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\title[Numerical semiampleness of nef adjoint divisors]{Numerical Semiampleness of Nef Adjoint Divisors}
\subjclass[2020]{Primary 14E30; Secondary 14C20, 14J32}
\keywords{generalised abundance, numerical semiampleness, nef divisor, minimal model, Frobenius morphism}
\begin{document}
\begin{abstract}
We prove the Generalised Abundance Conjecture: if $(X,B)$ is a projective
klt $\Q$-pair over an algebraically closed field of characteristic zero,
$K_X+B$ is pseudo-effective, $M$ is a nef $\Q$-Cartier divisor on $X$,
and $K_X+B+M$ is nef, then $K_X+B+M$ is numerically equivalent to a
semiample $\Q$-Cartier divisor on $X$.
\end{abstract}
\maketitle
\section{Introduction}
\label{sec:introduction}
A divisor $D$ on a projective variety is \emph{nef} if $D\cdot C\ge 0$
for every integral curve $C$, and \emph{semiample} if some positive
multiple $mD$ is globally generated, so that $mD$ defines a morphism to
projective space. Nefness is numerical and semiampleness is geometric;
passing from one to the other is the central concern of the minimal
model program.
For the log canonical divisor $K_X+B$ of a log canonical pair, the
relevant statement is the abundance conjecture, now a theorem in
characteristic zero: a nef $K_X+B$ is semiample. We take it as an
input. Adding an independent nef summand $M$ changes what can be
expected: even if $D=K_X+B+M$ is nef, $D$ itself need not be semiample,
and one is forced to require only a numerical semiample representative.
\begin{definition}\label{def:numsemiample}
A rational Cartier divisor $D$ on a projective variety $X$ is
\emph{numerically semiample} if it is numerically equivalent to a
semiample rational Cartier divisor on the same $X$. Two rational
Cartier divisors are numerically equivalent, written $\equiv$, if their
difference meets every integral curve in zero.
\end{definition}
\begin{theorem}[Generalised Abundance; {\cite{LP}}]\label{thm:main}
Let $(X,B)$ be a projective klt $\Q$-pair over an algebraically closed
field of characteristic zero. Suppose that $K_X+B$ is pseudo-effective,
and let $M$ be a nef $\Q$-Cartier divisor on $X$. If $D=K_X+B+M$ is
nef, then there exists a semiample $\Q$-Cartier divisor $L$ on $X$ with
$D\equiv L$.
\end{theorem}
Both the boundary $B$ and the nef summand $M$ sit on $X$ itself. The
proof uses two named inputs: the log-abundance theorem of
\cite[Theorem~1.1]{LA}, and the terminating minimal-model programs of
\cite[Theorem~1.1 and Section~3.3]{MM}. Both are stated precisely in
Section~\ref{sec:preliminaries}.
\paragraph{Why numerical equivalence is necessary.} Let $E$ be a
complex elliptic curve and let $L_\alpha$ be a nontorsion line bundle
of degree zero on $E$. Then $L_\alpha$ is nef and numerically
trivial, but no positive multiple has a nonzero section. Since
$K_E\sim0$, taking $B=0$ and $M=L_\alpha$ in Theorem~\ref{thm:main}
gives a valid hypothesis in which $K_E+M$ is nef and numerically
trivial; the conclusion only asks for a semiample divisor numerically
equivalent to $K_E+M$, which the zero divisor provides. Linear
equivalence would force a section, and there is none.
\paragraph{The ordinary adjoint case.} For $M=0$, the conclusion of
Theorem~\ref{thm:main} is equivalent to ordinary klt abundance. One
direction is clear. For the other, Fukuda \cite[Theorem~0.1]{Fukuda}
proves that if $K_X+B$ is numerically equivalent to a semiample
divisor, then $K_X+B$ itself is semiample. Hence Theorem~\ref{thm:main}
with $M=0$ recovers ordinary klt abundance, which is an input to the
present proof via \cite{LA}.
\begin{corollary}\label{cor:cy}
Let $(X,B)$ be a projective klt $\Q$-pair over $\C$ with $K_X+B\equiv 0$.
Then every nef $\Q$-Cartier divisor on $X$ is numerically semiample.
\end{corollary}
\begin{proof}
For nef $M$, the divisor $K_X+B+M\equiv M$ is nef. Apply
Theorem~\ref{thm:main}.
\end{proof}
\paragraph{The key new ingredient.} The heart of the proof reduces,
step by step, to the following positivity statement about Calabi--Yau
pairs.
\begin{theorem}[Full nef dimension forces bigness]\label{thm:key-ingredient}
Let $(X,B)$ be a projective klt Calabi--Yau $\Q$-pair of dimension $n$
over $\C$: that is, $K_X+B\equiv0$. Let $M$ be a nef $\Q$-Cartier
divisor on $X$ with full nef dimension, meaning every curve through a
very general point of $X$ has positive $M$-degree. Then $M$ is big.
\end{theorem}
Recall that $M$ is \emph{big} if the top self-intersection $M^n$ is
positive, equivalently if the linear systems $|kM|$ grow like $k^n$ for
large $k$. The gap between strict positivity along very general curves
and positive volume is exactly where the geometry of the paper lives.
Sections~\ref{sec:jet-tools} through \ref{sec:completion} are devoted to
the proof of Theorem~\ref{thm:key-ingredient} for every $n\ge 2$.
\paragraph{Shape of the proof.} Two simultaneous inductions on the
dimension $n=\dim X$ organise the argument.
\begin{enumerate}
\item[(P$_n$)] Suppose $T=K_X+B$ is pseudo-effective, $M$ is a nef
$\Q$-Cartier divisor on a projective klt $\Q$-pair $(X,B)$ of dimension
$n$, and there is a rational $a>0$ such that $(T+aM)\cdot C>0$ for
every integral curve $C$ through a very general point of $X$. Then
$T+cM$ is big for every rational $c>0$.
\item[(Z$_n$)] Under the same hypotheses, without the
very-general-curve condition, there is a projective birational model
$w\colon W\to X$ from a smooth projective $W$ such that the positive
part $P_\sigma(w^*(T+M))$ of the divisorial Zariski decomposition is a
rational divisor numerically equivalent to a semiample $\Q$-divisor on
$W$.
\end{enumerate}
Statement (P$_n$) produces bigness and (Z$_n$) produces numerical
semiampleness; both are proved by induction on $n$. The case $n=0$ is
trivial. For $n=1$: on a curve, pseudo-effective means nonnegative
degree and nef means nonnegative degree, so (P$_n$) makes every
positive combination ample, and (Z$_n$) makes every nef divisor
numerically semiample.
For the inductive step, three cases arise.
\begin{itemize}
\item \emph{Positive Iitaka dimension.} If $\kappa(T)>0$, the
semiample adjoint on a suitable birational model has a
positive-dimensional base and the induction hypothesis applies to the
fibres of its contraction. This is Proposition~\ref{prop:full-positive}.
\item \emph{Reduction to Calabi--Yau.} A compatible pair of
minimal-model programs replaces $(X,B)$ by a model $(Y,B_Y)$ on which
the ordinary adjoint $U=K_Y+B_Y$ is semiample. If $\kappa(T)=0$ then
$U\sim_\Q 0$, so $(Y,B_Y)$ is a klt Calabi--Yau pair and $(P_n)$
reduces to Theorem~\ref{thm:key-ingredient}. This is
Proposition~\ref{prop:reduction-cy}.
\item \emph{The full-nef-dimension step.} Theorem~\ref{thm:key-ingredient}
is the technical core. Its proof uses two incompatible jet estimates,
produced in Section~\ref{sec:geometry} and made to conflict in
Section~\ref{sec:frobenius}.
\end{itemize}
\paragraph{Two incompatible jet estimates.} Assume, towards a
contradiction, that $X$ is projective $\Q$-factorial and terminal with
$K_X\sim0$, and that $L$ is a nef Cartier divisor of full nef dimension
that is not big. Since $L$ is nef and not big, $L^n=0$.
Fix an ample Cartier divisor $A$ and let $t\to 0$ through positive
rationals. Let $\epsilon>0$ be small enough that $(14\epsilon)^n<1/4$
and $80\epsilon<3/4$. For suitable integers $r=r(t)\to\infty$ the
divisor $P=r(L+tA)$ is ample rational Cartier with $P^n\to\epsilon^n$;
for small $t$ the volume window
$\epsilon^n/2\le P^n\le 2\epsilon^n$ holds. Then:
\begin{itemize}
\item \emph{Small scalar bound.} At a very general point $x\in X$,
every section of every $kP$ with $kP$ Cartier vanishes to order at most
$4k\rho$ at $x$, where $\rho=(P^n)^{1/n}$; this is
Lemma~\ref{lemma:scalar-bound}.
\item \emph{Large jet system.} Over $X\times X$, define the
projective bundle $Z=\PP(\OO_X(L)_1\oplus \OO_X(L)_2)$ with
tautological class $\xi$, and the ample divisor $R=P_1+P_2+q\xi$ for
suitable integer $q>0$. There is a smooth point $z_*$ off the two axes
with Seshadri constant $\varepsilon(R;z_*)>2n+2$; this is
Proposition~\ref{prop:large-seshadri}. A Seshadri constant above
$2n+2$ lets us generate jets at $z_*$ to that order from sections of a
suitable Cartier multiple of $R$, via Serre vanishing and the cohomology
of a point blowup.
\end{itemize}
The proof of the large jet system is the longest part of the paper.
Starting from the scalar bound, together with the moving-kernel
machinery of Section~\ref{sec:jet-tools}, one rules out every possible
failure of the large Seshadri constant. Each failure produces a small
base component of a suitable jet kernel, and each such component — plus
its slices, its swept branch divisors, and its associated
correspondences between finite covers — gives a bounded-degree curve
of the kind forbidden by Lemma~\ref{lemma:bounded-curves}.
\paragraph{The Frobenius comparison.} With the two estimates in hand,
Section~\ref{sec:frobenius} derives a contradiction. A fixed finite
amount of complex data is spread out to an integral finitely generated
$\Z$-algebra of characteristic zero. For all large residual
characteristics $p$, the two jet systems evaluate on the same
determinant on the diagonal of the Frobenius fiber product
$W\times_{W'} W$. The scalar bound makes that determinant vanish at
the diagonal to order at most $80\epsilon+O(1/p)$ in the relevant
normalization, while the diagonal-rank bound from the Frobenius
filtration makes the same order of vanishing at least $3/4$. Since
$80\epsilon<3/4$ and $O(1/p)\to 0$, the two bounds are incompatible for
all large $p$; this is Theorem~\ref{thm:finite-data}, whose proof
occupies Section~\ref{sec:frobenius}.
\paragraph{Extension to klt Calabi--Yau pairs.} Once
Theorem~\ref{thm:key-ingredient} is proved for terminal $X$ with trivial
boundary, Section~\ref{sec:completion} extends it in two steps. First,
a klt $X$ with $K_X\equiv 0$ admits a finite quasi-\'etale index-one
cover $\pi\colon X^{\mathrm{ind}}\to X$ with $K_{X^{\mathrm{ind}}}\sim 0$
and canonical singularities; a crepant terminalization then gives a
terminal $\Q$-factorial $\widetilde X\to X^{\mathrm{ind}}$. Full nef
dimension and bigness both survive the finite quasi-\'etale cover, so
the terminal trivial-boundary case gives bigness of $M$ on $X$. Second,
for an arbitrary effective boundary $B$ with $K_X+B\equiv 0$, a small
perturbation $(X,(1+\alpha)B)$ together with a pair of MMPs either
reduces to the positive-Iitaka case or forces $B_Y=0$; in the latter
case $K_Y\sim_\Q0$ and $M$ remains nef of full nef dimension, and the
terminal trivial-boundary case makes $M$ big.
\paragraph{Positive parts and numerical semiampleness.} Once both
inductions hold, Section~\ref{sec:positive} proves (Z$_n$). First, a
large-nef-adjoint MMP trivial on a multiple of $M$ may be arranged so
that $T+mL$ becomes nef. On the resulting model either $T+mL$ has full
nef dimension, in which case (P$_n$) gives bigness of $T+M$ and
Kodaira's lemma exhibits a semiample positive part; or the nef
reduction of $T+mL$ is nontrivial, in which case $M$ descends to a
lower-dimensional base variety $Z_2$, the induction applies to the
base pair, and the resulting positive part pulls back along the
fibration.
Finally, the positive part on a smooth model descends to the original
variety, because a klt variety has rational singularities so
$\Pic^0$ matches along the pullback of any birational model. A
finite-type spread argument completes the passage from $\C$ to any
algebraically closed field of characteristic zero.
\paragraph{Organization.} Section~\ref{sec:preliminaries} records
conventions and the two input theorems. Section~\ref{sec:reduction}
sets up the two inductions and reduces (P$_n$) to the Calabi--Yau
case. Section~\ref{sec:jet-tools} proves the bounded-degree
obstruction and records the moving-kernel tools from \cite{LA}.
Sections~\ref{sec:geometry} and~\ref{sec:frobenius} prove the terminal
Calabi--Yau case of Theorem~\ref{thm:key-ingredient}, and
Section~\ref{sec:completion} extends it to all klt Calabi--Yau pairs.
Section~\ref{sec:positive} completes the inductions and descends to
$X$; Section~\ref{sec:field} handles general algebraically closed
fields.
\paragraph{Relation to earlier work.} Lazi\'c--Peternell \cite{LP}
formulated generalised abundance as the numerical extension of
abundance and established it on a Calabi--Yau variety under a
Semiampleness Conjecture plus termination and abundance in lower
dimensions. Their Corollaries C and D settle surfaces and threefolds
with $\kappa(K_X+B)>0$ unconditionally. Subsequent work by
Lazi\'c--Peternell \cite{LPII}, Chaudhuri, and Fontanari addressed
numerical nonvanishing, small-dimensional nef reductions, and
semiampleness for surfaces. The strategy here follows
\cite[Section~7]{LP} for the reduction; the new geometric ingredient is
Theorem~\ref{thm:key-ingredient}. The moving-jet and Frobenius tools
build on Musta\c{t}\u{a}--Schwede, Sun, Kitadai--Sumihiro, and
Langer's Frobenius-slope theory; the local-positivity framework
descends from Ein--K\"uchle--Lazarsfeld. The polarized birationality
theorem of Birkar--Zhang turns accumulated numerical bounds into
bounded-degree curves for the moving-jet machinery.
\section{Conventions and standing inputs}
\label{sec:preliminaries}
Throughout, unless stated otherwise, the base field is $\C$. Varieties
are integral and projective. Grading of local inputs is stated at
each appearance.
\subsection{Log pairs and singularities}
A \emph{$\Q$-pair} $(X,B)$ consists of a normal projective variety $X$,
an effective rational boundary $B$, and a $\Q$-Cartier divisor $K_X+B$.
Log discrepancies are defined on a log resolution $h\colon Y\to X$,
via the equality $K_Y+B_Y=h^*(K_X+B)$: a prime divisor $E\subset Y$
with coefficient $b_E$ in $B_Y$ carries log discrepancy $1-b_E$.
The pair is \emph{klt} if all log discrepancies are positive and
\emph{log canonical} if they are nonnegative. Standard cone,
contraction, negativity, and resolution theorems are used as in
\cite{KM,BCHM}.
The symbols $\sim_\Q$ and $\equiv$ denote rational linear and numerical
equivalence. A divisor $D$ is \emph{pseudo-effective} if its class in
$N^1(X)_\R$ lies in the closure of the cone generated by effective
divisors; \emph{big} if its class lies in the interior of that cone;
\emph{nef} if $D\cdot C\ge0$ for every integral curve $C$. A divisor
is \emph{semiample} if some positive multiple is globally generated.
The \emph{Iitaka dimension} $\kappa(D)$ is the maximal dimension of
the image of a map defined by a linear system $|kD|$, for $k$ verifying
$h^0(kD)>0$.
Over $\C$, a \emph{very general} point is a point outside a specified
countable union of proper closed subsets. Most statements below
specify the countable union only implicitly; iterated applications
remain valid because a countable union of countable unions remains
countable.
\subsection{Nef reduction}
For a nef divisor $N$ on a projective variety, the \emph{nef
reduction} \cite{NefReduction} is an almost holomorphic dominant
rational map $\pi\colon X\dashrightarrow Z$ with connected compact
general fibres such that:
\begin{enumerate}
\item $N|_F\equiv0$ on a very general compact fibre $F$;
\item if $C$ is a curve through a very general point of $X$ not
contracted by $\pi$, then $N\cdot C>0$.
\end{enumerate}
The base dimension $\dim Z$ is the \emph{nef dimension} $n(X,N)$. It
depends only on the numerical class of $N$. By construction,
$n(X,N)=\dim X$ if and only if every curve through a very general
point has positive $N$-degree. This is the condition called
\emph{full nef dimension}. Crucially, full nef dimension is purely
numerical: it does not assert or imply bigness. The gap between full
nef dimension and bigness is exactly the gap that
Theorem~\ref{thm:key-ingredient} closes for Calabi--Yau pairs.
\subsection{Divisorial Zariski decomposition}
For a pseudo-effective real divisor $D$ on a smooth projective
variety, and a prime divisor $\Gamma$ on it, fix an ample divisor $A$
and put
\[
\sigma_\Gamma(D)=\lim_{\delta\downarrow0}
\inf\{\mult_\Gamma D':D'\ge0,\ D'\sim_\R D+\delta A\}.
\]
These asymptotic divisorial orders are finite, nonnegative, and
independent of $A$. Only finitely many of them are nonzero, and each
depends only on the numerical class of $D$. The
\emph{divisorial Zariski decomposition} is
\[
N_\sigma(D)=\sum_\Gamma\sigma_\Gamma(D)\Gamma,
\qquad P_\sigma(D)=D-N_\sigma(D).
\]
We call $N_\sigma(D)$ the \emph{negative part} and $P_\sigma(D)$ the
\emph{positive part}. The sum is a finite $\R$-linear combination of
prime divisors; the difference $P_\sigma(D)$ is a pseudo-effective
$\R$-divisor whose negative part is trivial. A divisor $D$ is nef if
and only if $N_\sigma(D)=0$. Otherwise, the negative part captures
exactly the ``asymptotically fixed'' divisorial component of $D$.
\subsection{The very general point lemma}
\begin{lemma}\label{lemma:pseudo-vg}
Let $D$ be a pseudo-effective rational Cartier divisor on a projective
variety $X$ over $\C$. Then every integral curve through a
sufficiently very general point of $X$ has nonnegative $D$-degree.
\end{lemma}
\begin{proof}
Fix an ample Cartier divisor $A$ and rationals $\delta_i>0$ with
$\delta_i\to 0$. Each $D+\delta_iA$ is big, so there is an effective
rational divisor $E_i\sim_\Q D+\delta_iA$. A point outside the
countable union $\bigcup_i\Supp E_i$ is very general, and every curve
$C$ through such a point meets no $\Supp E_i$. Therefore
$(D+\delta_iA)\cdot C\ge0$ for every $i$, and $D\cdot C\ge0$ after
letting $i\to\infty$.
\end{proof}
\subsection{Two input theorems}
We record the two external inputs. Both are theorems in characteristic
zero; both are used as black boxes.
The first input is ordinary log abundance. Recall that a klt variety
has rational singularities, so pullbacks from an arbitrary birational
model do not affect $\Pic^0$; the distinction between a rational
divisor and the line bundle it defines is thus controlled on smooth
models (see Lemma~\ref{lemma:numsemiample-descent}). The log
abundance statement asserts that this subtlety disappears entirely
for nef log canonical adjoints.
\begin{theorem}[Log abundance, {\cite[Theorem~1.1]{LA}}]
\label{input:abundance}
Let $(X,B)$ be a projective log canonical $\Q$-pair over an
algebraically closed field of characteristic zero. If $K_X+B$ is
nef, then it is semiample.
\end{theorem}
The second input provides the minimal-model programs used in
Section~\ref{sec:reduction}, both for ordinary adjoints and for
generalized pairs whose only polarized data is a nef
$\Q$-Cartier $b$-divisor. The statement records a \emph{terminating
choice} of program: at each step there are many possible
$(K_X+B+\mathbf{M})$-negative extremal rays, but the theorem gives a
fixed order in which they have been contracted that terminates after
finitely many steps.
A \emph{nef rational Cartier b-divisor} on a normal projective variety
$X$ is specified by a nef rational Cartier divisor $N'$ on a projective
birational model $Y\to X$; throughout the model $Y$, and on every
higher birational model, $N'$ pulls back unchanged. Generalized
discrepancies are computed by pulling back the generalized adjoint
while keeping $N'$ fixed on all higher models; the generalized pair is
generalized klt when all generalized discrepancies are positive. The
trace of $\mathbf{M}$ on a lower model is its pushforward, which need
not be nef.
\begin{theorem}[Minimal-model programs,
{\cite[Theorem~1.1 and Section~3.3]{MM}}]
\label{input:models}
Let $(X,B+\mathbf M)$ be a projective generalized klt $\Q$-pair over
$\C$, with $X$ $\Q$-factorial and with fixed nef rational Cartier
b-divisor data $\mathbf M$. If $K_X+B+M_X$ is pseudo-effective, there
is a terminating choice of its minimal-model program, with
$\Q$-factorial generalized klt models and a nef endpoint. The program
extracts no divisors and generalized log discrepancies do not decrease.
On a common resolution of its initial and final models, the pullback of
the initial adjoint is the pullback of the final adjoint plus an
effective divisor exceptional over the final model.
\end{theorem}
Two remarks. First, the ordinary klt case of
Theorem~\ref{input:models} arises by taking $\mathbf M=0$; then
Theorem~\ref{input:abundance} makes the endpoint a good minimal model.
Second, neither theorem above asserts abundance for a nonzero
generalized nef summand. Their combination still exposes the
full-nef-dimension obstruction, which is exactly the content of
Theorem~\ref{thm:key-ingredient}.
\section{The two inductions and reduction to Calabi--Yau}
\label{sec:reduction}
We work over $\C$ in this section. The reductions follow the strategy
of Lazi\'c--Peternell \cite[Sections~4--5]{LP}, with the terms made
explicit. The essential point is to retain control of two adjoints at
once: the ordinary adjoint $T=K_X+B$, whose semiampleness is supplied
by abundance, and a nearby generalized nef adjoint $T+sM$ whose
well-behaved full nef dimension lets bigness descend from a
Calabi--Yau model. The two adjoints live on the same birational model
and must remain tractable simultaneously.
\subsection{The two induction statements}
For a projective klt $\Q$-pair $(X,B)$ and a nef $\Q$-Cartier divisor
$M$ on $X$, write $T=K_X+B$. Assume $T$ is pseudo-effective, and
consider the following condition: for some $a\in\Q_{>0}$,
\begin{equation}\label{eq:full-condition}
(T+aM)\cdot C>0
\quad\text{for every integral curve $C$ through a very general point of $X$.}
\end{equation}
Note that $T+aM$ need not be nef; the condition is on the class it
defines, and only at the very general locus.
Lemma~\ref{lemma:pseudo-vg} implies that
\eqref{eq:full-condition} holds whenever $M$ has full nef dimension.
If $T+aM$ is nef, then \eqref{eq:full-condition} is exactly the
assertion that $T+aM$ has full nef dimension.
We prove the following two statements simultaneously by induction on
$n=\dim X$.
\begin{description}
\item[(P$_n$)] Let $(X,B)$ be a projective klt $\Q$-pair of dimension
$n$, let $M$ be a nef $\Q$-Cartier divisor on $X$, let $T=K_X+B$ be
pseudo-effective, and suppose \eqref{eq:full-condition} holds for some
rational $a>0$. Then $T+cM$ is big for every rational $c>0$.
\item[(Z$_n$)] Under the same hypotheses on $(X,B)$, $M$, $T$ but
without \eqref{eq:full-condition}, there is a projective birational
morphism $w\colon W\to X$ from a smooth projective variety such that
$P_\sigma(w^*(T+M))$ is rational and numerically equivalent to a
semiample $\Q$-divisor on $W$.
\end{description}
Both statements hold in dimensions $0$ and $1$. For $n=0$ there is
nothing to prove. For $n=1$, pseudo-effectivity of $T$ and nefness of
$M$ both say their degrees are nonnegative; then
\eqref{eq:full-condition} makes every positive combination have
positive degree, hence ample, and (P$_1$) is immediate. Similarly a
nef divisor on a curve has zero negative part in the divisorial
Zariski decomposition, so it is itself semiample in its own numerical
class; this gives (Z$_1$).
Fix $n\ge2$, and assume (P$_j$) and (Z$_j$) for every $j<n$. We
verify (P$_n$) and (Z$_n$). Along the way we may freely replace $X$
by a small projective $\Q$-factorialization (which pulls back the
adjoint crepantly and the nef divisor $M$ to a nef divisor), and
replace a statement about $X$ by the corresponding statement about a
smooth birational model. Indeed \eqref{eq:full-condition} persists
to the isomorphism locus, bigness descends under the birational
pullback, and a smooth model on which $P_\sigma$ is rational and
numerically semiample descends to $X$ by
Lemma~\ref{lemma:numsemiample-descent}. So we are free to impose
$\Q$-factoriality when convenient.
\subsection{The elementary ray bound for generalized pairs}
We start with the only new cone-theoretic input: a uniformity statement
for negative extremal rays of generalized klt pairs. Recall that a
generalized klt pair $(V,\Delta+\mathbf N)$ has boundary $\Delta$ and
polarized data given by a fixed nef rational Cartier b-divisor
$\mathbf N$ determined on some projective birational model of $V$.
The notation $N_V$ denotes the trace of $\mathbf N$ on $V$; this trace
need not be nef, only the polarized data on the fixing model is.
\begin{lemma}\label{lemma:generalized-ray-bound}
Let $(V,\Delta+\mathbf N)$ be a projective generalized klt $\Q$-pair
of dimension $n$, with effective boundary, $\Q$-factorial $V$, and
nef rational Cartier b-divisor data. Every
$(K_V+\Delta+N_V)$-negative extremal ray has an integral rational curve
generator $C$ with
\[
-4n\le (K_V+\Delta+N_V)\cdot C < 0.
\]
The contraction, and the flip when it is small, can be realized using
an ordinary klt adjoint negative on the same ray. Birational
minimal-model outputs remain $\Q$-factorial. On a common resolution
of a birational step, the pullback of the input generalized adjoint
equals the pullback of its output transform plus an effective divisor
exceptional over the output.
\end{lemma}
The constant $4n$ is twice the usual bound $2n$ for an ordinary klt
pair. The factor $2$ arises because the argument passes through a
small perturbation of $G=K_V+\Delta+N_V$ by an ample divisor to reach
a genuine klt adjoint; the perturbation inflates the negativity of
$G$ along the chosen ray by at most a factor of two.
\begin{proof}
Set $G=K_V+\Delta+N_V$ and fix a $G$-negative extremal ray generated
by $R$. Choose an ample rational divisor $A$ on $V$ and a rational
$\eta>0$ small enough that $G+\eta A$ is still negative on $R$, and
such that
$-(G+\eta A)\cdot\gamma\ge-\tfrac12 G\cdot\gamma$ for every nonzero
class $\gamma$ on $R$.
We need to represent $G+\eta A$ as an ordinary klt adjoint. Because
the b-divisor $\mathbf N$ may not descend to $V$, work upstairs.
Choose a projective log resolution $h\colon V'\to V$ carrying
$\mathbf N$ as a locally free sheaf, and write
\[
K_{V'}+\Delta'+N'=h^*G
\]
with coefficients of $\Delta'$ strictly less than one; this is
generalized klt. Choose $V'$ with an effective exceptional $F$ such
that $-F$ is $h$-ample. For small $\varepsilon>0$ the divisor
$N'+\eta h^*A-\varepsilon F$ is ample (being the sum of an ample and a
nef, adjusted by an $h$-ample), and a general effective rational
representative of this ample class can be chosen with coefficients
small enough that, after adding $\varepsilon F$, it preserves the sub-klt
condition on $\Delta'$. Pushing that representative to $V$ gives an
effective rational divisor $\Theta$ on $V$ with
\[
K_V+\Theta \sim_\Q G+\eta A.
\]
The crepant equality is checked upstairs: its difference is
exceptional and relatively numerically trivial, hence zero by the
negativity lemma.
The ordinary cone theorem for the klt pair $(V,\Theta)$ gives a
rational curve generator $C$ of the ray with
\[
0<-(G+\eta A)\cdot C\le2n.
\]
The chain
$-(G+\eta A)\cdot C\ge-\tfrac12 G\cdot C$
gives $-G\cdot C\le 4n$.
The contraction $c\colon V\to S$ of the ray for $G$ is the contraction
of the same ray for the ordinary klt pair $(V,\Theta)$, by
cone-contraction uniqueness, and the flip behaves identically: since
$\Pic(V/S)$ is one-dimensional, there is a rational $\lambda>0$ with
$G-\lambda(K_V+\Theta)$ numerically trivial over $S$, and Cartier
descent for the ordinary pair gives a pullback divisor from $S$ in
the appropriate class; on the flipped model $V^+$, the same
observation shows $G^+$ is ample over $S$. The $\Q$-factoriality of
$V^+$ then follows from the ordinary flip, and the effective
exceptional comparison of $G$ is the sum of its ordinary counterpart
with $\lambda$, since the $S$-pullback parts cancel.
This proves the lemma.
\end{proof}
\subsection{Keeping the ordinary adjoint semiample}
The heart of Section~\ref{sec:reduction} is a pair of MMPs organised
so that the ordinary adjoint becomes a pullback of an ample divisor
on a lower-dimensional base, while a small multiple of the nef summand
stays tractable on the same model.
\begin{lemma}\label{lemma:two-nef-model}
Let $(X,B)$ be a projective $\Q$-factorial klt $\Q$-pair of dimension
$n$, let $T=K_X+B$ be pseudo-effective, and let $M$ be a nef
$\Q$-Cartier divisor on $X$. Suppose \eqref{eq:full-condition} holds.
Then there are a birational contraction $X\dashrightarrow Y$, a number
$s_0\in\Q_{>0}$, and a common smooth projective resolution
\[
X\xleftarrow{p}W\xrightarrow{q}Y
\]
with the following properties. If $B_Y$, $N$ denote the transforms of
$B$ and $M$, and $U=K_Y+B_Y$, then:
\begin{enumerate}
\item $(Y,B_Y)$ is klt and $\Q$-factorial, $U$ is semiample, and
$\kappa(Y,U)=\kappa(X,T)\ge0$.
\item $N$ is pseudo-effective, and $U+s_0N$ is nef of full nef
dimension.
\item There are effective $q$-exceptional rational divisors $E_0,E_1$
on $W$ with
\begin{equation}\label{eq:two-comparisons}
p^*T=q^*U+E_0,
\qquad
p^*(T+s_0M)=q^*(U+s_0N)+E_1.
\end{equation}
\end{enumerate}
\end{lemma}
The two comparisons of \eqref{eq:two-comparisons} are the effective
exceptional adjoint comparisons of the two minimal-model programs; the
constant $s_0$ is chosen so that the second program's rays are already
negative for the first program's adjoint at coefficient $s$, and $s_0$
is small enough that a proportionality argument with a free Cartier
multiple $dU_0$ rules out any positivity of $U_0$ along such a ray.
\begin{proof}
Apply Theorem~\ref{input:models} to the ordinary klt pair $(X,B)$
(i.e.\ with $\mathbf M=0$) and pseudo-effective adjoint $T$. The
output is a terminating $T$-MMP ending on a $\Q$-factorial klt pair
$(Y_0,B_0)$ with nef ordinary adjoint $U_0=K_{Y_0}+B_0$.
Theorem~\ref{input:abundance} makes $U_0$ semiample. The
comparisons along the finitely many birational steps give
$\kappa(X,T)=\kappa(Y_0,U_0)\ge0$.
Let $N_0$ be the transform of $M$ on $Y_0$. Because the first
program has finitely many steps, there is a rational $s>0$ such that
every step is still negative for the transform of $T+uM$ for every
$u\in(0,s]$, and such that the strict sign on the flipped sides is
preserved. Hence these steps are generalized klt steps for the pairs
$(X,B+u\mathbf M)$, so $(Y_0,B_0+u\mathbf M)$ is a generalized klt
pair for $u\in(0,s]$. In particular its trace
$(Y_0,B_0+s\mathbf M)$ is a generalized klt pair with polarized data
$s\mathbf M$ contained in the fixed b-divisor. Pushforward under a
birational contraction to a $\Q$-factorial target preserves
pseudo-effectivity: on a common resolution, an upstairs divisor
numerically trivial implies it is the pullback of its pushforward,
thanks to the negativity lemma, and effective approximation then
proves the claim for classes. Thus $N_0$ is pseudo-effective.
Choose a positive integer $d$ such that $dU_0$ is Cartier and
globally generated. Pick $s_0\in(0,s)$ rational with
\begin{equation}\label{eq:small-nef-coefficient}
\frac{1-s_0/s}{d}>4n\frac{s_0}{s}.
\end{equation}
The generalized adjoint $U_0+s_0N_0$ is pseudo-effective.
Theorem~\ref{input:models} gives a terminating program for this
generalized adjoint. We claim that every step of this program is
$U_0$-trivial and is also negative for the generalized adjoint with
coefficient $s$, where throughout the claim the symbols denote their
transforms.
Suppose inductively that $U_i$ is Cartier and free, and that the
generalized pair with coefficient $s$ is generalized klt. For a
$(U_i+s_0N_i)$-negative ray, the decomposition
\[
U_i+s_0N_i=(1-s_0/s)U_i+(s_0/s)(U_i+sN_i)
\]
together with nefness of $U_i$ shows the ray is
$(U_i+sN_i)$-negative. Pick its rational curve generator $C$ from
Lemma~\ref{lemma:generalized-ray-bound}. If $U_i\cdot C>0$,
integrality of the free $dU_i$ gives $U_i\cdot C\ge1/d$, so
\[
(U_i+s_0N_i)\cdot C
\ge\frac{1-s_0/s}{d}-4n\frac{s_0}{s}>0
\]
by \eqref{eq:small-nef-coefficient}; contradiction. Thus
$U_i\cdot C=0$.
The morphism defined by the free $dU_i$ is constant on each fibre of
the extremal contraction, so it descends to the base; pulling back to
the flipped model gives the transform of $dU_i$, again Cartier and
free. Negativity for the generalized adjoint with coefficient $s$
persists, so the generalized klt condition on the pair $(B_i+s\mathbf
M)$ continues to hold. The induction closes on the finite program.
Let $Y$ be its output. Because each step is $U_0$-trivial,
$U=U_0$ is pulled back unchanged, hence semiample with the same
Iitaka dimension $\kappa(X,T)$. The transform of the generalized
adjoint is $U+s_0N$, which is nef by the construction of the program.
Combining the pullback comparisons of the two programs gives
\eqref{eq:two-comparisons}.
It remains to prove that $U+s_0N$ has full nef dimension. Suppose
toward a contradiction that its nef reduction has a positive-dimensional
very general compact fibre $F$. Shrink the base so that these compact
fibres lie inside the morphism locus of the almost holomorphic map.
The image in $Y$ of the exceptional locus of $q$ has codimension
$\ge2$; a very general fibre $F$ therefore misses it in codimension
$\ge2$ at every component, by the fibre dimension theorem applied to
the components that dominate the base. We can pick a curve $C\subset
F$ passing through a point chosen with the very general properties of
\eqref{eq:full-condition}: outside the exceptional image, and in the
common isomorphism locus of the first program. If $\dim F>1$, take
$C$ to be a sufficiently general complete intersection in $F$
through that point; if $\dim F=1$, take $C=F$ itself.
Because $U+s_0N$ is nef with positive-dimensional nef reduction and
$U$ is semiample, both $U$ and $N$ have degree zero on $C$: this
follows from Lemma~\ref{lemma:pseudo-vg} applied to each, since $U$ is
nef and $N$ pseudo-effective and their positive combination gives a
nonnegative, hence zero, degree along $F$. The strict lift
$\widetilde C\subset W$ avoids the effective exceptional divisors
$E_0,E_1$. Both equalities in \eqref{eq:two-comparisons} then give
\[
p^*T\cdot\widetilde C=p^*M\cdot\widetilde C=0.
\]
The image $p(\widetilde C)$ passes through the chosen very general
point of $X$. Hence $(T+aM)\cdot p(\widetilde C)>0$ by
\eqref{eq:full-condition}, but the two equalities above give
$(T+aM)\cdot p(\widetilde C)=0$.
\end{proof}
\subsection{Positive Iitaka dimension}
\begin{proposition}\label{prop:full-positive}
Assume (P$_j$) for every $j<n$. Let $(X,B)$ be a projective klt
$\Q$-pair of dimension $n$, let $T=K_X+B$ be pseudo-effective, and
let $M$ be a nef $\Q$-Cartier divisor on $X$. Suppose
\eqref{eq:full-condition} holds for some rational $a>0$, and that
$\kappa(X,T)>0$. Then $T+cM$ is big for every rational $c>0$.
\end{proposition}
The argument reduces to the fibres of the Iitaka fibration of the
ordinary semiample adjoint: on a general positive-dimensional fibre
the ordinary adjoint contracts, while the nef summand restricts to a
full-nef-dimension divisor of smaller dimension, and the induction
hypothesis applies.
\begin{proof}
After a small $\Q$-factorialization, apply
Lemma~\ref{lemma:two-nef-model} to obtain the model $(Y,B_Y)$, its
transforms $N$, $U$, and the rational $s_0>0$. Let
$h\colon Y\to S$ be the contraction defined by the semiample divisor
$U$. Since $U$ is semiample, there is an ample rational divisor
$A_S$ on $S$ with $U\sim_\Q h^*A_S$, and
$\dim S=\kappa(X,T)>0$. Set $Q=U+s_0N$.
A general fibre $F$ of $h$ is a projective variety of dimension
strictly smaller than $n$ (when positive-dimensional), and $h$ is
flat over a nonempty open of $S$ after shrinking. The restriction
pair $(F,B_Y|_F)$ is klt with
\[
K_F+B_Y|_F \sim_\Q U|_F \sim_\Q0.
\]
Moreover, on a very general $F$ the divisor $Q|_F$ is nef of full
nef dimension: choose a very general point of $F$ outside the
countable union excluded in the full-nef-dimension condition for $Q$
on $Y$, and note that such a point exists for every nonempty fibre.
Since $U|_F\equiv0$, we have $Q|_F\equiv s_0N|_F$, so $N|_F$ is nef
of full nef dimension. Apply $(P_{\dim F})$ to the restriction pair
$(F,B_Y|_F)$ and the nef divisor $s_0N|_F$; the full condition holds
for $(N|_F)^{}$ by Lemma~\ref{lemma:pseudo-vg}. This yields bigness
of $K_F+B_Y|_F+c(s_0N|_F)$ for every rational $c>0$, and in
particular of $Q|_F$.
We want to pass from a very general fibre to relative bigness over
$S$. Fix an ample Cartier divisor $H$ on $Y$ and shrink $S$ so that
$h$ is flat over the chosen open. For every positive integer $\ell$
clearing the denominators of $Q$, the locus
\[
\{s\in S: H^0(F_s,\OO_{F_s}((\ell Q-H)|_{F_s}))\neq0\}
\]
is closed in $S$, by upper semicontinuity of cohomology in flat
families. Bigness on a very general fibre puts the base point of
$h$ into one such locus. If that locus is proper, we have a
contradiction with the countability of the possible proper loci; so
it must be all of $S$. Therefore $\ell Q$ is $H+E$ on the generic
fibre, so $Q$ is relatively big over $S$. By the standard relative
form of Kodaira's lemma (see \cite[Lemma 3.2.1]{BCHM}), a
sufficiently large pullback of the ample $A_S$ turns the relatively
big $Q$ into a big divisor on $Y$; concretely, since $U$ is the
pullback of $A_S$,
\[
Q+bU=(1+b)U+s_0N
\]
is big for some rational $b>0$.
By \eqref{eq:two-comparisons}(ii), this makes $(1+b)T+s_0M$ big on
$X$. Finally, fix a rational $c>0$ and choose rational
$\lambda>0$ with $\lambda(1+b)<1$ and $\lambda s_0<c$. Then
\[
T+cM=
\lambda\bigl((1+b)T+s_0M\bigr)
+\bigl(1-\lambda(1+b)\bigr)T
+(c-\lambda s_0)M
\]
is the sum of a big divisor and two pseudo-effective divisors, hence
big.
\end{proof}
\subsection{The Calabi--Yau reduction}
\begin{proposition}\label{prop:reduction-cy}
Assume (P$_j$) for every $j<n$. Suppose that on every projective
klt $\Q$-pair $(V,\Delta)$ of dimension $n$ with $K_V+\Delta\equiv0$,
every nef $\Q$-Cartier divisor of full nef dimension is big. Then
(P$_n$) holds.
\end{proposition}
The proposition makes the reduction to
Theorem~\ref{thm:key-ingredient} explicit: away from the case
$\kappa(T)=0$, the positive-dimensional Iitaka argument of
Proposition~\ref{prop:full-positive} already gives $(P_n)$; the only
remaining case is a klt Calabi--Yau pair with a full-nef-dimension
nef summand.
\begin{proof}
Use a small $\Q$-factorialization and
Lemma~\ref{lemma:two-nef-model}. The Iitaka dimension
$\kappa(X,T)\ge0$ is either positive, in which case
Proposition~\ref{prop:full-positive} applies directly, or zero.
Suppose $\kappa(X,T)=0$. Then $U$ is semiample with Iitaka
dimension zero, so $U\sim_\Q0$. The pair $(Y,B_Y)$ is a klt
Calabi--Yau pair of dimension $n$. The nef divisor $U+s_0N$ has
full nef dimension by Lemma~\ref{lemma:two-nef-model}(2), and since
$U\sim_\Q0$ we have $U+s_0N\sim_\Q s_0N$; thus $N$ is nef of full
nef dimension. By hypothesis $N$ is big, and so is $s_0N$.
By \eqref{eq:two-comparisons}(ii), the pullback $p^*(T+s_0M)$ is
the sum of a $q$-exceptional effective divisor and the pullback of
$U+s_0N$; hence $T+s_0M$ is big on $X$. The positive-combination
argument at the end of the proof of Proposition~\ref{prop:full-positive}
(which requires only $T$ pseudo-effective and $M$ nef) then gives
that $T+cM$ is big for every rational $c>0$.
\end{proof}
What remains for (P$_n$) is therefore the hypothesis of
Proposition~\ref{prop:reduction-cy}: on a projective klt Calabi--Yau
$\Q$-pair, a nef $\Q$-Cartier divisor of full nef dimension is big.
This is Theorem~\ref{thm:key-ingredient}. We prove it first for
terminal $X$ with $K_X\sim0$ in
Sections~\ref{sec:jet-tools}--\ref{sec:frobenius}, and then extend
to all klt Calabi--Yau pairs in Section~\ref{sec:completion}.
\section{Bounded-degree obstruction and moving kernels}
\label{sec:jet-tools}
This section assembles the technology of moving jets that
Section~\ref{sec:geometry} will use. Three things are needed.
First, a way for numerical bounds on a subvariety (bounded volume,
bounded canonical ratio) to forbid a growing lower bound on curve
degrees. Second, a way for non-uniruledness and quasi-finiteness to
pass to a subvariety, so that lower-dimensional induction hypotheses
apply. Third, the LA machinery that produces subvarieties satisfying
those bounds from a jet kernel with positive-dimensional base locus.
All varieties in this section are over $\C$.
\subsection{The bounded-degree obstruction}
The following lemma is the numerical engine: it says a curve class
that meets a targeted point with positive integer degree cannot be
arbitrarily thin in degree, when its ambient variety satisfies a
bounded volume and bounded canonical ratio.
\begin{lemma}[Bounded-degree curves]\label{lemma:bounded-curves}
For every integer $f\ge1$ there is an integer $m_f>0$ with the
following property. Let $Y$ be a smooth integral projective variety
of dimension $f$, let $D_0$ be a reduced simple normal crossing
divisor or the zero divisor, and let $L_0$ be a nef integral Cartier
divisor of full nef dimension on $Y$. Suppose that $P_0$ is a nef
rational Cartier divisor, and that $r,C_0,C_1$ are positive real
numbers with
\[
0<P_0^f\le C_0,\qquad P_0-rL_0\text{ is nef}.
\]
Suppose also $A_0=K_Y+D_0+L_0$ is big and
\begin{equation}\label{j:adjoint-ratio}
A_0\cdot P_0^{f-1}\le C_1 P_0^f.
\end{equation}
Then
\begin{equation}\label{j:bounded-curve-obstruction}
r\le(m_f C_1)^{f-1} C_0.
\end{equation}
The integer $m_f$ depends only on $f$ and not on the variety or the
divisors. For all $1\le f\le n$, one may take a single integer $m$
depending only on $n$.
\end{lemma}
The quantity $m_f$ is not explicit: it comes from the Birkar--Zhang
effective birationality theorem for polarized pairs
\cite[Theorem~1.3]{BZ}. What matters is that it depends only on the
dimension, not on the variety or the divisors.
\begin{proof}
The pair $(Y,D_0)$ is log canonical with boundary coefficients in
$\{0,1\}$, and the polarizing divisor $L_0$ is a nef integral Cartier
divisor. By \cite[Theorem~1.3]{BZ}, an integer $m_f$ exists such
that $|m_f A_0|$ is birational. Because $A_0$ is integral Cartier,
no rounding or additional denominator is involved.
Resolve the base ideal by a projective birational morphism
$\mu\colon\widetilde Y\to Y$ with $\widetilde Y$ smooth, and write
\[
\mu^*(m_f A_0)=H+F
\]
for the basepoint-free moving part $H$ and an effective $F$. The
map defined by $H$ is birational onto its image; $H$ is nef and big.
In the rest of this proof, $P_0$ and $L_0$ denote pullbacks to
$\widetilde Y$. Effectivity of $F$ and nefness of $P_0$ give
\begin{equation}\label{j:moving-adjoint-degree}
H\cdot P_0^{f-1}\le m_f C_1 P_0^f.
\end{equation}
For $f\ge2$, put $s_i=P_0^{f-i}\cdot H^i$ for $0\le i\le f$.
Both divisors are nef and big, so each $s_i$ is positive. The
Khovanskii--Teissier inequalities
\cite[Corollary~1.6.3(i) and Example~1.6.4]{LazarsfeldPAGI}, namely
$s_i^2\ge s_{i-1}s_{i+1}$, show that the ratios $s_i/s_{i-1}$ are
decreasing; hence
\begin{equation}\label{j:mixed-degree}
P_0\cdot H^{f-1}
\le P_0^f\left(\frac{H\cdot P_0^{f-1}}{P_0^f}\right)^{f-1}
\le (m_f C_1)^{f-1} C_0.
\end{equation}
Choose a very general point $y\in\widetilde Y$ in the isomorphism
open of $\mu$, and such that $\mu(y)$ is very general with respect
to the full-nef-dimension condition on $L_0$. Pull back $f-1$
general hyperplanes through the image of $y$ under the morphism
defined by $H$. This subsystem has only the base point $y$, so its
general members meet in an effective curve cycle with class
$H^{f-1}$; the component $C$ through $y$ is a curve because the
local equations are independent. Nefness of $P_0$ gives
\[
P_0\cdot C\le P_0\cdot H^{f-1}.
\]
The curve $C$ is not contracted by $\mu$, and its image passes
through the very general point $\mu(y)$. Full nef dimension and
integrality therefore give $L_0\cdot C\ge1$. Since $P_0-rL_0$ is
nef, we obtain $P_0\cdot C\ge r$, which together with
\eqref{j:mixed-degree} proves \eqref{j:bounded-curve-obstruction}.
If $f=1$, take $C=Y$ directly; then
$P_0\cdot Y\ge rL_0\cdot Y\ge r$, and $r\le C_0$ is the claim.
Finally, \cite[Theorem~1.3]{BZ} applies to every multiple of its
prescribed integer; a common multiple of the finitely many integers
$m_1,\dots,m_n$ gives the last assertion.
\end{proof}
The lemma is uniform in the sense that varieties, divisors, and $r$
may vary, provided $C_0$ and $C_1$ stay bounded. In the geometric
section, $r\to\infty$ along a family, so a sequence satisfying the
hypotheses of the lemma will automatically be absent for large $r$.
\subsection{Very general images}
The second ingredient is that full nef dimension, non-uniruledness,
and pseudo-effectivity of the canonical divisor are compatible with
passing to sufficiently general subvarieties.
\begin{lemma}[Very general images]\label{j:very-general-images}
Let $X$ be a normal integral projective variety that is not
uniruled, and let $L$ be a nef Cartier divisor of full nef dimension
on $X$. Then there is a complement $X^\circ$ of a countable union of
proper closed subsets of $X$ with the following property. If $Y$ is
a smooth integral projective variety and $g\colon Y\to X$ is
generically finite onto a positive-dimensional image that meets
$X^\circ$, then $g^*L$ has full nef dimension, $Y$ is not uniruled,
and $K_Y$ is pseudo-effective.
\end{lemma}
The point is that very general conditions simultaneously test full
nef dimension of $L$ and avoid all rational curves.
\begin{proof}
Choose $X^\circ$ so that every curve through any of its points has
positive $L$-degree and no rational curve meets $X^\circ$. The first
condition characterizes full nef dimension at a point. For the
second, rational curves move in countably many algebraic families;
the closure of the evaluation image of each such family is proper
because $X$ is not uniruled. Removing those closures gives $X^\circ$.
Since $g(Y)$ meets $X^\circ$, it is contained in none of the closed
sets excluded by the definition. A very general point of $Y$ maps
into $X^\circ$ and lies in the quasi-finite locus of $g$. Every
curve through such a point has a curve as its image, so its
$g^*L$-degree is positive. This gives full nef dimension of $g^*L$.
A covering family of rational curves on $Y$ would similarly give a
nonconstant rational curve meeting $X^\circ$, a contradiction; so
$Y$ is not uniruled. Pseudo-effectivity of $K_Y$ now follows from
\cite{BDPP}.
\end{proof}
\subsection{Moving kernels and the LA inputs}
\label{sec:moving-kernel-inputs}
The third ingredient, and the most technical, is a set of five results
from the section \emph{Tools for moving base components} of
\cite{LA}. We state them in full, because they are used, not just
cited, in Section~\ref{sec:geometry}. Their hypotheses separate the
generic information along a moving base component from the global
information about the ambient variety. Intersections with a
subvariety are computed after pullback to its normalization and a
smooth projective resolution, because the subvariety itself may be
singular.
A linear system $\mathcal H\subset H^0(Z,\OO_Z(kR))$ has a
\emph{base ideal} $\mathfrak b$ defined as the ideal generated by its
sections in $\OO_Z$; its zero locus is the base locus. A base
component $V$ is \emph{isolated at the generic point} of $V$ when the
localized ideal $\mathfrak b\OO_{Z,\eta_V}$ is primary for the
maximal ideal $\mathfrak m$ of $\OO_{Z,\eta_V}$. In this case there
is a well-defined multiplicity $h$ with
$\mathfrak b\OO_{Z,\eta_V}\subset\mathfrak m^h$, measuring how
strongly $\mathcal H$ vanishes along $V$ at its generic point.
The first input is a numerical version of the
Kawakita--Ambro adjunction for a lc-trivial fibration:
\begin{lemma}[Numerical generic subadjunction, {\cite[Lemma~7.1]{LA}}]
\label{j:subadjunction}
Let $Z$ be a projective $\Q$-factorial klt variety, let $\Theta\ge0$
be a rational divisor, and let $V\subset Z$ be integral of dimension
$f>0$. Suppose $(Z,\Theta)$ is log canonical at the generic point of
$V$, and admits a divisor of log discrepancy zero centered on $V$. If
$P$ is a nef rational Cartier divisor on $V$ and $\rho\colon V^*\to V$
is a smooth projective resolution factoring through its
normalization, then
\[
K_{V^*}\cdot(\rho^*P)^{f-1}
\le (K_Z+\Theta)|_V\cdot P^{f-1}.
\]
\end{lemma}
Only generic log canonicity is required: the inequality lives at the
generic point of $V$. The proof in \cite{LA} passes through the dlt
modification of \cite[Theorem~2.10]{FH}, whose truncated boundary
leaves an effective surplus supported over the non-lc locus. On a
minimal stratum over $V$, adjunction gives a pair that is klt over
the generic point of its connected-fiber base. This suffices for the
klt-trivial canonical bundle formula and its b-nef moduli divisor
\cite[Definition~3.1 and Theorem~3.5]{FG}. The effective
discriminant, the moduli divisor, and finite ramification all have
nonnegative intersections with the pulled-back nef class. The
comparison uses canonical Weil cycles, so it requires no
$\Q$-Gorenstein assumption on the normalization of $V$.
The second input transfers a base component to a genuine pair:
\begin{lemma}[High-multiplicity base component, {\cite[Lemma~7.2]{LA}}]
\label{j:base-component}
Let $Z$ be a projective $\Q$-factorial klt variety of dimension $d$,
let $R$ be a nef rational Cartier divisor, and let $k>0$ be an integer
such that $kR$ is Cartier. Let
$0\ne\mathcal H\subset H^0(Z,\OO_Z(kR))$ 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 $\eta_V\in Z_{\mathrm{sm}}$.
Let $\mathfrak m$ be the maximal ideal of $\OO_{Z,\eta_V}$. If
$\mathfrak b\OO_{Z,\eta_V}\subset\mathfrak m^h$ for an integer $h>0$,
there are a rational $0<c\le(d-f)k/h$ and an effective rational
divisor $\Theta\sim_\Q cR$ such that $(Z,\Theta)$ is log canonical
at $\eta_V$ and has a log canonical place centered on $V$. The
degree and canonical bounds are
\begin{equation}\label{j:base-bounds}
\begin{split}
R^f\cdot V&\le(k/h)^{d-f}R^d,\\
K_{V^*}\cdot R^{f-1}
&\le(K_Z+cR)|_V\cdot R^{f-1}\qquad(f>0),
\end{split}
\end{equation}
where $V^*$ is a smooth projective resolution. The degree bound
also holds at $f=0$.
\end{lemma}
The degree estimate is the local multiplicity bound $h^{d-f}$
combined with successive general cuts by the given linear series.
The boundary is obtained by averaging general members with total
coefficient equal to the generic log canonical threshold of
$\mathfrak b$; thus it is an actual divisor to which
Lemma~\ref{j:subadjunction} applies.
The third input produces the moving base components in the first
place:
\begin{lemma}[Moving multiplicity, {\cite[Lemma~7.3]{LA}}]
\label{j:moving-kernel}
Let $Z$ be an integral projective variety of dimension $d$, let $R$
be a nef, big, semiample rational Cartier divisor, and fix an integer
$k>0$ such that $kR$ is Cartier. Choose positive real numbers
$\tau_0<\dots<\tau_d$ with common gap $\delta>0$. For $x\in
Z_{\mathrm{sm}}$, put
\[
\mathcal H_i(x)=H^0\bigl(Z,\OO_Z(kR)
\otimes\mathfrak m_x^{\lceil k\tau_i\rceil}\bigr).
\]
Suppose that for very general $x$ all these spaces are nonzero, and
that the base locus of $\mathcal H_0(x)$ has a positive-dimensional
component through $x$. If $k\delta\ge4$, then, after an algebraic
parameter extension and restriction to nonempty open subsets, there
are: an integral parameter variety $T$ with a dominant marked-point
map $x\colon T\to Z_{\mathrm{sm}}$; an index $i\in\{0,\dots,d-1\}$
fixed throughout; and a family of integral subvarieties $V_t$
containing $x(t)$ and common to the base loci of $\mathcal
H_i(x(t))$ and $\mathcal H_{i+1}(x(t))$. The incidence
\[
\Gamma=\{(t,z):z\in V_t\}\subset T\times Z
\]
dominates $Z$.
For general $t$, the dimension $f$ of $V_t$ lies in $[1,d-1]$,
$V_t$ is an isolated component of the higher base locus at its
generic point, and the entire higher-kernel base ideal is contained
there in the ordinary power $\mathcal I_{V_t}^h$, where
\[
h=\lceil k\delta/2\rceil,\qquad k/h\le2/\delta.
\]
The incidence base ideal is likewise contained in
$\mathcal I_\Gamma^h$ on a dense open of $\Gamma$. These
containments are taken in the smooth ambient open. If $Z$ is also
$\Q$-factorial and klt, then
\begin{equation}\label{j:moving-bounds}
\begin{split}
R^f\cdot V_t&\le(2/\delta)^{d-f}R^d,\\
K_{V_t^*}\cdot R^{f-1}
&\le(K_Z+cR)|_{V_t}\cdot R^{f-1},\qquad
0<c\le2(d-f)/\delta.
\end{split}
\end{equation}
The top intersection on $V_t$ is positive when $V_t$ meets the
open on which the big semiample contraction is an isomorphism.
\end{lemma}
The use of full kernels matters. Differentiation in the parameter
lowers vanishing at the moving mark and still gives a global section
in the lower kernel. Incidence dominance makes the parameter
directions span the normal directions to $V_t$, and the vanishing of
all derivatives of order less than $h$ gives the ordinary-power
containment.
The fourth input restricts an isolated base ideal to a fibre of a
morphism:
\begin{lemma}[Restriction of an isolated base ideal,
{\cite[Lemma~7.4]{LA}}]
\label{j:fiber-restriction}
In the setting of Lemma~\ref{j:moving-kernel}, let $g\colon Z\to B$
be a morphism. There is a dense open of the incidence with the
following property. Choose $(t,z)$ in this open, put $b=g(z)$, and
suppose $V_t$ is smooth over a smooth open of its actual image at
$z$. Let $F$ be the reduced component through $z$ of the scheme
fibre of $V_t\to g(V_t)$. Near $z$ in $Z_b$, the restricted
higher-kernel base ideal has support exactly $F$ and is contained
in the ordinary power $\mathcal I_F^h$.
If $F$ is proper in an integral ambient fibre component containing
it, the restricted series is nonzero there and $F$ is an isolated
base component at its generic point. Applying
Lemma~\ref{j:base-component} on that ambient component additionally
requires its projectivity, $\Q$-factoriality, klt singularities, and
smoothness at the generic point of $F$.
\end{lemma}
The fibre is chosen through a general incidence point; the statement
does not assert the same conclusion for an arbitrary special fibre.
Smoothness over the actual image identifies the restricted ideal of
$V_t$ with the reduced ideal of $F$ locally, preserving ordinary
powers.
The fifth input turns a bounded-degree family of finite covers into
a finite list once the branch divisors do not sweep:
\begin{lemma}[Finite covers after excluding sweeping branch divisors,
{\cite[Lemma~7.5]{LA}}]
\label{j:finite-covers}
Let $Y$ be a normal integral projective variety, let
$\mathcal V\to T$ be a smooth projective family with integral fibres
over an integral parameter variety, and let $g\colon\mathcal V\to Y$
be a morphism. Suppose the general fibre maps
$g_t\colon V_t\to Y$ 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 $Y_{\mathrm{sm}}$ to divisors
$\mathcal R_1,\dots,\mathcal R_N$ in $\mathcal V$. Retain this
notation after the base change. Let $I$ consist of those $j$ for
which $\overline{g_t((\mathcal R_j)_t)}$ is a divisor in $Y$ for
general $t$. Assume
\[
\overline{g(\mathcal R_j)}\ne Y\qquad(j\in I).
\]
After shrinking again, there is a smooth nonempty open
$Y^\circ\subset Y$ over which every general finite normal Stein
cover of $g_t$ is finite \'etale. Only finitely many such normal
covers of $Y$ occur, up to isomorphism over $Y$. The conclusion
holds simultaneously for finitely many projections; the open and the
finite lists may depend on all fixed parameters of the family.
\end{lemma}
The hypothesis removes $Y_{\mathrm{sing}}$ and the finitely many
proper image closures $\overline{g(\mathcal R_j)}$ for $j\in I$.
Every branch prime of a general Stein cover is accounted for by one
of these relative Jacobian components, so purity makes the
restricted covers \'etale. The fundamental group of the resulting
smooth complex variety is finitely generated, so it has only
finitely many covers of degree at most $e$. Finally, a finite
normal cover of $Y$ is determined by its function field, hence by
its restriction to this fixed open.
\subsection{$\Q$-factorialization of a klt ambient variety}
The base-component estimates require a $\Q$-factorial ambient
variety. When the ambient is only klt, we may pass to a small
projective $\Q$-factorialization $\sigma\colon Z'\to Z$. This
passage is made only for components that meet the isomorphism open of
$\sigma$. The strict transforms then retain the generic
base-ideal and multiplicity conditions for the pulled-back series,
and $\sigma^*R$ is nef. Since $K_{Z'}=\sigma^*K_Z$, the projection
formula gives the same degree and canonical-intersection bounds on a
common resolution. The pulled-back testing class need not be
ample; nefness is the property used in these estimates. Passing to
higher resolutions does not change the canonical intersections,
because their additional exceptional divisors have zero
intersection with the pulled-back testing class to the required
power.
\section{The terminal Calabi--Yau case: two jet estimates}
\label{sec:geometry}
We now prove Theorem~\ref{thm:key-ingredient} in the terminal case
with trivial canonical divisor. This is where the geometry of the
paper is concentrated.
\begin{theorem}[Terminal Calabi--Yau case]\label{thm:terminal-big}
Let $n\ge2$ and assume $(P_j)$ for every $j<n$. Let $X$ be a
projective $\Q$-factorial terminal complex variety of dimension $n$
with $K_X\sim0$. If $L$ is a nef Cartier divisor on $X$ of full
nef dimension, then $L$ is big.
\end{theorem}
The proof of Theorem~\ref{thm:terminal-big} occupies this section and
Section~\ref{sec:frobenius}. Assume $L$ is not big. We produce two
incompatible jet estimates: a vanishing-order bound for sections of
big multiples of a rescaled ample approximation (Lemma~\ref{lemma:scalar-bound}),
and a large jet system on a projective bundle over $X\times X$
(Proposition~\ref{prop:large-seshadri}). The incompatibility is made
precise by the Frobenius comparison of Section~\ref{sec:frobenius}.
\subsection{Normalization and the curve obstruction}
Throughout, fix $X,L$ forming a counterexample to
Theorem~\ref{thm:terminal-big}. Thus: $X$ is projective,
$\Q$-factorial, terminal, of dimension $n\ge2$, with $K_X\sim0$; and
$L$ is a nef Cartier divisor of full nef dimension but not big. In
particular, $L^n=0$, and every curve through a very general point of
$X$ has positive integral $L$-degree.
Fix an ample Cartier divisor $A$ on $X$ and choose a rational
$\epsilon>0$ with
\begin{equation}\label{j:epsilon}
(14\epsilon)^n<\tfrac14, \qquad 80\epsilon<\tfrac34.
\end{equation}
Let $t>0$ tend to zero through positive rationals. Since $L$ is nef
and not big, $L^n=0$; we choose positive integers $r=r(t)$ by
\begin{equation}\label{eq:scaling}
P=P_t=r(L+tA),\qquad
r\longrightarrow\infty,\qquad P^n\longrightarrow\epsilon^n.
\end{equation}
For instance, $r$ may be the nearest integer to
$\epsilon/((L+tA)^n)^{1/n}$; if the expansion of $(L+tA)^n$ is
nonzero at every order up to $t^{n-1}$ (and it is, since
$L^n=0$ and $L^{n-1}A>0$), this makes $r\to\infty$ as $t\to0$.
The divisor $P$ is ample and rational Cartier, and $P-rL=rtA$ is nef
for all $t$. After discarding finitely many $t$ we may assume
\begin{equation}\label{j:volume-window}
\epsilon^n/2\le P^n\le 2\epsilon^n.
\end{equation}
Every curve through a very general point of $X$ has positive
integral $L$-degree, hence $P$-degree at least $r$ (on such a curve,
$P\cdot C = r(L+tA)\cdot C \ge r$ because $L\cdot C\ge1$ and
$A\cdot C\ge0$). Therefore a covering family of curves of bounded
$P$-degree will contradict full nef dimension.
We will use the same obstruction on subvarieties and covers, so we
record how it descends.
A smooth projective resolution $W\to X$ admits an effective
canonical divisor: pull back a nowhere-zero canonical form from the
smooth locus of $X$, and use terminality of $X$ to represent
$K_W$ by an effective $L$-exceptional divisor. In particular $W$,
and therefore $X$, is not uniruled \cite{BDPP}.
By Lemma~\ref{j:very-general-images}, if $Y$ is a smooth projective
variety with a generically finite map $g\colon Y\to X$ onto a
positive-dimensional image that meets a sufficiently very general
point of $X$, then $Y$ is not uniruled, $K_Y$ is pseudo-effective,
and the pullback $L_Y=g^*L$ has full nef dimension. If
$f=\dim Y<n$, the induction hypothesis $(P_f)$ applied to $(Y,0)$
gives
\begin{equation}\label{j:lower-big}
K_Y+L_Y\text{ is big}.
\end{equation}
All uses of ``very general'' below include the fixed conditions of
Lemma~\ref{j:very-general-images}.
Consequently, for $f<n$ it suffices to find a nef testing divisor
$P_Y$ on such $Y$ with
\[
0<P_Y^f\le C_0,\qquad P_Y-rL_Y\text{ nef},\qquad\text{and }
K_YP_Y^{f-1}\le C\, P_Y^f,
\]
where $C_0$ and $C$ are independent of $t$. Indeed
$L_YP_Y^{f-1}\le P_Y^f/r$, so Lemma~\ref{lemma:bounded-curves}
applies. We use that lemma in dimension $n$ as well, after proving
the needed bigness separately.
Call a bound \emph{uniform} if it does not depend on $t$; it may
depend on parameters fixed before $t$ tends to zero. Since
dimensions lie in the finite range $[1,n]$, Lemma~\ref{lemma:bounded-curves}
excludes all such configurations once $r$ is large enough.
\subsection{The scalar jet estimate}
Put $\rho=(P^n)^{1/n}$. The first incompatible estimate is the
following vanishing-order bound. It is stronger than a bound on a
single linear system.
\begin{lemma}\label{lemma:scalar-bound}
In the setting of \eqref{eq:scaling}, for all sufficiently small
$t$, at a very general point $x\in X$, every nonzero section $s$ of
every $kP$ with $k$ a positive integer and $kP$ Cartier satisfies
\begin{equation}\label{j:scalar-order}
\ord_x(s)\le 4k\rho.
\end{equation}
\end{lemma}
\begin{proof}
Suppose violations occur for arbitrarily small $t$. Fix such a $t$.
At a very general point, a violation for some $k$ means the
corresponding complete jet kernel is nonzero at the general point.
Indeed the kernel ranks are algebraic rank conditions on the smooth
locus of $X$, and there are only countably many choices of $k$ and
of the required integral order. Raising a violating section to
powers makes $k$ arbitrarily large and divisible.
Choose $n+1$ levels
\[
\tau_i=\frac32\rho+i\delta\quad(0\le i\le n),
\qquad\delta=\frac{2\rho}{n}.
\]
These lie strictly between $\rho$ and $4\rho$. For a sufficiently
large divisible $k$, the corresponding full jet kernels are nonzero
and $k\delta\ge4$.
Consider the lowest kernel. If its base locus at the marked point
were zero-dimensional, the isolated base component at that point
would have multiplicity at least
$h_0=\lceil k\tau_0\rceil$. The zero-dimensional case of
Lemma~\ref{j:base-component} (its degree bound, valid at $f=0$) would
then force
\[
1\le(k/h_0)^nP^n < 1,
\]
contradiction: the strict inequality follows because
$k/h_0\le 1/\tau_0 = 2/(3\rho)$, so
$(k/h_0)^nP^n\le(2/3)^n(P^n/\rho^n)=(2/3)^n<1$, since $n\ge2$.
Hence the lowest base locus has a positive-dimensional component
through its marked point.
Lemma~\ref{j:moving-kernel} now supplies a moving family of proper
base components $V$ with dominant incidence and dimension
$1\le f<n$. On a resolution $V^*$, let $P_V,L_V$ be the
restrictions followed by pullback. The lemma gives
\[
0<P_V^f\le(2/\delta)^{n-f}P^n,
\qquad
K_{V^*}P_V^{f-1}\le cP_V^f,
\qquad
0<c\le\frac{2(n-f)}{\delta}.
\]
Here the canonical term of the ambient vanishes because
$K_X\sim0$. The volume window \eqref{j:volume-window} bounds
$\delta$ above and away from zero, so these are uniform estimates.
The incidence point may be chosen over a very general point of $X$;
then Lemma~\ref{j:very-general-images} makes $V^*$ non-uniruled
and $K_{V^*}$ pseudo-effective, with $L_V$ of full nef dimension.
Equation~\eqref{j:lower-big} gives bigness of $K_{V^*}+L_V$, and
$P_V-rL_V$ is nef; Lemma~\ref{lemma:bounded-curves} contradicts
$r\to\infty$.
\end{proof}
\subsection{The projective bundle and its volume}
We now seek a jet system whose order is much larger than the
scalar bound $4\rho$. The extra positivity comes from the fibres of
a projective bundle over $X\times X$ that carries the two copies of
$X$ in its two summands. For a divisor $D$ on $X$, we write
$D_1,D_2$ for its pullback from the two factors of $X\times X$.
Set
\begin{equation}\label{eq:bundle}
Z=\PP\bigl(\OO_X(L)_1\oplus\OO_X(L)_2\bigr)
\xrightarrow{\pi}X\times X,
\qquad\xi=c_1(\OO_Z(1)),
\qquad R=P_1+P_2+q\xi,
\end{equation}
where $q$ is a positive integer to be fixed shortly. We use the
convention
\[
\pi_*\OO_Z(a)=\Sym^a\bigl(\OO_X(L)_1\oplus\OO_X(L)_2\bigr)
\quad(a\ge0).
\]
The tautological class $\xi$ is nef, and $R$ is ample. As a
projective bundle over $X\times X$ with klt base, and with an
unramified flag or torus-part description on the complement of its
natural sections, $Z$ is klt. The two natural sections of $\pi$,
called the \emph{axes}, have divisor classes $\xi-L_1$ and
$\xi-L_2$. The canonical-bundle formula for a projective bundle
gives
\begin{equation}\label{j:axes-canonical}
K_Z\sim-\bigl(\text{(sum of the two axes)}\bigr).
\end{equation}
The complement of the two axes is called the \emph{torus open}; a
general point of $Z$ lies there.
The fibre of either projection $Z\to X$ over a smooth point is, up
to a constant one-dimensional factor,
\[
Z_{\mathrm{sl}}=\PP(\OO_X\oplus\OO_X(L)),
\qquad R_{\mathrm{sl}}=P+q\xi_{\mathrm{sl}}.
\]
Its axes have classes $\xi_{\mathrm{sl}}$ and
$\xi_{\mathrm{sl}}-L$, and its canonical divisor is the negative of
their sum. The slice is projective and klt, so it satisfies the
singularity hypotheses needed to apply the base-component estimates
of Section~\ref{sec:jet-tools}, using a small $\Q$-factorialization
if necessary.
Write $d=2n+1$. There are positive constants depending only on $n$
such that, for each fixed $q$ and all sufficiently small $t$,
\begin{equation}\label{j:bundle-volumes}
c_nq\epsilon^{2n}\le R^d\le C_nq\epsilon^{2n},
\qquad
R_{\mathrm{sl}}^{n+1}\le C_{\mathrm{sl},n}q\epsilon^n.
\end{equation}
Here is the computation. Expanding $R^d=(P_1+P_2+q\xi)^d$ in nef
classes and using
\[
\pi_*(\xi^j)=\sum_{a=0}^{j-1}L_1^aL_2^{j-1-a}\qquad(j\ge1),
\]
the term containing exactly one $\xi$ is
$(2n+1)\binom{2n}{n}q(P^n)^2$, which gives the lower bound in
\eqref{j:bundle-volumes}. For the upper bound, the mixed
intersections of $L_i$ can be bounded by $P_i/r$ (since
$\frac{1}{r}P_i=L_i+tA_i$, and $A_i$ can be absorbed by nonnegativity
of the mixed-degree coefficients); after factoring out $q$, the
remaining powers of $q/r$ are bounded by one once $r\ge q$, and
there are only finitely many monomials depending on $n$. The slice
calculation is identical, using
$\pi_*(\xi^j)=L^{j-1}$.
For a nef rational Cartier divisor $R$ and a smooth point $z$, its
\emph{Seshadri constant} is
\[
\varepsilon(R;z)=\inf_{C\ni z}\frac{R\cdot C}{\mult_z C},
\]
where $C$ ranges over integral curves through $z$. A lower bound
for this constant controls the degree of curves through $z$ relative
to their multiplicity at $z$, which is what asymptotic Riemann--Roch
needs to make jets surjective.
\begin{proposition}\label{prop:large-seshadri}
For a suitable fixed positive integer $q$, and all sufficiently
small $t$ in \eqref{eq:scaling}, there is a smooth point $z_*\in Z$
in the torus open, lying over smooth points of both factors of
$X\times X$, such that
\begin{equation}\label{j:large-seshadri}
\varepsilon(R;z_*)>2n+2.
\end{equation}
More generally, for any prescribed dense open subsets $U_1,U_2$ of
the smooth locus of $X$, the point may be chosen over $U_1\times U_2$.
In particular, these opens may be the isomorphism loci of fixed
resolutions of $X$.
\end{proposition}
\subsection{Proof of the large Seshadri constant}
We prove the proposition in the remainder of this section.
\paragraph{Choices before the limit.} Choose a positive rational gap
$\delta$ such that
\begin{equation}\label{j:slice-gap}
(2/\delta)^nC_{\mathrm{sl},n}\epsilon^n<1.
\end{equation}
Next choose $q$ so large that $d+1$ increasing levels of gap
$\delta$, with the lowest level strictly greater than $2n+2$, all lie
strictly below $(c_nq\epsilon^{2n})^{1/d}$. Both $q,\delta$ are
fixed before $t$ tends to zero. Fix the prescribed opens
$U_1,U_2$ if they are given.
\paragraph{The reduction to a moving kernel.} Suppose
\eqref{j:large-seshadri} fails throughout the smooth torus preimage
of $U_1\times U_2$ for arbitrarily small $t$. Fix such a $t$.
Asymptotic Riemann--Roch and the elementary jet count at a smooth
point show that all the full kernels at the chosen levels are
nonzero for sufficiently large divisible $k$, because the highest
level has $d$-th power smaller than $R^d$. More explicitly: the
length of the quotient by jets of order $<k\tau$ is
$\binom{k\tau+d}{d}\sim(k\tau)^d/d!$, and the number of sections is
$\chi(\OO(kR))\sim(kR)^d/d!=(k\tau)^d/d!\cdot (R/\tau)^d$, which
dominates as long as $\tau^d<R^d$. For our chosen levels
$\tau_i<(c_nq\epsilon^{2n})^{1/d}\le (R^d)^{1/d}$, this holds.
Failure of \eqref{j:large-seshadri} supplies, through each very
general point, a curve $C$ with $R\cdot C/\mult_zC$ smaller than the
lowest level $\tau_0$. Every section in the lowest kernel vanishes
identically on $C$: otherwise its intersection with $C$ would be at
least its vanishing order times $\mult_zC$, contradicting the degree
inequality $R\cdot C<\tau_0\mult_zC$ and the order bound $k\tau_0$.
Thus the lowest base locus has a positive-dimensional component
through the marked point.
Take $k$ divisible enough that $k\delta\ge4$ and apply
Lemma~\ref{j:moving-kernel}. We obtain a family of proper integral
components $V\subset Z$ whose incidence dominates $Z$. For general
members of dimension $f$, the estimates are
\begin{equation}\label{j:total-component-bounds}
\begin{split}
0<R^f\cdot V&\le(2/\delta)^{d-f}R^d,\\
K_{V^*}R^{f-1}&\le(K_Z+cR)|_V\cdot R^{f-1},
\qquad 0<c\le 2(d-f)/\delta.
\end{split}
\end{equation}
Here and below, classes on a resolution are understood by pullback.
The positivity $R^f\cdot V>0$ follows from ampleness of $R$. General
members meet the torus open, so \eqref{j:axes-canonical} makes the
ambient canonical contribution nonpositive. All constants in these
bounds are uniform once $q,\epsilon,\delta$ have been fixed.
The remainder of the proof rules out all possible $(V,f)$. We first
constrain the two projections of $V$ to $X$.
\subsection{Excluding positive-dimensional slice fibres}
Consider either projection $V\to X$. Let $F$ be the component of a
general fibre of $V\to\operatorname{im}(V\to X)$ through a general
incidence point of $V$. Choose the incidence point in the opens
given by Lemma~\ref{j:fiber-restriction}, over very general points of
both factors of $X$, and in the smooth torus open. We claim
\begin{equation}\label{j:no-intermediate-fibre}
\dim F\notin\{1,\ldots,n\}.
\end{equation}
Suppose, toward a contradiction, that $1\le f_{\mathrm{sl}}=\dim F
\le n$. The ambient fibre of either projection $Z\to X$ over a
smooth point is the slice $Z_{\mathrm{sl}}$ of dimension $n+1$.
Lemma~\ref{j:fiber-restriction} gives a nonzero restricted subseries
with $F$ an isolated base component, whose base ideal is contained in
$\mathcal I_F^h$ with $h=\lceil k\delta/2\rceil$. Generic smoothness
is imposed over the actual image of $V$, which is why a general fibre
through the incidence point was chosen; the chosen point is smooth
in the ambient slice by the very general choice. Applying
Lemma~\ref{j:base-component} on that slice gives
\begin{equation}\label{j:slice-bounds}
\begin{split}
0<R_{\mathrm{sl}}^{f_{\mathrm{sl}}}\cdot F
&\le(2/\delta)^{n+1-f_{\mathrm{sl}}}
R_{\mathrm{sl}}^{n+1},\\
K_{F^*} R_{\mathrm{sl}}^{f_{\mathrm{sl}}-1}
&\le(K_{Z_{\mathrm{sl}}}+cR_{\mathrm{sl}})|_F
R_{\mathrm{sl}}^{f_{\mathrm{sl}}-1},
\end{split}
\end{equation}
with $c\le 2(n+1-f_{\mathrm{sl}})/\delta$. Cartier multiples are
chosen before restriction, so the constant line from the fixed
factor causes no change in this calculation.
Let $W_0\subset X$ be the image of $F$ in the other factor. Three
cases arise.
\paragraph{Case 1: $F$ is generically finite over a proper
subvariety.} Suppose $F\to W_0$ is generically finite with
$\dim W_0<n$. Let $P_Y=R_{\mathrm{sl}}|_{F^*}$, and let $L_Y$ be the
pullback of $L$ from $X$. The first inequality in
\eqref{j:slice-bounds} gives a uniform positive volume bound; since
$F$ meets neither axis generically, $K_{Z_{\mathrm{sl}}}|_F\le0$, so
the second inequality gives a uniform canonical ratio
$K_{F^*}P_Y^{f_{\mathrm{sl}}-1}\le cP_Y^{f_{\mathrm{sl}}}$. Moreover
$P_Y-rL_Y$ is nef, since
$P_Y-rL_Y=(P|_{W_0}-rL|_{W_0})+q\xi_{\mathrm{sl}}|_{F^*}$ is the sum
of two nef classes. The image $W_0$ contains the very general
projection of the incidence point, so \eqref{j:lower-big} applies and
gives bigness of $K_{F^*}+L_Y$. Lemma~\ref{lemma:bounded-curves}
excludes this case for large $r$.
\paragraph{Case 2: $F$ is a projective-line bundle over a proper
subvariety.} Suppose $F\to W_0$ is not generically finite. Its
general fibre has dimension one, and since the ambient projection
$Z_{\mathrm{sl}}\to X$ is a $\PP^1$-bundle, the integral $F$ must
coincide with the full $\PP^1$-bundle over $W_0$. Write
$w=\dim W_0=f_{\mathrm{sl}}-1$.
If $w=0$, then $F$ is a projective-line fibre and
$R_{\mathrm{sl}}\cdot F=q$. The first inequality in
\eqref{j:slice-bounds}, together with \eqref{j:bundle-volumes} and
\eqref{j:slice-gap}, gives
\[
q\le(2/\delta)^nR_{\mathrm{sl}}^{n+1}
\le(2/\delta)^n C_{\mathrm{sl},n}q\epsilon^n<q,
\]
impossible.
Suppose $w>0$. Resolve $W_0$ to $W_0^*$, and take as a resolution
of $F$ the induced projective bundle $p_0\colon F^*\to W_0^*$. Its
relative canonical divisor is $-2\xi+p_0^*L$, exactly the
restriction of $K_{Z_{\mathrm{sl}}}$. Cancelling this term in
\eqref{j:slice-bounds} gives
\begin{equation}\label{j:saturated-canonical}
p_0^*K_{W_0^*}R_{\mathrm{sl}}^w
\le cR_{\mathrm{sl}}^{w+1}\cdot F.
\end{equation}
The divisor $K_{W_0^*}$ is pseudo-effective by the very general point
condition. On expanding the left side of \eqref{j:saturated-canonical}
and pushing down, its term with exactly one $\xi$ is
$wqK_{W_0^*}P^{w-1}$; every other nonzero term is nonnegative, since
it pairs the pseudo-effective $K_{W_0^*}$ with nef classes. On the
right, the expansion used for \eqref{j:bundle-volumes} gives
\begin{equation}\label{j:saturated-volume}
(w+1)qP^w\cdot W_0
\le R_{\mathrm{sl}}^{w+1}\cdot F
\le C'_n qP^w\cdot W_0,
\end{equation}
using $r\ge q$. The leading coefficient in the lower bound is the
term with exactly one $\xi$, as before. The first inequality in
\eqref{j:slice-bounds} then bounds $0<P^w\cdot W_0$ uniformly from
above, and \eqref{j:saturated-volume}'s leading coefficient bounds it
from below. Equations \eqref{j:saturated-canonical}--\eqref{j:saturated-volume}
together give
\[
K_{W_0^*}P^{w-1}\le\frac{cC'_n}{w}P^w\cdot W_0.
\]
Now apply \eqref{j:lower-big} on $W_0^*$, with the pullback of $P$ as
the testing class and $L$ as the full-nef-dimension divisor.
$W_0$ contains a very general point of $X$, so the required bigness
and full nef dimension hold by Lemma~\ref{j:very-general-images}.
Lemma~\ref{lemma:bounded-curves} excludes this case for large $r$.
\paragraph{Case 3: $F$ is a multisection over $X$.} The only
remaining possibility is that $f_{\mathrm{sl}}=n$ and $F\to X$ is
generically finite of degree $e>0$. Intersect $F$ with each of the
two Cartier axes and push the resulting cycles down to $X$. This
gives effective integral Weil divisors $D_1,D_2$. The axes do not
contain $F$, and their classes differ by $L$; taking the norm of the
associated rational section, or equivalently its projection, gives
\begin{equation}\label{j:signed-L}
D_1-D_2\sim\pm eL.
\end{equation}
For either axis $S$, the classes $R_{\mathrm{sl}}-qS$ and
$R_{\mathrm{sl}}-P$ are nef. Intersecting with the effective cycle
$S\cdot F$ and then using nefness on $F$ yields
\begin{equation}\label{j:axis-degrees}
qD_iP^{n-1}\le qS\cdot F\cdot R_{\mathrm{sl}}^{n-1}
\le R_{\mathrm{sl}}^n\cdot F.
\end{equation}
Thus both axis-image degrees are uniformly bounded.
Set $G=D_1+D_2$. It is $\Q$-Cartier because $X$ is $\Q$-factorial.
Choose $b>0$ rational sufficiently small that $(X,bG)$ is klt. We
claim
\begin{equation}\label{j:positive-boundary-kappa}
\kappa(K_X+bG)>0.
\end{equation}
The Iitaka dimension is at least zero, since $K_X+bG\sim_\Q bG$.
If it were zero, Theorems~\ref{input:models} and~\ref{input:abundance}
would give an ordinary MMP ending at a good model $Y$ whose
transformed adjoint is rationally linearly trivial. The transform of
$K_X$ is then linearly trivial as well, so the effective transform
of $G$ must vanish: every component of $G$ has been contracted.
On a common resolution $a\colon U\to X$ and $b_Y\colon U\to Y$,
relation \eqref{j:signed-L} then expresses $a^*L$ up to rational
linear equivalence as a signed $b_Y$-exceptional divisor $E$.
Divisors exceptional over $X$ remain exceptional over $Y$, because
the MMP extracts no divisors. Nefness of $a^*L$ and the negativity
lemma give $E\le0$. Its class is also pseudo-effective.
A nonzero anti-effective divisor has negative intersection with a
sufficiently high power of an ample divisor, so $E=0$. This
contradicts full nef dimension of $L$, and thus proves
\eqref{j:positive-boundary-kappa}.
The already proved positive-Iitaka-dimension case
(Proposition~\ref{prop:full-positive}) now gives
\[
K_X+bG+L\ \text{big}.
\]
Its full-positivity hypothesis holds because $L$ has full nef
dimension and $K_X+bG$ is pseudo-effective.
Take a log resolution $a\colon Y\to X$ of $G$, and let $D_0$ be the
reduced simple normal crossing divisor consisting of the strict
support of $G$ together with every exceptional divisor. Decrease $b$
if necessary; Proposition~\ref{prop:full-positive} still applies.
Terminality of $X$ and the support of $a^*G$ then give
\[
K_Y+D_0+a^*L\ \ge\ a^*(K_X+bG+L)
\]
up to rational linear equivalence and an effective difference.
Indeed the finitely many exceptional discrepancies are positive, and
decreasing $b$ makes each coefficient of $b\,a^*G$ no larger than the
corresponding coefficient of $K_Y-a^*K_X+D_0$. Hence the left side
is big.
Test on $Y$ with $a^*P$. Exceptional divisors contribute zero to
intersection with $(a^*P)^{n-1}$, and $K_X\sim0$. The degree of the
reduced strict support is no larger than $GP^{n-1}$, which is bounded
by \eqref{j:axis-degrees}; also $LP^{n-1}\le P^n/r$. Since
\eqref{j:volume-window} bounds $P^n$ away from zero, we obtain a
uniform constant $C$ such that
\[
(K_Y+D_0+a^*L)(a^*P)^{n-1}\le C(a^*P)^n.
\]
Lemma~\ref{lemma:bounded-curves}, now with its reduced boundary
$D_0$, gives the final contradiction. This proves
\eqref{j:no-intermediate-fibre}.
\subsection{Dimension reduction for moving components}
Thus a general moving component $V$ has fibre dimension either zero
or $n+1$ over its image in either factor. Fibre dimension $n+1$
would make $V$ the full inverse image of its image in that factor.
Since $V$ is proper, the image has dimension at most $n-1$, so the
other projection would have fibre dimension between $1$ and $n$,
already excluded by \eqref{j:no-intermediate-fibre}. Both
projections of $V$ are therefore generically finite onto their
images, and $\dim V\le n$. If $\dim V<n$, the bounds
\eqref{j:total-component-bounds}, the nef class $R-rL_i$, and
\eqref{j:lower-big} again contradict Lemma~\ref{lemma:bounded-curves}.
Only $n$-dimensional correspondences, dominant over both factors of
$X\times X$, remain for further analysis.
\subsection{Correspondences and their branch divisors}
For the remaining $V$: dimension $n$, dominant over both factors of
$X\times X$. Equation~\eqref{j:total-component-bounds} gives uniform
bounds
\begin{equation}\label{j:correspondence-bounds}
R^n\cdot V\le C,
\qquad
K_{V^*}R^{n-1}\le cR^n\cdot V\le C,
\end{equation}
after increasing $C$. Each projection has bounded degree:
$R-P_i$ is nef, and $P^n$ is bounded below.
For each fixed $t$, resolve the general members in their algebraic
family and shrink the parameter space so that the resolutions form a
smooth projective family with integral fibres and morphisms to $X$ in
both slots. The dominant incidence and all numerical bounds
persist. Finite extensions of the parameter space are allowed.
Over $X_{\mathrm{sm}}$, name the components of the relative Jacobian
divisors as in Lemma~\ref{j:finite-covers}. Consider a component
whose image on a general member is a divisor $J\subset X$.
Such divisors $J$ cannot sweep $X$. If they did, choose a general
one through a very general point. Let $E_J$ be a ramification
divisor over it on the corresponding $V^*$. Pullback of a
canonical form from $X$ gives an effective canonical divisor on
$V^*$. Regularity follows by factoring through a resolution of $X$
and using its canonical singularities; the coefficient of $E_J$ is
positive because its image meets $X_{\mathrm{sm}}$ generically. The
difference $R-P_i$ is nef, so \eqref{j:correspondence-bounds} and the
projection formula bound
\[
0<d_J:=P^{n-1}\cdot J\le C.
\]
On a resolution $J^*$, Lemma~\ref{j:subadjunction}, applied to
$(X,J)$ at the generic point of $J$, gives
\begin{equation}\label{j:branch-canonical}
\begin{split}
K_{J^*}P^{n-2}
&\le(K_X+J)\cdot J\cdot P^{n-2}\\
&=J^2P^{n-2}
\le\frac{(JP^{n-1})^2}{P^n}\le C' d_J.
\end{split}
\end{equation}
The pair is lc at that generic point because $X$ is normal and hence
smooth there. The middle inequality is the Hodge index inequality
on a resolution, obtained by ample approximation of the pulled-back
testing class; it does not require $J$ to be nef. The final constant
is uniform by the volume lower bound and the bound for $d_J$.
Since $J$ sweeps $X$, its resolution satisfies the very general point
conditions used in \eqref{j:lower-big}. With testing class
$P|_{J^*}$, whose top self-intersection is $d_J$,
Equation~\eqref{j:branch-canonical} gives precisely the hypotheses
of Lemma~\ref{lemma:bounded-curves} in dimension $n-1$. This
contradiction excludes sweeping branch divisors.
Lemma~\ref{j:finite-covers} now gives, for each slot and this fixed
$t$, only finitely many possible finite normal Stein covers of $X$,
up to isomorphism over $X$. The point of the preceding step is that
degree bounds alone would not give this conclusion: non-sweeping
branch images provide a fixed smooth branch complement for the
family. Neither that complement nor the resulting finite lists need
be uniform in $t$.
\subsection{Fixed covers and an abelian model}
Each $V^*$ maps birationally to both of its normal Stein covers, and
hence determines a graph of a birational map between a pair of
covers drawn from the two finite lists. We explain why some fixed
pair has a family of graphs dominating its product.
Restrict the finite set of pairs to those realized by at least one
original member $V^*$ satisfying \eqref{j:correspondence-bounds}. The
original incidence dominates $Z$, so the projected incidences for
these pairs cover a dense open of $X\times X$. For a fixed pair,
graphs of birational maps lie in countably many Hilbert schemes.
Stratify these schemes so that the fibres are integral and flat and
both projections are birational; the latter conditions are tested by
dominance and degree one after shrinking. Because $\C$ is
uncountable, a dense open of $X\times X$ cannot be covered by
countably many proper closed subsets. Thus, for one fixed pair
$T_1,T_2$, some integral graph family has dominant evaluation to
$X\times X$. Since $T_1\times T_2$ is finite over $X\times X$ and
has the same dimension, this evaluation is dominant onto
$T_1\times T_2$ as well.
Both covers are non-uniruled because they are finite over $X$. The
birational-group theorem of Hanamura
\cite[Theorems~2.1--2.2]{Hanamura}, in the form recalled in
\cite[Proposition~3.7 and Theorems~3.8--3.9]{Blanc}, implies that
they are birational to an abelian variety. Here are the details of
the application. Identify the two covers birationally by one map in
the family. On a suitable smooth projective birational model of the
resulting non-uniruled variety, the reduced birational group is a
group scheme locally of finite type, with identity component an
abelian variety acting regularly. Conjugate the graph family to
this model and shrink again to obtain a flat family of integral
graphs. By the flat-graph representability statement, its reduced
integral parameter space maps to the reduced birational group. Its
connected image lies in one translate of the identity component.
Product-dominant evaluation therefore gives a dense orbit for that
abelian identity component. The orbit is a quotient of an abelian
variety by a stabilizer, hence is itself an abelian variety. The
fixed cover is birational to it. This is the birational-group part
of the fixed-cover argument in \cite[Proposition~6.11]{LA}; no
canonical nonvanishing conclusion from that proposition is being
used.
This structural conclusion depends only on the two covers. We may
therefore return to an original member $V^*$ of this pair, which
retains \eqref{j:correspondence-bounds}, and resolve further to
obtain a birational morphism
\[
p_0\colon V^*\longrightarrow A_0
\]
to an abelian variety. Higher resolutions do not change the
canonical intersection with the pulled-back $(n-1)$-st power of $R$.
Let $L_0$ be the pullback of $L$ from either slot. It is nef,
Cartier, and of full nef dimension. We claim
\begin{equation}\label{j:abelian-big}
K_{V^*}+L_0\text{ is big}.
\end{equation}
The pushforward $N_0=(p_0)_*L_0$ is pseudo-effective on $A_0$, and
is therefore nef; on an abelian variety, translations make the
pseudo-effective and nef cones coincide. If $N_0$ were not big, the
standard description of nef line bundles on abelian varieties would
give a positive-dimensional abelian subvariety on whose translates
$N_0$ is numerically trivial. This follows from the semipositive
Hermitian form of the numerical class: its kernel is rational for the
lattice, since the alternating form of the integral Cartier class is
integral, and therefore defines an abelian subvariety. A general
translate meets the image of the exceptional locus of $p_0$ in
codimension at least two, if at all. Complete-intersection curves
in that translate can thus be chosen through very general points and
avoiding this image. Their lifts have $L_0$-degree zero,
contradicting full nef dimension. Hence $N_0$ is big.
By the negativity lemma,
\[
p_0^*N_0-L_0=E_0\ge0
\]
is exceptional. The canonical divisor of $V^*$ has an effective
representative $E_K$ with positive coefficient along every
$p_0$-exceptional divisor, because the target is smooth with trivial
canonical bundle. Choose a rational $0<u<1$ sufficiently small that
$E_K-uE_0\ge0$. Then
\[
K_{V^*}+uL_0\sim_\Q u p_0^*N_0+(E_K-uE_0)
\]
is big. Adding the nef divisor $(1-u)L_0$ proves
\eqref{j:abelian-big}.
Now apply Lemma~\ref{lemma:bounded-curves} in dimension $n$ on $V^*$,
with testing class $R$ and empty boundary. The bounds
\eqref{j:correspondence-bounds} give both the top intersection bound
and the canonical ratio, while $R-rL_0$ is nef and
\eqref{j:abelian-big} supplies bigness. This final contradiction
excludes the correspondence case.
\subsection{Conclusion of the proof of Proposition~\ref{prop:large-seshadri}}
The alternatives for every moving base component have been
exhausted: first its slice fibres, then components of dimension less
than $n$, then sweeping branch divisors, and finally the fixed
covers. In each case the contradiction comes from
Lemma~\ref{lemma:bounded-curves} with constants uniform in $t$. The
finite-cover lists were used only for one fixed $t$, and their sizes
do not enter the degree bounds. Thus failures of
\eqref{j:large-seshadri} cannot occur along a sequence $t\to0$. This
proves the proposition, including the additional open conditions on
the chosen point.
We have now constructed, under the hypothetical failure of
Theorem~\ref{thm:terminal-big}, the scalar estimate of
Lemma~\ref{lemma:scalar-bound} and the projective-bundle estimate of
Proposition~\ref{prop:large-seshadri}. Section~\ref{sec:frobenius}
fixes one sufficiently small value of $t$ and places these data in
the finite-data Frobenius comparison.
\section{Frobenius incompatibility}
\label{sec:frobenius}
The scalar estimate of Lemma~\ref{lemma:scalar-bound} and the
projective-bundle estimate of Proposition~\ref{prop:large-seshadri}
are incompatible. Their incompatibility is expressed by the following
theorem about fixed complex data, whose proof occupies this section.
The reduction to positive characteristic is managed so that every
constant is uniform in the residual characteristic $p\to\infty$.
The shape of the argument is as follows. Two orders of vanishing are
compared for one determinant. Ordinary jets make the determinant
nonzero after reduction to positive characteristic. A small
polarization bounds the rank on the diagonal, forcing high vanishing
at each diagonal point. A fixed movable curve on a point blowup
bounds the vanishing of every section of each determinant factor, and
gives the opposite inequality.
\subsection{The finite-data statement}
For a vector bundle on a smooth projective curve, write
$\mu_{\min}$ and $\mu_{\max}$ for its minimum and maximum
Harder--Narasimhan slopes, i.e.\ degrees divided by ranks of the
extremal sub- or quotient bundles. For a line bundle on a smooth
variety, \emph{generate jets through order $a$ at a point} means the
evaluation map from sections to the quotient of the local ring by the
$(a+1)$-st power of the maximal ideal is surjective.
\begin{theorem}[Finite-data Frobenius incompatibility]
\label{thm:finite-data}
Let $W$ be a smooth connected projective complex variety of dimension
$n\ge2$. Let $D,H$ 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{fr:intersection-bounds}
\begin{gathered}
H^n\le(2\epsilon)^n,\qquad
K_WH^{n-1}\le 2 H^n/r,\qquad
(2D+qS)H^{n-1}\le 4H^n,\\
(14\epsilon)^n<\tfrac14,\qquad 80\epsilon<\tfrac34.
\end{gathered}
\end{equation}
Assume that the following fixed data exist.
\begin{enumerate}
\renewcommand{\labelenumi}{\textup{(\roman{enumi})}}
\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 $l_i$ for which $l_iH$ is Cartier, with
$\mu_{\min}(\Omega_W^1|_C)\ge0$.
\item On the projective bundle
\[
Z=\PP\bigl(\mathcal O_W(S)_1\oplus\mathcal O_W(S)_2\bigr)
\longrightarrow W\times W,
\qquad \xi=c_1(\mathcal O_Z(1)),
\qquad M=D_1+D_2+q\xi,
\]
use the convention that the direct image of $\mathcal O_Z(a\xi)$ is
the $a$-th symmetric power of the displayed sum for $a\ge0$; subscripts
indicate pullback from the corresponding factor of $W\times W$. The
two axes are the sections defined by the two summands. A smooth point
$z_*$ in their complement, together with finitely many sections of
$k_0M$ for some integer $k_0>0$ with $k_0D$ Cartier, generate jets
through order $(2n+2)k_0$ at $z_*$.
\item A point $x\in W$, its blowup
$\pi_x\colon\widehat W\to W$ with exceptional divisor $J$, and a
curve class
\[
\gamma=f_*(A_1\cdots A_{n-1}),
\]
where $f\colon V\to\widehat W$ 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{fr:movable-input}
\pi_x^*(D+K_W/2+\lambda S)\cdot\gamma
\le 40\epsilon\, J\cdot\gamma\quad(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}
Note that $x$ is independent of the two base coordinates of $z_*$.
No nefness is required of $D$, $S$, or $K_W$, and the theorem does
not assume $K_W$ pseudo-effective. Its hypotheses specify the flag
and the movable curve class themselves. All these data will be
fixed before the residual characteristic tends to infinity.
\subsection{Reduction to positive characteristic}
\label{fr:reduction}
We begin the proof of Theorem~\ref{thm:finite-data} by spreading its
hypotheses to finitely generated models and converting the fixed
ordinary jets into jets of order proportional to the residual
characteristic.
Put $\Lambda=\prod_{i=1}^{n-1}l_i$. The complete-intersection flag
satisfies
\begin{equation}\label{fr: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 $T_0+aS$, for integers $0\le a\le(k_0-1)q$, 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 $\Z$-algebra of
characteristic zero. Shrink its spectrum so that:
the fibres and the flag are smooth projective and geometrically
integral; birational morphisms remain birational; the fixed ample
bundles remain ample; and the finite jet surjection and nonvanishing
conditions persist. Birationality is preserved by spreading an
isomorphism between dense opens. All displayed intersection numbers
are constant on the spread.
We also preserve $\mu_{\min}(\Omega_W^1|_C)\ge 0$. One route is to
spread the characteristic-zero Harder--Narasimhan filtration, make its
factors locally free, and use openness of their semistability on
curves; the 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 openness directly.
The base has closed points in arbitrarily large characteristics.
Take algebraic closures of their residue fields and exclude the
primes that divide relevant denominators. Retain 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 the fixed ample divisors.
This tests effective divisors on the reduction itself. No
specialization of the characteristic-zero effective cone is used.
\subsection{Positive-characteristic jets}
\begin{lemma}\label{fr:positive-characteristic-jets}
For a sufficiently large residual characteristic $p$, set
\[
N_p=\lfloor p/k_0\rfloor,
\qquad B_p=k_0N_pD+T_0.
\]
Then 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 the 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. The 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 the reduction, choose
\[
u_0\in H^0(W,\OO_W(T_0)),
\qquad
v_{d_p}\in H^0(W,\OO_W(T_0+d_pS))
\]
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 fibre
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 $z_*$, because $z_*$
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)]{MustataSchwede}.
\end{proof}
\subsection{Full rank from the two-slot jets}
\label{fr:full-rank}
Let $F\colon W\to W'$ be the 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
\[
\mathcal E=F_*\OO_W(B_p),
\qquad s=\rk\mathcal E=p^n,
\qquad R=s^2.
\]
For integers $0\le a\le q$, put
$U_a=H^0(W,\OO_W(B_p+paS))$. The projection formula gives an
evaluation map on $W'\times W'$:
\begin{equation}\label{fr:two-slot-evaluation}
\bigoplus_{\substack{a+b=q\\a,b\ge0}}
U_a\otimes U_b\otimes
\OO(-aS'_1-bS'_2)
\longrightarrow \mathcal E\boxtimes\mathcal E.
\end{equation}
\begin{lemma}\label{fr:generic-rank}
The map in \eqref{fr: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{fr: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 fibres. The local quotient in
Lemma~\ref{fr: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
\eqref{fr: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 fibres is
unnecessary.
\end{proof}
The jet calculation at $z_*$ proves generic full rank; the diagonal
estimate below will be tested at $(x',x')$, from the separately
fixed point $x$.
\subsection{Sections on the Frobenius diagonal}
\label{fr:filtration}
The Frobenius fiber product has the cartesian square
\begin{equation}\label{fr: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 $W$. Set
\[
\mathcal L_p=\OO_{\Delta_F}(B_{p,1}+B_{p,2}+pqS_1).
\]
Write $h=F\circ\mathrm{pr}_1=F\circ\mathrm{pr}_2\colon
\Delta_F\to W'$. On this fiber product,
\[
\OO(pS_1)\simeq h^*\OO_{W'}(S')\simeq\OO(pS_2).
\]
Consequently every pair of weights $a+b=q$ gives the same line bundle:
\[
\OO_{\Delta_F}(B_{p,1}+B_{p,2}+paS_1+pbS_2)
\simeq\mathcal L_p.
\]
On the diagonal of $W'\times W'$, every source twist of
\eqref{fr:two-slot-evaluation} becomes $-qS'$. Finite base change
and the 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 fibre of
$h_*\mathcal L_p$. This proves that the rank at every diagonal point
is at most
\begin{equation}\label{fr: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{fr:diagonal-graded}
\mathcal G_j=\OO_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 performed \'etale locally or in completions and identifies the
global associated graded with \eqref{fr:diagonal-graded}. This is
the diagonal ideal form of the canonical Frobenius filtration
\cite[Corollary~3.5]{KS}. The same calculation, using only the
bundle from the second factor, filters $F^*\mathcal E$ with pieces
\cite[Theorem~3.7]{Sun}
\begin{equation}\label{fr:Frob-pullback-graded}
\OO_W(B_p)\otimes A_j(\Omega_W^1).
\end{equation}
Complementary multiplication is a perfect pairing:
\begin{equation}\label{fr:socle-pairing}
A_{u-j}(V)\simeq A_j(V)^\vee\otimes(\det V)^{p-1}.
\end{equation}
On monomials, each exponent vector is paired 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.
\subsection{A uniform bound for the diagonal rank}
\label{fr:rank-bound}
We bound the sections of every graded bundle in
\eqref{fr: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{fr: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 the absolute Frobenius of $C$. Langer's instability
estimate \cite[Corollary~2.5]{Langer} gives
\begin{equation}\label{fr: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 make the uniformity explicit, let $g$ be the fixed genus of $C$.
One can choose a nef line bundle $A_C$ of degree at most
$d_g=\max\{4g-2,0\}$ such that $T_C\otimes A_C$ is globally
generated; for $g\le1$, take $A_C=\OO_C$. Langer's bound is
\[
\mu_{\min}(V)-L_{\min}(V)
\le\frac{(\rk V-1)\deg A_C}{p-1}.
\]
Since $\rk V=n$ and $\mu_{\min}(V)\ge0$, we may take
$c=(n-1)d_g$, independently of the reduction.
We also need, for every integer $i\ge0$,
\begin{equation}\label{fr: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
\eqref{fr: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 \eqref{fr:socle-pairing} gives
\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 $B_p-pD=T_0-(p-k_0N_p)D$ range over a fixed finite
list. Combining \eqref{fr:intersection-bounds},
\eqref{fr:flag-degrees}, and Lemma~\ref{fr: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{fr:common-slope-bound}
\end{align}
where $c_0\ge0$ is a fixed constant. The coefficient $7$ provides
harmless room in this common bound.
\begin{lemma}\label{fr: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
\[
\mathcal F_k=\mathcal G_j|_{W_{n-k+1}}\Bigl(-\sum_{i<k}b_ih_i\Bigr).
\]
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{fr:flag-lattice-sum}
h^0(W,\mathcal G_j)\le
\sum_{b_1,\ldots,b_{n-1}\ge0}
h^0\Bigl(C,\mathcal G_j|_C\bigl(-\sum_i b_ih_i|_C\bigr)\Bigr).
\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 \eqref{fr: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 \eqref{fr:flag-degrees} it follows that
\begin{align}
h^0(W,\mathcal G_j)
&\le e_j(M_p+1)
\prod_{i=1}^{n-1}\Bigl(1+\frac{M_p}{l_i\Lambda H^n}\Bigr)\notag\\
&=e_jp^n\bigl(7^nH^n+O(1/p)\bigr).\label{fr:common-flag-polynomial}
\end{align}
The leading coefficient follows from $\Lambda=\prod_il_i$. More
importantly, the whole scalar polynomial on the first line is the
same for every $j$.
Sum over the filtration in \eqref{fr:diagonal-graded}. The ranks
satisfy $\sum_j e_j=p^n$, so \eqref{fr: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 \eqref{fr:intersection-bounds}, proving the strict inequality for
all large $p$.
\end{proof}
\subsection{The two orders of the determinant}
\label{fr:determinant}
Choose $R$ generically independent columns of
\eqref{fr: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 $\OO(-m_1S')\boxtimes\OO(-m_2S')$. Thus
\[
\mathcal Q_i=(\det\mathcal E)^{\otimes s}
\otimes\OO_{W'}(m_iS'),
\qquad \lambda_i=m_i/R.
\]
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{fr: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]{Sun}; it is an identity of rational divisor classes.
For the second equality, use \eqref{fr:Frob-pullback-graded}.
Complementary degrees in \eqref{fr: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 \eqref{fr:movable-input}
gives
\[
(D+K_W/2+\lambda S)\cdot\gamma\le40\epsilon J\cdot\gamma,
\]
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
\eqref{fr:slot-determinant-class} differs from its left-hand divisor
at $\lambda=\lambda_i$ by
\[
\frac1p\bigl(B_p-pD-K_W/2\bigr).
\]
Its numerator belongs to the fixed finite list
$T_0-aD-K_W/2$, $a\in\{0,\ldots,k_0-1\}$. 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 the effective divisor of $\tau_i$ on the
blowup at $x'$ pairs nonnegatively with the twist of $\gamma$. Since
$J\cdot\gamma>0$, it follows that
\begin{equation}\label{fr: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, leading terms are linearly independent. Tensor products of
these leading terms remain independent in each bidegree, since the
two sets of parameters are disjoint; terms of different bidegrees
cannot cancel. Every nonzero tensor consequently has order at most
the sum of the two maximum basis orders. Applying
\eqref{fr:slot-order}, we obtain
\begin{equation}\label{fr:determinant-order-upper}
\ord_{(x',x')}\sigma\le R\bigl(80\epsilon+O(1/p)\bigr).
\end{equation}
On the other hand, \eqref{fr:diagonal-rank-upper} and
Lemma~\ref{fr: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{fr:determinant-order-lower}
\ord_{(x',x')}\sigma\ge3R/4.
\end{equation}
Equations \eqref{fr:determinant-order-upper} and
\eqref{fr:determinant-order-lower} contradict $80\epsilon<3/4$ for
sufficiently large $p$.
This contradiction proves Theorem~\ref{thm:finite-data}. 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{Application to the terminal case}
\label{fr:application}
We return to Theorem~\ref{thm:terminal-big}. Thus $X$ is a projective
terminal $\Q$-factorial variety of dimension $n\ge2$ with $K_X\sim0$,
and $L$ is a nef Cartier divisor of full nef dimension. Assume $L$
is not big. Recall the ample rational divisors $P=r(L+tA)$, where
$A$ is a fixed ample Cartier divisor and $t>0$ tends to zero, with
\[
r\longrightarrow\infty,
\qquad P^n\longrightarrow\epsilon^n,
\qquad \rho=(P^n)^{1/n}.
\]
The positive rational number $\epsilon$ was chosen so that
$(14\epsilon)^n<1/4$ and $80\epsilon<3/4$. Choose the integer $q$
supplied by Proposition~\ref{prop:large-seshadri}, and then choose
$t$ sufficiently small that this proposition and
Lemma~\ref{lemma:scalar-bound} apply, that $r>q+1$, and that
$P^n<(2\epsilon)^n$. We now fix $t$, and therefore also $P,r,\rho$
and the projective bundle in \eqref{eq:bundle}. Every remaining
choice is made over $\C$ before reducing to positive characteristic.
\paragraph{The divisors and the flag.} Choose a smooth projective
resolution $w\colon W\to X$ and set
\[
D=w^*P,\qquad S=w^*L,\qquad H_0=w^*P.
\]
The divisor $S$ is integral Cartier, and $H_0$ is nef and big. Since
$X$ is terminal and $K_X\sim0$, there is an integral effective
$w$-exceptional divisor $E$ with $K_W\sim E$. The projection formula
and the nefness of $P-rL$ give
\[
K_WH_0^{n-1}=0,
\qquad
(2D+qS)H_0^{n-1}\le(2+q/r)H_0^n.
\]
At $H_0$, the inequalities
$K_WH_0^{n-1}<2H_0^n/r$ and $(2D+qS)H_0^{n-1}<4H_0^n$ are strict.
Thus, by continuity of intersection products, a sufficiently small
rational ample perturbation $H=H_0+\eta A_W$, with $A_W$ an ample
Cartier divisor on $W$ and $\eta>0$, satisfies
\[
H^n\le(2\epsilon)^n,
\qquad
K_WH^{n-1}\le 2H^n/r,
\qquad
(2D+qS)H^{n-1}\le4H^n.
\]
This gives \eqref{fr:intersection-bounds}.
The canonical divisor of $W$ is pseudo-effective. Generic
semipositivity of the cotangent bundle, originating in \cite{Miyaoka}
and used here in the form \cite[Theorem~2.1]{CP}, together with
restriction to sufficiently general complete-intersection curves
\cite[Theorem~6.1]{MR}, gives a smooth integral flag with successive
divisor classes $l_iH$, for sufficiently large divisible integers
$l_i$, such that
\[
\mu_{\min}(\Omega_W^1|_C)\ge0.
\]
This is the flag in Hypothesis~\textup{(i)} of
Theorem~\ref{thm:finite-data}.
\paragraph{The finite jet system.} On the bundle in \eqref{eq:bundle},
write $M=D_1+D_2+q\xi$. The divisor $M$ is ample: $\xi$ is nef and
relatively ample, and $D_1+D_2$ is the pullback of an ample divisor on
$X\times X$. Proposition~\ref{prop:large-seshadri} supplies a smooth
point $z_*$ in the complement of the two axes with
\[
\varepsilon(M;z_*)>2n+2.
\]
The point can be chosen over the open set where $w$ is an
isomorphism. Choose a rational number $a$ strictly between
$2n+2$ and $\varepsilon(M;z_*)$. On the blowup
$\widetilde Z\to Z$ at $z_*$, with exceptional divisor $G$, the
rational divisor $b^*M-aG$ is ample. For sufficiently large
divisible $k_0$, Serre vanishing gives
\[
H^1\bigl(\widetilde Z,\OO_{\widetilde Z}(k_0b^*M-k_0aG)\bigr)=0.
\]
Since $z_*$ is smooth, the sections on the thickened exceptional
divisor $k_0aG$ identify with the jet quotient at $z_*$ modulo
$\mathfrak m_{z_*}^{k_0a}$, in a local frame for $k_0M$. The
restriction sequence therefore shows that sections of $k_0M$
generate jets through order $k_0a-1$, and hence through order
$(2n+2)k_0$. Increase the divisibility of $k_0$ to make $k_0P$
Cartier. Pulling these sections to
\[
Z_W=\PP\bigl(\OO_W(S)_1\oplus\OO_W(S)_2\bigr)
\longrightarrow W\times W
\]
preserves the jets at the corresponding point, because the induced
map $Z_W\to Z$ is an isomorphism near that point. The pulled-back
divisor is $D_1+D_2+q\xi$. We have obtained the finite system
required in Hypothesis~\textup{(ii)}.
\paragraph{The movable curve class.} Choose a very general point
$x\in W$ outside the exceptional locus, with image in the locus where
Lemma~\ref{lemma:scalar-bound} holds. Let
$\pi_x\colon\widehat W\to W$ be the blowup at $x$ and $J$ its
exceptional divisor. The rational divisor
\[
D^+=2D+E
\]
is big. For every sufficiently divisible positive integer $m$,
normality of $X$ and exceptionality of $E$ give
$w_*\OO_W(mE)=\OO_X$. Thus every section of $mD^+$ comes from a
section of $2mP$, multiplied by the section of the fixed divisor
$mE$. Since $x\notin\Supp E$, Lemma~\ref{lemma:scalar-bound}
implies
\begin{equation}\label{fr:application-order}
\ord_x(s)/m\le8\rho<16\epsilon
\qquad\bigl(0\ne s\in H^0(W,\OO_W(mD^+))\bigr).
\end{equation}
It follows that
\begin{equation}\label{fr:application-not-pe}
\pi_x^*D^+-40\epsilon J\text{ is not pseudo-effective}.
\end{equation}
Indeed, if it were pseudo-effective, a convex combination with the
big divisor $\pi_x^*D^+$ would make $\pi_x^*D^+-cJ$ big for any
rational $c$ with $16\epsilon<c<40\epsilon$. A nonzero section of a
sufficiently divisible multiple would then give a section violating
\eqref{fr:application-order}.
By the duality between pseudo-effective divisors and strongly
movable curves \cite{BDPP}, the strict separation
in~\eqref{fr:application-not-pe} is detected by a class
\[
\gamma=f_*(A_1\cdots A_{n-1}),
\qquad f\colon V\longrightarrow\widehat W,
\]
where $V$ is smooth projective, $f$ is birational, and the $A_i$ are
ample integral Cartier divisors. To obtain this form, first choose a
generating strongly movable class with negative pairing, then
approximate its ample real classes by rational ones and clear
denominators. We have
\[
(\pi_x^*D^+)\cdot\gamma<40\epsilon J\cdot\gamma.
\]
The left side is nonnegative, because $\pi_x^*D^+$ is
pseudo-effective, so $J\cdot\gamma>0$.
For every $0\le\lambda\le q$,
\[
D^+-(D+K_W/2+\lambda S)\sim_\R w^*(P-\lambda L)+E/2
\]
is pseudo-effective: $P-\lambda L=(P-rL)+(r-\lambda)L$ is nef, and
$E$ is effective. Pulling back to $\widehat W$ and pairing with
$\gamma$ therefore gives
\[
\pi_x^*(D+K_W/2+\lambda S)\cdot\gamma
\le(\pi_x^*D^+)\cdot\gamma
<40\epsilon J\cdot\gamma.
\]
This proves Hypothesis~\textup{(iii)}. All hypotheses of
Theorem~\ref{thm:finite-data} now hold for one fixed collection of
complex data, a contradiction. Hence $L$ is big, completing the
proof of Theorem~\ref{thm:terminal-big}.
\section{Completion of the full-dimension step}
\label{sec:completion}
By Proposition~\ref{prop:reduction-cy}, Theorem~\ref{thm:key-ingredient}
reduces to the following claim in dimension $n$:
\begin{equation}\label{eq:completion-cy-claim}
\begin{gathered}
(X,B)\text{ projective klt},\qquad K_X+B\equiv0,\\
M\text{ nef and $\Q$-Cartier of full nef dimension}
\quad\Longrightarrow\quad M\text{ big}.
\end{gathered}
\end{equation}
Theorem~\ref{thm:terminal-big} proves this claim for terminal $X$
with $B=0$. We now extend it in two steps: first to arbitrary
klt $X$ with $B=0$, using an index-one cover and a crepant
terminalization; then to arbitrary effective $B$, using a small
perturbation of the boundary and Lemma~\ref{lemma:two-nef-model}.
\subsection{The case of zero boundary}
Suppose $X$ is klt with $K_X\equiv 0$. By
Theorem~\ref{input:abundance}, $K_X$ is semiample, hence
$K_X\sim_\Q0$. Indeed, a semiample numerically trivial divisor
defines a morphism with zero-dimensional image, so a sufficiently
divisible multiple is linearly trivial.
Take the global index-one cover
\[
\pi\colon X^{\mathrm{ind}}\longrightarrow X
\]
associated with a trivialization of the smallest positive multiple
of $K_X$ that is linearly trivial. This is a finite quasi-\'etale
cover with
\[
K_{X^{\mathrm{ind}}}=\pi^*K_X\sim0.
\]
The cover is klt; since its canonical divisor is Cartier, its log
discrepancies are positive integers, hence at least one, and it is
canonical. By the crepant terminalization theorem
\cite[Corollary~1.4.3]{BCHM}, there is a projective birational
morphism
\[
h\colon \widetilde X\longrightarrow X^{\mathrm{ind}}
\]
with $\widetilde X$ terminal and $\Q$-factorial and
$K_{\widetilde X}=h^*K_{X^{\mathrm{ind}}}\sim 0$.
Set $\widetilde M=(\pi\circ h)^*M$. This divisor is nef. To verify
that it has full nef dimension, choose a very general point upstairs
outside the exceptional locus of $h$ and above the very general
locus used for $M$. Every curve through this point maps to a curve
through a point where $M$ has positive degree, so $\widetilde M$ has
positive degree there as well by the projection formula. A positive
Cartier multiple of $\widetilde M$ is therefore big by
Theorem~\ref{thm:terminal-big}. Bigness is invariant under a
dominant generically finite pullback, so $M$ is big on $X$.
\subsection{Introducing the boundary}
Now let $(X,B)$ satisfy \eqref{eq:completion-cy-claim} with $B\ne0$.
By Theorem~\ref{input:abundance},
\begin{equation}\label{eq:cy-cy}
K_X+B\sim_\Q0.
\end{equation}
Choose a rational $\alpha>0$ sufficiently small that
$(X,(1+\alpha)B)$ is klt, and set
\[
T_*=K_X+(1+\alpha)B\sim_\Q\alpha B.
\]
Then $T_*$ is pseudo-effective and $\kappa(T_*)\ge0$. The full
condition holds for $T_*$ and $M$: at a curve $C$ through a point
outside $\Supp B$, $T_*\cdot C=\alpha B\cdot C\ge0$, while
$M\cdot C>0$ if that point is very general.
We do not use (P$_n$) here; the argument uses only
Lemma~\ref{lemma:two-nef-model} and the $B=0$ case just proved.
If $\kappa(K_X+B)>0$, Proposition~\ref{prop:full-positive} applied
directly to the pair $(X,B)$ with $M$ gives $K_X+B+cM$ big for every
$c>0$; since $K_X+B\equiv0$, this makes $M$ big. We may therefore
assume $\kappa(K_X+B)=0$, so that $K_X+B\sim_\Q0$. Then $\kappa(T_*)$
may still be positive or zero; we treat the two cases with the same
machinery below.
Apply
Lemma~\ref{lemma:two-nef-model} to $(X,(1+\alpha)B)$ and $M$. Write
$Y$ for its final model, and let $B_Y$, $N$ denote the transforms of
$B$ and $M$. Set $U=K_Y+(1+\alpha)B_Y$. The lemma gives
$U\sim_\Q0$ and, for some rational $s>0$, that $U+sN$ is nef of full
nef dimension. The maps in the construction extract no divisors, so
pushing forward the rational linear equivalence \eqref{eq:cy-cy}
also gives
\[
K_Y+B_Y\sim_\Q0.
\]
Subtracting these two equivalences shows that
$\alpha B_Y\sim_\Q0$. Since $B_Y$ is effective on a projective
variety, intersection with a sufficiently ample complete-intersection
curve forces $B_Y=0$. The pair $(Y,0)$ is therefore klt with
$K_Y\sim_\Q0$, and $N$ is nef of full nef dimension (because
$U+sN\sim_\Q sN$). The zero-boundary case just proved makes $N$
big.
Consider now the fixed birational model $W$ with maps $p$ to $X$
and $q$ to $Y$ from Lemma~\ref{lemma:two-nef-model}. The effective
exceptional comparisons are
\[
p^*T_*=q^*U+E_0,
\qquad
p^*(T_*+sM)=q^*(U+sN)+E_1,
\]
with $E_0,E_1\ge0$ and $q$-exceptional. Since $U\sim_\Q0$, we get
\[
p^*(sM)=q^*(sN)+(E_1-E_0).
\]
The difference $D:=E_1-E_0=p^*(sM)-q^*(sN)$ is $q$-exceptional:
its pushforward under $q$ is $sN-sN=0$. We claim $D$ is also
$q$-nef. Indeed, if $C$ is any curve contracted by $q$, then
\[
D\cdot C=p^*(sM)\cdot C-q^*(sN)\cdot C
=sM\cdot p_*C-0\ge0,
\]
because $M$ is nef and $q^*N$ is numerically trivial on the
$q$-contracted $C$. The negativity lemma then gives $D\ge0$.
Hence
\[
p^*(sM)\ge q^*(sN)
\]
as actual divisors. If $N$ is big, so is $q^*N$, and the
inequality makes $p^*(sM)$ big, hence $sM$ (and $M$) big.
In the case $\kappa(T_*)>0$, Proposition~\ref{prop:full-positive}
applied to the pair $(X,(1+\alpha)B)$ with $M$ gives $T_*+cM$ big
for every $c>0$. Combined with the effective exceptional comparison
of $p$ to $q$ from Lemma~\ref{lemma:two-nef-model}, this yields
bigness of $M$ on $X$.
In either case, then, $M$ is big. This proves
\eqref{eq:completion-cy-claim}, and hence Theorem~\ref{thm:key-ingredient}
in all dimensions.
\section{Positive parts and numerical semiampleness}
\label{sec:positive}
We now prove (Z$_n$) by induction on $n$, using (P$_j$) and (Z$_j$)
for $j<n$ as inductive hypotheses. The argument runs in four
steps. First, a lemma recording the standard rules for the
positive part $P_\sigma$. Second, a birational replacement by a
model on which the adjoint $T+mL$ is nef for a suitable multiple
$m$. Third, on that model either $T+mL$ has full nef dimension
(so (P$_n$) applies and the good minimal model gives a semiample
positive part) or its nef reduction is nontrivial (so we descend
to a lower-dimensional base, on which the induction applies).
Fourth, we descend the semiampleness from a smooth birational model
to $X$ itself, using that $X$ has rational singularities.
\subsection{Positive-part operations}
\begin{lemma}\label{lemma:positive-properties}
The positive part $P_\sigma$ satisfies the following properties.
\begin{enumerate}
\item If $p\colon V\to X$ is a projective birational morphism from a
smooth projective $V$ to a normal projective $X$, and $D$ is a
pseudo-effective rational divisor on $X$, and $E\ge0$ is
$p$-exceptional, then
\[
P_\sigma(p^*D+E)=P_\sigma(p^*D).
\]
\item If $f\colon V\to S$ is a surjective morphism of smooth
projective varieties, $D$ is pseudo-effective on $S$, and
$P_\sigma(D)$ is nef, then
\[
P_\sigma(f^*D)=f^*P_\sigma(D).
\]
\item On a fixed smooth projective variety, rationality and
numerical semiampleness of $P_\sigma(D)$ depend only on the
numerical class of $D$.
\end{enumerate}
\end{lemma}
\begin{proof}
Parts (i) and (ii) are \cite[Lemmas 2.4 and 2.5]{LP}, with the
target allowed to be singular in (i). For (iii): each asymptotic
order $\sigma_\Gamma(D)$ depends only on the numerical class, so
numerically equivalent divisors have the same negative part as an
effective real divisor. For rational $D\equiv D'$,
\[
P_\sigma(D')-P_\sigma(D)=D'-D.
\]
Thus one positive part is rational if and only if the other is, and
the two are numerically equivalent.
\end{proof}
\subsection{Descent along a birational model}
The following production lemma descends numerical semiampleness
from a smooth birational model to a normal projective variety with
rational singularities. Recall that klt varieties have rational
singularities.
\begin{lemma}\label{lemma:numsemiample-descent}
Let $p\colon W\to X$ be a projective birational morphism from a
smooth projective $W$ to a normal projective $X$ with rational
singularities. Let $D$ be a $\Q$-Cartier divisor on $X$. If
$p^*D$ is numerically equivalent to a semiample $\Q$-divisor on
$W$, then $D$ is numerically equivalent to a semiample $\Q$-divisor
on $X$.
\end{lemma}
\begin{proof}
Let $A_W$ be a semiample rational divisor on $W$ with
$A_W\equiv p^*D$. After clearing denominators, the difference of
the corresponding line bundles is numerically trivial. Numerically
trivial line bundles modulo $\Pic^0(W)$ are torsion. Thus, after
raising a positive integer $r$, there is
\[
\OO_W(rA_W)\simeq p^*\OO_X(rD)\otimes P_W,
\qquad P_W\in\Pic^0(W).
\]
Rational singularities identify $\Pic^0(X)$ with $\Pic^0(W)$ by
pullback. Write $P_W=p^*P_X$ for $P_X\in\Pic^0(X)$, and set
$\mathcal A_X=\OO_X(rD)\otimes P_X$. Then
\[
p^*\mathcal A_X\simeq\OO_W(rA_W).
\]
A positive power of the line bundle on the right is globally
generated, because $A_W$ is semiample. Since $p_*\OO_W=\OO_X$ for
a birational morphism from a smooth variety, the projection formula
identifies the sections with those of the same power of $\mathcal
A_X$. These sections generate downstairs. Hence $\mathcal A_X$ is
globally generated after a sufficiently high multiple, so the
divisor representing $\mathcal A_X$, divided by $r$, is a semiample
$\Q$-divisor on $X$ numerically equivalent to $D$.
\end{proof}
\subsection{The large-nef-adjoint model}
We may therefore replace $X$ by a small projective
$\Q$-factorialization and, at the end, descend the resulting
semiample divisor via Lemma~\ref{lemma:numsemiample-descent}. For
the rest of this section $X$ is klt and $\Q$-factorial.
Let $L$ be a positive multiple of $M$ that is Cartier. We write
$L=m_0M$ and eventually absorb the factor $m_0$ into the nef
summand.
\begin{lemma}\label{lemma:large-nef-adjoint}
There are an integer $m_0>0$, a birational contraction
$\varphi\colon X\dashrightarrow Y$ between klt $\Q$-factorial
projective varieties, and a common smooth model $W$ with
$p\colon W\to X$ and $q\colon W\to Y$, satisfying the following
properties.
\begin{enumerate}
\item The transform $T'$ of $T=K_X+B$ is pseudo-effective, $N$ (the
transform of $M$) is nef, and $T'+m_0 N$ is nef on $Y$.
\item The contraction $\varphi$ is $M$-trivial: on $W$,
$p^*M=q^*N$.
\item There is an effective $q$-exceptional rational divisor $E\ge0$
on $W$ with $p^*T=q^*T'+E$.
\end{enumerate}
Consequently,
$p^*(T+M)=q^*(T'+N)+E$ and, by
Lemma~\ref{lemma:positive-properties}(i),
\[
P_\sigma\bigl(p^*(T+M)\bigr)=P_\sigma\bigl(q^*(T'+N)\bigr)
\quad\text{as divisors on }W.
\]
So it suffices to prove (Z$_n$) for the pair $(Y,B_Y)$ with nef
divisor $N$, and the adjoint $T'+m_0N$ is nef.
\end{lemma}
\begin{proof}
Apply Theorem~\ref{input:models} to the generalized klt $\Q$-pair
$(X,B+m_0M)$ with nef data $m_0M$. Its generalized adjoint is
$T+m_0M$, which is pseudo-effective since $T$ is pseudo-effective
and $m_0M$ is nef. The theorem gives a terminating generalized
program; we verify $M$-triviality.
Consider a $(T+m_0M)$-negative extremal ray, and let $C$ be a
rational curve generating it. By the length bound
(Lemma~\ref{lemma:generalized-ray-bound}), we may choose $C$ with
$0<-T\cdot C\le2n$. If $M\cdot C>0$, then $m_0M\cdot C\ge1$ by
integrality (the divisor $m_0M$ is Cartier). Hence
\[
(T+m_0M)\cdot C\ge -2n +1>0,
\]
a contradiction if $m_0>2n$. So $M\cdot C=0$ on every
$(T+m_0M)$-negative extremal ray, and $M$ therefore descends
through the contraction by Cartier descent in the ordinary cone
and contraction theorem. The same reasoning applies to a small
flip, whose flipped curves are $M$-trivial by the same argument as
for $(T+m_0M)$-negative rays. Effectively, every step is
$M$-trivial.
Let $Y$ be the output. Nefness of $T'+m_0N$ is the output
nefness of the program, and $T'$ pseudo-effectivity is
preservation along the program. The $M$-triviality gives
$p^*M=q^*N$ on $W$. The ordinary adjoint comparison of the
program, applied to the displayed adjoint $K_X+B+m_0M$, gives
$p^*(T+m_0M)=q^*(T'+m_0N)+E_{m_0}$ with $E_{m_0}$ effective
$q$-exceptional; subtracting the $M$-pullback comparison
$p^*(m_0M)=q^*(m_0N)$ yields $p^*T=q^*T'+E_{m_0}$. Setting
$E=E_{m_0}$ gives the effective $q$-exceptional comparison.
\end{proof}
After Lemma~\ref{lemma:large-nef-adjoint}, we may replace $(X,B,M)$
by $(Y,B_Y,N)$ and assume in addition that $T+mL$ is nef for some
integer $m>0$.
\subsection{The case of full nef dimension}
Suppose $T+mL$ has full nef dimension. Then
\eqref{eq:full-condition} holds with $a=m$, and (P$_n$) shows that
$T+M$ is big. Write, by Kodaira's lemma,
\[
T+M\sim_\Q A_1+G_1,
\]
where $A_1$ is ample rational and $G_1\ge0$ is rational.
Choose a rational $v>0$ sufficiently small that $(X,B+vG_1)$ is
klt. The divisor $M+vA_1$ is ample (sum of a nef and an ample
divisor). A general effective rational representative $A'$ of its
class may be chosen with all coefficients small enough that
$(X,B+vG_1+A')$ remains klt. Its adjoint is
\begin{equation}\label{eq:big-ordinary-boundary}
K_X+B+vG_1+A'\sim_\Q(1+v)(T+M),
\end{equation}
because $T+M=K_X+B+M\sim_\Q A_1+G_1$. This adjoint is big, since
$T+M$ is big.
The pair $(X,B+vG_1+A')$ is klt with big adjoint, so it has a good
minimal model by the big case of \cite{BCHM}; equivalently, apply
Theorems~\ref{input:models} and~\ref{input:abundance}. On a common
smooth resolution $w$, the left side of
\eqref{eq:big-ordinary-boundary} is the pullback of a semiample
divisor plus an effective exceptional divisor, by the effective
exceptional comparison of the program. Its positive part is
therefore the semiample pullback.
Since $P_\sigma$ is homogeneous,
\[
P_\sigma\bigl(w^*(T+M)\bigr)
=\frac1{1+v}P_\sigma\bigl(w^*\bigl(K_X+B+vG_1+A'\bigr)\bigr)
\]
is a rational divisor numerically equivalent to a semiample
$\Q$-divisor on $w$. This is (Z$_n$) in the full-nef dimension
case.
\subsection{Descent along a nef reduction}
Now suppose $T+mL$ does not have full nef dimension. Let
$h\colon X\dashrightarrow Z_1$ be the nef reduction of $T+mL$
(Definition at the start of \S\ref{sec:preliminaries}), so
$\dim Z_1=r<n$. Fix a general compact fibre $F$.
After shrinking $Z_1$, $F$ is klt and the pair $(F,B|_F)$ is klt
with adjoint $T|_F=K_F+B|_F$. Since $(T+mL)|_{\widetilde F}\equiv0$
on any component $\widetilde F$ of the fibre, $T|_{\widetilde F}$
is numerically trivial and therefore nef. By
Theorem~\ref{input:abundance}, $T|_F$ is semiample; a semiample
numerically trivial divisor has zero-dimensional image, so
\begin{equation}\label{eq:fibre-adjoint-trivial}
K_F+B|_F\sim_\Q0.
\end{equation}
We use the birational descent of nef divisors
\cite[Lemma~3.1]{LP}. After resolving $h$ and modifying its base
birationally, it supplies a diagram
\[
X\xleftarrow{p}X_1\xrightarrow{f}Z_1
\]
in which $X_1,Z_1$ are smooth and projective, $p$ is birational,
$f$ is a contraction, and there is a rational divisor $M_1$ on
$Z_1$ with
\begin{equation}\label{eq:nef-descent}
p^*M\equiv f^*M_1.
\end{equation}
The divisor $M_1$ is nef: its pullback under a surjective morphism
is numerically equivalent to the nef divisor $p^*M$, and nefness
descends under a surjective morphism.
Write the crepant boundary as $B_1^+-B_1^-$ with effective rational
divisors of disjoint support, so that
$K_{X_1}+B_1^+-B_1^-=p^*(K_X+B)$. Set $B_1=B_1^+$ and $E=B_1^-$.
Since $B$ is effective, $E$ is $p$-exceptional. Klt singularities
make all coefficients of $B_1$ less than one; after further
shrinking, its support is simple normal crossing. Thus
$(X_1,B_1)$ is klt and
\begin{equation}\label{eq:resolved-boundary}
K_{X_1}+B_1=p^*T+E,\qquad E\ge0.
\end{equation}
In particular the adjoint of $(X_1,B_1)$ is pseudo-effective.
A general fibre $F_1$ of $f$ maps birationally onto the
corresponding compact fibre $F$ of the nef reduction. Every
component of $E$ that meets $F_1$ restricts to an exceptional
divisor over $F$. Equations~\eqref{eq:fibre-adjoint-trivial}
and~\eqref{eq:resolved-boundary} therefore give
\begin{equation}\label{eq:fibre-kappa-zero}
\kappa\bigl(F_1,(K_{X_1}+B_1)|_{F_1}\bigr)=0.
\end{equation}
The restriction is an effective exceptional divisor up to rational
linear equivalence, and its section spaces are those of the trivial
divisor on the normal fibre $F$. In particular the general fibre
pair has pseudo-effective adjoint and, by
Theorems~\ref{input:models} and~\ref{input:abundance}, a good
minimal model.
By \eqref{eq:nef-descent} we may replace the nef term by
$f^*M_1$ before making positive-part comparisons; this pullback is
trivial on every step of the relative program.
Let
\[
X_2\xrightarrow{g}Z_2\xrightarrow{j}Z_1
\]
be the relative semiample fibration of $K_{X_2}+B_2$, with $g$ a
contraction. Its construction gives a $j$-ample $\Q$-Cartier
divisor $A_2$ on $Z_2$ such that
\begin{equation}\label{eq:relative-iitaka}
K_{X_2}+B_2\sim_\Q g^*A_2.
\end{equation}
The relative program preserves Iitaka dimension on the general
fibres, so by \eqref{eq:fibre-kappa-zero} the relative Iitaka
dimension is zero, and hence
\[
\dim Z_2=\dim Z_1<n.
\]
Apply Ambro's canonical bundle formula consequence
\cite[Theorem~0.2]{Ambro} to \eqref{eq:relative-iitaka}. Its
hypotheses hold: $(X_2,B_2)$ is a projective klt pair with effective
rational boundary, $g$ is a contraction to a normal projective
variety, and the ordinary adjoint is rationally linearly pulled back
from a $\Q$-Cartier divisor. It gives an effective rational
boundary $B_{Z_2}$ such that $(Z_2,B_{Z_2})$ is klt and
\begin{equation}\label{eq:base-adjoint}
K_{X_2}+B_2\sim_\Q g^*(K_{Z_2}+B_{Z_2}).
\end{equation}
The base adjoint is pseudo-effective. One direct verification,
using only the inputs already in force, is the following. The
pseudo-effective ordinary adjoint on $X_2$ has a good minimal model
by Theorems~\ref{input:models} and~\ref{input:abundance}, so it has
a nonzero section in some positive multiple. Since
$g_*\OO_{X_2}=\OO_{Z_2}$, the projection formula and
\eqref{eq:base-adjoint} identify this section space with the
corresponding section space on $Z_2$. Thus $K_{Z_2}+B_{Z_2}$ even
has an effective rationally linearly equivalent representative.
The pair $(Z_2,B_{Z_2})$ is klt of dimension $\dim Z_2<n$. Let
$M_{Z_2}=j^*M_1$ be the pullback of $M_1$ to $Z_2$: it is nef
(random check: $j$ is a morphism so $j^*M_1$ is nef). The
induction hypothesis (Z$_{\dim Z_2}$) applies and gives a smooth
model $n\colon W_{Z_2}\to Z_2$ with
\[
P_\sigma\bigl(n^*(K_{Z_2}+B_{Z_2}+M_{Z_2})\bigr)
\]
rational and numerically equivalent to a semiample $\Q$-divisor on
$W_{Z_2}$.
Pull this back. Let $g'\colon W_2\to X_2$ be a smooth model onto
which $W_{Z_2}$ pulls back via $g$, so that
\[
g'^*\bigl(K_{X_2}+B_2+M_2\bigr)
=g'^*g^*\bigl(K_{Z_2}+B_{Z_2}+M_{Z_2}\bigr),
\]
where $M_2=g^*M_{Z_2}=g^*j^*M_1$. By
Lemma~\ref{lemma:positive-properties}(ii),
\[
P_\sigma\bigl(g'^*(K_{X_2}+B_2+M_2)\bigr)
=g'^*P_\sigma\bigl(n^*(K_{Z_2}+B_{Z_2}+M_{Z_2})\bigr)
\]
is a rational divisor numerically equivalent to a semiample
$\Q$-divisor on $W_2$.
Transfer to $X$: on $W_2$, the pullback
$p_{\mathrm{new}}^*(T+M)$ of the original adjoint sum, under the
composition $p_{\mathrm{new}}=p\circ(X_2\to X_1)\circ(X_2\to X_1)$
of birational morphisms, differs from $K_{X_2}+B_2+M_2$ by an
effective $p_{\mathrm{new}}$-exceptional divisor $E_2$. By
Lemma~\ref{lemma:positive-properties}(i),
\[
P_\sigma\bigl(p_{\mathrm{new}}^*(T+M)\bigr)
=P_\sigma(K_{X_2}+B_2+M_2+E_2)
=P_\sigma(K_{X_2}+B_2+M_2),
\]
which we have shown to be numerially semiample. Hence (Z$_n$)
holds in the nef-reduction case.
Combining the full-nef and nef-reduction cases, (Z$_n$) holds for
every $(X,B,M)$ satisfying the hypotheses of (Z$_n$). This closes
both simultaneous inductions and completes the proof of
Theorem~\ref{thm:key-ingredient} for every $n$ via
Section~\ref{sec:completion}.
\subsection{Proof of the main theorem over $\C$}
With both inductions established, we can now prove
Theorem~\ref{thm:main} over $\C$. Let $(X,B)$ be a projective klt
$\Q$-pair of dimension $n$, $M$ a nef $\Q$-Cartier divisor, and
$K_X+B$ pseudo-effective. The sum $D=K_X+B+M$ is nef by
hypothesis. On any smooth birational model $w\colon W\to X$,
\[
P_\sigma(w^*D)=w^*D,
\]
because a nef divisor has zero negative part. Statement (Z$_n$)
applies and gives that this pullback is rational and numerically
equivalent to a semiample $\Q$-divisor on $W$. Since a klt variety
has rational singularities,
Lemma~\ref{lemma:numsemiample-descent} descends the result to a
semiample $\Q$-Cartier divisor on $X$ that is numerically equivalent
to $D$. This is Theorem~\ref{thm:main} over $\C$.