==== TABLE OF CONTENTS ====
Lifting-sections-from-the-reduced-support-of-an-adjoint//abundance.tex
Lifting-sections-from-the-reduced-support-of-an-adjoint//bibliography.tex
Lifting-sections-from-the-reduced-support-of-an-adjoint//covers.tex
Lifting-sections-from-the-reduced-support-of-an-adjoint//curve-base.tex
Lifting-sections-from-the-reduced-support-of-an-adjoint//fields.tex
Lifting-sections-from-the-reduced-support-of-an-adjoint//frames.tex
Lifting-sections-from-the-reduced-support-of-an-adjoint//hodge.tex
Lifting-sections-from-the-reduced-support-of-an-adjoint//introduction.tex
Lifting-sections-from-the-reduced-support-of-an-adjoint//lifting.tex
Lifting-sections-from-the-reduced-support-of-an-adjoint//main.tex
Lifting-sections-from-the-reduced-support-of-an-adjoint//preamble.tex
===========================
==== BEGIN Lifting-sections-from-the-reduced-support-of-an-adjoint//abundance.tex ====
\section{Abundance after nonvanishing}
\label{bl:sec:abundance}
Over $\C$, the supported theorem closes the zero-Iitaka-dimension
case once lower-dimensional abundance generates the whole reduced
floor. For positive Iitaka dimension, the induction also needs
effective minimal models of fiber adjoints, which need not be nef.
We organize these two uses in one proposition, then verify its
hypotheses in dimension four and under the all-dimensional premise.
Every effective representative below is effective up to actual
$\Q$-linear equivalence. Ordinary log minimal models supply nef
adjoints; semiampleness is the conclusion to be proved. The first
lemma records exactly which section spaces survive passage to such
a model.
\subsection{Minimal models and the whole reduced floor}
\begin{lemma}[Comparison with an ordinary log minimal model]
\label{bl:lem:abundance-model-comparison}
Let $(W,\Delta_W)$ be a projective lc rational pair and let
$(V,\Delta_V)$ be a $\Q$-factorial dlt log minimal model. Use the
log birational model convention in which divisors extracted on $V$
have coefficient one. On a common resolution
$\alpha:U\to W$, $\beta:U\to V$, with compatible canonical divisors,
\begin{equation}
\alpha^*(K_W+\Delta_W)
=\beta^*(K_V+\Delta_V)+E,
\qquad E\geq0,\quad E\text{ is }\beta\text{-exceptional}.
\label{bl:eq:abundance-model-comparison}
\end{equation}
Consequently the adjoints have the same section spaces in sufficiently
divisible degrees, and hence the same Kodaira dimension.
\end{lemma}
\begin{proof}
Set $E$ equal to the difference in
\eqref{bl:eq:abundance-model-comparison}. It is anti-nef over $W$, since
$K_V+\Delta_V$ is nef. The discrepancy condition for a log minimal
model gives $\alpha_*E\geq0$: only prime divisors of $W$ contracted
by the model map can contribute, and their discrepancies improve.
The negativity lemma therefore gives $E\geq0$.
At a prime divisor on $V$ which is also a divisor on $W$, the
coefficient of $E$ is zero. At a prime divisor on $V$ extracted over
$W$, it is the new log discrepancy minus the old log discrepancy.
The former is zero and the latter is nonnegative. Its coefficient
is thus at most zero, and the effectivity just proved makes it zero.
This proves exceptionality over $V$.
For divisible $m$, adding the effective exceptional divisor $mE$
to $\beta^*m(K_V+\Delta_V)$ creates no additional sections.
Indeed a rational function allowed by the former divisor has the
required pole bounds at every prime divisor on the normal variety
$V$, and therefore is a section downstairs; the converse follows
by pullback. Birational pullback from $W$ preserves sections as
well. These identifications prove the assertion.
\end{proof}
We may take the $\Q$-factorial dlt models in this lemma in the
ordinary minimal-model existence statements below. A crepant dlt
modification, when needed, has the same adjoint pullback and preserves
nefness and all section spaces. The dlt modification theorem is,
for example, \cite[Theorem~2.8]{FujinoGongyo2014}.
\begin{lemma}[Semiampleness on the whole floor]
\label{bl:lem:abundance-floor}
Assume abundance for projective lc rational pairs of dimension less
than $d$. If $(V,C)$ is a projective $\Q$-factorial dlt pair of
dimension $d$ and $A=K_V+C$ is nef, then the actual restricted
rational line bundle $A|_P$, where $P=\lfloor C\rfloor$, is
semiample. Its restriction to any reduced union of components of
$P$ is also semiample.
\end{lemma}
\begin{proof}
There is nothing to prove if $P$ is empty. Whole-floor dlt
adjunction gives an effective rational boundary $\Delta_P$ for
which $(P,\Delta_P)$ is semi-divisorial log terminal, in particular
semi-log-canonical, and
\begin{equation}
K_P+\Delta_P=A|_P.
\label{bl:eq:abundance-floor-adjunction}
\end{equation}
Here the equality is an adjunction identification of rational line
bundles on the entire reduced scheme $P$; see
\cite[Remark~2.7]{FujinoGongyo2014}. In particular, it specifies the
gluing along the double locus.
For clarity, if $\zeta:\coprod P_i\to P$ is the normalization,
its pullback is the disjoint union of the adjunction formulas
$K_{P_i}+\Delta_i=A|_{P_i}$. The conductor has coefficient one in
$\Delta_i$ and the normalization pairs are dlt. The whole-floor
statement includes the $S_2$ and nodal-in-codimension-one properties
of $P$. The adjunction maps are pluriresidue maps: at a generic
double point, the two branch residues have the usual opposite
subsequent residues of an ambient logarithmic form. Taking an
even divisible power removes the sign convention and gives the
dualizing-sheaf gluing. Thus \eqref{bl:eq:abundance-floor-adjunction}
is not merely a collection of componentwise equivalences; it
identifies a divisible restricted invertible sheaf on $P$ itself.
Each $K_{P_i}+\Delta_i$ is nef and is semiample by the assumed
lower-dimensional abundance. Semiampleness descends from the
normalization of a projective slc pair by
\cite[Theorem~4.3, equivalently Theorem~1.5]{FujinoGongyo2014}.
Applied to \eqref{bl:eq:abundance-floor-adjunction}, this gives
generation of a power on the whole $P$. Restricting its evaluation
map to a reduced closed union of components preserves surjectivity,
which proves the last assertion. This also treats extra
coefficient-one components meeting the selected union and its strata.
\end{proof}
\subsection{The induction step}
\begin{proposition}[Abundance from effective models and lower-dimensional abundance]
\label{bl:prop:abundance-induction}
Fix $d\geq1$ and assume the following two statements over $\C$:
\begin{enumerate}
\item every projective lc rational pair of dimension at most $d$
whose adjoint has an effective $\Q$-linearly equivalent divisor
has an ordinary log minimal model;
\item every nef projective lc rational adjoint in dimension less
than $d$ is semiample.
\end{enumerate}
Then, for a projective lc rational pair $(X,B_X)$ of dimension $d$,
\[
D=K_X+B_X\text{ nef},\qquad \kappa(X,D)\geq0
\quad\Longrightarrow\quad D\text{ semiample}.
\]
If $\kappa(X,D)=0$, then $D\sim_{\Q}0$.
\end{proposition}
\begin{proof}
We first treat Kodaira dimension zero, then the Iitaka fibers in
the positive-dimensional case, and finally the lc centers. The
second assumption is full lower-dimensional abundance; no
nonvanishing hypothesis is imposed on the lower-dimensional
restrictions in Lemma~\ref{bl:lem:abundance-floor}.
\smallskip\noindent
\emph{Kodaira dimension zero.}
Choose $M\geq0$ with $D\sim_{\Q}M$. Suppose that $M\ne0$.
Take a projective log resolution $p:W\to X$ of the pair together
with $\Supp M$. Start with the strict transform of $B_X$, raise
the coefficient to one along every strict transform of a component
of $M$, and put coefficient one on every exceptional prime divisor.
Call the resulting snc boundary $C_W$, and let $T_W\geq0$ be the
sum of the coefficient increases on nonexceptional divisors.
Using log discrepancies, we have
\begin{equation}
\begin{split}
K_W+C_W&\sim_{\Q}M_W,\\
M_W&=p^*M+\sum_{E_i\text{ exceptional}}
a(E_i;X,B_X)E_i+T_W\geq0.
\end{split}
\label{bl:eq:abundance-raised-boundary}
\end{equation}
The pair $(W,C_W)$ is lc, and
$\Supp M_W\subset\Supp\lfloor C_W\rfloor$.
Raising the coefficients here preserves Kodaira dimension zero.
To see this directly, a rational function in any divisible system
$|mM_W|$ has, on $X$, poles only along $\Supp M$. There is a
constant $c>0$, independent of $m$, such that the permitted pole
order at every such prime is at most $cm$ times its coefficient
in $M$: this is a finite list of positive coefficients. Increasing
$c$ and clearing denominators identifies this rational function
with a section of some $|nM|$. Since $M$ is effective and
$\kappa(M)=0$, every such section is constant in the rational-function
description of $H^0(X,\OO_X(nM))$. Thus
$h^0(W,mM_W)=1$ for every sufficiently divisible $m$, and
$\kappa(W,K_W+C_W)=0$.
By the first assumption, take an ordinary log minimal model
$(V_*,C_*)$ of $(W,C_W)$. On a common resolution
$\alpha:U\to W$, $\beta:U\to V_*$ set
\[
M_*:=\beta_*\alpha^*M_W.
\]
Lemma~\ref{bl:lem:abundance-model-comparison} shows that
$M_*\geq0$, $M_*\sim_{\Q}K_{V_*}+C_*$, and that the latter
adjoint has Kodaira dimension zero. The support of $M_*$ lies
in $\lfloor C_*\rfloor$: surviving components of $M_W$ already
have coefficient one, and any component extracted over $W$ has
coefficient one by the model convention.
Crucially, $M_*\ne0$. Otherwise $\alpha^*M_W$ would be
$\beta$-exceptional. The nonzero effective divisor
\[
Q=(p\alpha)^*M\leq\alpha^*M_W
\]
would then be exceptional over $V_*$ as well. But
$Q\sim_{\Q}(p\alpha)^*D$ is nef. The negativity lemma applied
to this effective exceptional, relatively nef divisor forces
$Q=0$, a contradiction. This is the place where nefness of the
original adjoint ensures that the effective representative
survives on the new model.
Choose $q>0$ sufficiently divisible that $G=qM_*$ is Cartier and
$G\sim q(K_{V_*}+C_*)$. Lemma~\ref{bl:lem:abundance-floor} makes
$\OO_{V_*}(G)|_{G_{\mathrm{red}}}$ semiample. All the supported
pair hypotheses now hold, with $G\ne0$. Proposition~\ref{bl:prop:complex-supported}
gives $\kappa(V_*,K_{V_*}+C_*)>0$, a contradiction. Therefore
$M=0$ and $D\sim_{\Q}0$.
\smallskip\noindent
\emph{A non-nef adjoint on an Iitaka fiber.}
Put $k=\kappa(X,D)>0$. Resolve an Iitaka map of a sufficiently
divisible multiple of $D$ and take its Stein factorization,
using a simultaneous projective log resolution $\pi:W\to X$:
\[
f:W\longrightarrow Z,\qquad \dim Z=k.
\]
With $\Gamma$ the strict transform of $B_X$ plus the reduced
exceptional divisor, there is an actual equality
\begin{equation}
L:=K_W+\Gamma=\pi^*D+N,
\qquad N\geq0\text{ exceptional over }X.
\label{bl:eq:abundance-nonnef-adjoint}
\end{equation}
In particular $\kappa(W,L)=k$. The resolved linear system also
gives, for a positive integer $u$ and an ample rational divisor
$A_Z$,
\begin{equation}
uL\sim_{\Q}f^*A_Z+E_0,\qquad E_0\geq0.
\label{bl:eq:abundance-iitaka-bound}
\end{equation}
The ample divisor is obtained by pulling back the hyperplane
bundle through the finite map in the Stein factorization.
Let $F$ be a very general smooth fiber. Generic smoothness for
the finitely many horizontal snc strata and avoidance of the
vertical strata give the log smooth lc pair $(F,\Gamma_F)$, with
\[
L|_F=K_F+\Gamma_F.
\]
It has an effective rational representative: restrict any fixed
nonzero divisible section of $L$, choosing $F$ outside the locus
where the section vanishes identically. Moreover
$\kappa(F,L|_F)=0$. Here is a justification that does not require
$L|_F$ to be nef. Work with a Cartier multiple $\mathcal L$ of
$L$ and clear \eqref{bl:eq:abundance-iitaka-bound} to write
$v\mathcal L\sim f^*A+E_1$, with $A$ ample Cartier and $E_1\geq0$.
Choose $F$ outside the countable union of the proper closed
sets needed for base change of $f_*\OO_W(t\mathcal L)$, for
all $t>0$. If some fiber system has positive-dimensional image,
then, after twisting that direct image by a sufficiently positive
multiple $aA$, its global sections generate the fiber system at
the general point of $Z$. Multiplying by sections of a further
very ample multiple $bA$ gives a system in
\[
|t\mathcal L+(a+b)f^*A|
\]
which detects both the base and a nonconstant fiber ratio. Its
image has dimension at least $k+1$. Multiplication by the section
of $(a+b)E_1$ embeds this system in
$|(t+(a+b)v)\mathcal L|$, contradicting $\kappa(W,L)=k$.
Assume $s:=\dim F>0$. The pair $(F,\Gamma_F)$ has dimension
$s=d-k<d$ and an effective adjoint, so it has an ordinary log
minimal model $(F_*,\Gamma_*)$ by the first assumption.
Its nef adjoint has Kodaira dimension zero by
Lemma~\ref{bl:lem:abundance-model-comparison}. The second assumption
therefore makes it semiample, and hence $\Q$-linearly trivial.
Notice that this applies lower-dimensional abundance on $F_*$;
the divisor $K_F+\Gamma_F$ on the original fiber may be non-nef.
On a smooth common resolution $\tau:\widetilde F\to F$ and
$\mu:\widetilde F\to F_*$, the comparison gives
\begin{equation}
\begin{gathered}
P+N_F\sim_{\Q}E_F,\qquad P:=\tau^*(\pi^*D|_F)\text{ nef},\\
N_F:=\tau^*(N|_F)\geq0,\qquad
E_F\geq0\text{ exceptional over }F_*.
\end{gathered}
\label{bl:eq:abundance-fiber-comparison}
\end{equation}
If $H$ is the pullback of an ample divisor on $F_*$, the
projection formula and effectivity yield the full inequality
\begin{equation}
0\leq P\cdot H^{s-1}
=-N_F\cdot H^{s-1}\leq0.
\label{bl:eq:abundance-fiber-sandwich}
\end{equation}
Since $H$ is nef and big, write $H\sim_{\Q}\epsilon A+T$ with
$A$ ample, $\epsilon>0$, and $T\geq0$. Intersections of $T$
with products of nef classes are nonnegative. Replacing one
factor $H$ at a time in the zero intersection in
\eqref{bl:eq:abundance-fiber-sandwich} thus gives
$P\cdot A^{s-1}=0$. The Hodge index theorem for a nef class
then gives $P\equiv0$; when $s=1$ this is the degree criterion.
It follows that $\pi^*D|_F\equiv0$. For zero-dimensional $F$
the same conclusion is automatic.
\smallskip\noindent
\emph{Numerical dimension and lc centers.}
For a nef divisor, $\kappa(D)\leq\nu(D)$. We prove the reverse
inequality here. If $\nu(D)>k$, a general sufficiently ample
complete intersection $T\subset W$ of dimension $k+1$ has
\[
(\pi^*D|_T)^{k+1}>0.
\]
The restriction is nef and big. The map $T\to Z$ is dominant
and its very general fiber is an integral curve, by Bertini
on the generic fiber of $W\to Z$. The degree of $\pi^*D|_T$
on this curve is zero by the preceding argument. On the other
hand, a big decomposition
$\pi^*D|_T\sim_{\Q}\epsilon A_T+E_T$, with $A_T$ ample and
$E_T\geq0$, gives strictly positive degree on a very general
fiber curve: the curves cover $T$, so the general one is not
contained in $\Supp E_T$. This contradiction proves
$\nu(X,D)=\kappa(X,D)$.
Take a projective crepant $\Q$-factorial dlt modification
$\xi:(X',B')\to(X,B_X)$. Its adjoint $D'=\xi^*D$ is nef
and abundant. Lemma~\ref{bl:lem:abundance-floor} makes its
restriction to $\lfloor B'\rfloor$ semiample. Every lc center
of the dlt pair is contained in this floor; restriction to the
center and then pullback to its normalization remain semiample.
Thus $D'$ is log abundant in the precise sense of
\cite[Definition~4.1]{FujinoGongyo2014}. The nef and log abundant
semiampleness theorem
\cite[Theorem~4.2, equivalently Theorem~1.6]{FujinoGongyo2014}
applies to $(X',B')$. Finally, semiampleness descends along
$\xi$, since $\xi_*\OO_{X'}=\OO_X$ and a divisible Cartier
multiple of $D'$ is the pullback of that of $D$. This proves
the proposition.
\end{proof}
\subsection{The two applications}
\begin{proof}[Proof of Theorem~\ref{thm:abundance-after-nonvanishing} over $\C$]
Projective lc abundance through dimension three is the classical
threefold theorem \cite[Theorem~1.1]{KMM1994}, with its correction
\cite{KMM2004}. A direct surface reference is
\cite[Theorem~7.2]{Fujino2012}; curves are elementary. These
statements concern effective rational boundaries and the actual
log canonical divisor.
Ordinary log minimal models exist for effective lc fourfolds by
\cite[Corollary~1.6]{Birkar2010}. Equivalently, one may use
\cite[Corollary~1.7]{Birkar2011} together with the known
\emph{relative} three-dimensional minimal-model theorem for
pseudo-effective $\Q$-factorial dlt pairs. The relative theorem,
not merely absolute threefold abundance, is the premise of that
corollary. Its effectivity hypothesis is satisfied by the
$\Q$-linear effectivity used here. The lower-dimensional
minimal-model statements are known as well.
Proposition~\ref{bl:prop:abundance-induction} with $d=4$ now proves
semiampleness under $\kappa\geq0$. In Kodaira dimension zero
its proof gives $D\sim_{\Q}0$. The lower-dimensional cases
follow from the cited abundance theorems. The complex assertion is now complete. Section~\ref{bl:sec:fields}
proves its field extension as a separate corollary.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{bl:thm:conditional} over $\C$]
Assume that every smooth projective complex variety $W$ with
pseudo-effective $K_W$ has a nonzero section of $mK_W$ for some
$m>0$. In particular $K_W$ is not required to be nef.
Hashizume's theorem \cite[Theorem~1.4]{Hashizume2018} yields
lc nonvanishing and existence of ordinary log minimal models
in all dimensions. The smooth premise is exactly Conjecture~1.3
of that paper; the output is its Conjectures~1.1 and~1.2.
For rational data, its real-linear nonvanishing gives rational-linear
nonvanishing. Apply the independent finite rational-feasibility lemma
of \cite[Lemma~8.1]{OpenAINonvanishing2026}
to the finite coefficients of $D$ and of the principal divisors in an
expression $D\sim_{\R}M_{\R}\geq0$. That lemma uses only rational
linear algebra, and no nonvanishing or abundance theorem. It produces
$M_{\Q}\geq0$ with $D\sim_{\Q}M_{\Q}$ on the same normal variety;
denominators may be cleared together with any specified Cartier index.
Induct on the dimension, starting with dimension zero. For a
nef lc adjoint in dimension $d$, Hashizume supplies nonvanishing,
because nefness implies pseudo-effectivity, and supplies all
the effective ordinary minimal models required through dimension
$d$. The induction hypothesis supplies full nef lc abundance
in every smaller dimension. Proposition~\ref{bl:prop:abundance-induction}
therefore applies. This proves the conditional abundance
statement over $\C$, without adding semiampleness to Hashizume's
minimal-model conclusion. The ground-field transfer follows next.
\end{proof}
==== END Lifting-sections-from-the-reduced-support-of-an-adjoint//abundance.tex ====
==== BEGIN Lifting-sections-from-the-reduced-support-of-an-adjoint//bibliography.tex ====
\begin{thebibliography}{99}
\raggedright
\bibitem{AndreiniJiangTseng2015}
Elena Andreini, Yunfeng Jiang and Hsian-Hua Tseng.
\newblock \emph{{Gromov--Witten} theory of root gerbes {I}: structure of genus \(0\) moduli spaces}.
\newblock Journal of Differential Geometry \textbf{99} (2015), no.~1, 1--45.
\newblock \href{https://doi.org/10.4310/jdg/1418345536}{doi:10.4310/jdg/1418345536}.
\newblock \href{https://arxiv.org/abs/0907.2087}{Referenced arXiv version}.
\bibitem{BierstoneMilman2008}
E.~Bierstone and P.~D.~Milman,
\emph{Functoriality in resolution of singularities},
Publ. Res. Inst. Math. Sci. \textbf{44} (2008), no.~2, 609--639.
\href{https://doi.org/10.2977/PRIMS/1210167338}{doi:10.2977/PRIMS/1210167338}.
\bibitem{Birkar2010}
C.~Birkar,
\emph{On existence of log minimal models},
Compos. Math. \textbf{146} (2010), no.~4, 919--928.
\href{https://doi.org/10.1112/S0010437X09004564}{doi:10.1112/S0010437X09004564}.
\bibitem{Birkar2011}
C.~Birkar,
\emph{On existence of log minimal models II},
J. Reine Angew. Math. \textbf{658} (2011), 99--113.
\href{https://doi.org/10.1515/crelle.2011.062}{doi:10.1515/crelle.2011.062}.
\bibitem{Birkar2012}
Caucher Birkar.
\newblock \emph{Existence of log canonical flips and a special {LMMP}}.
\newblock Publications Math{\'e}matiques de l'IH{\'E}S \textbf{115} (2012), 325--368.
\newblock \href{https://doi.org/10.1007/s10240-012-0039-5}{doi:10.1007/s10240-012-0039-5}.
\newblock \href{https://arxiv.org/abs/1104.4981v3}{Referenced arXiv version}.
\bibitem{ChanChoi2021}
Tsz On Mario Chan and Young-Jun Choi.
\newblock Extension with log-canonical measures and an improvement to
the plt extension of Demailly--Hacon--P\u{a}un.
\newblock \href{https://arxiv.org/abs/1912.08076v5}{arXiv:1912.08076v5}, 17 March 2021.
\newblock \href{https://doi.org/10.1007/s00208-021-02152-3}{doi:10.1007/s00208-021-02152-3}.
Theorem locators refer to the specified preprint version.
\bibitem{Coste2007}
Michel Coste.
\newblock \emph{Real algebraic sets}.
\newblock In \emph{Arc Spaces and Additive Invariants in Real Algebraic and Analytic Geometry}, Panoramas et Synth{\`e}ses, vol.~24, Soci{\'e}t{\'e} Math{\'e}matique de France, 2007, pp.~1--32.
\newblock \href{https://perso.univ-rennes1.fr/michel.coste/polyens/RASroot.pdf}{Public text}.
\newblock Theorem numbering follows the \href{https://perso.univ-rennes1.fr/michel.coste/polyens/RASroot.pdf}{author's course notes}.
\bibitem{DHP2013}
J.-P.~Demailly, C.~D.~Hacon, and M.~P\u{a}un,
\emph{Extension theorems, non-vanishing and the existence of good minimal models},
Acta Math. \textbf{210} (2013), no.~2, 203--259.
\href{https://doi.org/10.1007/s11511-013-0094-x}{doi:10.1007/s11511-013-0094-x}.
\bibitem{EsnaultViehweg1992}
H.~Esnault and E.~Viehweg,
\emph{Lectures on vanishing theorems}, DMV Seminar, vol.~20,
Birkh\"auser, Basel, 1992.
\href{https://page.mi.fu-berlin.de/esnault/books/esvibuch.pdf}{Author copy}.
\bibitem{Fujino2000}
Osamu Fujino.
\newblock Abundance theorem for semi log canonical threefolds.
\newblock {\em Duke Mathematical Journal}, 102(3):513--532, 2000.
\bibitem{Fujino2012}
O.~Fujino,
\emph{Minimal model theory for log surfaces},
Publ. Res. Inst. Math. Sci. \textbf{48} (2012), no.~2, 339--371.
\href{https://ems.press/journals/prims/articles/11294}{Publisher's article}.
\bibitem{FujinoGongyo2014}
Osamu Fujino and Yoshinori Gongyo.
\newblock Log pluricanonical representations and the abundance conjecture.
\newblock {\em Compositio Mathematica}, 150(4):593--620, 2014.
\newblock \href{https://doi.org/10.1112/S0010437X13007495}{doi:10.1112/S0010437X13007495}.
\newblock Theorem locators refer to the final published version.
\bibitem{FujinoVanishing}
O.~Fujino,
\emph{Canonical bundle formula and vanishing theorem},
preprint (2009), version of 19 December 2009.
\href{https://www.math.kyoto-u.ac.jp/~fujino/cano-vani-bessatsu2.pdf}{Author manuscript}.
\bibitem{Griffiths1970}
Phillip A. Griffiths.
\newblock \emph{Periods of integrals on algebraic manifolds, {III} (Some global differential-geometric properties of the period mapping)}.
\newblock Publications Math\'ematiques de l'IH\'ES \textbf{38} (1970), 125--180.
\newblock \href{https://doi.org/10.1007/BF02684654}{doi:10.1007/BF02684654}.
\bibitem{Hashizume2018}
Kenta Hashizume.
\newblock On the non-vanishing conjecture and existence of log minimal models.
\newblock {\em Publications of the Research Institute for Mathematical Sciences}, 54(1):89--104, 2018.
\newblock Theorem 1.4, p. 90.
\bibitem{Kawamata1992}
Yujiro Kawamata.
\newblock Abundance theorem for minimal threefolds.
\newblock {\em Inventiones Mathematicae}, 108:229--246, 1992.
\newblock \href{https://doi.org/10.1007/BF02100604}{doi:10.1007/BF02100604}.
\bibitem{KMM1994}
Sean Keel, Kenji Matsuki, and James McKernan.
\newblock Log abundance theorem for threefolds.
\newblock {\em Duke Mathematical Journal}, 75(1):99--119, 1994.
\bibitem{KMM2004}
Sean Keel, Kenji Matsuki, and James McKernan.
\newblock Corrections to: ``log abundance theorem for threefolds''.
\newblock {\em Duke Mathematical Journal}, 122(3):625--630, 2004.
\bibitem{Kollar2013}
J{\'a}nos Koll{\'a}r.
\newblock {\em Singularities of the Minimal Model Program}, volume 200 of {\em Cambridge Tracts in Mathematics}.
\newblock Cambridge University Press, Cambridge, 2013.
\newblock With the collaboration of S{\'a}ndor Kov{\'a}cs.
\bibitem{Kollar2023}
J{\'a}nos Koll{\'a}r.
\newblock \emph{Families of Varieties of General Type}.
\newblock Cambridge Tracts in Mathematics, vol.~231, Cambridge University Press, 2023.
\newblock \href{https://doi.org/10.1017/9781009346115}{doi:10.1017/9781009346115}.
\newblock With the collaboration of Klaus Altmann and S{\'a}ndor J. Kov{\'a}cs. Section numbering follows the \href{https://web.math.princeton.edu/~kollar/FromMyHomePage/modbook-final.pdf}{author's manuscript dated 8 August 2022}.
\bibitem{KollarKovacs2010}
J{\'a}nos Koll{\'a}r and S{\'a}ndor J. Kov{\'a}cs.
\newblock \emph{Log canonical singularities are {Du Bois}}.
\newblock Journal of the American Mathematical Society \textbf{23} (2010), no.~3, 791--813.
\newblock \href{https://doi.org/10.1090/S0894-0347-10-00663-6}{doi:10.1090/S0894-0347-10-00663-6}.
\bibitem{KollarLocal2011}
J{\'a}nos Koll{\'a}r.
\newblock \emph{A local version of the {Kawamata--Viehweg} vanishing theorem}.
\newblock Pure and Applied Mathematics Quarterly \textbf{7} (2011), no.~4, 1477--1494.
\newblock \href{https://doi.org/10.4310/PAMQ.2011.v7.n4.a18}{doi:10.4310/PAMQ.2011.v7.n4.a18}.
\newblock \href{https://arxiv.org/abs/1005.4843}{Referenced arXiv version}.
\bibitem{KollarMori1998}
J{\'a}nos Koll{\'a}r and Shigefumi Mori.
\newblock \emph{Birational Geometry of Algebraic Varieties}.
\newblock Cambridge Tracts in Mathematics, vol.~134, Cambridge University Press, 1998.
\bibitem{LazicTsakanikas2022}
Vladimir Lazi{\'c} and Nikolaos Tsakanikas.
\newblock \emph{On the existence of minimal models for log canonical pairs}.
\newblock Publications of the Research Institute for Mathematical Sciences \textbf{58} (2022), no.~2, 311--339.
\newblock \href{https://doi.org/10.4171/PRIMS/58-2-3}{doi:10.4171/PRIMS/58-2-3}.
\newblock \href{https://arxiv.org/abs/1905.05576v3}{Referenced arXiv version}.
\bibitem{LiuXu2025}
J.~Liu and Z.~Xu,
\emph{Non-vanishing implies numerical dimension one abundance},
\href{https://arxiv.org/abs/2505.05250v2}{arXiv:2505.05250v2} (2025).
For the conormal--period method, the locators explicitly marked v1 refer to
\href{https://arxiv.org/abs/2505.05250v1}{arXiv:2505.05250v1}.
\bibitem{MustataPopa2019}
M.~Musta\c{t}\u{a} and M.~Popa,
\emph{Hodge ideals},
Mem. Amer. Math. Soc. \textbf{262} (2019), no.~1268.
\href{https://people.math.harvard.edu/~mpopa/papers/SI.pdf}{Author manuscript}.
\bibitem{OpenAINonvanishing2026}
OpenAI.
\newblock \emph{Fourfold nonvanishing by minimal metrics and moving jets}.
\newblock OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets-September-27-2026/paper.pdf}{OAI:Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets-September-27-2026},
2026.
\bibitem{Popa2016}
M.~Popa,
\emph{Kodaira--Saito vanishing and applications},
Enseign. Math. \textbf{62} (2016), no.~1--2, 49--89.
\href{https://doi.org/10.4171/LEM/62-1/2-5}{doi:10.4171/LEM/62-1/2-5}.
\bibitem{RamisRuget1970}
Jean-Pierre Ramis and Gabriel Ruget.
\newblock \emph{Complexe dualisant et th{\'e}or{\`e}mes de dualit{\'e} en g{\'e}om{\'e}trie analytique complexe}.
\newblock Publications Math{\'e}matiques de l'IH{\'E}S \textbf{38} (1970), 77--91.
\newblock \href{https://doi.org/10.1007/BF02684652}{doi:10.1007/BF02684652}.
\bibitem{Saito1988}
M.~Saito,
\emph{Modules de Hodge polarisables},
Publ. Res. Inst. Math. Sci. \textbf{24} (1988), no.~6, 849--995.
\href{https://doi.org/10.2977/PRIMS/1195173930}{doi:10.2977/PRIMS/1195173930}.
\bibitem{Saito1990}
M.~Saito,
\emph{Mixed Hodge modules},
Publ. Res. Inst. Math. Sci. \textbf{26} (1990), no.~2, 221--333.
\href{https://doi.org/10.2977/PRIMS/1195171082}{doi:10.2977/PRIMS/1195171082}.
\bibitem{Saito2013}
M.~Saito,
\emph{On the definition of mixed Hodge modules},
\href{https://arxiv.org/abs/1307.2140v5}{arXiv:1307.2140v5} (2013).
\bibitem{Saito1989}
Morihiko Saito.
\newblock \emph{Induced {$\mathcal D$}-modules and differential complexes}.
\newblock Bulletin de la Soci{\'e}t{\'e} Math{\'e}matique de France \textbf{117} (1989), no.~3, 361--387.
\newblock \href{https://doi.org/10.24033/bsmf.2128}{doi:10.24033/bsmf.2128}.
\bibitem{Saito2000}
Morihiko Saito.
\newblock \emph{Mixed {Hodge} Complexes on Algebraic Varieties}.
\newblock Mathematische Annalen \textbf{316} (2000), no.~2, 283--331.
\newblock \href{https://doi.org/10.1007/s002080050014}{doi:10.1007/s002080050014}.
\newblock Locators refer to the primary preprint \href{https://arxiv.org/abs/math/9906088v1}{arXiv:math/9906088v1}.
\bibitem{Schmid1973}
Wilfried Schmid.
\newblock \emph{Variation of {Hodge} structure: The singularities of the period mapping}.
\newblock Inventiones Mathematicae \textbf{22} (1973), 211--319.
\newblock \href{https://doi.org/10.1007/BF01389674}{doi:10.1007/BF01389674}.
\bibitem{Schnell2014}
C.~Schnell,
\emph{An overview of Morihiko Saito's theory of mixed Hodge modules},
\href{https://arxiv.org/abs/1405.3096v1}{arXiv:1405.3096v1}, 13 May 2014.
\bibitem{StacksProject}
The Stacks Project Authors,
\emph{The Stacks Project},
\href{https://stacks.math.columbia.edu}{stacks.math.columbia.edu},
Tags \href{https://stacks.math.columbia.edu/tag/02KH}{02KH} and
\href{https://stacks.math.columbia.edu/tag/0C3M}{0C3M},
accessed 16 September 2026.
\bibitem{Wlodarczyk2005}
J.~W\l{}odarczyk,
\emph{Simple Hironaka resolution in characteristic zero},
J. Amer. Math. Soc. \textbf{18} (2005), no.~4, 779--822.
\href{https://www.math.purdue.edu/~wlodarcz/singularities/Resolution.pdf}{Author's copy of the published article}.
\end{thebibliography}
==== END Lifting-sections-from-the-reduced-support-of-an-adjoint//bibliography.tex ====
==== BEGIN Lifting-sections-from-the-reduced-support-of-an-adjoint//covers.tex ====
\section{Moving frames, cyclic covers, and residues}
\label{bl:cov:section}
\label{bl:sec:covers}
The first step in Proposition~\ref{bl:prop:complex-supported} is to
embed the entire lifting obstruction into logarithmic residue
cohomology. Moving to a frame bundle supplies compatible roots without
choosing roots separately on open sets. The complete cyclic covers
retain the character summand needed for an injective residue map,
including the poles along the extra coefficient-one boundary.
Work over $\C$, and put $A=K_V+C$ and $S=G_{\mathrm{red}}$.
Replace $(q,G)$ by
$(bq,bG)$, for a sufficiently divisible positive integer $b$, so that
semiampleness on the entire reduced scheme $S$ gives fixed identifications
\begin{equation}\label{bl:cov:initial-data}
L=\OO_V(G)\simeq\OO_V(qA),\qquad
L|_S\simeq f^*J,\qquad
f:S\longrightarrow T=\PP^\ell,\quad J=\OO_T(1).
\end{equation}
Indeed, a globally generated power of the original restriction defines
$f$ by its complete linear system; surjectivity onto $T$ is unnecessary.
Further powers preserve this property. We also clear all Cartier indices
and boundary denominators that occur below. Let $s\in H^0(V,L)$ be the
section with divisor $G$, and write
\[
G=\sum_i d_iS_i,\qquad S=\sum_iS_i,\qquad C=S+H+C_0.
\]
Here $d_i>0$, $H$ is the reduced sum of the remaining coefficient-one
components, and $C_0$ has coefficients in $[0,1)$ and shares no component
with $S+H$. Choose $r$ divisible by $q$ and all $d_i$, and put $a=r/q$.
The singularity properties used in this section, and in the remainder of
the supported-pair argument, are log canonicity and klt singularities away
from $S+H$. A dlt pair has both properties. Apart from normality, the
Cartier hypotheses used are that $K_V$ and the indicated boundary
components are $\Q$-Cartier, together with the actual identifications
in~\eqref{bl:cov:initial-data}.
\subsection{The moving frame and the first cover}
Let $p:P=L^\times\to V$ and $B=J^\times\to T$ denote the bundles of
nonzero vectors. The tautological frame $w$ trivializes $p^*L$ on $P$.
For $\lambda\in\Gm$, let $\lambda$ multiply the vectors in both bundles
by $\lambda^r$. All weights below are pullback weights: a tensor $u$
has weight $m$ if $\lambda^*u=\lambda^m u$. In particular $w$ has
weight $r$. The fixed restriction isomorphism gives
\[
S_P:=p^{-1}(S)=S\times_TB\longrightarrow B,
\]
an equivariant projective morphism.
There is an invariant canonical comparison
$\omega_P\simeq p^*\omega_V$, understood reflexively on the normal
spaces. On a bundle chart with nonzero fiber coordinate $t$, it sends a
top form $\eta$ on $V$ to $p^*\eta\wedge dt/t$. Replacing $t$ by
$ut$, with $u$ invertible on the base, adds $du/u$ to $dt/t$; its wedge
with $\eta$ vanishes. Thus the comparison glues and is invariant under
scaling. The same construction applies to divisible adjoint powers.
Moreover, the pullback pair $(P,p^*C)$ is lc and is klt off $S_P+p^*H$:
locally a log resolution is the product of a log resolution on $V$ with
$\Gm$, with the same discrepancy coefficients.
The ratio $u=p^*s/w$ is a regular function on $P$. Form the whole finite
algebra $\OO_P[y]/(y^r-u)$ and let
\begin{equation}\label{bl:cov:first-cover}
\pi:Z\longrightarrow P
\end{equation}
be its normalization. This is the cyclic-cover construction of
\cite[Section~3.5]{EsnaultViehweg1992}, applied to the trivial line
bundle; we retain its full-algebra convention. Throughout, normalization of a generically
reduced algebra means normalization in its full total ring of fractions.
Thus $Z$ is a disjoint union of normal varieties if the generic algebra
splits; no component is discarded. The polynomial is separable over
every generic field, and normalization is finite since the schemes are
of finite type over $\C$. The function $y$ remains regular, and the
action lifts by
\[
\lambda^*y=\lambda^{-1}y.
\]
It commutes with the deck action $y\mapsto\zeta y$, $\zeta\in\mu_r$.
\begin{lemma}\label{bl:cov:reduced-cover}
The divisor $E=(y=0)$ on $Z$ is reduced and Cartier, with ideal
$I=(y)=\OO_Z(-E)$. Put
$H_Z=\pi^*p^*H$ and $C_{0,Z}=\pi^*p^*C_0$. With compatible canonical
divisors one has
\begin{equation}\label{bl:cov:crepant}
K_Z+E+H_Z+C_{0,Z}=\pi^*(K_P+p^*C).
\end{equation}
The pair $(Z,E+H_Z+C_{0,Z})$ is lc and is klt off $E+H_Z$.
The map $E\to S_P$ induces an equivariant projective morphism
$g:E\to B$.
\end{lemma}
\begin{proof}
At the generic point of $p^{-1}(S_i)$, write $u=vx^{d_i}$ in the
corresponding discrete valuation ring, with $v$ a unit. After an
unramified extension extracting a root of $v$, the normalized Kummer
algebra has ramification index $e_i=r/d_i$ on each branch, since $d_i$
divides $r$. Consequently
\[
\ord_F(y)=e_id_i/r=1
\]
at every prime $F$ lying over this divisor. These are all the zero
divisors of $y$. On every normal component of $Z$, the nonzero function
$y$ is a nonzerodivisor and its principal ideal is radical. For the
latter assertion, if $v^N\in(y)$, the height-one valuation inequalities
give $\ord_F(v)\geq\ord_F(y)$ at every prime; normality then implies
$v/y\in\OO_Z$. This also proves that $E$ is reduced as a scheme.
Off $S_P$ the first root algebra is finite \'{e}tale. At a prime above
$S_P$ the canonical ramification coefficient is $e_i-1$, whereas the
pullback coefficient of $S_P$ is $e_i$. This proves
\eqref{bl:cov:crepant} in codimension one, and hence as an identity of
$\Q$-Cartier divisors. In particular $H_Z$ is reduced and the
coefficients of $C_{0,Z}$ are those of $C_0$.
We record why this calculation controls all divisorial valuations,
including exceptional ones. For a finite dominant map of normal
varieties $X'\to X$ in characteristic zero and a comparison
$K_{X'}+\Delta'=\phi^*(K_X+\Delta)$, let $v'$ be a normalized divisorial
valuation upstairs. Its restriction is $e v$ for a normalized
divisorial valuation $v$ downstairs: the residue extension is finite,
so its transcendence degree is still $\dim X-1$. Choose a model of $X$
on which $v$ is a divisor and normalize it in the upstairs function
field. At the resulting discrete valuation rings the different has
order $e-1$ (the extension is tame). Comparing the canonical
coefficients on these models yields
\begin{equation}\label{bl:cov:finite-discrepancy}
A_{X',\Delta'}(v')=e A_{X,\Delta}(v),
\end{equation}
where $A$ denotes log discrepancy. The same argument applies to each
field factor of a disconnected cover. Applied to
\eqref{bl:cov:crepant}, it gives lc singularities everywhere and klt
singularities over $P\setminus(S_P+p^*H)$. Finally, the reduced scheme
$E$ maps into $S_P$ because its image is supported there. The resulting
map is finite, so its composite with $S_P\to B$ is projective.
\end{proof}
\subsection{The root of the adjoint tensor}
The nowhere-vanishing frame $w$, the fixed isomorphism in
\eqref{bl:cov:initial-data}, and the canonical comparison produce a
meromorphic $q$-canonical tensor on $P$. As an adjoint section it has
no zeros, so its divisor as a meromorphic tensor is $-qp^*C$.
Its differential pullback to $Z$ is a tensor $\tau$, nonzero on every
component, satisfying
\begin{equation}\label{bl:cov:tau}
\Div(\tau)=-q\Delta_Z,\qquad
\Delta_Z=E+H_Z+C_{0,Z},\qquad \wt(\tau)=r.
\end{equation}
The canonical ramification formula used in
\eqref{bl:cov:crepant} proves this equality, including its integral
coefficients. Thus the construction depends on the given rational
linear equivalence, not merely on a numerical equivalence.
Let $\nu:\widetilde Z\to Z$ be the full normalized $q$-th root of
$\tau$. Explicitly, for a meromorphic canonical frame $\theta$ on a
component with function field $K$, normalize in the finite reduced
$K$-algebra
\begin{equation}\label{bl:cov:second-algebra}
K[c]/\bigl(c^q-\tau/\theta^{\otimes q}\bigr).
\end{equation}
Changing $\theta$ to $v\theta$ changes $c$ to $c/v$, so the algebras
and the tautological form $c\theta$ glue. Pullbacks of forms in this
description are differential pullbacks. The deck group $\mu_q$ acts
by $c\mapsto\zeta c$, and the tautological form has its character
$\chi(\zeta)=\zeta$. Even if the algebra splits, its invariant
subalgebra is exactly $K$: the basis $1,c,\ldots,c^{q-1}$ has all the
distinct characters. This observation will be needed for descent of
ratios of forms.
The $\Gm$-action also lifts algebraically. If on a meromorphic chart
$\lambda^*\theta=j_\lambda\theta$, using differentials relative to the
parameter $\lambda$, prescribe
\begin{equation}\label{bl:cov:second-action}
\lambda^*c=\lambda^a j_\lambda^{-1}c.
\end{equation}
This respects the equation because $aq=r$ and
$\lambda^*\tau=\lambda^r\tau$; the factors $j_\lambda$ satisfy the
cocycle identity. These formulas define the action on the generic
algebra and on its integral closure. More explicitly, normalization
commutes with the smooth base extension by $\Gm$, and the isomorphism
between the two pullback algebras over $\Gm\times Z$ therefore induces
an isomorphism of their normalizations. This gives a regular action,
not just a separate lift of each scalar automorphism. It commutes
with $\mu_q$, and the tautological form has weight $a$.
Take a projective $(\Gm\times\mu_q)$-equivariant log resolution of
$\widetilde Z$ and the inverse images of the boundary supports, and
write its composite with $\nu$ as
\[
\rho:Y\longrightarrow Z.
\]
Functorial resolution and principalization in characteristic zero
give this resolution by invariant centers and projective blowups
\cite[Theorem~1.1]{BierstoneMilman2008},
\cite[Theorem~1.0.1]{Wlodarczyk2005}.
Indeed, functoriality for the smooth action and projection morphisms
$(\Gm\times\mu_q)\times\widetilde Z\to\widetilde Z$ lifts the action
and its group identities at every step.
We use the pulled-back tautological top form
$\sigma$ on $Y$; it has deck character $\chi$ and weight $a$.
We may choose $Y$ smooth and quasi-projective with a $\Gm$-linearized
ample line bundle. To see the latter point, compactify $P$ as
$\PP_V(\OO_V\oplus L)$, with its scaling action. A sufficiently
positive twist of the tautological bundle by an ample bundle from
$V$ is ample and linearized. Restrict to $P$ and pull back through
the finite covers, which preserves ampleness and linearization.
For every invariant blowup center its ideal, and hence the
tautological relatively ample bundle on the blowup, is linearized.
Tensoring these with sufficiently high powers of the preceding ample
bundles at successive steps gives the assertion for $Y$.
Define effective divisors on $Y$ by
\begin{equation}\label{bl:cov:resolved-boundary}
\widehat E=\rho^*E,\qquad
D_h=(\Supp\rho^*H_Z)_{\mathrm{red}},\qquad
R=\sum_{\substack{F\text{ prime in }\Supp\widehat E\\
F\not\subset D_h}} F.
\end{equation}
The union $R+D_h$ is reduced simple normal crossing. Both $R$ and
$\widehat E$ map scheme-theoretically to $E$: the function $y$ vanishes
on them. Write $q_R:R\to E$ and $q_{\widehat E}:\widehat E\to E$ for
these projective maps, and set $h=g\circ q_R:R\to B$.
The spaces and morphisms are summarized in Figure~\ref{fig:boundary-covers}.
The form $\sigma$ lives on $Y$; its residue will live on $R$ and will
be pushed through the top row to $B$.
\begin{figure}[ht]
\centering
\begin{tikzcd}[column sep=large,row sep=large]
R \arrow[r,"q_R"] \arrow[d,hook] &
E \arrow[r] \arrow[d,hook] &
S_P \arrow[r] \arrow[d,hook] & B\\
Y \arrow[r,"\rho"'] & Z \arrow[r,"\pi"'] & P &
\end{tikzcd}
\caption{The reduced boundary maps to $B$ through the top row, whose
composite is $h$. The lower row contains the first normalized root
cover $\pi$ and the second normalized cover followed by resolution,
combined in $\rho$. Only the boundary is equipped with a map to $B$.}
\label{fig:boundary-covers}
\end{figure}
\begin{lemma}\label{bl:cov:poles}
The tautological form and its residue satisfy
\begin{equation}\label{bl:cov:residue-form}
\sigma\in H^0\bigl(Y,\omega_Y(R+D_h)\bigr),\qquad
\Omega=\res_R(\sigma)\in H^0\bigl(R,\omega_R(D_h)\bigr).
\end{equation}
Here $\omega_R$ is the absolute dualizing sheaf of the reduced
simple-normal-crossing divisor $R$. Both forms have weight $a$ and
deck character $\chi$.
\end{lemma}
\begin{proof}
Let $F$ be any prime divisor on $Y$, and restrict its valuation to a
function-field component of $Z$, obtaining $e v$. On a downstairs
model carrying $v$, \eqref{bl:cov:tau} gives order
$q(A_{Z,\Delta_Z}(v)-1)$ for $\tau$. Differential pullback adds
$q(e-1)$ to $e$ times this order. Since $\sigma^{\otimes q}=\rho^*\tau$,
\begin{equation}\label{bl:cov:sigma-order}
\ord_F(\sigma)+1=e A_{Z,\Delta_Z}(v).
\end{equation}
Log canonicity makes the right side nonnegative. If the center lies
outside $E+H_Z$, it is strictly positive by the klt property.
The order on the left is integral, so $\sigma$ has at most a simple
pole, and every pole is over $E+H_Z$. Because $E$ and $H_Z$ are
effective $\Q$-Cartier divisors, a prime whose center lies in this
union occurs in the support of their pullback. That reduced support
is exactly $R+D_h$. This proves the first assertion. Adjunction for
the effective Cartier divisor $R$ on the smooth variety $Y$ gives
the residue sequence
\begin{equation}\label{bl:cov:residue-sequence}
0\longrightarrow\omega_Y(D_h)
\longrightarrow\omega_Y(R+D_h)
\longrightarrow\omega_R(D_h)\longrightarrow0.
\end{equation}
It gives $\Omega$ and is equivariant, so residue preserves the stated
weight and character.
\end{proof}
The definition of $D_h$ is essential here. An exceptional divisor
over $E\cap H_Z$ belongs to $D_h$ and is omitted from $R$, but its
possible simple pole is still allowed in~\eqref{bl:cov:residue-form}.
No disjointness condition on $H$ and the strata of $S$ has been used.
\subsection{The residue summand}
\begin{lemma}[Residue injection]\label{bl:lem:residue-injection}
Multiplication by $\Omega$ after pullback induces a split morphism
\begin{equation}\label{bl:cov:derived-split}
\OO_E\longrightarrow \mathbf R(q_R)_*\omega_R(D_h)
\end{equation}
in the derived category of $\OO_E$-modules. Consequently, for every
$i\geq0$, it induces an injection
\begin{equation}\label{bl:cov:injections}
R^ig_*\OO_E\lhook\joinrel\longrightarrow
R^ih_*\omega_R(D_h),\qquad e'\longmapsto\Omega e'.
\end{equation}
In particular this holds for $i=1$. The map increases pullback
weight by $a$.
\end{lemma}
\begin{proof}
\emph{The character summand.}
We first identify the $\chi$-eigensheaf:
\begin{equation}\label{bl:cov:eigensheaf}
\bigl(\rho_*\omega_Y(D_h)\bigr)_\chi
=\sigma\cdot\OO_Z(-E).
\end{equation}
A meromorphic $\chi$-form divided by $\sigma$ is deck-invariant, so it
is the pullback of a meromorphic function $v$ on $Z$, by the invariant
algebra calculation following~\eqref{bl:cov:second-algebra}. At a prime
of $Z$ with coefficient $d$ in $\Delta_Z$, and at a prime of the cover
above its generic point, \eqref{bl:cov:sigma-order} becomes
\[
\ord_F(v\sigma)=e\ord(v)+e(1-d)-1.
\]
Over $E$ no pole is permitted in $\omega_Y(D_h)$, so $d=1$ forces
$\ord(v)\geq1$. Over $H_Z$ a simple pole is permitted and forces
$\ord(v)\geq0$. Elsewhere $0\leq d<1$, and the integer
$e(1-d)$ lies between $1$ and $e$; hence holomorphy is equivalent
to $\ord(v)\geq0$. These necessary inequalities, and normality,
give $v\in\OO_Z(-E)$. Conversely such a function pulls back to a
regular function vanishing along $\widehat E$. Since
$\widehat E\geq R$, it cancels every pole of $\sigma$ outside $D_h$
on the whole resolution, including exceptional divisors. This
proves~\eqref{bl:cov:eigensheaf}.
\smallskip\noindent
\emph{Vanishing and the thickened residue.}
We next prove the required vanishing:
\begin{equation}\label{bl:cov:vanishing}
R^j\rho_*\omega_Y(D_h)=0\qquad(j>0).
\end{equation}
Choose a rational $\epsilon>0$ so small that every nonzero coefficient
of $F=\epsilon\rho^*H_Z$ lies strictly between $0$ and $1$.
Then $D_h-F$ is an effective simple-normal-crossing boundary with
all coefficients less than one, and
\[
(K_Y+D_h)-\bigl(K_Y+(D_h-F)\bigr)=F.
\]
The divisor $F$ is $\rho$-nef, being a pullback from $Z$, and
$\rho$-big: $\rho$ is projective and generically finite on every
component, and every divisor restricts to a big divisor on its
zero-dimensional generic fiber. Relative Kawamata--Viehweg vanishing
therefore applies to the Cartier divisor $K_Y+D_h$ and the klt
boundary $D_h-F$; see \cite[Theorem~3.3]{FujinoVanishing}.
If $H=0$, take $F=D_h=0$; the same relative bigness observation
includes this case. This proves~\eqref{bl:cov:vanishing} without
asserting any absolute nefness of an auxiliary adjoint.
Since $E$ is Cartier, the projection formula gives the same vanishing
for $\omega_Y(\widehat E+D_h)$ and, by
\eqref{bl:cov:eigensheaf}, identifies their derived eigensummands as
\begin{equation}\label{bl:cov:two-derived-images}
\begin{split}
\bigl(\mathbf R\rho_*\omega_Y(D_h)\bigr)_\chi
&\simeq\OO_Z(-E),\\
\bigl(\mathbf R\rho_*\omega_Y(\widehat E+D_h)\bigr)_\chi
&\simeq\OO_Z.
\end{split}
\end{equation}
Both are concentrated in degree zero; the isomorphisms are
multiplication by $\sigma$. The inclusion of the two sheaves on $Y$
corresponds to the ordinary ideal inclusion $\OO_Z(-E)\hookrightarrow
\OO_Z$.
For the possibly nonreduced effective Cartier divisor $\widehat E$,
adjunction still gives
\[
\omega_{\widehat E}(D_h)
=\omega_Y(\widehat E+D_h)/\omega_Y(D_h).
\]
Taking the quotient in~\eqref{bl:cov:two-derived-images} shows that
\begin{equation}\label{bl:cov:thick-summand}
\bigl(\mathbf R(q_{\widehat E})_*
\omega_{\widehat E}(D_h)\bigr)_\chi
\simeq\OO_E
\end{equation}
in the derived category on $E$. Here this conclusion can be checked
on cohomology sheaves after the exact faithful pushforward by the
closed immersion $E\hookrightarrow Z$: the higher eigencohomology
vanishes and the degree-zero map is the residue of multiplication by
$\sigma$. There is no need to assert full faithfulness of derived
pushforward for that immersion.
\smallskip\noindent
\emph{Retraction from the reduced residue.}
We construct the splitting map on $E$ itself.
Pullback of functions and multiplication by $\Omega$ give
\[
\alpha:\OO_E\longrightarrow\mathbf R(q_R)_*\omega_R(D_h).
\]
The inequality $R\leq\widehat E$ gives an inclusion of residue
quotients on $Y$, hence a morphism
\[
\beta:\mathbf R(q_R)_*\omega_R(D_h)
\longrightarrow\mathbf R(q_{\widehat E})_*
\omega_{\widehat E}(D_h).
\]
These are $\OO_E$-linear: multiplication by the pulled-back equation
$y$ kills both quotients. If a local function on $E$ is lifted to
$v\in\OO_Z$, the composite $\beta\alpha$ is represented by the
class of $\rho^*v\,\sigma$ modulo $\omega_Y(D_h)$.
Changing $v$ by a multiple of $y$ does not change this class, since
$y\sigma\in\omega_Y(D_h)$ by $\widehat E\geq R$.
Thus projection onto the $\chi$-summand followed by
\eqref{bl:cov:thick-summand} sends $\beta\alpha$ to the identity.
The projection is available in the derived category because
averaging over $\mu_q$ gives the idempotent
$q^{-1}\sum_{\zeta\in\mu_q}\chi(\zeta)^{-1}\zeta^*$.
This supplies a retraction of $\alpha$ and proves
\eqref{bl:cov:derived-split}. Finally apply $\mathbf Rg_*$ and take
cohomology to obtain~\eqref{bl:cov:injections}. The weight statement
follows from $\wt(\Omega)=a$.
\end{proof}
\subsection{The obstruction to be detected}
\label{bl:cov:obstruction-contract}
Put $A_d=|g|_*(\OO_Z/I^d)$ for $d\geq1$, pushing the underlying
sheaves of complex vector spaces in the Zariski topology along
$|g|:|E|\to|B|$. This uses no morphism from a thickening to $B$.
Fix $k\geq1$ and suppose that $A_{d+1}\to A_d$ is surjective
for $1\leq d<k$.
The current connecting map is
$\partial_k:A_k\to R^1g_*(I^k/I^{k+1})$.
Section~\ref{bl:sec:lifting} will show that it factors, after framing
$I^k/I^{k+1}$ by $y^k$, through a derivation
\[
\delta_k:g_*\OO_E\longrightarrow R^1g_*\OO_E,
\qquad \partial_k(\widetilde F)=y^k\delta_k(F).
\]
Here $\widetilde F\in A_k$ is any local lift of $F\in g_*\OO_E$.
The factorization and derivation rule will be proved there from the
previous surjections. Since $y$ has weight $-1$, this derivation
has degree $k$ for the frame action.
Restrict $\delta_k$ to pullbacks of functions on $B$. In local
coordinates $t_i$, the residue injection produces a global section
of $R^1h_*\omega_R(D_h)\otimes T_B$, given by
\[
v_k=\sum_i\Omega\delta_k(g^*t_i)\otimes\partial_{t_i}
,\qquad \wt(v_k)=a+k>0.
\]
The derivation rule makes this tensor independent of coordinates.
The next two sections develop a differential-operator test for it.
The graph identity that makes the test applicable, and that then
annihilates $\delta_k$ on all of $g_*\OO_E$, is proved with the
induction in Section~\ref{bl:sec:lifting}.
==== END Lifting-sections-from-the-reduced-support-of-an-adjoint//covers.tex ====
==== BEGIN Lifting-sections-from-the-reduced-support-of-an-adjoint//curve-base.tex ====
\section{A conormal--period argument in numerical dimension two}
\label{nu2:sec:method}
This section gives an independent use of the boundary: an alternative
proof of the following restricted canonical statement.
\begin{theorem}[The curve-base alternative]\label{nu2:thm:alternative}
Let $X$ be a smooth connected projective complex fourfold. If its
canonical divisor $K_X$ is nef and $\kappa(X,K_X)=0$, then
$\nu(K_X)\ne2$.
\end{theorem}
Here $\nu$ denotes the intersection-theoretic numerical dimension of a
nef divisor. The effective divisor used below comes from the section
supplied by $\kappa=0$. No condition on the irregularity or the Euler
characteristic is imposed. The statement concerns the actual
canonical divisor on a smooth variety; it is not a nonvanishing
assertion for $\kappa=-\infty$ or a statement for arbitrary klt pairs.
The proof has three parts. First, a dimension-preserving effective
MMP puts the pluricanonical support on a reduced boundary whose
adjoint defines a pencil. Second, root covers turn the obstruction
to lifting through a nilpotent thickening into a period equation on
that curve. Positivity of the source line kills its moving part;
a separate trace test kills classes supported at exceptional
parameters. Finally, finite invariant filtrations convert lifting
into a quadratic lower bound for plurisections, contradicting
$\kappa=0$.
The use of cyclic covers, infinitesimal neighborhoods and Hodge duality
continues a method in Kawamata's alternative proof of the numerical-
dimension-one case for minimal threefolds
\cite[Section~4, Theorem~4.1 and Lemmas~4.2--4.3]{Kawamata1992}.
That result supplies the ancestry of this method, rather than the
fourfold assertion of Theorem~\ref{nu2:thm:alternative}.
\subsection{The reduced boundary and a genuine pencil}\label{nu2:subsec:reduction}
Suppose that $\nu(K_X)=2$. A nonzero pluricanonical section gives
$0\ne D_X\sim_{\mathbb Q}K_X$, with $D_X\ge0$. On a log resolution
$p:Y\to X$, let $T$ be its reduced strict-transform support, let $E_Y$
be the reduced exceptional divisor, and write $K_Y=p^*K_X+F$, with
$F\ge0$ exceptional. Then
\[
K_Y+T+E_Y\sim_{\mathbb Q}D_Y:=p^*D_X+F+T+E_Y,
\qquad \operatorname{Supp}D_Y=T+E_Y.
\]
For some rational $c>0$,
$p^*D_X\le D_Y\le(1+c)p^*D_X+F+E_Y$.
Monotonicity, birational invariance and exceptional invariance of
$\kappa$ and Nakayama's $\kappa_\sigma$ therefore give dimensions
$0$ and $2$ for this effective log adjoint
\cite[v1, Lemma~2.5 and proof of Lemma~4.1]{LiuXu2025}.
Effective fourfold minimal-model existence and termination with scaling
give a projective $\mathbb Q$-factorial dlt model
\cite[Corollary~D(i), Lemma~2.9(i)]{LazicTsakanikas2022},
\cite[Theorem~4.1(ii),(iii)]{Birkar2012}:
\begin{equation}\label{nu2:eq:model}
A=K_V+B\sim_{\mathbb Q}D\ge0,\quad
B=B_{\mathrm{red}}=\operatorname{Supp}D,\quad
A\text{ nef},\quad \kappa(V,A)=0,\quad\nu(A)=2.
\end{equation}
Choose the scaling boundary to contain a fixed ample rational divisor.
For a positive limiting scaling parameter, the dlt pair and this
ampleness hypothesis allow the first termination clause; limit zero with every parameter positive uses the
already obtained log minimal model and the second clause; an attained
zero means the current adjoint is nef. The nonextracting MMP preserves
the displayed support and both dimensions. The underlying $V$ is klt.
The sheaves $\mathcal O_V$ and $\mathcal O_V(-B)$ are
Cohen--Macaulay by the local vanishing theorem for integral
$\mathbb Q$-Cartier divisors on a klt variety
\cite[Theorem~2 and Example~4.1]{KollarLocal2011}. Their quotient shows
that $B$ is a Cohen--Macaulay threefold. Adjunction on the whole union,
including conductor gluing on its normalization, gives an slc pair
$(B,\Delta_B)$ with $K_B+\Delta_B=A|_B$.
Threefold slc abundance makes this restriction semiample
\cite[Theorem~0.1]{Fujino2000}.
Write $D=\sum c_iB_i$, $c_i>0$, and choose an ample divisor $H$. Then
\[
0=A^3H=\sum_i c_iA^2B_iH,
\qquad 0<A^2H^2=\sum_i c_iAB_iH^2.
\]
All summands are nonnegative. The image of a generated multiple on
each $B_i$ consequently has dimension at most one, and one component
has one-dimensional image. Two general linear forms with common
center disjoint from this image generate the restricted line bundle
everywhere, including components mapped to points. After fixing one
divisible integer $q$, we obtain
\begin{equation}\label{nu2:eq:pencil}
G=\sum_i d_iB_i\sim qA,\quad d_i\in\mathbb Z_{>0},\qquad
L=\mathcal O_V(G),\qquad
L|_B\simeq f^*\mathcal O_{\mathbb P^1}(1),
\end{equation}
where $f:B\to\mathbb P^1$ is surjective. Its fibers need not be
connected. This is a morphism of the boundary; no ambient fibration
is required.
\subsection{Root covers and the two conormal frames}\label{nu2:subsec:roots}
The pencil exists only on $B$. We now construct local covers on which its conormal line and absolute dualizing line have compatible frames. The construction must retain the entire reduced divisor, including components over individual parameter values.
Fix $r$ divisible by $q$ and every $d_i$, and put $a=r/q>0$.
These integers will not vary with the infinitesimal order.
Near a compact reduced fiber of $f$, choose a disk $U$ and a frame
$\ell$ of $\mathcal O(1)|_U$. There is an analytic neighborhood
$W$ of $B_U$, a root $P^r\simeq L|_W$, and a boundary frame $p$
with $p^r=f^*\ell$. Indeed choose a simultaneous triangulation compatible with the fiber
and boundary \cite[Theorem~1.10]{Coste2007}. In successive barycentric
subdivisions take the open stars of the compact fiber subcomplex.
Their regular-neighborhood retractions preserve the boundary
subcomplex. These stars are cofinal among neighborhoods of the
fiber, giving arbitrarily small $W$ for which $W$ and $W\cap B$
retract compatibly onto it. The retractions are the elementary
regular-neighborhood consequence of the triangulation, not an
additional assertion of the cited triangulation theorem.
The Kummer obstruction in $H^2(W,\mu_r)$ vanishes because $L$ is
trivial on that fiber. Twisting the root by a $\mu_r$-torsor,
using the corresponding $H^1$ isomorphisms, gives the prescribed
boundary frame. Properness then permits shrinking $U$.
Put $Q=\omega_V(B)$ and $M=Q\otimes P^{-a}$, with reflexive powers
understood; $M^{[q]}\simeq\mathcal O_W$. Normalize the full fiber
product of the cyclic algebras
\[
\bigoplus_{i=0}^{r-1}P^{-i},\qquad
\bigoplus_{j=0}^{q-1}M^{[-j]}.
\]
The first multiplication uses the section defining $G$, and the second
uses this specified power trivialization. Retaining every component
gives a finite cover $\pi:Z\to W$ with group
$\mu_r\times\mu_q$ and quotient $W$. Its tautological section $y$
has reduced Cartier zero divisor $E$; the ramification over $B_i$ is
$r/d_i$, and the second cover is unramified in codimension one.
The two Hurwitz formulas are
\[
K_Z=\pi^*\left(K_V+\sum_i(1-d_i/r)B_i\right),\qquad
K_Z+E=\pi^*(K_V+B).
\]
The finite-map discrepancy rule gives klt $Z$ and lc $(Z,E)$
\cite[Proposition~5.20]{KollarMori1998}. In this setting it follows by exposing
any divisorial valuation downstairs, normalizing its finite pullback,
and using the DVR Hurwitz coefficient $e-1$:
$A_{\mathrm{up}}(F)=eA_{\mathrm{down}}(E_0)$ for a divisor
$F$ above $E_0$, where $A$ denotes log discrepancy for the
corresponding Hurwitz pairs. For the second displayed formula
these are $(Z,E)$ and $(V,B)$. For the first they are $Z$ with
zero boundary and $V$ with boundary
$\sum_i(1-d_i/r)B_i$; the latter is klt because every coefficient
of the reduced dlt boundary has been lowered. Thus the lc and
klt inequalities hold for all divisorial valuations.
The torsion root gives actual
isomorphisms
\begin{equation}\label{nu2:eq:adjunction}
\omega_Z(E)\simeq(\pi^*P)^a\simeq\mathcal O_Z(aE),
\qquad\omega_E\simeq\mathcal O_E(aE).
\end{equation}
Klt Cohen--Macaulayness makes $Z$ Gorenstein; Cartier adjunction
makes $E$ Gorenstein. The normalized adjunction pair has conductor
coefficient one at nodes and coefficient zero at regular
codimension-one points, so $E$ is slc
\cite[Lemma~11.16 and Theorem~11.17]{Kollar2023}. As a reduced union of lc
centers it is Du Bois \cite[Theorem~1.4]{KollarKovacs2010}.
The reduced families used here are algebraic, also through exceptional
parameters. One explicit construction performs the two normalized
cyclic constructions over the root gerbe
$\sqrt[r]{L/V}=[L^\times/\mathbb G_m]$, where
$\lambda$ acts by $\lambda^r$. An object over a map $h:S\to V$
is a line bundle $P_S$ together with an isomorphism
$P_S^r\simeq h^*L$. Over $B_U$, the trivial line with power
identification given by $f^*\ell$ defines the selected morphism
$\sigma:B_U\to\sqrt[r]{L/V}|_{B_U}$. Pull back along $\sigma$
the reduced divisor of the already normalized cover. Thus the
normalization precedes this restriction. The root-gerbe description is standard
\cite[Definition~2.2 and Proposition~2.3]{AndreiniJiangTseng2015};
on the possibly singular $V$ it follows from the same local line-bundle
trivializations. The normalization and singularity comparisons used
here are given by the following local construction. Locally the substitution $v=w^r$ in the frame parameter
identifies the construction with a fixed normalized cover times that
parameter; normalization commutes with this étale localization.
It does not require normalization to commute with restriction to $B$.
A specified boundary root gives a finite étale section of the root
gerbe: its pullback along any other root is the finite étale torsor
of power-compatible isomorphisms. Pulling the reduced divisor back
by this section preserves its total-space singularity properties.
The resulting $E_U\to U$ comes from a projective algebraic family
with reduced pure three-dimensional Gorenstein Du Bois total space.
Globally take a finite cover $\rho:T'\to\mathbb P^1$ with a positive
line $R$ satisfying $R^r=\rho^*\mathcal O(1)$; the degree-$r$ power
map with $T'=\mathbb P^1$ and $R=\mathcal O(1)$ suffices.
The same construction produces a reduced algebraic family
$g':E'\to T'^\circ$ on a nonempty open $T'^\circ$. Remove branch values, vertical-component
images and the finitely many parameters needed for flatness, slc
surface fibers, coherent base change and topological local systems.
Keep the original families through the removed points. In what
follows, a good disk, annulus or locus lies inside the specified open
on which these flatness, base-change and local-system properties
hold, or its corresponding open in a local chart family.
An isomorphism between specified boundary roots extends uniquely as
a germ near a compact fiber, since its isomorphism sheaf is a finite
$\mu_r$-torsor and the retractions preserve its sections. These germ
isomorphisms respect every actual ideal power and its connecting
map. They give the cocycles comparing all local root choices with
the one family on $T'^\circ$.
Set $I=\mathcal O_Z(-E)$ and $\mathcal A_j=I^j/I^{j+1}$ for
every $j\in\mathbb Z$. The frame $p$ gives the graded conormal
frame $u=y/p$ of $\mathcal A_1$; no ambient lift of $u$ is chosen.
Under \eqref{nu2:eq:adjunction}, $u^{-a}$ is an absolute dualizing
frame $\Omega$ of $\omega_E$. A change of base root frame has
$p'=\lambda p$, where $\lambda$ is a nowhere-vanishing holomorphic
function of the base coordinate. Under this change,
\begin{equation}\label{nu2:eq:frames}
u'=\lambda^{-1}u,\qquad\Omega'=\lambda^a\Omega.
\end{equation}
At branch parameters the factors comparing an extending frame of $R$
with these root frames are full-disk holomorphic units, bounded with
their inverses on a smaller disk.
\subsection{The Hodge test, including classes supported at a point}\label{nu2:subsec:hodge}
The moving period equation alone cannot detect a class supported over one parameter. The following test supplies both the growth estimate needed on the punctured curve and detection at the omitted points.
\begin{lemma}[Curve Hodge and supported trace test]\label{nu2:lem:hodge-test}
Let $p:Y\to T$ be projective, where $T$ is a smooth complex
algebraic curve and $Y$ is a reduced, pure three-dimensional,
Cohen--Macaulay Du Bois variety.
Components of $Y$ may map to points. On an open $T^\circ$ with flat
Du Bois surface fibers, coherent base change and topological local
systems, put $\mathbb H=R^1p_*\mathbb Q$. Then
\begin{equation}\label{nu2:eq:hodge}
R^1p_*\omega_Y=\omega_T\otimes F^0\mathcal H^\vee,
\qquad F^1\mathcal H^\vee=0
\quad\text{on }T^\circ,
\end{equation}
where $\mathcal H=\mathbb H\otimes\mathcal O$. For a flat local
section $\alpha$ of $\mathbb H\otimes\mathbb C$, let
$\alpha_{\mathcal O}$ be its image in $R^1p_*\mathcal O_Y$.
The functional corresponding to $s$ is
$\alpha\mapsto\langle s/dt,\alpha_{\mathcal O}\rangle$ under
relative Serre duality.
The variation is graded-polarizable and admissible, with quasi-unipotent
monodromy. A section $s$ extending over a missing parameter obeys
\begin{equation}\label{nu2:eq:one-pole}
\langle s/dt,\alpha_{\mathcal O}\rangle
=O\bigl(|t|^{-1}(1+|\log|t||)^N\bigr)
\end{equation}
in flat frames on bounded-argument sectors. Moreover, a germ of
$R^1p_*\omega_Y$ supported at a point is zero if its absolute trace
annihilates the image of
$H_c^2(Y_U,\mathbb C)\to H_c^2(Y_U,\mathcal O_Y)$ on one
sufficiently small disk $U$. Neither assertion at the missing point
requires a flat map or a reduced Du Bois central fiber.
\end{lemma}
\begin{proof}
We verify the filtered comparison first, then the growth estimate and
the supported-class assertion. Use
increasing filtrations on right $\mathcal D_T$-modules. The symbol
$\mathbb Q_Y^H$ denotes the unshifted constant object in the derived
category of mixed Hodge modules, with rational realization
$\mathbb Q_Y$; it need not be a single perverse Hodge module.
Write $\mathbb D$ for Verdier duality in that category and set
$N=Rp_*\mathbb Q_Y^H$ and $P=\mathbb DN$. Let $\mathcal M_i$ be
the right module underlying the $i$th Hodge-module cohomology of $P$.
For this calculation we reset the convention used for the earlier
boundary-graph modules: there is no dimension-dependent Tate twist
of the constant object here. The lowest index computed below comes
from this unshifted constant object and its dual, and the variations
retain their ordinary decreasing Hodge filtration. For a point $x\in T$, write $i_x:\{x\}\hookrightarrow T$.
The constant/Du Bois comparison and filtered proper duality
\cite{Saito1990,Saito1989,Schnell2014}, with the Du Bois comparison
\cite[preprint, Corollary~0.3, Section~5.3, (5.3.2), and Section~5.4, (5.4.1)]{Saito2000}, give
\[
\operatorname{gr}_j^F\operatorname{DR}(P)=0\ (j<0),\qquad
\operatorname{gr}_0^F\operatorname{DR}(P)=Rp_*\omega_Y[3].
\]
Only finitely many $\mathcal M_i$ occur locally, and their good
filtrations have a common lower bound. The truncation spectral sequence of coherent cohomology sheaves is
\[
E_2^{u,v}=\mathcal H^u\bigl(\operatorname{gr}_j^F
\operatorname{DR}(\mathcal M_v)\bigr)
\ \Longrightarrow\ \mathcal H^{u+v}
\bigl(\operatorname{gr}_j^F\operatorname{DR}(P)\bigr).
\]
At the lowest possible filtration index, the curve de Rham complex
has only its degree-zero term $F_j\mathcal M_v$. The sole column
$u=0$ therefore survives unchanged. The abutment is zero for $j<0$,
so these lowest pieces vanish. Ascending through the negative grades
repeats this argument; at $j=0$ the abutment in degree $v$ is
$R^{v+3}p_*\omega_Y$. We obtain
\begin{equation}\label{nu2:eq:lowest}
F_{<0}\mathcal M_i=0,\qquad
F_0\mathcal M_i=R^{i+3}p_*\omega_Y.
\end{equation}
No splitting of $P$ is used. On $T^\circ$, duality turns the
variation of $\mathcal M_{-2}$ into $\mathbb H^\vee(1)$; the right
module shift gives \eqref{nu2:eq:hodge}. Degree-one proper
cohomology has only types $(0,0),(1,0),(0,1)$, proving $F^1=0$.
Put $\mathcal M=\mathcal M_{-2}$ and fix a coordinate $t$ at an
exceptional parameter $x$. We use the increasing right-module
$V$-filtration, so $t$ lowers its index and $\partial_t$ raises it.
First, for every $\alpha>0$, strict specializability gives
\[
F_0\operatorname{gr}_\alpha^V\mathcal M
=\partial_t\bigl(F_{-1}\operatorname{gr}_{\alpha-1}^V
\mathcal M\bigr)=0,
\]
where the last equality follows from \eqref{nu2:eq:lowest}.
The exhaustive, locally discrete $V$-filtration then gives
$F_0\mathcal M\subset V_0\mathcal M$: any positive leading
$V$-grade of an $F_0$ section would contradict the displayed vanishing.
The localized image lies in the Deligne lattice whose flat coefficients
have the form $t^{-1+\lambda}(\log t)^jh(t)$, with
$0\le\lambda<1$ and $h$ holomorphic. This proves
\eqref{nu2:eq:one-pole}.
At grade zero, the required strictness has a different formulation;
no surjectivity of the exceptional map is assumed. Write
\[
\Psi=\bigl(\operatorname{gr}_{-1}^V\mathcal M,
F_{\bullet-1}\bigr),\qquad
\Phi=\bigl(\operatorname{gr}_0^V\mathcal M,F_\bullet\bigr).
\]
These underlie the unipotent nearby and vanishing mixed Hodge modules.
Their canonical morphisms are
\[
\mathrm{can}=\partial_t:\Psi\longrightarrow\Phi,
\qquad
\mathrm{var}=t:\Phi\longrightarrow\Psi(-1).
\]
For increasing filtrations our Tate convention is
$F_p(\mathcal H(k))=F_{p-k}\mathcal H$. Thus
$F_p\Psi(-1)=F_p\operatorname{gr}_{-1}^V\mathcal M$;
the Tate twist belongs to $\mathrm{var}$, not to $\mathrm{can}$.
The mixed-Hodge-module construction and strictness apply without a
pure decomposition \cite{Saito1990}; the precise filtered formulas
are \cite[v1, (9.3), Sections~20--21, and (30.3)--(30.4)]{Schnell2014}.
Strictness of $\mathrm{can}$ says, for each integer $p$,
\[
F_p\operatorname{gr}_0^V\mathcal M\cap\operatorname{im}(\partial_t)
=\partial_t\bigl(F_{p-1}\operatorname{gr}_{-1}^V\mathcal M\bigr).
\]
Taking $p=0$ and again using $F_{-1}\mathcal M=0$ proves
\begin{equation}\label{nu2:eq:strict}
F_0\operatorname{gr}_0^V\mathcal M\cap
\operatorname{im}\bigl(\partial_t:
\operatorname{gr}_{-1}^V\mathcal M\to
\operatorname{gr}_0^V\mathcal M\bigr)
=\partial_t F_{-1}\operatorname{gr}_{-1}^V\mathcal M=0.
\end{equation}
We now apply this identity to a point-supported germ
$s\in F_0\mathcal M$. Some power of $t$ kills $s$. Its image in
$\operatorname{gr}_0^V\mathcal M$ is nonzero unless $s=0$, since
$t$ is injective on $V_{<0}\mathcal M$.
If $s=u\partial_t$ were a de Rham boundary, consider a nonzero leading
$V$-grade $\beta>-1$ of $u$. The isomorphism
$\partial_t:\operatorname{gr}_\beta^V\mathcal M\to
\operatorname{gr}_{\beta+1}^V\mathcal M$ would give a nonzero
positive grade of $s$, contradicting $s\in V_0\mathcal M$.
Descending through the finitely many jumps between an index containing
$u$ and $-1$ puts $u$ in $V_{-1}\mathcal M$. The grade-zero class
of $s$ would then belong to the intersection in
\eqref{nu2:eq:strict}, which is impossible for nonzero $s$.
Equivalently, the local inverse-image complex is
\[
i_x^*\operatorname{DR}(\mathcal M)\simeq
\bigl[\operatorname{gr}_{-1}^V\mathcal M
\xrightarrow{\ \partial_t\ }
\operatorname{gr}_0^V\mathcal M\bigr],
\qquad\text{in degrees }-1,0,
\]
with the source and target filtrations of $\Psi$ and $\Phi$
\cite[(30.3)]{Schnell2014}. A supported $F_0$ germ therefore injects
into $H^0(i_x^*\operatorname{DR}\mathcal M)$.
Finally, the perverse-cohomology spectral sequence is
\[
E_2^{p,q}=H^p(i_x^*\operatorname{DR}\mathcal M_q)
\ \Longrightarrow\ H^{p+q}(i_x^*\operatorname{DR}P),
\qquad p\in\{-1,0\}.
\]
All other columns vanish on a curve, so no higher differential enters
or leaves them. For $\mathcal M=\mathcal M_{-2}$ its edge sequence is
\[
0\longrightarrow H^0(i_x^*\operatorname{DR}\mathcal M_{-2})
\longrightarrow H^{-2}(i_x^*\operatorname{DR}P)
\longrightarrow H^{-1}(i_x^*\operatorname{DR}\mathcal M_{-1})
\longrightarrow0.
\]
This supplies the desired injection without splitting $P$.
We specify the comparison with trace, since detection of a supported
class requires the actual pairing. Let $U$ be a sufficiently small
disk. The Du Bois comparison identifies the natural map
$\iota:Rp_*\mathbb C_Y\to Rp_*\mathcal O_Y$ with the roof
\[
\operatorname{DR}(N)\ \xleftarrow{\ \simeq\ }
F_0\operatorname{DR}(N)\longrightarrow
\operatorname{gr}_0^F\operatorname{DR}(N).
\]
Filtered transpose duality identifies its transpose with
\begin{equation}\label{nu2:eq:trace-comparison}
\gamma:Rp_*\omega_Y[3]\simeq
\operatorname{gr}_0^F\operatorname{DR}(P)
\simeq F_0\operatorname{DR}(P)
\longrightarrow\operatorname{DR}(P).
\end{equation}
The middle identification uses the vanishing of the negative graded
pieces, so it is an identification in the filtered derived
comparison, not a choice of a splitting. Under proper duality and
Verdier duality the induced map is
\[
H^1(Y_U,\omega_Y)\xrightarrow{\ \gamma\ }
H^{-2}(U,\operatorname{DR}(P))
\simeq H_c^2(Y_U,\mathbb C)^\vee,
\]
and its pairing is exactly
\begin{equation}\label{nu2:eq:trace-pairing}
\langle\gamma(s),\alpha\rangle_{\mathrm{Verdier}}
=\operatorname{Tr}_{\mathrm{coh}}
(s\smile\iota(\alpha)).
\end{equation}
This is the compatibility of filtered transpose duality with coherent
and topological trace
\cite[Sections~2.6--2.12 and 3.7--3.13]{Saito1989}.
Shrink $U$ within a constructibility neighborhood of its center.
The preceding supported-germ injection, the two-row perverse-stalk
spectral sequence, and \eqref{nu2:eq:trace-comparison} then identify a
nonzero supported $s$ with a nonzero class in the displayed
finite-dimensional constructible cohomology group. Verdier duality
detects it by some $\alpha\in H_c^2(Y_U,\mathbb C)$. Thus annihilating
all the stated coherent images forces $s=0$ by
\eqref{nu2:eq:trace-pairing}. This proof neither asserts that arbitrary
coherent compact-support cohomology is Hausdorff nor that the map
from constant-sheaf cohomology onto it is surjective.
\end{proof}
The annular form of the same pairing will produce the differential
equation. On a good annulus $S$ in a simply connected good disk, compact-support
Leray gives
\[
H_c^2(Y_S,\mathbb C)\twoheadrightarrow
H_c^1(S,R^1p_*\mathbb C),
\]
with kernel $H_c^2(S,R^0p_*\mathbb C)$. Its coherent image
factors through $H_c^2(S,p_*\mathcal O_Y)=0$: the annulus is a
Stein curve, and this coherent compact-support group vanishes
above degree one. Compatibility of the Leray pairings with
trace therefore kills the kernel. The lift of a circle-dual
class tensored with a flat $\alpha$ consequently tests $s$ by
$\oint\langle s/dt,\alpha_{\mathcal O}\rangle\,dt$, up to one
fixed nonzero normalization. The test is independent of its lift.
\subsection{The period equation and all-order lifting}\label{nu2:subsec:lifting}
Let $g:E\to U$ be any one of the local full covers above. Push
$I^j/I^{j+k}$ forward by the underlying continuous map, as sheaves
of complex vector spaces. This convention does not posit a
holomorphic map from a thickening to $U$. We prove simultaneously
for every $j\in\mathbb Z$, every $k\ge1$, every parameter point
and every permitted root choice that
\begin{equation}\label{nu2:eq:lifting}
g_*(I^j/I^{j+k+1})\longrightarrow g_*(I^j/I^{j+k})
\quad\text{is surjective}.
\end{equation}
At each fixed order neighborhoods may shrink.
Assume the earlier orders. The current connecting map factors
uniquely through the leading layer as
$\delta_k^j:g_*\mathcal A_j\to R^1g_*\mathcal A_{j+k}$:
the kernel of the leading-layer map lifts by the preceding induction
step. Shorter truncation images have finite filtrations by coherent
$g_*\mathcal A_i$. Acyclicity on small disks and annuli gives
leading-section lifts on those entire opens. The overlap difference
of products has double-error term of order $2k\ge k+1$; hence the collection
$\delta_k=(\delta_k^j)_{j\in\mathbb Z}$ is one
$\mathbb C$-linear graded derivation.
This remains true in negative degrees because $I$ is invertible.
Define $e,b\in R^1g_*\mathcal O_E$ on the full disk by
\[
\delta_k(t)=e u^k,\qquad\delta_k(u)=b u^{k+1},
\qquad m=a+k>0.
\]
On the good locus the derivation rule, evaluated in each locally
free fiber, gives $\delta_k(h)=h'e u^k$ for holomorphic $h$.
Consequently
\begin{equation}\label{nu2:eq:obstruction}
\delta_k(hu^{-m})=(h'e-mhb)\Omega.
\end{equation}
Every value of this last obstruction annihilates the constant-sheaf
tests, on good annuli and on disks through exceptional points.
Indeed its exact sequence is
\[
0\to\omega_E\to\omega_{(k+1)E}
\to I^{-a-k}/I^{-a}\to0,
\qquad\omega_{(k+1)E}=I^{-a-k}/I^{1-a}.
\]
The first map is dual to reduction of structure sheaves. An
obstruction class maps to zero in the middle term; absolute coherent
trace is functorial for this closed embedding
\cite[\S\S3--5]{RamisRuget1970}. Since
$\mathbb C_E\to\mathcal O_{(k+1)E}\to\mathcal O_E$ lifts the
constant test, the pairing is zero. Only the reduced algebraic
family is subjected to Hodge theory; the nilpotent thickening enters
through this analytic duality.
Define functional-valued periods
$\beta_e(\alpha)=\langle e\Omega/dt,\alpha_{\mathcal O}\rangle$
and $\beta_b(\alpha)=\langle b\Omega/dt,\alpha_{\mathcal O}\rangle$.
The annular trace of \eqref{nu2:eq:obstruction} is
\[
\oint(h'\beta_e-mh\beta_b)\,dt=0.
\]
Functions $h$ may have poles at the omitted center because the
required lifts exist on the annulus. Integration by parts and the
test $h=(t-c)^{-1}$ give the decisive equation
\begin{equation}\label{nu2:eq:period}
\nabla\beta_e=-m\beta_b\,dt.
\end{equation}
Write $\dot\lambda=d\lambda/dt$ for a base-frame change as in
\eqref{nu2:eq:frames}. The derivation gives
\[
e'=\lambda^ke,\quad b'=\lambda^kb-\lambda^{k-1}\dot\lambda e,
\quad\beta_e'=\lambda^m\beta_e,\quad
\beta_b'=\lambda^m\beta_b-\lambda^{m-1}\dot\lambda\beta_e.
\]
Coordinate Jacobians cancel between $\delta_k(t)$ and $\Omega/dt$.
The actual root-germ cocycles therefore glue the obstruction to
\begin{equation}\label{nu2:eq:positive-map}
R^m|_{T'^\circ}\longrightarrow
F^0\bigl((R^1g'_*\mathbb Q)^\vee\otimes\mathcal O_{T'^\circ}\bigr).
\end{equation}
Both periods lie in $F^0$, so \eqref{nu2:eq:period} says this map
is killed by the Higgs field.
Before current-order vanishing, $b\Omega$ is already a holomorphic
section of $R^1g_*\omega_E$ on each original full disk. Comparison
with its algebraic family, valid also for arbitrary holomorphic
frames, gives \eqref{nu2:eq:one-pole} for $\beta_b$.
Radial integration of \eqref{nu2:eq:period} gives
\[
\beta_e=O\bigl((1+|\log|t||)^{N+1}\bigr).
\]
Finite substitutions and the full-disk unit factors preserve this
bound in a frame of $R^m$ extending over each puncture. At a branch
point, compare on the good punctured overlap, where $t=t(w)$ is
unramified. Write $\Omega_t$ for the absolute residue form in
the original family and $\Omega_w$ for its transported class
there. Then $\Omega_w/dw=(dt/dw)\Omega_t/dt$, whereas the
derivation coefficient satisfies $e_w=(dt/dw)^{-1}e_t$.
The factors cancel. Thus the bound from the original full disk,
after finite substitution and the full-disk unit comparison of
source frames, introduces no extra pole.
We now use the positivity of the line $R^m$ on the compact curve.
More generally, a Higgs-killed map from a positive line to $F^0$ of a
graded-polarizable rational mixed variation with $F^1=0$ and
quasi-unipotent monodromy is zero when its coefficients have the
logarithmic-power bound in frames of that line extending over every
puncture. To prove this assertion, suppose the map is nonzero and choose the least $w$ for which the image is generically
in $W_w$; holomorphicity makes this containment global. Projection
to $\operatorname{gr}_w^W$ is nonzero. Flatness of $W_w$ preserves
the derivative condition, and a flat basis adapted to $W$ preserves
the coefficient bound. Equip the chosen pure highest Hodge bundle
with its Hodge metric $H$, and denote its Chern connection by $D$.
For a local image section $s$, let $\pi_s$ be orthogonal projection
onto the line spanned by $s$, away from its zeros. Highest-step
curvature gives
\[
\partial_t\partial_{\bar t}\log\|s\|_H^2
=\frac{\|(1-\pi_s)D_{\partial_t}s\|_H^2}{\|s\|_H^2}\ge0
\]
\cite[Theorem~(5.2), Lemmas~(4.9),(4.13)]{Griffiths1970}.
The incoming Higgs term is absent and the outgoing term vanishes;
no flatness of the image line is asserted. A constant Tate twist
handles negative Hodge indices. Schmid's estimate after killing
the finite monodromy part bounds flat-vector Hodge norms by powers
of logarithms \cite[Theorem~(6.6$'$)]{Schmid1973}.
Thus, relative to a positive-curvature metric on the source line,
the logarithmic norm ratio $v$ has
$\sqrt{-1}\partial\bar\partial v\ge\omega_R>0$ and upper growth
$O(\log(1+\log(1/|t|)))$.
Subtract a local potential for $\omega_R$ and then
$\epsilon\log(1/|t|)$. The maximum principle on shrinking annuli,
followed by $\epsilon\downarrow0$, gives a subharmonic extension
preserving this inequality. Interior zeros contribute nonnegative
point masses. Integration over the compact curve contradicts
$\deg R^m>0$.
Perfect relative Serre duality now gives $e=0$ on the good locus;
\eqref{nu2:eq:period} and $m>0$ give $b=0$. There
$g_*\mathcal O_E$ is locally a product of copies of $\mathcal O_U$:
the primitive idempotents of $R^0g_*\mathbb C$ split the proper
reduced fibers into connected pieces, and base change identifies
each piece's functions with $\mathcal O_U$. A derivation kills
idempotents and, by $e=0$, curve functions. It therefore kills
every leading section in every degree.
At an exceptional parameter, each value of $\delta_k^{-m}$ is now
point-supported and annihilates the constant tests. The Hodge test
above makes it zero. Applying this to $u^{-m}$ first gives $b=0$;
applying it to $f_0u^{-m}$ for an arbitrary
$f_0\in g_*\mathcal O_E$ gives $\delta_k(f_0)=0$.
This includes functions on vertical components, not only pullbacks
from $U$. The Laurent-power rule kills every degree and completes
the simultaneous induction \eqref{nu2:eq:lifting}.
\subsection{Finite descent and the quadratic count}\label{nu2:subsec:count}
All conormal lifting obstructions have vanished. We descend only finite collections of layers and count their global sections. This avoids any assertion about convergence of a formal splitting.
For every signed integer $j$ define
\[
J_j=\mathcal O_V\left(-\left\lceil jG/r\right\rceil\right),
\qquad\mathcal B_j=J_j/J_{j+1},\qquad F_j=f_*\mathcal B_j.
\]
On the full cover with deck group $H$,
$(\pi_*I^j)^H=J_j$: the valuation condition is
$(r/d_i)\operatorname{ord}_{B_i}(h)\ge j$, equivalently
$\operatorname{ord}_{B_i}(h)\ge\lceil jd_i/r\rceil$.
Normality checks these divisorial sheaves in codimension one.
Since $0<d_i/r\le1$, consecutive ceiling differences are zero or
one; hence $\mathcal B_j$ is a coherent sheaf on the reduced $B$.
Exactness of finite pushforward and of finite-group invariants,
and averaging of the lifts in \eqref{nu2:eq:lifting}, give finite
filtrations of $f_*(J_l/J_m)$ with full coherent quotients $F_j$,
for every $l<m$. The total filtered sheaf is only required to be
a sheaf of complex vector spaces.
Cartier periodicity and \eqref{nu2:eq:pencil} give
$J_{j-r}=J_j\otimes\mathcal O_V(G)$ and $F_{j-r}=F_j(1)$.
For a coherent $Q$ on $\mathbb P^1$, a point outside its torsion
support gives
$0\to Q(b-1)\to Q(b)\to\mathbb C_x^{\operatorname{rk}Q}\to0$;
thus $\chi(Q(b))=\chi(Q)+b\operatorname{rk}Q$ for every integer $b$.
Finite-filtered cohomology is additive, vanishes above degree one,
and satisfies $h^0\ge\chi$. Consequently
\begin{align}\label{nu2:eq:quadratic-count}
h^0(V,J_{-nr}/J_0)
&\ge\sum_{j=-nr}^{-1}\chi(F_j)\notag\\
&=n\sum_{\ell=0}^{r-1}\chi(F_\ell)
+\frac{n(n+1)}2\sum_{\ell=0}^{r-1}\operatorname{rk}F_\ell.
\end{align}
Here $\mathcal B_0=\mathcal O_B$ and the surjectivity of $f$ gives
$\operatorname{rk}F_0\ge1$. Passing through
$0\to\mathcal O_V\to\mathcal O_V(nG)\to J_{-nr}/J_0\to0$
loses at most the fixed number $h^1(V,\mathcal O_V)$.
In particular, the coefficient of $n^2$ in this lower bound is exactly
$\tfrac12\sum_{\ell=0}^{r-1}\operatorname{rk}F_\ell>0$; all remaining
terms are linear or constant. Thus $h^0(V,\mathcal O_V(nG))$ has positive quadratic growth,
contradicting $G\sim qA$ and $\kappa(V,A)=0$ in
\eqref{nu2:eq:model}. Every count uses only finitely many
infinitesimal orders; no convergent formal neighborhood or one
neighborhood valid at all orders is needed. This proves
Theorem~\ref{nu2:thm:alternative}.
==== END Lifting-sections-from-the-reduced-support-of-an-adjoint//curve-base.tex ====
==== BEGIN Lifting-sections-from-the-reduced-support-of-an-adjoint//fields.tex ====
\section{Transfer of the conclusions to characteristic-zero fields}
\label{bl:sec:fields}
The remaining task is to recover generation of an actual line bundle
on the original variety over an arbitrary algebraically closed field
of characteristic zero. We descend the specified divisor data to a
countable algebraically closed field, embed that field into $\C$,
and detect generation by the evaluation cokernel. Faithful flatness
then transfers the same Cartier multiple back. The auxiliary minimal
models used over $\C$ play no role in this descent.
\begin{lemma}[Transfer of specified divisor data]
\label[lemma]{bl:lem:field-transfer}
Let $\kk$ be an algebraically closed field of characteristic zero.
A finite collection of projective varieties over $\kk$, rational
divisors, specified Cartier multiples, morphisms, line bundles,
sections, and identities between these data can be defined over
a countable algebraically closed subfield $\kk_0\subset\kk$.
The field $\kk_0$ admits an embedding into $\C$.
For the data considered here, one can include a log resolution and
its discrepancy identities in this descent. Log canonicity, and
klt singularities outside a specified reduced boundary, then hold
after extension from $\kk_0$ to either $\kk$ or $\C$. For a
rational Cartier divisor on a projective variety over $\kk_0$,
nefness, Kodaira dimension, and generation of any fixed Cartier
multiple are unchanged under algebraically closed field extension.
\end{lemma}
\begin{proof}
Choose finitely many equations and transition functions defining
the schemes, morphisms, line bundles, and sections in question.
Equalities and isomorphisms among them also have finite
presentation data. They are defined over a finitely generated
subfield $\kk_1\subset\kk$; enlarge it finitely whenever another
one of the specified data is added. Take for $\kk_0$ the algebraic
closure of $\kk_1$ inside $\kk$. It is algebraically closed and
countable. Choosing images in $\C$ of a transcendence basis of
$\kk_1/\Q$, and then extending over its algebraic closure, gives
an embedding $\kk_0\hookrightarrow\C$.
Include the individual prime divisors, a projective log resolution,
its exceptional divisors, and all the required Cartier powers in
this construction. The resulting varieties over $\kk_0$ are
normal or smooth as appropriate; these properties descend along
the faithfully flat extension to $\kk$. Over the algebraically
closed field $\kk_0$, normal varieties are geometrically normal
(see \cite[Tag~0C3M]{StacksProject}), and integral varieties are
geometrically integral. The divisors, simple normal crossing
conditions on the resolution, and birationality of the chosen
morphisms consequently persist under extension. One can also
include an open set on which a birational morphism is an
isomorphism.
For canonical divisors, choose compatible rational top forms on
the smooth loci and on the resolution, and descend them together
with the rational functions specifying the divisor comparisons.
Pullback identities for the selected rational Cartier divisors
then remain identities after extension. In particular a formula
for the log discrepancies on the chosen resolution is unchanged.
The finitely many coefficient inequalities on its snc boundary
test lc singularities. On the inverse image of the complement
of the specified floor, the strict inequalities test klt
singularities. They therefore give the asserted singularity
properties on the complex extension as well. Only the specified
Cartier powers are being transferred; this argument asserts no
general preservation of $\Q$-factoriality.
We give the nefness argument, since it involves all curves and
is not itself a finite list of witnesses. Let $X_0$ be projective
over $\kk_0$, let $L_0$ be a line bundle, and let $\kk'$ be an
algebraically closed extension. An integral curve over $\kk_0$
remains integral after extension and retains its $L_0$-degree.
Thus nefness over $\kk'$ implies nefness over $\kk_0$.
Conversely, suppose an integral curve $C\subset X_{\kk'}$ has
negative degree. It is defined over a finitely generated field
extension of $\kk_0$. Spread it as a closed subscheme of
$X_0\times T$, where $T$ is an integral finite-type
$\kk_0$-scheme with that function field. After shrinking $T$,
the family is flat with geometrically integral one-dimensional
fibers, and the degree of $L_0$ on the fibers is constant. These
assertions follow from geometric integrality of the generic
fiber, generic flatness, and constancy of the relevant Hilbert
polynomials. Every nonempty such $T$ has a closed
$\kk_0$-point. Its fiber is therefore a negative curve on
$X_0$, a contradiction. Apply this argument to a Cartier
multiple for rational divisors.
For every integer $m$ making $mD_0$ Cartier, flat base change
gives
\begin{equation}
H^0(X_0,\OO_{X_0}(mD_0))\otimes_{\kk_0}\kk'
\simeq H^0(X_{\kk'},\OO_{X_{\kk'}}(mD_{\kk'}));
\label{bl:eq:fields-sections}
\end{equation}
see \cite[Tag~02KH]{StacksProject}. The complete linear-system
maps therefore base change, as do their image closures, so their
image dimensions and the Kodaira dimension are unchanged.
Alternatively, their spaces of sections have the same dimensions
in every divisible degree. The evaluation map for $mD_0$ also
base changes to the evaluation map for $mD_{\kk'}$. Its
cokernel vanishes exactly when its faithfully flat pullback
vanishes. This proves generation in a fixed degree, and thus
semiampleness, in both directions. The same argument applies to
line bundles on the whole reduced boundary scheme, even when
that scheme is reducible.
\end{proof}
\begin{proof}[Completion of Theorem~\ref{bl:thm:supported}]
Start with its given data over $\kk$. In
Lemma~\ref{bl:lem:field-transfer}, include the pair, every component
of its boundary and of $G$, the exact identity
$G\sim q(K_V+C)$, and the section defining the nonzero effective
Cartier divisor $G$. Include also a generated positive power of
$\OO_V(G)|_S$ with finitely many generating sections on the
entire $S=G_{\mathrm{red}}$. Reduced supports and their
inclusions are part of these data, so $G$ stays nonzero and
$\Supp G\subset\Supp\lfloor C\rfloor$ after extension.
Because the original variety is $\Q$-factorial, $K_V$ and each
of the finitely many boundary components have individual Cartier
multiples; include these witnesses. The descended discrepancy
data give lc singularities and klt singularities outside the
floor over $\C$. These are exactly the weaker complex
hypotheses in Proposition~\ref{bl:prop:complex-supported}; global
$\Q$-factoriality of the complex extension is unnecessary.
That proposition gives $\kappa(K_V+C)>0$ on the complex
extension. Equation~\eqref{bl:eq:fields-sections} transfers this
conclusion first to $\kk_0$ and then to $\kk$, proving the
supported theorem as stated.
\end{proof}
\begin{corollary}[Abundance after nonvanishing over characteristic-zero fields]
\label{cor:abundance-after-nonvanishing-fields}
Let $(X,\Delta)$ be a projective lc rational pair of dimension at most
four over an algebraically closed field $\kk$ of characteristic zero.
Suppose that $D=K_X+\Delta$ is $\Q$-Cartier and nef and
$\kappa(X,D)\geq0$. Then $D$ is semiample, and if $\kappa(X,D)=0$,
then $D\sim_{\Q}0$.
\end{corollary}
\begin{proof}[Proof of Corollary~\ref{cor:abundance-after-nonvanishing-fields} and completion of Theorem~\ref{bl:thm:conditional}]
For the four-dimensional assertion, descend the original lc pair
and its rational Cartier adjoint $D$, with a log resolution and
its discrepancy identities. No separate Cartier property for
$K_X$ or the boundary components is needed for this application.
The complex extension is lc, $D$ is nef, and its Kodaira
dimension is the same. The complex case proved in
Section~\ref{bl:sec:abundance} makes one positive Cartier multiple
of $D$ generated. Generation in this particular degree descends
to $\kk_0$ and then extends to $\kk$. If $\kappa(D)=0$, the
generated line bundle has a one-dimensional space of sections;
its nonzero generating section is nowhere vanishing. Hence it
is trivial, and $D\sim_{\Q}0$ over $\kk$.
For the conditional assertion, keep its smooth canonical
nonvanishing premise over $\C$ exactly as stated. Descend a
given nef projective lc rational pair over $\kk$, extend to
$\C$, and apply the conditional complex result there. Its
proof may use very general complex fibers and complex minimal
models. Only the resulting generation of a fixed positive
multiple of the original adjoint is transferred back by
Lemma~\ref{bl:lem:field-transfer}. This needs neither very general
points over the countable field $\kk_0$ nor descent of the
auxiliary minimal models, and completes both theorems.
\end{proof}
\begin{proof}[Proof of \Cref{thm:full-abundance}]
First work over $\C$. Fix a Cartier index $r>0$ for the given
nef adjoint $D$. The exact original-variety lc nonvanishing result
\cite[Corollary~1.2]{OpenAINonvanishing2026}
produces a nonzero section of $mrD$ for some $m>0$. Thus
$\kappa(X,D)\geq0$, and Theorem~\ref{thm:abundance-after-nonvanishing}
proves semiampleness, with $\Q$-linear triviality when $\kappa=0$.
The cited nonvanishing corollary uses smooth fourfold nonvanishing,
Hashizume's reduction, and the independent finite rational-feasibility
lemma \cite[Lemma~8.1]{OpenAINonvanishing2026}.
It supplies a section on the original normal variety and does not use
any supported lifting or abundance theorem from this paper.
Over an arbitrary algebraically closed characteristic-zero field,
descend the original pair, a specified Cartier multiple of its
adjoint, and a log resolution with its discrepancy identities as in
\Cref{bl:lem:field-transfer}. The complex extension is lc and its
adjoint is nef. The complex conclusion makes one positive Cartier
multiple generated. The evaluation-cokernel argument of that lemma
transfers this precise generation to the original field. Kodaira
dimension transfers as well. If it is zero, the generated multiple
has a one-dimensional space of sections; its nonzero generating
section is nowhere vanishing, hence trivializes that line bundle.
This proves the stated $\Q$-linear triviality on the original variety.
\end{proof}
==== END Lifting-sections-from-the-reduced-support-of-an-adjoint//fields.tex ====
==== BEGIN Lifting-sections-from-the-reduced-support-of-an-adjoint//frames.tex ====
\section{Positive frame weight and the Euler direction}
\label{bl:sec:frames}
We now combine the graph module with the scaling action on the
frame bundle. A positive weight has two consequences: after a
transverse cover it gives an ample source for the horizontal symbol,
and in the vertical direction it gives a nonzero Euler eigenvalue.
The proof keeps these two operations separate until both have been
identified on the same right differential module.
Return to Section~\ref{bl:sec:covers}. Thus $T=\PP^\ell$,
$J=\OO_T(1)$, and $B=J^\times$ is the bundle of
nonzero vectors, with projection $p_B:B\to T$. The parameter
$\lambda\in\Gm$ acts on $B$ by multiplication by $\lambda^r$.
Its action on $Y$ preserves $R$ and $D_h$, and $h:R\to B$ is
equivariant. All the modules in~\eqref{bl:hod:modules} inherit their
natural geometric action.
Let $\varepsilon$ be the infinitesimal generator on $B$. In a
bundle chart with fiber coordinate $\beta$,
\begin{equation}\label{bl:frm:euler}
\varepsilon=r\beta\frac{\partial}{\partial\beta}.
\end{equation}
It is nowhere zero and spans the kernel of
$T_B\to p_B^*T_T$. By weight $m$ we mean pullback weight:
$\lambda^*s=\lambda^m s$.
\begin{proposition}[Positive-weight obstruction]
\label{bl:prop:frame-positivity}
For the filtered module $\mathcal N$ of~\eqref{bl:hod:modules}, the
following assertions hold.
\begin{enumerate}[label=\textup{(\roman*)}]
\item If
$v\in H^0(B,F_0\mathcal N\otimes T_B)$ is homogeneous of weight
$m>0$ and has zero symbol in $\gr_1^F\mathcal N$, then its image
in $F_0\mathcal N\otimes p_B^*T_T$ is zero.
\item If $s\in H^0(B,F_0\mathcal N)$ is homogeneous of weight $m$,
then its actual right differential-operator action satisfies
\begin{equation}\label{bl:frm:euler-action}
s\varepsilon=-ms.
\end{equation}
The same assertion is valid on invariant bundle charts.
\end{enumerate}
\end{proposition}
The first assertion concerns the symbol, and the second concerns the
operator itself. Their combination will annihilate the obstruction.
We prove them separately, keeping track of both canonical factors
and filtration shifts.
The horizontal directions cannot in general be suppressed. For
example, if $V=\PP^2$ and $C$ is a smooth quartic, then
$A=K_V+C\sim\OO_{\PP^2}(1)$ and $G=C\sim4A$ satisfy the
supported hypotheses, while $\OO_V(G)|_C$ is ample and nontrivial.
The construction must therefore retain the full map $f:S\to T$.
\subsection{A transverse power cover}
Suppose first that $\ell>0$. Take the coordinate $r$-th power map of
$\PP^\ell$ and compose it with a general target automorphism, obtaining
\[
\phi:T'=\PP^\ell\longrightarrow T,
\qquad J'=\OO_{T'}(1),\qquad (J')^r\simeq\phi^*J.
\]
It is finite flat. Put $B'=(J')^\times$, with its ordinary scalar
action, and let $p':B'\to T'$ be the projection. Raising frames to
the $r$-th power gives a finite flat equivariant map $\eta$ in
\begin{equation}\label{bl:frm:bundle-square}
\begin{tikzcd}[column sep=large,row sep=large]
B'=(J')^\times \arrow[r,"\eta"] \arrow[d,"p'"'] &
B=J^\times \arrow[d,"p_B"]\\
T' \arrow[r,"\phi"'] & T.
\end{tikzcd}
\end{equation}
Along each nonzero fiber its differential is invertible.
Choose $\phi$ transverse to the maps to $T$ from every closed
intersection of components of $R+D_h$ involving a component of $R$.
These intersections are smooth and there are finitely many of them.
Here is the generic-transversality justification. For such an
intersection $L$, the matching-pair space for the map $L\to T$
and the translated power map is smooth: variation of the target
automorphism supplies every target tangent direction. A general
fiber over the automorphism group is smooth by generic smoothness
in characteristic zero. This is exactly the required transversality.
The finitely many conditions can be imposed simultaneously.
Since the vertical differential in~\eqref{bl:frm:bundle-square} is
invertible, this also gives transversality to the corresponding
maps $L\to B$. If a point of $T$ is prescribed, one may in addition
require that the branch divisor avoid it, an open condition on
the target automorphism.
Let $Q'=Q\times_B B'\subset Y\times B'$, and set
\begin{equation}\label{bl:frm:basechanged-modules}
\mathcal M'=\mathcal H^{b+1}_{[Q']}
\bigl(\omega_{Y\times B'}(*D_h)\bigr),
\qquad
\mathcal N'=\mathcal H^1\operatorname{pr}_{B',+}\mathcal M'.
\end{equation}
These are again mixed Hodge modules with projective support.
\begin{lemma}[Filtered transverse comparison]\label{bl:frm:comparison}
For every $j$ there are natural equivariant isomorphisms
\begin{equation}\label{bl:frm:filtered-comparison}
F_j\mathcal N'
\simeq \eta^*F_j\mathcal N\otimes\omega_{B'/B}.
\end{equation}
Under these isomorphisms, symbol action transforms by the tangent
map $d\eta:T_{B'}\to\eta^*T_B$.
\end{lemma}
\begin{proof}
Use the local coordinates in the proof of
Lemma~\ref{bl:lem:lowest-piece}. The pulled-back graph equations are
\[
\psi'_i=t_i\circ\eta-v_i.
\]
At a point of $Q'$, transversality to the intersection of all
boundary components through that point says that the differentials
of the $\psi'_i$, together with those boundary coordinates, are
independent. They can therefore be used as part of a local smooth
coordinate system. In particular the local support calculation
remains a normal crossing calculation.
Flat pullback preserves the localization quotient and its local
cohomology. More specifically, the explicit description following
Lemma~\ref{bl:lem:lowest-piece} identifies its filtration level by
level: the fractions with total extra denominator exponent at most
$j$ pull back to precisely the corresponding fractions for $\mathcal M'$.
For right modules one must also replace the absolute top-form
frame on $Y\times B$ by that on $Y\times B'$. Thus the filtered
comparison before direct image is
\[
F_j\mathcal M'
\simeq (\id_Y\times\eta)^*F_j\mathcal M
\otimes\operatorname{pr}_{B'}^*\omega_{B'/B}.
\]
This is an isomorphism with the abstract relative canonical line;
it does not multiply sections by a Jacobian that vanishes at
ramification.
The relative de Rham differentials differentiate only in $Y$.
They commute with this comparison and with its relative canonical
factor. Flat base change therefore applies to the entire filtered
relative de Rham complex. One can check it on a finite affine
cover and its \v{C}ech complex; flat pullback commutes with all
cohomology in question. Projective-support strictness identifies
the cohomology filtration with that of $\mathcal N$ and
$\mathcal N'$, giving~\eqref{bl:frm:filtered-comparison} for all $j$.
The chain rule in the parameter coordinates gives the assertion
about symbols. Terms arising from a change of canonical frame
have order zero and do not change that assertion.
\end{proof}
One can also express the transversality in this proof as a
non-characteristic condition. The characteristic variety of the
normal crossing graph module is contained in the union of
conormals to the graph strata involving $R$. A characteristic
covector killed by $d(\id_Y\times\eta)$ has zero $Y$ component;
its remaining component annihilates both the stratum tangent
image and $d\eta(T_{B'})$. Transversality forces it to be zero.
Finite flatness also makes the higher $\operatorname{Tor}$ groups of
every graded filtration piece vanish. These are the two
non-characteristic conditions of
\cite[Section~3.5.1 and Lemmas~3.5.3, 3.5.5]{Saito1988}; in relative dimension
zero they give the same filtration index and right canonical factor
as the explicit calculation above.
\subsection{The adjugate and the projective quotient}
For vector bundles of equal rank the adjugate of $d\eta$ defines
a regular morphism
\begin{equation}\label{bl:frm:adjugate}
\operatorname{adj}(d\eta):
\eta^*T_B\longrightarrow\omega_{B'/B}\otimes T_{B'}.
\end{equation}
Indeed $\det(d\eta)$ is a section of $\omega_{B'/B}$, so this
is the usual determinant-twisted adjugate. It involves no division.
If $v$ has zero symbol, applying~\eqref{bl:frm:adjugate} to its tangent
factor and then~\eqref{bl:frm:filtered-comparison} gives a section
\begin{equation}\label{bl:frm:transported-symbol}
v'\in H^0(B',F_0\mathcal N'\otimes T_{B'})
\end{equation}
with zero symbol. In local matrices this is just
$d\eta\operatorname{adj}(d\eta)=\det(d\eta)\id$.
The construction preserves the weight of $v$.
Suppose the horizontal image of $v$ is nonzero. Choose a point
where that coherent-sheaf section is nonzero, and choose $\phi$
as above with branch divisor avoiding its image in $T$.
On an unbranched neighborhood the tangent map is an isomorphism
preserving the vertical tangent line. Faithful flatness there
shows that the horizontal image of $v'$ is still nonzero.
Now take the diagonal quotient
\[
\overline X=(Y\times B')/\Gm\longrightarrow T'.
\]
It exists as an associated algebraic variety: a local section of
$B'\to T'$ identifies it with $Y$ times that base chart, and changes
of section glue these products through the action on $Y$.
It is smooth. The linearized ample bundle on $Y$ descends to a
relatively ample bundle on this associated variety, so
$\overline X$ is quasi-projective. The invariant divisor $D_h$
descends to a divisor $\overline D_h$, and the invariant closed
support $Q'$ descends to $\overline Q$.
The support map $\overline Q\to T'$ is projective. Properness
descends from the projective map $Q'\to B'$, and a descended
relatively ample bundle gives projectivity. The quotient changes
both the ambient dimension and the support dimension by one;
thus $\overline Q$ still has codimension $b+1$. Define
\begin{equation}\label{bl:frm:quotient-modules}
\overline{\mathcal M}=
\mathcal H^{b+1}_{[\overline Q]}
\bigl(\omega_{\overline X}(*\overline D_h)\bigr),
\qquad
\overline{\mathcal N}
=\mathcal H^1\overline{\operatorname{pr}}_+
\overline{\mathcal M}.
\end{equation}
Use the constant twist by $\dim\overline X$ in this formula.
These are algebraic mixed Hodge modules, and the second has
projective support over the projective base $T'$.
Choose a frame of $J'$ on $U\subset T'$ and write a point of
$Y\times B'|_U$ as $(z,t,u)$, with $z\in Y$, $t\in U$, and
$u\in\Gm$. The coordinate change
\begin{equation}\label{bl:frm:untwisting}
(z,t,u)\longmapsto(u^{-1}\!\cdot z,t,u)
\end{equation}
turns the diagonal action
$(z,t,u)\mapsto(\lambda\!\cdot z,t,\lambda u)$ into the action
on $u$ alone. Since $Q'$ and $D_h$ are invariant, the support
and divisor become products with $\Gm$ as well. Localization
and support commute with this product. Their normalizations give
\begin{equation}\label{bl:frm:quotient-filtration}
F_j\mathcal N'
\simeq (p')^*F_j\overline{\mathcal N}
\otimes\omega_{B'/T'}.
\end{equation}
To check the index, ordinary right smooth pullback of relative
dimension one changes $F_j$ to $F_{j+1}$ and tensors with the
relative canonical line. The normalized constant upstairs has
one additional Tate twist. This cancels the index change: the
normalized smooth inverse image is $[1](1)$, and gives exactly
\eqref{bl:frm:quotient-filtration}. See~\cite[Section~30]{Schnell2014}.
Relative direct image has the same relative dimension before and
after this product, so its cohomological index stays one.
The relative canonical line in~\eqref{bl:frm:quotient-filtration}
has invariant frame $du/u$. This frame is intrinsic as a relative
form: a change of bundle trivialization changes it only by a
form from the base. In the product description the vertical
symbol is zero and the horizontal symbol is the pullback of that
on $\overline{\mathcal N}$. In particular
$F_{<0}\overline{\mathcal N}=0$, by~\eqref{bl:frm:filtered-comparison},
\eqref{bl:frm:quotient-filtration} and faithful flatness.
Project $v'$ horizontally and use~\eqref{bl:frm:quotient-filtration}.
A homogeneous section of weight $m$ on the nonzero vectors of
$J'$ corresponds to a section twisted by $(J')^{-m}$. Explicitly,
if local frames satisfy $e_j=g_{ij}e_i$ and a vector is $u_ie_i$,
then $u_j=g_{ij}^{-1}u_i$; equality of expressions $u_i^m s_i$
forces $s_j=g_{ij}^m s_i$, the transition rule for a homomorphism
from $(J')^m$. Thus the projected zero-symbol section gives
\begin{equation}\label{bl:frm:ample-map}
(J')^m\longrightarrow
\ker\left[F_0\overline{\mathcal N}\otimes T_{T'}
\longrightarrow\gr_1^F\overline{\mathcal N}\right].
\end{equation}
It is nonzero if the original horizontal projection was nonzero.
For $m>0$ its source is ample, contradicting
Lemma~\ref{bl:lem:symbol-vanishing}. This proves assertion
\textup{(i)} of Proposition~\ref{bl:prop:frame-positivity}.
For $\ell=0$ there is no horizontal tangent direction and
assertion~\textup{(i)} is immediate.
\subsection{The actual Euler action}
It remains to prove assertion~\textup{(ii)}, which does not follow
merely from the vanishing of the vertical symbol. Work on a
bundle chart of $B$ and make the finite \emph{\'etale} fiber cover
\[
\beta=u^r,\qquad u\ne0.
\]
The generator~\eqref{bl:frm:euler} becomes $u\partial_u$.
The coordinate change~\eqref{bl:frm:untwisting} again identifies the
supported geometric construction with a product in the $u$ direction. This is an
identification of the underlying differential modules, with their
natural equivariance, not just an identification of coherent
filtration pieces.
A weight-$m$ section is consequently $u^m$ times a section
independent of $u$, in the invariant relative frame $du/u$.
Right action on top forms is negative Lie differentiation.
Since $du/u$ is invariant,
\[
(u^m s_1\otimes du/u)\,(u\partial_u)
=-m(u^m s_1\otimes du/u).
\]
The product direct image retains this action, proving
\eqref{bl:frm:euler-action} on the \'{e}tale cover and hence downstairs.
This completes the proof of Proposition~\ref{bl:prop:frame-positivity}.
\begin{remark}\label{bl:frm:operator-order}
If $v=s_0\otimes\varepsilon$ and
$\varepsilon=\sum_i c_i\partial_{t_i}$, then the relevant
right operator is
\[
\sum_i(s_0c_i)\partial_{t_i}=s_0\varepsilon.
\]
The coefficients precede the derivatives. Moving them to the
other side would introduce a divergence correction; the displayed
identity is simply associativity in the ring of differential
operators.
\end{remark}
==== END Lifting-sections-from-the-reduced-support-of-an-adjoint//frames.tex ====
==== BEGIN Lifting-sections-from-the-reduced-support-of-an-adjoint//hodge.tex ====
\section{A filtered module on the boundary graph}
\label{bl:sec:hodge}
The tensor $v_k$ of Section~\ref{bl:cov:obstruction-contract} has
coefficients in $R^1h_*\omega_R(D_h)$. We construct a filtered right
$\mathcal D_B$-module whose lowest piece is this entire coherent
sheaf, so that differentiation in base coordinates gives a first
symbol for the tensor. We then prove a vanishing statement for the
kernel of that symbol on a projective base. Section~\ref{bl:sec:frames}
will apply it to the positive frame weight $a+k$.
Temporarily isolate the graph calculation from the covers. Let $Y$ be
a smooth complex quasi-projective variety, let $R+D_h$ be
a reduced simple normal crossing divisor with no common component
between $R$ and $D_h$, and let $h:R\to B$ be a projective morphism
to a smooth quasi-projective variety. Write $b=\dim B$, and let
$Q\subset Y\times B$ be its graph. The projection is not required
to be proper, but its restriction to $Q$ is projective:
\[
\begin{tikzcd}[column sep=large,row sep=large]
R \arrow[r,hook,"{(\iota,h)}"] \arrow[dr,"h"'] &
Y\times B \arrow[d,"\operatorname{pr}_B"]\\
& B.
\end{tikzcd}
\]
Throughout this section $D_h$ also denotes its pullback to the product.
\subsection{Conventions and the lowest filtration piece}
We use right $\mathcal D$-modules underlying graded-polarizable
algebraic mixed Hodge modules with rational structures. On a smooth
$d$-fold we twist the constant Hodge module by $d$, so that its
underlying canonical sheaf starts at $F_0$, rather than $F_{-d}$.
All subsequent localization and support functors carry this
normalization. The required functorial statements are the
localization and support operations, exterior products, smooth
inverse images, and projective direct-image strictness in
\cite{Saito1988,Saito1990,Saito2013}; conventions are also explained
in~\cite{Schnell2014,Popa2016}.
Define
\begin{equation}\label{bl:hod:modules}
\mathcal M=\mathcal H^{b+1}_{[Q]}
\bigl(\omega_{Y\times B}(*D_h)\bigr),
\qquad
\mathcal N=\mathcal H^1\operatorname{pr}_{B,+}\mathcal M.
\end{equation}
Here the brackets denote algebraic local cohomology, and the plus
sign denotes differential-module direct image. Localization and
cohomology with supports show that $\mathcal M$ is the underlying
filtered module of a mixed Hodge module. Its direct image has
projective support, as above.
For a right module, the relative filtered de Rham complex for the
projection has term
\begin{equation}\label{bl:hod:relative-dr}
F_{j-i}\mathcal M\otimes\bigwedge^i T_Y
\quad\text{in degree }-i.
\end{equation}
The tangent sheaf here is pulled back from $Y$. In particular its
last term, in degree zero, is $F_j\mathcal M$.
\begin{lemma}[Lowest piece and strict insertion]\label{bl:lem:lowest-piece}
With the normalization above,
\begin{align}
F_{<0}\mathcal M&=0,&
F_0\mathcal M&=(\iota,h)_*\omega_R(D_h),
\label{bl:hod:lowest-source}\\
F_{<0}\mathcal N&=0,&
F_0\mathcal N&=R^1h_*\omega_R(D_h).
\label{bl:hod:lowest-target}
\end{align}
The equality for $F_0\mathcal N$ identifies it with a subsheaf of $\mathcal N$.
It is induced by simple graph residues and the inclusion of the
last term of the relative de Rham complex. The latter inclusion
commutes with right differentiation in base coordinates.
\end{lemma}
\begin{proof}
Work locally along $Q$. Take simple normal crossing coordinates
for $R+D_h$, write $r_0$ for the reduced equation of $R$, and take
coordinates $t_1,\ldots,t_b$ on the base. The functions $h^*t_i$ on
$R$ lift locally to functions $v_i$ on $Y$. Thus $Q$ is defined by
\[
r_0=0,\qquad \psi_i=t_i-v_i=0\quad(1\leq i\leq b).
\]
The $\psi_i$, together with the boundary coordinates, are independent.
The support sequence is a hypersurface followed by $b$ smooth
support coordinates, which explains the index $b+1$ in
\eqref{bl:hod:modules}.
Before adding the graph coordinates, the boundary module is
\begin{equation}\label{bl:hod:boundary-module}
\omega_Y(*(R+D_h))/\omega_Y(*D_h).
\end{equation}
In one coordinate, localization of the normalized constant begins
with simple poles at $F_0$; $F_j$ permits $j$ further derivatives.
The quotient by regular forms is the residue delta module, whose
simple residue is again at level zero. This is the usual
localization--residue exact sequence with its strict Hodge filtration.
Taking exterior products gives the same rule for a normal crossing
divisor: one begins with simple poles in all the indicated
coordinates, and allows a total of at most $j$ extra denominator
exponents; see also~\cite[Section~3 and Proposition~8.2]{MustataPopa2019}.
Quotienting out the part regular in the $R$-support
direction gives~\eqref{bl:hod:boundary-module}. The $b$ independent
graph residues add free normal derivatives with the same order rule.
It follows that $F_{<0}\mathcal M=0$, and its lowest piece is
\[
\omega_Y(R+D_h)/\omega_Y(D_h)=\omega_R(D_h),
\]
inserted along the graph by the class
\begin{equation}\label{bl:hod:simple-graph}
\alpha\longmapsto
\left[\alpha\,
\frac{dt_1\wedge\cdots\wedge dt_b}
{\prod_{i=1}^b(t_i-v_i)}\right],
\qquad \alpha\in\omega_Y(R+D_h).
\end{equation}
A form regular in the $R$-support direction gives zero, even though
poles on $D_h$ are allowed. This computes the absolute dualizing
sheaf of the whole reduced divisor, including its crossings.
The map is independent of the lifts. If $v_i$ changes by a multiple
of $r_0$, the correction to the simple graph fraction becomes
regular in the support direction and vanishes. Under a base
coordinate change the Jacobian in the top differential cancels
the determinant in the ordered graph residues. These are also the
usual transformation rules for a complete-intersection residue.
Thus~\eqref{bl:hod:simple-graph} is intrinsic.
For completeness, strictness applies here to a projective morphism,
despite the quasi-projective ambient notation. Compactify the
projection projectively over $B$. Since $Q\to B$ is proper, its
image in that compactification remains closed and acquires no
points in the relative boundary. Extend the object with this
unchanged support. Projective filtered strictness applies to it
and hence computes the original direct image. The mixed statement
is Saito's Theorem~2.14 and Proposition~2.15
in~\cite{Saito1990}, with the algebraic formulation in its
Section~4.1; no decomposition of a mixed object is needed.
At level zero,~\eqref{bl:hod:relative-dr} has only the last term.
Its first cohomological direct image is therefore
$R^1h_*\omega_R(D_h)$. Strictness identifies this with $F_0\mathcal N$
as a subsheaf of $\mathcal N$, and gives the assertion at negative
levels as well. Finally, base differentiation commutes with the
relative differential in the product. It therefore commutes with
the map from the last term to differential-module direct image.
\end{proof}
The same calculation describes every filtration level locally:
$F_j\mathcal M$ consists of the graph fractions with simple initial
denominators in the boundary and graph coordinates, and at most
$j$ extra denominator exponents in total, modulo the fractions
regular in the $R$-support direction. We will use this explicit
description across a ramified base change.
\subsection{The consequence of Kodaira--Saito vanishing}
For a mixed Hodge module $\mathcal N_0$ on a smooth projective variety
$T_0$ and an ample line bundle $H_0$, Kodaira--Saito vanishing gives
\begin{equation}\label{bl:hod:ksv}
\mathbb H^i\!\left(T_0,
\gr_j^F\DR(\mathcal N_0)\otimes H_0^{-1}\right)=0
\qquad(i<0).
\end{equation}
We use perverse normalization, so the right de Rham complex ends
in degree zero. The mixed version follows from the pure strict-support decomposition
and the strict finite weight filtration; see~\cite[Proposition~2.33]{Saito1990}
and~\cite[Theorem~28]{Popa2016}.
Tate twists do not affect the assertion, which holds for every $j$.
\begin{lemma}[No ample line in the lowest symbol kernel]
\label{bl:lem:symbol-vanishing}
Suppose $F_{<0}\mathcal N_0=0$. Then
\begin{equation}\label{bl:hod:symbol-vanishing}
\Hom\!\left(H_0,
\ker\left[F_0\mathcal N_0\otimes T_{T_0}
\longrightarrow\gr_1^F\mathcal N_0\right]\right)=0.
\end{equation}
The arrow is the symbol of the right differential-operator action.
\end{lemma}
\begin{proof}
At $j=1$ all terms in degrees at most $-2$ vanish, so the graded
de Rham complex is the two-term complex
\[
[F_0\mathcal N_0\otimes T_{T_0}\longrightarrow\gr_1^F\mathcal N_0]
\quad\text{in degrees }-1,0.
\]
Its hypercohomology in degree $-1$, after tensoring by $H_0^{-1}$,
is exactly the space of global sections of the twisted kernel.
Equation~\eqref{bl:hod:ksv} proves the claim.
\end{proof}
\begin{remark}\label{bl:hod:whole-sheaf}
Both Lemma~\ref{bl:lem:lowest-piece} and
Lemma~\ref{bl:lem:symbol-vanishing} concern complete coherent sheaves,
including sections on smaller supports. They require no local
freeness and no restriction to the open set carrying a variation
of Hodge structure. This is essential for the obstruction calculation.
\end{remark}
==== END Lifting-sections-from-the-reduced-support-of-an-adjoint//hodge.tex ====
==== BEGIN Lifting-sections-from-the-reduced-support-of-an-adjoint//introduction.tex ====
\section{Introduction}
\label{bl:sec:introduction}
A nonzero section of an adjoint divisor gives an effective representative;
abundance requires enough sections to define a morphism. Adjunction
suggests looking first on the support of the zero divisor, where a
multiple may already be generated. We prove that, when this support lies
in the coefficient-one boundary, generation on the entire reduced support
forces positive ambient Iitaka dimension.
\begin{theorem}[Supported boundary lifting]\label{bl:thm:supported}
Let $(V,C)$ be a normal projective $\mathbb Q$-factorial dlt pair over an
algebraically closed field of characteristic zero, with $C$ an effective
rational boundary, and put $A=K_V+C$. Suppose that
\[
0\ne G\geq0,\qquad G\sim qA,\qquad
\operatorname{Supp}G\subseteq\operatorname{Supp}\lfloor C\rfloor,
\]
where $G$ is Cartier and $q>0$ is a sufficiently divisible integer.
If $\mathcal O_V(G)|_{G_{\mathrm{red}}}$ is semiample, then
$\kappa(V,A)>0$.
\end{theorem}
The restriction is a line bundle on the whole reduced scheme
$S=G_{\mathrm{red}}$. Sections on its separate components must agree
along their intersections. The support inclusion may be strict: in
$C=S+H+C_0$, the extra coefficient-one divisor $H$ may meet $S$ and
any of its strata, while $C_0$ has coefficients less than one.
No additional nefness hypothesis is needed. Curves in $S$ have
nonnegative $G$-degree by semiampleness, and curves outside $S$ have
nonnegative degree by effectivity. The rational coefficients of $C$
may be arbitrary in $[0,1]$.
The first application separates abundance from the task of obtaining a
first section.
\begin{theorem}[Abundance after nonvanishing]
\label{thm:abundance-after-nonvanishing}
Let $(X,\Delta)$ be a projective log canonical pair of dimension at most
four over $\C$, with
effective rational boundary. If $D=K_X+\Delta$ is $\mathbb Q$-Cartier
and nef and $\kappa(X,D)\geq0$, then $D$ is semiample. If
$\kappa(X,D)=0$, then $D\sim_{\mathbb Q}0$.
\end{theorem}
The proof uses lower-dimensional abundance and ordinary minimal models
for effective log canonical pairs. Theorem~\ref{thm:abundance-after-nonvanishing}
is independent of the existence of a first section in dimension four.
It applies to arbitrary rational boundary coefficients and supplies a
qualitative generated Cartier multiple, with no uniform index.
Section~\ref{bl:sec:fields} transfers the same conclusion to every
algebraically closed field of characteristic zero.
The same supported argument gives a reduction in all dimensions.
\begin{theorem}[Reduction to smooth canonical nonvanishing]
\label{bl:thm:conditional}
Assume that every smooth connected projective complex variety $W$ with
pseudo-effective $K_W$, in every dimension, has a nonzero section of
$mK_W$ for some $m>0$. Let $(X,\Delta)$ be a normal projective
log canonical pair over an algebraically closed field of characteristic
zero, with $\Delta$ an effective rational boundary and
$D=K_X+\Delta$ a nef $\Q$-Cartier divisor. Then $D$ is semiample.
\end{theorem}
The premise concerns pseudo-effective canonical divisors and includes
those that are not nef. Hashizume's reduction supplies nonvanishing
and ordinary log minimal models under this premise; it does not
supply the semiampleness proved here.
A separate companion provides the first section needed in dimension four.
\begin{corollary}[Fourfold log abundance]\label{thm:full-abundance}
Let $(X,\Delta)$ be a normal projective log canonical pair of dimension
at most four over an algebraically closed field of characteristic zero.
If $\Delta$ is an effective rational boundary and $D=K_X+\Delta$ is
$\Q$-Cartier and nef, then $D$ is semiample. If $\kappa(X,D)=0$,
then $D\sim_{\Q}0$.
\end{corollary}
This resolves rational log abundance positively in the stated range,
using Theorem~\ref{thm:abundance-after-nonvanishing} and the nonvanishing
result in OpenAI, \emph{Fourfold nonvanishing by minimal metrics and moving jets}
(September 27, 2026), Corollary~1.2
\cite{OpenAINonvanishing2026}. The proof of the supported theorem and
of abundance after nonvanishing uses no result from that companion.
\subsection*{Boundary deformation and extension}
The relation between boundary geometry and ambient sections already
appears in dimension three. Kawamata's alternative proof of Miyaoka's
numerical-dimension-one theorem constructs compatible infinitesimal
boundary deformations using finite covers and Hodge theory, and then
produces a moving family \cite[Section 4]{Kawamata1992}. The question
of how infinitesimal boundary data yield ambient positivity also
motivates the lifting problem considered here.
Log abundance through dimension three rests on the surface and threefold
theorems, including the corrected Keel--Matsuki--McKernan argument
\cite{KMM1994,KMM2004,Fujino2012} and semi-log-canonical gluing
\cite{Fujino2000}. Liu--Xu proved abundance after nonvanishing through
dimension five under the additional numerical-dimension bound
$\nu\leq1$, together with a corresponding conditional reduction from
smooth canonical nonvanishing in dimension $d-1$ and in numerical
dimension one in dimension $d$
\cite[Theorems~5.1 and~5.4]{LiuXu2025}. Theorem~\ref{thm:abundance-after-nonvanishing}
removes that numerical-dimension bound in dimension at most four;
the lifting theorem itself holds in every dimension.
On the analytic side, Demailly--Hacon--P\u{a}un proved extension for
plt pairs without requiring an ample boundary part and explained its
connection with nonvanishing and good minimal models
\cite[Theorem 1.7 and Corollary 1.8]{DHP2013}. Their nef plt
corollary gives restriction surjectivity in the presence of a suitable
effective supported representative. Chan--Choi subsequently removed
one support-containment restriction in the log-smooth plt setting
\cite[Theorem 1.6.1]{ChanChoi2021}. Theorem~\ref{bl:thm:supported}
starts with semiampleness on an entire reducible support and allows
additional coefficient-one components to meet it. Compatibility at
those intersections, and obstructions concentrated on smaller strata,
are central to its proof.
The passage to abundance also needs distinct results about gluing and
minimal models. Birkar's effective log minimal model results provide
the birational models used in the induction
\cite{Birkar2010,Birkar2011}. Fujino--Gongyo show that semiampleness
on the normalization of a semi-log-canonical pair, with its conductor
boundary, descends to the original pair
\cite[Theorem 4.3]{FujinoGongyo2014}. Their nef log-abundance theorem
requires abundance on the ambient variety and on the normalizations of
all log canonical centers \cite[Theorem 4.2]{FujinoGongyo2014}.
Both requirements enter the proof of Theorem~\ref{thm:abundance-after-nonvanishing}.
\subsection*{The obstruction on a finite neighborhood}
After replacing $G$ by a multiple, its generated restriction gives
$f:S\to\mathbb P^\ell$ with
$\mathcal O_V(G)|_S\simeq f^*\mathcal O(1)$. We want to extend
functions and sections through successive infinitesimal neighborhoods
of $S$. There is an immediate difficulty: $f$ itself need not extend
to any neighborhood. The proof must therefore measure the lifting
obstruction using only the morphism on the reduced boundary.
Pass to the bundle of nonzero frames of $\mathcal O_V(G)$, where the
pulled-back line bundle has a tautological trivialization. Scalar
multiplication of a frame gives an action of $\mathbb G_m$. A function
or form has weight $a$ if this action multiplies it by the character
$\lambda^a$. Two cyclic-cover constructions, retaining all components
of each normalized cover, produce a reduced Cartier divisor $E=(y=0)$.
After an equivariant resolution there is a logarithmic top form $\sigma$
whose residue along the reduced resolved boundary has positive weight. Poles over the extra
boundary $H$ remain present, including above its intersections with $S$.
Locally, the finite neighborhoods are defined by $(y^k)$.
At the first stage where a lift from $y^k=0$ to $y^{k+1}=0$ is not yet
known, differences of local lifts are multiples of $y^k$. Their
coefficients form a cohomology class on $E$: this is the obstruction.
Because products of two such errors vanish at that order, the connecting
map on functions satisfies the Leibniz rule. Thus the obstruction is a
derivation. A suitable character part of the residue map embeds its
cohomology group into the cohomology of logarithmic residues.
The identity $\mathcal O_V(G)|_S=f^*\mathcal O(1)$ lifts $f$ to a
map between the bundles of nonzero frames. On a chart trivializing
$\mathcal O(1)$, the target of this lifted map has projective-base
coordinates and an invertible frame scalar. Changing the base coordinates
gives a horizontal direction; rescaling that scalar gives the vertical
direction. The proof first eliminates the
horizontal part of the obstruction, then uses the nonzero weight of the
frame action to eliminate the vertical part. This distinction explains
why positivity of the residue weight is useful.
To retain classes supported on singular fibers or smaller strata, the
residue comparison is made in a filtered direct image of a Hodge module
on the graph of the boundary morphism. Its lowest filtration term is the
whole coherent residue cohomology sheaf. Its first symbol records the
effect of differentiating in coordinates on the target of the lifted map.
Saito's strictness and
Kodaira--Saito vanishing provide the required control of these filtered
objects \cite{Saito1988,Saito1990,Popa2016}; restricting to the smooth
locus of a variation of Hodge structure would discard some of the
obstructions we need to test.
A local calculation on the graph expresses the derivation and this
first symbol in a single residue identity. The symbol has positive frame
weight. If its horizontal part were nonzero, a finite cover transverse
to that part, together with the adjugate of its differential, would
produce a nonzero map from an ample line bundle into the kernel of the
first symbol. Kodaira--Saito vanishing excludes that map. The remaining
vertical term is the actual Euler derivative of frame scaling. The
residue identity now says that a nonzero scalar, determined by its weight,
times the vertical obstruction is zero. Hence that obstruction vanishes
as well. Applying the same identity
to arbitrary boundary functions removes the complete obstruction. This
proves lifting at every finite order.
Taking finite-cover invariants and the degree-zero part of the frame
action descends the lifted layers to divisorial sheaves on $V$.
These layers recur with positive twists of $\mathcal O_{\mathbb P^\ell}(1)$.
Their accumulated sections grow while higher cohomology stays bounded;
the loss in passing from a finite neighborhood to ambient sections is
bounded by a fixed cohomology group. Hence the section spaces of
multiples of $G$ are unbounded, proving positive Iitaka dimension.
If the boundary morphism has zero-dimensional image, the growing number
of layers supplies this increase. The argument uses finite filtrations
and does not require convergence of a formal neighborhood.
\subsection*{From lifting to abundance and other boundary methods}
For Theorem~\ref{thm:abundance-after-nonvanishing}, lower-dimensional abundance and
conductor gluing generate the adjoint on the whole reduced floor.
Theorem~\ref{bl:thm:supported} then excludes a nonzero effective
representative in the case $\kappa=0$. For positive Iitaka dimension,
effective minimal models handle the generally non-nef adjoints on the
Iitaka fibers. Numerical triviality there gives abundance, including
on log canonical centers, and the preceding semiampleness and descent
results apply.
The complex lifting argument also works under a weaker formulation:
the canonical and boundary divisors are individually $\mathbb Q$-Cartier,
the pair is log canonical, and it is klt off the reduced floor. This
formulation permits transfer to other algebraically closed fields of
characteristic zero. One descends finitely many defining data to a countable algebraically closed
field embeddable in $\mathbb C$ and transfers sections and generation
by base change of the original line bundle and its evaluation map.
No preservation of $\mathbb Q$-factoriality under that descent is needed.
A further method illuminates the boundary obstruction with different
hypotheses. A complete conormal--period argument proves that a smooth
projective complex fourfold with $K_X$ nef and $\kappa(K_X)=0$ cannot
have $\nu(K_X)=2$, without an Euler-characteristic hypothesis.
Its period equation and trace tests at exceptional parameters kill
successive obstructions; a quadratic count of invariant layers then
forces growth. This method is not needed for
Theorem~\ref{bl:thm:supported}.
\subsection*{Reading the proof}
Section~\ref{bl:sec:covers} constructs the two covers and a split residue
insertion. Section~\ref{bl:sec:hodge} identifies the lowest filtered
piece and proves the symbol-kernel vanishing. Section~\ref{bl:sec:frames}
separates horizontal symbols from the actual Euler action, and
Section~\ref{bl:sec:lifting} uses these two conclusions to lift and
count all finite layers. Section~\ref{bl:sec:abundance} proves the
abundance implications over $\C$; Section~\ref{bl:sec:fields} returns
the resulting sections and generated multiples to the original field.
The independent curve-base method is developed in
Section~\ref{nu2:sec:method}.
\subsection*{Conventions and the complex formulation}
All boundaries are effective with rational coefficients in $[0,1]$.
An adjoint is an actual $\Q$-Cartier divisor, and $\sim_{\Q}$ denotes
rational linear equivalence. A rational Cartier divisor is semiample
if a positive Cartier multiple is generated by global sections; its
Iitaka dimension is the dimension of the images of all sufficiently
divisible nonempty complete systems, with value $-\infty$ if no such
system exists. For a nef divisor $D$ on a projective $d$-fold,
\[
\nu(D)=\max\{j:D^jH^{d-j}>0\}
\]
for an ample divisor $H$; these intersections may be computed on a
smooth resolution. We use log discrepancies, so coefficient-one
divisors have log discrepancy zero. Standard conventions for lc,
klt, dlt and dlt adjunction follow \cite{Kollar2013}.
The complex proof uses only individual Cartier witnesses, rather than
preservation of $\Q$-factoriality under a field extension.
\begin{proposition}[Complex supported assertion]
\label{bl:prop:complex-supported}
Let $V$ be normal and projective over $\C$, and let $C$ be an
effective rational boundary. Suppose that $K_V$ and every component
of $C$ are individually $\Q$-Cartier, that $(V,C)$ is lc, and that
it is klt outside $\Supp\lfloor C\rfloor$. If a nonzero effective
Cartier divisor $G\sim q(K_V+C)$ has support in
$\Supp\lfloor C\rfloor$, for a sufficiently divisible positive
integer $q$, and $\OO_V(G)|_{G_{\mathrm{red}}}$ is semiample on the
whole reduced scheme, then $\kappa(V,K_V+C)>0$.
\end{proposition}
This proposition is proved in Sections~\ref{bl:sec:covers}--\ref{bl:sec:lifting}.
Its hypotheses follow from those of Theorem~\ref{bl:thm:supported}
over $\C$, and its individual Cartier requirements are the finite
witnesses used in the final field transfer.
==== END Lifting-sections-from-the-reduced-support-of-an-adjoint//introduction.tex ====
==== BEGIN Lifting-sections-from-the-reduced-support-of-an-adjoint//lifting.tex ====
\section{Infinitesimal lifting and section growth}\label{bl:sec:lifting}
We now apply the residue insertion and the two frame assertions to
prove lifting through every finite order. Finite-cover invariants
and the weight-zero part of the frame action will then return the
lifted layers to \(V\), where their positive twists force unbounded
section spaces.
Retain the complex algebraic setup of the preceding sections.
In particular, \(I=\OO_Z(-E)=y\OO_Z\), the pullback weight of \(y\) is \(-1\), and
\(\Omega=\res(\sigma)\) has weight \(a=r/q>0\). Write
\begin{equation}\label{bl:lift:insertion}
\iota:R^1g_*\OO_E\hookrightarrow
R^1h_*\omega_R(D_h)=F_0\mathcal N\hookrightarrow\mathcal N
\end{equation}
for the injections of Lemmas~\ref{bl:lem:residue-injection}
and~\ref{bl:lem:lowest-piece}. Thus \(\iota(e)=\Omega e\), with pullback and
residue understood. If \(F\in g_*\OO_E\), the expression \(\Omega Fe\)
means \(\iota(Fe)\): multiplication takes place on \(E\), before the
insertion. This convention does not require \(F\) to come from \(B\).
For all integers \(j<d\), we push the underlying sheaf of complex vector
spaces of \(I^j/I^d\) along \(|g|\) using the Zariski-topology convention
of Section~\ref{bl:cov:obstruction-contract}; negative powers mean
invertible fractional ideals. No morphism from an infinitesimal
thickening to \(B\) is assumed. In contrast, multiplication by \(y^j\) identifies
\(I^j/I^{j+1}\) with the coherent sheaf \(\OO_E\), with an equivariant
weight shift. Its additive higher direct images are therefore the
underlying sheaves of the usual coherent higher direct images under the
projective morphism \(g\).
\begin{proposition}[Infinitesimal lifting]\label{bl:prop:lifting}
For every \(k\geq1\), the truncation map
\begin{equation}\label{bl:lift:successive}
g_*(\OO_Z/I^{k+1})\longrightarrow g_*(\OO_Z/I^k)
\end{equation}
is surjective as a map of additive sheaves on \(B\). More generally, for
all integers \(j<d\), the maps
\[
g_*(I^j/I^{d+1})\longrightarrow g_*(I^j/I^d),
\qquad
g_*(I^j/I^d)\longrightarrow g_*(I^j/I^{j+1})
\]
are surjective. Each of these surjections is also surjective on sections
over every affine open subset of \(B\).
\end{proposition}
\begin{proof}
\emph{The obstruction derivation.}
We first prove \eqref{bl:lift:successive} by induction on \(k\).
Put \(A_d=g_*(\OO_Z/I^d)\). The obstruction to surjectivity at the
\(k\)-th step is the connecting homomorphism
\[
\partial_k:A_k\longrightarrow R^1g_*(I^k/I^{k+1}).
\]
The previous steps give a surjection \(A_k\to A_1\). For \(k>1\), its
kernel is \(g_*(I/I^k)\), by left exactness. This kernel lifts to
\(g_*(I/I^{k+1})\): multiplication by \(y\) identifies the required
surjection with the already established map \(A_k\to A_{k-1}\).
Consequently \(\partial_k\) vanishes on that kernel. For \(k=1\) the
kernel is zero. After framing the target layer by \(y^k\), we obtain in
either case a uniquely defined map
\begin{equation}\label{bl:lift:derivation}
\delta:g_*\OO_E\longrightarrow R^1g_*\OO_E,
\qquad
\partial_k(\widetilde F)=y^k\delta(F).
\end{equation}
Here \(\widetilde F\) denotes any local lift of \(F\) to \(A_k\);
we suppress the index \(k\) in the notation \(\delta_k\) of
Section~\ref{bl:cov:obstruction-contract}.
The map \(\delta\) is a \(\C\)-linear derivation, where
\(R^1g_*\OO_E\) is a module over \(g_*\OO_E\) by multiplication on
\(E\). To check this assertion, work locally on the base, choose lifts
of \(F,G\) to \(A_k\), and represent them on an open cover along \(E\)
by ambient functions \(F_\alpha,G_\alpha\). We use the \v{C}ech
convention \((dc)_{\alpha\beta}=c_\beta-c_\alpha\).
Both differences \(F_\beta-F_\alpha\) and
\(G_\beta-G_\alpha\) are divisible by \(y^k\). In
\[
F_\beta G_\beta-F_\alpha G_\alpha
=F_\alpha(G_\beta-G_\alpha)
+G_\alpha(F_\beta-F_\alpha)
+(F_\beta-F_\alpha)(G_\beta-G_\alpha),
\]
the last term vanishes modulo \(y^{k+1}\), since \(2k\geq k+1\).
Divide by \(y^k\) and restrict to \(E\). The resulting cocycles give
\begin{equation}\label{bl:lift:leibniz}
\delta(FG)=F\delta(G)+G\delta(F),\qquad \delta(1)=0.
\end{equation}
The factorization just proved ensures independence of the chosen shorter
lifts.
\smallskip\noindent
\emph{The graph residue identity.}
We relate this derivation to right differentiation on \(\mathcal N\).
Choose a coordinate chart in \(B\), with coordinates
\(t_1,\ldots,t_b\), and put \(e_i=\delta(t_i)\), applying \(\delta\)
to their pullbacks to \(E\). For every local section
\(F\in g_*\OO_E\), we claim that
\begin{equation}\label{bl:lift:variable-identity}
\iota\bigl(\delta(F)\bigr)
+\sum_{i=1}^b\iota(Fe_i)\,\partial_{t_i}=0
\qquad\text{in }\mathcal N.
\end{equation}
It is essential here that \(F\) is arbitrary.
This can be checked at base germs. Shrink the base so that the induction
provides lifts of \(F,t_1,\ldots,t_b\) modulo \(I^k\), and choose
ambient representatives \(F_\alpha,v_{\alpha i}\) on an open cover
along \(E\). Pull them back to \(Y\), still writing \(y\) for its
pullback. On overlaps write
\begin{equation}\label{bl:lift:overlap-functions}
v_{\beta i}-v_{\alpha i}=y^k u_{\alpha\beta,i},
\qquad F_\beta-F_\alpha=y^k b_{\alpha\beta}.
\end{equation}
The restrictions of \(u_{\alpha\beta,i}\) and
\(b_{\alpha\beta}\) to \(E\) represent \(e_i\) and \(\delta(F)\).
The same assertions hold if representatives are specified only modulo
\(y^{k+1}\). On the product with the coordinate chart of \(B\), put
\(\psi_{\alpha i}=t_i-v_{\alpha i}\) and
\(dt=dt_1\wedge\cdots\wedge dt_b\). In the full local-cohomology
module \(\mathcal M\) of Lemma~\ref{bl:lem:lowest-piece}, consider
the zero-cochain of ordered generalized fractions
\begin{equation}\label{bl:lift:cochain}
C_\alpha=
\left[\frac{\sigma F_\alpha\wedge dt}
{y^k\prod_i\psi_{\alpha i}}\right].
\end{equation}
The brackets include the divisor support direction as well as the
\(b\) graph directions, in a fixed order. Poles along \(D_h\) are
allowed. The divisor pole bound for \eqref{bl:lift:cochain} is
\(R+D_h+k\widehat E\), so these classes belong to \(\mathcal M\);
they need not belong to \(F_0\mathcal M\).
Here is a calculation valid also at crossings and at components with
multiplicity in \(\widehat E\). Locally invert an equation of \(D_h\)
and let \(r_0\) be a reduced equation for \(R\). In this localization,
\(\sigma=\alpha/r_0\), where \(\alpha\) is a regular top form,
\(r_0\mid y\), and the zero support of \(y\) is exactly \(R\).
The divisor denominator in \eqref{bl:lift:cochain} is \(r_0y^k\).
In the quotient by this denominator,
\begin{equation}\label{bl:lift:nilpotent-modulus}
y^{2k}=0,\qquad y^{k+1}=0
\quad\text{in }\OO_Y(*D_h)/(r_0y^k),
\end{equation}
because \(r_0\mid y\) and \(k\geq1\). Thus the ideal generated by
\(y^k\) is square zero there. Localizations by
\(\psi_{\alpha i}\) and \(\psi_{\beta i}\) agree canonically,
since the two elements differ by a nilpotent element, and the exact
first-order formula is
\[
\psi_{\beta i}^{-1}
=\psi_{\alpha i}^{-1}
+y^k u_{\alpha\beta,i}\psi_{\alpha i}^{-2}.
\]
Multiplying these identities and using
\(F_\beta=F_\alpha+y^kb_{\alpha\beta}\), all terms containing two
differences vanish by \eqref{bl:lift:nilpotent-modulus}. After dividing
by the divisor denominator, the result in local cohomology is
\begin{align}\label{bl:lift:cech-expansion}
C_\beta-C_\alpha
={}&\left[
\frac{\sigma b_{\alpha\beta}\wedge dt}
{\prod_j\psi_{\alpha j}}\right] \notag\\
&+\sum_i\left[
\frac{\sigma F_\alpha u_{\alpha\beta,i}\wedge dt}
{\psi_{\alpha i}^{2}\prod_{j\ne i}\psi_{\alpha j}}
\right].
\end{align}
One can formalize this passage by first computing graph local cohomology
over \(\OO_Y(*D_h)/(r_0y^k)\), and then mapping to divisor local
cohomology by the fraction with denominator \(r_0y^k\).
Changing the graph equations by the indicated nilpotents gives the same
localizations at the first stage. Divisor local cohomology is the direct
limit of these denominator calculations, so the resulting identity is
an identity in the original support module. This also explains why
ordinary reduced-support division, without the factor \(y^k\) in the
modulus, would not justify the computation.
The coefficients remaining in \eqref{bl:lift:cech-expansion} have only the
simple divisor pole bound \(R+D_h\). Their residues are respectively
\(\Omega\delta(F)\) and \(\Omega Fe_i\) as \v{C}ech classes under
\eqref{bl:lift:insertion}. Indeed division by \(y^k\) has already
cancelled in the displayed coefficients. Changing a coefficient
representative by a multiple of \(y\) changes its product with
\(\sigma\) by a form regular in the \(R\)-direction after the
localization, because \(r_0\mid y\). Its residue is therefore zero.
This is the product-residue map on the entire divisor, including its
crossings.
For completeness, the sign in \eqref{bl:lift:cech-expansion} agrees with
the right-module action. If \(\eta\) is a form pulled back from
\(Y\), right differentiation of top forms is minus differentiation of
their coefficients. Hence
\begin{equation}\label{bl:lift:right-sign}
\left[\frac{\eta\wedge dt}{\prod_j\psi_{\alpha j}}\right]
\partial_{t_i}
=\left[\frac{\eta\wedge dt}
{\psi_{\alpha i}^{2}\prod_{j\ne i}\psi_{\alpha j}}\right].
\end{equation}
The rightmost-term map from \(\mathcal M[0]\) to the relative right
de Rham complex is a map of complexes and commutes with target
differentiation. Compute its derived pushforward with an affine
\v{C}ech refinement. The total differential of the zero-cochain
\eqref{bl:lift:cochain}, which lies in relative degree zero, is its
\v{C}ech coboundary, since there is no subsequent relative term.
Its image in \(\mathcal N=\mathcal H^1\operatorname{pr}_{B,+}
\mathcal M\) is zero. Equations~\eqref{bl:lift:cech-expansion}
and~\eqref{bl:lift:right-sign}, and the lowest-piece identification in
Lemma~\ref{bl:lem:lowest-piece}, prove \eqref{bl:lift:variable-identity}.
No ambient lift of the map \(g\) on a neighborhood of a whole fiber
was used: the chosen ambient representatives and their overlaps suffice.
\smallskip\noindent
\emph{Vanishing of the obstruction.}
Apply \eqref{bl:lift:variable-identity} first with \(F=1\), and take the
first symbol. The tensor \(v_k\), written \(v\) at this fixed order, satisfies
\begin{equation}\label{bl:lift:symbol-vector}
v=\sum_i\iota(e_i)\otimes\partial_{t_i}
\in H^0(B,F_0\mathcal N\otimes T_B),
\qquad \operatorname{symbol}(v)=0.
\end{equation}
The restriction of \(\delta\) to \(\OO_B\) factors through the
algebraic K\"ahler differentials \(\Omega_B^1\), by
\eqref{bl:lift:leibniz}. Thus \eqref{bl:lift:symbol-vector} is independent
of coordinates and is a global algebraic tensor.
The connecting map before the framing in \eqref{bl:lift:derivation} is
equivariant. Since \(\lambda^*y=\lambda^{-1}y\), the framed map
satisfies
\[
\lambda^*\delta(F)=\lambda^k\delta(\lambda^*F).
\]
Consequently the derivation tensor has weight \(k\): a coordinate's
weight in its coefficient is cancelled by the weight of its tangent
vector. Multiplication by \(\Omega\) adds \(a\), so \(v\) has
weight
\begin{equation}\label{bl:lift:positive-weight}
m=a+k>0.
\end{equation}
Proposition~\ref{bl:prop:frame-positivity} forces the horizontal projection
of \(v\) to vanish. The vertical tangent line of \(B\to T\) is
freely generated by the invariant, nowhere-zero Euler field
\(\varepsilon=r\beta\partial_\beta\). Thus
\(v=s_0\otimes\varepsilon\) for a global section
\(s_0\in H^0(B,F_0\mathcal N)\) of weight \(m\).
Write \(\varepsilon=\sum_i c_i\partial_{t_i}\). The identity with
\(F=1\), before taking the symbol, reads
\[
0=\sum_i\iota(e_i)\partial_{t_i}
=\sum_i(s_0c_i)\partial_{t_i}=s_0\varepsilon.
\]
Coefficients occur before the derivatives, so the last equality is
associativity of the right action; no coefficient is commuted through a
derivative. The Euler assertion of
Proposition~\ref{bl:prop:frame-positivity} now gives
\(0=s_0\varepsilon=-ms_0\), whence \(s_0=0\).
It follows that every \(\iota(e_i)\), and hence every \(e_i\), is
zero. Return to \eqref{bl:lift:variable-identity} for arbitrary \(F\).
It yields \(\iota(\delta(F))=0\), and injectivity of
\eqref{bl:lift:insertion} gives \(\delta(F)=0\). This proves the
induction step and hence \eqref{bl:lift:successive}. The same argument
includes \(\dim T=0\), when there is no horizontal tangent direction.
\smallskip\noindent
\emph{Signed powers and affine sections.}
For any integers \(j<d\), multiplication by the global meromorphic
function \(y^j\) identifies
\[
\OO_Z/I^{d-j}\simeq I^j/I^d.
\]
It transports the established surjections to all successive Laurent
truncations. Iteration gives the surjection to the leading layer.
Moreover, each single successive truncation fits into an exact sequence
of additive sheaves
\begin{equation}\label{bl:lift:coherent-kernel}
0\longrightarrow g_*(I^d/I^{d+1})
\longrightarrow g_*(I^j/I^{d+1})
\longrightarrow g_*(I^j/I^d)\longrightarrow0.
\end{equation}
The kernel is coherent on \(B\). On an affine open it has vanishing
first cohomology, so the additive-sheaf cohomology sequence gives
surjectivity on sections. Finitely many such steps give the asserted
affine-section surjectivity to every leading layer. Only the kernels
in \eqref{bl:lift:coherent-kernel} were assigned coherent \(\OO_B\)-module
structures.
\end{proof}
\begin{lemma}[Descent and section growth]\label{bl:lem:descent-growth}
For \(j\in\Z\), define the divisorial sheaf and its layer by
\[
J_j^V=\OO_V\bigl(-\lceil jG/r\rceil\bigr),
\qquad \mathcal B_j=J_j^V/J_{j+1}^V,
\qquad Q_j=f_*\mathcal B_j.
\]
The sheaves \(\mathcal B_j\) are coherent on \(S\), and for every
\(d>j\) the map
\begin{equation}\label{bl:lift:descent-surjection}
f_*(J_j^V/J_d^V)\longrightarrow Q_j
\end{equation}
is surjective as a map of additive sheaves on \(T\). Here the left
side is pushed along \(|f|\) on its topological support \(|S|\).
Furthermore,
\[
h^0\bigl(V,\OO_V(NG)/\OO_V\bigr)\longrightarrow\infty
\qquad\text{as }N\longrightarrow\infty.
\]
\end{lemma}
\begin{proof}
\emph{The invariant layers.}
Since \(r\) is divisible by every \(d_i\), we have \(0<d_i/r\leq1\).
Thus the coefficientwise difference
\(\lceil(j+1)G/r\rceil-\lceil jG/r\rceil\) has coefficients zero
or one. A function in the reduced ideal of \(S\) has order at least
one along each \(S_i\). The divisorial inequalities therefore show
that this ideal carries \(J_j^V\) into \(J_{j+1}^V\).
Consequently \(\mathcal B_j\) is a coherent \(\OO_S\)-module.
Let \(\mu_r\) denote the deck group of the first, full root cover
\(\pi:Z\to P\). We have
\begin{equation}\label{bl:lift:invariant-ideals}
(\pi_*I^j)^{\mu_r}=p^*J_j^V
\qquad (j\in\Z).
\end{equation}
Indeed an invariant meromorphic function in the full root algebra
descends to \(P\), even when that algebra is disconnected.
At a prime over \(p^{-1}(S_i)\), its downstairs order \(n\) becomes
\((r/d_i)n\), whereas \(y\) has order one. Membership in \(I^j\)
is thus equivalent there to
\[
(r/d_i)n\geq j
\quad\Longleftrightarrow\quad n\geq\lceil jd_i/r\rceil.
\]
At other primes it means ordinary regularity. The codimension-one
test on the normal schemes proves \eqref{bl:lift:invariant-ideals}, also
for negative \(j\). The identification with the pullback divisorial
sheaf can be checked on the local product charts of \(P\to V\).
Finite pushforward is exact, and averaging over \(\mu_r\) is exact
in characteristic zero. Hence
\begin{equation}\label{bl:lift:invariant-quotients}
\bigl(\pi_*(I^j/I^d)\bigr)^{\mu_r}
=p^*(J_j^V/J_d^V) \qquad(j<d).
\end{equation}
\smallskip\noindent
\emph{Descending the lifting maps.}
To prove \eqref{bl:lift:descent-surjection}, let \(U\subset T\) be
affine and put \(B_U=B\times_T U\). The morphism \(B\to T\) is
affine, so \(B_U\) is affine. Define the open neighborhood
\[
V_U=V\setminus\bigl(S\setminus f^{-1}(U)\bigr),\qquad
P_U=p^{-1}(V_U),\qquad Z_U=\pi^{-1}(P_U).
\]
The complement defining \(V_U\) is closed in \(S\), hence in \(V\).
Moreover \(E\cap Z_U=g^{-1}(B_U)\) as topological spaces.
Since the relevant quotients are supported on \(E\), the
affine-section assertion of Proposition~\ref{bl:prop:lifting} gives
\[
\Gamma(Z_U,I^j/I^d)\longrightarrow
\Gamma(Z_U,I^j/I^{j+1})
\quad\text{surjectively}.
\]
Taking \(\mu_r\)-invariants and using
\eqref{bl:lift:invariant-quotients} gives a surjection between sections
over \(P_U\) of the corresponding pulled-back quotient sheaves.
These are canonically linearized pullbacks: their linearizations in
\eqref{bl:lift:invariant-ideals} are the ones obtained by descending
meromorphic functions.
We next take Laurent degree zero for the frame action. For a coherent
sheaf \(\mathcal A\) on \(V_U\), the affine projection of the
nonzero-vector bundle gives
\begin{equation}\label{bl:lift:laurent-descent}
p_*p^*\mathcal A
=\bigoplus_{n\in\Z}\mathcal A\otimes L^{-n}.
\end{equation}
The degree \(n\) term has pullback weight \(rn\). Sections commute
with this direct sum on the quasi-compact separated open \(V_U\);
this also follows from a finite affine cover. The preceding
surjection respects the displayed grading. Its degree-zero map is
therefore surjective, and is exactly
\[
\Gamma(V_U,J_j^V/J_d^V)\longrightarrow
\Gamma(V_U,\mathcal B_j).
\]
These are the sections on \(U\) of the two additive pushforwards in
\eqref{bl:lift:descent-surjection}. Affine opens form a basis, proving
that assertion.
\smallskip\noindent
\emph{Counting the descended layers.}
The sheaves \(Q_j\) are coherent, since \(f\) is projective.
Ceiling arithmetic and the fixed identification \(L|_S=f^*J\) give
\begin{equation}\label{bl:lift:periodicity}
J_{j-r}^V=J_j^V\otimes L,\qquad
\mathcal B_{j-r}=\mathcal B_j\otimes f^*J,\qquad
Q_{j-r}=Q_j\otimes J.
\end{equation}
Also \(J_0^V=\OO_V\) and \(J_1^V=\OO_V(-S)\), so
\(\mathcal B_0=\OO_S\) and \(Q_0=f_*\OO_S\ne0\), because
\(S\ne\varnothing\).
Set
\[
\mathcal A_N=f_*(J_{-Nr}^V/J_0^V).
\]
By \eqref{bl:lift:descent-surjection} and left exactness, for each
\(-Nr\leq j<0\) there is an exact sequence of additive sheaves
\[
0\longrightarrow f_*(J_{j+1}^V/J_0^V)
\longrightarrow f_*(J_j^V/J_0^V)
\longrightarrow Q_j\longrightarrow0.
\]
Thus \(\mathcal A_N\) has a finite filtration whose coherent
quotients, by \eqref{bl:lift:periodicity}, are precisely
\begin{equation}\label{bl:lift:positive-layers}
Q_t\otimes J^n,\qquad 0\leq t<r,\quad 1\leq n\leq N.
\end{equation}
The middle additive sheaves need not be coherent \(\OO_T\)-modules;
the following estimates use only these exact sequences and their
coherent quotients.
Choose a common integer \(n_0\geq1\) such that, for all
\(0\leq t<r\) and \(n\geq n_0\), Serre vanishing and generation
give
\[
H^i(T,Q_t\otimes J^n)=0\ (i>0),\qquad
Q_t\otimes J^n\text{ is generated by global sections}.
\]
Write \(\ell=\dim T\). The cohomology sequences of the finite
filtration show that all cohomology of \(\mathcal A_N\) is
finite-dimensional, that it vanishes in degrees greater than \(\ell\),
and, for \(i>0\), that
\begin{equation}\label{bl:lift:higher-bound}
h^i(T,\mathcal A_N)
\leq\sum_{t=0}^{r-1}\sum_{n=1}^{n_0-1}
h^i(T,Q_t\otimes J^n)=:C_i.
\end{equation}
The constants \(C_i\) are independent of \(N\). Euler
characteristic is additive for the same exact sequences, whence
\[
\chi(T,\mathcal A_N)
=\sum_{t=0}^{r-1}\sum_{n=1}^N\chi(T,Q_t\otimes J^n).
\]
For \(n\geq n_0\) all summands are nonnegative, and the summand
with \(t=0\) is at least one: a nonzero globally generated sheaf
has a nonzero global section. Therefore
\(\chi(T,\mathcal A_N)\to+\infty\). Together with
\eqref{bl:lift:higher-bound}, this implies
\(h^0(T,\mathcal A_N)\to\infty\). This argument also covers
zero-dimensional support of the \(Q_t\), and \(\ell=0\): an
individual twist need not grow, since the number of positive layers
already grows with \(N\).
Finally \(J_{-Nr}^V=\OO_V(NG)\), and the definition of the
additive pushforward identifies
\[
H^0(T,\mathcal A_N)
=H^0\bigl(V,\OO_V(NG)/\OO_V\bigr).
\]
This proves the claimed growth.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{bl:prop:complex-supported}]
Apply Lemma~\ref{bl:lem:descent-growth}. From the exact sequence
\[
0\longrightarrow\OO_V\longrightarrow\OO_V(NG)
\longrightarrow\OO_V(NG)/\OO_V\longrightarrow0
\]
we obtain
\[
h^0(V,\OO_V(NG))
\geq h^0\bigl(V,\OO_V(NG)/\OO_V\bigr)-h^1(V,\OO_V).
\]
The right side is unbounded. In particular some \(\OO_V(NG)\)
has two linearly independent sections. On the integral variety \(V\),
their ratio is a nonconstant rational function, so the corresponding
linear system has positive-dimensional image. Hence
\(\kappa(V,G)>0\). The actual linear equivalence
\(G\sim qA\) yields \(\kappa(V,A)=\kappa(V,G)>0\), as required.
\end{proof}
==== END Lifting-sections-from-the-reduced-support-of-an-adjoint//lifting.tex ====
==== BEGIN Lifting-sections-from-the-reduced-support-of-an-adjoint//main.tex ====
\documentclass[11pt]{article}
\input{preamble}
\hypersetup{pdftitle={Lifting sections from the reduced support of an adjoint},pdfauthor={OpenAI},bookmarksdepth=2}
\title{Lifting sections from the reduced support\\of an adjoint}
\author{OpenAI}
\date{September 27, 2026}
\begin{document}
\maketitle
\begin{abstract}
For a projective \(\Q\)-factorial dlt pair with effective rational boundary
over an algebraically closed field of characteristic zero, we prove that the
adjoint has positive Iitaka dimension whenever a nonzero effective Cartier
multiple is supported on the coefficient-one boundary and restricts to a
semiample line bundle on its whole reduced support. This gives log abundance
after nonvanishing in dimension at most four over \(\C\).
\end{abstract}
\setcounter{tocdepth}{1}
\begingroup\small\tableofcontents\endgroup
\clearpage
\input{sections/introduction}
\input{sections/covers}
\input{sections/hodge}
\input{sections/frames}
\input{sections/lifting}
\input{sections/abundance}
\input{sections/fields}
\appendix
\input{sections/curve-base}
\input{bibliography}
\end{document}
==== END Lifting-sections-from-the-reduced-support-of-an-adjoint//main.tex ====
==== BEGIN Lifting-sections-from-the-reduced-support-of-an-adjoint//preamble.tex ====
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