrees, and nonzero jet kernels in
each degree are detected by algebraic rank loci. Thus failure at a very
general point gives a degree in which a violating section exists on a
common constant-rank marked-point open. Taking powers makes this degree
\(k\) arbitrarily large and divisible. Choose \(n+1\) levels strictly
between \(\rho\) and \(4\rho\), with equal gaps comparable to \(\rho\).
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
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
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 \(t\);
the gap is bounded above and away from zero because
\(\rho\) is comparable to the fixed \(\epsilon\).
The resulting bounded-genus, bounded-normalized-degree covering
curves contradict Proposition~\ref{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}

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
 \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}.
The torus open in either bundle is the complement of its two axes.

\begin{lemma}[Models, volume, and weights]\label{jet:bundle-models}
Both the total class \(M\) and the slice class have big semiample
positive parts \(P\) on \(\Q\)-factorial klt models \(Z\), with
\[
 K_Z+\Delta\sim_{\Q}c_tP,\qquad \Delta\ge0,
\]
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}
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}
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
\(\widetilde Y\times\widetilde Y\) 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
factorwise pullbacks. Push this relation down to \(Y\times Y\).
The resulting factor divisors are \(\Q\)-Cartier because \(Y\) is
\(\Q\)-factorial, and hence so is the original divisor. Products of
canonical singularities are canonical, and the projective bundles
are klt.  The two disjoint Cartier axes form a plt boundary \(D\),
and
\[
 K_{Z_0}+D=K_{\rm sum},\qquad
 D\sim2\xi-mK_{\rm sum}.
\]
The notation \(K_{\rm sum}\) means \(K_1+K_2\), or \(K\) on a slice.

For \(0<\sigma\le1\), put
\[
 \gamma=2\sigma+\frac{q(1+\sigma m)}r,\qquad
 \Xi_t\sim_{\Q}\xi+
       \frac{(1+\sigma m)t}{\gamma}A_{\rm sum}.
\]
Both endpoint weight systems are big, so choose an effective rational
representative of \(\Xi_t\) containing neither axis, and denote the
representative by \(\Xi_t\) as well.
To make the adjoint construction uniform, we first obtain a threshold
bound independent of \(\sigma\).
Restrict to \(0<t\le\sigma/(1+m)\).  Since \(\gamma\ge2\sigma\),
the coefficient
\[
 a=\frac{(1+\sigma m)t}{\gamma}
\]
lies in \((0,1]\).  Thus the numerical classes
\(\xi+aA_{\rm sum}\), and their restrictions to either fixed axis,
range over bounded segments independent of \(\sigma,q,r,t\).
On one fixed smooth resolution the effective pullbacks of these chosen
divisors have uniformly bounded degree against a fixed very ample class.
That degree bounds their multiplicity at every point, including
the coefficients of exceptional components.  Rational denominators
do not enter this estimate.  The smooth lc threshold is at least
the reciprocal of maximal multiplicity.

To transfer this estimate to the fixed klt space, write the crepant
boundary on its resolution as an SNC divisor with coefficients below one.
The minimum of
one and the positive numbers one minus its positive coefficients
bounds its log discrepancies below by a fixed positive multiple
of smooth log discrepancies.  Therefore the preceding smooth
bound gives a common klt threshold \(\eta>0\) for \(\Xi_t\) on the
original space. The same reasoning on each axis gives, after decreasing
\(\eta\), a klt threshold bound for the restricted divisor there.
The first bound will be used away from the axes; the restricted bounds
allow inversion of adjunction along the coefficient-one axes.

Now choose rational \(0<\sigma<\min\{1,\eta/4\}\), independently of
\(q,t\), and then take \(t\) small enough that
\(t\le\sigma/(1+m)\) and \(q(1+\sigma m)/r<\sigma\).
It follows that \(\gamma<3\sigma<\eta\).
Inversion of adjunction with the disjoint coefficient-one axes
gives plt near them; away from them the threshold bound gives klt.
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
\(q\), then sufficiently small \(t\) with \(r\) as in
Equation~\eqref{jet:normalization}.
A direct calculation gives its adjoint class
\[
 K_{Z_0}+(1-\sigma)D+\gamma\Xi_t
   \sim_{\Q}\frac{1+\sigma m}{r}M.
\]
The big klt adjoint has a good minimal model by
\cite{BCHM2010}.  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}
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
and, in particular, proves bigness used above.
In the range of Equation~\eqref{jet:uniform-normalized-volume},
the bounds needed when choosing \(q\) are explicitly
\[
 \vol(M)\ge \frac{(2n+1)!}{4(n!)^2}\,q\epsilon^{2n},
 \qquad
 \vol(L+q\xi)\le2(n+1)q\epsilon^n.
\]
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 \(F_{\mathrm{full}}\)
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
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
individual sufficiently divisible degrees shows that every section
at that weight, after the full fixed subtraction, contains the
corresponding multiple of \(E_b\).

At a generic base-divisor point the small base-model maps are
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
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
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
\[
 \varepsilon(P;x)=
 \inf_{\substack{C\subset Z\ \mathrm{integral\ curve}\\x\in C}}
       \frac{P\cdot C}{\mult_x C}.
\]

\begin{proposition}[Large two-slot Seshadri constant]\label{prop:large-seshadri}
For each fixed sufficiently small \(\epsilon>0\), 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
\[
 \varepsilon(P;x)>2n+2
\]
at a very general smooth point of its torus open.
In particular, for some sufficiently divisible integer \(k>0\),
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}
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
\[
 2(n+1)\epsilon^n(2/\delta)^n<1.
\]
This inequality will exclude a component that is a whole projective-line
fiber of a slice. Now choose \(q\) sufficiently large that
\[
 \left(\frac{(2n+1)!}{4(n!)^2}q\epsilon^{2n}\right)^{1/d}
 >2n+3+(d+1)\delta.
\]
For all sufficiently small \(t\), the \(d+1\) levels
\(\tau_i=2n+3+(i+1)\delta\), \(0\le i\le d\), then lie below
the volume root of the total positive part. Both choices precede
the choice of \(t\).
Assume that the Seshadri assertion fails for arbitrarily small \(t\).
Because the highest level is below the volume root, jet counts show
that the full kernels at all these levels are nonzero in sufficiently
large divisible degree. A curve through the marked
point with degree-to-multiplicity ratio below the lowest level is
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
moving proper component \(V\) and both estimates
\eqref{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
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
contradiction.  To treat the remaining components, we begin with
their fibers over a base factor.

\paragraph{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
part; their dimension and volume are \(n+1\) and
Equation~\eqref{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
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
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
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
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
subseries.
The model \(Z_{\mathrm{sl}}\) is projective,
\(\Q\)-factorial and klt, and \(P_{\mathrm{sl}}\) 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
\[
 \begin{aligned}
 P_{\mathrm{sl}}^f\cdot F
 &\le(2/\delta)^{n+1-f}P_{\mathrm{sl}}^{n+1},\\
 K_{F^*}P_{\mathrm{sl}}^{f-1}
 &\le(K_{Z_{\mathrm{sl}}}+cP_{\mathrm{sl}})|_F
       P_{\mathrm{sl}}^{f-1},
 \qquad 0<c\le2(n+1-f)/\delta.
 \end{aligned}
\]

The restricted components must also move sufficiently to apply curve
exclusion.  These \(F\)'s may be chosen in families sweeping the opposite
slice.  Indeed the incidence of the \(V\)-family dominates the
total torus.  Choose a general incidence point, not necessarily
the original marked point, and then a general fiber of its first
slot evaluation.  Dominance gives the required dominant evaluation
onto that fixed opposite slice. Choose a component with dominant
evaluation there; generic flatness permits the fiber-component choices
just made. For each fixed \(q,t\), very general choices
also avoid the exceptional sets for all rational weights.
Choose this point also in the isomorphism open of the slice's big
semiample contraction and outside the relevant fixed differences.
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.}
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}.
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.

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
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
\(qN_b^w.W\), 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
choice of \(\delta\) makes the latter strictly less than \(q\).

Suppose \(w>0\).  The ruled component \(F\) itself need not be of
general type, so we seek the adjunction estimate on its base \(W\).
We obtain it from the coefficients of the slice sections. Use the
section-space identification to view the subseries on the original
slice, where the toric weight decomposition is defined. The full fixed
factors are units at the chosen generic torus point, so this identification
preserves the order \(h\). In degree \(k\), expand each of these sections
into its toric weights. At the generic point of \(W\), let \(I_l\) be
the ideal generated by the coefficients of the nonzero weight \(l\)
over all these sections.  Put \(h=\lceil k\delta/2\rceil\).
All \(I_l\) lie in \(\mathfrak m_W^h\): vanishing along the generic
torus fiber says that the coefficient of every Laurent monomial
vanishes to this order.  Their sum is
\(\mathfrak m_W\)-primary.  Otherwise its larger common zero germ,
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
\(W\) 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.
Work in the regular local ring at the generic point of \(W\), of
dimension \(c=n-w\). Resolve the finitely many coefficient ideals
simultaneously. The exponents for which their product defines an lc
ideal pair form the rational polytope
\[
 u_l\ge0,\qquad
 \sum_lu_l\ord_E(I_l)\le a(E;Y,0).
\]
Here \(E\) runs through the divisors needed to test log canonicity on
the simultaneous resolution, and \(a(E;Y,0)\) is the log discrepancy.
Include the exceptional divisor of the blowup of the closed point.
Since every \(I_l\subset\mathfrak m_W^h\), it gives
\(\sum_lu_l\le c/h\). The polytope is compact, and \(\sum_lu_l\)
has a positive rational maximum on it. Fix a rational maximizing point.
For each coordinate \(l\), some equality at this point must involve
a valuation with \(\ord_E(I_l)>0\); otherwise \(u_l\) could be
increased while staying in the polytope. Such a valuation has log
discrepancy zero for the ideal pair. Its center \(C_l\) is contained
in \(V(I_l)\) and contains the generic point of \(W\). Their intersection
is contained in \(V(\sum_l I_l)\), so it is exactly \(W\) locally.
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
\(\Theta=\sum_l(u_l/M_0)\sum_{j=1}^{M_0}D_{l,j}\).
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
point of \(W\), and every active valuation remains an lc place.
Each \(C_l\) is consequently an lc center of this pair. The intersection
property \cite[Theorem~1.7]{KollarKovacs2010}, applied to the identity
morphism on its lc open, shows that \(W\) is an lc center generically.
This also applies when some
maximizing coordinate \(u_l\) is zero: the active valuation blocking that
coordinate still has center in \(V(I_l)\).

Set
\[
 b=\frac{\sum_lu_l(l/k)}{\sum_lu_l},
 \qquad c'=k\sum_lu_l\le\frac{kc}{h}.
\]
Since a weight-\(l\) coefficient divisor has class \(kL_{l/k}\),
the constructed boundary has class \(c'L_b\). Also
\(c'\le kc/h\le2(n-w)/\delta\). Transform it and \(W\) to the small
scaling model \(Y(b)\); our general incidence choice places the
generic point of \(W\) in their common isomorphism open. On this
model the boundary has class \(c'N_b\). Numerical subadjunction
and \(N_b=(r+bm)K_{Y(b)}+rtA_b\) give
\[
 K_{W^*}N_b^{w-1}
 \le\left(\frac1{r+bm}+c'\right)N_b^w.W.
\]
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,
\[
 0<N_b^w.W
 \le\frac{(2/\delta)^{n-w}}q P_{\mathrm{sl}}^{n+1}
 \le2(n+1)\epsilon^n(2/\delta)^{n-w}.
\]
These are precisely the two bounds required in
Equation~\eqref{jet:curve-input}, with constants independent of \(t\).
The variety \(W\) is of general type by
Proposition~\ref{prop:general-type}; normalized curve exclusion
again gives a contradiction.

\paragraph{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
\[
 eN_b^n\le P_{\mathrm{sl}}^n.F,
\]
so \(e\) is bounded uniformly in \(t\).
Here the two axes provide the boundary needed to use log general
type on the base.  The closure of \(F\) on the original bundle is
neither axis.
Intersect this closure with each Cartier axis and push the resulting
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}.
Intersecting with \(N_b^{n-1}\) and using mixed nef comparison yields
\[
 qN_b^{n-1}.D_i\le P_{\mathrm{sl}}^n.F.
\]
The other divisor has bounded degree for the same \(N_b\), because
of Equation~\eqref{jet:axis-signed} and
\[
 0\le K_{Y(b)}N_b^{n-1}\le\frac{N_b^n}{r+bm}.
\]
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
the reduced strict support of \(D_1+D_2\) and all exceptional divisors
to the canonical divisor. Proposition~\ref{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.
Its intersection with the pulled-back \(N_b^{n-1}\) is bounded:
exceptionals vanish under projection, and reduced support has
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.}
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
one slot would mean that \(V\) contains the whole opposite slice
over its first image \(W\).  Its fiber dimension over the other
slot would then be \(\dim W+1\), strictly between zero and \(n+1\),
unless \(V\) were the full total space.  That is impossible.
Hence \(V\) projects generically finitely in both slots and
\(\dim V\le n\).  If either image is proper, general type and the
total-space estimates give the previous curve contradiction.
The same is true if \(V\) is of general type.
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.}
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
resolution \(V^*\), choose a ramified divisor \(R\) above its generic
point.  There is a rational weight \(b\) for which the effective
difference \(E_b|_{V^*}\) does not contain \(R\).
Indeed take a fixed free divisible degree of \(P\); some section
does not vanish generically on \(R\).  Monomial-weight sections
span that degree, so one of them does not vanish there.
Only finitely many weights are tested in this fixed family; their
model comparisons can be resolved simultaneously.

Let \(J\) be the branch divisor on the corresponding small axis
model. Restrict the effective difference to \(R\); this is permitted
because \(R\) is not in its support. Mixed nef intersections and
projection to \(J\) bound its \(N_b\)-degree by \(P^{n-1}.R\).
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
estimate.

To turn this degree bound into bounded curves, we still need a
canonical intersection bound on \(J\).  The required adjunction
does not assume that \((Y(b),J)\) is globally lc.
Terminal \(Y(b)\) is smooth in codimension
two.  Thus normalization and conductor adjunction along \(J\)
give
\[
 K_{J^\nu}+C=(K_{Y(b)}+J)|_{J^\nu},\qquad C\ge0
\]
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}
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
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}
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.}
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
the two evaluation maps simultaneously over a smooth parameter open
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
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
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,
not by deleting their possibly dominant images in $Y$. Both $Y^0$ and
the finite cover lists may depend on this fixed $t$; no common complement
or list as $t\to0$ is used.

For a general member \(V_a^*\) of the resolved family, factor the two
projections through their finite normal Stein covers. The preceding
finiteness lets us fix these as \(T_1\to Y\) and \(T_2\to Y\) after
restricting to a family whose images still dominate \(Y\times Y\).
Each map \(\beta_{i,a}:V_a^*\to T_i\) is birational, so the two
projections determine a birational map
\[
 \phi_a=\beta_{2,a}\circ\beta_{1,a}^{-1}:T_1\dashrightarrow T_2.
\]
The graphs of these maps form an algebraic family after choosing a
Hilbert-scheme component and shrinking for flatness and birationality
of both projections. Such a component with dominant evaluation exists:
there are only countably many graph Hilbert schemes and finitely many
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
family of birational maps between the two fixed covers.

Every possible moving component \(V\) has now led to a contradiction,
so the Seshadri bound holds.  To obtain the assertion about jets,
observe that at a very general point the big semiample
contraction is an isomorphism onto its image near that point.
The jet interpretation of the Seshadri constant for the ample
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}

The scalar and two-slot estimates lead to a contradiction through a
comparison in positive characteristic. We first prove that certain
fixed divisors, jets, and a movable-curve witness cannot coexist.
This theorem and its proof are independent of nonvanishing, abundance,
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}

We now work with arbitrary fixed data, independent of the scaling
models and divisors of Section~\ref{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.
A fixed movable curve on a blowup bounds every section in each
determinant factor, producing the opposite estimate.

For a vector bundle on a smooth projective curve, its slope is
degree divided by rank. The least and greatest Harder--Narasimhan
slopes are denoted by $\mu_{\min}$ and $\mu_{\max}$.
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}
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
 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}
Assume that the following fixed data exist.
\begin{enumerate}[label=\textup{(\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
$\mu_{\min}(\Omega_W^1|_C)\ge0$.
\item On the projective bundle
\[
 Z=\mathbf P\bigl(\mathcal O_W(S)_1\oplus
                         \mathcal O_W(S)_2\bigr)\longrightarrow W\times W,
 \qquad \xi=c_1(\mathcal O_Z(1)),\qquad M=L_1+L_2+q\xi,
\]
use the convention that the direct image of $\mathcal O_Z(a\xi)$
is the $a$-th symmetric power of the displayed sum. A smooth
point $z_*$ in the complement of the two axes and finitely many
sections of $k_0M$, for some integer $k_0>0$ with $k_0L$ Cartier,
generate jets through order $(2n+2)k_0$ at $z_*$.
\item A point $x\in W$, its blowup
$\pi_x:\widehat W\to W$ with exceptional divisor $J$, and a
curve class
\[
 \gamma=f_*(A_1\cdots A_{n-1}),
\]
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}
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
nef, and $K_W$ is not assumed pseudo-effective. The theorem
starts with the actual flag and movable witness, rather than
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}

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}
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 $z_*$.
Fix these finitely many sections. All models, morphisms, divisors,
points, the flag, the complete-intersection witness for $\gamma$,
and the jet and filler sections spread over an integral finitely
generated $\mathbf Z$-algebra of characteristic zero. Shrink its
spectrum so that the fibers and flag are smooth projective and
geometrically integral, the birational morphisms remain birational,
the fixed ample bundles remain ample, and the finite jet
surjection and nonvanishings persist. Birationality is preserved
by spreading an isomorphism between dense opens. All displayed
intersection numbers are constant.

We also preserve $\mu_{\min}(\Omega_W^1|_C)\ge0$.
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
fixed. Equivalently, a single relative ample twist globally
generates the curve bundle. Every quotient of rank $a$ then has
degree bounded below by $-ad$ for a fixed integer $d$.
A negative-degree quotient has one of finitely many ranks and
degrees. The proper relative Quot schemes for those Hilbert
polynomials have images missing the generic point: after removal
of torsion, a negative-degree coherent quotient there would
give a negative-degree vector-bundle quotient. Removing those
finitely many images proves the same openness assertion directly.

This base has closed points in arbitrarily large characteristics.
Take algebraic closures of their residue fields and exclude
denominator primes. Retain the notation for the reductions.
The curve $\gamma$ continues to pair nonnegatively with every
effective divisor: pull that divisor back under the fixed
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
\[
 N_p=\lfloor p/k_0\rfloor,\qquad B_p=k_0N_pL+T_0.
\]
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}
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
monomials gives a triangular system, ordered by total degree.
Starting at degree zero and removing the error in each successive
degree proves the required jet surjection. This argument takes
place in the local ring and never divides by factorials.

Put $d_p=(p-k_0N_p)q$, so $0\le d_p\le(k_0-1)q$.
Among the sections fixed before reduction, choose
\[
 u_0\in H^0(W,\cO_W(T_0)),\qquad
 v_{d_p}\in H^0(W,\cO_W(T_0+d_pS))
\]
nonzero at the first and second base coordinates of $z_*$,
respectively. Their tensor product belongs to the summand of
first-slot weight zero in the projective-bundle formula; its
fiber monomial is the $d_p$-th power of the second coordinate.
It defines a section of
$d_p\xi+T_{0,1}+T_{0,2}$ nonzero at $z_*$, since $z_*$ lies
off both axes. Multiplying the product jet system by this section
changes its divisor from $N_pk_0M$ to
$pq\xi+B_{p,1}+B_{p,2}$. Multiplication is invertible on the
local jet algebra. Only the finitely many previously fixed
values of $d_p$ occur.

Since $\dim Z=2n+1$, the
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}

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
of $S$, so $F^*S'=pS$, and define
\[
 \mathcal E=F_*\cO_W(B_p),\quad s=\rk\mathcal E=p^n,\quad R=s^2,
 \qquad U_a=H^0(W,\cO_W(B_p+paS)).
\]
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
 \cO(-aS'_1-bS'_2)
 \longrightarrow \mathcal E\boxtimes\mathcal E.
\end{equation}

\begin{lemma}\label{common:generic-rank}
The map in Equation~\eqref{common:two-slot-evaluation} has generic rank $R$.
\end{lemma}

\begin{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 $z$
with weight-$j$ coefficients in
\[
 H^0(W,B_p+jS)\otimes
 H^0(W,B_p+(pq-j)S),\qquad 0\le j\le pq.
\]
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
\[
 A_*[z]/(z^p-z_0^p),
\]
free over $A_*$ with basis $1,z,\ldots,z^{p-1}$.  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
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 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"]\\
 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 $h=F\circ\mathrm{pr}_1=F\circ\mathrm{pr}_2:
\Delta_F\to W'$. On this fiber product,
\[
 \cO(pS_1)\simeq h^*\cO_{W'}(S')\simeq\cO(pS_2).
\]
Consequently every pair of weights $a+b=q$ gives the same line bundle:
\[
 \cO_{\Delta_F}(B_{p,1}+B_{p,2}+paS_1+pbS_2)
 \simeq\mathcal L_p.
\]
On the diagonal of $W'\times W'$, every source twist of
Equation~\eqref{common:two-slot-evaluation} becomes $-qS'$.
Finite base change and projection formula identify the twisted target as
\[
 (\mathcal E\otimes\mathcal E)(qS')\simeq h_*\mathcal L_p.
\]
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
\[
 u=n(p-1),\qquad e_j=\rk A_j(V),\qquad
 \sum_{j=0}^u e_j=p^n.
\]
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),
 \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
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}

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}
On monomials, each exponent vector pairs with its complement to
$(p-1,\ldots,p-1)$.  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.

\subsubsection{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
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
\[
 \mu_{\max}(A_j(V))\le(p-1)K_W\cdot C+nc.
\]
\end{lemma}

\begin{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}
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
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}
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
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.
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 $p,i$;
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*}
 \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+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}
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}
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 $p$,
\[
 h^0(\Delta_F,\mathcal L_p)
 \le R\bigl(7^nH^n+O(1/p)\bigr)<R/4.
\]
The error constant is independent of the filtration index $j$.
\end{lemma}

\begin{proof}
Write the fixed flag as
\[
 W=W_n\supset W_{n-1}\supset\cdots\supset W_1=C,
 \qquad W_{n-k}\in|h_k|_{W_{n-k+1}}
 \quad(1\le k\le n-1).
\]
At step $k$, fix the preceding nonnegative integers
$b_1,\ldots,b_{k-1}$ and put
\[
 \mathcal F_k=
 \mathcal G_j|_{W_{n-k+1}}
 \Bigl(-\sum_{i<k}b_i h_i\Bigr).
\]
For each $b_k\ge0$, the divisor restriction sequence is
\[
 0\longrightarrow\mathcal F_k(-(b_k+1)h_k)
 \longrightarrow\mathcal F_k(-b_kh_k)
 \longrightarrow
 \mathcal F_k(-b_kh_k)|_{W_{n-k}}
 \longrightarrow0.
\]
Iterating it for $b_k=0,1,\ldots$ 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}
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
$\sum_i b_i(h_i\cdot C)>M_p$, 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}
 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}
The leading coefficient follows from
$\Lambda=\prod_i l_i$.  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
the asserted bound with $R=p^{2n}$.  There is no extra factor for
the number of graded pieces.  Finally
\[
 7^nH^n\le(14\epsilon)^n<1/4
\]
by Equation~\eqref{common:intersection-bounds}, proving the strict inequality
for all large $p$.
\end{proof}

\subsubsection{The determinant has incompatible orders}

Choose $R$ generically independent columns of
Equation~\eqref{common:two-slot-evaluation} from bases of its source vector spaces.
Their determinant is a nonzero section
\[
 0\ne\sigma\in H^0(W'\times W',\mathcal Q_1\boxtimes\mathcal Q_2)
\]
of an actual external-product line bundle. To specify its factors,
let $m_i$ be the sum of the weights of the $R$ chosen columns in
slot $i$. The target determinant is
$(\det\mathcal E)^s\boxtimes(\det\mathcal E)^s$, and the
chosen source determinant is
$\cO(-m_1S')\boxtimes\cO(-m_2S')$. Thus
\[
 \mathcal Q_i=(\det\mathcal E)^{\otimes s}
                 \otimes\cO_{W'}(m_iS'),\qquad
 \lambda_i=m_i/R.
\]
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,
 \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
$e_j(p-1)K_W$.  Summing gives
\[
 \sum_jc_1(A_j(\Omega_W^1))=\frac{s(p-1)}2K_W.
\]
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
\[
 (L+K_W/2+\lambda S)\cdot\gamma\le40\epsilon J\cdot\gamma,
\]
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
left-hand divisor at $\lambda=\lambda_i$ by
\[
 \frac1p\bigl(B_p-pL-K_W/2\bigr).
\]
Its numerator belongs to the fixed finite list
$T_0-aL-K_W/2$, $0\le a<k_0$. The resulting error after
intersection with $\gamma$ is therefore $O(p^{-1})$, uniformly
over the selected determinant columns.

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}
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
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}

On the other hand, Equation~\eqref{common:diagonal-rank-upper}, Lemma~\ref{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 $p$.


This contradiction proves Theorem~\ref{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}

We return to the notation and the hypothetical counterexample of
Section~\ref{sec:jets}. Theorem~\ref{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
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}
Assume the lower-dimensional real-boundary good-model hypothesis of
Assumption~\ref{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
\(h^0(X,K_X)=g(X)>0\). We therefore suppose \(n\geq2\) and argue by
contradiction. We use the late terminal model \(Y\), the divisors
\(K=K_Y\) and \(A=A_Y\), and the scaling models of the preceding
section. Fix \(m>0\) with \(mK\) Cartier, and write
\[
 \mu_t=\vol(K_X+tA_X)^{1/n},\qquad
 L=r(K+tA).
\]
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
so that
\begin{equation}
\label{fr:epsilon}
 (14\epsilon)^n<\frac14,\qquad
 80\epsilon<\frac34.
\end{equation}
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
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 ample polarization and a complete-intersection flag will
control the number of sections on the Frobenius thickening of the
diagonal. We construct them from the positive part on a scaling model.

Choose a smooth common resolution \(W\) of \(Y\) and the scaling model
for \(K+tA\). In this application \(K,A,L\) also denote their pullbacks to
\(W\), and
\[
 K_W=K+E,\qquad E\geq0.
\]
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
 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}
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
over \(Y_t\).  The differences discarded when passing to the moving
parts are exceptional over the scaling model, and the transform of
\(A\) is effective up to rational linear equivalence.  Intersecting
with \(H_0^{n-1}\) gives the displayed equalities and inequality.
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
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
\(\eta>0\) 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
 (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
semipositivity, applied with empty boundary
\cite[Theorem~2.1]{CampanaPaun2015}, and restriction to a sufficiently
general complete-intersection curve give the following fixed data.
Apply restriction to the Harder--Narasimhan filtration and its factors
\cite[Theorem~6.1 and Remark~6.2]{MehtaRamanathan1982}. Choose successively
large divisible integers $l_i$ such that $h_i=l_iH$ are integral very ample
classes, and choose a smooth integral complete-intersection flag
\[
 W=W_n\supset W_{n-1}\supset\cdots\supset W_1=C,
 \qquad W_{i-1}\in |h_{n-i+1}|_{W_i},
\]
for which
\begin{equation}
\label{fr:generic-nef}
 \mu_{\min}(\Omega_W^1|_C)\geq0.
\end{equation}
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 \(\Omega_W^1\).

\subsubsection{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
\(\pi_x:\widehat W\to W\) be its blowup, with exceptional
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
section spaces, since the added exceptional part does not change them.
If \(\rho=\vol(2r(K+2tA))^{1/n}\), then
\[
 \rho\leq4r\mu_t\leq8\epsilon
\]
for our sufficiently small \(t\). Thus every section of a sufficiently
divisible multiple of \(D^+\) has normalized order at \(x\) at most
\(4\rho\leq32\epsilon\).

We convert this order bound into a curve inequality.
It follows that \(\pi_x^*D^+-40\epsilon J\) 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.
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}
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
intersection of very ample divisors on one fixed smooth birational
model of \(\widehat W\). Fix this model, its divisors, and the resulting
covering family.  Thus the choice consists of finite algebraic data,
rather than a limiting real movable class.  Positivity of
\(J\cdot\gamma\) follows from the strict inequality and
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 $W$;
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
\[
 \pi_x^*Q_\lambda\cdot\gamma
 \le \pi_x^*D^+\cdot\gamma
 <40\epsilon J\cdot\gamma,
 \qquad J\cdot\gamma>0.
\]
These are the required movable inequalities. Only the two endpoint
inequalities need to be retained as finite data, since the pairing is
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}

We must similarly realize the two-slot jet estimate by finitely many
sections before reducing.  Their products will then supply the jet
orders needed for arbitrarily large characteristics.

On \(W\times W\), let
\[
 Z=\PP(mK_1\oplus mK_2),\qquad
 \xi=\OO_Z(1),\qquad
 M=L_1+L_2+q\xi,
\]
with the symmetric-power convention for the projective bundle.
Proposition~\ref{prop:large-seshadri} gives a semiample big model whose
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
an integer \(k_0>0\) such that \(|k_0M|\) generates jets of order
at least \((2n+2)k_0\) at a fixed smooth torus point \(z_*\in Z\).
Indeed, on the ample model choose a rational number \(c\) strictly
between \(2n+2\) and its Seshadri constant. On the blowup of the
selected smooth point, the pullback of the ample class minus \(c\)
times the exceptional divisor is ample. Serre vanishing and the
exceptional-divisor sequence then give jets of order \(kc-1\) for
all sufficiently divisible large \(k\). Pulling sections back on the
isomorphic open gives the asserted bound.
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}

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
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
theorem also fixes its auxiliary divisor $T_0$ and finitely many
filler sections before choosing any residual characteristic.

The common theorem now rules out these fixed data. Its contradiction
compares two orders of the same nonzero determinant. After reduction,
the Frobenius direct image \(\mathcal E\) used there has rank \(p^n\),
so \(\mathcal E\boxtimes\mathcal E\) has rank \(R=p^{2n}\). The fixed
jets give full evaluation rank \(R\), while the small polarization
bounds the rank on the diagonal by \(R/4\). Thus a nonzero maximal
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 \(p\).
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}.

\section{Completion and change of ground field}\label{sec:completion}

Smooth nonvanishing is now available in the order required by the
induction. We first finish the complex proof, including descent from
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 \(\C\)]
Use dimension induction in the real-boundary category of
Proposition~\ref{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
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}
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
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
\[
 p^*(K_X+B)=a^*(K_{X_d}+B_d)=q^*(K_V+B_V).
\]
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.
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
\(m(K_X+B)\) whose pullback is globally generated. The projection
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
conclusion over every algebraically closed field of characteristic zero.
\end{proposition}

\begin{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
\((X_{\C},B_{\C})\) for the complex base change.

The chosen log resolution tests log canonicity after either
algebraically closed field extension: its boundary remains simple
normal crossing and all discrepancy coefficients remain unchanged.
The descended pair is consequently lc over both \(k_0\) and \(\C\).

Nefness is also invariant under such extensions. Indeed, a curve
over an extension is represented by a point of a relative Hilbert
scheme after its finite defining data have been spread over a
finite-type parameter scheme. The degree of a fixed line bundle is
constant in the resulting flat family. Specializing to a closed
point over the algebraically closed smaller field preserves a
negative degree, if one existed; some irreducible component of the
specialized curve would then have negative degree. This contradicts
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.

The complex case supplies an integer \(m>0\), enlarged to a multiple
of the Cartier index fixed over \(k_0\), for which the Cartier
line bundle \(M=\OO_{X_0}(m(K_{X_0}+B_0))\) becomes
globally generated over \(\C\). Proper flat base change for sections
identifies
\[
 H^0(X_0,M)\otimes_{k_0}\C
   = H^0(X_{\C},M_{\C}).
\]
Tensoring the evaluation map for \(M\) with \(\C\) therefore gives
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}

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}.
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}
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
dimension zero and proves the full real-boundary conclusion of
Proposition~\ref{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}
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 \(X\dashrightarrow V\)
such that \(K_V\) is \(\Q\)-Cartier and semiample. On a common smooth
resolution \(p:W\to X\), \(q:W\to V\), compatible canonical divisors
satisfy
\[
 p^*K_X=q^*K_V+E,
 \qquad E\geq0\quad\text{and }E\text{ is }q\text{-exceptional}.
\]
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 \((X,0)\).
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
klt, and has the displayed effective exceptional comparison.
Its boundary is zero, being the strict transform of the zero boundary
with no extracted divisors.
Discrepancies strictly improve at contracted divisors, which is the
\(K_X\)-negative condition; semiampleness is the good-model conclusion.
For sufficiently divisible \(m>0\), the effective \(q\)-exceptional
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}
Choose a positive integer \(m\) such that the line bundle
\(L=\OO_Z(mA)\) is globally generated. Its complete system gives
\[
 \phi:Z\longrightarrow\PP(H^0(Z,L)^*),\qquad
 L\simeq\phi^*\OO(1).
\]
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}.
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
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}

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 \(\Theta\)
defined by
\[
 K_{X^\nu}+\Theta=\nu^*(K_X+\Delta)
\]
includes the conductor boundary, and \((X^\nu,\Theta)\) 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}
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}
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
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
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}
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 \(D\),
\[
 K_X+\Delta\sim_{\Q}D\geq0,
 \qquad S\subseteq\Supp D.
\]
Then, for every integer \(m\geq2\) for which \(m(K_X+\Delta)\) is Cartier,
the restriction map
\[
 H^0\bigl(X,\OO_X(m(K_X+\Delta))\bigr)
 \longrightarrow
 H^0\bigl(S,\OO_X(m(K_X+\Delta))|_S\bigr)
\]
is surjective.
\end{corollary}
\begin{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.
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
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
restriction-surjectivity statement is asserted.

\section{Finite generation of rational log canonical rings}
\label{sec:canonical-rings}

The preceding results also settle finite generation for rational lc pairs.
The finite-generation conjecture asks whether finitely many rounded
pluricanonical sections generate the log canonical ring
\cite[Conjecture~A]{FujinoGongyoRings2014}. The assertion below concerns
the full rounded ring, not merely a sufficiently divisible subring, and
requires no nefness assumption.
For the projective absolute statement, we combine the ordinary minimal-model
theorem with log abundance and keep track of every rounded degree. For the
proper relative statement over \(\C\), we apply Fujino--Gongyo's published
reduction from good minimal models.
In that relative statement the morphism is only required to be proper;
the source need not be projective over \(\C\).

For an integral Weil divisor \(G\) on a normal variety \(X\),
\(\OO_X(G)\) denotes its divisorial sheaf. For a rational divisor \(D\)
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}
\leavevmode
\begin{enumerate}[label=\textup{(\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
\[
 R(X,D)=\bigoplus_{m\geq0}
 H^0\bigl(X,\OO_X(\lfloor mD\rfloor)\bigr)
\]
is a finitely generated \(k\)-algebra.

\item Let \((X,\Delta)\) be a log canonical pair over \(\C\), with
\(\Delta\geq0\) rational and \(K_X+\Delta\) \(\Q\)-Cartier. For every
proper morphism \(f:X\to S\) onto an algebraic variety \(S\), the relative
log canonical algebra
\[
 \mathcal R(X/S,K_X+\Delta)
 =\bigoplus_{m\geq0}f_*\OO_X\bigl(\lfloor m(K_X+\Delta)\rfloor\bigr)
\]
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 \(S\).
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
section \(s\) in degree \(m>0\) would give
\[
 \operatorname{div}(s)+mD
 =\bigl(\operatorname{div}(s)+\lfloor mD\rfloor\bigr)
   +\bigl(mD-\lfloor mD\rfloor\bigr)\geq0.
\]
This would make \(D\) pseudo-effective. Thus all positive graded pieces
vanish in this case. Since \(H^0(X,\OO_X)=k\), we have \(R(X,D)=k\).

Suppose now that \(D\) is pseudo-effective. The ordinary-pair conclusion
of the companion minimal-model theorem
\cite[Corollary~1.2]{OpenAIMinimalModels2026} gives a normal projective lc
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.

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}
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
divisors, \(F\) is the strict transform of a divisor on \(X\). The
pushed-forward boundary gives equality of these discrepancies. This proves
the asserted support.

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
\[
 \operatorname{div}_Z(s)+mG\geq0.
\]
This is equivalent to the usual inequality with \(\lfloor mG\rfloor\)
because the coefficients of \(\operatorname{div}_Z(s)\) 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}
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,
\[
 H^0\bigl(X,\OO_X(\lfloor mD\rfloor)\bigr)
 =H^0\bigl(Y,\OO_Y(\lfloor mD_Y\rfloor)\bigr)
 \qquad(m\geq0).
\]
These identifications preserve multiplication, so they identify the full
graded rings.

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
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
\[
 \lfloor(\ell r+i)D_Y\rfloor
 =\ell(rD_Y)+\lfloor iD_Y\rfloor\qquad(\ell\geq0),
\]
projection formula identifies the part in degrees congruent to \(i\)
modulo \(r\), as an \(A\)-module, with
\[
 \bigoplus_{\ell\geq0}H^0\bigl(\PP^N_k,\mathcal F_i(\ell)\bigr).
\]
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)}.

For \textup{(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}
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
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
be projective over \(\C\).
Their Theorem~1.1 also gives \textup{(i)} directly when \(k=\C\).
\end{proof}

The good-model input allows real boundaries, but both finite-generation
statements retain rational boundaries. In the absolute proof, rationality
is what leaves finitely many residue-class section modules. The relative
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}

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
but not uniruled.

\begin{corollary}[Uniruledness and Mumford's criterion]
\label{cor:uniruled-mumford}
Let \(X\) be a smooth connected projective complex variety.
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(X\) is uniruled if and only if \(\kappa(X)=-\infty\).
\item \(X\) is rationally connected if and only if
\[
 H^0\bigl(X,(\Omega_X^1)^{\otimes m}\bigr)=0
 \quad\text{for every integer }m>0.
\]
\end{enumerate}
\end{corollary}
\begin{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 \(\kappa(X)\geq0\):
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)}.

For \textup{(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
of positive dimension at most \(n\) whose canonical class is
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 \(\C\);
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
\(\Omega_X^0=\OO_X\).

\begin{corollary}[Pluriforms and the rational quotient]
\label{cor:pluriforms}
\begin{enumerate}[label=\textup{(\roman*)}]
\item For every smooth connected projective complex variety \(X\),
\[
 \dim R(X)=
 \max_{\substack{p\in\Z_{\geq0},\,k\in\Z_{>0}\\
 H^0(X,\operatorname{Sym}^k(\Omega_X^p))\ne0}} p.
\]
\item A compact connected K\"ahler manifold \(X\) is rationally
connected if and only if
\[
 H^0\bigl(X,\operatorname{Sym}^k(\Omega_X^p)\bigr)=0
 \quad\text{for every pair of integers }p,k>0.
\]
\end{enumerate}
\end{corollary}
\begin{proof}
Write \(r=\dim R(X)\) in \textup{(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
\[
 0\longrightarrow\OO_F^{\oplus r}
 \longrightarrow\Omega_{X'}^1|_F
 \longrightarrow\Omega_F^1\longrightarrow0.
\]
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
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
antisymmetrization and symmetrization over \(\C\). Rational
connectedness of \(F\) makes all their sections vanish, by
Corollary~\ref{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 \(kK_{R(X)}\)
for some \(k>0\) pulls back to a nonzero section of
\(\operatorname{Sym}^k(\Omega_X^r)\). 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
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
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.
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
smooth projective variety, its double dual is a line bundle; put
\(\kappa(\mathcal L)=\kappa(\mathcal L^{**})\). Define
\[
 \omega(X)=
 \max_{\substack{m>0\\
 \mathcal L\subset(\Omega_X^1)^{\otimes m}\\
 \operatorname{rank}\mathcal L=1}}
 \kappa(\mathcal L),
\]
where \(m\) is an integer and the value is \(-\infty\) 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 \(H^0(X,(\mathcal L^{**})^{\otimes a})\),
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
\[
 \kappa^+(X)=
 \max_{\substack{p>0\\0\ne\mathcal F\subset\Omega_X^p}}
 \kappa(\det\mathcal F).
\]
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}.

\begin{corollary}[Cotangent invariants and the MRC base]
\label{cor:cotangent-mrc}
Let \(X\) be a smooth connected projective complex variety. Then
\[
 \omega(X)=\omega(R(X))=
 \begin{cases}
 -\infty,&\text{if }R(X)\text{ is a point},\\
 \kappa(R(X)),&\text{if }\dim R(X)>0.
 \end{cases}
\]
In particular, if \(X\) is positive-dimensional and non-uniruled, then
\(\omega(X)=\kappa(X)\). If \(X\) is positive-dimensional and
\(\kappa(X)\geq0\), then
\[
 \kappa^+(X)=\kappa(X).
\]
\end{corollary}
\begin{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
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
resolving the MRC map. If \(R(X)\) is a point, the displayed formula
therefore follows without changing our convention \(\kappa(\mathrm{pt})=0\).

Suppose that \(R=R(X)\) has positive dimension. It is non-uniruled, so
\(\kappa(R)\geq0\) by Corollary~\ref{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
\(V_F\) 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
such a normal birational model implies \(\omega(F)=0\); it does not
require \(K_F\) itself to be trivial. Property~4 of the same appendix
and birational invariance now give
\(\omega(R)=\omega(R')\leq\dim Z=\kappa(R)\).
If \(F\) is a point, this upper bound is simply
\(\omega(R)\leq\dim R=\kappa(R)\). The reverse inequality
\(\omega(R)\geq\kappa(R)\) is one of the defining cotangent inequalities
recorded in the same appendix. This proves the piecewise formula.
A non-uniruled \(X\) has \(R(X)\) birational to \(X\), giving the
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
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}
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 \(\kappa^+(X)=0\).
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}

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}
Let \(X\) be a connected normal projective complex variety.
\begin{enumerate}[label=\textup{(\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
\(A\) is an abelian variety and the fiber is simply connected and projective.
\item If \(\widetilde X\to X\) is an infinite \'etale Galois cover
and \(\widetilde X\) is biholomorphic to a quasi-projective variety,
there are a finite \'etale Galois cover \(X'\to X\), a locally trivial
holomorphic fiber bundle \(\alpha:X'\to A\) over an abelian variety,
and an \'etale cover \(\widetilde A\to A\) with no positive-dimensional
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
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}

The simply connected fiber condition belongs to \textup{(i)}, not to
the general-cover statement \textup{(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}

The complex rational case of Theorem~\ref{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
analytic spaces are Hausdorff and second-countable, and a complex variety
means a reduced and irreducible complex analytic space, as in
\cite[p.~5]{FujinoRelativeAnalytic2025}.

Here a projective analytic morphism is proper and possesses a relatively
ample line bundle. Rational Cartierness is local on the source; a single
global Cartier index is not part of that definition. A divisor is
\(\pi\)-nef over \(Y\) if it has nonnegative intersection with every
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}
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 \(D=K_X+\Delta\)
\(\Q\)-Cartier. Assume that \(D\) is \(\pi\)-nef over all of \(Y\).
For every compact subset \(W\subset Y\), there exist an analytic open
neighborhood \(U\supset W\) and a positive integer \(m\) such that
\(mD|_{X_U}\) is Cartier, where \(X_U=\pi^{-1}(U)\), and the line bundle
\[
 L=\OO_{X_U}(mD|_{X_U})
\]
is generated relative to \(\pi_U:X_U\to U\). Equivalently, the evaluation
map
\[
 \pi_U^*(\pi_U)_*L\longrightarrow L
\]
is surjective.
\end{corollary}
\begin{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
open neighborhood \(V\supset W\) with \(\pi^{-1}(V)\subset O\).
Thus \(m_0D\) is Cartier over \(V\).

Fix \(P\in W\). Following
\cite[the proof of Theorem~1.10, p.~30]{FujinoRelativeAnalytic2025},
work over a sufficiently small neighborhood of \(P\) contained in \(V\)
and take a crepant dlt blow-up
\(p:(X',\Delta')\to(X,\Delta)\). Put \(\pi'=\pi\circ p\) and
\(D'=K_{X'}+\Delta'=p^*D\). For every log canonical stratum \(S\)
of \((X',\Delta')\), including \(X'\) itself, adjunction gives an
effective rational boundary \(\Delta_S\) with
\[
 K_S+\Delta_S=D'|_S.
\]
The strata are normal, and this adjoint is nef over \(\pi'(S)\).
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
and numerical dimensions agree. Thus \(D'|_S\) is abundant over
\(\pi'(S)\) for every stratum: this is precisely
\(\pi'\)-log abundance in
\cite[Definitions~2.10 and~2.13]{FujinoRelativeAnalytic2025}.

The nef-and-log-abundant theorem
\cite[Theorem~1.4]{FujinoRelativeAnalytic2025} now gives a relatively
generated multiple of \(D'\) over a neighborhood \(U_P\subset V\)
of \(P\). Enlarge its integer to a multiple \(m_P\) of \(m_0\);
tensor powers preserve relative generation. Normality gives
\(p_*\OO_{X'}=\OO_X\), and projection formula identifies the direct
images over \(U_P\) of \(\OO_X(m_PD)\) and
\(\OO_{X'}(m_PD')\). Their evaluation maps are related by pullback.
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 \(U_{P_1},\ldots,U_{P_s}\)
covering \(W\), and let \(m\) be a common multiple of
\(m_0,m_{P_1},\ldots,m_{P_s}\). Put \(U=\bigcup_i U_{P_i}\).
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,
the morphism, and \(W\); no single Cartier index or relatively generated
multiple over all of a noncompact base is asserted. The nefness premise is
over all of \(Y\), not only over \(W\). Projectivity cannot be replaced here
by mere properness or by a K\"ahler hypothesis: when \(Y\) is a point it
requires \(X\) to be projective. The statement does not include real
boundaries or semi-log-canonical analytic pairs.


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%   eprint = {0907.1506},
%   archivePrefix = {arXiv},
%   primaryClass = {math.AG},
%   url = {https://www.math.kyoto-u.ac.jp/~fujino/MMP21.pdf},
%   note = {Version 6.01, January 8, 2009; Theorem 3.74(3), p. 109}
% }

% @article{GongyoLehmann2013,
%   author = {Gongyo, Yoshinori and Lehmann, Brian},
%   title = {Reduction maps and minimal model theory},
%   journal = {Compositio Mathematica},
%   year = {2013}, volume = {149}, number = {2}, pages = {295--308},
%   doi = {10.1112/S0010437X12000553},
%   url = {https://www.ms.u-tokyo.ac.jp/~gongyo/papers/redmapsmmp-final.pdf},
%   note = {Read with the numerical-dimension erratum on Lehmann's research page}
% }

% @article{Fujino2019Numerical,
%   author = {Fujino, Osamu},
%   title = {Corrigendum to ``On subadditivity of the logarithmic {Kodaira} dimension''},
%   journal = {Journal of the Mathematical Society of Japan},
%   year = {2020}, volume = {72}, number = {4}, pages = {1181--1187},
%   doi = {10.2969/jmsj/82568256},
%   eprint = {1904.11639}, archivePrefix = {arXiv}, primaryClass = {math.AG},
%   url = {https://www.jstage.jst.go.jp/article/jmath/72/4/72_1181/_article/-char/ja}
% }

% @article{KovacsPatakfalvi2017,
%   author = {Kov{\'a}cs, S{\'a}ndor J. and Patakfalvi, Zsolt},
%   title = {Projectivity of the moduli space of stable log-varieties and subadditivity of log-{Kodaira} dimension},
%   journal = {Journal of the American Mathematical Society},
%   year = {2017}, volume = {30}, number = {4}, pages = {959--1021},
%   doi = {10.1090/jams/871},
%   url = {https://sites.math.washington.edu/~kovacs/current/papers/Kovacs_Patakfalvi__Projectivity.pdf},
%   note = {Theorem 9.9 in the author version}
% }

% @article{BDPP2013,
%   author = {Boucksom, S{\'e}bastien and Demailly, Jean-Pierre and P{\u a}un, Mihai and Peternell, Thomas},
%   title = {The pseudo-effective cone of a compact {K{\"a}hler} manifold and varieties of negative {Kodaira} dimension},
%   journal = {Journal of Algebraic Geometry},
%   year = {2013}, volume = {22}, number = {2}, pages = {201--248},
%   doi = {10.1090/S1056-3911-2012-00574-8},
%   eprint = {math/0405285}, archivePrefix = {arXiv}, primaryClass = {math.AG},
%   url = {https://arxiv.org/abs/math/0405285},
%   note = {Theorem locators refer to arXiv version 1}
% }

% @article{LazicPeternell2017,
%   author = {Lazi{\'c}, Vladimir and Peternell, Thomas},
%   title = {Rationally connected varieties---on a conjecture of {Mumford}},
%   journal = {Science China Mathematics},
%   year = {2017}, volume = {60}, number = {6}, pages = {1019--1028},
%   doi = {10.1007/s11425-016-9039-2},
%   eprint = {1608.04706}, archivePrefix = {arXiv}, primaryClass = {math.AG},
%   url = {https://arxiv.org/abs/1608.04706},
%   note = {Theorem locators refer to arXiv version 1}
% }

% @misc{BrunebarbeCampana2015,
%   author = {Brunebarbe, Yohan and Campana, Fr{\'e}d{\'e}ric},
%   title = {Fundamental group and pluridifferentials on compact {K{\"a}hler} manifolds},
%   year = {2015},
%   eprint = {1510.07922}, archivePrefix = {arXiv}, primaryClass = {math.AG},
%   url = {https://arxiv.org/abs/1510.07922},
%   note = {Version 2}
% }

% @article{Campana1995,
%   author = {Campana, Fr{\'e}d{\'e}ric},
%   title = {Fundamental group and positivity of cotangent bundles of compact {K{\"a}hler} manifolds},
%   journal = {Journal of Algebraic Geometry},
%   year = {1995}, volume = {4}, number = {3}, pages = {487--502}
% }

% @article{CampanaPaun2015,
%   author = {Campana, Fr{\'e}d{\'e}ric and P{\u a}un, Mihai},
%   title = {Orbifold generic semi-positivity: an application to families of canonically polarized manifolds},
%   journal = {Annales de l'Institut Fourier},
%   year = {2015}, volume = {65}, number = {2}, pages = {835--861},
%   doi = {10.5802/aif.2945},
%   url = {https://www.numdam.org/item/AIF_2015__65_2_835_0/}
% }

% @article{Langer2004,
%   author = {Langer, Adrian},
%   title = {Semistable sheaves in positive characteristic},
%   journal = {Annals of Mathematics},
%   year = {2004}, volume = {159}, number = {1}, pages = {251--276},
%   doi = {10.4007/annals.2004.159.251},
%   url = {https://annals.math.princeton.edu/2004/159-1/p04}
% }

% @article{Sun2008,
%   author = {Sun, Xiaotao},
%   title = {Direct images of bundles under {Frobenius} morphism},
%   journal = {Inventiones Mathematicae},
%   year = {2008}, volume = {173}, number = {2}, pages = {427--447},
%   doi = {10.1007/s00222-008-0125-y},
%   eprint = {math/0611360v2}, archivePrefix = {arXiv}, primaryClass = {math.AG},
%   url = {https://arxiv.org/abs/math/0611360v2},
%   note = {Theorem 3.7 and Lemma 4.2 in arXiv:math/0611360v2}
% }

% @article{FujinoHashizume2023,
%   author = {Fujino, Osamu and Hashizume, Kenta},
%   title = {Existence of log canonical modifications and its applications},
%   journal = {European Journal of Mathematics},
%   year = {2023}, volume = {9}, number = {1},
%   note = {Paper No. 13, 32 pp.; Theorem 2.10 in the accepted author version},
%   doi = {10.1007/s40879-023-00598-0},
%   url = {https://link.springer.com/article/10.1007/s40879-023-00598-0}
% }

% @article{FujinoGongyoModuli2014,
%   author = {Fujino, Osamu and Gongyo, Yoshinori},
%   title = {On the moduli b-divisors of lc-trivial fibrations},
%   journal = {Annales de l'Institut Fourier},
%   year = {2014}, volume = {64}, number = {4}, pages = {1721--1735},
%   doi = {10.5802/aif.2894},
%   url = {https://www.numdam.org/item/AIF_2014__64_4_1721_0/}
% }

% @incollection{HMX2018,
%   author = {Hacon, Christopher D. and McKernan, James and Xu, Chenyang},
%   title = {Boundedness of varieties of log general type},
%   booktitle = {Algebraic Geometry: Salt Lake City 2015, Part 1},
%   series = {Proceedings of Symposia in Pure Mathematics},
%   volume = {97.1}, pages = {309--348},
%   publisher = {American Mathematical Society}, year = {2018},
%   doi = {10.1090/pspum/097.1/01677},
%   url = {https://www.claymath.org/wp-content/uploads/2022/03/Hacon-AG2015.pdf},
%   note = {Published Theorem 4.0.1; Theorem 3.0.1 in arXiv version 2}
% }

% @article{Saito1990,
%   author = {Saito, Morihiko},
%   title = {Mixed {Hodge} modules},
%   journal = {Publications of the Research Institute for Mathematical Sciences},
%   year = {1990}, volume = {26}, number = {2}, pages = {221--333},
%   doi = {10.2977/PRIMS/1195171082},
%   url = {https://ems.press/journals/prims/articles/3552}
% }

% @article{Saito2000,
%   author = {Saito, Morihiko},
%   title = {Mixed {Hodge} complexes on algebraic varieties},
%   journal = {Mathematische Annalen},
%   year = {2000}, volume = {316}, number = {2}, pages = {283--331},
%   doi = {10.1007/s002080050014},
%   eprint = {math/9906088}, archivePrefix = {arXiv},
%   url = {https://arxiv.org/abs/math/9906088v1},
%   note = {Theorem 0.2, Corollary 0.3 and Proposition 5.3 in arXiv:math/9906088v1}
% }

% @incollection{Coste2007,
%   author = {Coste, Michel},
%   title = {Real Algebraic Sets},
%   booktitle = {Arc spaces and additive invariants in real algebraic and analytic geometry},
%   series = {Panoramas et Synth{\`e}ses},
%   volume = {24},
%   publisher = {Soci{\'e}t{\'e} Math{\'e}matique de France},
%   year = {2007},
%   pages = {1--32},
%   note = {Theorem 1.10 in the author's March 23, 2005 notes},
%   url = {https://indico.ictp.it/event/a04204/session/10/contribution/7/material/0/0.pdf}
% }

% @book{SGA1,
%   author = {Grothendieck, Alexander and others},
%   title = {Rev{\^e}tements {\'e}tales et groupe fondamental ({SGA} 1)},
%   series = {Lecture Notes in Mathematics},
%   volume = {224},
%   publisher = {Springer-Verlag},
%   year = {1971},
%   eprint = {math/0206203v2},
%   archivePrefix = {arXiv},
%   url = {https://arxiv.org/abs/math/0206203v2},
%   note = {Expos{\'e} X, Theorem 3.1, and Expos{\'e} XII, Theorem 5.1}
% }

% @article{MustataPopa2022,
%   author = {Musta{\c t}{\u a}, Mircea and Popa, Mihnea},
%   title = {{Hodge} filtration on local cohomology, {Du Bois} complex and local cohomological dimension},
%   journal = {Forum of Mathematics, Pi},
%   year = {2022}, volume = {10},
%   doi = {10.1017/fmp.2022.15},
%   note = {Paper No. e22; Theorem C and Proposition 13.1},
%   url = {https://people.math.harvard.edu/~mpopa/papers/Filtration_LC_Pi.pdf}
% }

% @article{KollarKovacs2010,
%   author = {Koll{\'a}r, J{\'a}nos and Kov{\'a}cs, S{\'a}ndor J.},
%   title = {Log canonical singularities are {Du Bois}},
%   journal = {Journal of the American Mathematical Society},
%   year = {2010}, volume = {23}, number = {3}, pages = {791--813},
%   doi = {10.1090/S0894-0347-10-00663-6},
%   url = {https://sites.math.washington.edu/~kovacs/current/papers/Kollar_Kovacs__Log_canonical_singularities_are_Du_Bois.pdf}
% }

% @incollection{BauerEtAl2002,
%   author = {Bauer, Thomas and Campana, Fr{\'e}d{\'e}ric and Eckl, Thomas and Kebekus, Stefan and Peternell, Thomas and Rams, S{\l}awomir and Szemberg, Tomasz and Wotzlaw, Lorenz},
%   title = {A reduction map for nef line bundles},
%   booktitle = {Complex Geometry: Collection of Papers Dedicated to {Hans Grauert}},
%   publisher = {Springer}, address = {Berlin},
%   year = {2002}, pages = {27--36},
%   doi = {10.1007/978-3-642-56202-0_2},
%   url = {https://arxiv.org/abs/math/0106147v1},
%   note = {Theorem 2.1 in arXiv:math/0106147v1}
% }

% @misc{StacksExistence,
%   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}},
%   note = {Tag 087V, Lemmas 30.24.1 and 30.24.3; accessed September 26, 2026}
% }

% @article{Taji2014,
%   author = {Taji, Behrouz},
%   title = {Birational positivity in dimension $4$},
%   journal = {Annales de l'Institut Fourier},
%   year = {2014}, volume = {64}, number = {1}, pages = {203--216},
%   doi = {10.5802/aif.2845},
%   url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2845/},
%   note = {With an appendix by Fr{\'e}d{\'e}ric Campana}
% }

% @article{Hanamura1988,
%   author = {Hanamura, Masaki},
%   title = {Structure of birational automorphism groups, {I}: non-uniruled varieties},
%   journal = {Inventiones Mathematicae},
%   year = {1988}, volume = {93}, pages = {383--403},
%   doi = {10.1007/BF01394338},
%   note = {Theorems 2.1--2.2; statements also reproduced in Blanc, Theorems 3.8--3.9}
% }

% @incollection{Blanc2017,
%   author = {Blanc, J{\'e}r{\'e}my},
%   title = {Algebraic structures of groups of birational transformations},
%   booktitle = {Algebraic Groups: Structure and Actions},
%   series = {Proceedings of Symposia in Pure Mathematics},
%   volume = {94}, year = {2017}, pages = {17--30},
%   publisher = {American Mathematical Society},
%   doi = {10.1090/pspum/094/01620},
%   url = {https://algebra.dmi.unibas.ch/blanc/articles/structurebir.pdf},
%   note = {Theorems 3.8--3.9}
% }

% @article{Siu1974,
%   author = {Siu, Yum-Tong},
%   title = {Analyticity of sets associated to {Lelong} numbers and the extension of closed positive currents},
%   journal = {Inventiones Mathematicae},
%   year = {1974}, volume = {27}, pages = {53--156},
%   doi = {10.1007/BF01389965}
% }

% @book{Demailly2012,
%   author = {Demailly, Jean-Pierre},
%   title = {Analytic Methods in Algebraic Geometry},
%   series = {Surveys of Modern Mathematics},
%   volume = {1}, year = {2012},
%   publisher = {International Press and Higher Education Press},
%   address = {Somerville, MA, and Beijing},
%   isbn = {978-1-57146-234-3},
%   url = {https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/analmeth_book.pdf},
%   note = {Theorem numbering refers to the author manuscript dated July 2011}
% }

% @article{MiyaokaMori1986,
%   author = {Miyaoka, Yoichi and Mori, Shigefumi},
%   title = {A numerical criterion for uniruledness},
%   journal = {Annals of Mathematics},
%   year = {1986}, volume = {124}, number = {1}, pages = {65--69},
%   doi = {10.2307/1971387},
%   url = {https://annals.math.princeton.edu/1986/124/p03}
% }

% @article{Paun2007,
%   author = {P{\u a}un, Mihai},
%   title = {{Siu}'s invariance of plurigenera: a one-tower proof},
%   journal = {Journal of Differential Geometry},
%   year = {2007}, volume = {76}, number = {3}, pages = {485--493},
%   url = {https://archive.ymsc.tsinghua.edu.cn/pacm_download/21/685-JDG-76-3-485-493.pdf},
%   note = {Theorem 1}
% }

% @article{Miyaoka1988,
%   author = {Miyaoka, Yoichi},
%   title = {Abundance conjecture for 3-folds: case {$\nu=1$}},
%   journal = {Compositio Mathematica},
%   volume = {68}, number = {2}, year = {1988}, pages = {203--220},
%   url = {https://www.numdam.org/item/CM_1988__68_2_203_0/}
% }

% @article{Kawamata1992,
%   author = {Kawamata, Yujiro},
%   title = {Abundance theorem for minimal threefolds},
%   journal = {Inventiones Mathematicae},
%   volume = {108}, year = {1992}, pages = {229--246},
%   doi = {10.1007/BF02100604},
%   url = {https://link.springer.com/article/10.1007/BF02100604}
% }

% @article{KeelMatsukiMcKernan1994,
%   author = {Keel, Sean and Matsuki, Kenji and McKernan, James},
%   title = {Log abundance theorem for threefolds},
%   journal = {Duke Mathematical Journal},
%   volume = {75}, number = {1}, year = {1994}, pages = {99--119},
%   doi = {10.1215/S0012-7094-94-07504-2},
%   note = {Read together with the authors' 2004 corrections}
% }

% @article{KeelMatsukiMcKernan2004Correction,
%   author = {Keel, Sean and Matsuki, Kenji and McKernan, James},
%   title = {Corrections to ``Log abundance theorem for threefolds''},
%   journal = {Duke Mathematical Journal},
%   volume = {122}, number = {3}, year = {2004}, pages = {625--630},
%   doi = {10.1215/S0012-7094-04-12236-5},
%   url = {https://doi.org/10.1215/S0012-7094-04-12236-5}
% }

% @misc{Matsuki2003Correction,
%   author = {Matsuki, Kenji},
%   title = {A correction to the paper ``Log abundance theorem for threefolds''},
%   year = {2003},
%   eprint = {math/0302360v1}, archivePrefix = {arXiv}, primaryClass = {math.AG},
%   url = {https://arxiv.org/abs/math/0302360v1},
%   note = {arXiv:math/0302360v1}
% }

% @article{Fujino2000,
%   author = {Fujino, Osamu},
%   title = {Abundance theorem for semi log canonical threefolds},
%   journal = {Duke Mathematical Journal},
%   volume = {102}, number = {3}, year = {2000}, pages = {513--532},
%   doi = {10.1215/S0012-7094-00-10237-2},
%   url = {https://www.math.kyoto-u.ac.jp/~fujino/Abundance.pdf}
% }

% @article{Gongyo2013,
%   author = {Gongyo, Yoshinori},
%   title = {Abundance theorem for numerically trivial log canonical divisors of semi-log canonical pairs},
%   journal = {Journal of Algebraic Geometry},
%   volume = {22}, number = {3}, year = {2013}, pages = {549--564},
%   doi = {10.1090/S1056-3911-2012-00593-1},
%   eprint = {1005.2796v2}, archivePrefix = {arXiv}, primaryClass = {math.AG},
%   url = {https://arxiv.org/abs/1005.2796v2}
% }

% @article{DemaillyHaconPaun2013,
%   author = {Demailly, Jean-Pierre and Hacon, Christopher D. and P{\u a}un, Mihai},
%   title = {Extension theorems, non-vanishing and the existence of good minimal models},
%   journal = {Acta Mathematica},
%   volume = {210}, number = {2}, year = {2013}, pages = {203--259},
%   doi = {10.1007/s11511-013-0094-x},
%   url = {https://www-fourier.univ-grenoble-alpes.fr/~demailly/source_files/dhp/dhp_final_corr.pdf}
% }

% @book{EsnaultViehweg1992,
%   author = {Esnault, H{\'e}l{\`e}ne and Viehweg, Eckart},
%   title = {Lectures on Vanishing Theorems},
%   series = {DMV Seminar}, volume = {20}, year = {1992},
%   publisher = {Birkh{\"a}user}, address = {Basel},
%   doi = {10.1007/978-3-0348-8600-0},
%   url = {https://page.mi.fu-berlin.de/esnault/books/esvibuch.pdf}
% }

% @article{Deligne1971,
%   author = {Deligne, Pierre},
%   title = {Th{\'e}orie de {Hodge}. {II}},
%   journal = {Publications Math{\'e}matiques de l'IH{\'E}S},
%   volume = {40}, year = {1971}, pages = {5--57},
%   doi = {10.1007/BF02684692},
%   url = {https://www.numdam.org/item/10.1007/BF02684692.pdf}
% }

% @misc{OpenAIMinimalModels2026,
%   author = {{OpenAI}},
%   title = {{Minimal models and Mori fibre spaces for generalized log canonical $\mathbb{Q}$-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}},
%   year = {2026},
%   note = {Corollary 1.2}
% }

% @misc{LiuXu2025,
%   author = {Liu, Jihao and Xu, Zheng},
%   title = {Non-vanishing implies numerical dimension one abundance},
%   year = {2025},
%   eprint = {2505.05250v2}, archivePrefix = {arXiv}, primaryClass = {math.AG},
%   url = {https://arxiv.org/abs/2505.05250v2},
%   note = {arXiv:2505.05250v2, July 30, 2025; Theorems 3.1, 5.1 and 5.4}
% }

% @article{KitadaiSumihiro2008,
%   author = {Kitadai, Yukinori and Sumihiro, Hideyasu},
%   title = {Canonical filtrations and stability of direct images by {Frobenius} morphisms},
%   journal = {Tohoku Mathematical Journal},
%   series = {2}, volume = {60}, number = {2}, year = {2008},
%   pages = {287--301}, doi = {10.2748/tmj/1215442876},
%   url = {https://doi.org/10.2748/tmj/1215442876},
%   note = {Definition 3.1, Remark 3.2, and Corollary 3.5}
% }

% @article{MustataSchwede2014,
%   author = {Musta{\c t}{\u a}, Mircea and Schwede, Karl},
%   title = {A {Frobenius} variant of {Seshadri} constants},
%   journal = {Mathematische Annalen},
%   volume = {358}, number = {3-4}, year = {2014}, pages = {861--878},
%   doi = {10.1007/s00208-013-0976-4},
%   eprint = {1203.1081v1}, archivePrefix = {arXiv}, primaryClass = {math.AG},
%   url = {https://arxiv.org/abs/1203.1081v1},
%   note = {Section 2 and Proposition 2.12 in arXiv:1203.1081v1}
% }

% @article{JoshiRamananXiaYu2006,
%   author = {Joshi, Kirti and Ramanan, S. and Xia, Eugene Z. and Yu, Jiu-Kang},
%   title = {On vector bundles destabilized by {Frobenius} pull-back},
%   journal = {Compositio Mathematica},
%   volume = {142}, number = {3}, year = {2006}, pages = {616--630},
%   doi = {10.1112/S0010437X05001788},
%   url = {https://doi.org/10.1112/S0010437X05001788},
%   note = {Section 5.3}
% }

% @misc{ChanChoi2021,
%  author = {Chan, Tsz On Mario and Choi, Young-Jun},
%  title = {Extension with log-canonical measures and an improvement to the plt extension of {Demailly--Hacon--P\u{a}un}},
%  year = {2021}, eprint = {1912.08076v5}, archivePrefix = {arXiv},
%  url = {https://arxiv.org/abs/1912.08076v5},
%  doi = {10.1007/s00208-021-02152-3},
%  note = {arXiv:1912.08076v5; Theorem 1.6.1}
% }


% @book{Kollar2013,
%   author = {Koll{\'a}r, J{\'a}nos},
%   title = {Singularities of the Minimal Model Program},
%   series = {Cambridge Tracts in Mathematics}, volume = {200},
%   publisher = {Cambridge University Press}, address = {Cambridge}, year = {2013},
%   note = {With the collaboration of S{\'a}ndor Kov{\'a}cs}
% }

% @article{MehtaRamanathan1982,
%   author = {Mehta, V. B. and Ramanathan, A.},
%   title = {Semistable sheaves on projective varieties and their restriction to curves},
%   journal = {Mathematische Annalen}, volume = {258}, year = {1982},
%   pages = {213--224}, note = {Theorem 6.1 and Remark 6.2}
% }

% @misc{StacksSectionModules,
%   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}},
%   note = {Tag 01YS, Lemma 30.14.1(5); accessed September 27, 2026}
% }

% @misc{FujinoRelativeAnalytic2025,
%   author = {Fujino, Osamu},
%   title = {On finiteness of relative log pluricanonical representations},
%   year = {2025},
%   howpublished = {Author manuscript},
%   url = {https://www.math.kyoto-u.ac.jp/~fujino/analytic-abundance7.pdf},
%   note = {Version 0.14, May 29, 2025; Theorem 1.10 and its proof on p. 30}
% }

% @misc{FujinoAnalyticMMP2022,
%   author = {Fujino, Osamu},
%   title = {Minimal model program for projective morphisms between complex analytic spaces},
%   year = {2022},
%   eprint = {2201.11315v1}, archivePrefix = {arXiv}, primaryClass = {math.AG},
%   url = {https://arxiv.org/abs/2201.11315v1},
%   note = {Definitions 2.28, 2.32 and 2.48}
% }

% @misc{CaoPaunAbelian2016,
%   author = {Cao, Junyan and P{\u a}un, Mihai},
%   title = {{Kodaira} dimension of algebraic fiber spaces over abelian varieties},
%   year = {2016},
%   eprint = {1504.01095v3},
%   archivePrefix = {arXiv},
%   url = {https://arxiv.org/abs/1504.01095v3},
%   note = {Version 3; Theorem 1.1 gives the abelian-base case}
% }

% @misc{HaconPopaSchnell2017,
%   author = {Hacon, Christopher D. and Popa, Mihnea and Schnell, Christian},
%   title = {Algebraic fiber spaces over abelian varieties: around a recent theorem by {Cao} and {P{\u a}un}},
%   year = {2017},
%   eprint = {1611.08768v2},
%   archivePrefix = {arXiv},
%   url = {https://arxiv.org/abs/1611.08768v2},
%   note = {Version 2; Theorem 1.1}
% }
% \end{filecontents*}