==== TABLE OF CONTENTS ====
Minimal-metrics-and-interior-injectivity-for-nef-adjoints//adjoint-metrics.tex
Minimal-metrics-and-interior-injectivity-for-nef-adjoints//bibliography.tex
Minimal-metrics-and-interior-injectivity-for-nef-adjoints//euler-route.tex
Minimal-metrics-and-interior-injectivity-for-nef-adjoints//interior-injectivity.tex
Minimal-metrics-and-interior-injectivity-for-nef-adjoints//introduction.tex
Minimal-metrics-and-interior-injectivity-for-nef-adjoints//main.tex
Minimal-metrics-and-interior-injectivity-for-nef-adjoints//preamble.tex
===========================
==== BEGIN Minimal-metrics-and-interior-injectivity-for-nef-adjoints//adjoint-metrics.tex ====
\section{Zero Lelong numbers for nef klt adjoints}
\label{sec:adjoint-metrics}
We prove \Cref{thm:adjoint-metric}. The construction uses sections
only after one fixed twist pulled back from an ample bundle on the
normal base. Its central estimate must reduce a
normalized section's mass without changing its value at a point of
positive Lelong number. We first recall the top-degree $L^2$ estimate,
then construct the normalized extremal sections. The affine localization
lemma turns their small sublevel mass into the required competitor.
\subsection{Weights and the complete \texorpdfstring{$L^2$}{L2} estimate}
A local weight $\varphi$ of a Hermitian metric gives the squared norm
$|\xi|^2e^{-\varphi}$ in its local frame. Semipositive curvature means
that the local weights are plurisubharmonic. A metric on a rational
line bundle is defined by taking a positive integral tensor power and
dividing its weights by that integer. All identifications below are
identifications of rational line bundles, rather than of numerical
classes.
If a weight $\eta$ is a weight on $K_V$, the expression $e^\eta$ denotes
a density: in a canonical coordinate frame it is multiplied by the
corresponding coordinate volume. Likewise, for an $F$-valued top form
$v$, the expression $|v|^2e^{-\psi}$, with $\psi$ a weight on $F$,
denotes a density. These densities are invariant under changes of
coordinates and frames. In particular, the integral of the latter
does not depend on an auxiliary K\"ahler metric.
We shall use the following form of the complete $L^2$ theorem. The
matrix observation in its statement will be responsible for the
uniform constants.
\begin{lemma}\label[lemma]{lem:top-degree-l2}
Let $(M,\omega_M)$ be a complete K\"ahler manifold of dimension $n$,
and let $F$ be a holomorphic line bundle with a smooth Hermitian metric
of local weight $\psi$ and strictly positive curvature $\theta$.
Write $\Lambda_{\omega_M}$ for the adjoint of wedging with
$\omega_M$. If $\beta$ is an
$F$-valued, $\dbar$-closed $(n,1)$ form which is square integrable and
satisfies
\[
I(\beta)=\int_M
\left\langle[\theta,\Lambda_{\omega_M}]^{-1}\beta,
\beta\right\rangle_{\psi,\omega_M}\,dV_{\omega_M}<\infty,
\]
then there is an $F$-valued $(n,0)$ form $v$ with
\[
\dbar v=\beta,\qquad \int_M |v|^2e^{-\psi}\le I(\beta).
\]
In local coordinates, the inverse-curvature density in $I(\beta)$ is
the inverse Hermitian curvature matrix acting on the remaining
$(0,1)$ covector, multiplied by the coordinate top-form density. It
is independent of the complete background metric. Consequently:
\begin{enumerate}[label=\textup{(\roman*)}]
\item if $\theta\ge\omega_0>0$, this density is bounded by the
$(n,1)$ norm density computed with $\omega_0$;
\item if $g$ is a smooth real function and
$\theta\ge a\,i\partial g\wedge\bar\partial g$ with $a>0$,
the inverse-curvature squared norm of $\bar\partial g$ is at most
$a^{-1}$.
\end{enumerate}
\end{lemma}
\begin{proof}
The existence statement is the complete K\"ahler $L^2$ theorem with
positive curvature operator and finite inverse-curvature integral
\cite[Theorem~5.1, p.~33]{Demailly2012}. It requires pointwise
positivity and the displayed integral, not a uniform lower spectral
bound relative to $\omega_M$.
For the density calculation, write a top-form-valued $(0,1)$ form as
$\sum_j\beta_j\,dz_1\wedge\cdots\wedge dz_n\wedge d\bar z_j$.
The determinant in the squared norm of the top form cancels the volume
determinant. Diagonalizing the curvature relative to $\omega_M$
shows that applying the inverse commutator leaves
$\sum_{j,k}(\theta^{-1})^{j\bar k}\beta_j\overline{\beta_k}$
against the coordinate volume and line weight. All formulas use the
same standard normalization of forms and volume. Matrix order gives
$\theta^{-1}\le\omega_0^{-1}$ in (i). For (ii), conjugate the
rank-one inequality $a\,\partial g\otimes\overline{\partial g}
\le\theta$ by $\theta^{-1/2}$.
\end{proof}
\subsection{The fixed twist and normalization}
The zero-dimensional case is
immediate, so put $n=\dim V>0$ and fix a K\"ahler form $\omega$ on
$V$. Choose compatible canonical divisors
and write
\begin{equation}\label{eq:adjoint-boundary}
N=K_V+B.
\end{equation}
The support of $B$ is simple normal crossing, every coefficient is
strictly less than one, and the negative part $B^-$ is exceptional over
$H$. The last assertion follows because the nonexceptional
coefficients are those of the effective boundary $\Theta$.
Let $b=[B]$ be the divisor weight, a rational linear combination of
logarithms of squared absolute values of local divisor equations. In
crossing coordinates it has the form
\[
b=\sum_{j=1}^{a}\beta_j\log|z_j|^2+O(1),\qquad \beta_j<1.
\]
Thus $e^{-b}$ is locally integrable. Negative coefficients cause no
failure of this integrability statement.
Fix a very ample Cartier divisor $H_0$ on $H$, and set
$A_H=(n+1)H_0$ and $A=\pi^*A_H$. Give $A$ a smooth semipositive
metric with weight $a$. For every positive integer $m$ for which
$mD_H$ is Cartier, the line bundle $mD_H+A_H$ is globally generated.
Indeed, for $1\le j\le n$,
\[
mD_H+A_H-jH_0-(K_H+\Theta)
=(m-1)D_H+(n+1-j)H_0
\]
is ample. Klt Kawamata--Viehweg vanishing, in the form of
\cite[Theorem~3.2]{Fujino2011}, gives the cohomology vanishings for
Castelnuovo--Mumford regularity. The global-generation conclusion of
\cite[Theorem~1.8.5(i)]{Lazarsfeld2004I} then applies. Pullback proves
that $mN+A$ is globally generated as well.
Since $N$ is nef, it is pseudo-effective. Choose a semipositive metric
$\phi$ on $N$ with minimal singularities. Such a metric is obtained
as the upper envelope, relative to a smooth reference weight, of the
normalized plurisubharmonic weights; see
\cite[Definition~1.4 and Theorem~1.5]{DPS2001}. Its defining property is
that every other semipositive weight $\psi$ on $N$ satisfies
\begin{equation}\label{eq:minimal-weight-comparison}
\psi\le\phi+C_\psi
\end{equation}
for a constant $C_\psi$. We shall show that $\phi$ has zero Lelong
numbers.
Suppose, to the contrary, that its Lelong number at $p\in V$ is
positive. For each allowed $m$ define
\begin{equation}\label{eq:extremal-gauge}
Q_m(s)=\int_V |s|^{2/m}e^{-a/m-b},
\qquad s\in H^0(V,mN+A).
\end{equation}
The integrand is a canonical density by \Cref{eq:adjoint-boundary}.
The integral is finite, continuous in $s$, positive for $s\ne0$, and
homogeneous of degree $2/m$. These facts also show that $Q_m(s)=1$ is
a compact subset of the finite-dimensional section space: restrict
$Q_m$ first to the unit sphere of any vector-space norm and use its
positive minimum. Choose $s_m$ maximizing the absolute evaluation at
$p$ on this compact set, using any fixed norm on the one-dimensional
fiber for this $m$. Global generation implies
\begin{equation}\label{eq:extremal-value}
Q_m(s_m)=1,\qquad s_m(p)\ne0.
\end{equation}
Put $\tau_m=m^{-1}\log|s_m|^2$, a weight on $N+A/m$.
\begin{claim}\label{clm:extremal-compactness}
After passage to an unbounded subsequence of allowed integers, the
weights $\tau_m-a/m$ are uniformly locally bounded above and converge
locally in $L^1$ and almost everywhere to a semipositive weight
$\tau_\infty$ on $N$. This compactness assertion holds for every
sequence of sections with $Q_m(s_m)=1$; it does not require their
extremality at $p$.
\end{claim}
\begin{proof}
Choose a smooth reference weight $\phi_{\rm ref}$ on $N$ and write
$u_m=\tau_m-a/m-\phi_{\rm ref}$. These are global quasi-psh functions
whose complex Hessians have one common lower bound $-C\omega$ on the
compact connected manifold $V$. Normalize them by subtracting
$M_m=\sup_Vu_m$. The compactness theorem for sup-normalized quasi-psh
functions \cite[Proposition~2.7]{GuedjZeriahi2005} gives, along a
subsequence, local $L^1$ and almost-everywhere
convergence of $u_m-M_m$ to a quasi-psh function $u_\infty$ that is
finite almost everywhere. In particular the normalized functions do
not converge identically to $-\infty$.
The fixed density $e^{\phi_{\rm ref}-b}$ is integrable. Since
$u_m-M_m\le0$, dominated convergence gives
\[
J_m:=\int_V e^{u_m-M_m}e^{\phi_{\rm ref}-b}
\longrightarrow
J_\infty:=\int_V e^{u_\infty}e^{\phi_{\rm ref}-b}\in(0,\infty).
\]
The positivity uses finiteness of $u_\infty$ almost everywhere, not a
pointwise lower bound. The normalization in
\Cref{eq:extremal-value} says $e^{M_m}J_m=1$. Thus $M_m$ converges to
a finite real number, proving the asserted upper bounds and
convergence without the sup normalization. Finally,
$i\partial\bar\partial(\tau_m-a/m)\ge
-m^{-1}i\partial\bar\partial a$; the error tends to zero. The limit
weight is therefore semipositive.
\end{proof}
Define the globally scalar difference, almost everywhere,
\[
g_m=\phi+a/m-\tau_m.
\]
Its values on the zero divisor of $s_m$ may be assigned arbitrarily
when taking integrals; that divisor has measure zero.
By \Cref{eq:minimal-weight-comparison},
$\tau_\infty\le\phi+C_\infty$. Choose once and for all
$R>C_\infty+1$. The preceding convergence and domination imply
\begin{equation}\label{eq:extremal-tail}
\varepsilon_m:=\int_{\{g_m<-R\}}e^{\tau_m-a/m-b}
\longrightarrow0.
\end{equation}
Indeed, the indicators tend to zero almost everywhere, because
$\phi-\tau_\infty\ge-C_\infty$ where both weights are finite; the
densities have one integrable majorant on a fixed finite coordinate
cover.
It is now enough to construct sections $w_m\in H^0(V,mN+A)$ such that
\[
w_m(p)=s_m(p),\qquad Q_m(w_m)^m\le C_0\varepsilon_m,
\]
where $C_0$ is independent of $m$. For large $m$ this gives
$Q_m(w_m)<1$, so scalar normalization would produce a section of
gauge one with larger absolute evaluation at $p$. We next construct
these competitors and prove the two displayed properties.
\subsection{A singular weighted equation with a fixed constant}
We will cut off $s_m$ on $\{g_m<-R\}$ and correct the resulting
$\dbar$ error. The weight for this equation must serve two purposes.
Its singularity near $p$ must force the correction to vanish at $p$
after the holomorphic competitor has been extended. Its growth where $g_m$
is positive must also allow a uniform conversion from the quadratic
estimate to $Q_m$.
The first requirement determines its slope $c$ near $-\infty$; a fixed
upper slope bound $d$ will give the second.
Positive Lelong number gives local coordinates at $p$ and a sufficiently
small fixed $\alpha>0$ with
$\phi(z)\le\alpha\log|z|^2+O(1)$. On a cone of positive angular
measure on which all crossing coordinates have modulus comparable to
$|z|$, the weight $e^{-c\phi-b}$ is bounded below by a positive
constant times
$|z|^{-2(c\alpha+\sum\beta_j)}$. Choose a fixed $c>0$ large enough
that $c\alpha+\sum\beta_j\ge n$. Then
\begin{equation}\label{eq:nonintegrable-weight}
e^{-c\phi-b}\quad\hbox{is not locally integrable at }p.
\end{equation}
This choice is valid even if some $\beta_j$ are negative.
Choose a smooth function $0\le\chi\le1$ equal to one on
$(-\infty,-R-1]$ and to zero on $[-R,\infty)$. Choose a smooth
convex increasing function $f$ which is affine of slope $c$ near
$-\infty$, has $f''>0$ on the closed transition interval
$[-R-1,-R]$, and satisfies $c\le f'\le d$ for one fixed $d$. Such a
function is obtained by prescribing a nonnegative compactly supported
second derivative positive on that interval. In particular there is
a fixed constant $C_f$ with
\begin{equation}\label{eq:convex-weight-bound}
f(t)\le d\max(0,t)+C_f\qquad(t\in\R).
\end{equation}
All of $R,c,\chi,f,d,C_f$ remain fixed as $m$ varies.
For a fixed $m$, choose a dense smooth affine open $U=U_m\subset V$
avoiding the supports of $B$ and $\operatorname{div}(s_m)$. This open
set need not contain $p$. We will recover the value at $p$ only after
extending the resulting holomorphic section to all of $V$. On $U$
the function $g=g_m$ is psh, since the weight $\tau_m$ is locally
pluriharmonic there and the metrics $\phi,a$ are semipositive. Put
\begin{equation}\label{eq:local-adjoint-weight}
F_m=mN+A-K_V,\qquad
S=(m-1)\tau_m+a/m+b.
\end{equation}
This is a smooth semipositive weight on $F_m|_U$. Divisor weights and
$\tau_m$ are pluriharmonic on $U$, so its curvature is
$m^{-1}i\partial\bar\partial a$. We regard $s_m$ as an
$F_m$-valued top form.
\begin{lemma}[Localization of top forms]\label[lemma]{mm:lem:localization}
Let $U$ be a smooth connected affine complex variety of dimension $n\ge1$.
Let $F$ be a holomorphic line bundle on $U$ with a smooth semipositive
metric of weight $S$, let $g$ be a global plurisubharmonic function
not identically $-\infty$, and let $s$ be a holomorphic $F$-valued
$(n,0)$-form. Fix $R\in\R$, a smooth cutoff $0\le\chi\le1$ equal
to one for $t\le-R-1$ and zero for $t\ge-R$, and a smooth convex
function $f$ that is affine of positive slope $c$ near $-\infty$,
with $c\le f'\le d$ for some $d$ and $f''>0$ on $[-R-1,-R]$.
If
\[
M=\int_{\{g<-R\}}|s|^2e^{-S}<\infty,
\]
there is a measurable $F$-valued top form $u$ such that
\begin{equation}\label{mm:eq:localization-estimate}
\dbar u=\dbar(\chi(g)s),\qquad
\int_U|u|^2e^{-S-f(g)}\le C_{\chi,f}M,
\end{equation}
where the equation is distributional and
\begin{equation}\label{mm:eq:localization-constant}
C_{\chi,f}=\max_{[-R-1,-R]}
\frac{|\chi'|^2e^{-f}}{f''}.
\end{equation}
The constant is independent of $U,F,S,g,s$ and of the auxiliary
complete K\"ahler metrics. At $g=-\infty$ we set $\chi(g)=1$ and
$f(g)=-\infty$; that locus has measure zero.
\end{lemma}
\begin{proof}
Embed $U$ as a closed complex submanifold of some $\C^a$ and let
$\rho=|z|^2|_U$. The tubular-neighborhood theorem supplies a
holomorphic retraction of a neighborhood onto $U$
\cite{DocquierGrauert1960}; see also the formulation in
\cite[Theorem~3.1]{Forstneric2010}. Compose $g$ with the retraction.
Radial convolution in that neighborhood, at radii decreasing to zero,
gives smooth psh functions $g_j\downarrow g$ on common smaller
domains. More precisely, choose increasing regular exhaustion values
$a_j\to\infty$ and $\Omega_j=\{\rho<a_j\}$. Use one fixed nonnegative smooth radial averaging kernel of total mass
one. The convolution radius at stage $j$ is at most $1/j$, is small
enough to work near $\overline{\Omega_j}$, and is smaller than all
previous radii. Radial
averages of a psh function are monotone in the radius, so these choices
give monotonicity wherever two of the functions are defined. They
also give $g_j\ge g$.
Each $\Omega_j$ has a complete K\"ahler metric dominating
$i\partial\bar\partial\rho$, for example the metric with potential
$\rho-\log(a_j-\rho)$. For a decreasing positive sequence
$\delta_j\to0$, use the line weight
\[
P_j=S+f(g_j)+\delta_j\rho.
\]
It is smooth near $\overline{\Omega_j}$, its curvature is strictly
positive, and
\[
i\partial\bar\partial P_j
\ge f''(g_j)i\partial g_j\wedge\bar\partial g_j.
\]
The smooth closed datum
$\beta_j=\chi'(g_j)\bar\partial g_j\wedge s$ is square
integrable for the complete metric. To see this even near the boundary
of $\Omega_j$, use the top-degree density calculation: enlarging the
background metric decreases the $(n,1)$ density, and all coefficients
and line weights are smooth on a neighborhood of the relatively
compact closure.
By \Cref{lem:top-degree-l2}, the inverse-curvature integral is at most
\[
C\int_{\Omega_j\cap\{g_j<-R\}} |s|^2e^{-S}
\le C\int_{\{g<-R\}}|s|^2e^{-S},\qquad
C=\sup_{[-R-1,-R]}\frac{|\chi'|^2e^{-f}}{f''}.
\]
The nonnegative term $\delta_j\rho$ only decreases the integrand.
The last integral is $M$ by definition.
Thus we obtain solutions $u_j$ on $\Omega_j$ with
$\int_{\Omega_j}|u_j|^2e^{-P_j}\le CM$. The background
complete metrics have introduced no comparison factor.
For completeness, the singular passage uses fixed Hilbert spaces.
If $k\le l$ are fixed and $j\ge l$, then on $\Omega_k$ we have
$P_j\le P_l$, and hence
\[
\int_{\Omega_k}|u_j|^2e^{-P_l}\le CM.
\]
Take a diagonal weakly convergent subsequence in these countably many
fixed earlier smooth weighted spaces. Their limits agree as
distributions on overlaps, and define $u$ on $U$. Weak lower
semicontinuity, then monotone convergence as $l\to\infty$, gives
\[
\int_{\Omega_k}|u|^2e^{-S-f(g)}\le CM.
\]
Exhausting $U$ proves the required global inequality. Finally
$\chi(g_j)s\to\chi(g)s$ locally in $L^2$ by bounded convergence,
where $\chi(-\infty)=1$. Thus the equations pass to distributions
and give \Cref{mm:eq:localization-estimate}.
\end{proof}
We apply the lemma to the adjoint data already fixed on $U$.
The exact density identity is
\begin{equation}\label{eq:normalized-top-density}
|s_m|^2e^{-S}=e^{\tau_m-a/m-b},
\end{equation}
so its sublevel mass is $M=\varepsilon_m$. Applying
\Cref{mm:lem:localization} with $F=F_m$, $g=g_m$, and $s=s_m$ gives an
$F_m$-valued top form $u$ on $U$ with
\begin{equation}\label{eq:singular-cutoff-estimate}
\dbar u=\dbar(\chi(g)s_m),\qquad
\int_U |u|^2e^{-S-f(g)}\le C\varepsilon_m,
\end{equation}
where $C=C_{\chi,f}$ is independent of $m$ and of $U$.
\subsection{Extension, evaluation, and the extremal contradiction}
For $m>d+1$ set
\[
G=S+d\max(0,g),\qquad w_m=\chi(g)s_m-u\quad\hbox{on }U.
\]
\Cref{eq:singular-cutoff-estimate} shows that $w_m$ is holomorphic. Since $G\ge S$, the cutoff term has $G$-norm squared at
most $\varepsilon_m$. Since $S+f(g)\le G+C_f$, the $G$-norm squared
of $u$ is at most $e^{C_f}C\varepsilon_m$. Therefore
\begin{equation}\label{eq:competitor-norm}
\int_U |w_m|^2e^{-G}\le C'\varepsilon_m,
\qquad C'=2+2e^{C_f}C.
\end{equation}
This constant is independent of $m$.
To extend the section, rewrite its weight as
\begin{equation}\label{eq:extension-weight}
G=b+a/m+(m-1-d)\tau_m
+d\max\{\tau_m,\phi+a/m\}.
\end{equation}
It is locally bounded above away from $\Supp B^-$: the coefficient
$m-1-d$ is positive, the psh weights in the last two terms are locally
bounded above, and $b$ has that property away from its negative
components. Consequently \Cref{eq:competitor-norm} gives ordinary
local $L^2$ bounds there. Holomorphic $L^2$ removal across analytic
sets extends $w_m$ to $V\setminus\Supp B^-$.
There is an open $H^\circ\subset H$, with complement of codimension
at least two, over which $\pi$ is an isomorphism. It misses the image
of $B^-$. The section just obtained descends on $H^\circ$ to a
section of the actual Cartier bundle
$\cO_H(mD_H+A_H)$. Normality extends it across $H\setminus
H^\circ$: after trivializing that line bundle, this is the Hartogs
extension of regular holomorphic functions on a normal space.
Pullback now gives a global section, again denoted $w_m$, of $mN+A$.
By projectivity and GAGA it is an algebraic global section as well.
It agrees with the constructed section on $U$ by the identity theorem.
In particular \Cref{eq:competitor-norm} continues to concern the same
section. No estimate across the negative exceptional divisor was
needed for this extension.
Near $p$, the weight $\tau_m$ is smooth by
\Cref{eq:extremal-value}, whereas $g\to-\infty$. Thus $\chi(g)=1$
and $f(g)=cg+\text{constant}$ near $p$ on $U$, so
\[
u=s_m-w_m,\qquad S+f(g)=b+c\phi+O_m(1).
\]
If $s_m(p)-w_m(p)$ were nonzero, its holomorphic coefficient would
have a positive lower bound on a smaller neighborhood. The finite
weighted integral in \Cref{eq:singular-cutoff-estimate} would then
contradict \Cref{eq:nonintegrable-weight}. Removing the analytic
null set $V\setminus U$ does not change this divergence. This also
applies when $p$ lies on $B^-$, since both sections are now holomorphic
at $p$. We conclude
\begin{equation}\label{eq:competitor-value}
w_m(p)=s_m(p)\ne0.
\end{equation}
The bounded factor denoted $O_m(1)$ is used only to test integrability
for a fixed $m$; it never enters \Cref{eq:competitor-norm}.
It remains to convert that quadratic estimate to the original gauge.
H\"older's inequality gives
\begin{equation}\label{eq:gauge-holder-first}
Q_m(w_m)^m\le C'\varepsilon_m
\left(\int_U e^{(G-a-mb)/(m-1)}\right)^{m-1}.
\end{equation}
There is no unbounded factor hidden in the second integral. Put
$\eta=d/(m-1)\in(0,1)$. Direct expansion yields
\begin{equation}\label{eq:canonical-density-mixture}
\frac{G-a-mb}{m-1}
=(1-\eta)(\tau_m-a/m-b)
+\eta\bigl(\max\{\tau_m-a/m,\phi\}-b\bigr).
\end{equation}
Both exponentials on the right are canonical densities. The first
has integral one; the second has integral at most $1+I$, where
\[
I=\int_V e^{\phi-b}<\infty.
\]
Finiteness follows from local upper bounds for $\phi$ and local
integrability of $e^{-b}$ on a fixed finite coordinate cover.
Applying H\"older to \Cref{eq:canonical-density-mixture} gives
\[
\left(\int_U e^{(G-a-mb)/(m-1)}\right)^{m-1}
\le (1+I)^{\eta(m-1)}=(1+I)^d.
\]
All integrals are unchanged by the omitted analytic null set.
Together with \Cref{eq:extremal-tail,eq:gauge-holder-first}, this proves
\[
Q_m(w_m)^m\le C'\varepsilon_m(1+I)^d\longrightarrow0.
\]
For large $m$, therefore, $0<Q_m(w_m)<1$. Multiply $w_m$ by
$Q_m(w_m)^{-m/2}>1$. The new section has gauge one and, by
\Cref{eq:competitor-value}, strictly larger absolute evaluation than
$s_m$. This contradicts the choice of $s_m$, regardless of the rate
at which $s_m(p)$ or $\varepsilon_m$ tends to zero. Hence $\phi$ has
no positive Lelong number at any point. This proves
\Cref{thm:adjoint-metric}.
\begin{corollary}\label[corollary]{cor:minimal-multiplier}
Under the hypotheses of \Cref{thm:adjoint-metric}, let $\phi$ be a
minimal weight on the actual rational line bundle $N$. Then
\[
\mathcal I(t\phi)=\cO_V\qquad\text{for every real }t>0,
\]
where the ideal is defined by local integrability of $|f|^2e^{-t\phi}$.
The conclusion is independent of the choice of minimal metric and of
the Cartier multiple used to normalize its weights.
\end{corollary}
\begin{proof}
Skoda's integrability theorem gives local integrability of
$e^{-t\phi}$ for every fixed $t>0$ when the Lelong numbers vanish
\cite[Proposition~7.1]{Skoda1972}; for this formulation for arbitrary
plurisubharmonic weights, see \cite[Property~1.4(8)]{DemaillyKollar2001}
and \cite[Lemma~5.6(a)]{Demailly2012}. Every holomorphic germ is bounded
on a smaller neighborhood, so the multiplier ideal is the structure
sheaf. Two metrics with minimal singularities have weights differing
by a bounded function, directly from their defining comparison.
Dividing the weight of a positive Cartier multiple by that multiple
preserves the stated normalization. These observations prove the
independence assertions.
\end{proof}
==== END Minimal-metrics-and-interior-injectivity-for-nef-adjoints//adjoint-metrics.tex ====
==== BEGIN Minimal-metrics-and-interior-injectivity-for-nef-adjoints//bibliography.tex ====
\begin{thebibliography}{99}
\raggedright
\bibitem{BermanDemailly2012}
R.~Berman and J.-P.~Demailly,
\emph{Regularity of plurisubharmonic upper envelopes in big cohomology classes},
in \emph{Perspectives in Analysis, Geometry, and Topology},
Progress in Mathematics, vol.~296, Birkh\"auser/Springer, 2012, 39--66.
\href{https://doi.org/10.1007/978-0-8176-8277-4_3}{doi:10.1007/978-0-8176-8277-4\_3}.
Locators refer to \href{https://arxiv.org/abs/0905.1246v1}{arXiv:0905.1246v1}.
\bibitem{DPS2001}
J.-P.~Demailly, T.~Peternell, and M.~Schneider,
\emph{Pseudo-effective line bundles on compact K\"ahler manifolds},
International Journal of Mathematics \textbf{12} (2001), no.~6, 689--741.
Locators refer to \href{https://arxiv.org/abs/math/0006205v2}{arXiv:math/0006205v2}.
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==== END Minimal-metrics-and-interior-injectivity-for-nef-adjoints//bibliography.tex ====
==== BEGIN Minimal-metrics-and-interior-injectivity-for-nef-adjoints//euler-route.tex ====
\section{Abundance for fourfolds with nonzero Euler characteristic}
\label{sec:euler-route}
The metric theorem gives abundance for klt fourfolds when
the Euler characteristic is nonzero. A published nonvanishing
criterion uses this Euler hypothesis to produce the first
section; a second criterion then gives semiampleness. We state
both inputs in terms of the actual adjoint bundle on a resolution,
so that the same zero-Lelong metric supplies their analytic
hypotheses.
\begin{corollary}\label[corollary]{cor:euler-abundance}
Let $(X,\Delta)$ be a projective complex klt fourfold with effective
rational boundary and nef adjoint $D=K_X+\Delta$.
If $\chi(X,\cO_X)\ne0$, then $D$ is semiample.
\end{corollary}
This proof combines \Cref{thm:adjoint-metric} with published
nonvanishing and semiampleness criteria. It isolates the use of
$\chi(X,\cO_X)\ne0$: that condition enters the nonvanishing
criterion, whereas the metric theorem has no Euler-characteristic
hypothesis. Related work on multiplier-ideal approximations and
supercanonical currents studies this analytic setting
\cite[Theorems~A--B]{Lazic2024Metrics}.
\subsection*{The two algebraic inputs}
For a $\mathbb Q$-Cartier divisor $P$ on a normal projective variety,
the metric condition in \cite[Definition~2.12]{LazicPeternell2020}
requires a positive integer $r$, with $rP$ Cartier, and a resolution
$f:Z\to X$ carrying a singular metric on $f^*\mathcal O_X(rP)$ whose
curvature has the form
\begin{equation}\label{mm:eq:generalized-algebraic}
T=T_0+\sum_j a_j[E_j],\qquad
T_0\geq0,\quad \nu(T_0,z)=0\ (z\in Z),\quad a_j\in\mathbb Q_{\geq0}.
\end{equation}
The divisor sum is finite. This is called a metric with
\emph{generalized algebraic singularities}. The sum may be zero;
in particular, every semipositive metric with zero Lelong numbers
everywhere meets this definition.
\begin{theorem}[Lazi\'c--Peternell, fourfold nonvanishing]
\label{mm:thm:lp-fourfold}
Let $(X,\Delta)$ be a normal projective complex klt pair of dimension
four, with effective rational boundary, and put $D=K_X+\Delta$.
Suppose $D$ is pseudoeffective and $N$ is a nef $\mathbb Q$-Cartier
divisor. If both $D$ and $D+N$ satisfy the metric condition
\eqref{mm:eq:generalized-algebraic} and $\chi(X,\mathcal O_X)\ne0$,
then some rational $t_0>0$ satisfies
\[
\kappa(X,D+tN)\geq0\qquad
\text{for every rational }t\in[0,t_0].
\]
\end{theorem}
This is \cite[Corollary~D]{LazicPeternell2020}, the unconditional
four-dimensional consequence of their Theorem~C(i). Its interval includes
$t=0$. We will take $N=0$, so both metric assumptions concern $D$.
\begin{theorem}[Gongyo--Matsumura, fourfold semiampleness]
\label{mm:thm:gm-fourfold}
Let $(X,\Delta)$ be a normal projective complex klt pair of dimension
four, with effective rational boundary, and set $D=K_X+\Delta$.
Suppose that for a projective birational morphism $f:Z\to X$ with
$Z$ smooth and some positive integer $r$ with $rD$ Cartier, the bundle
$f^*\mathcal O_X(rD)$ carries a semipositive singular Hermitian metric
with zero Lelong numbers at every point of $Z$.
If $\kappa(X,D)\geq0$, then $D$ is semiample.
\end{theorem}
This is the form of \cite[Corollary~5.3]{GongyoMatsumura2017}
used here. Positivity of the Cartier multiple is also explicit in
their Theorem~5.1, from which that corollary follows. Neither of these
fourfold results requires $X$ to be $\mathbb Q$-factorial or
non-uniruled. Their dimension-three inputs are already incorporated
in the stated corollaries.
\subsection*{Application to the minimal metric}
\begin{proof}[Proof of Corollary~\ref{cor:euler-abundance}]
Choose a projective log resolution $\pi:Y\to X$, a positive integer
$r$ such that $rD$ is Cartier, and a metric $h_{\min}$ with minimal
singularities on $L=\pi^*D$. Nefness makes $D$ pseudoeffective,
so such a metric exists. By Theorem~\ref{thm:adjoint-metric},
\[
\nu(h_{\min},y)=0\qquad(y\in Y).
\]
The tensor power $h_{\min}^{\otimes r}$ is a metric on the actual
line bundle $\pi^*\mathcal O_X(rD)$. Its local weights are $r\phi$,
so its curvature is semipositive and its Lelong numbers are
$r\nu(\phi,y)=0$.
Consequently, \eqref{mm:eq:generalized-algebraic} holds with
$T_0$ equal to this curvature current and with zero divisor sum.
Apply Theorem~\ref{mm:thm:lp-fourfold} with $N=0$ and $t=0$ to obtain
\[
\kappa(X,D)\geq0.
\]
This is nonvanishing for $D$ itself: a positive Cartier multiple
of $D$ has a nonzero global section. It is therefore the Kodaira
dimension hypothesis of Theorem~\ref{mm:thm:gm-fourfold}. The same
resolution and the same metric $h_{\min}^{\otimes r}$ supply its
remaining hypothesis, and that theorem proves that $D$ is semiample.
\end{proof}
The argument imposes no restriction on the numerical dimension of $D$.
The Euler-characteristic hypothesis is used only to obtain the first
section; the interior-injectivity theorem of this article is not an
input to this particular application.
==== END Minimal-metrics-and-interior-injectivity-for-nef-adjoints//euler-route.tex ====
==== BEGIN Minimal-metrics-and-interior-injectivity-for-nef-adjoints//interior-injectivity.tex ====
\section{An interior injectivity theorem}
\label{sec:interior-injectivity}
We prove \Cref{thm:interior-injectivity} from its two endpoint
metrics, using the top-degree $L^2$ estimate in
\Cref{lem:top-degree-l2}. The proof does not use
\Cref{thm:adjoint-metric} or its extremal-section construction. The output is the natural map on ordinary coherent cohomology.
The proof therefore needs both a weighted estimate on the complement
of the crossing divisor and a comparison that retains the original
ordinary class. The fixed-cover construction below supplies this
comparison together with a uniform bound for primitives.
For a class in the kernel, we will construct weighted harmonic
representatives whose images have norms tending to zero. The endpoint
metrics make multiplication a contraction and supply the curvature
comparison for this estimate. Local solutions of the $\dbar$ equation
leave holomorphic differences on overlaps; controlling a splitting of
these differences on one fixed cover gives uniformly bounded global
primitives. The extended differences form a cocycle representing the
ordinary cohomology class of the original form. Finally, comparison in
one fixed Hilbert space lets the small images force the original
ordinary class to vanish.
Classical injectivity for roots of divisors, including an additional
reduced boundary, is proved in \cite[Theorem~5.1]{EsnaultViehweg1992}.
The local and uniform-primitive methods have precedents in
\cite[Sections~5.2--5.3]{Matsumura2018}; the precise comparison needed
for the present line bundles is proved here.
\subsection{Endpoint regularization and the comparison weights}
Fix a K\"ahler form $\omega$ on $V$, and put $n=\dim V$. The
zero-dimensional assertion is trivial, so assume $n>0$. We use the
following consequence of regularization and exponential integrability.
\begin{lemma}\label[lemma]{lem:zero-lelong-regularization}
Let $F$ be a rational line bundle on a smooth projective complex
variety $V$ equipped with a K\"ahler form $\omega$, and suppose that
$F$ has a semipositive singular metric $\psi$ having zero Lelong
numbers everywhere. There are smooth weights $\psi_k$ on $F$ and
numbers $\epsilon_k\downarrow0$ such that
\[
i\partial\bar\partial\psi_k\ge-\epsilon_k\omega.
\]
In fixed local frames these weights are uniformly bounded above on
relatively compact coordinate neighborhoods, and, for every fixed
$b>0$, their exponentials $e^{-b\psi_k}$ have uniformly bounded local
$L^1$ norms.
\end{lemma}
\begin{proof}
Choose a smooth reference metric on $F$. Apply the attenuation form of
Demailly's regularization theorem to the specified current
$T=i\partial\bar\partial\psi$, using the global difference between
$\psi$ and the reference weight as its potential
\cite[Main Theorem~1.1]{Demailly1992}. Fix a positive attenuation
level $c$. Adding the reference weight back to the approximating
potentials gives weights $\psi_k$ on the same rational bundle, with
$\psi_k\downarrow\psi$. They are smooth outside the Lelong level set
$E_c(T)=\{x:\nu(T,x)\ge c\}$, which is empty. Their curvature lower bounds have the form
$-\min(\lambda_k,c)u-\delta_k\omega$, where $u$ is one fixed
smooth nonnegative form, $\delta_k\to0$, and the continuous functions
$\lambda_k$ decrease to the Lelong-number function. That function
is zero here. Dini's theorem on the compact manifold gives
$\sup_V\lambda_k\to0$, and $u\le C\omega$ gives the stated
loss $\epsilon_k\to0$. Passing to a subsequence makes the losses
decrease. Work first on a common integral tensor power of $F$ and
divide all weights by that integer afterwards.
On each fixed coordinate neighborhood the decreasing approximants
satisfy $\psi\le\psi_k\le\psi_1$. The upper bound is smooth on a
slightly larger neighborhood, and hence is bounded on the chosen
compact subset. For every fixed $b>0$,
\[
e^{-b\psi_k}\le e^{-b\psi}.
\]
The right side is locally integrable by the zero-Lelong case of Skoda's
integrability criterion \cite{Skoda1972}; see
\cite[Property~1.4(8)]{DemaillyKollar2001} for the formulation for
arbitrary plurisubharmonic weights. This proves the uniform
fixed-exponent bounds directly.
\end{proof}
Apply the lemma simultaneously to $L-C_0$ and $L-C_2$, obtaining
weights $\psi_{0,k},\psi_{2,k}$ with a common sequence
$0<\epsilon_k\le1$ tending to zero. Let
\[
D=\Supp(C_0+C_2)_{\rm red},\qquad U=V\setminus D.
\]
For a rational divisor $C$, write $[C]$ for its divisor weight.
The endpoint weights $\psi_{i,k}$ live on the rational bundles
$L-C_i$. Adding $[C_i]$ gives weights on $L$. We first raise the
weight at the first endpoint above that at the second, so that the
natural inclusion of the eventual integral bundles will be a
contraction. The symbols $\Phi_{i,k}$ below are weights on $L$,
whereas $h_{i,k}$ are weights on $L_i=L-\lfloor C_i\rfloor$.
On $U$ set
\begin{align}
\psi'_{0,k}
&=\log\bigl(e^{\psi_{0,k}}
+e^{\psi_{2,k}+[C_2-C_0]}\bigr),
\label{eq:logsum-endpoint}\\
\Phi_{0,k}&=\psi'_{0,k}+[C_0],\qquad
\Phi_{2,k}=\psi_{2,k}+[C_2],\qquad
\Phi_{1,k}=(1-\lambda)\Phi_{0,k}+\lambda\Phi_{2,k},
\label{eq:interior-weights}\\
h_{i,k}&=\Phi_{i,k}-[\floor{C_i}],\qquad i=0,1.
\label{eq:integral-line-weights}
\end{align}
The two summands in \Cref{eq:logsum-endpoint} are weights on the same
rational line bundle $L-C_0$, so the expression is intrinsically
well-defined. The logarithm of a sum of exponentials preserves the
common curvature lower bound: locally add a potential for
$\epsilon_k\omega$ to both summands and use the psh log-sum
inequality. Moreover $\psi'_{0,k}\ge\psi_{0,k}$. Effectivity of
$C_2-C_0$ gives a uniform local upper bound for its divisor weight, so
$\psi'_{0,k}$ retains both the upper bounds and all the fixed-exponent
integrability bounds of \Cref{lem:zero-lelong-regularization}.
The weights $h_{i,k}$ are smooth metrics on $L_i|_U$. They have
uniform local upper bounds in frames extending across $D$, and
\begin{equation}\label{eq:weight-integral-bounds}
\sup_k\int_{B\cap U}e^{-h_{i,k}}\,d\lambda_B<\infty
\quad(i=0,1)
\end{equation}
on every relatively compact coordinate neighborhood $B$, where
$d\lambda_B$ is coordinate volume. To verify the last assertion,
write
\[
h_{0,k}=\psi'_{0,k}+[C_0-\floor{C_0}],\qquad
h_{1,k}=(1-\lambda)\psi'_{0,k}+\lambda\psi_{2,k}
+[C_1-\floor{C_1}].
\]
Every coefficient of either fractional divisor lies in $[0,1)$.
Choose one H\"older exponent $a>1$ so close to one that its product
with each of these finitely many coefficients is still less than one.
The exponential of the fractional divisor weight is in $L^a$ by the
crossing-coordinate integral. The conjugate H\"older exponent for
the remaining weights is allowed by the fixed-exponent bounds; a
second H\"older inequality handles the two summands in $h_{1,k}$.
This proves \Cref{eq:weight-integral-bounds} uniformly in $k$.
Let $E=\floor{C_1}-\floor{C_0}\ge0$, and let
$s:L_1\to L_0$ be multiplication by the canonical section of $E$.
Since $\Phi_{2,k}\le\Phi_{0,k}$, we have
$\Phi_{1,k}\le\Phi_{0,k}$ and
\begin{equation}\label{eq:multiplication-contraction}
|s|^2e^{h_{1,k}-h_{0,k}}
=e^{\Phi_{1,k}-\Phi_{0,k}}\le1.
\end{equation}
Thus multiplication is a contraction on all form degrees for any one
fixed background metric. On $U$ the curvatures are
$\theta_{i,k}=i\partial\bar\partial\Phi_{i,k}$; divisor weights
contribute no curvature there.
\subsection{A complete metric with bounded local potentials}
The singular set must be removed to use smooth weighted harmonic
forms. We choose the complete metric carefully so that local
solvability still has a uniform comparison with the given weights.
The bounded-potential construction follows the complete-metric method
of \cite[Lemma~3.1]{FujinoInjectivity2012}; we include the calculation
because both bounded potentials and completeness are needed in the
uniform primitive estimate.
\begin{lemma}\label{lem:bounded-potential-complete-metric}
There is a complete K\"ahler metric $\omega_c\ge\omega$ on $U$ such
that, on sufficiently small coordinate neighborhoods in $V$, it has
potentials bounded even on approach to $D$.
\end{lemma}
\begin{proof}
If $D$ is empty, take $\omega_c=\omega$. Otherwise choose a smooth
metric on $\cO_V(D)$ and scale it so that
$t=\log|s_D|_{\rm sm}^2<-e^2$ on $U$. Set $f_0(t)=1/\log(-t)$.
For $x=-t$,
\[
f_0'(t)=\frac1{x(\log x)^2},\qquad
f_0''(t)=\frac1{x^2}\left(\frac1{(\log x)^2}
+\frac2{(\log x)^3}\right).
\]
In particular $f_0'$ is bounded and
$f_0''(t)\ge(t\log(-t))^{-2}$. Off $D$ the form
$i\partial\bar\partial t$ is the negative of a fixed smooth
curvature form, so it is bounded by a multiple of $\omega$. For a
sufficiently large fixed $M$ the form
\[
\omega_c=M\omega+i\partial\bar\partial f_0(t)
\]
is positive and dominates both $\omega$ and
$i\partial t\wedge\bar\partial t/(t\log(-t))^2$.
A path leaving every compact subset of $U$ approaches $D$, because
$V$ is compact; along such an approach $t\to-\infty$. The displayed
radial term bounds its length below by a positive constant times the
total variation of $\log\log(-t)$, which diverges. Hence the metric
is complete, including at crossings of components of $D$. Finally,
if $\omega=i\partial\bar\partial\rho$ on a small coordinate
neighborhood, then $M\rho+f_0(t)$ is a local potential for $\omega_c$.
The smooth function $\rho$ is bounded on a smaller relatively compact
neighborhood, and $f_0(t)$ is bounded and tends to zero at $D$.
\end{proof}
Fix this metric for the rest of the section. When $D\ne\varnothing$,
the function $r=\log\log(-t)$ is a smooth proper exhaustion of $U$
with bounded gradient in $\omega_c$. If $\zeta$ is a smooth function
equal to one on $(-\infty,1]$ and zero on $[2,\infty)$, then
$\zeta(r/R)$, as $R\to\infty$, gives compactly supported smooth
cutoffs tending to one and with gradient $O(R^{-1})$. When $D$ is
empty, the constant cutoff one suffices.
The curvature comparison
following \Cref{eq:interior-weights} becomes
\begin{equation}\label{eq:curvature-interior-comparison}
\theta_{1,k}+\epsilon_k\omega_c
\ge(1-\lambda)(\theta_{0,k}+\epsilon_k\omega_c)\ge0,
\end{equation}
since $\theta_{2,k}+\epsilon_k\omega_c\ge0$ as well.
\subsection{A fixed-cover primitive estimate and the ordinary class map}
Fix a holomorphic line bundle $F$ on $V$. The local $L^2$ theorem
gives primitives on coordinate neighborhoods,
but a global primitive requires their holomorphic differences on
overlaps to split. We use one finite cover and its fixed spaces of
holomorphic sections to make that splitting estimate uniform in the
weights. The extended differences also define a class in the ordinary
group $H^1(V,K_V\otimes F)$. Thus the class map below has ordinary
coherent cohomology as its target, and the primitive estimate shows
that its kernel consists of actual weighted exact forms.
This local-solutions and holomorphic-cocycle comparison follows the
method of \cite[Section~3, Claim~1 and Lemma~3.2]{FujinoInjectivity2012}.
For a smooth metric $h$ on
$F|_U$, write $L^2_h(n,q)$ for the Hilbert space of $F$-valued
$(n,q)$ forms using $h$ and $\omega_c$. The Dolbeault operator is its
maximal distributional realization, and
$Z^1_h=\ker(\dbar:L^2_h(n,1)\to L^2_h(n,2))$.
\begin{proposition}\label[proposition]{prop:uniform-primitives}
Let $F$ be a fixed holomorphic line bundle on $V$, and let $h_k$ be
smooth metric weights on $F|_U$. Suppose that, for a fixed constant
$A_0\ge0$,
\[
i\partial\bar\partial h_k\ge-A_0\omega_c,
\]
and that in local frames on $V$ the weights are uniformly bounded
above and the functions $e^{-h_k}$ have uniformly bounded local
integrals on punctured coordinate neighborhoods. Then:
\begin{enumerate}[label=\textup{(\roman*)}]
\item There is a continuous linear map
\[
\kappa_k:Z^1_{h_k}\longrightarrow H^1(V,K_V\otimes F)
\]
agreeing with the ordinary Dolbeault class on smooth closed forms on
$V$. It annihilates the closure in $L^2_{h_k}(n,1)$ of the image of
$\dbar:L^2_{h_k}(n,0)\to L^2_{h_k}(n,1)$.
\item There is a constant $C$ independent of $k$ such that every
$f\in Z^1_{h_k}$ with $\kappa_k(f)=0$ has a global primitive on $U$
satisfying
\[
\dbar v=f,\qquad \|v\|_{h_k}\le C\|f\|_{h_k}.
\]
No continuous linear choice of these global primitives is asserted.
\end{enumerate}
\end{proposition}
\begin{proof}
Choose a finite collection of coordinate and frame balls
\[
V_j\Subset U_j\Subset\widehat U_j
\]
such that the $V_j$ still cover $V$ and $\omega_c$ has a bounded local
potential $\rho_j$ on $\widehat U_j\cap U$. These sets and potentials
are independent of $k$.
\emph{Local solutions.}
On $U_j\cap U$, replace the line weight by
$h_k+(A_0+1)\rho_j$. Its curvature dominates $\omega_c$, and the
bounded potential compares its top-form norms with those of $h_k$ by
fixed constants. Equip $U_j\cap U$ with the complete metric
$\omega_c+\omega_{\rm ball}$, where $\omega_{\rm ball}$ is a complete
metric for the coordinate ball. This sum is complete at both the
deleted divisor and the ball boundary. An $(n,1)$ datum square
integrable for $\omega_c$ remains so for the larger background metric:
its density uses the inverse background matrix on its one remaining
covector. By \Cref{lem:top-degree-l2}, the inverse-curvature density
for the modified weight is bounded by the original $\omega_c$ datum
density. The top-form solution density is independent of the
auxiliary complete metric. Consequently there is a local solution
operator $T_{j,k}$ with
\begin{equation}\label{eq:local-primitive-bound}
\dbar T_{j,k}f=f,\qquad
\|T_{j,k}f\|_{h_k,U_j\cap U}
\le C_j\|f\|_{h_k,U_j\cap U},
\end{equation}
where $C_j$ is independent of $k$. For each fixed $k$, choose the
solution of smallest norm in the modified metric. Orthogonal
projection onto the complement of the holomorphic kernel makes this
a linear continuous choice. The affine solution set is closed because
the distributional operator is closed.
\emph{Holomorphic cocycles.}
Given $f\in Z^1_{h_k}$, put $w_j=T_{j,k}(f|_{U_j\cap U})$.
On $(U_i\cap U_j)\cap U$, the difference $w_j-w_i$ is holomorphic.
Uniform upper bounds for $h_k$ turn
\Cref{eq:local-primitive-bound} into ordinary local $L^2$ bounds in
frames on $V$. Holomorphic $L^2$ removal therefore extends the
differences uniquely across $D$. One can see the removal locally by
Laurent expansion in the crossing coordinates and Fubini: a nonzero
negative power is not square integrable. The extended differences
$c_{ij}=w_j-w_i$, sections of $K_V\otimes F$ on $U_i\cap U_j$, form a holomorphic
\v Cech cocycle.
For later use, this family of cocycles is bounded in every fixed
compact-convergence seminorm whenever $\|f\|_{h_k}$ is bounded,
uniformly in $k$. Indeed, for any compact set in an overlap, choose a
slightly larger compact neighborhood still in that overlap. The
ordinary $L^2$ bound there and the holomorphic mean-value inequality
bound the supremum on the smaller compact. Each such compact has
positive separation from the ball boundaries. Its constant may
depend on that compact, but not on $k$. No bound on the supremum over
an entire open overlap is required.
\emph{The ordinary class.}
Let $\mathcal Z^1$ be the Fr\'echet space of holomorphic cocycles on
this finite cover, with compact-convergence topology. The cocycle
equations make it a closed subspace of the finite product of overlap
section spaces. Choose a smooth partition of unity $(\eta_i)$ with
$\Supp\eta_i\Subset U_i$. For a cocycle $c$ put
$b_j=\sum_i\eta_i c_{ij}$ on $U_j$, extending each summand by zero
outside its overlap. Then $b_j-b_i=c_{ij}$, so the forms $\dbar b_j$
agree and define a global smooth form $\beta_c$. This defines the
continuous linear map
\[
\kappa:\mathcal Z^1\longrightarrow H^1(V,K_V\otimes F),
\qquad c\longmapsto[\beta_c].
\]
Continuity follows from the compact support of the partition and
Cauchy estimates for derivatives on slightly larger compact sets;
ordinary cohomology is finite-dimensional and Hausdorff on the compact
manifold. Define $\kappa_k(f)=\kappa((c_{ij}))$.
Changing the local primitives
changes the cocycle by a holomorphic coboundary: the differences of
two primitives extend by the same ordinary $L^2$ argument. The local
solution estimates and the compact-seminorm estimates just proved
make $\kappa_k$ continuous and linear.
If $f$ is smooth on $V$, choose smooth local primitives on the larger
balls. On the smaller balls they have finite weighted norms, because
their coefficients are bounded and the local integrals of $e^{-h_k}$
are finite. Comparison with these primitives shows that $\kappa_k$
agrees with the ordinary Dolbeault class. If instead $f=\dbar v$ for
a global weighted $L^2$ primitive on $U$, then $w_j-v$ is holomorphic
on $U_j\cap U$ and ordinary locally $L^2$. Its extension gives a
holomorphic splitting of $c_{ij}$, so $\kappa_k(f)=0$. Continuity
therefore annihilates the closure of the actual exact forms, as in
(i).
\emph{A uniformly bounded splitting.}
Let
$\mathcal C^0=\prod_jH^0(U_j,K_V\otimes F)$, again with
compact-convergence topology, and let $\mathcal B^1$ be the image of
its coboundary map $\delta$ in $\mathcal Z^1$. We need the equality
\begin{equation}\label{eq:closed-coboundary-kernel}
\mathcal B^1=\ker\kappa.
\end{equation}
The forward inclusion is immediate. For the reverse inclusion,
construct smooth local cochains from a partition of unity for a
cocycle in $\ker\kappa$. Their common $\dbar$ is a global smooth
form with zero ordinary Dolbeault class. Subtract a global smooth
primitive from every cochain. The resulting cochains are
holomorphic and split the original cocycle. This proves
\Cref{eq:closed-coboundary-kernel} without any assumption that the
coordinate cover is a Leray cover.
It follows that $\mathcal B^1$ is a closed Fr\'echet subspace. The
continuous surjection
$\delta:\mathcal C^0\to\mathcal B^1$ is open. Prescribe only the
single continuous seminorm
\[
q((a_j))=\max_j\sup_{\overline{V_j}}|a_j|,
\]
where the norms are those of fixed smooth frames or a fixed smooth
bundle metric. Openness says that the image of $\{q<1\}$ contains a
zero neighborhood of $\mathcal B^1$ described by finitely many compact
seminorms. The cocycles arising from all $k$ and all data of norm at
most one are bounded in those seminorms. A single rescaling therefore
gives, for every such cocycle in $\mathcal B^1$, a splitting
$(a_j)$ with $q((a_j))\le C_{\rm split}$ for a fixed
$C_{\rm split}$. This uses no
assertion that a bounded subset has a lift bounded in every
Fr\'echet seminorm.
When $\kappa_k(f)=0$, choose this splitting and set
$v=w_j-a_j$ on $U_j\cap U$. The expressions agree on the full
overlaps, and hence define a global primitive on $U$. Estimate its
norm on the smaller $V_j$, which cover $V$. The local primitives
are controlled by \Cref{eq:local-primitive-bound}; the corrections
are controlled by their supremum on $\overline{V_j}$ and the uniform
integral of $e^{-h_k}$ there. Summing over the finite cover proves a
uniform bound for unit-norm data. Scaling proves (ii).
\end{proof}
Both sequences $h_{0,k}$ and $h_{1,k}$ satisfy this proposition, by
\Cref{eq:weight-integral-bounds,eq:curvature-interior-comparison}.
For each weight, parts (i) and (ii) give
\[
\ker\kappa_k=\im\dbar=\overline{\im\dbar}
\quad\text{inside }Z^1_{h_k}.
\]
Indeed, part (i) kills the closure of the exact forms, while part (ii)
provides a primitive for every form in the kernel. We will keep track
of the closures explicitly when comparing different weighted spaces.
The proposition also applies to a fixed smooth line metric from $V$: its curvature
is bounded below by $-A_0\omega_c$ for some fixed $A_0$, and its
local upper and integral bounds are automatic. This last observation
will allow us to compare all approximation parameters in one Hilbert
space.
\subsection{Harmonic representatives and the Bochner comparison}
Take a class in the kernel of the map in
\Cref{thm:interior-injectivity}, and represent it by a smooth closed
$L_1$-valued $(n,1)$ form $\gamma$ on $V$. All norms in this
subsection use $\omega_c$; a subscript $i,k$ indicates the line
weight $h_{i,k}$, and integrals of pointwise inner products use
$dV_{\omega_c}$. The norms $\|\gamma\|_{1,k}$ are bounded
uniformly. Indeed, enlarging $\omega$ to $\omega_c$ decreases the
top-form-valued $(0,1)$ density, and the coefficients of $\gamma$ are
bounded on a finite cover, where \Cref{eq:weight-integral-bounds}
applies.
In the maximal $L^2_{1,k}$ Dolbeault complex, the closure of the image
of degree-zero forms lies in the closed kernel of the degree-one
operator. Subtract its orthogonal projection from $\gamma$, obtaining
$u_k$. Thus
\begin{equation}\label{eq:reduced-harmonic-representative}
\gamma-u_k\in\overline{\im\dbar},\qquad
\dbar u_k=0,\qquad \dbar^*_{1,k}u_k=0,\qquad
\|u_k\|_{1,k}\le\|\gamma\|_{1,k}.
\end{equation}
The adjoint assertion follows from orthogonality to the actual range;
it does not require that range to be closed. Local elliptic
regularity for the smooth weighted Dolbeault Laplacian on $U$ makes
$u_k$ smooth there.
Write $\Lambda=\Lambda_{\omega_c}$ and let $D'_{i,k}$ be the
$(1,0)$ Chern derivative. In bidegree $(n,1)$ put
\[
A_{i,k}=[\theta_{i,k}+\epsilon_k\omega_c,\Lambda].
\]
These are pointwise nonnegative operators, and
$A_{1,k}\ge(1-\lambda)A_{0,k}$. The smooth compact-support
Bochner--Kodaira identity is
\begin{equation}\label{eq:bochner-top-degree}
\|\dbar v\|_{i,k}^2+\|\dbar^*_{i,k}v\|_{i,k}^2
=\|D_{i,k}^{\prime *}v\|_{i,k}^2
+\int_U\langle[\theta_{i,k},\Lambda]v,v\rangle_{i,k}.
\end{equation}
Here $D'_{i,k}v=0$ in top holomorphic degree, and
$[\omega_c,\Lambda]$ is the identity in this bidegree. The same
identity on complete manifolds, with the indicated domain
interpretation and lower curvature bound, is discussed in
\cite[Proposition~2.4]{Matsumura2018}.
To justify its use without already knowing the extra terms are
integrable, use the compactly supported smooth cutoffs constructed above, whose
gradients tend uniformly to zero in $\omega_c$. Apply
\Cref{eq:bochner-top-degree} to the cutoff multiples of $u_k$ and add
$\epsilon_k$ times their squared norms. The left-hand derivative
terms tend to zero by \Cref{eq:reduced-harmonic-representative} and
the gradient bound. The two terms retained on the right are
nonnegative, so lower semicontinuity gives
\begin{equation}\label{eq:source-bochner-energy}
\|D_{1,k}^{\prime *}u_k\|_{1,k}^2
+\int_U\langle A_{1,k}u_k,u_k\rangle_{1,k}
\le\epsilon_k\|u_k\|_{1,k}^2.
\end{equation}
The commutators with the cutoffs contain only their derivatives; they
do not introduce constants from derivatives of the varying line
metrics.
The K\"ahler identity
$D^{\prime *}=i[\Lambda,\dbar]$ shows that this formal operator
commutes with holomorphic multiplication by $s$. Combining
\Cref{eq:multiplication-contraction,eq:curvature-interior-comparison}
with \Cref{eq:source-bochner-energy} therefore gives
\begin{equation}\label{eq:target-bochner-energy}
\|D_{0,k}^{\prime *}(su_k)\|_{0,k}^2
+\int_U\langle A_{0,k}su_k,su_k\rangle_{0,k}
\le C_\lambda\epsilon_k\|u_k\|_{1,k}^2.
\end{equation}
For the curvature term one first uses
$A_{1,k}\ge(1-\lambda)A_{0,k}$ and then the line-norm contraction.
One may take $C_\lambda=(1-\lambda)^{-1}$: this factor bounds both
the derivative term and the curvature term in the source energy.
The strict interior hypothesis keeps this comparison factor finite.
We also need the target $\dbar$ adjoint. The form $su_k$ is smooth,
closed, and square integrable. Apply the compact-support identity to
cutoff multiples of this form. On its right-hand side use
\Cref{eq:target-bochner-energy}, replace the curvature commutator by
the larger nonnegative $A_{0,k}$, and let the gradient errors tend to
zero. It follows that the formal adjoint is square integrable and
\begin{equation}\label{eq:small-target-adjoint}
\|\dbar^*_{0,k}(su_k)\|_{0,k}
\le C\sqrt{\epsilon_k}.
\end{equation}
This also verifies the Hilbert-adjoint domain condition. In detail,
complete-metric cutoffs and local smooth approximation give density in
the maximal graph norm, so integration by parts against compactly
supported forms extends to that domain. The formal adjoint, now
known to be $L^2$, is its Hilbert adjoint. Thus
\Cref{eq:small-target-adjoint} has not assumed the existence of a
global primitive or an adjoint-domain statement in advance.
We have shown that the image of the harmonic representative has a
small adjoint norm. The fixed-cover proposition supplies a uniformly
bounded primitive for this image; pairing these two statements will
make its entire target norm tend to zero.
\subsection{From a small image to vanishing of the ordinary class}
Multiplication by $s$ is bounded in degrees zero and one and commutes
with the distributional Dolbeault operator. Hence
\Cref{eq:reduced-harmonic-representative} implies that
$s\gamma-su_k$ is in the target closure of the actual exact forms.
Apply the target class map in \Cref{prop:uniform-primitives}. It
annihilates that closure and agrees with ordinary cohomology on
$s\gamma$. The latter class is zero by the choice of $\gamma$.
Consequently $\kappa_{0,k}(su_k)=0$.
The norms of $su_k$ are uniformly bounded by
\Cref{eq:multiplication-contraction,eq:reduced-harmonic-representative}.
Part (ii) of \Cref{prop:uniform-primitives} gives global forms $v_k$
on $U$ with
\[
\dbar v_k=su_k,\qquad \sup_k\|v_k\|_{0,k}<\infty.
\]
The adjoint-domain conclusion above now justifies
\begin{equation}\label{eq:small-target-form}
\|su_k\|_{0,k}^2
=\langle\dbar v_k,su_k\rangle_{0,k}
=\langle v_k,\dbar^*_{0,k}(su_k)\rangle_{0,k}
\longrightarrow0.
\end{equation}
To retain the original class while passing to a limit, fix a smooth
metric on $L_1$ over all of $V$, with local weight $h_*$ and use the same complete
background $\omega_c$ on $U$. Uniform upper bounds in a finite
framed cover give the global metric comparison
\begin{equation}\label{eq:fixed-hilbert-comparison}
h_{1,k}\le h_*+C_*,\qquad
e^{-h_*}\le e^{C_*}e^{-h_{1,k}}.
\end{equation}
It gives one bounded inclusion from each varying weighted space into
the fixed space, in both degrees zero and one. In particular the
$u_k$ have uniformly bounded fixed norms.
On any compact subset $K\subset U$ the multiplier $s$ is nowhere
zero, and the target weights $h_{0,k}$ have uniform upper bounds.
Thus \Cref{eq:small-target-form} implies that $u_k\to0$ in the fixed
$L^2$ norm on $K$. This local strong convergence and the global bound
imply global weak convergence to zero. Indeed, approximate any fixed
$L^2$ test form by one supported in a compact subset of $U$; its
pairing tends to zero by local convergence, and the norm of the
remaining tail controls its pairing uniformly in $k$. Concentration
of norm near $D$ does not affect this weak convergence.
For each $k$, choose actual exact forms $\dbar z_{k,l}$ converging to
$\gamma-u_k$ in \Cref{eq:reduced-harmonic-representative}. By
\Cref{eq:fixed-hilbert-comparison}, their primitives belong to the
fixed degree-zero space and their derivatives converge in the fixed
degree-one norm. Therefore $\gamma-u_k$ belongs to the fixed closure
of actual exact forms. Apply \Cref{prop:uniform-primitives} once
more, now to the fixed smooth metric with weight $h_*$. Its continuous class
map $\kappa_*$ annihilates that fixed closure, so
\[
\kappa_*(u_k)=\kappa_*(\gamma)=[\gamma]
\quad\hbox{for every }k.
\]
The fixed closed-form space is a closed Hilbert subspace, and a
continuous linear map from it to finite-dimensional ordinary
cohomology is weakly continuous. Since $u_k\rightharpoonup0$, the
last displayed identity forces $[\gamma]=0$.
This proves analytic Dolbeault injectivity. Dolbeault theory and GAGA
identify it with the natural map in algebraic coherent cohomology,
proving \Cref{thm:interior-injectivity}.
==== END Minimal-metrics-and-interior-injectivity-for-nef-adjoints//interior-injectivity.tex ====
==== BEGIN Minimal-metrics-and-interior-injectivity-for-nef-adjoints//introduction.tex ====
\section{Introduction}\label{sec:introduction}
A nef line bundle admits smooth metrics whose curvatures approach
semipositivity. A limit of those metrics may acquire singularities.
For an adjoint bundle, the question is whether the canonical and boundary
terms prevent logarithmic poles in a least singular semipositive metric.
We prove that they do for projective klt pairs, and use zero-Lelong
endpoint metrics to establish an injectivity theorem in ordinary
coherent cohomology.
We use the convention that a local metric weight $\phi$ gives squared
norm $|\xi|^2e^{-\phi}$. Semipositivity means that its curvature current
$i\partial\bar\partial\phi$ is nonnegative. Such a metric has
\emph{minimal singularities} if every other semipositive weight $\psi$
on the same line bundle satisfies $\psi\le\phi+C_\psi$ for a constant
$C_\psi$. Weights on rational line bundles are defined by clearing
denominators and dividing by the same positive integer. Lelong numbers
measure the logarithmic singularities of these weights; in particular,
zero Lelong numbers imply local integrability of $e^{-t\phi}$ for every
fixed $t>0$.
For a normal variety $H$ and an effective rational divisor $\Theta$,
the klt condition means that $K_H+\Theta$ is $\Q$-Cartier and, on a
log resolution $\pi:V\to H$, the crossing divisor $B$ in
$\pi^*(K_H+\Theta)=K_V+B$ has every coefficient less than one.
The actual rational line bundle, rather than its numerical class,
is the object in the following theorem.
\begin{theorem}\label{thm:adjoint-metric}
Let $(H,\Theta)$ be a normal connected projective complex klt pair, with
$\Theta$ an effective rational divisor. Suppose that
$D_H=K_H+\Theta$ is a nef $\Q$-Cartier divisor. For every projective log
resolution $\pi:V\to H$, the rational line bundle $N=\pi^*D_H$ has the following property: every semipositive singular Hermitian
metric with minimal singularities on $N$ has Lelong number zero
at every point of $V$. Such a metric exists. Neither $K_H$ nor $\Theta$ is required to be separately $\Q$-Cartier.
\end{theorem}
Consequently, for every minimal weight $\phi$ on $\pi^*(K_H+\Theta)$,
\[
\mathcal I(t\phi)=\cO_V\qquad(t>0),
\]
where $\mathcal I(t\phi)$ consists of holomorphic germs $f$ for which
$|f|^2e^{-t\phi}$ is locally integrable. We record the precise independence
of the metric and the Cartier multiple in
\Cref{cor:minimal-multiplier}.
Demailly--Peternell--Schneider constructed minimal-singularity metrics
on pseudoeffective line bundles and proved zero Lelong numbers in the
big and nef case \cite[Theorem~1.5 and Proposition~1.7]{DPS2001}.
Nefness alone does not give this conclusion. For example, on a ruled
surface over an elliptic curve, a nef divisor bundle can have its
divisor metric as a minimal metric, with a positive Lelong number
along the divisor \cite[Corollary~1.2 and Example~3.5]{Koike2015}.
Gongyo--Matsumura asked how to construct a zero-Lelong semipositive
metric on a nef log canonical bundle in their study of injectivity and abundance
\cite[Question~5.6]{GongyoMatsumura2017}.
\Cref{thm:adjoint-metric} gives a positive answer in the projective klt
setting on every projective log resolution, without bigness or
nonvanishing. Once one zero-Lelong semipositive metric exists, every
minimal metric is no more singular and therefore also has zero Lelong
numbers. Thus the universal assertion about minimal metrics expresses
the same existence conclusion on each pullback bundle.
\subsection{Interior boundaries and restriction of sections}
Our second result concerns a family of effective rational divisors
between two fixed endpoints. Their coefficients may cross integers.
The resulting changes in the integral part of the divisor give natural
inclusions of line bundles. Zero-Lelong metrics at the endpoints
control the induced map on ordinary $H^1$.
\begin{theorem}\label{thm:interior-injectivity}
Let $V$ be a smooth projective complex variety, let $L$ be an integral
divisor, and let $0\le C_0\le C_2$ be effective rational divisors whose
combined support is simple normal crossing. Suppose that both
rational line bundles $L-C_0$ and $L-C_2$ admit singular Hermitian
metrics of semipositive curvature with zero Lelong numbers at every
point. For a rational number $0<\lambda<1$, set
\[
C_1=(1-\lambda)C_0+\lambda C_2,\qquad
L_i=L-\floor{C_i}\quad(i=0,1).
\]
Then the natural inclusion of line bundles induces an injection
\[
H^1\bigl(V,\cO_V(K_V+L_1)\bigr)
\longrightarrow H^1\bigl(V,\cO_V(K_V+L_0)\bigr).
\]
\end{theorem}
The strict inequality $\lambda<1$ provides a positive coefficient in
the curvature comparison. Effectivity and the common crossing support
provide integrability for the fractional boundary weights. The theorem
concerns the displayed integral line bundles and the natural sheaf map;
its endpoint assumptions are on the actual rational bundles $L-C_i$.
One direct consequence explains the connection with restriction of
sections. Put $E=\lfloor C_1\rfloor-\lfloor C_0\rfloor$, an effective
integral divisor, which may be nonreduced. If $E\ne0$, then
\begin{equation}\label{eq:restriction-surjective}
H^0\bigl(V,\cO_V(K_V+L_0)\bigr)
\longrightarrow
H^0\bigl(E,\cO_V(K_V+L_0)|_E\bigr)
\quad\text{is surjective}.
\end{equation}
Indeed, the connecting homomorphism for
$0\to\cO_V(K_V+L_1)\to\cO_V(K_V+L_0)
\to\cO_V(K_V+L_0)|_E\to0$ has image equal to the kernel of
the injective $H^1$ map. Thus the comparison lifts sections from the
entire divisor scheme $E$ at once.
Classical root-cover injectivity, including a reduced boundary, appears
in Esnault--Viehweg \cite[Theorem~5.1]{EsnaultViehweg1992}.
Fujino's transcendental approach develops complete metrics on an
analytic complement and comparison with coherent cohomology
\cite[Section~3, Lemmas~3.1--3.2 and Claim~1]{FujinoInjectivity2012}.
Matsumura extends analytic injectivity to metrics with transcendental
singularities and obtains uniform estimates for primitives
\cite[Theorem~1.3 and Sections~5.2--5.3]{Matsumura2018}.
Our proof uses the same harmonic-form and local-primitive methods, with
a comparison of two endpoint weights. We prove the required uniform
primitive estimate and the return to ordinary cohomology explicitly.
\subsection{The two analytic arguments}
For the metric theorem, write $N=\pi^*(K_H+\Theta)=K_V+B$.
The adjoint hypothesis enters twice. The klt coefficients make the
adjoint density integrable, even where $B$ has negative exceptional
coefficients. Klt vanishing and nefness then give one ample bundle
$A_H$ on $H$ whose pullback $A=\pi^*A_H$ makes $mN+A$ globally
generated for every positive integer $m$ with $m(K_H+\Theta)$ Cartier.
Normalize a section $s$ by its integral
\[
Q_m(s)=\int_V |s|^{2/m}e^{-a/m-b},
\]
where $a$ is a smooth weight of $A$ and $b$ the divisor weight of $B$.
Choose a normalized section maximizing its value at a proposed
positive-Lelong point.
The $L^{2/m}$ normalization has its origin in Narasimhan--Simha's
work on ample canonical bundles \cite{NarasimhanSimha1968}; its role
in the later envelope construction is explained in
\cite[Section~5]{BermanDemailly2012}.
Tsuji constructs canonical singular metrics using normalized sections
of $A+mK_X$ with a fixed sufficiently ample twist
\cite[Sections~1.2 and~2]{Tsuji2007}.
Berman--Demailly develop integral-normalized adjoint envelopes and
their algebraic approximation
\cite[Definition~5.3 and Proposition~5.19]{BermanDemailly2012}.
Their general pseudoeffective approximation allows twists $p_mA$ with
$p_m\to\infty$ and $p_m/m\to0$
\cite[Equation~(5.24)]{BermanDemailly2012}.
Here the fixed twist and the klt density support an extremal argument:
a localization estimate produces a better normalized section whenever
the minimal weight has a positive Lelong number.
The normalized extremal sections have a semipositive limit after the
twist is divided by $m$. Minimality bounds that limit by the chosen
minimal weight. Consequently a fixed sublevel set of the difference
of the weights has adjoint mass tending to zero. A localized
$\bar\partial$ equation, solved on an affine open set, produces a
holomorphic competitor whose weighted norm is bounded by that mass.
The exact localization constant is independent of $m$ and of the
affine open set. Positive Lelong number forces this competitor to have
the same value at the chosen point. A H\"older estimate whose final bound is independent of $m$ makes
its normalization strictly smaller, contradicting extremality.
Two details keep the construction uniform and valid on a log resolution.
The weak limits in the localization lemma are taken in fixed weighted
Hilbert spaces. Extension across negative exceptional coefficients is
performed on the normal base $H$ in the actual Cartier bundle
$m(K_H+\Theta)+A_H$, and then pulled back. The complete K\"ahler
$L^2$ estimate supplies the local analytic solution
\cite[Theorem~5.1]{Demailly2012}; the localization and extension
arguments are given in \Cref{sec:adjoint-metrics}.
For injectivity, regularize both endpoint metrics, retaining arbitrarily
small curvature loss and uniform integrability at every fixed exponent.
A logarithm of a sum of endpoint weights makes the inclusion of line
bundles a contraction. On the complement of the crossing divisor,
a complete K\"ahler metric with bounded local potentials allows
local $L^2$ solutions. Their holomorphic differences extend across the
divisor and define ordinary \v Cech classes. An open-mapping argument
on one fixed finite cover gives a uniform bound for primitives of
forms whose ordinary class vanishes.
A Bochner comparison then makes the image of a harmonic representative
small. All approximating representatives embed into a single fixed
Hilbert space; weak convergence there preserves the original ordinary
cohomology class. This is the final step from weighted estimates on the
complement to the coherent-cohomology injection. The argument is
independent of the adjoint metric theorem: it uses only the endpoint
metrics stated in \Cref{thm:interior-injectivity}.
\subsection{A fourfold application and organization}
The metric theorem also gives a short abundance argument for a
projective complex klt fourfold $(X,\Delta)$ with nef rational adjoint
and $\chi(X,\cO_X)\ne0$. Lazi\'c--Peternell's nonvanishing theorem
uses generalized algebraic singularities of a semipositive metric
\cite[Corollary~D]{LazicPeternell2020}; zero Lelong numbers satisfy
that condition with no divisorial part. With a first section obtained,
Gongyo--Matsumura's criterion gives semiampleness
\cite[Corollary~5.3]{GongyoMatsumura2017}.
The precise application is \Cref{cor:euler-abundance}; it has no
numerical-dimension restriction. For nef adjoints of numerical dimension at most one,
Liu--Xu already obtain good minimal models for projective klt pairs
of dimension at most four with nonzero Euler characteristic
\cite[Corollary~5.2]{LiuXu2025}; in the nef case this gives
semiampleness. Lazi\'c's work supplies a semiampleness criterion using
uniform multiplier-ideal comparisons, and an algebraic approximation
theorem for supercanonical currents, under their respective hypotheses
\cite[Theorems~A--B]{Lazic2024Metrics}.
In the compact K\"ahler setting, H\"oring--Lazi\'c--Lehn obtain
nonvanishing for non-uniruled $\Q$-factorial klt pairs with nef adjoint
of numerical dimension one and nonzero Euler characteristic
\cite[Theorem~A]{HoringLazicLehn2025}. Under these hypotheses in
dimension four, they obtain semiampleness when the minimal pullback
current has zero Lelong numbers
\cite[Corollary~B]{HoringLazicLehn2025}.
\Cref{sec:adjoint-metrics} proves the metric theorem, the arbitrary-data
localization estimate and the multiplier-ideal consequence.
\Cref{sec:interior-injectivity} proves the independent interior
injectivity theorem. \Cref{sec:euler-route} states the two published
fourfold criteria and applies them to the same minimal metric.
==== END Minimal-metrics-and-interior-injectivity-for-nef-adjoints//introduction.tex ====
==== BEGIN Minimal-metrics-and-interior-injectivity-for-nef-adjoints//main.tex ====
\documentclass[11pt]{article}
\input{preamble}
\hypersetup{pdftitle={Minimal metrics and interior injectivity for nef adjoints},pdfauthor={OpenAI},bookmarksdepth=2}
\title{Minimal metrics and interior injectivity\\for nef adjoints}
\author{OpenAI}
\date{September 27, 2026}
\begin{document}
\maketitle
\begin{abstract}
Let \((H,\Theta)\) be a projective complex klt pair with effective rational
boundary and nef \(\Q\)-Cartier adjoint. On any projective log resolution,
minimal semipositive metrics on the pulled-back adjoint exist and have zero
Lelong numbers everywhere. On smooth projective complex varieties, we also
prove an \(H^1\)-injectivity
theorem for rational interior boundaries with simple-normal-crossing support
when both endpoint bundles carry zero-Lelong semipositive metrics.
\end{abstract}
\setcounter{tocdepth}{1}
\tableofcontents
\input{sections/introduction}
\input{sections/adjoint-metrics}
\input{sections/interior-injectivity}
\input{sections/euler-route}
\input{bibliography}
\end{document}
==== END Minimal-metrics-and-interior-injectivity-for-nef-adjoints//main.tex ====
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