64c3814
==== TABLE OF CONTENTS ====
Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//main.tex
Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//preamble.tex
Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//references.tex
Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//sections/blocks.tex
Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//sections/completion.tex
Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//sections/degeneration.tex
Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//sections/fibration.tex
Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//sections/introduction.tex
Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//sections/mmp.tex
Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//sections/preliminaries.tex
Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//sections/weights.tex
===========================
==== BEGIN Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//main.tex ====
\documentclass[11pt]{article}
\input{preamble}
\title{Uniform log Iitaka fibrations\\and bounded moduli denominators}
\author{OpenAI}
\date{October 4, 2026}
\begin{document}
\maketitle
\begin{abstract}
For normal projective log canonical pairs over algebraically closed fields of
characteristic zero, of fixed dimension $d\geq5$ and with effective boundary
coefficients in a fixed finite rational set, we prove that one complete rounded
pluricanonical system generates the full Iitaka field whenever the
$\Q$-Cartier log canonical divisor has nonnegative Kodaira dimension.
Its degree depends only on the dimension and coefficient set.
For the paper's normalized canonical bundle formulae over $\C$, we also bound
the Cartier denominators of the moduli divisors on smooth projective determining
models in terms of the dimension and coefficient set.
\end{abstract}
\begingroup
\small
\tableofcontents
\endgroup
\clearpage
\input{sections/introduction}
\input{sections/preliminaries}
\input{sections/mmp}
\input{sections/weights}
\input{sections/blocks}
\input{sections/degeneration}
\input{sections/fibration}
\input{sections/completion}
\input{references}
\end{document}
==== END Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//main.tex ====
==== BEGIN Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//preamble.tex ====
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==== BEGIN Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//references.tex ====
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==== END Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//references.tex ====
==== BEGIN Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//sections/blocks.tex ====
\section{Product covers and semilinear factor actions}\label{sec:blocks}
We now prepare the klt case of the curve weight bound. A finite
cover decomposes a klt log Calabi--Yau pair into factors with controlled
holomorphic forms. The cover need not be Galois, however, and its
Galois closure need not be a product. The purpose of this section is
to retain the factor directions on that closure and to show that a
bounded power of each finite transformation acts regularly on suitable
finite covers of the individual factors. This is the comparison
construction of \cite[Lemma~3.5]{U}, extended to a rationally connected
factor carrying the boundary.
\subsection{The factors and their automorphisms}
A finite surjective morphism between normal varieties is
\emph{quasi-\'etale} if it is \'etale in codimension one. Over an
algebraically closed field of characteristic zero, purity implies that
such a morphism is \'etale over the smooth locus of its target.
For a normal variety $F$, we write $\Omega_F^{[j]}$ for the reflexive
extension of $j$-forms from its smooth locus.
\begin{proposition}[Product decomposition]\label{prop:product-cover}
Let $(V,B)$ be a projective klt pair over $\C$ with $B\geq0$,
$K_V$ rational Cartier, and $K_V+B\simq0$. There is a finite
quasi-\'etale cover $\pi:P\to V$ and a decomposition
\[
(P,\pi^*B)
\simeq (F,H)\times A\times\prod_j Y_j\times\prod_k Z_k,
\]
where $(F,H)$ is a rationally connected klt log Calabi--Yau pair,
$A$ is an abelian variety, and the $Y_j$ and $Z_k$ are respectively
irreducible Calabi--Yau and irreducible holomorphic symplectic
varieties in the singular Beauville--Bogomolov decomposition.
All factors are $\Q$-Gorenstein. Point factors are omitted, and
the boundary is pulled back entirely from $F$.
\end{proposition}
\begin{proof}
The product decomposition, including its assertion about the boundary,
is \cite[Theorem~1.3]{MW}. Its boundary-free factors have the stated
properties by the singular Beauville--Bogomolov theorem
\cite[Definition~1.4 and Theorem~1.5]{HP}. Since $K_V$ is rational
Cartier, quasi-\'etaleness makes $K_P$ rational Cartier. Restricting
the canonical sheaf in product charts, with the complementary points
smooth, gives the same property on every factor. Restriction of
$K_P+\pi^*B\simq0$ gives $K_F+H\simq0$.
\end{proof}
We use two properties of the nonabelian boundary-free factors. They
and all their connected normal finite quasi-\'etale covers have
canonical singularities and Cartier trivial canonical divisor. On
each such cover the reflexive form algebra is generated by a top form
in the Calabi--Yau case and by the pulled-back symplectic form in the
symplectic case \cite[Definition~1.4]{HP}. In particular there are no
reflexive one-forms or vector fields: contraction with a volume form,
or with the symplectic form, proves the latter assertion. Connected
normal finite quasi-\'etale covers of an abelian variety are \'etale
by purity and are again abelian varieties after an origin is chosen.
For the rationally connected factor, vector fields must be required
to preserve the boundary. If $D$ is a reduced Weil divisor on a
normal variety, $\mathcal T_F(-\log D)$ denotes the subsheaf of
derivations preserving the reduced ideal of every component of $D$.
This condition can be checked at height-one primes and then extended
reflexively.
\begin{lemma}[Automorphisms of a rationally connected pair]
\label{lem:rc-aut}
Let $(F,H)$ be a rationally connected projective klt pair over $\C$
with $H\geq0$ and $K_F+H\simq0$. Then
\[
H^j(F,\OO_F)=0\quad(j>0),\qquad
H^0(F,\Omega_F^{[1]})=0,\qquad
H^0\bigl(F,\mathcal T_F(-\log\Supp H)\bigr)=0.
\]
For every ample line bundle $L$, the group of automorphisms of
$(F,H)$ preserving $L$ is finite.
\end{lemma}
\begin{proof}
A smooth projective resolution of $F$ is rationally connected; one
may lift rational curves through general smooth points, or use the
rational connectedness results for klt varieties in \cite{HM}.
Its positive-degree structure-sheaf cohomology vanishes. Klt
singularities are rational, so the same is true on $F$. The
extension theorem for klt differential forms \cite{GKKP} then gives
the asserted vanishing of reflexive one-forms.
Choose $a>0$ and a rational $a$-canonical form $\theta$ with
\[
\Div(\theta)+aH=0.
\]
On the smooth locus, the density $|\theta|^{2/a}$ defines a positive
measure. Pull it to a log resolution. Every divisorial order of
the pulled-back form is strictly greater than $-a$, because the pair
is klt. In SNC coordinates this is exactly the local integrability
condition for the density. Thus it defines a finite nonzero measure
$\mu$ on $F$, with no mass on proper algebraic subsets.
An automorphism preserving $H$ multiplies $\theta$ by a nonzero
constant. Its pullback therefore multiplies $\mu$ by a positive
constant; comparison of total masses makes that constant one.
Consequently every automorphism of the pair preserves $\mu$.
There can be no nontrivial additive or multiplicative algebraic
one-parameter subgroup of such automorphisms. Indeed, iterate the
element $1\in\mathbb G_a$ or $2\in\mathbb G_m$ on a nonfixed
point. The orbit map extends across infinity by projectivity, so
these iterates converge to a subgroup-fixed point. The nonfixed
locus has full $\mu$-measure. Such convergence contradicts
Poincar\'e recurrence for the resulting invertible transformation of
the finite measure space $(F,\mu)$.
The group preserving $L$ and $H$ is a linear algebraic group: a
sufficiently high power of $L$ realizes it as a closed subgroup of a
projective linear group. Every positive-dimensional linear
algebraic group over $\C$ contains a copy of $\mathbb G_a$ or
$\mathbb G_m$, so this group is finite. Moreover, the identity
component of the automorphism group preserving $H$ preserves $L$.
Indeed its variation of $L$ lies in $\operatorname{Pic}^0(F)$,
which is zero because $H^1(F,\OO_F)=0$. Its Lie algebra therefore
vanishes. In characteristic zero this Lie algebra is precisely the
space of vector fields tangent to $\Supp H$: a connected group
preserving the support fixes its components and their coefficients.
This proves the last vanishing.
\end{proof}
\begin{lemma}[A choice stable under further covers]\label{lem:minimal-rc}
The cover in Proposition~\ref{prop:product-cover} can be chosen so
that every connected normal finite quasi-\'etale cover of $F$ is
rationally connected. On every such cover, with the pulled-back
boundary, the conclusions of Lemma~\ref{lem:rc-aut} hold.
\end{lemma}
\begin{proof}
Among all covers in Proposition~\ref{prop:product-cover}, choose
one with $\dim F$ minimal, taking this dimension to be zero when
the factor is absent. Let $F'\to F$ be a connected normal finite
quasi-\'etale cover. The pulled-back pair is klt and log
Calabi--Yau, and $K_{F'}$ is rational Cartier. Apply
Proposition~\ref{prop:product-cover} to it. If $F'$ were not
rationally connected, the resulting product would have a
positive-dimensional boundary-free part: otherwise rational
connectedness would descend from its rationally connected factor
along a finite surjection. Replacing $F$ by this product would
therefore give a product cover of $V$ with a smaller rationally
connected factor, a contradiction. Lemma~\ref{lem:rc-aut} now
applies to $(F',H')$.
\end{proof}
\subsection{Separating finite actions on a comparison cover}
We refer to the positive-dimensional factors just described as
\emph{blocks}, collecting the whole abelian part into one block.
The rationally connected block is always chosen as in
Lemma~\ref{lem:minimal-rc}. For a block over a nonclosed field,
the corresponding properties under connected normal finite
quasi-\'etale covers, including the form and vector-field
vanishings above, are imposed after algebraic closure. In the
application the field is a finite extension of a
complex curve function field. Its algebraic closure is abstractly
isomorphic to $\C$, so the preceding complex projective results
apply, and their finite algebraic data descend to a finite extension.
A \emph{semilinear automorphism} of a variety over a field $k$ is an
automorphism together with an automorphism of $k$ over which it lies;
equivalently, it is a $k$-isomorphism to the corresponding field
conjugate. Differentials below are relative to $k$.
\begin{proposition}[Semilinear comparison with the blocks]
\label{prop:semilinear}
Let $k$ be a characteristic-zero field, and let
$P=\prod_{i\in I}P_i$ be a product of geometrically integral
projective blocks as above. Let $H_P$ be the pullback of the
boundary on its rationally connected block, or zero if that block
is absent. Suppose
\[
\rho:R\longrightarrow P
\]
is a finite quasi-\'etale morphism from a geometrically integral
normal variety, and a finite group $G$ acts semilinearly on
$(R,H_R)$, where $H_R=\rho^*H_P$. Put $s=\dim R>0$.
There are normal geometrically integral varieties $R_i$, finite
quasi-\'etale maps $R_i\to P_i$, and a finite quasi-\'etale map
$R\to\prod_iR_i$ such that every $g^{s!}$, for $g\in G$, induces
regular semilinear automorphisms of the pairs $(R_i,H_i)$.
Here $H_i$ is the pulled-back boundary on the rationally connected
block and is zero on the other blocks. The projections from $R$
are equivariant for these automorphisms. Each geometric $R_i$
has the same block properties as $P_i$.
\end{proposition}
\begin{proof}
We first show that a bounded power preserves the factor directions on $R$.
We then recover their constant fields and show that the resulting
birational actions on the finite factor covers are regular.
\smallskip\noindent\emph{The factor directions.}
On a smooth big open where $\rho$ is \'etale, the product tangent
directions pull back to a splitting. Reflexive extension gives
\[
\mathcal T_R=\bigoplus_{i\in I}\mathcal E_i.
\]
Consider the finite-dimensional $k$-algebra
\[
\mathcal A=
\{u\in\operatorname{End}_R(\mathcal T_R):
u(\mathcal T_R(-\log\Supp H_R))
\subseteq\mathcal T_R(-\log\Supp H_R)\}.
\]
Every projection onto $\mathcal E_i$ belongs to $\mathcal A$.
We claim that every member of $\mathcal A$ has zero entries
between distinct summands.
This may be checked after extending $k$ to an algebraic closure.
Fix general smooth points in all but one factor, say $P_i$, and
take the corresponding slice of $R$. After shrinking the space of
complementary points, its connected components are normal and
finite quasi-\'etale over $P_i$. To see this, the generic slice is
normal by localization and geometrically normal in characteristic
zero. The projection to the complementary factors is projective,
so generic flatness and openness of geometric normality give
normality of the entire fibres after shrinking that base. Every
component has dimension $\dim P_i$:
the fibre-dimension inequality gives the lower bound, and finiteness
over $P_i$ gives the upper bound. Its finite image is therefore all
of $P_i$. The closed subset omitted from the \'etale product
charts has codimension at least two on a general slice: its
components not dominating the complementary factors can be avoided,
and the others have fibre dimension at most $\dim P_i-2$.
On its smooth big open, $\mathcal E_i$ is the tangent sheaf of
the slice, and every complementary summand is trivial.
If either of two distinct summands is nonabelian and boundary-free,
slice in that direction. A homomorphism in either direction gives
reflexive one-forms or vector fields on a finite quasi-\'etale
cover of that block, and hence vanishes. The only remaining
possibility is an abelian and a rationally connected summand.
Slice in the rationally connected direction. An entry from this
summand to the abelian one gives reflexive one-forms. An entry in
the reverse direction gives vector fields tangent to the pulled-back
boundary, because the endomorphism preserves the log tangent
subsheaf. Both vanish by Lemmas~\ref{lem:rc-aut} and
\ref{lem:minimal-rc}. These slices cover a dense open of $R$,
which proves the claim.
The block projections are therefore central idempotents of
$\mathcal A$. Its primitive central idempotents give nonzero
direct summands of $\mathcal T_R$, each of positive generic rank,
so there are at most $s$ of them. Conjugation by $G$ acts on
$\mathcal A$, since $G$ preserves the pair; it is a ring
automorphism even for a semilinear action. It permutes these
primitive central idempotents. Thus $g^{s!}$ fixes every one of
them and consequently every block projection. This argument takes
place over $k$ itself and requires no extension of $G$ to an
algebraic closure.
\smallskip\noindent\emph{The finite factor covers.}
Let $k(R_i)$ be the relative algebraic closure of $k(P_i)$ in
$k(R)$, and let
\[
R\longrightarrow R_i\longrightarrow P_i
\]
be the Stein factorization of the projection. At the generic point
of $R$, the complementary directions span
$\operatorname{Der}_{k(P_i)}k(R)$. In characteristic zero their
common constants are exactly $k(R_i)$. Preservation of the
directions therefore gives semilinear birational transformations of
$R_i$ under every $g^{s!}$.
The fields $k(R_i)$ are regular over $k$, since they lie in the
regular extension $k(R)/k$; hence $R_i$ is geometrically integral.
The extension obtained by adjoining the complementary product
factors to $k(P_i)$ is regular and linearly disjoint from
$k(R_i)/k(P_i)$. Consequently the normal product
$R_i\times\prod_{j\ne i}P_j$ is an intermediate finite cover of
$P$. Ramification over a prime of $P_i$ would persist on this
product and on a prolongation to $R$, contradicting
quasi-\'etaleness of $R\to P$. This proves that $R_i\to P_i$
is quasi-\'etale.
The product map $R\to\prod_iR_i$ is proper with finite fibres,
because its composite to $P$ is finite. It is therefore finite.
Its image has the dimension of the integral target, so it is
surjective, and the same ramification argument makes it
quasi-\'etale. In particular every fibre of $R\to R_i$ has
dimension $s-\dim R_i$.
\smallskip\noindent\emph{Regularity of the factor actions.}
Fix a transformation preserving the factor directions. The two
morphisms from $R$ to $R_i$---the projection and its composition
with the transformation---map $R$ onto the closed graph
$\Gamma_i$ of the induced birational factor transformation.
Their target is replaced by its field conjugate in the semilinear
case. Every fibre of $R\to\Gamma_i$ has dimension at least
$s-\dim R_i$. A positive-dimensional fibre of either graph
projection would therefore give a fibre of $R\to R_i$ of larger
dimension. Both graph projections are thus finite and birational;
normality of their targets makes them isomorphisms. This proves
regularity.
Finally, equality of divisors on $\prod_iR_i$ can be checked after
pullback along the finite surjection from $R$. Since its entire
boundary is pulled back from the rationally connected factor,
invariance of $H_R$ implies invariance of that factor's boundary.
Hence the factor actions preserve their boundaries. The asserted
geometric properties of the $R_i$ follow
from their being quasi-\'etale covers of the chosen blocks.
\end{proof}
After the bounded power, the action on $\prod_iR_i$ is the product
of its regular factor actions. The factor degrees and the orders
of these automorphisms may be unbounded.
\subsection{Block forms over a curve field}
We now collect the exact data needed for degeneration, including
the degrees of the log pluriforms on the factors.
\begin{corollary}[Comparison cover and block forms]\label{cor:block-forms}
Let $K=\C(C)$ for a smooth projective curve $C$. Let $(V,B)$
be a geometrically integral normal projective $s$-fold pair over
$K$, with $s>0$, geometrically klt, $B\geq0$, and $K_V$
rational Cartier. Suppose that, for an integer $m>0$, a rational $m$-canonical form
$\phi$ satisfies $\Div(\phi)+mB=0$.
There is a finite Galois extension $K_1/K$, a geometrically integral
normal projective variety $R/K_1$, and finite quasi-\'etale maps
\[
\begin{tikzcd}[column sep=large]
R \arrow[r] & \displaystyle\prod_{i\in I}R_i
\arrow[r] & \displaystyle\prod_{i\in I}P_i
\arrow[r] & V_{K_1}
\end{tikzcd}
\]
with the following properties. The field $K_1(R)$ is finite Galois
over $K(V)$; its Galois group acts semilinearly on $R$, maps onto
$\operatorname{Gal}(K_1/K)$, and preserves the pulled-back pair.
For each element $g$ of this group, $g^{s!}$ induces regular
semilinear actions on the pairs $(R_i,H_i)$. There is at most one
rationally connected block, chosen with the cover-stability of
Lemma~\ref{lem:minimal-rc}, and all other blocks are boundary-free
of the types in Proposition~\ref{prop:product-cover}.
Put $p_i=m$ on the rationally connected block and $p_i=1$ on the
other blocks. There are rational $p_i$-canonical forms $\eta_i$
over $K_1$ satisfying
\begin{equation}\label{eq:block-trivializations}
\Div(\eta_i)+p_iH_i=0.
\end{equation}
On $R$, after pulling back all forms, one has
\begin{equation}\label{eq:block-product-form}
\phi=a\bigwedge_{i\in I}\eta_i^{\otimes m/p_i}
\qquad\text{for some }a\in K_1^*.
\end{equation}
The wedge uses any fixed ordering of the blocks and the natural
identification of their relative canonical lines.
\end{corollary}
\begin{proof}
Apply Proposition~\ref{prop:product-cover} and
Lemma~\ref{lem:minimal-rc} to the geometric generic pair, and
define the resulting product cover over a finite extension of $K$.
Take a Galois closure $L$ of its total function field over the
original field $K(V)$, and let $K_1$ be the relative algebraic
closure of $K$ in $L$. The extension $K(V)/K$ is regular.
It follows that $K_1/K$ is finite Galois and that restriction maps
$\operatorname{Gal}(L/K(V))$ onto
$\operatorname{Gal}(K_1/K)$. Let $R$ be the normalization of
$V_{K_1}$ in $L$. Since $K_1$ is algebraically closed in $L$,
$R$ is geometrically integral.
Over an algebraic closure of $K$, the conjugate product covers are
all \'etale over the smooth locus of the original variety. Their
compositum, and hence the Galois closure, has the same property.
Thus $R$ is finite quasi-\'etale over the product and over
$V_{K_1}$. Its Galois transformations preserve the crepant
pullback of $B$. Proposition~\ref{prop:semilinear} now constructs
the $R_i$ and supplies the factor actions.
The boundary-free blocks and their covers have Cartier trivial
canonical divisor over the algebraic closure, so they possess
degree-one volume forms there. Still over the algebraic closure,
on the rationally connected factor,
restrict the pulled-back degree-$m$ trivialization from $V$ to
the product factor, choosing the complementary points smooth.
This gives a degree-$m$ log trivialization; pull it back to its
Stein block $R_i$.
Here $p_iH_i$ is integral: on the rationally connected block this
follows from the integrality of $mB$, quasi-\'etale pullback, and
restriction to the product factor; on the other blocks $H_i=0$.
Thus $p_iK_{R_i}+p_iH_i$ is an integral Weil divisor whose
divisorial sheaf becomes trivial after algebraic closure.
Lemma~\ref{lem:trivialization-descent} descends this trivialization
in degree $p_i$, giving \eqref{eq:block-trivializations}. The product
in \eqref{eq:block-product-form} and the pullback of $\phi$ have
the same divisor on the normal projective variety $R$. Their
ratio has neither zeros nor poles and is a global unit. Since
$R$ is geometrically integral, that unit belongs to $K_1^*$.
\end{proof}
For later use, put $L=K_1(R)$ and let $K_2\supset K_1$ be a finite extension Galois
over $K$, and choose a branch over a fixed point of $C$ in the
corresponding tower of curves. The inertia groups are cyclic and
the upper one surjects onto the lower one. Choose generators of
these inertia groups, viewed as elements
$\tau_2\in\operatorname{Gal}(K_2/K)$ and
$\tau_1\in\operatorname{Gal}(K_1/K)$ with
$\tau_2|_{K_1}=\tau_1$, and choose a lift
$g\in\operatorname{Gal}(L/K(V))$ of $\tau_1$.
Since $L/K_1$ is regular, it is linearly disjoint from $K_2/K_1$.
The compatible pair $(g,\tau_2)$ therefore defines a finite-order
automorphism $g_2$ of $LK_2$. The base change $R_{K_2}$ is
geometrically integral, and $g_2^{s!}$ acts regularly on every
base-changed block by the action of $g^{s!}$ together with
$\tau_2^{s!}$ on its constants. This supplies the compatible
inertia lift and block actions after the further base changes used
for semistable reduction.
==== END Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//sections/blocks.tex ====
==== BEGIN Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//sections/completion.tex ====
\section{Effective systems and the field of section ratios}
\label{sec:completion}
The moduli denominator now allows effective birationality on the base
of a log Calabi--Yau fibration. We first make two section comparisons
explicit. They will retain the actual subfield of the function field
through both the fibration and the passage to a good minimal model.
\begin{lemma}[Rounded section comparisons]\label{lem:rounded-comparison}
All varieties in this statement are normal and projective over a field
of characteristic zero.
\begin{enumerate}
\item
Let $f:X\to Z$ be a contraction and let $D$ and $D_Z$ be
$\Q$-Cartier divisors such that
\[
D+\frac1a\Div(\psi)=f^*D_Z
\]
for an integer $a>0$ and $\psi\in k(X)^*$. For every positive $\ell$ divisible
by $a$, the spaces of divisorial sections, regarded as rational
functions, satisfy
\begin{equation}\label{eq:section-pullback}
H^0\bigl(X,\OO_X(\floor{\ell D})\bigr)
=\psi^{\ell/a}f^*H^0\bigl(Z,\OO_Z(\floor{\ell D_Z})\bigr).
\end{equation}
\item
Let $X\dashrightarrow X'$ be birational, let $D$ and $D'$ be
$\Q$-Cartier divisors, and suppose that on a common resolution
$p:T\to X$, $q:T\to X'$ one has
\[
p^*D=q^*D'+E,
\]
where $E\geq0$ is $q$-exceptional. Under the identification of the
function fields, for every integer $\ell>0$ one has
\begin{equation}\label{eq:section-birational}
H^0\bigl(X,\OO_X(\floor{\ell D})\bigr)
=H^0\bigl(X',\OO_{X'}(\floor{\ell D'})\bigr).
\end{equation}
\end{enumerate}
\end{lemma}
\begin{proof}
For a rational function $u$, the inequality
$\Div(u)+\floor{\ell D}\geq0$ is equivalent to
$\Div(u)+\ell D\geq0$, since function orders are integral.
Also, effectivity of a $\Q$-Cartier divisor is preserved by surjective
pullback. It is detected by such a pullback: over the generic point of
each prime divisor downstairs, a local defining parameter pulls back
with positive order along some divisor upstairs.
For (i), divide a section upstairs by $\psi^{\ell/a}$. The result
$v$ satisfies
\[
\Div(v)+\ell f^*D_Z\geq0.
\]
It has no pole on the normal projective generic fibre, so is a base
function. For $v\in k(Z)$ the displayed divisor equals
$f^*(\Div(v)+\ell D_Z)$. The effectivity comparison just noted proves
both inclusions in~\eqref{eq:section-pullback}.
For (ii), pull an effective divisor $\Div(u)+\ell D$ to $T$ and push
it to $X'$. Since $q_*E=0$, this gives
$\Div(u)+\ell D'\geq0$. Conversely, pull the latter inequality to
$T$, add $\ell E$, and push to $X$. This yields the original
inequality. The argument uses $\Q$-Cartier pullbacks, so it does not
require $\ell D$ or $\ell D'$ to be Cartier.
\end{proof}
\begin{proposition}[Effective degree on the base]
\label{prop:effective-fibration}
Fix $d\geq1$ and a finite set $\Phi\subset[0,1]\cap\Q$.
There is an integer $N=N(d,\Phi)>0$ with the following property.
Let $f:X\to Z$ be a contraction of normal projective complex varieties
with $\dim X\leq d$ and $\dim Z>0$, and let $(X,B)$ be lc with
$B\geq0$, coefficients in $\Phi$, and $D=K_X+B$ $\Q$-Cartier.
Suppose that $D\simq f^*L$ for a big $\Q$-Cartier divisor $L$ on $Z$.
Then $|\floor{ND}|$ is nonempty, and its section ratios generate
exactly $f^*\C(Z)$ inside $\C(X)$. This is also the field $K(D)$
generated by section ratios in all positive degrees.
\end{proposition}
\begin{proof}
Use Proposition~\ref{prop:exact-presentation} to obtain
\eqref{eq:exact-presentation}, and choose a smooth projective
determination $\tau:W\to Z$ of its moduli b-divisor. After a further
resolution we may suppose that the strict transform of $B_Z$ and the
exceptional divisor of $\tau$ have SNC support. Theorem~
\ref{thm:moduli-denominator} gives a uniform $p$ for which $pM_W$ is
nef Cartier. Let
\[
A=\tau_*^{-1}B_Z+\operatorname{Exc}(\tau)_{\mathrm{red}}.
\]
Then $(W,A)$ is log smooth and lc. Its coefficients lie in the fixed
DCC set for the discriminant, enlarged by $1$. Moreover,
\begin{equation}\label{eq:base-effective-exceptional}
K_W+A+M_W=\tau^*D_Z+E,\qquad E=A-B_W\geq0,
\end{equation}
where $E$ is exceptional: at nonexceptional primes $A$ and $B_W$
agree, and at exceptional primes $A$ has coefficient one whereas
$B_W$ has coefficient at most one.
The divisor in~\eqref{eq:base-effective-exceptional} is big.
The effective birationality theorem for polarized pairs
\cite[Theorem~1.3]{BZ} now applies to $(W,A)$ and $M_W$: the pair is
projective lc, its boundary coefficients belong to a fixed DCC set,
$pM_W$ is nef Cartier, and the adjoint sum is big. Consequently
\[
\left|\floor{n(K_W+A+M_W)}\right|
\]
is birational for every $n$ divisible by an integer depending only on
$\dim W$, the DCC set, and $p$. The notation in that theorem uses
round down, as in the displayed system.
Pushing the rational section inequalities to $Z$ shows that this
birational system is a subsystem of $|\floor{nD_Z}|$ under the
identification $\C(W)=\C(Z)$. Thus the latter system is nonempty
and its ratios generate $\C(Z)$. Choose a common such $n=N$ over
$1\leq\dim Z\leq d$, also divisible by $p_0$.
Equation~\eqref{eq:section-pullback} then proves the asserted
nonemptiness and ratio-field equality upstairs.
It remains to compare with all degrees. If $s,s_0$ are nonzero
sections in degree $\ell$, their ratio can be represented in every
multiple degree $b\ell$ as
\begin{equation}\label{eq:ratio-multiple}
\frac{s}{s_0}=\frac{s\,s_0^{b-1}}{s_0^b}.
\end{equation}
These products are sections in degree $b\ell$ because
$b\floor{\ell D}\leq\floor{b\ell D}$.
Choose $b$ so that $p_0\mid b\ell$ and apply
\eqref{eq:section-pullback}. Every such ratio belongs to
$f^*\C(Z)$, proving $K(D)=f^*\C(Z)$.
\end{proof}
The proof of the main theorem first treats the ground field $\C$.
For the final change of ground field we use the following elementary
observation, which keeps track of equality of subfields rather than only
the associated rational maps.
\begin{lemma}[Descent of the field of ratios]\label{lem:field-descent}
Let $k\subset k'$ be an extension of algebraically closed fields, let
$X$ be a normal integral projective variety over $k$, and let $D$ be a
$\Q$-divisor on $X$. Write $X'=X_{k'}$ and $D'=D_{k'}$.
For each $\ell>0$ having nonzero sections, let $F_\ell\subset k(X)$
be the field generated over $k$ by ratios of sections of
$\OO_X(\floor{\ell D})$, and define $F'_\ell$ similarly on $X'$.
Then
\[
F'_\ell=k'F_\ell,\qquad K(D')=k'K(D),
\]
where the composita are taken in $k'(X')$. In particular, for a fixed
$m>0$, nonemptiness of $|\floor{mD}|$ and the equality
$F_m=K(D)$ hold if and only if they hold after extension to $k'$.
\end{lemma}
\begin{proof}
The rounded divisorial sheaves commute with these field extensions.
One may check this on the smooth big open set where the prime divisors
are Cartier, and then use reflexive extension across its complement.
Base change for global sections gives
\begin{equation}\label{eq:section-base-change}
H^0\bigl(X',\OO_{X'}(\floor{\ell D'})\bigr)
=k'\otimes_k H^0\bigl(X,\OO_X(\floor{\ell D})\bigr).
\end{equation}
Choose a nonzero section $s_0$ over $k$. Dividing any section over
$k'$ by $s_0$ expresses it as a finite $k'$-linear combination of
ratios over $k$. This proves $F'_\ell=k'F_\ell$; taking all degrees
proves the analogous assertion for $K(D)$.
For descent of the equality, we use the following intersection fact.
If $F$ is an intermediate field $k\subset F\subset k(X)$, then
\begin{equation}\label{eq:field-intersection}
k(X)\cap k'F=F
\end{equation}
inside $k'(X')$. Indeed $X$ is geometrically integral, so
$k(X)\otimes_k k'$ injects into $k'(X')$. If $u\in k(X)\cap k'F$,
write
\[
u=\frac{\sum_{i=1}^a c_i a_i}{\sum_{i=1}^a c_i b_i},
\qquad a_i,b_i\in F,\quad c_i\in k',
\]
where the $c_i$ are linearly independent over $k$ and the denominator
is nonzero. Such an expression is obtained by collecting a finite
list of constants into a $k$-basis. After clearing the denominator,
injectivity of the tensor product map and linear independence give
$ub_i=a_i$ for every $i$. Some $b_i$ is nonzero, so $u\in F$.
Applying~\eqref{eq:field-intersection} to $F=F_m$ proves descent of
$F'_m=K(D')$. The forward implication follows from the compositum
identities, and nonemptiness follows in either direction from
\eqref{eq:section-base-change}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
First let the ground field be $\C$, and put $D=K_X+B$.
The hypothesis $\kappa(X,D)\geq0$ implies that $D$ is
pseudo-effective. Take a crepant $\Q$-factorial dlt modification
$\mu:(Y,\Delta)\to(X,B)$. Its adjoint is $\mu^*D$, and its
boundary coefficients belong to $\Phi\cup\{1\}$.
Theorem~\ref{thm:good-models} and termination with ample scaling
\cite[Theorem~A]{TX} give a terminating $(K_Y+\Delta)$-MMP to a
$\Q$-factorial dlt pair $(X',B')$. The nef case of the good-model
theorem makes its adjoint semiample. The MMP extracts no divisors,
so the coefficients of $B'$ still belong to $\Phi\cup\{1\}$.
With compatible canonical divisors, crepancy of $\mu$ and the
discrepancy comparison for this MMP give, on a common resolution,
\[
p^*(K_X+B)=q^*(K_{X'}+B')+E,
\qquad E\geq0\quad\text{$q$-exceptional};
\]
see the minimal-model discrepancy comparison and negativity lemma
in~\cite{KM}. Lemma~\ref{lem:rounded-comparison}(ii) identifies all
rounded section spaces as subspaces of the common function field.
It therefore suffices to prove the result on $(X',B')$.
The adjoint $D'=K_{X'}+B'$ is semiample. Its semiample contraction
$f:X'\to Z$ satisfies $D'\simq f^*L$ with $L$ an ample
$\Q$-Cartier divisor on the normal projective base. If $\dim Z>0$,
Proposition~\ref{prop:effective-fibration}, with coefficient set
$\Phi\cup\{1\}$, gives a uniform nonempty degree whose ratios
generate the entire field $K(D')$.
If $Z$ is a point, then $D'\simq0$, and
Theorem~\ref{thm:normal-index} gives a uniform principal multiple.
Its system is nonempty and all ratios in that degree are constants.
For any other degree, pass to a common multiple by
\eqref{eq:ratio-multiple}; ratios in the latter degree are again
constants. Hence $K(D')=\C$ in this case. A common multiple of the
two uniform degrees works in both cases, again by
\eqref{eq:ratio-multiple}. Denote it by $m(d,\Phi)$. The rounded
comparison transfers the conclusion to $X$.
Now let $k$ be any algebraically closed field of characteristic zero.
Descend $X$, its prime boundary components and coefficients, the
canonical and $\Q$-Cartier data, and a log resolution to a finitely
generated subfield of $k$. Let $k_0\subset k$ be the algebraic
closure of that field inside $k$. Enlarging the finitely generated
field if necessary, the descended variety $X_0$ is geometrically
integral and normal, its pair $(X_0,B_0)$ is lc, and its adjoint
$D_0$ is $\Q$-Cartier. These conditions can be checked after the
faithfully flat extension to $k$, using the chosen resolution for
the discrepancy inequalities. The field $k_0$ embeds into $\C$.
Equation~\eqref{eq:section-base-change} shows that nonzero sections in
some positive degree, and hence nonnegative Kodaira dimension, descend
from $k$ to $k_0$ and persist after extension to $\C$.
The complex case gives the conclusion in the same integer
$m(d,\Phi)$ for $(X_0,B_0)_\C$.
Lemma~\ref{lem:field-descent} first descends the nonemptiness and exact
equality of ratio fields to $k_0$, and then extends them to $k$.
This proves the theorem over the stated ground field.
\end{proof}
==== END Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//sections/completion.tex ====
==== BEGIN Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//sections/degeneration.tex ====
\section{Degeneration characters and the klt weight bound}
\label{sec:degeneration}
For a klt generic pair, Section~\ref{sec:blocks} supplies a finite
comparison cover and separates the action of a bounded power of inertia
into actions on its factors. We now pass to semistable models of these
factors. The degree of the necessary curve cover is unrestricted.
The point of this section is to bound the characters on normalized
logarithmic forms; those bounds will cancel the unrestricted ramification
in the weight calculation.
\subsection{Equivariant semistable models}
We first record the form of semistable reduction that we use. A marked
horizontal divisor in this statement may include both a boundary and
exceptional divisors of a generic log resolution.
\begin{lemma}[Marked semistable reduction]\label{lem:marked-semistable}
Let $C_1$ be a smooth projective complex curve with a marked point $c_1$,
and let a finite cyclic group act on $C_1$, fixing $c_1$.
Consider finitely many geometrically integral normal projective varieties
over $\C(C_1)$, each with a compatible semilinear action of this group
and an invariant finite set of horizontal divisor markings.
After a finite extension of curve fields and replacement by compatible
finite-order lifts of the actions, there are equivariant smooth
projective models for which the marked fibre is reduced simple normal
crossings and its union with the horizontal markings is simple normal
crossings. The generic fibres may be log resolutions.
If a given generic fibre is already smooth and has no markings requiring
resolution, it can be left unchanged.
\end{lemma}
\begin{proof}
Take equivariant projective models by closing graphs of the finitely
many translates, and resolve equivariantly, including the reduced
marked fibre and the horizontal markings. Near the marked fibre the
map to the curve is toroidal: in local coordinates a uniformizer is a
monomial in the vertical coordinates, up to a unit. The horizontal
markings are among the remaining coordinates. Root extraction on the
base, normalization, and the projective subdivision theorem of
\cite[Chapter~IV, Section~3]{KKMSD} give the reduced SNC fibre.
The horizontal coordinates remain transverse throughout this
construction.
Here equivariance can be retained in the subdivision step. First
barycentrically subdivide the finite vertical cone complex so that
stabilizers fix each ray of any stabilized cone. They then act trivially
on that cone's lattice. After removing face self-identifications by
further such subdivision, choose compatible projective subdivisions on
orbit representatives and pull them back. The lattices used on the
combinatorial quotient are the original cone lattices, not the
invariant lattices of a geometric quotient. The regular height-one
subdivision theorem after sufficiently divisible ramification applies
to this finite complex. It applies with identical subdivisions to the
copies of strata that split after normalization.
This is the equivariant marked construction used in
\cite[Section~4]{U}; the marked resolution and finite-group extension
are also described in \cite[Paragraphs~16 and~21]{KNX}.
Taking a common further extension handles the finite list of models.
One may take a Galois closure and further roots of an original local
parameter, so that the extension remains Galois over any specified
original curve field. Compatible automorphisms extend to the
compositum; they have finite order. Equivariant resolution away from
the marked fibre gives projective models over the complete curve.
\end{proof}
Apply this lemma to the factors from Corollary~\ref{cor:block-forms}.
Let $C'\to C$ be the resulting Galois curve cover, let $c'$ lie over $c$,
and write
\[
K'=\C(C'),\qquad l=e(c'/c),\qquad b=s!.
\]
The compatible lift constructed after Corollary~\ref{cor:block-forms}
gives a finite-order automorphism $g$ of the comparison cover whose
base action generates inertia. Its power $\sigma=g^b$ acts separately
on the factors.
Choose a rational uniformizer $u$ at $c'$. Its leading transformation
under $g$ is
\begin{equation}\label{eq:inertia-parameter}
g^*u\sim\zeta u,
\end{equation}
where $\zeta$ is a primitive $l$th root of unity.
Only the leading coefficient is used; it is unnecessary to require
$g^*u=\zeta u$ as an equality of rational functions.
For a factor $(R_i,H_i)$, let $p_i=m$ on the rationally connected factor
and $p_i=1$ on the other factors. We have a rational $p_i$-canonical form
$\eta_i$ satisfying
\[
\Div(\eta_i)+p_iH_i=0
\]
on its generic normal model. On the semistable resolution choose the
horizontal boundary $\widehat H_i$ to equal the strict transform of
$H_i$ plus the positive parts of the horizontal crepant exceptional
coefficients. These coefficients are less than one. On the abelian
and nonabelian Beauville--Bogomolov factors we have $\widehat H_i=0$,
because their singularities are canonical.
\begin{lemma}[Normalization and products]\label{lem:normalized-product}
Multiplying each $\eta_i$ by an integral power of $u$, one can arrange
$w_u(\eta_i)=0$. For these normalized forms, the exterior product in
common degree $m$ has weight zero. On the comparison cover there is a
base function $a\in K'^*$ such that
\begin{equation}\label{eq:product-weight}
\phi=a\bigwedge_i\eta_i^{\otimes m/p_i},
\qquad
l\,w_z(\phi)=\frac{\ord_{c'}a}{m}.
\end{equation}
Here the exterior product denotes the tensor power of the relative
canonical product identification, with a fixed order of factors.
\end{lemma}
\begin{proof}
On a semistable model every component of the marked fibre has
multiplicity one. The computation of weights on a log-smooth model
in Lemma~\ref{lem:snc-weight} shows that $p_iw_u(\eta_i)$ is an
integer. Multiplication by $u^n$ changes the weight by $n/p_i$, so
normalization is possible. The logarithmic divisor
\[
\Div\bigl(\eta_i\wedge(du/u)^{\otimes p_i}\bigr)
+p_i(T_i+\widehat H_i)
\]
is then effective near the marked fibre, and its coefficient on at
least one component of $T_i$ is zero.
We check the product assertion before passing to the comparison
cover. On the fibre product of the semistable models, the local
vertical equations have the form
\[
\prod_j x_{ij}=u
\quad\text{for each factor }i.
\]
Horizontal coordinates are independent. Include their SNC markings
in the toroidal boundary. Relative logarithmic top forms multiply
over the logarithmic curve with exactly one base differential
$du/u$ in the absolute form. The product therefore acquires no
additional power of $u$. In logarithmic coordinates the resulting
form has at most full logarithmic poles. This remains true on
toroidal resolutions, since their logarithmic canonical generators
pull back to logarithmic generators. Thus its weight is nonnegative.
For equality, choose on each factor a fibre component of zero
logarithmic order and a general point away from all other markings.
The morphism to the curve is smooth there. The product of these
open subsets supplies a fibre component of zero logarithmic order,
so the weight is zero. Finite pullback to the comparison cover
preserves this weight by Lemma~\ref{lem:weight-rules}.
On its normal proper generic fibre, the pulled-back product and
$\phi$ have the same divisor: both trivialize the same degree-$m$
log canonical divisor. Their ratio has zero divisor and hence lies
in the constant field $K'$. The scalar rule and the ramification
rule for weights now give \eqref{eq:product-weight}.
\end{proof}
Since $\sigma$ preserves each factor pair, write
\[
\sigma^*\eta_i=d_i\eta_i,\qquad d_i\in K'^*.
\]
Both sides have weight zero, hence $\ord_{c'}d_i=0$. Put
$\lambda_i=d_i(c')$. Finite order of $\sigma$ and the fact that it
fixes $c'$ imply that each $\lambda_i$ is a root of unity.
The form $\phi$ descends to the original field and is fixed by $g$.
Taking leading coefficients of its transformation in
\eqref{eq:product-weight} yields
\begin{equation}\label{eq:character-cancellation}
\zeta^{b\ord_{c'}a}
\prod_i\lambda_i^{m/p_i}=1.
\end{equation}
There is no permutation sign in this formula: $\sigma$ preserves
each ordered factor.
We have reduced the weight problem to one concrete question:
can the orders of the $\lambda_i$ be bounded using only $s$ and $m$?
The following subsections treat abelian factors and the remaining
factors separately.
\subsection{Abelian factors}
\begin{lemma}[The abelian character]\label{lem:abelian-character}
For an abelian factor of dimension $h$, the order of its normalized
character $\lambda_i$ is bounded in terms of $h$ alone.
\end{lemma}
\begin{proof}
Here $p_i=1$. On the semistable model the normalized form is a regular
relative logarithmic top form. It is not divisible by $u$ as such a
section, since its logarithmic order is zero on some fibre component.
Consequently it frames the extended top Hodge line. This identification
is the semistable comparison between logarithmic de Rham cohomology
and Deligne's canonical extension, with nilpotent residue
\cite{Deligne,Steenbrink}. The action on the central fibre of this line
is $\lambda_i$.
The integral local system in degree $h$, modulo torsion, has rank
$\binom{2h}{h}$. Indeed its smooth fibres are abelian varieties; our
models were unchanged on the smooth generic fibre. In a small disc
choose an analytic coordinate linearizing the finite base action.
On the universal cover of the punctured disc the action, combined
with parallel transport, is an integral matrix $A$ commuting with
the unipotent monodromy $T$. Since the geometric action has finite
order, a power of $A$ is a power of $T$.
Passing from flat frames to canonical-extension frames changes $A$
by a commuting factor of the form $\exp(c\log T)$, which is unipotent.
The eigenvalues are therefore unchanged. Since the Hodge line is
invariant, $\lambda_i$ is an eigenvalue of the integral matrix $A$.
If its order is $n$, its cyclotomic polynomial divides the
characteristic polynomial of $A$, and
\[
\varphi(n)\leq\binom{2h}{h}.
\]
Only finitely many positive integers satisfy this inequality.
Their least common multiple bounds and annihilates the character
orders in the stated dimension.
\end{proof}
\subsection{An equivariant model with a log generator}
For the remaining factors the ordinary Betti numbers are not bounded
by the argument. Instead we use the much smaller structure-sheaf
cohomology and a normal component of a special fibre. This requires
a model on which the normalized logarithmic form generates everywhere
near that fibre.
An exact log trivialization in degree $p_i$ then has a residue in the
same degree on a normal fibre component. The coefficients of its
different must consequently lie in $\{0,1/p_i,\ldots,1\}$, allowing
the normal index theorem to bound the residue character.
\begin{lemma}[Equivariant good model]\label{lem:equivariant-good}
Let $(R_i,H_i)$ be a rationally connected or nonabelian
Beauville--Bogomolov factor of dimension $h>0$ above, with
$p_i\in\{1,m\}$ and normalized form $\eta_i$.
There is a projective equivariant model $Y\to C'$ such that, near $c'$,
\begin{enumerate}
\item $Y$ is klt and its fibre $T=Y_{c'}$ is reduced Cartier;
\item with $J$ the transformed horizontal boundary,
$(Y,T+J)$ is dlt and $K_Y+J$ is semiample over $C'$;
\item for $\Omega_i=\eta_i\wedge(du/u)^{\otimes p_i}$,
\begin{equation}\label{eq:log-generator}
\Div_Y(\Omega_i)+p_i(T+J)=0.
\end{equation}
\end{enumerate}
The geometric generic fibre is birational to $R_i$.
\end{lemma}
\begin{proof}
Start with the equivariant smooth semistable model $Q$ from
Lemma~\ref{lem:marked-semistable}. The pair $(Q,\widehat H_i)$ is klt;
adding the reduced fibre $Q_{c'}$ gives a dlt pair. Since this fibre
is a base pullback, a $(K_Q+\widehat H_i)$-negative program over the
curve is also negative for the adjoint with this fibre added.
We construct the program equivariantly. Let $G$ be the finite cyclic
group generated by $\sigma$ and take the quotient
$\pi:Q\to\overline Q=Q/G$. Define the branch-corrected boundary
$\overline H$ by
\[
K_Q+\widehat H_i
=\pi^*(K_{\overline Q}+\overline H).
\]
Its coefficients are $1-(1-\delta)/e$, where $\delta<1$ is an
upstairs coefficient and $e$ a ramification index. They lie in $[0,1)$,
and finite-map discrepancy comparison makes
$(\overline Q,\overline H)$ klt. A finite quotient of a smooth variety
is $\Q$-factorial.
On the generic fibre upstairs the adjoint has a nonzero section in
a sufficiently divisible degree: the log-resolution error is effective
and exceptional over the log Calabi--Yau factor. Multiplying the
finitely many translates of this section gives an invariant section
in a divisible degree downstairs. The horizontal section inequalities
descend by the finite-map formula. Thus Lemma~\ref{lem:curve-mmp}
applies to the quotient pair over $C'/G$ and gives a terminating
program with relatively semiample output.
Lift each step by normalization in the upstairs function field,
using Stein factorization for contractions. These are the usual
finite-group equivariant MMP diagrams; see also
\cite[Complement~3 and Paragraph~21]{KNX}.
They remain over $C'$, since functions integral over the downstairs
curve extend on the normalizations. Finite pullback preserves relative
negativity and relative ampleness. The canonical pullback equality
continues in codimension one because no step extracts divisors.
In particular the lifted pairs remain klt. The invariant divisor $J$
is rational Cartier: it descends rationally to the $\Q$-factorial
quotient, and its finite pullback is $J$. The displayed canonical
pullback equality makes $K_Y+J$ rational Cartier as well, and their
difference makes $K_Y$ rational Cartier.
For completeness, ordinary $\Q$-factoriality upstairs is unnecessary
for dlt preservation here. The lifted negative contraction or flip
has the same discrepancy comparison as an ordinary MMP step.
Adding the marked fibre changes the adjoint by a pullback and leaves
this comparison unchanged. Discrepancies strictly improve for centres
in the affected locus; hence every lc centre on the output meets the
unchanged locus of an input lc centre. It therefore meets its SNC
locus. This is the generic-SNC characterization of dlt, and gives dlt
on each output; compare \cite[Section~3, before Lemma~16]{dFKX}.
Since no divisors are extracted, every surviving marked fibre
component still has multiplicity one. The fibre is Cartier, and the
klt total space is Cohen--Macaulay. It has no embedded fibre components,
so the fibre is reduced. Relative semiampleness pulls back from the
quotient output.
It remains to prove the exact equality \eqref{eq:log-generator}.
The generic adjoint has Kodaira dimension zero, and the steps preserve
its divisible section spaces. The relative semiample contraction
therefore has zero-dimensional generic image. Connected fibres
identify its Stein image with $C'$, so $K_Y+J\sim_{\Q,C'}0$.
The logarithmic divisor on the left of \eqref{eq:log-generator}
is effective near $c'$ by normalization, and stays effective by
pushforward along the program. Its horizontal part is effective
and rationally trivial on the proper generic fibre, hence zero.
Its remaining vertical part is relatively rationally principal.
A function with vertical divisor is constant on the normal proper
generic fibre, and so comes from $K'$. Thus this divisor is a
rational multiple of $T$ near $c'$.
If the multiple were positive, the dlt discrepancy calculation would
make every weight quotient positive, contradicting
$w_u(\eta_i)=0$. It is therefore zero, proving
\eqref{eq:log-generator}.
\end{proof}
\subsection{Fixed points and normal component characters}
\begin{lemma}[A character on a normal log Calabi--Yau pair]
\label{lem:normal-character}
Fix $h\geq0$ and $p\geq1$. Let $(A,\Delta)$ be a normal integral
projective lc pair over $\C$, and let $\omega$ be a nonzero rational
$p$-canonical form with
\[
\Delta\geq0,\qquad \Div(\omega)+p\Delta=0.
\]
If a finite-order automorphism $\gamma$ preserves the pair and
$\gamma^*\omega=\lambda\omega$, the order of $\lambda$ divides
an integer depending only on $h=\dim A$ and $p$.
\end{lemma}
\begin{proof}
The divisor of $\omega$ is integral, so the coefficients of $\Delta$
belong to $\{0,1/p,\ldots,1\}$. On the normal quotient
$A/\langle\gamma\rangle$ the crepant branch boundary has coefficients in
\[
\Psi_p=
\left\{1-\frac{1-\delta}{e}:
\delta\in\{0,1/p,\ldots,1\},\ e\in\Z_{>0}\right\}.
\]
This is a rational DCC subset of $[0,1]$. An invariant tensor power
of $\omega$ descends and shows that the quotient log canonical divisor
is rational Cartier and rationally linearly trivial. Finite-map
discrepancy comparison makes the quotient pair lc.
Apply Theorem~\ref{thm:normal-index} in dimension $h$ with coefficient
set $\Psi_p$. Choose its principalizing degree $a$ divisible also by
$p$. The pullback of a degree-$a$ trivialization downstairs is invariant
and has the same divisor as $\omega^{\otimes a/p}$. Their ratio is
constant on the normal proper variety $A$. Consequently
$\lambda^{a/p}=1$. This gives the asserted uniform integer.
\end{proof}
\begin{lemma}[Characters of the other factors]\label{lem:other-characters}
For the normalized rationally connected, Calabi--Yau, and symplectic
factors, the orders of the $\lambda_i$ are bounded in terms of $s$ and $m$.
\end{lemma}
\begin{proof}
Use the model of Lemma~\ref{lem:equivariant-good} and write
$h=\dim R_i$. It is flat over the smooth curve, since its integral
total space is torsion-free over the local discrete valuation rings.
The central fibre $T$ is a union of lc centres of the dlt pair
$(Y,T+J)$ and is therefore Du Bois. Cohomology and base change for a
proper flat family with Du Bois special fibre imply local constancy
of $h^j(\OO)$ near that fibre
\cite[Theorem~2.62 and Corollary~2.64]{KollarFamilies}.
After shrinking, the other fibres are klt, as can be checked on a
resolution. Rational singularities and birational invariance identify
their structure-sheaf cohomology with that of a resolution of the
generic factor.
For the rationally connected factor this gives
\[
H^j(T,\OO_T)=0\quad(j>0),\qquad h^0=1.
\]
For a Calabi--Yau factor the only nonzero groups are in degrees $0$
and $h$, each of dimension one. For a symplectic factor there is
one dimension in every even degree and none in odd degree.
These identifications follow from extension of reflexive forms,
the defining form algebras of the factors, and Hodge symmetry on
resolutions \cite{GKKP,HP}.
We use the following consequence of coherent Lefschetz:
a finite-order automorphism of a projective scheme with nonzero
alternating trace on $H^\bullet(\OO)$ has a fixed point.
It applies to this possibly singular fibre. One can either use
\cite{BFQ}, or embed the fibre equivariantly in a smooth projective
space and apply \cite[Corollary~5.5]{Donovan} to its pushed-forward
structure sheaf. If the fibre has no fixed point, the sheaf restricts
to zero along each ambient fixed component, and its Lefschetz
contribution vanishes.
In the rationally connected case the alternating trace of $\sigma$
is $1$, so $\sigma$ has a fixed point. In the symplectic case write
$\mu_1,\ldots,\mu_N$ for the eigenvalues on its nonzero cohomology
groups, where $N=h/2+1$. They are nonzero. Their first $N$ power sums
cannot all vanish: Newton's identities would then give
$\prod_j\mu_j=0$. Thus $\sigma^a$ has a fixed point for some
$1\leq a\leq N$.
In the Calabi--Yau case $p_i=1$ and $J=0$.
Equation~\eqref{eq:log-generator} makes $K_Y$ Cartier near $T$.
The klt Cohen--Macaulay total space, and hence its Cartier fibre,
is Gorenstein there. Adjunction trivializes $\omega_T$ by the
residue of $\Omega_i$. The induced scalar on this generator is
$\lambda_i$: the action on $du/u$ has residue one.
Serre duality therefore gives alternating trace
\[
1+(-1)^h\lambda_i^{-1}.
\]
If $\lambda_i$ has order at most two it is already bounded.
Otherwise the trace is nonzero, and $\sigma$ has a fixed point.
In every remaining case a power $\sigma^a$, with $a\leq h+1$, fixes
a point of $T$. At most $h+1$ components of the coefficient-one part
of a dlt boundary on an $(h+1)$-fold meet at one point.
Indeed their common intersection
is a union of lc centres, and at the generic point of such a centre
the pair is SNC. A further power of exponent at most $(h+1)!$
therefore preserves a prime component $A$ through that fixed point.
By dlt adjunction $A$ is normal and carries an effective lc different
$\Delta_A$. The rational $p_i$-pluriresidue $\omega_A$ of $\Omega_i$
satisfies
\[
\Div(\omega_A)+p_i\Delta_A=0.
\]
This is an equality of divisors: at the generic smooth point it is
the ordinary residue, and at every prime it follows by taking a
Cartier tensor power in adjunction and dividing the resulting
identity. No change of the residue's degree is required.
Naturality of residue shows that the character of the chosen power
on $\omega_A$ is the corresponding power of $\lambda_i$.
Apply Lemma~\ref{lem:normal-character} with $p_i=1$ or $m$ and
$\dim A=h\leq s$. The power used to stabilize $A$ was bounded in
terms of $s$, so the order of $\lambda_i$ is bounded in terms
of $s$ and $m$. Taking least common multiples supplies a single
annihilating integer for all factors.
\end{proof}
\subsection{Completion of the klt case}
\begin{proposition}[The klt weight bound]\label{prop:klt-weight}
For every $s\geq1$ and $m\geq1$ there is a positive integer
$q_{\mathrm{klt}}(s,m)$ with the following property.
Let $K=\C(C)$ for a smooth projective complex curve, let $z$ be a
uniformizer at $c\in C$, and let $(V,B)$ be a geometrically integral
normal projective geometrically klt $s$-fold pair over $K$.
Assume that $K_V$ is rational Cartier, $B\geq0$, and that a rational
$m$-canonical form $\phi$ satisfies $\Div(\phi)+mB=0$.
Then
\[
q_{\mathrm{klt}}(s,m)w_z(\phi)\in\Z.
\]
\end{proposition}
\begin{proof}
Use the product and comparison constructions of
Section~\ref{sec:blocks}, and then the normalized models above.
Lemmas~\ref{lem:abelian-character} and~\ref{lem:other-characters}
give a common integer $Q=Q(s,m)>0$ with $\lambda_i^Q=1$ for every $i$.
Equation~\eqref{eq:character-cancellation} implies
\[
l\mid bQ\,\ord_{c'}a .
\]
Together with \eqref{eq:product-weight}, this gives
\[
mbQ\,w_z(\phi)=\frac{bQ\,\ord_{c'}a}{l}\in\Z.
\]
Thus $q_{\mathrm{klt}}(s,m)=m\,s!\,Q(s,m)$ works.
The reduced generic presentation in Section~\ref{sec:weights}
is rationally Gorenstein, so this is exactly the klt assertion
needed there. The residue induction therefore proves
Theorem~\ref{thm:weight}.
\end{proof}
==== END Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//sections/degeneration.tex ====
==== BEGIN Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//sections/fibration.tex ====
\section{Denominators in the canonical bundle formula}\label{sec:fibration}
The curve weight theorem controls a coefficient of the moduli divisor after
restriction to a transverse curve. To use that control for a linear system,
we must keep track of the actual divisors in the canonical bundle formula,
including their principal parts. We first choose a presentation with a
uniform pluricanonical degree, and then show that its moduli divisor has a
uniform denominator. Throughout this section the ground field is $\C$.
\begin{proposition}[An exact presentation]\label{prop:exact-presentation}
Fix an integer $d\geq 1$ and a finite set
$\Phi\subset [0,1]\cap\Q$. There is an integer $p_0=p_0(d,\Phi)>0$,
clearing the denominators of $\Phi$, with the following property.
Let $f:X\to Z$ be a contraction of normal projective varieties with
$\dim X\leq d$, and let $(X,B)$ be lc with $B\geq0$ and nonzero
coefficients in $\Phi$. Suppose that $K_X+B$ is $\Q$-Cartier and
\[
K_X+B\simq f^*L
\]
for a $\Q$-Cartier divisor $L$ on $Z$. Then there are
$\psi\in\C(X)^*$ and a $\Q$-Cartier divisor $D_Z\simq L$ such that
\begin{equation}\label{eq:exact-presentation}
K_X+B+\frac{1}{p_0}\Div(\psi)=f^*D_Z.
\end{equation}
\end{proposition}
\begin{proof}
Put $K=\C(Z)$ and let $F=X\times_Z\Spec K$ be the generic fibre.
Since $f$ is a contraction and the characteristic is zero, $F$ is
geometrically integral. It is normal, its induced pair $(F,B_F)$ is
geometrically lc, and $K_F+B_F\simq0$. These assertions may be checked
on a log resolution: after shrinking the base, the resolution and its
marked strata have the required generic smoothness, and the discrepancy
formula restricts to the fibres. We choose the canonical divisor on $F$
by dividing a rational top form on $X$ by a rational top form on $Z$.
Theorem~\ref{thm:normal-index}, applied to the geometric generic fibre,
gives a principal multiple in a degree depending only on its dimension
and $\Phi$. Taking a common multiple over dimensions at most $d$, and
also clearing $\Phi$, gives $p_0$. The theorem applies over the algebraic
closure of $K$ by characteristic-zero comparison: all data descend to an
algebraically closed field admitting an embedding into $\C$.
By Lemma~\ref{lem:trivialization-descent}, this trivialization descends
to $K$ in the same degree. Consequently we may choose
$\psi\in\C(X)^*$ such that
\[
V:=p_0(K_X+B)+\Div(\psi)
\]
has no component dominating $Z$.
It remains to show that this particular vertical divisor is a pullback.
Choose $n>0$ and $h\in\C(X)^*$ with
$n(K_X+B-f^*L)=\Div(h)$. Then
\[
n(V-p_0f^*L)=\Div(h^{p_0}\psi^n).
\]
The divisor on the right is vertical. Its defining function has zero
divisor on the normal projective generic fibre and hence belongs to
$K^*$, because $H^0(F,\OO_F)=K$. Write
$h^{p_0}\psi^n=f^*a$ for $a\in K^*$. We obtain
\[
V=f^*\left(p_0L+\frac1n\Div(a)\right).
\]
Thus $D_Z=L+(np_0)^{-1}\Div(a)$ has all the asserted properties.
\end{proof}
We recall the part of the canonical bundle formula needed below. Fix a
presentation~\eqref{eq:exact-presentation}. For a prime divisor $P$ on
$Z$, let $t_P$ be the log canonical threshold of $f^*P$ over the generic
point of $P$; here $P$ is Cartier after restricting to a neighborhood of
that point. Set
\begin{equation}\label{eq:cbf-definition}
B_Z=\sum_P(1-t_P)P,\qquad M_Z=D_Z-K_Z-B_Z.
\end{equation}
On a higher model $\tau:W\to Z$, use the crepant induced sub-pair to
define the thresholds and put
\begin{equation}\label{eq:cbf-higher}
D_W=\tau^*D_Z=K_W+B_W+M_W,
\end{equation}
with compatible canonical divisors. A sub-pair here permits negative
boundary coefficients. The traces $M_W$ form the moduli b-divisor
$\mathbf M$. We say that it is \emph{determined on $W$} if its trace on
every higher model is the pullback of $M_W$.
The lc-trivial fibration theorem gives a projective model on which
$\mathbf M$ is determined and its trace is nef and $\Q$-Cartier
\cite[Theorem~3.6]{FG}, extending the klt-trivial theory of
Ambro~\cite{AmbroBoundary,Ambro}. The lc formulation
uses the discrepancy b-divisor $\mathbf A^*$ with the discrepancy $-1$
terms omitted. Its rank-one hypothesis holds here: on a resolution,
$\lceil\mathbf A^*\rceil$ has only effective exceptional terms over the
generic fibre, since $B$ is an effective boundary. Its pushforward on
that normal fibre is the structure sheaf, and the contraction condition
gives rank one. In particular, horizontal components of coefficient one
are allowed. A further resolution gives a smooth projective
determination. We use the representative~\eqref{eq:cbf-definition};
changing $D_Z$ by a rational principal divisor changes $\mathbf M$ by
the corresponding principal b-divisor and preserves these qualitative
properties.
The original discriminant $B_Z$ is effective and its coefficients lie in
a rational DCC set depending only on $d$ and $\Phi$. To see this, lc
gives $t_P\geq0$. A component of $f^*P$ with multiplicity $a\geq1$
and boundary coefficient $b\geq0$ gives
$t_P\leq(1-b)/a\leq1$. Near the generic point of $P$, the testing
divisor $f^*P$ has positive integral coefficients. A log resolution,
followed by removing proper closed subsets of $P$, realizes its generic
threshold as an ordinary threshold on an open subset of $X$.
The ACC theorem for log canonical thresholds
\cite[Theorem~1.1]{HMX}, with testing coefficient set
$\Z_{>0}$, therefore puts the numbers $1-t_P$ in the claimed DCC set.
On higher models, $B_W$ need not be effective, but its coefficients
remain at most one, since the induced sub-pair is crepant and sub-lc.
\begin{theorem}[A uniform denominator for the moduli divisor]
\label{thm:moduli-denominator}
Fix $d\geq1$ and a finite set $\Phi\subset[0,1]\cap\Q$, and let
$p_0=p_0(d,\Phi)$ be as in
Proposition~\ref{prop:exact-presentation}. There is a positive integer
$p=p(d,\Phi)$ divisible by $p_0$ with the following property.
Let $f:X\to Z$ be a contraction of normal projective complex varieties
with $\dim X\leq d$ and $\dim Z>0$, and let $(X,B)$ be lc with
$B\geq0$, coefficients in $\Phi$, and $K_X+B$ $\Q$-Cartier.
For any exact presentation
\[
K_X+B+\frac1{p_0}\Div(\psi)=f^*D_Z
\]
with $\psi\in\C(X)^*$ and $D_Z$ $\Q$-Cartier, let $\mathbf M$ be
its moduli b-divisor.
On every smooth projective determination $W$ of $\mathbf M$,
the divisor $pM_W$ is nef Cartier.
\end{theorem}
The normalization by $p_0$ fixes the scale at which principal changes
are allowed. The next lemma computes one coefficient of this actual
moduli divisor by a curve weight. It does not require the crepant
boundary on a resolution to be effective.
\begin{lemma}[A transverse curve and its weight]\label{lem:slice}
Let $f:X\to Z$ be a contraction of normal projective complex varieties,
let $(X,B)$ be lc with $B\geq0$, and suppose that
\[
K_X+B+\frac1m\Div(\psi)=f^*D_Z
\]
for an integer $m>0$, a function $\psi\in\C(X)^*$, and a
$\Q$-Cartier divisor $D_Z$.
Let $\tau:W\to Z$ be a smooth projective birational model, and let
$P$ be a prime divisor of $W$. Write
$\alpha=\operatorname{coeff}_P(\tau^*D_Z)$, and let $t_P$ be the
threshold for the crepant induced sub-pair over the generic point of
$P$. Put $s=\dim X-\dim Z$.
Then there are a smooth projective curve $C$, a point $c\in C$, a
rational uniformizer $z$ at $c$, and a regular function-field extension
of $\C(C)$ admitting a geometrically integral projective normal
effective $s$-fold pair $(V,B_V)$ that is geometrically lc, and a
rational relative $m$-canonical form $\phi$ such that
\begin{equation}\label{eq:slice-weight}
\Div(\phi)+mB_V=0,\qquad w_z(\phi)=\alpha-1+t_P.
\end{equation}
\end{lemma}
\begin{proof}
Choose a smooth projective common resolution $U$ mapping to $X$ and to
$W$, and denote the latter morphism by $g$. With a compatible rational
top form $\theta$ on $U$, write
\begin{equation}\label{eq:slice-upstairs}
\Div(\theta)+B_U^c+\frac1m\Div(\psi)=g^*D_W,
\qquad D_W=\tau^*D_Z.
\end{equation}
Here $B_U^c$ is the crepant sub-boundary. Resolve so that its support,
the exceptional divisors over $X$, and the support of $g^*P$ are SNC.
We may include the finitely many canonical and principal supports
occurring in~\eqref{eq:slice-upstairs} among the markings.
Write $r=\dim Z$. If $r>1$, take a sufficiently general smooth
complete-intersection curve
\[
C=L_1\cap\cdots\cap L_{r-1}\subset W
\]
of very ample members, and choose a transverse point $c\in C\cap P$
over a general point of $P$. If $r=1$, take $C=W$ and $c=P$.
Here are the avoidance and smoothness conditions used in this choice.
The fibre-dimension jumping locus of $g$ has codimension at least two:
a jump over a divisor would give a proper inverse image of dimension
$\dim U$. There are also finitely many smooth marked intersections
on $U$. We avoid their images when those images have codimension at
least two. For each intersection whose image is a divisor, generic
smoothness gives a dense smooth open of that divisor over which its
map is smooth; we avoid the complement. All these excluded closed
sets have codimension at least two in $W$, so a general complete
intersection curve misses them. Choose $c$ also away from the other
divisor supports.
For global smoothness of the cut, apply Bertini successively to the
basepoint-free systems $g^*|L_j|$ on $U$ and simultaneously on its
smooth marked intersections. This gives a smooth variety
$\widetilde U=g^{-1}(C)$ with restricted SNC markings. This argument
uses the smoothness of $U$, not smoothness or flatness of $g$ along
$P$.
The morphism $g$ has connected fibres, as follows from Stein
factorization and the fact that $\C(W)$ is relatively algebraically
closed in $\C(U)$. Its restriction $h:\widetilde U\to C$ has the
same fibres over points of $C$. Thus $\widetilde U$ is connected and,
being smooth, is integral; $h$ is a contraction. In particular its
function field is regular over $\C(C)$.
We record why the threshold is retained in this cut. On the SNC model,
the threshold over the generic point of $P$ is
\[
\min_i\frac{1-b_i}{a_i},
\]
where the marked prime divisors dominating $P$ have crepant boundary
coefficients $b_i$ and multiplicities $a_i$ in $g^*P$.
Transversality gives the same coefficients and multiplicities in the
fibre over $c$. Components with image a proper closed subset of $P$
are avoided by the choice of $c$. The SNC criterion for sub-lc pairs
then shows that this minimum is also the threshold on the cut.
Exact adjunction fixes the coefficient of the base divisor as well.
For each $j$ choose $L'_j\sim L_j$ avoiding $c$, and choose a rational
function on $W$ with divisor $L'_j-L_j$. Multiply $\theta$ by the
pullbacks of these functions and take iterated residues along the cuts.
The resulting rational top form $\widetilde\theta$ on
$\widetilde U$ has the canonical divisor prescribed by adjunction,
namely the restriction of
$\Div(\theta)+\sum_jg^*L'_j$. Restricting
\eqref{eq:slice-upstairs} therefore gives
\begin{equation}\label{eq:slice-exact}
\Div(\widetilde\theta)+\widetilde B^c+
\frac1m\Div(\widetilde\psi)=h^*D'_C,
\qquad
D'_C=\left(D_W+\sum_j L'_j\right)\big|_C.
\end{equation}
All restrictions are defined by generality. Since $L'_j$ avoid $c$
and $C$ meets $P$ transversely,
$\operatorname{coeff}_cD'_C=\alpha$. When $r=1$, the sums and the
residue operations are empty.
To obtain the effective generic presentation, return to the original
model $X$. Its normal generic fibre over $Z$ is geometrically normal
over its perfect characteristic-zero ground field, and is geometrically
integral since $f$ is a contraction. Generic flatness and openness of
geometric normality and integrality therefore give a dense open subset
of $W$ where $W\to Z$ is an isomorphism and the original fibres have
these properties. Shrink this open set further using generic smoothness
of the resolution strata, so that the restricted resolution computes
the fibre discrepancies also after algebraic closure.
Exceptional centres that do not dominate the base disappear
there; centres that dominate it retain codimension at least two in
the fibres by the dimension formula. These are conditions at the
generic point of $C$, which is chosen in this open set even when the
marked point $c$ is outside it. Hence the generic
fibre $V$ is a normal projective geometrically lc pair with the
effective boundary $B_V$ induced by $B$, and $\widetilde B^c$ compares
crepantly with it.
Choose a rational uniformizer $z$ at $c$ and define the relative form
\[
\phi=(\widetilde\theta/dz)^{\otimes m}\widetilde\psi.
\]
Restriction of~\eqref{eq:slice-exact} to the generic fibre gives
$\Div(\phi)+mB_V=0$. The absolute form used to compute its weight is
\[
\Omega=\phi\wedge(dz/z)^{\otimes m}
=z^{-m}\widetilde\theta^{\otimes m}\widetilde\psi.
\]
For a divisor $E$ over the cut lying above $c$, put
$a_E=\ord_E(z)>0$ and let $b_E$ be its crepant boundary coefficient.
Equation~\eqref{eq:slice-exact} gives
\[
\frac1m\ord_E(\Omega)+1
=(\alpha-1)a_E+(1-b_E).
\]
Taking the infimum after division by $a_E$ proves
\eqref{eq:slice-weight}, since the infimum of
$(1-b_E)/a_E$ is exactly the preserved threshold $t_P$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:moduli-denominator}]
Let $P$ be any prime divisor of the smooth determination $W$, and put
$\alpha=\operatorname{coeff}_P D_W$. From
\eqref{eq:cbf-higher},
\[
\operatorname{coeff}_P M_W
=\alpha+t_P-1-\operatorname{coeff}_P K_W.
\]
Lemma~\ref{lem:slice} realizes $\alpha-1+t_P$ as the weight of an
exact degree-$p_0$ form with an effective lc generic presentation of
dimension $s=\dim X-\dim Z$. Theorem~\ref{thm:weight} consequently
gives
\[
q(s,p_0)\operatorname{coeff}_P M_W\in\Z,
\]
because $K_W$ is integral. Take $p$ to be a common multiple of $p_0$
and $q(s,p_0)$ for $0\leq s\leq d-1$. This choice works for every
$P$ without any bound on the complexity of $W$. Thus $pM_W$ is an
integral Weil divisor on the smooth variety $W$, hence Cartier.
It is nef by the lc-trivial fibration theorem.
\end{proof}
==== END Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//sections/fibration.tex ====
==== BEGIN Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//sections/introduction.tex ====
\section{Introduction}\label{sec:introduction}
Let $(X,B)$ be a normal projective log canonical pair over an algebraically
closed field $k$ of characteristic zero, and put
$D=K_X+B$. When $\kappa(X,D)\geq0$, the sections of multiples of $D$
determine the Iitaka fibration. The effective problem asks whether one
degree, depending only on the dimension and the allowed boundary
coefficients, already determines that fibration. A uniform degree must
control both the birational geometry of its base and the information
lost in passing between different pluricanonical degrees.
We formulate the conclusion directly as an equality of subfields of
the function field. For an integer $\ell>0$, write
\[
V_\ell(D)=H^0\bigl(X,\OO_X(\floor{\ell D})\bigr).
\]
When $V_\ell(D)\neq0$, the ratios of its nonzero sections generate a
field $F_\ell(D)\subset k(X)$. Define
\[
K(D)=k\left(\frac{s}{t}:
s,t\in V_\ell(D)\setminus\{0\},\
\ell>0,\ V_\ell(D)\neq0\right).
\]
Thus $F_\ell(D)$ is the function field of the image of the complete
degree-$\ell$ system, viewed inside $k(X)$, and $K(D)$ collects these
fields over all degrees. The reflexive sheaf
$\OO_X(\floor{\ell D})$ makes this definition meaningful even when
$\ell D$ is not Cartier.
\begin{theorem}[Uniform log Iitaka degree]\label{thm:main}
For every integer $d\geq5$ and every finite set
$\Phi\subset[0,1]\cap\Q$, there is an integer $m=m(d,\Phi)>0$
with the following property. Let $k$ be any algebraically closed field
of characteristic zero, and let $(X,B)$ be a normal integral projective
lc $d$-fold pair over $k$. Assume that $B\geq0$, every nonzero coefficient
of $B$ belongs to $\Phi$, $D=K_X+B$ is rational Cartier, and
$\kappa(X,D)\geq0$. Then
\[
V_m(D)\neq0,\qquad F_m(D)=K(D)
\quad\text{inside }k(X).
\]
\end{theorem}
The bound is existential. The proof uses the good-model theorem of
\cite{LA}, the normal log canonical index theorem of \cite{U}, and
the arithmetic Stein-degree theorem of \cite{SD}. Their precise
statements are recalled in Section~\ref{sec:preliminaries}.
The contribution developed here is the passage from these inputs to
uniform denominators for log fibrations with horizontal boundary,
followed by the exact comparison of rounded section fields.
We state the higher-dimensional range $d\geq5$ to complement the
lower-dimensional results discussed below; the intermediate arguments
are formulated in all fibre dimensions.
\subsection{History and the denominator problem}
Iitaka's work established the asymptotic fibration associated with
pluricanonical systems \cite{Iitaka}. Uniform effectivity asks for a
degree independent of the particular variety. In the general-type
case this became uniform birationality of pluricanonical maps,
proved in arbitrary dimension by Hacon--McKernan, Takayama, and
Tsuji \cite{HaconMcKernan,Takayama,Tsuji}.
Hacon--McKernan--Xu extended effective birationality to big log
canonical adjoints with coefficients in a DCC set
\cite[Theorem~1.3]{HMX}. That theorem includes round-down systems,
the convention used here.
When the Kodaira dimension is smaller than the dimension, the
induced log adjoint on the generic fibre has Kodaira dimension zero.
The canonical bundle formula
transfers the adjoint to the base, where the variation of the fibres
contributes a moduli divisor. Fujino--Mori developed this approach
to effective Iitaka fibrations \cite{FujinoMori}; Viehweg--Zhang
obtained effective results with a surface base and controlled fibre
invariants \cite{ViehwegZhang}. Birkar--Zhang's general effectivity
theorem depends on the dimension, the least nonvanishing pluricanonical
degree of the general fibre, and the middle Betti number of a smooth
model of its canonical cover
\cite[Theorem~1.2]{BZ}. Their effective birationality theorem for
polarized pairs \cite[Theorem~1.3]{BZ} is also the final birational
input in the present proof.
The logarithmic problem introduces horizontal boundary on the
generic fibre. Earlier results include the low-dimensional klt
results of Todorov and Todorov--Xu \cite{Todorov,TodorovXu}, and the
boundedness and birationality results of Hacon--Xu for klt pairs whose
boundary is big over the generic point of the Iitaka base
\cite{HaconXu}. Chen--Han--Liu formulate the effective log
Iitaka problem for lc pairs with DCC coefficients and relate it to
good minimal models and complements
\cite[Conjecture~1.2 and Theorem~1.4]{ChenHanLiu}.
They obtain the dimension-at-most-three case in
\cite[Corollary~1.5]{ChenHanLiu}. Our coefficient set is finite,
and the higher-dimensional argument here uses the three companion
theorems stated above.
A bounded index of the generic fibre does not by itself give the
needed Cartier denominator for the moduli divisor. After a curve
base change one can obtain semistable reduction, but its ramification
degree is generally uncontrolled. The relevant question is whether
the action of inertia on a suitably normalized logarithmic volume
form has bounded order. Only this character needs to be bounded;
the covering degree and the order of the entire automorphism need not be.
Fujino--Mori already control finite characters through the cohomology
of the fibre \cite[Theorem~3.1 and Sections~3.6--3.8]{FujinoMori}.
The companion \cite{U} develops the character method for
boundary-free factors using the singular Beauville--Bogomolov
decomposition. The curve and effective-system arguments of \cite{U4}
treat log Calabi--Yau fibrations in total dimension at most four,
using the index of the whole reduced slc fibre.
We adapt these methods to horizontal boundary in arbitrary fibre
dimension. The geometric structure theorem of Matsumura--Wang
\cite[Theorem~1.3]{MW} supplies a rationally connected factor carrying
the boundary, together with abelian and nonabelian
Beauville--Bogomolov factors. The additional work is to retain the
factor decomposition under semilinear actions and to control the
logarithmic character of the rationally connected factor.
\subsection{The main intermediate result and the proof}
The reusable intermediate statement is a bound for weights over
curves. Let $C$ be a smooth projective complex curve, put $K=\C(C)$,
let $z$ be a uniformizer at $c\in C$, and
let $(V,\Delta)$ be a geometrically integral normal projective lc
$s$-fold pair over $K$, with $\Delta\geq0$. For a positive integer $a$,
a degree-$a$ rational
log trivialization is a nonzero rational $a$-canonical form $\phi$
satisfying $\Div(\phi)+a\Delta=0$.
Its weight at $c$ is the infimum
\[
w_z(\phi)=\inf_E
\frac{\ord_E\bigl(\phi\wedge(dz/z)^{\otimes a}\bigr)+a}
{a\,\ord_E z},
\]
where $E$ ranges over divisorial valuations of the total function
field centred over $c$. These are ordinary canonical orders;
the additive $a$ records the logarithmic pole along the base.
Theorem~\ref{thm:weight} proves that an integer depending only on
$s$ and $a$ clears this weight. Once a degree-$a$ trivialization is
given, no further coefficient parameter is needed.
There are two parts to the weight argument. If a dlt model has a
horizontal coefficient-one component, taking a rational pluriresidue
reduces the fibre dimension. Its Stein factorization may change the
curve field. The arithmetic bound of \cite{SD} controls that change,
so residue gives an induction with bounded denominators.
In the klt case the product theorem of \cite{MW} applies, but the
product cover need not carry the inertia action. We pass to a Galois
comparison cover and prove that a bounded power acts separately on
suitable finite covers of the factors. The proof uses the algebra of
endomorphisms of the tangent sheaf of the comparison cover that preserve
its subsheaf of vector fields tangent to the boundary. The required
vanishing of global vector fields tangent to the boundary on the
rationally connected factor follows from a finite invariant log-volume
measure.
After equivariant semistable reduction, each factor form can be
normalized to have weight zero. Their product has weight zero as
well. If the original form differs from this product by a base function $v$
on a curve cover, and $c'$ is a point over $c$ of ramification index $e$, then
\[
e\,w_z(\phi)=\frac{\ord_{c'}(v)}{a}.
\]
The inertia identity forces $e$ to divide a bounded multiple of
$\ord_{c'}(v)$ once the factor characters have bounded order. This
cancels the uncontrolled ramification and bounds the original weight
denominator.
For abelian factors the character acts on a bounded-rank integral
cohomology group. For the other factors we instead use a dlt special
fibre on a good model. Du Bois base change determines its
structure-sheaf cohomology, and coherent Lefschetz gives a fixed
point of a bounded power of the action. A bounded further power
preserves a normal component. The residue there is a log trivialization
in the original degree, so its different has coefficients in a fixed
finite set. The normal lc index theorem of \cite{U}, applied to the
cyclic quotient of this component, bounds the character. This step
works on one normal component and is the reason an index theorem
for a reducible special fibre is unnecessary.
To finish the proof over $\C$, let $f:Y\to Z$ be the semiample contraction of a good model
$(Y,B_Y)$ of $(X,B)$. The normal index theorem gives a bounded positive
integer $p_0$ and a rational function $\psi\in\C(Y)^*$ such that the
canonical bundle formula has the exact presentation
\[
K_Y+B_Y+\frac1{p_0}\Div(\psi)=f^*(K_Z+B_Z+M_Z).
\]
Here $B_Z$ is the discriminant defined by log canonical thresholds,
and the equality specifies the moduli representative $M_Z$.
Fixing the degree $p_0$ is essential: arbitrary rational principal
shifts can introduce new denominators. Slicing the base by a curve
identifies a coefficient of this moduli divisor with the curve weight,
up to an integer. This proves Theorem~\ref{thm:moduli-denominator}:
a uniform multiple of the moduli trace on a suitable smooth birational
model of $Z$ is nef Cartier.
Birkar--Zhang's theorem then gives a birational system on the base.
The remaining step is to identify complete rounded section spaces
upstairs with those downstairs. This yields equality with $K(D)$
inside the original function field, including ratios initially
appearing in other degrees, and permits descent from $\C$ to any
algebraically closed characteristic-zero field.
Section~\ref{sec:mmp} records the relative minimal-model construction.
Section~\ref{sec:weights} proves the residue reduction for weights.
Sections~\ref{sec:blocks} and~\ref{sec:degeneration} establish the
klt weight bound. Section~\ref{sec:fibration} converts that bound
into moduli denominators, and Section~\ref{sec:completion} proves
Theorem~\ref{thm:main}.
==== END Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//sections/introduction.tex ====
==== BEGIN Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//sections/mmp.tex ====
\section{Minimal models over a curve}\label{sec:mmp}
We will use good minimal models for several degenerations of log
Calabi--Yau pairs. Theorem~\ref{thm:good-models} gives absolute good
models. The following consequence supplies the relative form needed
here, even when the original adjoint is not pseudo-effective on the
total space.
\begin{lemma}\label{lem:curve-mmp}
Let $f:X\to C$ be a contraction from a projective variety over $\C$ to
a smooth projective curve, and let $(X,\Delta)$ be a $\Q$-factorial dlt
pair with effective rational boundary. Suppose that, for some sufficiently
divisible integer $r>0$,
\[
H^0\bigl(X_\eta,\OO_{X_\eta}(r(K_{X_\eta}+\Delta_\eta))\bigr)\ne0,
\qquad \eta=\Spec\C(C).
\]
There is a $(K_X+\Delta)$-MMP over $C$ that terminates with a
$\Q$-factorial dlt pair $(X',\Delta')$ for which $K_{X'}+\Delta'$ is
semiample over $C$. If $(X,\Delta)$ is klt, its output is klt.
\end{lemma}
\begin{proof}
A generic section extends after a sufficiently positive twist from $C$.
Choose an ample rational divisor $A$ on $C$ such that
\[
K_X+\Delta+f^*A\quad\hbox{is pseudo-effective},
\qquad \deg A>2\dim X.
\]
We may represent $A$ by many general points with small positive
coefficients. Then $(X,\Delta+f^*A)$ is dlt, and is klt if
$(X,\Delta)$ is klt. Choose an effective ample rational scaling
divisor $G$, represented by general members with small coefficients,
such that $(X,\Delta+f^*A+G)$ is lc and
$K_X+\Delta+f^*A+G$ is nef. The good-model theorem just recalled and
\cite[Theorem~A]{TX} give a terminating MMP for $K_X+\Delta+f^*A$
with scaling of $G$. The resulting log adjoint is nef and hence
semiample by the nef case of \cite[Theorem~11.1]{LA}.
Every step of this program is over $C$. Indeed, suppose inductively
that the current variety $X_i$ has a morphism $f_i:X_i\to C$, and let
$R$ be the negative extremal ray chosen at that step. Since $f_i^*A$
is nef, $R$ is also negative for $K_{X_i}+\Delta_i$. The lc length
bound \cite[Theorem~1.1(5)]{FujinoMMP} supplies a rational curve
$\Gamma$ spanning $R$ with
\[
0<-(K_{X_i}+\Delta_i)\cdot\Gamma\le2\dim X.
\]
If $\Gamma$ dominated $C$, then $f_i^*A\cdot\Gamma\ge\deg A$, giving
\[
(K_{X_i}+\Delta_i+f_i^*A)\cdot\Gamma
\ge-2\dim X+\deg A>0,
\]
a contradiction. Thus the ray is vertical. The morphism to $C$
descends through its contraction, and any flip is also over $C$.
On a vertical ray the two adjoints have the same intersection numbers,
so these are also steps of a $(K_X+\Delta)$-MMP over $C$.
The transform of the added divisor remains $f_i^*A$. Dlt preservation
for the augmented pairs, followed by decreasing the boundary, keeps
$(X_i,\Delta_i)$ dlt throughout; the usual discrepancy comparison
preserves klt in the klt case. Finally, subtracting a divisor pulled
back from $C$ does not change relative semiampleness. Hence the
semiampleness of $K_{X'}+\Delta'+f'^*A$ proves the assertion.
\end{proof}
==== END Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//sections/mmp.tex ====
==== BEGIN Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//sections/preliminaries.tex ====
\section{Conventions and companion theorems}\label{sec:preliminaries}
We assume familiarity with discrepancies, singularities of pairs, and
the minimal model program, as in \cite{KM,KollarSing}. We recall the
conventions and precise theorem inputs needed for the argument.
The argument is carried out over $\C$ until the final descent to an
arbitrary algebraically closed field of characteristic zero. Varieties
are integral and projective unless a different setting is specified.
A \emph{contraction} is a projective surjective morphism
$f:X\to Z$ of normal varieties satisfying $f_*\OO_X=\OO_Z$.
In characteristic zero its generic fibre is geometrically integral.
All boundaries in our argument are rational.
For a normal variety, a rational section of a divisorial sheaf is
identified with a rational function. Thus, for a rational Weil divisor $D$,
\begin{equation}\label{eq:section-inequality}
H^0(X,\OO_X(\floor{D}))
=\{a\in k(X):\Div(a)+D\geq0\}\cup\{0\}.
\end{equation}
The equality uses the integrality of the orders of $a$. It does not
require $\floor{D}$ to be Cartier. We use compatible canonical divisors
on birational models, obtained from one rational top differential.
A rational $m$-canonical form means an element of the $m$th tensor power
of the top differential line of a function field; its divisor is computed
on normal models at codimension-one regular points.
We use log discrepancies. If $\pi:Y\to X$ is a birational model and
$K_Y+B_Y=\pi^*(K_X+B)$, then the log discrepancy of a prime divisor $E$
on $Y$ is $1-\operatorname{coeff}_E B_Y$. In particular, log canonicity
means that all these numbers are nonnegative. Crepant boundaries on
higher models may have negative coefficients. Effectiveness will always
be specified when it is needed. We use dlt adjunction in its usual form:
a component $S$ of the coefficient-one part of a dlt boundary is normal,
and
\[
(K_Y+B_Y)|_S=K_S+\operatorname{Diff}_S(B_Y-S)
\]
with an effective lc different when $B_Y$ is an effective boundary
\cite{KM,KollarSing}.
\subsection{The three companion inputs}
The following statements are the substantive inputs from companion
manuscripts. We recall their full scope because the coefficient and
ground-field conditions matter at different places in the proof.
\begin{theorem}[Normal log canonical indices {\cite[Theorem~1.2]{U}}]
\label{thm:normal-index}
For every integer $n\geq0$ and every DCC set
$\Psi\subset[0,1]\cap\Q$, there is an integer $a(n,\Psi)>0$
such that every normal integral projective lc pair $(V,\Delta)$ over
$\C$ with $\dim V=n$, $\Delta\geq0$, coefficients in $\Psi$, and
$K_V+\Delta\simq0$ satisfies
\[
a(n,\Psi)(K_V+\Delta)\sim0.
\]
Here the displayed multiple is an integral principal divisor.
\end{theorem}
A DCC set is a set with no infinite strictly decreasing sequence.
We will apply Theorem~\ref{thm:normal-index} both to a finite coefficient
set and to the DCC set produced by finite cyclic quotients. Its
principal-divisor conclusion supplies an actual rational pluriform
in a bounded degree.
\begin{theorem}[Good models {\cite[Theorem~11.1]{LA}}]
\label{thm:good-models}
Every projective lc pair $(V,\Delta)$ over $\C$ with effective
real boundary and pseudo-effective real Cartier adjoint
$K_V+\Delta$ has a good log minimal model.
\end{theorem}
We use only the rational-boundary case, including semiampleness of a nef
adjoint. This theorem supplies existence; the terminating programs used
below also use the MMP with scaling \cite{TX}. Section~\ref{sec:mmp}
explains precisely how to apply these results over a projective curve.
\begin{theorem}[Arithmetic Stein degrees {\cite[Main Theorem]{SD}}]
\label{thm:stein-degree}
For every integer $n\geq1$ and every real number $t>0$, there is
an integer $N(n,t)$ with the following property. Let $F$ be a field of
characteristic zero, and let $(V,\Delta)$ be a normal integral projective
lc rational pair over $F$ with
\[
\dim V=n,\qquad H^0(V,\OO_V)=F,\qquad
\Delta\geq0,\qquad K_V+\Delta\simq0.
\]
If $S$ is a prime component of $\Delta$ of coefficient at least $t$,
the algebraic closure of $F$ in $F(S)$ has degree at most $N(n,t)$
over $F$.
\end{theorem}
For a proper normal component this algebraic closure is its field of
global functions. In our application $F$ is a complex curve field and
$t=1$; the theorem controls the finite curve in the Stein factorization
of a horizontal coefficient-one component.
\subsection{Trivializations and constants}
Two elementary facts will be useful repeatedly. First, a rational
function with zero divisor on a normal integral proper variety is an
invertible global function. Second, the degree of a geometric
trivialization does not increase on descending the constants.
\begin{lemma}[Descent in the same degree]\label{lem:trivialization-descent}
Let $V$ be a geometrically integral normal projective variety over a
characteristic-zero field $F$. Let $D$ be an integral Weil divisor.
If $D_{\overline F}$ is principal, then $D$ is principal.
In particular, an integral principal multiple of a rational log
canonical divisor on $V_{\overline F}$ descends in the same degree to $V$.
\end{lemma}
\begin{proof}
The divisorial sheaf $\OO_V(D)$ becomes the trivial line bundle after
extension to $\overline F$. Divisorial sheaves commute with this extension,
as can be seen on a regular big open and by reflexive extension.
Faithfully flat descent makes $\OO_V(D)$ invertible.
Proper base change shows that its space of sections is one-dimensional
over $F$. The evaluation map from this space tensored with $\OO_V$
is an isomorphism after scalar extension, hence already an isomorphism.
A nonzero section therefore trivializes the line bundle and principalizes
$D$.
\end{proof}
The geometric instances of Theorem~\ref{thm:normal-index} over a complex
function field are legitimate: the algebraic closure of any finitely
generated extension of $\C$ is abstractly isomorphic to $\C$.
Transporting the variety and its divisor through such an isomorphism
preserves the stated algebraic hypotheses. The final section gives
a separate descent argument for the ground field in the main theorem.
==== END Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//sections/preliminaries.tex ====
==== BEGIN Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//sections/weights.tex ====
\section{Weights of logarithmic pluriforms}\label{sec:weights}
The denominator needed in the canonical bundle formula will be detected
by a rational pluriform over a curve. Its weight compares the corrected
canonical order with the multiplicity of the base parameter. The
main assertion of this section bounds the denominator using only the
relative dimension and the degree of the pluriform. Coefficient-one
horizontal boundary will permit induction by residue; the remaining
klt case is proved in Proposition~\ref{prop:klt-weight}.
Let $C$ be a smooth projective complex curve, let $c\in C$, and put
$K=\C(C)$. A rational uniformizer at $c$ is an element $z\in K$ with
$\ord_c z=1$. Suppose that $F/K$ is a finitely generated regular field
extension of transcendence degree $s$. A rational relative
$m$-canonical form is a nonzero element of
$(\bigwedge^s\Omega_{F/K})^{\otimes m}$. We use the canonical-line
identification given by a fixed wedge order to write its associated
absolute form as
\[
\Omega=\phi\wedge(dz/z)^{\otimes m}.
\]
Orders of this absolute pluriform are ordinary canonical orders on
models of $F$ over $\C$.
\begin{definition}\label{def:weight}
For a rational relative $m$-canonical form $\phi$, set
\begin{equation}\label{eq:weight-definition}
w_z(\phi)=\inf_E
\frac{\ord_E\bigl(\phi\wedge(dz/z)^{\otimes m}\bigr)+m}
{m\ord_E z}.
\end{equation}
Here $E$ runs through the normalized divisorial valuations of $F/\C$
whose restriction to $K$ is a positive multiple of $\ord_c$.
\end{definition}
The definition depends only on the field and form. In the log canonical
setting below the infimum is finite and is attained on a log resolution.
\begin{theorem}[Uniform weight denominator]\label{thm:weight}
For integers $s\ge0$ and $m\ge1$ there is an integer $q(s,m)>0$
with the following property. Let $C$ be a smooth projective complex
curve, $c\in C$, and $z$ a rational uniformizer at $c$. Let $V$ be a
normal geometrically integral projective $s$-fold over $K=\C(C)$,
and let $(V,B)$ be a geometrically log canonical pair with effective
rational boundary. If a rational relative $m$-canonical form $\phi$
satisfies
\begin{equation}\label{eq:weight-log-trivialization}
\Div_V(\phi)+mB=0,
\end{equation}
then
\[
q(s,m)w_z(\phi)\in\Z.
\]
The integer $q(s,m)$ is independent of the boundary coefficients and
of the curve, variety, and form.
\end{theorem}
Although the boundary is not prescribed in the theorem,
\eqref{eq:weight-log-trivialization} already makes $mB$ integral.
What remains unbounded on an arbitrary model is the multiplicity of a
vertical divisor. The preparation below makes its contribution visible
in a dlt fibre.
\subsection{Changes of fields and forms}
We first record the valuation rules used throughout the proof. They
extend the boundary-free rules of \cite[Lemma~4.2]{U}; the boundary does
not enter their proof.
\begin{lemma}\label{lem:weight-rules}
Weights of rational relative pluriforms have the following properties.
\begin{enumerate}
\item They do not depend on the rational uniformizer at the fixed point.
For $a\in K^*$ and $t\ge1$,
\[
w_z(a\phi)=w_z(\phi)+\frac{\ord_c a}{m},
\qquad w_z(\phi^{\otimes t})=w_z(\phi).
\]
\item If $F'/F$ is finite and $K$ is algebraically closed in $F'$, then
the pullback of $\phi$ has the same weight over $C$.
\item Let $K'/K$ be finite Galois, let $C'\to C$ be the associated map
of smooth projective curves, and let $c'\in C'$ lie above $c$ with
ramification index $l$. For a rational uniformizer $u$ at $c'$ and the
pullback $\phi'$ to $FK'/K'$, one has
\begin{equation}\label{eq:weight-base-change}
w_u(\phi')=l w_z(\phi).
\end{equation}
\end{enumerate}
\end{lemma}
\begin{proof}
If $z'$ is another uniformizer, $(dz'/z')/(dz/z)$ is a base unit at
$c$. This proves independence of $z$. For every valuation in
\eqref{eq:weight-definition},
$\ord_E a=(\ord_c a)\ord_E z$; this gives the first formula, and the
tensor-power formula follows by multiplying numerator and denominator.
For a divisorial valuation $E'$ above $E$ in a finite extension of
characteristic-zero fields, with ramification index $a$, the canonical
ramification formula gives
\begin{equation}\label{eq:weight-finite-orders}
\ord_{E'}(\Omega)+m
=a\bigl(\ord_E(\Omega)+m\bigr).
\end{equation}
Divisorial valuations restrict and prolong to divisorial valuations.
When the constant field is unchanged, $\ord_{E'}z=a\ord_E z$, so
the individual quotients, and hence their infima, agree.
For a curve extension, the pullback of $dz/z$ is a unit times $du/u$
at $c'$, and
\[
\ord_{E'}u=\frac{a}{l}\ord_E z.
\]
Equation~\eqref{eq:weight-finite-orders} therefore multiplies each
quotient by $l$. Regularity of $F/K$ makes $F$ and $K'$ linearly
disjoint over $K$. The Galois action on their compositum is thus
available to move a prolongation to the chosen branch $c'$. Every
valuation downstairs has such a prolongation, proving
\eqref{eq:weight-base-change}.
\end{proof}
\subsection{A model on which the weight is exact}
We adapt the curve preparation in \cite[Section~4]{U4} to obtain an
exact divisor equality on a dlt model in any relative dimension.
The elementary discrepancy calculation behind the preparation will
also be used on semistable models. Write $A(E;W,\Delta)$ for the log
discrepancy, so a component of coefficient one has discrepancy zero.
\begin{lemma}\label{lem:snc-weight}
Let $f:W\to C$ be a projective model of $F/K$ with $W$ smooth. Put
$F_W=f^*c$ and $T_W=(F_W)_{\mathrm{red}}$. Let $H_W$ be an effective
horizontal rational boundary whose coefficients are at most one, and
suppose that $T_W+H_W$ has simple normal crossings near $T_W$.
For $\Omega=\phi\wedge(dz/z)^{\otimes m}$, assume that the horizontal
part of $\Div_W(\Omega)/m+H_W$ is effective. Then
\begin{equation}\label{eq:weight-snc-minimum}
w_z(\phi)=\min_{D\subset T_W}
\frac{\ord_D(\Omega)+m}{m\ord_D z},
\end{equation}
where $D$ ranges over the prime components of $T_W$.
\end{lemma}
\begin{proof}
Denote the minimum on the right by $w$. After shrinking the curve
around $c$, the rational divisor
\begin{equation}\label{eq:weight-error}
E_W=\frac1m\Div_W(\Omega)+T_W+H_W-wF_W
\end{equation}
is effective. For every divisorial valuation over $c$, pullback of
this equality gives
\begin{equation}\label{eq:weight-discrepancy}
\frac{\ord_E(\Omega)}m+1
=w\ord_E z+A(E;W,T_W+H_W)+\ord_E E_W.
\end{equation}
The last two terms are nonnegative. Thus every quotient in
\eqref{eq:weight-definition} is at least $w$. A component realizing
the displayed minimum gives equality.
\end{proof}
\begin{proposition}\label{prop:prepared}
For the data of Theorem~\ref{thm:weight}, with $s>0$, there is a
projective contraction $f:N\to C$ and an effective horizontal rational
boundary $H$ such that:
\begin{enumerate}
\item $N$ is $\Q$-factorial, its generic fibre is geometrically integral
and birational to $V$, and $(N,T+H)$ is dlt, where
$T=(f^*c)_{\mathrm{red}}$;
\item $K_N+T+H\sim_{\Q,C}0$;
\item for the same field-theoretic form $\Omega$ and
$w=w_z(\phi)$, there is an equality of rational divisors near $T$,
\begin{equation}\label{eq:prepared-equality}
\frac1m\Div_N(\Omega)+T+H=w f^*c.
\end{equation}
\end{enumerate}
In particular, on the generic fibre,
$\Div(\phi)+mH_\eta=0$. If $H$ has no coefficient-one component,
then $(N_\eta,H_\eta)$ is geometrically klt and $K_{N_\eta}$ is
$\Q$-Cartier.
\end{proposition}
\begin{proof}
Choose a smooth projective model $W\to C$ whose generic fibre is a
log resolution of $V$. Resolve also the closures of the horizontal
boundary and exceptional divisors, together with the fibre over $c$.
Let $B^c_{W_\eta}$ be the crepant subboundary on the generic resolution,
and take $H_W$ to be the closure of its positive part. Its coefficients
are at most one, and the horizontal divisor
\[
\frac1m\Div_W(\Omega)+H_W
\]
is effective and exceptional over $V$ on the generic fibre. The
markings can be chosen so that $(W,T_W+H_W)$ is log smooth.
Lemma~\ref{lem:snc-weight} computes $w$ and gives the effective error
$E_W$ in \eqref{eq:weight-error} near $c$.
On the generic fibre, $K_W+T_W+H_W$ is rationally linearly equivalent
to an effective exceptional divisor over the normal projective variety
$V$. Its Kodaira dimension is zero: a rational function whose poles
are supported on that exceptional divisor descends to a regular
function on $V$, hence is constant. In particular it has a section
in a sufficiently divisible degree. Lemma~\ref{lem:curve-mmp}
therefore gives a terminating MMP over $C$ for $K_W+T_W+H_W$.
Write its output as $(N,T+H)$. No divisors are extracted, so the
transform of $T_W$ is precisely $(f^*c)_{\mathrm{red}}$.
The generic function field is unchanged and is regular over $K$;
Stein factorization consequently makes $f:N\to C$ a contraction.
Discrepancy comparison preserves the divisible generic-fibre section
spaces through these steps. Thus the generic adjoint on $N$ still
has Kodaira dimension zero. Its relative semiample contraction has
zero-dimensional image on the generic fibre. The image is therefore
a normal curve finite over $C$. Connectedness of the fibres makes
its function field equal to $K$, so this curve is $C$. This proves
$K_N+T+H\sim_{\Q,C}0$.
Pushforward of \eqref{eq:weight-error} now gives, near the marked fibre,
\begin{equation}\label{eq:prepared-error}
\frac1m\Div_N(\Omega)+T+H=w f^*c+E_N,
\qquad E_N\ge0.
\end{equation}
The restriction of $E_N$ to the generic fibre is effective and
$\Q$-linearly trivial, hence zero. Thus $E_N$ is vertical, and it
is $\Q$-linearly trivial over $C$. Such a divisor is locally a
rational multiple of a fibre. Indeed, a relative principal
trivialization has vertical divisor, so its defining rational function
has neither zeros nor poles on the normal projective generic fibre.
It belongs to $K^*$, because that fibre has no new global functions.
Consequently $E_N=a f^*c$ near $c$ for some rational number $a$.
Applying the discrepancy calculation
\eqref{eq:weight-discrepancy} to \eqref{eq:prepared-error} shows that
all valuation quotients are at least $w+a$: the pair $(N,T+H)$ is lc.
Each component of $T$ has coefficient one, so its quotient is exactly
$w+a$. The valuation definition still gives $w$, since the function
field and form have not changed. Hence $a=0$, proving
\eqref{eq:prepared-equality}.
The generic restriction of that equality gives the claimed
trivialization. A dlt pair with no coefficient-one boundary is klt;
generic restriction, checked on a resolution, preserves this statement
after algebraic closure in characteristic zero. Finally,
$\Q$-factoriality of $N$ makes $K_{N_\eta}$ $\Q$-Cartier.
\end{proof}
\subsection{Residue and the reduction to klt fibres}
Suppose the prepared model has a coefficient-one horizontal component.
Residue lowers the relative dimension, but that component may acquire
new constants. The arithmetic Stein-degree theorem controls precisely
this finite extension.
\begin{lemma}\label{lem:residue-reduction}
Let $f:(N,T+H)\to C$ be as in Proposition~\ref{prop:prepared}, with
relative dimension $s\ge1$, and suppose that $S$ is a coefficient-one
component of $H$. Factor $S\to C$ as
\[
S\xrightarrow{h}C_S\xrightarrow{\nu}C,
\]
where $h$ is a contraction and $\nu$ is finite. Then $C_S$ is a
smooth projective curve and
\[
\deg\nu\le D_s
\]
for an integer $D_s$ depending only on $s$. For every $c'\in C_S$
above $c$, with ramification index $e$, there are data satisfying
Theorem~\ref{thm:weight} in relative dimension $s-1$ and degree $m$
over $\C(C_S)$, with a rational form $\phi_S$ whose weight is
\begin{equation}\label{eq:residue-weight}
w_u(\phi_S)=e w_z(\phi)
\end{equation}
for a rational uniformizer $u$ at $c'$.
\end{lemma}
\begin{proof}
Dlt adjunction \cite{KM} makes $S$ normal and gives an effective rational
different $\Theta$ such that $(S,\Theta)$ is lc. Its Stein curve
is normal and hence smooth. The generic pair $(N_\eta,H_\eta)$ is
normal, projective and lc, has $H^0(\OO_{N_\eta})=K$, and has
$K_{N_\eta}+H_\eta\simq0$. Its prime boundary component $S_\eta$
has coefficient one. Theorem~\ref{thm:stein-degree}, applied with
coefficient lower bound $1$, bounds the relative algebraic closure
of $K$ in $K(S_\eta)$. This field is exactly $\C(C_S)$, so it gives
the asserted $D_s$.
At the generic point of $S$, the absolute form $\Omega$ has a
logarithmic pole of order $m$. Choose a local equation $x$ of $S$ and
a rational $s$-form $\beta$, regular at the generic point of $S$, whose
restriction generates its canonical line. Write
\[
\Omega=a\bigl(dx/x\wedge\beta\bigr)^{\otimes m},
\qquad a\text{ a unit along }S.
\]
Its degree-$m$ Poincar\'e residue is the rational $m$-canonical form
\[
\Omega_S=(a|_S)(\beta|_S)^{\otimes m}.
\]
This construction is independent of the local equation and commutes
with tensor powers. Adjunction and
\eqref{eq:prepared-equality} give the equality of actual divisors
\begin{equation}\label{eq:residue-divisor}
\frac1m\Div_S(\Omega_S)+\Theta
=w(f|_S)^*c
\end{equation}
near the marked fibre. To verify the identity in this exact degree,
take a tensor power for which the ambient adjunction is Cartier and
apply the pluricanonical adjunction isomorphism. The resulting divisor
identity is that same positive multiple of \eqref{eq:residue-divisor}, and can
be divided by it. This uses the rational degree-$m$ residue already
defined at the generic point; it does not require a uniform Cartier
index along $S$.
The generic fibre of $h$ is normal, projective and geometrically
integral. Its pair induced by $\Theta$ is geometrically lc, as one
sees by generic restriction of a resolution in characteristic zero.
Dividing $\Omega_S$ by $(du/u)^{\otimes m}$ in the canonical-line
identification gives a rational relative form $\phi_S$. The generic
restriction of \eqref{eq:residue-divisor} is exactly
\[
\Div(\phi_S)+m\Theta_{\eta_S}=0,
\qquad \eta_S=\Spec\C(C_S).
\]
It remains to compute the weight at $c'$.
Near $h^{-1}(c')$, the right side of
\eqref{eq:residue-divisor} is $ew h^*c'$. For every valuation over
$c'$ the corrected order is consequently
\[
\frac{\ord_E(\Omega_S)}m+1
=ew\ord_Eu+A(E;S,\Theta).
\]
Log canonicity gives $w_u(\phi_S)\ge ew$. The fibre of $h$ has a
prime divisor, and any such divisor is a component of the intersection
of $S$ with $T$. At its generic point the two coefficient-one
branches of the dlt pair are simple normal crossings; adjunction gives
coefficient one in $\Theta$. The quotient for that divisor is $ew$,
which proves \eqref{eq:residue-weight}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:weight}]
When $s=0$, geometric integrality gives $V=\Spec K$, and
$\phi\in K^*$. The definition gives
$w_z(\phi)=\ord_c(\phi)/m$, so $q(0,m)=m$ works.
Proceed by induction on $s$. Use Proposition~\ref{prop:prepared}.
If its horizontal boundary has a coefficient-one component,
Lemma~\ref{lem:residue-reduction} and the induction hypothesis give
\[
q(s-1,m)e w_z(\phi)\in\Z
\quad\hbox{for some }1\le e\le D_s.
\]
Thus $q(s-1,m)\operatorname{lcm}(1,\ldots,D_s)$ clears the weight
in this case. Otherwise the prepared generic pair is geometrically
klt with $\Q$-Cartier canonical divisor. Proposition~\ref{prop:klt-weight},
proved independently of this induction below, gives a clearing integer
depending only on $s,m$. Taking a common multiple of these two
integers completes the induction.
\end{proof}
==== END Uniform-log-Iitaka-fibrations-and-bounded-moduli-denominators//sections/weights.tex ====