==== TABLE OF CONTENTS ====
Uniform-Pluricanonical-Iitaka-Fibrations//figures/induction.tex
Uniform-Pluricanonical-Iitaka-Fibrations//main.tex
Uniform-Pluricanonical-Iitaka-Fibrations//references.tex
Uniform-Pluricanonical-Iitaka-Fibrations//sections/completion.tex
Uniform-Pluricanonical-Iitaka-Fibrations//sections/denominators.tex
Uniform-Pluricanonical-Iitaka-Fibrations//sections/diagonal.tex
Uniform-Pluricanonical-Iitaka-Fibrations//sections/index.tex
Uniform-Pluricanonical-Iitaka-Fibrations//sections/introduction.tex
Uniform-Pluricanonical-Iitaka-Fibrations//sections/preliminaries.tex
Uniform-Pluricanonical-Iitaka-Fibrations//sections/scalar.tex
===========================
==== BEGIN Uniform-Pluricanonical-Iitaka-Fibrations//figures/induction.tex ====
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[
box/.style={draw,rounded corners=2pt,align=center,font=\small,
text width=39mm,minimum height=12mm,inner sep=4pt},
arr/.style={-{Stealth[length=2mm]},semithick}]
\node[box] (lowL) at (0,0) {$L_j$, $j0$, Proposition~\ref{prop:denominator} applies. Its base
adjoint is rationally linearly equivalent to $D$, so it is big.
Corollary~\ref{cor:big-base} supplies a degree depending only on $n$
whose section ratios give all of $\C(Z)$. The all-degree comparison
transfers this conclusion to $X$.
There are only finitely many possible base dimensions. Choose a common
multiple of the degrees supplied for these cases and for the point
case. Passing to this common multiple preserves the full section
field: if $s_0\ne0$ lies in the earlier degree and the new degree is
$q$ times that degree, the ratio $s/s_0$ is also the ratio
$(s s_0^{q-1})/s_0^q$ in the new complete system. All ratios in the
new system still lie in the Iitaka field by its definition. This
proves $I_n$ and closes the simultaneous induction starting from
dimension zero.
\end{proof}
\subsection{Algebraically closed fields of characteristic zero}
Let $k$ be any algebraically closed field of characteristic zero.
A given variety or pair and its finite defining data descend to a
finitely generated subfield $k_0\subset k$. Let $\overline{k_0}$ be an
algebraic closure inside $k$ and embed it into $\C$. Smoothness,
integrality, singularity conditions, and the relevant section spaces
are unchanged by these algebraically closed extensions. For
Theorem~\ref{thm:lc-index}, include a rational principalization of a
sufficiently divisible multiple among the descended data. The
uniform trivialization over $\C$ descends by the line-bundle argument
in Section~\ref{sec:preliminaries} and extends to $k$.
For Theorem~\ref{thm:main}, include one nonzero pluricanonical section
among the defining data to preserve nonnegative Kodaira dimension.
The same integer $m(n)$ gives a nonempty system after every extension.
To check its field, choose a sufficiently large divisible degree $r$
which also is divisible by $m(n)$ and defines the Iitaka fibration.
The products used in the preceding proof give a dominant rational
map from the degree-$r$ image to the degree-$m(n)$ image. It is
birational if and only if it is birational after an algebraically
closed extension: equivalently, the corresponding function-field
extension has degree one, a property detected by faithful flatness.
It is birational after comparison with $\C$, hence over $k$.
The same dimension-dependent integers therefore work over every
algebraically closed field of characteristic zero.
==== END Uniform-Pluricanonical-Iitaka-Fibrations//sections/completion.tex ====
==== BEGIN Uniform-Pluricanonical-Iitaka-Fibrations//sections/denominators.tex ====
\section{Uniform denominators and the base of a fibration}
\label{sec:denominators}
The induction needs more than a bound for the index of the general
fibre. It also needs a bound for the Cartier denominator of the moduli
b-divisor in the canonical bundle formula. We obtain this bound from
degenerations of the individual Beauville--Bogomolov factors. The
absence of a horizontal boundary is essential here: it allows us to use
the form algebras of those factors and the structure-sheaf cohomology of
their degenerations.
Throughout this section, fix $n\geq1$ and assume $L_j$ for every
$j0$, $X$ is
klt, and $K_X$ is rationally linearly equivalent to the pullback of
a rational Cartier divisor on $Z$. Then there are a rational function
$\psi\in\C(X)^*$ and canonical bundle formula data satisfying the
equality of rational divisors
\begin{equation}\label{eq:den-exact}
K_X+\frac{1}{p_0}\Div(\psi)=f^*D_Z,
\qquad D_Z=K_Z+B_Z+M_Z.
\end{equation}
Here $B_Z\geq0$ has coefficients in a rational DCC set depending only
on $n$, the generalized pair $(Z,B_Z+M_Z)$ is generalized klt,
and the moduli b-divisor $\mathbf M$ is b-nef with $p\mathbf M$
b-Cartier.
\end{proposition}
We first construct all the data except the uniform Cartier denominator.
The geometric generic fibre is klt, has dimension less than $n$, and
has rationally trivial canonical divisor. The induction hypothesis
therefore gives a common $p_0$ trivializing its canonical divisor.
This trivialization descends in the same degree to the generic fibre:
Cartierness descends by faithful flatness, and the one-dimensional
space of sections commutes with extension of the ground field. A
nonzero descended section generates, since it does so after that
extension.
Choose $\psi$ so that $p_0K_X+\Div(\psi)$ has no horizontal
component. To obtain the exact pullback in \eqref{eq:den-exact},
choose an initial rational Cartier divisor $D^0$ with
$K_X\sim_{\Q}f^*D^0$. A sufficiently divisible multiple of
$p_0K_X+\Div(\psi)-p_0f^*D^0$ is principal and vertical. A
rational function with vertical divisor is constant on the projective
generic fibre and hence lies in $\C(Z)$, since $f_*\OO_X=\OO_Z$.
Dividing its divisor by the chosen multiple gives the required
rational Cartier divisor $D_Z$.
Ambro's klt-trivial fibration theorem \cite{Ambro2005} now supplies
$B_Z$ and $\mathbf M$. Its rank-one condition holds: on a
resolution, the rounded discrepancy divisor over the generic base
point is effective and exceptional, and allowing poles along an
effective exceptional divisor creates no additional rational
functions. Thus the relevant generic direct image has rank one.
The discriminant coefficient at a prime $P\subset Z$ is
$1-t_P$, where $t_P$ is the lc threshold over its generic point of
$f^*P$. Since the total-space boundary is zero, $00$ such that, writing
\[
(g^b)^*\eta_i=c_i\eta_i,
\qquad \lambda_i=c_i(c),
\]
each $c_i$ is a unit at $c$ and $\lambda_i^N=1$.
\end{lemma}
\begin{proof}
\smallskip\noindent\textit{Abelian blocks.}
Let the block have dimension $s$. On its projective semistable
model over a disc, the direct image of the relative log top line is
the extended Hodge line $F^s$ in degree $s$. This is the semistable
comparison theorem for logarithmic de Rham cohomology and the
canonical extension with nilpotent residue
\cite{Deligne1970,Steenbrink1976}. The normalized form is a local
frame: it is a regular relative log form but cannot be divided by
$u$ while remaining regular, because its order on one component is
zero. Its eigenvalue on the central fibre of $F^s$ is $\lambda_i$.
The integral local system in degree $s$ has rank
$\binom{2s}{s}$. Choose a coordinate in which the finite base
action is a rotation. On the universal cover of the punctured disc,
the lifted transformation, combined with parallel transport, acts
by an integral matrix $A$ commuting with the unipotent monodromy.
A power of $A$ is a power of monodromy, because the original
transformation has finite order. Passing from flat frames to
canonical-extension frames multiplies $A$ by the exponential of a
scalar multiple of the logarithm of monodromy. This is a commuting
unipotent factor, so it does not change the eigenvalues. Hence a
primitive root of the order of $\lambda_i$ occurs among the
eigenvalues of an integral matrix of rank $\binom{2s}{s}$. Its
cyclotomic degree is at most that rank. There are only finitely many
possible orders, uniformly for $s2\dim Q$. The good-model theorem of
\cite[Theorem~11.1]{LA}, followed by a terminating MMP with scaling
to that good model, now applies. Every step is over the curve:
otherwise the extremal-ray length bound for
$(\overline Q,\Delta)$ would give a horizontal generating curve
$C_0$ with
$-(K_{\overline Q}+\Delta)\cdot C_0\leq2\dim Q$, while
the pullback of $H$ has degree greater than this on $C_0$.
Inductively the base pullback and the morphism to the curve persist.
Lift each step by normalization in the upstairs function field,
using Stein factorization for the contractions. Finite pullback
preserves the relative negativity and ampleness needed for flips.
No divisors are extracted, and the branch-canonical equality remains
valid in codimension one. This gives the finite-group equivariant
MMP; invariant $\Q$-factoriality suffices upstairs
\cite[Theorems~3.3.2 and~3.4.2]{Prokhorov2021}. The steps factor
over the upstairs curve as well, by normalization in its function
field. They preserve dlt for the pair with the special fibre added.
Special-fibre multiplicities remain one because there is no
extraction. The klt total space is Cohen--Macaulay, so its Cartier
fibre has no embedded associated components; generic reducedness
therefore makes the fibre reduced scheme-theoretically.
Relative semiampleness pulls back from the
downstairs good model. This proves \eqref{eq:den-good-model}.
\smallskip\noindent\textit{The degree-one log generator.}
The logarithmic divisor of $\eta\wedge du/u$ on $Y$ is effective
near $T$: this held on the semistable model, and the program extracts
no divisors. The generic fibre has Kodaira dimension zero.
Consequently the semiample contraction of $K_Y$ has
zero-dimensional image on the generic fibre; its Stein image over
the curve is the curve itself. Thus $K_Y$, and also $K_Y+T$, is
rationally linearly trivial over a neighbourhood of $c$.
The effective log divisor has no horizontal part, since its
restriction to the generic fibre is effective and rationally
trivial. A principalization of a multiple uses a rational function
with vertical divisor, hence a base function. The log divisor is
therefore $aT$ for some $a\in\Q_{\geq0}$. If $a>0$, every
weight quotient is at least $a$: it is $a$ plus the log discrepancy
of $(Y,T)$ divided by the fibre multiplicity. This contradicts
$w_u(\eta)=0$. Hence
\begin{equation}\label{eq:den-log-generator}
\Div_{\log}(\eta\wedge du/u)=0.
\end{equation}
In particular $K_Y+T$, and therefore $K_Y$, is Cartier near $T$.
Adjunction makes $T$ Gorenstein with trivial dualizing line, generated
by the residue of \eqref{eq:den-log-generator}. The transformation
acts on that generator by $\lambda$: the residue of its action on
$du/u$ is one. If a prime component $S\subset T$ is preserved by
a power of the transformation, the residue on $S$ generates
$K_S+\Theta_S$, where $\Theta_S$ is its different. Dlt
adjunction makes $(S,\Theta_S)$ normal lc and effective. Moreover
$\Theta_S$ is integral, since the divisor of this degree-one
rational form is integral and equals $-\Theta_S$. Its coefficients
are at most one, so it is reduced. The character on this component
generator is the corresponding power of $\lambda$.
Thus a bounded power preserving one component will reduce the
character bound to Lemma~\ref{lem:finite-character}. We obtain
such a power from the structure-sheaf cohomology of $T$.
\smallskip\noindent\textit{Finding a preserved component.}
The family is flat over the smooth curve. Its central fibre is slc
by \eqref{eq:den-good-model}, and its nearby fibres are klt after
shrinking; these fibres are Du Bois. Cohomology and base change for
proper flat Du Bois families makes $h^j(\OO_{Y_t})$ locally
constant \cite[Corollary~2.64]{KollarFamilies2023}. The nearby fibres are
birational to the block and have rational singularities. Their
structure-sheaf cohomology is therefore that of a smooth resolution
of the block. Extension of differential forms on klt spaces,
together with Hodge symmetry, identifies these dimensions with the
form algebra of the block \cite{GKKP2011}.
For an irreducible Calabi--Yau block, the only nonzero groups
$H^j(T,\OO_T)$ occur in degrees $0$ and $s$, and each has
dimension one. Serre duality and the log generator give alternating
trace
\[
1+(-1)^s\lambda^{-1}.
\]
If $\lambda$ has order greater than two, this is nonzero. In the
symplectic case there is one dimension in each even degree and none
in odd degree. For some power of the transformation of exponent at
most $s+1$ the trace is nonzero. Indeed, if the power sums of the
finitely many nonzero eigenvalues all vanished through their number,
Newton's identities would force their product to vanish.
The coherent Lefschetz theorem implies that a finite-order
transformation with nonzero alternating trace has a fixed point
\cite{Donovan1969}. Its vanishing statement when there are no fixed
points applies to the possibly singular projective scheme $T$, for
example by equivariant embedding and coherent Lefschetz localization.
At a fixed point at most $s+1$ components of $T$ meet: intersections
of components of a dlt boundary are lc strata, with the generic SNC
codimension description. A further power of exponent dividing
$(s+1)!$ therefore preserves a prime component through that point.
Its reduced log Calabi--Yau pair has dimension $s0$
with the following property. Let $Z$ be a normal projective complex
variety of dimension $d$ with a rationally connected smooth
projective resolution. Suppose $(Z,B+M_Z)$ is generalized klt,
$B\geq0$ has coefficients in $I$, the b-nef data satisfy
$p\mathbf M$ b-Cartier, and
\[
D=K_Z+B+M_Z\sim_{\Q}0.
\]
Then $\ell D$ is an integral principal divisor.
\end{proposition}
\begin{proof}
We follow the generalized torsion argument of
\cite[Proposition~7.1]{R4}, keeping track of why no
dimension-four restriction is needed.
We need two uniform integers: one making $D$ integral, and one
killing the resulting torsion class in the Weil divisor class
group $\Cl(Z)$. Global ACC, after the extractions below, gives a
finite coefficient set and a uniform positive lower bound for
generalized log discrepancies. The latter permits boundedness;
topology of the regular loci and Kummer theory then give a common
torsion exponent.
\smallskip\noindent\textit{Extraction and global ACC.}
We first justify the small modifications and one-divisor extractions
used with generalized global ACC. Choose a high log resolution
$g:W\to Z$ on which the nef data descend and, if a valuation $E$
is marked, on which $E$ appears. Write
\[
K_W+B_W+M_W=g^*D.
\]
The boundary $B_W$ is sub-klt and need not be effective. There is
an effective exceptional Cartier divisor $F$ with $-F$ relatively
ample; construct the resolution as a sequence of blowups and take
suitable positive combinations of the successive exceptional
divisors. Resolve all supports. If $E$ has generalized log
discrepancy $a\in(0,1)$, choose $\delta>0$ small enough that
$B_W+\delta F$ remains sub-klt and
$\delta\operatorname{mult}_EF0$ for generalized
$\epsilon$-lc singularities. Otherwise choose marked valuations
whose discrepancies decrease strictly to zero. Extract them as
above. The new coefficients $1-a_i$ form a strictly increasing
sequence, so adjoining them to $I$ still gives a DCC set. Global
ACC would force them into a finite set, a contradiction.
\smallskip\noindent\textit{Bounded topology and class-group torsion.}
Birkar's boundedness theorem \cite[Theorem~1.7]{Birkar2025} applies
in every fixed dimension: the varieties are projective and
rationally connected, their generalized adjoints are real-linearly
trivial, and they are uniformly generalized $\epsilon$-lc.
Consequently they are bounded up to isomorphism in codimension one.
The bounded comparison models can be normalized, still in a bounded
family and still isomorphic to $Z$ in codimension one.
Put $U=Z_{\mathrm{reg}}$. The groups $H_1(U(\C),\Z)$ are
finitely generated and range over only finitely many isomorphism
types. First, this group is invariant under isomorphism in
codimension one of normal varieties. On the regular loci the
common open is obtained by removing subsets of complex codimension
at least two. After stratifying those subsets, transversality moves
paths and their homotopies off every stratum, whose real codimension
is at least four; the fundamental group is unchanged. Second,
stratify a finite-type parameter space for the bounded comparison
family until it is flat. The relative smooth locus then has fibres
equal to the regular loci of the selected normal fibres. Cover the
parameter strata by finitely many affine opens and embed the family
projectively over each. Complex projective space is a compact real
algebraic set via its realization as rank-one Hermitian projection
matrices. Thus the entire relative smooth locus, not just individual
affine fibre charts, maps semialgebraically to each parameter open.
Semialgebraic triviality gives a finite partition on which these
fibres are homeomorphic, and semialgebraic triangulation gives each
the homotopy type of a finite complex \cite{DelfsKnebusch1982}.
This proves the asserted finite list of finitely generated homology
groups.
For each individual $Z$, the class group $\Cl(Z)$ is finitely
generated. Indeed a rationally connected smooth resolution $Y$
has $\operatorname{Pic}^0(Y)=0$, so $\operatorname{Pic}(Y)$ is
finitely generated by Neron--Severi finiteness; divisor pushforward
surjects onto $\Cl(Z)$. Since $Z$ is normal and projective,
$\Gamma(U,\OO_U^*)=\C^*$ and
$\operatorname{Pic}(U)=\Cl(Z)$. Kummer theory and comparison
for finite etale covers \cite{SGA1} give, for every $a\geq1$,
\begin{equation}\label{eq:den-kummer}
\Cl(Z)[a]\simeq H^1_{\mathrm{\acute et}}(U,\mu_a)
\simeq\operatorname{Hom}\bigl(H_1(U(\C),\Z),\mu_a(\C)\bigr).
\end{equation}
The left side has bounded order as $a$ varies for this fixed $Z$.
Thus $H_1(U(\C),\Z)$ has no free summand and is finite. There
are only finitely many such finite groups in our bounded topological
list. A common multiple $T$ of their exponents kills every torsion
class in every $\Cl(Z)$, by \eqref{eq:den-kummer}.
\smallskip\noindent\textit{Clearing the actual divisor.}
The trace $pM_Z$ is an integral Weil divisor, being the pushforward
of Cartier data. Choose $q$ divisible by $p$ and by the
denominators of the finite set $I_0$. Then $qD$ is integral.
Its class in $\Cl(Z)$ is torsion because $D\sim_{\Q}0$.
Therefore $TqD$ is principal. Taking $\ell=Tq$ proves the
proposition.
\end{proof}
\begin{corollary}\label{cor:relative-rc-torsion}
Under the hypotheses of Proposition~\ref{prop:denominator}, suppose
$K_X\sim_{\Q}0$ and a smooth projective resolution of $Z$ is
rationally connected. There is an integer $r=r(n)>0$ such that
$rK_X$ is an integral principal divisor.
\end{corollary}
\begin{proof}
The divisor $D_Z$ in \eqref{eq:den-exact} is rationally linearly
trivial. Indeed a Cartier multiple whose pullback is trivial is
itself trivial by the projection formula and
$f_*\OO_X=\OO_Z$. Proposition~\ref{prop:rc-torsion} gives a
uniform $\ell$ and a function $v\in\C(Z)^*$ with
$\ell D_Z=\Div(v)$. Choose a common multiple $r$ of $\ell$
and $p_0$. Then
\[
rK_X=\Div\left((v\circ f)^{r/\ell}\psi^{-r/p_0}\right).
\]
All constants depend only on $n$, since the dimensions of $Z$
range over a fixed finite set and Proposition~\ref{prop:denominator}
supplies a fixed coefficient set and nef denominator.
\end{proof}
==== END Uniform-Pluricanonical-Iitaka-Fibrations//sections/denominators.tex ====
==== BEGIN Uniform-Pluricanonical-Iitaka-Fibrations//sections/diagonal.tex ====
\section{A quadratic jet bound on diagonal products}\label{sec:diagonal}
The scalar estimates control section orders on a canonical variety and
local positivity on its index cover. We next obtain a complementary
upper bound on products of that cover. The bound grows quadratically
with the number of factors and is independent of the covering degree.
We give the arbitrary-dimensional form of the diagonal argument in
\cite[Section~7]{U4}, including its positive-characteristic estimates.
For a divisor $L$ on a variety $V$, write
$L^{\boxplus t}=\sum_{i=1}^t\operatorname{pr}_i^*L$ on $V^t$.
The order invariant $\gamma$ and the very general Seshadri constant
$\varepsilon$ have the conventions used in Proposition~\ref{prop:scalar}.
\begin{proposition}[Diagonal jet bound]\label{prop:diagonal}
Let $V$ be a normal projective canonical complex variety of dimension
$n\ge2$, with $K_V\sim_{\Q}0$ and canonical index $r$. Let
$\pi:Y\to V$ be its canonical cyclic cover. Suppose that $L$ is an
ample rational Cartier divisor and $C_1>0$ is rational, with
\[
1\le L^n\le2,\qquad \gamma(L;V)\le C_1.
\]
Choose a positive rational number $b_0$ such that
\begin{equation}\label{eq:diag-b0}
3(2b_0)^n<\frac14,\qquad b_0(C_1+1)<\frac34.
\end{equation}
For every integer $t\ge2$, set
\[
Z_t=Y^t/\mu_{r,\mathrm{diag}},\qquad
\theta_t:Z_t\longrightarrow V^t,\qquad
P_t=\theta_t^*L^{\boxplus t}.
\]
Then $P_t$ is ample and
\begin{equation}\label{eq:diag-epsilon}
\varepsilon(P_t)\le\frac{(n+1)t^2}{b_0}.
\end{equation}
\end{proposition}
The proof compares two orders of vanishing of one determinant.
If \eqref{eq:diag-epsilon} fails, ordinary jets, specialized to large
positive characteristic, produce a column map of full generic rank.
Its rank drops on the diagonal, forcing a maximal minor to vanish
there to high order. A movable curve test, chosen in characteristic
zero from the order bound on $V$, gives a smaller upper bound for
that same minor. The next three estimates make both
bounds quantitative.
\subsection{Truncated powers, flags, and orders}
For a vector bundle $E$ on a smooth projective curve, let
$\mu_{\min}(E)$ and $\mu_{\max}(E)$ denote the extreme slopes in
its Harder--Narasimhan filtration. In characteristic $p>0$, let
$A_\bullet(E)$ be the symmetric algebra modulo the ideal generated
locally by $p$-th powers of degree-one elements. Thus in a frame of
a rank-$\rho$ bundle,
\[
A_\bullet(E)=\OO[v_1,\ldots,v_\rho]/(v_1^p,\ldots,v_\rho^p).
\]
This definition is independent of the frame.
The relevant local model is the fibre product of $t$ copies of a
smooth $n$-fold $W$ over its relative Frobenius $F:W\to W'$, where
$W'$ is the Frobenius twist. In \'etale coordinates, differences
from the first factor give $n(t-1)$ variables whose $p$th powers vanish.
The associated graded along the reduced diagonal is therefore the
truncated algebra of $(\Omega_W^1)^{\oplus(t-1)}$. Below we use
the slope estimate for these graded pieces, the flag estimate to
bound their global sections, and the external-product estimate to
bound a determinant's order.
\begin{lemma}[Truncated-power slopes]\label{lem:diag-slopes}
Let $C$ be a smooth projective curve of genus $g$ over an
algebraically closed field of characteristic $p>0$. If $E$ has rank
$\rho$ and $\mu_{\min}(E)\ge0$, then, with
\[
G=\max\{0,2g-2\},\qquad
u=\rho(p-1),\qquad c_2=\rho(\rho-1)G,
\]
every nonzero graded piece satisfies
\begin{equation}\label{eq:diag-truncated-slope}
\mu_{\max}(A_j(E))\le(p-1)\deg E+c_2,
\qquad 0\le j\le u.
\end{equation}
The sum of their ranks is $p^\rho$.
\end{lemma}
\begin{proof}
We give the canonical-connection argument underlying the Frobenius
instability estimates of Langer \cite[Section~2]{Langer2004}; in
particular, we do not assume that tensor products are semistable in
characteristic $p$. For any rank-$\rho$ bundle $U$, absolute
Frobenius $F_C$ satisfies
\begin{equation}\label{eq:diag-one-step}
\mu_{\min}(F_C^*U)
\ge p\mu_{\min}(U)-(\rho-1)G.
\end{equation}
Indeed, consider the successive slopes of $F_C^*U$. If a successive
gap is larger than $G$, take the last such gap and let $S$ be the
preceding saturated filtration step. The second fundamental map of
the canonical connection,
\[
S\longrightarrow ((F_C^*U)/S)\otimes\Omega_C^1,
\]
vanishes by the slope inequality. Consequently $S$ descends under
Frobenius. Here this descent can be checked directly: over $k(C)$,
put a matrix for $S$ in a chart with an identity minor. Differentiating
its columns gives vectors in $S\otimes\Omega_{k(C)/k}^1$ with zero
identity rows, hence all matrix entries have derivative zero. They
belong to $k(C)^p$. Extracting their roots and saturating gives a
subbundle $S_0\subset U$ with $S=F_C^*S_0$, since Frobenius is flat.
The quotient $Q=(F_C^*U)/S$ is therefore the pullback of a quotient
of $U$, so $\mu(Q)\ge p\mu_{\min}(U)$. All successive slope gaps
of $Q$ are at most $G$; its smallest slope is at least its average
minus $(\rho-1)G$. It is also the smallest slope of $F_C^*U$.
This proves \eqref{eq:diag-one-step}. If there is no large gap, use
$S=0$ and the same average-slope argument.
Iteration gives
\begin{equation}\label{eq:diag-iterated-slope}
\mu_{\min}((F_C^e)^*E)
\ge-(\rho-1)G\frac{p^e-1}{p-1}.
\end{equation}
Choose a line bundle $T_e$ of integer degree $d_e$ in the interval
\[
2g-\mu_{\min}((F_C^e)^*E)
\le d_e<2g+1-\mu_{\min}((F_C^e)^*E).
\]
The bundle $E_e=(F_C^e)^*E\otimes T_e$ is globally generated:
after subtracting one point, its minimum slope is at least $2g-1$;
Serre duality gives vanishing of $H^1$ and hence surjectivity of
evaluation. If $Q_i$ is a positive-rank torsion-free quotient of
$E^{\otimes i}$, then $(F_C^e)^*Q_i\otimes T_e^{\otimes i}$
is globally generated. Thus $p^e\mu(Q_i)+i d_e\ge0$.
Divide by $p^e$ and let $e$ tend to infinity, using
\eqref{eq:diag-iterated-slope}, to obtain
\begin{equation}\label{eq:diag-tensor-lower}
\mu(Q_i)\ge-\frac{i(\rho-1)G}{p-1}.
\end{equation}
Since $A_i(E)$ is a quotient of $E^{\otimes i}$, its minimum slope
is at least $-c_2$ for $0\le i\le u$.
Multiplication gives a perfect pairing
$A_j(E)\otimes A_{u-j}(E)\to A_u(E)$: a truncated monomial pairs
with the unique complementary monomial. The top line is
$A_u(E)=(\det E)^{p-1}$, as is checked on diagonal and elementary
changes of frame. Therefore
\[
A_j(E)^*\otimes(\det E)^{p-1}\simeq A_{u-j}(E).
\]
Its minimum slope is at least $-c_2$, proving
\eqref{eq:diag-truncated-slope}. Counting the $p^\rho$ monomials
with exponents between $0$ and $p-1$ gives the rank assertion.
\end{proof}
\begin{lemma}[Sections from a fixed flag]\label{lem:diag-flag}
Let $W$ be a smooth projective $n$-fold, let $H$ be an ample
rational Cartier divisor, and fix a smooth complete intersection
flag cut successively by divisors in $|l_iH|$, $1\le iM$, by its
negative maximum slope. Otherwise evaluation at $\lfloor M\rfloor+1$
distinct points is injective, again by the slope criterion, so the
summand is at most $e(M+1)$. There are at most
$\prod_i(1+M/d_i)$ possible tuples. The same proof for $n=1$
uses an empty product.
\end{proof}
\begin{lemma}[Orders in an external product]\label{lem:diag-orders}
Let $W_i$ be integral projective varieties, $x_i$ smooth points,
and $Q_i$ line bundles with nonzero section spaces, for $1\le i\le t$.
Set $a_i=\max_{0\ne s\in H^0(W_i,Q_i)}\ord_{x_i}s$.
Every nonzero section of $\bigotimes_i\operatorname{pr}_i^*Q_i$
has order at most $\sum_i a_i$ at $(x_1,\ldots,x_t)$.
\end{lemma}
\begin{proof}
Choose bases adapted to the order filtrations of the section spaces.
Initial forms with the same order are linearly independent. By
K\"unneth, a section of the external product is a linear combination
of tensor products of these bases. In its smallest occurring total
degree, terms with different slot multidegrees cannot cancel; within
a multidegree their initial forms are independent because the slot
parameters are disjoint. Its order is therefore the smallest total
order of a basis tensor with nonzero coefficient, at most
$\sum_i a_i$.
\end{proof}
\subsection{The Frobenius rank comparison}
\begin{proof}[Proof of Proposition~\ref{prop:diagonal}]
The map $\theta_t$ is finite, so $P_t$ is ample. Fix $V,L,r,t$;
all auxiliary data in this proof may depend on these fixed choices.
Put
\begin{equation}\label{eq:diag-bq}
b=b_0/t,\qquad q=2tr.
\end{equation}
Suppose that $\varepsilon(P_t)>(n+1)t/b$.
\medskip\noindent\emph{A fixed supply of ordinary jets.}
Choose a smooth very general $z\in Z_t$ whose coordinates lie over
$V_{\mathrm{reg}}$, and a rational number $c$ with
$(n+1)t0$ small rational, such that
\begin{equation}\label{eq:diag-H}
\begin{split}
H^n&\le3,\qquad LH^{n-1}\le H^n,\\
(q+t-1)K_WH^{n-1}&\le b_0H^n.
\end{split}
\end{equation}
The canonical divisor of $W$ is pseudo-effective. Generic
semipositivity \cite[Theorem~2.1]{CampanaPaun2015}, followed by
restriction of its Harder--Narasimhan factors
\cite[Theorem~6.1]{MehtaRamanathan1982}, gives a smooth complete
intersection flag in multiples $l_iH$ ending in a curve $C$ with
\begin{equation}\label{eq:diag-cotangent}
\mu_{\min}(\Omega_W^1|_C)\ge0.
\end{equation}
The flag may be chosen to avoid the codimension-two singular sets
of the filtration quotients.
Independently choose a very general $x\in W$, in the isomorphism
locus of $u$, where the all-degree order bound defining $\gamma(L)$
holds. Let $\sigma:\widehat W\to W$ blow up $x$, with exceptional
divisor $J$, and define
\[
D^+=bL+(q+1)K_W,\qquad d_0=b(C_1+1).
\]
In divisible degrees the sections of $D^+$ are those of $bL$
multiplied by a fixed exceptional section: $rK_V\sim0$, and
$u_*\OO_W(E)=\OO_V$ for effective integral exceptional $E$.
That fixed section has order zero at $x$. Hence the normalized
orders of sections of $D^+$ at $x$ are at most $bC_1$.
It follows that $\sigma^*D^+-d_0J$ is not pseudo-effective.
Otherwise interpolation with the big divisor $\sigma^*D^+$ would
make $\sigma^*D^+-dJ$ big for a rational $d$ strictly between
$bC_1$ and $d_0$, producing an excluded section. The projective
movable-cone duality theorem \cite[Theorem~2.2, preprint]{BDPP2013}
therefore gives a strongly movable curve class $\beta$ on
$\widehat W$ satisfying
\begin{equation}\label{eq:diag-test}
D^+\cdot\beta0,
\qquad L\cdot\beta\ge0,
\qquad K_W\cdot\beta\ge0.
\end{equation}
Divisors from $W$ are pulled back in these pairings. More precisely,
we may choose $\beta=v_*(A_1\cdots A_{n-1})$, where
$v:T\to\widehat W$ is a smooth projective birational model and
the $A_i$ are very ample. The strict separation inequality holds on
one such generator of the movable cone. Nonnegative pairing with
$L$ and $K_W$ then forces $J\cdot\beta>0$.
\medskip\noindent\emph{Specialization of the finite data.}
Spread all the preceding algebraic data over an integral finitely
generated $\Z$-subalgebra of $\C$. This includes the points, blow-up,
flag, testing morphism, ample divisors, torsor, its marked lift,
and the finitely many jet-spanning sections. After shrinking the
base, preserve smoothness, geometric integrality, ampleness,
dominance of $v$, and the intersection numbers in
\eqref{eq:diag-H} and \eqref{eq:diag-test}. The fixed jet evaluation
is a map of finite locally free modules, so its surjectivity persists.
Invert $r$ and the finitely many denominators in the chosen data.
We can also preserve \eqref{eq:diag-cotangent}. A fixed relative
twist makes $\Omega_W^1|_C$ globally generated; thus the degrees of
its positive-rank quotients are bounded below uniformly by rank.
There are only finitely many negative degrees and ranks to exclude.
For each associated Hilbert polynomial, the relative Quot scheme
is projective and has empty geometric generic fibre, by
\eqref{eq:diag-cotangent}. Removing their images excludes all
negative-degree quotients. Quotients with torsion cause no issue,
since their torsion-free quotients have no larger degree.
Consequently \eqref{eq:diag-cotangent} holds on every remaining
geometric fibre.
There are such fibres in arbitrarily large characteristics $p$.
Work over their algebraically closed residue fields, retaining the
notation. The test $\beta$ still pairs nonnegatively with every
effective divisor: pull back by the retained dominant morphism $v$
and intersect with the very ample $A_i$. We have specialized a
fixed test, not an assertion about the entire pseudo-effective cone.
\medskip\noindent\emph{Full rank at a general product point.}
Let $F:W\to W'$ be relative Frobenius, where the prime indicates
the base-field twist. Set
\begin{equation}\label{eq:diag-ranks}
N=\lfloor p/k_0\rfloor,\quad B=NQ,\quad
\mathcal E=F_*\OO_W(B),\quad s=p^n,\quad R=s^t.
\end{equation}
The finite flat map $F$ has degree $s$, so $\mathcal E$ has rank $s$.
Products of $N$ of the fixed jet sections span jets through degree
$(n+1)t k_0N$. To check this, factor each monomial of degree at
most that number into $N$ monomials of degree at most
$(n+1)t k_0$, and choose corresponding sections with these initial
forms. Their products give a triangular spanning set for the jet
filtration. For large $p$,
\[
(n+1)t k_0N\ge nt(p-1).
\]
Thus these products span the thick point defined by the $p$-th powers
of all $nt$ smooth parameters. This is the elementary comparison
of ordinary and Frobenius jets used in
\cite[Proposition~2.12]{MustataSchwede2014}. Etaleness identifies
the completed local rings at the torsor lift and its tuple in $W^t$,
and therefore identifies these thick points.
For an $N$-fold product of pure tensors, write $m_i'$ for the new
exponent in slot $i$. Then $0\le m_i'\le N(r-1)0$, independent of $\nu$, such that
\[
\varepsilon(L_\nu)\ge c,\qquad
\varepsilon(\pi_\nu^*L_\nu)\ge cr_\nu^{1/n}.
\]
For $t\ge2$, let $Z_{\nu,t}=Y_\nu^t/G_{\nu,\mathrm{diag}}$,
let $\theta_{\nu,t}:Z_{\nu,t}\to V_\nu^t$ be the finite
quotient map, and set $P_{\nu,t}=\theta_{\nu,t}^*L_\nu^{\boxplus t}$.
The final condition is
\[
\varepsilon(P_{\nu,t})\le C_2t^2
\]
for every fixed $t$, on a tail allowed to depend on $t$.
All Seshadri constants are at very general points.
\end{proposition}
\begin{proof}
Fix an integer $T$, eventually large in terms of $n,c,C_2$.
On a tail of the sequence, use the upper bounds for all
$2\le t\le T$ simultaneously. Suppress $\nu$ until the covering
degree must vary, and put $B=C_2T^2+1$.
Since $G$ is abelian, $\theta_t$ is the finite quotient by
$G^t/G_{\mathrm{diag}}$: its further quotient has function field
$\C(V^t)$, and normality identifies it with $V^t$.
\medskip\noindent\emph{Curves and projection-compatible leaves.}
For each $t$, the strict bound $\varepsilon(P_t)0,\qquad P_t^{h_t}\cdot H_t\le(h_tB)^{h_t}.
\end{equation}
Its finite-quotient functoriality shows that
$W_t=\theta_t(H_t)$ is the geometric generic leaf
of a rational quotient
$\xi_t:V^t\dashrightarrow S_t$ with geometrically integral
generic fibre. These quotients and distributions are invariant
under permutations of the slots.
The image of an upper leaf under a coordinate projection lies
in the corresponding lower leaf. To check the genericity in
this assertion, retain the parameter of every upper family when
forming its image family. Equality of the values of the lower quotient
at a mark and a generic endpoint is an identity on that generic
incidence. It remains valid with the additional history of any
upper chain. Every upper step therefore preserves that quotient value;
a stationary projected step does so immediately. This proves
the inclusion for generic chains and then for their leaf closures.
All comparisons below use a generic tuple and its projections,
with their quotient parameter fields extended to a common
algebraically closed field.
\medskip\noindent\emph{Finiteness of the single-slot projections.}
For fixed $T$ and sufficiently large $r$, each $H_t$ projects
generically finitely onto its image in every $V$-slot. Suppose
otherwise. For one slot write $I$ for the image and
$j=\dim I$, $a=h_t-j>0$. The marked point of $I$ is ambient
very general and generic on $I$ over the leaf's quotient parameter
field. Thus
\begin{equation}\label{eq:transport-image-bound-new}
L^j\cdot I\ge c^j.
\end{equation}
Here and below the degree bound follows by blowing up the marked
point and taking the top intersection of the nef class given
by the Seshadri bound on the strict transform of the subvariety.
The generic point is smooth on the subvariety and has multiplicity
one. For $j=0$ the assertion reads $1\ge1$.
Over a geometric generic point of this slot, the fibre of
$Z_t\to V$ identifies with $Y^{t-1}$: choose one lift in the
torsor to fix the first coordinate. The polarization restricts
to the external sum of the remaining copies of $\pi^*L$.
Projecting a curve through the marked tuple to a nonconstant
slot, using the preceding degree--multiplicity comparison,
shows that this restricted polarization has Seshadri constant
at least $cr^{1/n}$ there. For an $a$-dimensional component $J$
of the geometric generic fibre of $H_t\to I$ through the mark,
we obtain $P_t^a\cdot J\ge(cr^{1/n})^a$.
The projection formula and nefness of
$P_t-\operatorname{pr}^*L$ now give
\[
P_t^{h_t}\cdot H_t
\ge(\operatorname{pr}^*L)^jP_t^a\cdot H_t
\ge c^j(cr^{1/n})^a.
\]
This contradicts \eqref{eq:transport-chain-degree-new} as
$r\to\infty$, since $h_t\le nT$ and $T$ is fixed.
Only finitely many slots and sizes are involved, so the conclusion
holds simultaneously on a tail. In particular,
\begin{equation}\label{eq:transport-dimension-new}
1\le h_t\le n,
\end{equation}
and the same single-slot finiteness holds for $W_t$.
\medskip\noindent\emph{A degree-one forgetting map.}
Let $d_t$ be the degree of $W_t$ over its first-slot image.
The degree of $H_t$ over that image contains $d_t$ as a factor.
Combining \eqref{eq:transport-chain-degree-new} with
\eqref{eq:transport-image-bound-new} gives
\begin{equation}\label{eq:transport-degree-new}
d_t\le KT^{2n},\qquad
K=\max_{1\le h\le n}\left(\frac{h(C_2+1)}c\right)^h.
\end{equation}
The constant $K$ is independent of $T$.
Projection compatibility and finite first-slot projection imply
$h_t\le h_{t-1}$ for $t\ge3$. Whenever equality holds, the
upper leaf maps onto the integral lower leaf; their first-slot
images coincide and
\begin{equation}\label{eq:transport-ratio-new}
d_t/d_{t-1}=\deg(W_t\longrightarrow W_{t-1})\in\N.
\end{equation}
There is a constant stretch in the $T-1$ dimensions of length
at least $(T-1)/n$. Choose $T$ so large that
\[
(T-1)/n-1>\log_2(KT^{2n}).
\]
If all the adjacent degrees on that stretch were at least two,
\eqref{eq:transport-ratio-new} would contradict
\eqref{eq:transport-degree-new}. Hence for some $k\ge3$,
\begin{equation}\label{eq:transport-degree-one-new}
h_k=h_{k-1}=h>0,\qquad
W_k\longrightarrow W_{k-1}\ \text{is birational}.
\end{equation}
Permutation invariance gives the same conclusion for every
single-slot omission.
We now have a positive-dimensional leaf which determines each
missing coordinate rationally. The rest of the proof turns this
rational recovery into too many birational transformations of $V$.
\medskip\noindent\emph{Transport of tangent directions.}
The tangent distribution $\mathcal D_k=\ker(d\xi_k)$ has rank
$h$ at a generic tuple. It projects injectively to each single-slot
tangent space, and isomorphically to the lower distribution under
every omission. These assertions follow from separability and the
generically finite maps of geometric generic leaves.
Let $A_i\subset T_{V,x_i}$ be its slot-$i$ image. It depends
rationally only on $x_i$. Indeed, in a rational Grassmann chart
its coordinates belong to the function field of every
$(k-1)$-tuple containing slot $i$, so they belong to the
intersection of these fields, namely $\C(V)$ in slot $i$.
Likewise, the transport from $A_i$ to $A_j$, obtained by lifting
to $\mathcal D_k$ and projecting to slot $j$, belongs to the
field of the pair $(x_i,x_j)$.
The field intersection just used is elementary for independent
product variables. Replace an omitted variable by a fresh
independent generic variable. A rational function pulled from
the remaining slots is unchanged by this replacement. Repeating
and then specializing the fresh variables to general constants
shows that it is pulled from the common slots. Applied to the
Grassmann and matrix chart coordinates, this proves both descent
statements, including when $k=3$.
Permutation symmetry therefore gives one rational plane
distribution $A(x)$ and rational isomorphisms
$R(x,y):A(x)\to A(y)$. Lifting a vector to the same leaf tangent
space and using three slots gives
\begin{equation}\label{eq:transport-cocycle-new}
R(y,z)R(x,y)=R(x,z).
\end{equation}
Choose a general reference point $a\in V(\C)$ and transport a
basis of $A(a)$ by $R(a,x)$. This produces rational vector fields
$v_1,\ldots,v_h$, independent at a generic point, whose simultaneous
copies $(v_i,\ldots,v_i)$ span $\mathcal D_k$ by
\eqref{eq:transport-cocycle-new}. No commutativity is asserted
or needed.
\medskip\noindent\emph{Rational recovery and local flows.}
Write $\xi=\xi_k$ and let $D$ be the image closure of
\[
\Lambda=(\operatorname{pr}_{0$ and the $v_i(y_0)$ are
independent, the map $z\mapsto E_z(y_0)$ has nonzero derivative
and uncountable image. This produces uncountably many distinct
members of $\Bir(V)$, a contradiction.
\end{proof}
Apply this proposition to the sequence in
Proposition~\ref{prop:index-reduction}. Choose $L$ rationally
scaled so that $1\le L^n\le2$. Proposition~\ref{prop:scalar}
gives uniform $\gamma(L)\le C_1$, $\varepsilon(L)\ge c$, and
$\varepsilon(\pi^*L)\ge cr^{1/n}$, since
$(\pi^*L)^n=rL^n$. Proposition~\ref{prop:diagonal} supplies the
remaining bound with $C_2=(n+1)/b_0$, while countability of
$\Bir(V)$ is part of Proposition~\ref{prop:index-reduction}.
The high-index sequence is impossible. Together with the other
cases of that reduction, this proves the zero-boundary klt
canonical-index bound in dimension $n$.
==== END Uniform-Pluricanonical-Iitaka-Fibrations//sections/diagonal.tex ====
==== BEGIN Uniform-Pluricanonical-Iitaka-Fibrations//sections/index.tex ====
\section{Normal log canonical indices and finite covers}\label{sec:index}
Two preliminary arguments will keep the induction within normal varieties.
The first recovers an lc index from the index of one normalized boundary
component and its Stein degree. The second separates finite actions on
the factors of a quasi-\'etale product cover. Throughout this section,
the inductive assertions $I_j,L_j$ are available for $j0$ and a rational
$aq$-canonical form $\tau$ with $\Div(\tau)+aqB=0$. The function
$s^{\otimes a}/\tau$ has divisor $aH$, so the first assertion writes
it as $f^*h$ for some $h\in k(Z)^*$. Thus
$H=f^*((1/a)\Div(h))$, as required.
\end{proof}
\begin{proposition}[Reduction to the zero-boundary klt index]\label{prop:lc-reduction}
Assume $L_j$ for every $j0$ such that
\[
c(n,\Gamma)(K_X+B)\sim0
\]
for every normal projective lc $n$-fold pair $(X,B)$ with coefficients
in $\Gamma$ and $K_X+B\sim_{\Q}0$.
\end{proposition}
\begin{proof}
The assumptions give the zero-boundary klt index bounds through
dimension $n$. Xu's klt induction theorem
\cite[Theorem~1.8]{Xu2019} therefore supplies the assertion when $(X,B)$
is klt. We prove the non-klt case without an slc index statement.
\emph{A Mori fibre space and a horizontal lc divisor.}
Take a crepant $\Q$-factorial dlt modification, and enlarge $\Gamma$
by $1$. Write $T=\lfloor B\rfloor\ne0$. For sufficiently small
rational $\epsilon>0$, the pair $(X,B-\epsilon T)$ is klt and
\[
K_X+B-\epsilon T\sim_{\Q}-\epsilon T
\]
is not pseudo-effective: its intersection with the $(n-1)$st power
of an ample divisor is negative. The klt minimal model program
\cite[Corollary~1.3.3]{BCHM2010} ends in a Mori fibre space
$f:X'\to Z$.
The full log canonical divisor remains crepant through this program.
Indeed, fix compatible canonical divisors and write
$r(K_X+B)=\Div(h)$. Divisorial contractions push forward this same
identity, and flips preserve it under their codimension-one
identification. On a common resolution, both pullbacks equal
$r^{-1}\Div(h)$. Thus the full pairs on all models are lc, even
though they need not be dlt. The exact order of their log canonical
divisors is unchanged. We henceforth write $(X,B)$ for the last
model.
If $\dim Z=0$, the klt log Fano pair $(X,B-\epsilon T)$ shows that
$X$ is of Fano type. The bounded complement theorem
\cite[Theorem~1.7]{Birkar2019}, with a fixed hyperstandard set containing
$\Gamma$, gives a bounded integer $c$ and a complement $B^+\ge B$
with $c(K_X+B^+)\sim0$. The effective divisor $B^+-B$ is numerically
trivial, hence zero by intersection with an ample class. This settles
that case.
Suppose $\dim Z>0$. On the Mori fibre space the surviving transform
of $T$ is relatively ample, since $\epsilon T$ is relatively
$\Q$-linearly equivalent to $-(K_X+B-\epsilon T)$. In particular,
some prime component $S$ of $T$ dominates $Z$. Let
$\nu:S^\nu\to S$ be its normalization. Divisorial adjunction gives
\begin{equation}\label{eq:index-adjunction}
(K_X+B)|_{S^\nu}=K_{S^\nu}+\Theta.
\end{equation}
Here $(S^\nu,\Theta)$ is a normal lc pair and $\Theta$ is effective;
its coefficients belong to the rational DCC set of differents
determined by $\Gamma$ \cite{KollarMori1998}. Its adjoint is
$\Q$-linearly trivial. The lower-dimensional assertion $L_{n-1}$,
together with global ACC as in the conventions, therefore bounds a
trivializing multiple of $K_{S^\nu}+\Theta$.
\emph{A common form degree.}
The geometric generic fibre of $f$ is a normal projective lc pair
of dimension less than $n$, with the same coefficient set. The
assertions $L_j$, $j0$
with the following property. Let $(V,E)$ be a normal projective lc
$j$-fold pair, with $E$ reduced, and let $\omega$ be a rational top
form satisfying $\Div(\omega)+E=0$. If $\gamma$ is a finite-order
automorphism of the pair and
$\gamma^*\omega=\lambda\omega$, then $\lambda^{c_j}=1$.
\end{lemma}
\begin{proof}
Form the quotient $\pi:V\to W=V/\langle\gamma\rangle$. Define
the boundary $E_W$ by the finite adjunction formula
\[
K_V+E=\pi^*(K_W+E_W).
\]
At a prime divisor with ramification index $e$, its coefficient is
$1-1/e$ if the upstairs coefficient is zero, and $1$ if the
upstairs coefficient is one. Thus the coefficients belong to the
fixed rational DCC set
$\{1-1/e:e\in\N\}\cup\{1\}$. The finite discrepancy formula
makes $(W,E_W)$ lc \cite[Proposition~5.20]{KollarMori1998}. A power of
$\omega$ invariant under the finite cyclic group descends to a
rational pluricanonical form downstairs; the ramification formula
shows that it trivializes a multiple of $K_W+E_W$. This also
establishes its $\Q$-Cartierness and $\Q$-linear triviality.
The assertion $L_j$ and global ACC supply a uniform $c_j$ for this
DCC set such that $c_j(K_W+E_W)\sim0$. Pull back a trivializing
log pluricanonical form. It is an invariant generator upstairs,
necessarily a scalar multiple of $\omega^{\otimes c_j}$, since
$V$ is projective and integral. Therefore $\lambda^{c_j}=1$.
For $j=0$ the same conclusion holds with $c_0=1$.
\end{proof}
\subsection{Quasi-\'etale covers and factor actions}
A finite map of normal varieties is \emph{quasi-\'etale} if it is
\'etale outside a set of codimension at least two. We write
$\Omega_V^{[p]}$ for the reflexive extension of $p$-forms from the
smooth locus. The elementary character criterion below records
that a bound on a volume character controls the exact global
canonical index, including Cartierness.
\begin{lemma}[Canonical index and the volume character]\label{lem:index-character}
Let $\pi:R\to X$ be a finite Galois quasi-\'etale cover of normal
projective varieties, with $K_R\sim0$. Choose a generating reflexive
top form $\omega$ on $R$, and write
$g^*\omega=\chi(g)\omega$ for the Galois action. Then
\[
mK_X\sim0\quad\Longleftrightarrow\quad\chi^m=1.
\]
\end{lemma}
\begin{proof}
If $\omega^{\otimes m}$ is invariant, it descends at the function
field level to a rational $m$-canonical form on $X$. Quasi-\'etaleness
means that its divisor has order zero at every prime divisor of
$X$. It therefore trivializes $\OO_X(mK_X)$ as a divisorial sheaf,
so $mK_X$ is principal. Conversely, a trivializing form downstairs
pulls back to a scalar multiple of $\omega^{\otimes m}$ and is
invariant.
\end{proof}
For a projective klt variety $X$ with $K_X\sim_{\Q}0$, the cyclic
cover attached to a primitive trivialization of its canonical
multiple is connected and quasi-\'etale. It has Cartier trivial
canonical divisor and is klt by finite adjunction, hence canonical:
its discrepancies are integers greater than $-1$. The singular
Beauville--Bogomolov decomposition theorem then supplies a finite
quasi-\'etale cover
\begin{equation}\label{eq:index-product-cover}
P=A\times\prod_{i=1}^r B_i\longrightarrow X.
\end{equation}
Here $A$ is abelian and each $B_i$ is an irreducible Calabi--Yau or
irreducible holomorphic symplectic variety in the singular sense
\cite[Definition~1.4 and Theorem~1.5]{HoringPeternell2019}. This theorem combines
the splitting and holonomy results of Druel and
Greb--Guenancia--Kebekus \cite{Druel2018,GGK2019}.
We use the following part of the definition of these nonabelian
blocks. They and all their connected normal finite quasi-\'etale
covers are canonical with trivial canonical divisor. Their
reflexive form algebras are generated by a top form in the
Calabi--Yau case, or by a generically nondegenerate two-form in
the symplectic case. In particular they have no reflexive
one-forms and no reflexive forms in degree one below their
dimension. Contraction with a volume form also gives the absence
of vector fields.
The cover $P\to X$ need not be Galois, and its Galois closure
need not remain a product. The next lemma is the comparison needed
to use the factors nonetheless. The argument follows the product
direction construction in \cite[Section~4.1]{U4}; we include the
regularity and semilinear details required here.
\begin{lemma}[Equivariant comparison with the factors]\label{lem:cover-comparison}
Let $k$ be a field of characteristic zero and let
$P=\prod_{i=1}^t B_i$ be a product of geometrically integral
projective factors whose geometric fibres are positive dimensional
blocks as above; include the abelian part, when present, as a single
factor. Let $R\to P$ be a geometrically integral normal finite
quasi-\'etale cover, and put $d=\dim R$. If a finite group $G$ acts
on $R$, then
there is a subgroup $G_0$ of index at most $d!$ and connected
finite quasi-\'etale covers $R_i\to B_i$ with the following
properties:
\begin{enumerate}
\item $R\to\prod_iR_i$ is finite and quasi-\'etale;
\item $G_0$ acts regularly on each $R_i$, and the projections
$R\to R_i$ are equivariant;
\item a generating volume form on $R$ is proportional to the
wedge product of the pulled-back volume generators on the $R_i$.
\end{enumerate}
The conclusions also hold for finite semilinear actions, interpreting
a factor transformation as an isomorphism to the corresponding
field-conjugate variety. In particular the $d!$th power of any
finite-order transformation preserves all factor fields.
\end{lemma}
\begin{proof}
\emph{Intrinsic preservation of the product directions.}
Pullback on the common smooth big open, followed by reflexive
extension, gives
\begin{equation}\label{eq:index-tangent-splitting}
\mathcal T_R=\bigoplus_{i=1}^t\mathcal E_i.
\end{equation}
There are no global homomorphisms between two distinct summands.
This can be checked after extending $k$ to an algebraic closure.
At least one of the corresponding factors is nonabelian,
say $B_i$. Fix general smooth points in all complementary factors.
The resulting slices of $R$ are normal and finite over $B_i$:
generic flatness and geometric normality in characteristic zero
allow this after shrinking the complementary parameter space.
The locus omitted from the quasi-\'etale map has codimension at
least two on a general slice. Each connected component of the
slice is consequently a connected normal quasi-\'etale cover
of $B_i$.
On the smooth big open of that component, $\mathcal E_i$ is its
tangent sheaf, while the complementary summands are trivial.
A homomorphism in either direction gives a vector field or a
one-form, which extends reflexively and vanishes by the block
properties. These slices cover a dense open of $R$, proving
the claimed vanishing.
The block projections in \eqref{eq:index-tangent-splitting} are
therefore central idempotents of the finite-dimensional algebra
$\operatorname{End}_R(\mathcal T_R)$. Its primitive central
idempotents give nonzero direct summands of positive generic
rank, so there are at most $d$ of them. The action of $G$
permutes this finite set. Its kernel $G_0$ has index at most
$d!$ and fixes each block projection, since a central idempotent
is a sum of primitive central idempotents. This reasoning is
unchanged for semilinear algebra automorphisms. In particular, the
idempotent argument is carried out over $k$ itself; it does not
require extending the finite group action to an algebraic closure.
\emph{Recovering the finite factor covers.}
Let $F_i$ be the relative algebraic closure of $k(B_i)$ in $k(R)$,
and let $R_i$ be the normalization of $B_i$ in $F_i$. Equivalently,
$R\to R_i\to B_i$ is the Stein factorization of the projection.
The intermediate fields $F_i$ are regular over $k$, because
$k(R)$ is regular over $k$; hence the $R_i$ are geometrically
integral. Stein factorization commutes here with algebraic closure:
the extensions $k(R)/F_i$ are regular in characteristic zero.
At the generic point, the complementary directions in
\eqref{eq:index-tangent-splitting} span
$\operatorname{Der}_{k(B_i)}k(R)$. For a finitely generated
extension in characteristic zero, the common constants of these
derivations are exactly the elements algebraic over $k(B_i)$.
Consequently $G_0$ preserves $F_i$ and induces birational factor
transformations.
The extension obtained by adjoining the other product factors
to $k(B_i)$ is regular, and hence is linearly disjoint from the
finite extension $F_i/k(B_i)$. Thus the function field of the
normal product $R_i\times\prod_{j\ne i}B_j$ is intermediate
between $k(P)$ and $k(R)$. Normalization gives finite maps
\[
R\longrightarrow R_i\times\prod_{j\ne i}B_j
\longrightarrow P.
\]
Ramification over a divisor of $B_i$ would persist in this
product and then at a prolongation to $k(R)$, contradicting
quasi-\'etaleness. Thus $R_i\to B_i$ is quasi-\'etale. The
product map $R\to\prod_iR_i$ is proper with finite fibres,
because its composite to $P$ is finite. Its image has dimension
$d$, equal to the dimension of the integral target product, so
it is surjective and finite. The same ramification argument
shows it is quasi-\'etale. In particular every fibre of
$R\to R_i$ has dimension $d-\dim R_i$.
\emph{Why the factor transformations are regular.}
Fix $g\in G_0$, and let $\Gamma_i$ be the closed graph of its
induced birational transformation on $R_i$. The two morphisms
from $R$ to $R_i$ give a surjection $R\to\Gamma_i$. All its
fibres have dimension at least $d-\dim R_i$. If the first
projection $\Gamma_i\to R_i$ had a positive-dimensional fibre,
its preimage in $R$ would have dimension strictly larger than
$d-\dim R_i$, contradicting the preceding equidimensionality.
Thus this projection is finite and birational. Normality of
$R_i$ makes it an isomorphism. Applying the same argument to
$g^{-1}$ proves that the induced transformation is an
automorphism. For a semilinear transformation, replace the
target by its field conjugate; the argument is identical.
Finally, each geometric $R_i$ is a quasi-\'etale cover of its
geometric block. It is canonical and has trivial canonical
divisor. These properties descend to $k$; linear triviality
descends by the same one-dimensional-section argument used in
Proposition~\ref{prop:lc-reduction}. The
product of its volume generators pulls back under the finite
quasi-\'etale product map to a generator of $K_R$. Any two
generators differ by a nonzero constant. This proves all
three assertions.
\end{proof}
The semilinear version applies to families over a curve as follows.
After a finite extension of the curve function field, define the
geometric product cover and its blocks. Take the normal Galois
closure of the resulting finite extension over the \emph{original}
total function field, and enlarge the curve field to its relative
algebraic closure in that closure. The latter is a finite Galois
extension, because the original total field is regular over the
original curve field. The closure is then geometrically integral
over its new constant field, and its finite Galois transformations
act semilinearly. On geometric fibres the comparison cover is
quasi-\'etale: over the original smooth locus purity makes the
initial covers \'etale, and the same is true of their conjugates
and Galois closure. After any further finite base extension the
same construction can be repeated. Lemma~\ref{lem:cover-comparison}
therefore separates the factor actions after taking a bounded
power, without requiring a bound on the degree of the comparison
cover.
==== END Uniform-Pluricanonical-Iitaka-Fibrations//sections/index.tex ====
==== BEGIN Uniform-Pluricanonical-Iitaka-Fibrations//sections/introduction.tex ====
\section{Introduction}\label{sec:introduction}
For a smooth integral projective variety $X$, the pluricanonical
systems organize the birational information carried by its canonical
divisor. If some $H^0(X,\OO_X(mK_X))$ is nonzero, the maximum of the
dimensions of the resulting rational images is the Kodaira dimension
$\kappa(X)$. In sufficiently large divisible degrees these maps determine
the Iitaka fibration \cite{Iitaka1971}. The question here is whether divisibility and
largeness can be prescribed using only $\dim X$.
To specify the required conclusion, let $K(K_X)\subset k(X)$ be the field
generated by ratios of nonzero sections in the same positive
pluricanonical degree, over all such degrees. This is the function field
of the Iitaka base. A complete system \emph{defines the Iitaka fibration}
if it is nonempty and its section ratios generate $K(K_X)$. Requiring
only an image of dimension $\kappa(X)$ would be weaker: it would allow
a proper finite subfield of the Iitaka field.
\begin{theorem}[Uniform pluricanonical degree]\label{thm:main}
For each integer $d\geq1$ there is an integer $m(d)>0$ such that, over
every algebraically closed field of characteristic zero, the complete
system $|m(d)K_X|$ defines the Iitaka fibration of every smooth integral
projective $d$-fold $X$ with $\kappa(X)\geq0$.
\end{theorem}
Using the log-abundance input from \cite{LA}, this gives an affirmative
resolution of the effective Iitaka fibration conjecture
\cite[Conjecture~1.1]{BirkarZhang2016}. Every positive multiple of
$m(d)$ works as well, by the section-ratio argument in
Section~\ref{sec:completion}.
Thus the same degree works for all nonnegative Kodaira dimensions. In
the general-type case its map is birational. In Kodaira dimension zero
the conclusion is nonemptiness in that degree, with image a point.
The theorem is existential; no small or practical numerical value of
$m(d)$ is asserted.
The proof establishes a second uniform statement at the same time.
A pair in the next theorem is normal and integral; its boundary is
effective and its log canonical divisor is rational Cartier.
\begin{theorem}[Uniform log canonical index]\label{thm:lc-index}
Let $d\geq0$, and let $\Phi\subset[0,1]\cap\Q$ satisfy the descending
chain condition. There is an integer $a(d,\Phi)>0$ such that every
projective log canonical pair $(X,B)$ of dimension $d$, with boundary
coefficients in $\Phi$ and $K_X+B\sim_\Q0$, satisfies
\[
a(d,\Phi)(K_X+B)\sim0.
\]
Here the multiple is an integral principal divisor, not merely a
numerically trivial or rationally linearly trivial divisor.
\end{theorem}
\subsection{Context and preceding work}
Effective pluricanonical maps are a quantitative part of birational
classification. The general-type case asks for a uniform birational
degree and was established by Hacon--McKernan, Takayama, and Tsuji
\cite{HaconMcKernan2006,Takayama2006,Tsuji2007}.
Hacon--McKernan--Xu's theorem for big log canonical adjoints permits
boundary coefficients in a fixed DCC set \cite[Theorem~1.3]{HMX2014}.
The lower-Kodaira-dimensional problem has an additional feature:
the general fibre contributes torsion and monodromy data which do not
appear in the big case.
Fujino--Mori obtained a uniform pluricanonical degree for threefolds
of Kodaira dimension one using their canonical bundle formula
\cite[Corollary~6.2]{FujinoMori2000}. Their denominator argument
identifies finite characters on an extended Hodge line and bounds
their orders using the middle Betti number of a canonical cover
\cite[Theorem~3.1 and Sections~3.6--3.8]{FujinoMori2000}.
Viehweg--Zhang obtained effective results for Iitaka bases of dimension
two, with constants controlled by invariants of the general fibre
\cite[Theorem~0.2]{ViehwegZhang2009}. Birkar--Zhang proved a general
effective Iitaka theorem with dependence on the dimension, the least
nonvanishing canonical degree of the general fibre, and a middle Betti
number of a smooth model of its canonical cover
\cite[Theorem~1.2]{BirkarZhang2016}. Their effective birationality theorem
for polarized adjoints is the final effective input used here
\cite[Theorem~1.3]{BirkarZhang2016}. The present argument supplies the
required denominator control without assuming a uniform bound for
those middle Betti numbers.
The index problem has its own inductive structure. Xu relates
zero-boundary klt index bounds to bounds with boundary
\cite[Theorem~1.8]{Xu2019}. Birkar proves the index theorem for rationally
connected klt Calabi--Yau pairs in every fixed dimension
\cite[Corollary~1.8]{Birkar2025}; his boundedness theorem for generalized
Calabi--Yau pairs also controls torsion on rationally connected bases
\cite[Theorem~1.7]{Birkar2025}. We combine these results with an arithmetic
Stein-degree theorem \cite[Main Theorem]{SD} to pass from klt pairs to
normal lc pairs without an index theorem for nonnormal pairs.
\subsection{Inputs and contributions}
One major input must be distinguished from standard minimal-model
technology. We use the good-log-minimal-model theorem of the companion
\emph{Log abundance in characteristic zero}
\cite[Theorem~11.1]{LA}, stated precisely in
Theorem~\ref{thm:good-models}. It includes nonvanishing and is used in
all dimensions. It is not being inferred from finite generation for
big adjoints. The other companion inputs are the chain and tracking
lemmas in \cite{U4}, and the arithmetic Stein-degree bound in \cite{SD}.
The arguments of \cite{R4} provide the preceding fourfold treatment of
relative denominators and rationally connected torsion.
The proof extends the canonical-index method of \cite{U4} to arbitrary
dimension. There are two substantial extensions. First, we prove a
relative denominator bound by decomposing the geometric generic fibre,
degenerating its individual factors, and controlling their volume-form
characters on normal components of a dlt special fibre. This separates
the potentially unbounded ramification of semistable reduction from
the bounded characters that enter the moduli divisor. The argument
uses the absence of horizontal boundary and only lower-dimensional
normal lc index bounds. It is therefore a reusable relative statement,
not merely the final step of the Iitaka theorem.
Second, the small-volume argument in \cite{U4} is extended to Iitaka
bases of arbitrary dimension. Weak positivity supplies a fibre-to-total
volume comparison, while flattening ensures that codimension-two
losses on the base create no divisorial poles on the original smooth
model. This gives scalar bounds for section orders and Seshadri
constants. We then give the dimension-dependent calculations in the
Frobenius diagonal argument and in the compatible-chain argument.
Those constructions are adaptations of \cite{U4}; the new claim is
their use in the all-dimensional induction, not a new origin for the
constructions themselves.
\subsection{The induction}
We prove Theorems~\ref{thm:main} and~\ref{thm:lc-index} simultaneously.
Fix $n$ and suppose both are known in smaller dimensions. The relative
step starts with a contraction $f:X\to Z$ from a klt $n$-fold, with
$\dim Z>0$ and $K_X$ rationally linearly pulled back from $Z$.
The canonical bundle formula writes
\[
K_X\sim_{\Q}f^*(K_Z+B_Z+M_Z),
\]
where $B_Z$ is the discriminant divisor and $M_Z$ is the moduli
divisor. Section~\ref{sec:denominators} bounds a Cartier multiple of
the moduli divisor on a birational base model using only
lower-dimensional normal lc indices. When $K_X\sim_{\Q}0$ and the
base has a rationally connected smooth resolution, this denominator
bound combines with torsion control on the base to bound the canonical
index of $X$.
Section~\ref{sec:reductions} then reduces failure of the zero-boundary
klt index bound to terminal varieties $V$ with canonical indices
$r\to\infty$ and countable birational groups. Their canonical cyclic
covers $\pi:Y\to V$ have degree $r$ and deck group $G$. Every
$G$-equivariant rational fibration of $Y$ with positive-dimensional
base and fibre has general-type smooth geometric generic fibre.
We call such fibrations intermediate. The corresponding assertion holds for
intermediate rational fibrations of $V$.
Choose an ample rational Cartier divisor $L$ on $V$ with
$1\le L^n\le2$. For an ample rational Cartier divisor $P$, write
$\varepsilon(P)$ for its very general Seshadri constant: the infimum
of $P\cdot C/\operatorname{mult}_xC$ over integral curves through a
very general point $x$. Section~\ref{sec:scalar} uses lower-dimensional
effective Iitaka fibrations to bound section orders on $V$ and bound
$\varepsilon(L)$ below uniformly. The pullback polarization has
$(\pi^*L)^n=rL^n$.
For $t\ge2$, let $Z_t=Y^t/G_{\mathrm{diag}}$, where $G$ acts by
the same deck transformation in every slot. Let
$\theta_t:Z_t\to V^t$ be the induced finite map and set
$P_t=\theta_t^*(\sum_{i=1}^t\operatorname{pr}_i^*L)$.
Together, the scalar estimates and the diagonal argument of
Section~\ref{sec:diagonal} give
\[
\varepsilon(\pi^*L)\ge c r^{1/n},
\qquad \varepsilon(P_t)\le Ct^2,
\]
with $c,C>0$ depending only on $n$. The second bound supplies curves
of bounded $P_t$-degree divided by marked multiplicity.
Section~\ref{sec:transport} arranges these curves into chains compatible
with coordinate projections. The resulting positive-dimensional
leaves project generically finitely onto their images in each
coordinate, and comparison of their projection degrees gives a
degree-one coordinate-forgetting map.
Rational recovery of its missing coordinate turns local flows into
uncountably many birational self-maps of $V$, a contradiction. This
bounds the zero-boundary klt index. The normal lc reduction in
Section~\ref{sec:index} then proves Theorem~\ref{thm:lc-index} in
dimension $n$. Finally, Section~\ref{sec:completion} reuses the
denominator bound with polarized effective birationality to prove
Theorem~\ref{thm:main} in that dimension. Figure~\ref{fig:induction}
records the dependencies after the induction statements have been
formally introduced.
==== END Uniform-Pluricanonical-Iitaka-Fibrations//sections/introduction.tex ====
==== BEGIN Uniform-Pluricanonical-Iitaka-Fibrations//sections/preliminaries.tex ====
\section{Conventions and the induction hypotheses}\label{sec:preliminaries}
Until the final field-descent argument we work over $\C$. Varieties are
integral and projective unless another setting is stated. We use the
standard discrepancy conventions for klt, dlt, and lc pairs
\cite{Kollar2013}. All boundaries in the proof are rational. A
\emph{rational Cartier divisor} means a $\Q$-Cartier $\Q$-divisor.
Canonical divisors are chosen compatibly when forms are pulled back
or residues are taken. On a normal variety, sections of an integral
Weil divisor always mean sections of its rank-one reflexive divisorial
sheaf.
For rational divisors $D,E$ we write $D\preceq E$ if $E-D$ is rationally
linearly equivalent to an effective divisor. An index bound is a
\emph{common trivializing multiple}: a single integer kills all the
actual divisor classes under consideration and clears their
coefficients. A bound on the least possible integers implies a common
multiple by taking their least common multiple.
A \emph{contraction} is a projective surjective morphism
$f:X\to Z$ of normal varieties such that $f_*\OO_X=\OO_Z$. Its generic
fibre is geometrically integral in characteristic zero. A rational
fibration is understood through a relatively algebraically closed
subfield of the total function field; after resolution and Stein
factorization it has connected fibres. A smooth geometric generic
model is a smooth projective model of the geometric generic fibre.
For generalized pairs, the nef data consist of a nef rational Cartier
divisor on a higher birational model, together with its compatible
pullbacks. A moduli b-divisor is \emph{b-nef and b-Cartier} if it is the
Cartier closure of a nef divisor on one such model. The generalized
discrepancies are computed by pulling back the full adjoint, including
these nef data. We use the conventions of \cite{BirkarZhang2016}.
\begin{theorem}[Good minimal models; \cite{LA}]
\label{thm:good-models}
Let $(X,B)$ be a projective lc pair over $\C$ with effective rational
boundary and pseudo-effective $K_X+B$. Then $(X,B)$ has a good log
minimal model. In particular, its adjoint is nonvanishing, and on the
good model the adjoint is semiample.
\end{theorem}
This is the rational-boundary case of \cite[Theorem~11.1]{LA}.
We also use the standard klt MMP and its discrepancy comparisons
\cite{BCHM2010}. For klt pairs with a good minimal model, an MMP with
scaling can be chosen terminating at such a model. We will specify
relative or equivariant uses when they arise; they do not require a
separate unmentioned abundance assumption.
\begin{notation}[Induction statements]\label{not:induction}
Let $I_n$ denote Theorem~\ref{thm:main} in dimension $n$, and let $L_n$
denote Theorem~\ref{thm:lc-index} in dimension $n$, for every rational
DCC coefficient set. Let $K_n$ denote its zero-boundary klt case:
there is a common integer $a_n>0$ with $a_nK_X\sim0$ for all projective
klt $n$-folds satisfying $K_X\sim_\Q0$.
\end{notation}
The dimension-zero statements hold with degree one. In a fixed
induction step we assume $I_j,L_j$ for all $j0$. Resolving the composite map
from $P$, we obtain $u:W\to P$ and $f:W\to T$, with $W$ smooth.
Pullback of differentials gives a nonzero section
\[
s=f^*\eta\in H^0\bigl(W,(\Omega_W^j)^{\otimes m}\bigr).
\]
At the generic point it has the form $s=a\beta^{\otimes m}$, where
$a\in\C(P)^*$ and $\beta$ is a nonzero decomposable rational
$j$-form, obtained by pulling back a rational top form on $T$.
The divisorial zero locus of $s$ is exceptional over $P$. To prove
this, let $H=u^*H_P$ for an ample Cartier divisor $H_P$ on $P$.
Canonicality and $K_P\sim0$ imply that $K_W$ is rationally linearly
equivalent to an effective $u$-exceptional divisor. In particular,
\[
K_W H^{n-1}=0.
\]
Generic semipositivity of the cotangent bundle~\cite{CampanaPaun2015}
implies that every torsion-free quotient of
$\mathcal V=(\Omega_W^j)^{\otimes m}$ has nonnegative degree against
$H^{n-1}$. Here one first uses an ample perturbation $H+\epsilon H_0$.
On a sufficiently high complete-intersection curve, the
Harder--Narasimhan factors of $\Omega_W^1$ restrict to semistable
bundles of nonnegative slope~\cite{MehtaRamanathan1982}.
Tensor products of semistable bundles in characteristic zero are
semistable~\cite{RamananRamanathan1984}, and exterior powers are
quotients of tensor powers. This proves the assertion for
$\mathcal V$ with the perturbed polarization; passage to the limit
gives the assertion for $H$.
The saturation of the line generated by $s$ is $\OO_W(Z)$, where
$Z\ge0$ is its divisorial zero divisor. Since
$c_1(\mathcal V)$ is a positive multiple of $K_W$, the quotient
inequality gives $ZH^{n-1}\le0$. Every nonexceptional prime has
strictly positive $H^{n-1}$-degree. Thus $Z$ is exceptional.
At the generic point of a prime divisor of $P$, write $\beta$ in a
regular frame and let $b$ be the minimum order of its coefficients.
The absence of zeros and poles of $s$ there gives
\[
\ord(a)+mb=0.
\]
All coefficients of $\Div_P(a)$ are therefore divisible by $m$.
Normalize $P$ in a field component obtained by adjoining an $m$th
root $z$ of $a$. This is a finite quasi-\'etale cover: over a
divisorial discrete valuation, removing the uniformizer to its
$m$-divisible order leaves the equation for a root of a unit, which
is unramified in characteristic zero. The rational form $z\beta$
has no divisorial pole, hence is a nonzero reflexive $j$-form on the
cover. It remains decomposable.
This contradicts the defining form algebra of the block. In the
Calabi--Yau case there are no intermediate-degree reflexive forms on
any connected quasi-\'etale cover. In the symplectic case the only
possibilities are scalar multiples of $\sigma^q$ with $0<2q0$ and $D$ is big.
For $q=\pi\circ u$, one has
\begin{equation}\label{eq:exceptional-fibre-class}
q^*D=g^*H+E,
\end{equation}
with $E\ge0$ exceptional over $Y$. At every nonexceptional prime
the two pullbacks agree, because its image on $V$ is a prime at
which the original rational map is defined. After fixing the
resolution, choose $H$ not to contain the generic image of any of
the finitely many exceptional primes. Its pullback has zero
coefficient at each such prime, whereas $q^*D$ is effective. This
proves the effectivity assertion in
\eqref{eq:exceptional-fibre-class}.
Let $F$ be the smooth geometric generic fibre of $f$. The class
$q^*D|_F$ is big: decompose the big class $q^*D$ into an ample
rational class and an effective rational class, and restrict to the
generic fibre, which is not contained in the effective support.
Since $g^*H|_F\sim0$, the class $E|_F$ is big. Terminality gives
\[
K_U\sim_{\Q}\sum_a d_a E_a,
\qquad d_a>0,
\]
where the sum includes every $u$-exceptional prime. Some positive
rational multiple of $E$ is coefficientwise dominated by this sum.
Restriction and adjunction therefore make $K_F=K_U|_F$ big. The
proof for $V$ is the same with the finite cover omitted.
\end{proof}
The preceding arguments prove Proposition~\ref{prop:index-reduction}.
They reduce an unbounded index to a sequence with countable birational
groups but particularly strong restrictions on equivariant fibrations.
We next turn those restrictions into uniform estimates for section
orders and curve degrees.
\section{Scalar order and curve estimates}
\label{sec:scalar}
Throughout this section, $A\preceq B$ means that $B-A$ is rationally
linearly equivalent to an effective rational divisor. A restriction
of this relation is made only to a member not contained in that
effective divisor. For a big, nef, semiample rational Cartier divisor
$P$ on an integral projective variety $V$ and a smooth point $x\in V$,
define
\[
\gamma(P;V,x)=\sup_{\substack{k>0\text{ sufficiently divisible}\\
0\ne s\in H^0(\widetilde V,k\mu^*P)}}
\frac{\ord_x(s)}{k},
\]
where $\mu:\widetilde V\to V$ is a smooth projective resolution
that is an isomorphism near $x$, and the same letter denotes its lift.
Proper birational pushforward and the projection formula make this
independent of the resolution. We write $\gamma(P;V)$ for its very
general value. Taking powers permits any further fixed divisibility
in $k$. For an ample $P$, also put
\[
\varepsilon(P;x)=\inf_{\substack{x\in C\subset V\\C\text{ integral curve}}}
\frac{P\cdot C}{\operatorname{mult}_x C}.
\]
We use its very general value, denoted $\varepsilon(P)$.
Both invariants are homogeneous under positive rational rescaling.
Moreover, if $d=\dim V$, the asymptotic section count and the number
of jet conditions at a smooth point give
\begin{equation}\label{eq:gamma-volume}
\gamma(P;V)\ge(P^d)^{1/d}.
\end{equation}
Indeed, $h^0(kP)=P^d k^d/d!+O(k^{d-1})$ in divisible degrees,
whereas vanishing to order at least $l$ imposes at most
$\binom{d+l-1}{d}$ conditions.
\begin{proposition}[Scalar estimates]
\label{prop:scalar}
Assume $I_j$ for $j0$ with the
following property. Let $V$ be a normal projective canonical
$n$-fold with $K_V\sim_{\Q}0$, and let a finite group $G$ act on
$V$. Suppose that every $G$-equivariant rational fibration with
geometrically integral generic fibre and positive-dimensional base
and fibre has general-type smooth geometric generic fibre. If $P$
is an ample rational Cartier divisor with $g^*P\sim_{\Q}P$ for
every $g\in G$, then
\begin{equation}\label{eq:scalar-estimates}
\gamma(P;V)\le C_n(P^n)^{1/n},
\qquad \varepsilon(P)\ge c_n(P^n)^{1/n}.
\end{equation}
The constants are independent of $V$, $G$, and the denominator of $P$.
\end{proposition}
The proof has two parts. An excessive section order produces
smaller-dimensional covering members with small polarized canonical
degree. We pass to their Iitaka fibres and repeat until their section
orders become small. Their chain quotient then contradicts the
general-type hypothesis. Once the upper order bound is known, a
thickening of a small-degree fibre gives the lower curve bound.
\subsection{The chain and tracking inputs}
We state the dimension-independent inputs from U4 in the form used
here. A marked covering family on $V$ consists of an integral
parameter space $S$, a dominant mark map $u:S\to V$, and a reduced
incidence $\mathcal I\subset S\times V$. Its fibre $C_b$ over a
geometric generic parameter $b$ is integral of positive dimension
and contains $u(b)$; the second projection evaluates its points in
$V$. One may replace $S$ by a dense open or a dominant extension
to specify geometric components or auxiliary data.
Fix a field of definition $k_*$ for these data. A generic movement
from a geometric generic mark $a$ chooses a generic parameter $b$
on a geometric component of $u^{-1}(a)$, and then a generic endpoint
on $C_b$. At each step genericity is over the field generated by
$k_*$ and the entire preceding path, enlarged to define the chosen
geometric component. In particular, new parameters remain generic
after the earlier marks, parameters, and endpoints have been fixed.
\begin{lemma}[Generic chains and order bounds]
\label{lem:chain-input}
For a finite nonempty list of marked covering families on an integral
projective variety in characteristic zero, there is a rational
fibration $\xi:V\dashrightarrow M$ with geometrically integral
generic fibre. If its generic leaf has dimension $h$, then
$1\le h\le\dim V$, and every generic member is contained in its
leaf. From a geometric generic mark $a$, a path of at most $h$
generic movements has endpoint generic in that leaf over
$\overline{k_*(a)}$.
The construction is equivariant for any finite group preserving the
list. A dominant rational map carrying upper movements to lower
movements or stationary steps carries upper leaves into lower leaves.
For a finite group quotient $V\to V'$, the images of the leaves of
an invariant list are the generic leaves of a rational fibration on
$V'$.
Let $P$ be big, nef and semiample. Suppose each geometric generic
marked member satisfies either of the following bounds, with the
same number $B$:
\begin{enumerate}
\item its mark $u(b)$ is smooth on $C_b$ and
$\gamma(P|_{C_b};C_b,u(b))\le B$;
\item it is a curve and
$P\cdot C_b/\operatorname{mult}_{u(b)}C_b\le B$.
\end{enumerate}
Let $H$ be a smooth projective resolution of the geometric generic
leaf, and retain pullback notation for $P$. Then
\begin{equation}\label{eq:chain-input-bounds}
\gamma(P;H)\le hB,
\qquad P^hH\le(hB)^h.
\end{equation}
\end{lemma}
This is \cite[Lemmas 5.1 and 5.4 and Corollary 5.2]{U4}.
The bounds concern the entire section spaces on a resolution of the
member, not merely restrictions of ambient sections. This distinction
is important when we apply the lemma to successive Iitaka fibres.
\begin{lemma}[Tracking and covering]
\label{lem:tracking-input}
Let $Z$ be projective, $\Q$-factorial and klt of dimension $d\ge2$,
and let $N$ be big, nef and semiample. Suppose that $d+1$ positive
rational numbers with consecutive spacing $\Delta>0$ lie strictly
between $(N^d)^{1/d}$ and $\gamma(N;Z)$. Then there is a covering
family of integral subvarieties $S$ of dimension $00$. Let $(W_i,L_i)$ be a sequence of
smooth projective $e$-folds with $\kappa(W_i)\ge0$ and big, nef,
semiample rational Cartier divisors $L_i$, such that
\begin{equation}\label{eq:small-volume-hypotheses}
L_i^e\longrightarrow0,
\qquad K_{W_i}L_i^{e-1}\le C'L_i^e.
\end{equation}
Assume $I_e$. After a subsequence and resolution of the Iitaka
fibrations, their smooth geometric generic fibres $U_i$ have fixed
positive dimension, satisfy $\kappa(U_i)=0$, and obey
\begin{equation}\label{eq:small-fibre-volume}
\vol(K_{U_i}+L_i|_{U_i})\longrightarrow0.
\end{equation}
The restricted polarizations are big, nef and semiample. If the
Iitaka base is a point, $U_i$ denotes the entire smooth model.
\end{lemma}
\begin{proof}
We suppress the sequence index. First, for every fixed positive
rational $a$,
\begin{equation}\label{eq:adjoint-volume-upper}
\vol(aK_W+L)\le(aC'+1)^eL^e.
\end{equation}
Choose a divided general member of a sufficiently divisible free
multiple of $L/a$, giving a klt effective boundary
$\Gamma\sim_{\Q}L/a$. The big klt adjoint $K_W+\Gamma$ has a
good model by BCHM. On a common resolution, let $P_a$ be $a$ times
the pullback of its semiample adjoint. The negative-map comparison
gives
\[
aK_W+L\sim_{\Q}P_a+F_a,
\qquad F_a\ge0,
\]
where $F_a$ is exceptional over that model. Thus
$P_a^e=\vol(aK_W+L)$. With $L$ pulled back to this resolution,
nef mixed-intersection inequalities give
\[
(aC'+1)L^e\ge(aK_W+L)L^{e-1}
\ge P_aL^{e-1}\ge(P_a^e)^{1/e}(L^e)^{(e-1)/e},
\]
which proves \eqref{eq:adjoint-volume-upper}. Replacing $W$ by a
higher smooth model and retaining the pullback of $L$ changes
neither this volume nor the intersection estimates: additional
canonical terms are effective exceptional divisors.
By $I_e$, a fixed integer $m_0$ defines every Iitaka fibration under
consideration. Pass to a subsequence where $j=\kappa(W)$ is
constant. The case $j=e$ is impossible for large indices. The
resolved system $|m_0K_W|$ would have an integral free moving part
with generically finite map and positive integral top intersection;
hence $\vol(K_W)\ge m_0^{-e}$, contrary to
\eqref{eq:adjoint-volume-upper}. If $j=0$, the assertion follows
directly from that inequality with $a=1$.
Suppose henceforth that $00$ for which
$(\operatorname{Sym}^b E_l)^{**}\otimes\OO_T(bH)$ is generated
by global sections at the generic point. A big $H$ suffices for
this consequence of weak positivity: choose $kH=A+F$ with $A$
ample and $F\ge0$, apply weak positivity with symmetric exponent
$k\beta$ and twist $\beta A$, and multiply by the section of
$\beta F$.
For every multiple $q$ of $b$, multiplication of those global
sections, followed by fibrewise multiplication in the adjoint
ring, gives sections over a big open of $T$ of
\[
ql(K_{W/T}+L)+qf^*H
\]
spanning all degree-$ql$ sections on the generic fibre. To see
the asserted domain of definition, remove only the codimension-two
sets where the relevant torsion-free sheaves differ from their
reflexive hulls or are not locally free. The multiplication maps
are defined at every codimension-one point. Generation is needed
only at the generic point, so its possible failure along a divisor
does not require removing that divisor.
Choose a basis over $\C(T)$ among these sections. Its size is
$h^0(U,ql(K_U+L|_U))$. Multiplying it by a basis of
$H^0(T,qlH)$ gives linearly independent products over $\C$:
a relation would first contradict the generic-fibre independence
over $\C(T)$ and then the independence of the base sections.
If necessary, multiply by a fixed nonzero section of
$(ql-q)H$ to place these products in degree $ql$ of the left side
of \eqref{eq:relative-adjoint-domination}. Multiplying further by
the effective difference in that equation, after clearing
denominators, embeds them into sections of its right side over
the same big open.
Let $T^\circ\subset T$ be the big open just obtained by removing
the codimension-two defects of the relevant torsion-free sheaves.
The resulting rational pluriforms are regular on $f^{-1}(T^\circ)$.
Every prime divisor on $W$ in its complement is $\rho$-exceptional,
by the flattening construction. Thus the forms are regular at the
generic point of the strict transform of every prime divisor on
$W_0$. Push them to $W_0$ with their $L_0$-twist. They have no
divisorial pole and therefore extend, by smoothness, to sections of
\[
H^0\!\left(W_0,
\OO_{W_0}\!\left(ql\bigl((1+(j+3)m_0)K_{W_0}+L_0\bigr)\right)\right).
\]
Their independence is preserved by this birational identification.
Put $g=e-j$. With this fibration and $l,b$ fixed, let $q$ tend
to infinity through sufficiently divisible multiples of $b$.
The asymptotic section counts on $T$ and $U$ give
\begin{equation}\label{eq:adjoint-volume-lower}
\vol\bigl((1+(j+3)m_0)K_{W_0}+L_0\bigr)
\ge \frac{e!}{g!j!}\,H^j\vol(K_U+L|_U).
\end{equation}
The left side tends to zero along the original sequence by
\eqref{eq:adjoint-volume-upper}, whereas $H^j\ge1$.
This proves \eqref{eq:small-fibre-volume}. No uniform choice of
$l$ or $b$ is needed: the asymptotic estimate is obtained separately
for each fibration before taking the sequence limit.
\end{proof}
\subsection{The upper order estimate}
\begin{proof}[Proof of the first inequality in Proposition~\ref{prop:scalar}]
The assertion in dimension one follows directly from degree, so
assume $n\ge2$. If no constant $C_n$ exists, rational rescaling
gives a sequence with
\[
P^n\longrightarrow0,
\qquad\gamma(P;V)>1.
\]
Take small projective $\Q$-factorializations and pull back $P$.
We describe a dimension-decreasing construction. At a stage, after
passing to a subsequence, there is a fixed dimension $d$ and data
$(Z,N)$ such that
\begin{enumerate}
\item $Z$ is projective, $\Q$-factorial and klt; $N$ is big, nef
and semiample; and $N^d\to0$;
\item a smooth model of $Z$ has nonnegative Kodaira dimension;
\item $K_Z\preceq a_0N$ for a fixed positive rational $a_0$;
\item $\gamma(N;Z)$ has a positive lower bound.
\end{enumerate}
These conditions hold initially by canonicality and crepantness
of the small modification. They cannot hold in dimension one,
because the order of a section is bounded by its degree.
For $d\ge2$, choose $d+1$ fixed equally spaced positive rational
levels below the order lower bound. Eventually they lie above
$(N^d)^{1/d}$. Lemma~\ref{lem:tracking-input} gives a covering
family with a smooth projective model $W$ of its geometric generic
member, of dimension $0D_0$. For each fixed
fibration, in large divisible degrees the leading coefficient
in \eqref{eq:thick-fibre-bound} is
\[
\frac{D^s}{s!h!}(P^hF)\,l^n.
\]
It tends to zero along the sequence, while the ambient section
count has leading coefficient $P^n/n!\ge1/n!$. Thus, first
choosing a sufficiently late sequence member and then a
sufficiently large divisible $l$, there is a nonzero ambient
section vanishing in $\mathcal I_F^u$.
At any smooth point of $F$, this section has order at least $u$,
because $\mathcal I_F\subset\mathfrak m_x$. The fibre and a
point on it can be chosen very general in the original
birational isomorphism locus and outside all the countably many
order-exceptional sets. Birational pullback preserves the
ambient section spaces and their orders there. The resulting
normalized order is at least $D>D_0$, contradicting the upper
bound. This proves the second inequality and completes
Proposition~\ref{prop:scalar}.
\end{proof}
==== END Uniform-Pluricanonical-Iitaka-Fibrations//sections/scalar.tex ====