==== TABLE OF CONTENTS ====
Exact-orders-and-invariant-anticanonical-linear-systems//approximate-arc.tex
Exact-orders-and-invariant-anticanonical-linear-systems//approximate-weights.tex
Exact-orders-and-invariant-anticanonical-linear-systems//arc-corollary.tex
Exact-orders-and-invariant-anticanonical-linear-systems//controlled-cone.tex
Exact-orders-and-invariant-anticanonical-linear-systems//endpoint.tex
Exact-orders-and-invariant-anticanonical-linear-systems//exact-proof.tex
Exact-orders-and-invariant-anticanonical-linear-systems//exact-weights.tex
Exact-orders-and-invariant-anticanonical-linear-systems//glyph-unicode.tex
Exact-orders-and-invariant-anticanonical-linear-systems//introduction.tex
Exact-orders-and-invariant-anticanonical-linear-systems//invariant-field.tex
Exact-orders-and-invariant-anticanonical-linear-systems//isometric-descent.tex
Exact-orders-and-invariant-anticanonical-linear-systems//main.tex
Exact-orders-and-invariant-anticanonical-linear-systems//mrc-descent.tex
Exact-orders-and-invariant-anticanonical-linear-systems//nef-tools.tex
Exact-orders-and-invariant-anticanonical-linear-systems//null-field.tex
Exact-orders-and-invariant-anticanonical-linear-systems//order-weight.tex
Exact-orders-and-invariant-anticanonical-linear-systems//references.bib
===========================

==== BEGIN Exact-orders-and-invariant-anticanonical-linear-systems//approximate-arc.tex ====
\section{An approximate transfer theorem for formal-arc orders}
\label{V:sec-transfer}

An order along a generically dominant formal arc records vanishing of
effective Cartier divisors even when the arc's center is not a closed
point.  We will transfer bounded orders in a twisted sequence to
arbitrarily small normalized orders in anticanonical systems.

\begin{theorem}[Approximate arc transfer]\label{V:approximate-arc}
Let $W$ be smooth connected projective over $\mathbb C$, with
$q(W)=0$, and let $A=-K_W$ admit a smooth semipositive Hermitian
metric.  Let $M$ be a line bundle with $-M$ pseudoeffective.  Let $v$
be the order along a formal arc over an extension field of
$\mathbb C$ whose fraction point maps to the generic point of $W$;
the trivial order is allowed.  Suppose there are effective integral
divisors
\[
 N_i\sim M+m_iA,\qquad 0<m_1<m_2<\cdots,
 \qquad v(N_i)=O(1).
\]
For every real $\delta>0$ there are an integer $r>0$ and an effective
integral divisor $N\sim rA$ with $v(N)<\delta r$.
\end{theorem}

The exact scalar theorem proved earlier yields a stronger conclusion
under these hypotheses.  The proof here explains a different
mechanism.  A fixed part absorbs none of the prescribed arc order;
a nef reduction then leaves a base line whose first jets can be
extended.  Bigness of that line supplies sections with arbitrarily
small normalized order.  Both the assumption $q(W)=0$ and the
pseudoeffectivity of $-M$ have specific roles in this construction.

\begin{proof}[Proof of Theorem~\ref{V:approximate-arc}]
The assertion is immediate if $W$ is a point.  We assume positive
dimension.  The arc lifts uniquely to every proper birational model,
by the valuative criterion.  We use the same letter $v$ for orders
there; orders of effective rational Cartier divisors are finite and
nonnegative, and are preserved by pullback.

\subsection*{The common field and its rank-one fibers}

Call the indices $i\ge2$ interior.  For each $i$, let $K_i$ be the
subfield of $\mathbb C(W)$ generated by ratios of nonzero sections
of the same positive multiple of $N_i$.  We first show that $K_i$
is independent of the interior index.

For an interior $i$ and any $j$, there is a rational $c>0$ such that
$N_i-cN_j$ is rationally linearly equivalent to an effective rational
divisor.  For $j<i$ choose an index beyond $i$, and for $j>i$ choose
an earlier index, then express $m_i$ as the corresponding strict
rational convex combination.  The same combination of the effective
divisors has class $M+m_iA$.  The case $i=j$ is immediate.  Given
a ratio of sections of $kN_j$, choose a sufficiently divisible $d$
so that $dN_i-kN_j$ has a nonzero section.  Multiplying numerator
and denominator by that section realizes the same ratio in $|dN_i|$.
Thus $K_j\subset K_i$, and the fields for interior indices coincide;
write $K$ for their common value.

Choose finitely many of these ratios attaining the transcendence
degree of $K$ and place them in one complete system $|dN_{i_0}|$.
Resolve its map and take the Stein factor.  Its function field $F$
is relatively algebraically closed in $\mathbb C(W)$ and contains
$K$: the field of the selected image is contained in $K$ with the
same transcendence degree, so every element of $K$ is algebraic over
it.  Apply Lemma~\ref{V:flat-model} and then resolve to obtain
\[
 S\longrightarrow V,\qquad f:V\longrightarrow Y,
 \qquad p:V\longrightarrow W,
\]
where $Y,S$ are smooth projective, $V$ is normal equidimensional over
$Y$, $p$ is birational, and $\mathbb C(Y)=F$.  We also write $f,p$
for the maps from $S$.  Set $A_S=p^*A$ and
$E=K_S-p^*K_W\ge0$.  The divisor $E$ is exceptional.  Thus, for a
line bundle $Q$ on $W$ and an integer $u\ge0$,
\[
 H^0(S,p^*Q+uE)=s_E^u p^*H^0(W,Q),
\]
by normality of $W$ and extension across codimension two.

The pullback of the hyperplane bundle from the selected image is big
on its Stein factor and remains big after modifying the base.  It is
linearly dominated by $d p^*N_{i_0}$.  The interpolation above therefore
shows that a sufficiently divisible multiple of any interior
$p^*N_i$ linearly dominates the pullback of any fixed divisor on $Y$.
This remains true on further base modifications.

For every $l>0$ and every interior $i$, on the generic fiber one has
\begin{equation}\label{V:generic-rank-one}
 h^0(S_\eta,l(p^*N_i+E)|_{S_\eta})
 =h^0(S_\eta,lp^*N_i|_{S_\eta})=1.
\end{equation}
Both spaces contain the indicated effective section.  If either had
two independent sections, a sufficiently ample base twist would extend
them globally by generation of the twisted direct image.  The preceding
domination clears this twist using a common effective multiplier.
Their ratio would then be a ratio in a multiple of $p^*N_i+E$ and
would not belong to $F$, by independence over the generic base field.
Exceptional pushforward identifies its ratio field with that of $N_i$,
contradicting $K_i\subset F$.

Rank one now turns interpolation of line bundles into interpolation
of horizontal divisors.  If interior indices $i<j<k$ satisfy
$m_j=c m_i+(1-c)m_k$ with $0<c<1$ rational, then
\[
 N_j\sim_{\mathbb Q}cN_i+(1-c)N_k.
\]
Clear denominators and restrict to the generic fiber.  The two effective
sections belong to the same one-dimensional space in
\eqref{V:generic-rank-one}; their ratio is a scalar in $F$ and has
no horizontal divisor.  Consequently
\[
 (p^*N_j)^{\mathrm{hor}}
 =c(p^*N_i)^{\mathrm{hor}}+(1-c)(p^*N_k)^{\mathrm{hor}}.
\]
Fixing two interior indices gives rational horizontal divisors $D_0,C_0$
with fixed finite support such that
\[
 (p^*N_i)^{\mathrm{hor}}=m_iD_0+C_0\quad(i\ge2).
\]
Since the left side is effective for arbitrarily large $m_i$, each
coefficient of $D_0$ is nonnegative.

Take two interior indices $j>i$ and set $G=p^*(N_j-N_i)$ on $V$.
Its horizontal part is nonnegative.  For each prime divisor $Q$ of $Y$
take the minimum of
\[
 \frac{\operatorname{coeff}_{\Gamma}G}
      {\operatorname{coeff}_{\Gamma}f^*Q}
 \quad\hbox{over the components $\Gamma$ dominating $Q$.}
\]
There are only finitely many nonzero minima.  Subtracting their
pullback makes $G$ effective: the equidimensional normal model has
no other vertical primes.  After clearing denominators this gives
\begin{equation}\label{V:base-decomposition}
 a p^*A\sim D^V+f^*B,\qquad a>0,
\end{equation}
where $D^V$ is effective Cartier, $B$ is Cartier, and over every prime
of $Y$ some component has coefficient zero in $D^V$.  On $S$ put
$D=(D^V)|_S$ by pullback.  Its horizontal part is $aD_0$.

For fixed $l\ge0$ and sufficiently large $i$, the horizontal formula
gives an effective generic-fiber divisor
\[
 R_{i,l}=(p^*N_i-lD)|_{S_\eta}\ge0.
\]
Since $aA_S|_{S_\eta}\sim D|_{S_\eta}$, multiplication by
$s_{R_{i,l}}$ embeds the section spaces of $laA_S$ and $laA_S+E$
on the generic fiber into those of $p^*N_i$ and $p^*N_i+E$,
respectively.  The latter spaces have dimension one by
\eqref{V:generic-rank-one}.  The former contain $s_D^l$ and
$s_Es_D^l$ in base frames, so they too have dimension one.  For $l=0$
this also uses connectedness of the generic fiber.
In particular,
\begin{equation}\label{V:global-sections-base}
 H^0(W,laA)=s_D^l f^*H^0(Y,lB)
\end{equation}
under pullback to $S$.  Indeed any section has this form with a
rational base section by the generic-fiber statement.  On $V$, a pole
over a base prime would remain a pole on its coefficient-zero
component, which is impossible.  The rational base section is therefore
regular.  Conversely every section on the right is regular on $S$ and
descends to $W$.

\subsection*{Removing the fixed divisor from the arc order}

Pass to an infinite subsequence with $m_i$ in one residue class modulo
$a$, and fix an index $j$ in it.  For subsequent indices put
$l_i=(m_i-m_j)/a$.  Horizontal equality and
\eqref{V:base-decomposition} give actual divisor identities on $V$
\[
 p^*N_i=p^*N_j+l_iD^V+f^*V_i,
 \qquad V_i\sim l_iB.
\]
To justify the base divisor, subtract first $l_iB$ using the line
bundle identification.  The remaining principal divisor is vertical;
its rational function has zero divisor on the proper normal generic
fiber, and hence belongs to its constant field $F$.

The negative parts of the $V_i$ are bounded by a fixed effective
Cartier divisor $R$ on $Y$.  For a negative coefficient over $Q$, use
a component where $D^V$ has coefficient zero.  The fixed divisor
$p^*N_j$ must compensate that coefficient; moreover only finitely many
such $Q$ can occur.  Taking orders yields
\[
 v(N_i)\ge v(N_j)+l_i v(D)-v(f^*R).
\]
The left side is bounded and $l_i\to\infty$, so $v(D)=0$.
The remaining task is to find base sections of small normalized order.

\subsection*{The base line is nef}

The normalized section $s_D$ on the equidimensional model has order
zero on a component over every base prime.  The preceding generic
fiber calculation gives ordinary adjoint rank one for every
$l\ge1$, with generator $s_Es_D^l$ in base frames.  Thus all the
hypotheses of Lemma~\ref{V:bounded-maximum} hold with $L=A$,
$\tau=s_{D^V}$ and the base line $B$.  It follows that $B$ has a
semipositive metric with locally bounded weights, and in particular
is nef.  The fixed divisor has zero arc order; it remains to use the
nefness of $B$ to construct base sections of small normalized order.

\subsection*{Nef reduction and a linearly descended base divisor}

The base has $q(Y)=0$: pull holomorphic one-forms through a resolution
of the dominant rational map $W\dashrightarrow Y$, and use birational
invariance of irregularity.  Apply Lemma~\ref{V:nef-descent} to $B$.
Lift $S$ further to a smooth model $S'$ mapping to its $Y'$, and
compose to obtain $g:S'\to Z$.  Multiply $a,D,B,L$ by the common
integer from that lemma.  Retain $D$ for its pullback, put
$A'=p^*A$ and $E'=K_{S'}-p^*K_W$, and obtain
\[
 aA'\sim D+g^*L,
 \qquad L\text{ Cartier and nef},\qquad v(D)=0.
\]
Equation~\eqref{V:global-sections-base}, the projection formula,
and $h_*\mathcal O_{Y'}=\mathcal O_Z$ give for every $l\ge0$
\begin{equation}\label{V:adjoint-sections-base}
 H^0(S',K_{S'}+(1+la)A')
   =s_{E'}s_D^l g^*H^0(Z,lL).
\end{equation}
If $Z$ is a point, the divisor $D$ already proves the theorem.
Assume henceforth that $b=\dim Z>0$.

Choose a very ample $H$ on $Z$.  For some integer $m_0>0$,
\begin{equation}\label{V:pseudoeffective-base-domination}
 P=m_0A'-g^*H\quad\hbox{is pseudoeffective}.
\end{equation}
Indeed a multiple of one interior $p^*N_i$ linearly dominates $g^*H$,
because the original image polarization is big on the modified base.
Writing $N_i\sim M+m_iA$ and adding the pseudoeffective negative
multiple of $p^*M$ proves \eqref{V:pseudoeffective-base-domination}.

\subsection*{Extending first jets and proving bigness}

Take a singular semipositive metric $h_P$ on $P$.  Skoda integrability
and a finite covering of the compact variety $S'$ give a small
rational $\epsilon>0$ with
$\mathcal I(h_P^\epsilon)=\mathcal O_{S'}$.
Use Lemma~\ref{V:sesha} with $c=b+1$ to choose $k$, a very general
$z\in Z$, the ample rational divisor
$J=\tfrac12(kL+\epsilon H)$, its smooth metric of curvature $\eta$,
and $\psi$ with
\[
 \eta+i\partial\bar\partial\psi\ge0,
 \qquad\psi(w)\le(b+1)\log|w|^2+O(1).
\]
Choose $z$ also so that some $x\in g^{-1}(z)$ lies outside $D\cup E'$
and $g$ is submersive at $x$.  The image of this nonempty open subset
of $S'$ contains a dense base open, so the additional condition is
compatible with being very general.

For all sufficiently large $l\ge k$ put $m=1+la$ and use
\begin{equation}\label{V:jet-metric-decomposition}
 mA'\sim_{\mathbb Q}
 kD+g^*(2J)+\epsilon P+(m-ka-\epsilon m_0)A'.
\end{equation}
The last coefficient is nonnegative.  The divisor metric of $kD$,
the smooth metric of $2J$, the metric $h_P^\epsilon$, and the smooth
semipositive metric on the remaining multiple of $A'$ define a
singular metric $h_m$ on the integral line bundle $mA'$.  Rational
identifications are interpreted on a common multiple and then rooted.
Vanishing by $kD$ cancels its divisorial singularity, so
\[
 \mathcal O_{S'}(-kD)\subseteq\mathcal I(h_m).
\]
Put $\Psi=g^*\psi$.  Locally $\psi=c\log\sum_j|f_j|^2+u$
with $c>0$ and $u$ smooth; pullback has the same form with
$f_j\circ g$ and $u\circ g$.  Surjectivity of $g$ prevents it
from being identically $-\infty$, so it still has neat analytic
singularities.  The two copies of the base curvature in
\eqref{V:jet-metric-decomposition} give
\[
 i\Theta_{h_m}+(1+s)i\partial\bar\partial\Psi\ge0
 \qquad(0\le s\le1).
\]
The extension theorem of Cao--Demailly--Matsumura
\cite[Theorem~1.1]{CDM2017} applies on the compact K\"ahler manifold
$S'$: $h_m$ may be singular, and $\Psi$ has neat analytic
singularities.  Injectivity in degree one, together with the long
exact sequence, yields the surjection
\begin{equation}\label{V:quotient-surjection}
 H^0\!\left(S',\mathcal O_{S'}(K_{S'}+mA')\otimes\mathcal I(h_m)\right)
 \longrightarrow
 H^0\!\left(S',\mathcal O_{S'}(K_{S'}+mA')\otimes
 \frac{\mathcal I(h_m)}{\mathcal I(h_me^{-\Psi})}\right).
\end{equation}

The quotient is supported over $z$, since $\Psi$ is smooth elsewhere.
For any holomorphic local section $\sigma$ of $lL$ near $z$, the
section $s_{E'}s_D^l g^*\sigma$ lies in the larger ideal twist near
the entire fiber, because $l\ge k$.  Its quotient class glues with
zero off the fiber.  Surjectivity and
\eqref{V:adjoint-sections-base} produce $u\in H^0(Z,lL)$ for which
\[
 s_{E'}s_D^l g^*(u-\sigma)
\in \mathcal O_{S'}(K_{S'}+mA')\otimes\mathcal I(h_me^{-\Psi})
\]
near $x$.  The effective factors are units at $x$, and the local weight
of $h_m$ is bounded above.  Submersion coordinates and Fubini's
theorem therefore imply local integrability downstairs of
\[
 |(u-\sigma)(w)|^2|w|^{-2(b+1)}.
\]
If the first nonzero homogeneous term had degree $d\le1$, integration
over a cone of directions where it does not vanish would contain
$\int_0^\varepsilon r^{2d-2(b+1)+2b-1}\,dr$, which diverges.
Thus $u-\sigma$ vanishes to order at least two.  First jets of $lL$
are generated at $z$.

Choose a local frame and prescribe the jets $1,w_1,\ldots,w_b$.
The resulting global sections define ratios with independent
differentials at $z$, so the associated rational map has image
dimension $b$.  Consequently $L$ is big.

\subsection*{Finishing the order estimate}

Write $L\sim_{\mathbb Q}H_1+C$ with $H_1$ ample rational and $C\ge0$.
For every sufficiently small rational $\gamma>0$, the rational divisor
$(1-\gamma)L+\gamma H_1$ is ample.  A sufficiently divisible basepoint
free multiple has a section not vanishing at the center of the lifted
arc on $Z$, even if this center is not a closed point.  Indeed global
generation gives a section which is a unit in the local ring at that
scheme point.  Dividing its divisor by the multiple gives an effective
rational divisor $H_\gamma\sim_{\mathbb Q}(1-\gamma)L+\gamma H_1$
with $v(g^*H_\gamma)=0$.  Hence
\[
 D+g^*(H_\gamma+\gamma C)\sim_{\mathbb Q}aA',
 \qquad v\bigl(D+g^*(H_\gamma+\gamma C)\bigr)
       =\gamma v(g^*C).
\]
After clearing denominators this is the divisor of a section of a
positive multiple $rA'$; such sections descend to $W$.
Its order divided by $r$ is $\gamma v(g^*C)/a$, which is less than
$\delta$ for a suitable $\gamma$.  This completes the proof.
\end{proof}
==== END Exact-orders-and-invariant-anticanonical-linear-systems//approximate-arc.tex ====

==== BEGIN Exact-orders-and-invariant-anticanonical-linear-systems//approximate-weights.tex ====
\section{Extremal weights from twisted differential forms}
\label{V:sec-extremal}

The transfer theorem becomes equivariant when its arc is a generic
one-parameter orbit.  Its bounded input order comes from the fixed
shift between a cohomological weight and the twist of a differential
form.  The output sections approximate every exposed vertex of the
anticanonical weight polytope, after which multiplication clears the
weights.

The cohomological assumptions here are $q(W)=0$ and
$\chi(W,\mathcal O_W)\ne0$.  The virtual character determines the
source coefficient directly from this Euler characteristic.  The
rest of the argument follows the source-weight construction of
Section~\ref{subsec:valuation-extremal-weights}, using pseudoeffectivity of the
inverse determinant and approximate arc transfer to approach the
vertices.  Thus the proof does not require vanishing of all the higher
structure-sheaf cohomology groups or exact attainment of a source weight.

\begin{proposition}\label{V:extremal-invariants}
Let $W$ be smooth connected projective with $q(W)=0$ and
$\chi(W,\mathcal O_W)\ne0$.  Suppose $A=-K_W$ has a smooth
semipositive Hermitian metric.  Let an algebraic torus $T$ act
faithfully on $W$, and suppose an equivariantly linearized very ample
line bundle is available.  For the natural linearization on $A$,
there is an integer $r>0$ with $H^0(W,rA)^T\ne0$.
\end{proposition}

\begin{proof}
The point case is immediate.  In the real character space of $T$ put
\[
 \mathcal P=\operatorname{conv}
 \{\hbox{fiber characters of $A$ on the components of $W^T$}\}.
\]
The fixed locus is nonempty and has finitely many components, so this
is a rational polytope.  Fix an integral cocharacter
$\lambda:\mathbb C^*\to T$.  For a nontrivial action let $S_+$ be the
source component, whose attracting set for parameter tending to zero
is dense, and let $\mu_+$ be the $\lambda$-weight of $A$ there.
The Bia\l ynicki-Birula decomposition \cite{BB1973} shows that all
normal weights at $S_+$ are positive and that every other fixed
component has a negative normal direction.  Following such a direction
backwards along an orbit moves to a component of strictly larger
weight for the fixed very ample linearization.  For any nef
linearized line bundle, its weight at the zero end minus its weight
at the infinity end is the degree on the parametrized orbit
$\mathbb P^1$ and is nonnegative.  Iterating the backwards move
therefore terminates at the source and gives
\begin{equation}\label{V:source-support}
 \mu_+=\max_{\alpha\in\mathcal P}\langle\alpha,\lambda\rangle.
\end{equation}
Here $S_+$ is $T$-stable by uniqueness and contains a $T$-fixed point.
Faithfulness makes the $\lambda$-action nontrivial for every nonzero
cocharacter.  Since the natural anticanonical weight is the sum of
tangent weights, $\mu_+>0$ for every such cocharacter.  Rational
polyhedral separation implies that $0$ is in the interior of
$\mathcal P$.  If $T$ is trivial, use the zero cocharacter, $S_+=W$,
and $\mu_+=0$ in the argument below.

\subsection*{The largest cohomological weight}

Apply the equivariant holomorphic fixed-point formula
\cite{AtiyahSegal1968,AtiyahSinger1968} to the virtual character
$\sum_j(-1)^j H^j(W,mA)$.  A fixed component $U$ contributes
\[
 t^{m\mu_U}\int_U e^{m c_1(A)|_U}\operatorname{td}(T_U)
       \prod_\nu(1-t^{-d_\nu}e^{-x_\nu})^{-1},
\]
where $d_\nu\ne0$ are its normal weights and $x_\nu$ the corresponding
Chern roots.  Expand at infinity.  A positive $d_\nu$ contributes a
series with highest exponent zero, whereas a negative $d_\nu$
contributes only strictly negative exponents.  Thus for $m\ge0$ the
coefficient of $t^{m\mu_+}$ is exactly
\[
 Q(m)=\chi(S_+,mA|_{S_+}).
\]
Indeed every other component has a negative normal direction and
$\mu_U\le\mu_+$.  At $m=0$, the connected torus acts trivially on
$H^j(W,\mathcal O_W)$: it acts trivially on ordinary cohomology by
homotopy, hence on its Hodge summands.  The constant coefficient is
therefore $\chi(W,\mathcal O_W)$.  It follows that
$Q(0)=\chi(W,\mathcal O_W)\ne0$.
Riemann--Roch makes $Q$ a polynomial, so it is nonzero for all
sufficiently large integers $m$.

For some fixed $j$, infinitely many $H^j(W,mA)$ therefore contain
weight $m\mu_+$.  Average a K\"ahler form over the compact circle of
$\lambda$.  Semipositive hard Lefschetz
\cite{DPS2001}, applied to $(m+1)A$, gives the equivariant surjection
\[
 H^0(W,\Omega_W^{n-j}\otimes(m+1)A)
 \longrightarrow H^j(W,K_W+(m+1)A)=H^j(W,mA),
 \qquad n=\dim W.
\]
The equality uses the natural anticanonical linearization; its shift
is $m+1$.  Projecting to the desired circle weight gives a nonzero
holomorphic eigenform of weight $m\mu_+$ for each of infinitely
many $m$.

\subsection*{Saturated determinants and the fixed order}

We use the determinant method of Lazi\'c--Peternell
\cite[Lemma~4.1]{LP2018} and its anticanonical adaptation in
\cite[Lemma~5.1]{LMPTX2023}.  Those published statements assume
that the twisting line is numerically nontrivial.  Here the twists
are already positive and unbounded, so the elementary construction
below also handles the numerically trivial case.  The weights of
its wedges, including the rank shift, are computed locally.

Put $p=n-j$ and take the generic span of these forms as lines in
$\Omega_W^p$ over $\mathbb C(W)$.  Let its rank be $s>0$.  Wedge
$s$ generic basis forms.  The saturation of the resulting generic
line in $\bigwedge^s\Omega_W^p$ is a line bundle $M$, with its induced
$\lambda$-linearization.  More concretely, the wedge has some twist
$a_1A$; removing its common divisorial zero $D_1$ gives
$M=D_1-a_1A$ and a map into $\bigwedge^s\Omega_W^p$ nonvanishing in
codimension one.  The saturation is reflexive of rank one, hence
invertible on the smooth variety.  Its map extends across codimension
two, and regular generic-line sections factor through it because
their coefficients have no poles in codimension one.

The negative divisor $-M$ is pseudoeffective.  When $p>0$, the
exterior/tensor inclusion places this saturated line in a positive
tensor power of $\Omega_W^1$.  The theorem of
Lazi\'c--Matsumura--Peternell--Tsakanikas--Xie
\cite[Theorem~4.1]{LMPTX2023}, following Ou's movable-slope theorem
\cite[Theorem~1.4]{Ou2023}, applies to the smooth zero-boundary pair
with nef anticanonical divisor.  If $p=0$, the line is the underlying
trivial bundle, so the same conclusion holds directly.

Complete each eigenform of increasingly large twist to a generic
basis using $s-1$ members of the fixed finite basis, then take an
infinite subsequence with the same choice of these members.  This
gives nonzero eigensections and effective divisors
\begin{equation}\label{V:weighted-determinants}
 N_i\sim M+m_iA,
 \qquad\hbox{section weight }(m_i-s)\mu_+,
 \qquad 0<m_1<m_2<\cdots.
\end{equation}
The exponent $m_i$ is the sum of the $s$ form twists $m+1$; the
subtracted $s$ in the weight is therefore essential.

Let $v$ be the order along the generic orbit arc
$t\mapsto\lambda(t)\xi$, where $\xi$ is the generic point regarded
as a $\mathbb C(W)$-valued point.  Projectivity extends it to
$\operatorname{Spec}\mathbb C(W)[[t]]$, and its fraction point still
maps to the generic point.  Its center is in $S_+$.
Lemma~\ref{V:order-weight} applies with the declared left action on
sections.  The order is the fiber weight minus the section weight.

Writing $\mu_M$ for the fiber weight of $M$ on the source, equation~\eqref{V:weighted-determinants} and Lemma~\ref{V:order-weight} give
\[
 v(N_i)=\mu_M+m_i\mu_+-(m_i-s)\mu_+
       =\mu_M+s\mu_+.
\]
These are bounded unnormalized orders.  Theorem~\ref{V:approximate-arc}
therefore supplies sections of $rA$ with $v/r$ as small as desired.

\subsection*{Approximating vertices and multiplying characters}

Decompose each such section into $T$-weight vectors.  At least one
nonzero component has order no greater than the original section,
since the order of a sum is at least the minimum of its summands'
orders.  If its character is $\alpha$, Lemma~\ref{V:order-weight} gives
\[
 \frac{\langle\alpha,\lambda\rangle}{r}
     =\mu_+-\frac{v}{r}.
\]
Every normalized character $\alpha/r$ of an actual section belongs
to $\mathcal P$: applying nonnegativity of order to each integral
cocharacter gives all the rational supporting-half-space inequalities
from \eqref{V:source-support}.

Choose one integral exposing direction for each vertex of
$\mathcal P$.  In such a direction the maximum is unique.  Compactness
of the polytope then turns approach to the maximal value into approach
to that vertex.  Thus every vertex is a limit of normalized characters
of actual sections.  Since zero is interior, choose finitely many
of these normalized characters close enough to the vertices that
zero remains in their convex hull.  Lemma~\ref{V:convex-product} now gives a nonzero invariant section
of a positive multiple of $A$.
For the trivial torus the transfer theorem has already given the
required section.  This proves the proposition.
\end{proof}
==== END Exact-orders-and-invariant-anticanonical-linear-systems//approximate-weights.tex ====

==== BEGIN Exact-orders-and-invariant-anticanonical-linear-systems//arc-corollary.tex ====
\section{Exact zero order along a formal arc}
\label{sec:arc}

A generic formal arc on \(X\) is a morphism
\[
\gamma:\Spec F[[t]]\longrightarrow X
\]
over an extension field \(F/\C\), whose fraction point maps to the
generic point of \(X\). For a Cartier divisor \(D\), write
\(v_\gamma(D)\) for the order in \(t\) of its local equation pulled
back along the arc. Genericity ensures that a nonzero equation remains
nonzero.

\begin{corollary}
\label{cor:arc}
Let \(X\) be smooth
integral projective over \(\C\), let \(L=-K_X\) have a smooth
semipositive Hermitian metric, and let \(M\) be a line bundle such that
\(-M\) is pseudoeffective. Let \(\gamma\) be a generic formal arc on
\(X\). Suppose that effective integral divisors satisfy
\[
N_j\sim M+m_jL,\qquad
m_j\in\Z_{>0},\quad m_j\longrightarrow\infty,\qquad
v_\gamma(N_j)=o(m_j).
\]
Then, for some integer \(r>0\), there is an effective integral divisor
\[
D\sim rL,\qquad v_\gamma(D)=0.
\]
\end{corollary}

\begin{proof}
If \(X\) is a point, use its zero divisor. Otherwise set \(n=\dim X>0\)
and choose \(G_0\) and \(\ell\) as in
Theorem~\ref{V:exact}. Smooth semipositivity makes \(L\)
nef. If \(\nu<n\), the functional \(\ell\) is nonnegative on effective
divisors, since its remaining factors are nef; by continuity it is
nonnegative on the closed pseudoeffective cone. If \(\alpha\) is the
curvature form normalized to represent \(c_1(L)\), then
\(\alpha^{\nu+1}=0\), so \(\ell(L)=0\). Thus
\[
0\leq\ell(N_j)
=\ell(M)+m_j\ell(L)
=\ell(M)\leq0.
\]
The last inequality uses pseudoeffectivity of \(-M\). This proves
\(\ell(N_j)=0\). For \(\nu=n\), the same conclusion follows from
the definition \(\ell=0\).

The generic fraction point of the arc gives an injection
\[
\C(X)\hookrightarrow F((t)).
\]
Restrict \(\ord_t\) to \(\C(X)^\times\). Its image is either zero
or \(d\Z\) for an integer \(d>0\). Let \(\xi\) be the image of the
closed point of the arc. The local ring \(\OO_{X,\xi}\) maps into
\(F[[t]]\), its maximal ideal maps into \(tF[[t]]\), and its units
have order zero. Hence the restricted valuation is centered at \(\xi\).
If it is zero, this forces \(\xi\) to be the generic point; we use
the zero-valuation case of Theorem~\ref{V:exact}. Otherwise divide by
\(d\) to obtain a normalized discrete valuation centered at \(\xi\).
Orders of local divisor equations agree with arc orders, up to this
fixed normalization.

The assumption \(v_\gamma(N_j)=o(m_j)\) is exactly the required
normalized-order limit, and dividing by \(d\) leaves that limit unchanged.
Theorem~\ref{V:exact} therefore supplies an effective
rational divisor \(N\sim_{\Q}L\) with \(v_\gamma(N)=0\). Choose \(r>0\)
clearing the coefficients of \(N\) and the rational linear equivalence.
Then \(D=rN\) is effective integral, \(D\sim rL\), and
\(v_\gamma(D)=0\).
\end{proof}

In particular, bounded input orders imply the corollary, and the same
output \(D\) satisfies \(v_\gamma(D)<\delta r\) for every \(\delta>0\).
There is also an independent route to the weaker approximate conclusion.
It starts from the numerical nef reduction of Bauer et al.\ 
\cite[Theorem~2.1]{BCEKPRSW2002} and uses the multiplier-ideal quotient
extension theorem of Cao--Demailly--Matsumura
\cite[Theorem~1.1]{CDM2017} to obtain the required jets. The passage
from nef reduction to linear equivalence, and the construction of that
quotient, are additional steps; the corresponding anticanonical
argument assumes irregularity zero. The implication proved here uses
neither that assumption nor the extension theorem. It uses the stronger
exact valuation theorem, whose finite-generation preparation remains essential.

==== END Exact-orders-and-invariant-anticanonical-linear-systems//arc-corollary.tex ====

==== BEGIN Exact-orders-and-invariant-anticanonical-linear-systems//controlled-cone.tex ====
\section{Controlled prime divisors and invariant section rings}
\label{V:controlled-cone-section}

Exact order preservation studies one valuation at a time.  We now give a
different finite-generation argument, in which the entire character of a
section is retained.  The geometric input is a smooth connected rationally
connected projective variety $V$, an algebraic torus $T$ acting on $V$, and
a smooth semipositive metric on $L=-K_V$.  We always use the natural
linearization of $L$.  Rational connectedness supplies
$H^i(V,\mathcal O_V)=0$ for $i>0$ and $\operatorname{Pic}^0(V)=0$.
Put $n=\dim V$.
If $V$ is a point, its constant section proves the desired nonvanishing;
assume below that $\dim V>0$.

Our first aim is to put a finite bound on a collection of divisor
classes.  Call a prime divisor $D$ on $V$ \emph{controlled} if
$aL-D$ is pseudoeffective for some integer $a>0$.  An effective integral
divisor supported on controlled primes is itself dominated by a positive
multiple of $L$: add the finitely many domination relations, with their
multiplicities.  Conversely, each component of a dominated effective
integral divisor is controlled.

We retain linearizations in this bound.  Write
$\operatorname{Pic}^T(V)_{\mathbb Q}$ for the group of linearized line
bundles tensored with $\mathbb Q$.  It is finite dimensional.  Indeed,
$\operatorname{Pic}^0(V)=0$ makes the ordinary rational Picard group
finite dimensional, and the kernel of forgetting a linearization is the
character lattice of $T$.  Every line bundle becomes linearizable after
a positive power.  To see this first for an ample line, a very ample
power gives a projective representation of the torus, which lifts after
a further power.  Write an arbitrary line as a difference of ample lines
to obtain the general assertion.

For an invariant rational divisor $D$, denote its divisor linearization
by $[D]^T$; its canonical rational section is invariant.  We regard a
character $\eta$ as the corresponding linearization of the trivial line.
With the left action on sections, an eigensection of a linearized line
$B^T$ of character $\eta$ has zero divisor of linearized class
$B^T-\eta$.

\begin{proposition}[Finite cone of controlled invariant divisors]
\label{V:controlled-cone}
Let $V$ be smooth, connected, rationally connected and projective over
$\mathbb C$.  Suppose that $L=-K_V$ has a smooth semipositive metric,
and let an algebraic torus $T$ act on $V$.  There is a rational
polyhedral cone in $\operatorname{Pic}^T(V)_{\mathbb R}$ generated by
classes of invariant effective rational divisors and containing $[D]^T$
for every invariant prime $D$ for which $aL-D$ is pseudoeffective for
some integer $a>0$.
\end{proposition}

The generators are effective but need not be controlled.  To prove
Proposition~\ref{V:controlled-cone}, we first show that all but finitely
many controlled primes come from divisors on a common base.  We then
construct an adjoint ring whose invariant generators control the
rational pullbacks of every invariant effective divisor on that base.
An auxiliary boundary makes finite generation possible and disappears
under rational pullback to $V$.

\subsection{The field of functions with controlled poles}

Let $P$ consist of zero and the rational functions whose poles are
supported on controlled primes of $V$.  Sums and products stay in $P$.
If $u\ne0$ belongs to $P$, its zero divisor is linearly equivalent to
its polar divisor, hence is dominated by a multiple of $L$.  Thus
$u^{-1}\in P$ as well.  This proves that $P$ is a $T$-stable subfield
of $\mathbb C(V)$.

The field is relatively algebraically closed.  In fact, if $u$ is
algebraic over $P$ and had a pole at an uncontrolled prime, all
coefficients of a monic equation for $u$ would have nonnegative order
there.  Its leading power would then have strictly smaller order than
every other term, which is impossible.  Also $P$ is finitely generated
over $\mathbb C$.  For completeness, choose a transcendence basis
$t_1,\ldots,t_d$ for $P/\mathbb C$ and extend it to a transcendence
basis of $\mathbb C(V)$.  The algebraic closure of
$\mathbb C(t_1,\ldots,t_d)$ inside this finitely generated field is
finite over $\mathbb C(t_1,\ldots,t_d)$, so its intermediate algebraic
extension $P$ is finite too.

Choose controlled primes $D_1,\ldots,D_s$ spanning the classes of
all controlled primes in $\operatorname{Pic}(V)_{\mathbb Q}$.  If a
controlled prime $D$ is not among them, clearing denominators and
torsion gives a principal divisor supported on $D,D_1,\ldots,D_s$
whose coefficient at $D$ is nonzero.  Its defining function belongs to
$P$.  Thus the valuation of $D$ is nontrivial on $P$, and $D$ is
vertical for any rational map with target field $P$.  In particular,
only finitely many controlled primes can be horizontal.  If
$P=\mathbb C$, this proves Proposition~\ref{V:controlled-cone} directly:
there are only finitely many controlled primes, and we take the cone of
the invariant ones.

Suppose $\operatorname{trdeg}_{\mathbb C}P>0$.
Choose a smooth projective equivariant model $Y$
of $P$ and an equivariant diagram
\begin{equation}\label{V:controlled-model}
 V\xleftarrow{\pi}U\xrightarrow{f}Y,
 \qquad \mathbb C(Y)=P,
\end{equation}
with $U$ smooth projective and $\pi$ birational.  We require that the
strict transform of each prime of $V$ either dominate $Y$ or map onto a
prime divisor of $Y$.  Here is a construction with this property.

Finitely many generators of $P$ have a common controlled polar bound
$N\ge0$.  A sufficiently large full linear system of a multiple of
$N$ contains these generators as ratios, and every ratio of its sections
still has controlled poles.  After linearizing a multiple, its image is
an equivariant projective model with function field $P$.  Over a dense
open, the closures in $V$ of the fibers form a flat family.  Resolve
equivariantly the graph of its rational map to the Hilbert scheme and
pull back the universal family.  This uses the Hilbert-family and
flattening constructions of Grothendieck and Raynaud--Gruson
\cite{Grothendieck1961Hilbert,RaynaudGruson1971}.
Flatness with fixed Hilbert polynomial
bounds fiber dimension by $\dim V-\dim Y$.  The main component is
birational to $V$.  A divisor coming from $V$ therefore cannot map to
codimension at least two: that would force its generic fiber dimension
to exceed this bound.  Resolve the total space equivariantly; strict
transforms retain the asserted images.  Relative algebraic closedness of
$P$ makes $f$ have connected fibers by Stein factorization.  Pullback of
holomorphic one-forms, followed by birational invariance, gives
$\operatorname{Pic}^0(Y)=0$.

Define rational pullback of divisors and linearized line bundles by
\[
 p^\diamond=\pi_*f^*.
\]
For line bundles, one can equivalently pull back on the big open set
where $V\dashrightarrow Y$ is defined and extend reflexively across
codimension two.  Since $V$ is smooth, this extension is a line bundle;
the equivariant structure extends uniquely.  If
$\sigma:Y'\to Y$ is a smooth birational modification, resolve the
diagram above it and use the same notation.  Then $p^\diamond$
commutes with $\sigma^*$ and kills $\sigma$-exceptional divisors,
including their divisor linearizations.  Indeed, a prime of $V$ above
such a divisor would have image of codimension at least two on $Y$,
contrary to the model property.  That property persists over $Y'$:
a horizontal prime remains horizontal, and a prime mapping onto a
divisor of $Y$ maps onto its strict transform, because $\sigma$ is
an isomorphism at its generic point.

We can now reduce the proposition to a statement about divisors on
$Y$.  On a nonempty Zariski open subset of $Y$, the morphism $f$ is
smooth with geometrically connected, hence geometrically integral,
fibers.  If the generic point of a prime $Q$ lies in this open, exactly
one prime of $U$ dominates $Q$, and its multiplicity in $f^*Q$ is one.
Every other component of $f^*Q$ maps into a smaller subset of $Y$;
by the model property it is exceptional over $V$ and pushes to zero.
Consequently, if a prime $D$ of $V$ maps onto such a $Q$, then
\[
 p^\diamond Q=D.
\]
The controlled primes not covered by this statement form a finite
set: they are the finitely many horizontal controlled primes already
identified and the finitely many primes over the excluded base divisors.
If $D$ is invariant, its image $Q$ is invariant, and the equality also
holds for their divisor linearizations.

It therefore remains to find a rational polyhedral cone of invariant
effective classes containing $p^\diamond[R]^T$ for every invariant
effective integral divisor $R$ on $Y$.  Adding the classes of the
finitely many invariant controlled exceptions will finish the proof.
We may construct this cone on any smooth equivariant modification
$\sigma:Y'\to Y$: the effective total transform $\sigma^*R$ has its
canonical divisor linearization, and
$p^\diamond[\sigma^*R]^T=p^\diamond[R]^T$.

\begin{lemma}\label{V:controlled-rank-metric-data}
For the diagram \eqref{V:controlled-model} and any very ample divisor
$H$ on $Y$, some integer $a>0$ satisfies
\begin{equation}\label{V:controlled-domination}
 a\pi^*L-f^*H\quad\text{pseudoeffective}.
\end{equation}
If $E=K_{U/V}$, then
\begin{equation}\label{V:controlled-rank}
 \operatorname{rank}f_*\mathcal O_U(E)=1.
\end{equation}
Moreover there is an invariant effective integral divisor $J$ with
\begin{equation}\label{V:controlled-J}
 (f_*\mathcal O_U(E))^{**}=\mathcal O_Y(J),
 \qquad p^\diamond[J]^T=0,
\end{equation}
where the first identification uses the canonical inclusion of constants.
\end{lemma}

\begin{proof}
The homogeneous coordinate ratios for $H$ belong to $P$, so they have
a common controlled polar bound $N$.  On $U$ the corresponding sections
have a common factor and define $f$; hence
$\pi^*N-f^*H$ is linearly equivalent to an effective divisor.
Combine this with the pseudoeffectivity of $aL-N$.

Twist $f_*\mathcal O_U(E)$ by a large multiple $bH$ so that its global
sections span its generic fiber.  Divide these sections by the Jacobian
section of $E$ and by a fixed rational frame with divisor $bH$.
The resulting rational functions on $U$ have poles bounded by
$E+f^*(bH)$.  Their pushdowns to $V$ have no poles contributed by $E$;
the remaining polar divisor is bounded by $p^\diamond(bH)$.
The preceding common-factor argument shows that this effective divisor
is controlled.  Thus all these functions belong to $P=\mathbb C(Y)$,
and their span over $P$ has dimension at most one.  The Jacobian section
is nonzero, proving \eqref{V:controlled-rank}.

The inclusion of constants makes the rank-one reflexive hull
$\mathcal O_Y(J)$ for an effective integral divisor $J$.  Every
construction is equivariant, so $J$ is invariant.  If a base prime $Q$
has a component above it that is not $\pi$-exceptional, $E$ has
coefficient zero there.  A rational function on the base with a pole at
$Q$ cannot therefore represent a local section of
$f_*\mathcal O_U(E)$.  Its coefficient in $J$ is zero.  The remaining
components of $J$ have only exceptional preimages, which proves the
second identity in \eqref{V:controlled-J}.
\end{proof}

The field construction has now supplied a rank-one adjoint image and
an exceptional correction $J$.  We next use smooth semipositivity on
$V$ to produce a strictly positive metric on $-K_Y+J$.  Its integrable
object will be the canonical section of $J$, which is the precise input
needed for weighted approximation.

\subsection{The weighted metric and its disappearing boundary}

By \eqref{V:controlled-domination}, $\pi^*L$ has a singular metric
whose curvature dominates a positive multiple of $f^*\omega_H$.
Mix its weight with the pulled-back smooth semipositive weight, using a
sufficiently small positive coefficient for the singular term.
Skoda's local integrability theorem \cite{Skoda1972} and compactness
give a metric $h$
such that
\begin{equation}\label{V:controlled-upstairs-metric}
 \Theta_h(\pi^*L)\ge\epsilon f^*\omega_H,
 \qquad \mathcal J(h)=\mathcal O_U,
 \qquad \epsilon>0.
\end{equation}

The identity
\[
 K_{U/Y}+\pi^*L=E-f^*K_Y
\]
identifies the reflexive hull of the adjoint direct image with
$-K_Y+J$.  Apply the singular direct-image metric theorem of
P\u aun--Takayama \cite[Theorem~3.3.5]{PT2018}.  Its multiplier-ideal
inclusion is an isomorphism here by
\eqref{V:controlled-upstairs-metric}.  The rank-one fiber integral
therefore gives a singular metric with weights $\varphi$ on
$-K_Y+J$, extending across divisors and then across codimension two.
It has curvature at least $\epsilon\omega_H$: locally subtract a
potential of this form from the coefficient weight before applying
positivity, and restore it afterward.

For clarity, its required integrability follows directly from the same
integral.  The section induced by the Jacobian section of $E$ is
represented by $s_J$ in a local frame of the reflexive hull.  Integrating
its squared fiber norm against base coordinate volume gives the integral
upstairs of the corresponding canonical-valued form with coefficient
metric $h$.  This is finite by $\mathcal J(h)=\mathcal O_U$ and
properness.  Fiber integration on the smooth locus, which has full
measure, proves
\begin{equation}\label{V:controlled-weighted-integral}
 |s_J|^2e^{-\varphi}\in L^1_{\mathrm{loc}}(Y),
 \qquad \mathcal J(\varphi)\supseteq\mathcal O_Y(-J).
\end{equation}
This is the weighted input of \cite[Theorem~\textup{(Weighted anticanonical approximation)}]{CompanionE}.  It does not require an unweighted
integrability assertion for $e^{-\varphi}$.

We record the complete output of that theorem at this use.  A power of
an ample line on $Y$ is $T$-linearizable, since
$\operatorname{Pic}^0(Y)=0$; the zero divisor of an eigensection of a
very ample power is invariant and ample.  Choose a small positive
rational multiple $A$ of this divisor so that $B=-K_Y+J-A$ still has
a strict curvature bound.  Resolve the full
system of a sufficiently divisible $rB$ together with $J$ by an
equivariant sequence of smooth blowups $\sigma:Y'\to Y$ and write
\[
 \sigma^*|rB|=G+|\mathrm{Mov}|,
\]
with the mobile system base-point free.  Weighted approximation gives
\begin{equation}\label{V:controlled-signed-boundary}
 \Psi=G/r-\sigma^*J-K_{Y'/Y},\qquad
 \operatorname{coeff}\Psi<1.
\end{equation}
Writing $\Psi=\Psi^+-\Psi^-$ with disjoint positive and negative
parts, put $D_0=\Psi^-$.  Then
\begin{equation}\label{V:controlled-adjoint}
 D_0\sim_{\mathbb Q}K_{Y'}+\Psi^+
             +\mathrm{Mov}/r+\sigma^*A.
\end{equation}
Here $\Psi^+$ is an effective simple-normal-crossings klt boundary,
and the last two summands are nef and big together.  The support of
$D_0$ is contained in that of $\sigma^*J$ and the exceptional divisor
of $\sigma$.  The divisors $G,K_{Y'/Y},\Psi^+,D_0$ are invariant and
the free system $|\mathrm{Mov}|$ is torus-stable.  Identity
\eqref{V:controlled-adjoint} is an ordinary rational linear equivalence;
we construct the needed linearized classes separately below.

The support statement has the crucial additional consequence
\begin{equation}\label{V:controlled-D0-zero}
 p^\diamond[D_0]^T=0.
\end{equation}
Indeed $p^\diamond$ kills the effective divisor $\sigma^*J$ by
\eqref{V:controlled-J}, and it kills every $\sigma$-exceptional prime
by the model property.  An effective divisor supported on this union
has zero pushdown.  Its canonical invariant section also pushes to the
constant rational section, so the equality includes the linearization.
No character shift is lost in \eqref{V:controlled-D0-zero}.

\subsection{The multigraded ring and the finite cone}

We now build the cone of base pullbacks required above.  Let $R_0$
be any invariant effective integral divisor on $Y'$.  Since $[D_0]^T$
disappears under $p^\diamond$, the divisor $D_0+tR_0$ for a small
rational $t>0$ has the same rational pullback class as $tR_0$.
We will place these divisors in a finite multigraded ring and express
their invariant sections through its generators.  This is why we need
a whole neighborhood of $[D_0]^T$ in the linearized Picard space,
rather than just the single adjoint ring of $D_0$.

Because $\sigma$ is a sequence of smooth blowups, there is an effective
exceptional rational divisor $F$ with $-F$ relatively ample.  One
constructs it successively by giving sufficiently small positive
coefficients to the later exceptional divisors.  For small rational
$\delta>0$, $\sigma^*A-\delta F$ is ample and
$\Psi^++\delta F$ is still a klt boundary with simple normal
crossings.  If $\sigma$ is the identity, take $F=0$.  Thus
\eqref{V:controlled-adjoint} expresses $D_0$ as $K_{Y'}$ plus that
boundary and an ample rational class.

Choose a fixed small ample rational part $A_*$.  Every rational class
in a sufficiently small neighborhood of the underlying class of $D_0$
can be written
\[
 K_{Y'}+A_*+\Delta_i,
\]
with $\Delta_i\ge0$ and klt.  For a finite set of classes, choose
general members of sufficiently divisible very ample systems for the
remaining ample parts, divided by their multiples.  They can be chosen
transverse to one another and to the fixed boundary, with coefficients
less than one.  Consequently their union has simple normal crossings.
Small character changes only change linearizations, so the same
construction gives a neighborhood in
$\operatorname{Pic}^T(Y')_{\mathbb R}$.

Choose rational vertices $U_1^T,\ldots,U_s^T$ of a polytope with
$[D_0]^T$ in its interior.  After a common multiple $k$, let
$\mathcal E_i$ be integral linearized representatives of $kU_i^T$.
The ordinary ring
\begin{equation}\label{V:controlled-ring}
 R=\bigoplus_{\boldsymbol\ell\in\mathbb N^s}
 H^0\!\left(Y',\bigotimes_{i=1}^s
                      \mathcal E_i^{\otimes\ell_i}\right)
\end{equation}
is finitely generated by the multigraded adjoint-ring theorem of
Cascini--Lazi\'c \cite[Theorem~A]{CasciniLazic2012}; compare
\cite[Theorem~3.2(1)]{CortiLazic2013}.  We take $k$ sufficiently
divisible to turn the rational linear equivalences above into
isomorphisms of line bundles.  This identifies \eqref{V:controlled-ring}
with a divisible subring of the stated adjoint ring.  The linearizations
give an algebraic $T$-action on $R$, independently of the general
boundary representatives used to prove finite generation.

Here is the invariant-ring step explicitly.  Enlarge finitely many
multihomogeneous generators of $R$ to their finitely many character
components.  They define an equivariant multigraded surjection from a
polynomial ring to $R$.  Projection to weight zero is exact for a torus,
so the invariant polynomial ring surjects onto $R^T$.  If the variable
weights are $\eta_1,\ldots,\eta_N$, its invariant monomials have
exponents
\[
 (a_1,\ldots,a_N)\in\mathbb N^N,
 \qquad \sum a_i\eta_i=0.
\]
These are the lattice points of a rational polyhedral cone; Gordan's
lemma makes their semigroup finitely generated.  Thus $R^T$ has
finitely many nonzero multihomogeneous generators $u_1,\ldots,u_b$.

Let $\mathcal C_0$ be the cone generated by the rational pullbacks
$p^\diamond[\operatorname{div}(u_j)]^T$.  These are classes of
invariant effective divisors on $V$.  We claim that $\mathcal C_0$
contains $p^\diamond[R_0]^T$ for \emph{every} invariant effective
integral divisor $R_0$ on $Y'$.  Choose a small rational $t>0$ such
that $[D_0]^T+t[R_0]^T$ belongs to the vertex polytope.  Express it as
a rational convex combination of the $U_i^T$.  Clearing denominators
and torsion gives an equivariant isomorphism between a positive
multiple of $D_0+tR_0$ and one multidegree in
\eqref{V:controlled-ring}.  Its canonical section gives a nonzero
element of $R^T$ in that multidegree.  Express it as a polynomial in
the $u_j$ and retain the terms of that multidegree.  At least one
nonzero monomial occurs, and its degree is a nonnegative integral sum
of generator degrees.  Thus the corresponding linearized class lies
in their cone.  Only equality of homogeneous degrees is used here;
the selected monomial need not have the same zero divisor as the
original section.  Apply $p^\diamond$, use
\eqref{V:controlled-D0-zero}, and divide by the positive factor
containing $t$.  This proves the claim.

Apply this claim to the total transforms of invariant effective
divisors on the original base $Y$.  Compatibility of $p^\diamond$
with $\sigma^*$ gives the required cone of their rational pullbacks.
Add the invariant effective classes of the finitely many controlled
primes excluded in the initial comparison with base divisors.  The
resulting rational polyhedral cone proves
Proposition~\ref{V:controlled-cone}.

\subsection{A weight bound and character separation}

The cone gives finitely many effective generators.  To obtain an
invariant anticanonical section, we must still put the \emph{specified
natural linearization} of $L$ in their cone.  The following fixed-point
calculation supplies the sign needed for separation.  It is a bound on
some cohomology weight, rather than the weight-zero index theorem.

\begin{lemma}\label{V:bounded-above-weight}
Let $V$ be smooth connected projective with
$H^i(V,\mathcal O_V)=0$ for $i>0$, and let
$v:\mathbb G_m\to\operatorname{Aut}(V)$ be an algebraic action.
Give $L=-K_V$ its natural linearization.  There is a constant $C$ such
that, for every sufficiently large integer $m$, some
$H^q(V,mL)$ has a $v$-weight at most $C$.
\end{lemma}

\begin{proof}
If the action is trivial, all weights are zero and the ordinary
Riemann--Roch polynomial $\chi(V,mL)$ has constant term one, so the
assertion follows.  Otherwise use the holomorphic fixed-point formula
of Atiyah--Bott, in its fixed-component equivariant index form
\cite{AtiyahBott1968,AtiyahSegal1968,AtiyahSinger1968}.
For a fixed component $Z$, let $a$ be the nonzero
weights of its normal tangent bundle, with Chern roots $x_{a,j}$.
The anticanonical fiber weight is $b_Z=\sum_{a,j}a$.  Its contribution
to the alternating cohomology character is
\[
 t^{mb_Z}\int_Z e^{m c_1(L|_Z)}\operatorname{td}(Z)
          \prod_{a,j}(1-t^{-a}e^{-x_{a,j}})^{-1}.
\]
The integral is a finite cohomological expansion.  Denote the alternating
sum of the cohomology characters by $\chi_v(V,mL)$.  Grouping by $b_Z$
writes it as
\begin{equation}\label{V:weight-windows}
 \chi_v(V,mL)=\sum_b t^{mb}R_b(m,t),
\end{equation}
where each $R_b$ is polynomial in $m$ with rational functions of $t$
as coefficients.

At $m=0$, the left side is one.  As $t\to0$, each component having
a positive normal tangent weight has contribution tending to zero;
the other components have $b_Z\le0$.  Hence at least one $b\le0$
has $R_b(0,t)\ne0$.  Multiply \eqref{V:weight-windows} by one fixed
nonzero Laurent polynomial clearing the denominators of every
coefficient.  For the selected $b$, the exponents lie in
$mb+[-C_0,C_0]$.  Some coefficient is a nonzero polynomial in $m$,
so this block is nonzero for all large $m$.  The finitely many windows
for different $b$ are disjoint for large $m$.  Thus the multiplied
character has a nonzero exponent bounded above independently of $m$.
Removing a fixed Laurent-polynomial factor changes the minimum
exponent only by a fixed number.  The original virtual character
therefore has such an exponent too, and it occurs in at least one
actual cohomology group.
\end{proof}

We now obtain an independent proof of the rationally connected
special case of Theorem~\ref{V:invariant-main}. Its additional output
is the controlled-divisor cone just constructed. The proof uses that
cone and character separation, and will supply the fiberwise section
in the locally constant MRC application of Section~\ref{V:sec-mrc-descent}.

\begin{theorem}\label{V:controlled-invariant-nonvanishing}
Let $V$ be a smooth connected rationally connected projective complex
variety with smoothly semipositive $-K_V$.  For every algebraic torus
$T$ acting on $V$, there is an integer $m>0$ such that
\[
 H^0(V,-mK_V)^T\ne0
\]
with the natural anticanonical linearization.
\end{theorem}

\begin{proof}
Let $\mathcal C$ be the cone in Proposition~\ref{V:controlled-cone}.
Suppose $L^T\notin\mathcal C$.  Rational polyhedral separation gives
a rational functional $\lambda$ on the linearized Picard space with
\[
 \lambda(L^T)>0,\qquad \lambda(\mathcal C)\le0.
\]
After positive rescaling, its restriction to characters is pairing with
a cocharacter $v:\mathbb G_m\to T$, possibly zero.
Lemma~\ref{V:bounded-above-weight} gives cohomology classes with
$v$-weights uniformly bounded above and with unbounded positive $m$.

Average a K\"ahler form and the smooth metric over the maximal compact
torus.  Smooth-coefficient hard Lefschetz then gives the equivariant
surjection
\[
 H^0(V,\Omega_V^{n-q}\otimes(m+1)L)
       \longrightarrow H^q(V,K_V+(m+1)L)=H^q(V,mL).
\]
This is the smooth case of the Demailly--Peternell--Schneider theorem
\cite[Theorem~0.1]{DPS2001}.
The coefficient shift $m+1$ and its equivariant identification use the
natural action on $L$.  Pass to a subsequence with fixed
$p=n-q$, and choose $T$-eigensections $\alpha_m$ of
$\Omega_V^p\otimes(m+1)L$ whose $v$-weights are bounded above.

Take the generic span of their untwisted lines in $\Omega_V^p$,
of rank $s$, and saturate its determinant to a line bundle
$M\subseteq\bigwedge^s\Omega_V^p$.  This line inherits a
$T$-linearization.  The inverse-determinant pseudoeffectivity theorem
of Lazi\'c--Matsumura--Peternell--Tsakanikas--Xie
\cite[Theorem~4.1]{LMPTX2023}, following Ou's generic nefness theorem
\cite[Theorem~1.4]{Ou2023}, gives $-M$ pseudoeffective.  Its hypotheses apply because $V$ is
smooth projective and $-K_V$ is nef; the exterior construction is a
direct summand of a cotangent tensor power, and saturation of the line
persists in that summand.  If $p=0$, $M=\mathcal O_V$ instead.

Choose finitely many $\alpha_m$ forming a generic basis.  Each further
nonzero $\alpha_m$ can be completed to a generic basis using this
fixed set.  Their wedge gives a nonzero eigensection of $M+kL$, for
unbounded positive $k$, whose $v$-weight is still bounded above.  The
wedge lands in $M+kL$ by saturation in codimension one and reflexive
extension.  Let $D_k$ be its divisor and $\eta_k$ its character.
Then $kL-D_k$ has class $-M$, so each component of $D_k$ is controlled.
Each is also invariant, because $T$ is connected and $D_k$ is the
zero divisor of an eigensection.  Consequently $[D_k]^T\in\mathcal C$.
On the other hand,
\[
 \lambda([D_k]^T)=\lambda(M^T)+k\lambda(L^T)
                              -\langle\eta_k,v\rangle>0
\]
for large $k$, a contradiction.

Thus $L^T\in\mathcal C$.  A real nonnegative expression in its
finitely many rational generators admits a rational nonnegative
expression.  Clear denominators and torsion in the linearized Picard
group and multiply the canonical invariant sections of these effective
divisors.  The product is nonzero since $V$ is integral, and is a
section of a positive multiple of the naturally linearized $L$.
If $L^T$ is zero in the rational class space, clearing its torsion gives
the constant invariant section directly.
\end{proof}
==== END Exact-orders-and-invariant-anticanonical-linear-systems//controlled-cone.tex ====

==== BEGIN Exact-orders-and-invariant-anticanonical-linear-systems//endpoint.tex ====
\section{Selecting a generator on the boundary ray}
\label{sec:generator}

For a field \(K\) containing \(\C\), a real-valued valuation is a map
\(v:K^\times\to\R\) satisfying
\[
v(fg)=v(f)+v(g),\qquad
v(f+g)\geq\min\{v(f),v(g)\}\quad(f+g\ne0).
\]
All valuations in this section are trivial on \(\C^\times\). The zero
valuation is allowed. If a homogeneous polynomial has the form
\(f x^a y^b\), we apply \(v\) to its coefficient \(f\); the formal
variables \(x,y\) record degrees only.

The relevant boundary ray consists of degrees \((0,b)\). A sequence
approaches that ray after normalization when its first degree is
negligible compared with its total degree.

\begin{lemma}
\label{lem:endpoint}
Let \(R\subset K[x,y]\) be a finitely generated \(\C\)-subalgebra,
graded by \(\N^2\) with \(\deg x=(1,0)\), \(\deg y=(0,1)\), and
\(R_{0,0}=\C\). Let \(v\) be a real-valued valuation of \(K\), trivial
on \(\C^\times\). Suppose that nonzero homogeneous elements
\[
s_j=f_jx^{a_j}y^{b_j}\in R,\qquad d_j=a_j+b_j>0,
\]
satisfy
\[
\frac{a_j}{d_j}\longrightarrow0,
\qquad
\frac{v(f_j)}{d_j}\longrightarrow\gamma\in\R.
\]
Then every finite set of nonzero homogeneous generators of \(R\)
contains a generator \(f x^0y^b\), with \(b>0\), such that
\[
\frac{v(f)}b\leq\gamma.
\]
\end{lemma}

\begin{proof}
Discard constant generators and write the others as
\[
r_i=g_ix^{u_i}y^{w_i},
\qquad u_i,w_i\in\N,\qquad u_i+w_i>0.
\]
There are finitely many indices \(i\). Because the generators are
homogeneous, \(s_j\) can be expressed as a sum of monomials of degree
\((a_j,b_j)\). One nonzero monomial has coefficient valuation at most
\(v(f_j)\): otherwise the valuation inequality for the sum would
contradict its coefficient \(f_j\). Let \(e_{ij}\) be the exponents
of the generators in such a monomial. Nonzero scalar coefficients
have valuation zero, so
\begin{equation}
\label{eq:monomial}
\sum_i e_{ij}u_i=a_j,\qquad
\sum_i e_{ij}w_i=b_j,\qquad
\sum_i e_{ij}v(g_i)\leq v(f_j).
\end{equation}

We compare the factors with \(u_i>0\) and those with \(u_i=0\).
Every positive \(u_i\) is at least one; hence
\[
\sum_{u_i>0}e_{ij}\leq a_j.
\]
Since the generating set is finite, there are constants \(C_1,C_2\geq0\)
such that these factors contribute at most \(C_1a_j\) to the second
degree and at least \(-C_2a_j\) to the valuation. For example, take
\[
C_1=\max_i w_i,\qquad
C_2=\max_i\max\{0,-v(g_i)\}.
\]
The set of nonconstant generators is nonempty because \(d_j>0\).

Some generator must have \(u_i=0\). Otherwise
\eqref{eq:monomial} would give \(b_j\leq C_1a_j\), contrary to
\(a_j/d_j\to0\). For every such generator \(w_i>0\), so we may define
\[
\lambda=\min_{u_i=0}\frac{v(g_i)}{w_i}.
\]
Let \(c_j\) be the second degree contributed by the factors with
\(u_i>0\). Thus \(0\leq c_j\leq C_1a_j\). The remaining factors
contribute second degree \(b_j-c_j\), and their valuation is at least
\(\lambda(b_j-c_j)\). Equation~\eqref{eq:monomial} therefore gives
\[
v(f_j)\geq
\lambda b_j-\bigl(C_2+|\lambda|C_1\bigr)a_j.
\]
Divide by \(d_j\) and pass to the limit. Since \(b_j/d_j\to1\), we
obtain \(\gamma\geq\lambda\). A generator realizing the finite minimum
\(\lambda\) proves the assertion.
\end{proof}

The proof needs the finite list of actual coefficients \(g_i\).
It does not follow from knowing only a limiting cone of numerical
classes. This is the feature that will matter for divisor section
rings in the next subsection.

\subsection{Removing a birational correction}
\label{sec:geometric}

We now specify the geometric data to which the lemma applies. The main
point is that the correction on the base need not have valuation zero.
It will disappear under birational pushforward before the last
valuation comparison.

All varieties below are integral projective varieties over \(\C\).
For a Cartier divisor \(B\) on a normal variety \(Y\), we identify
sections with rational functions:
\[
H^0(Y,\OO_Y(B))
=\{h\in\C(Y)^\times:\divisor_Y(h)+B\geq0\}\cup\{0\}.
\]
For two Cartier divisors \(B_1,B_0\), their bigraded section ring is
\begin{equation}
\label{eq:section-ring}
R(Y;B_1,B_0)
=\bigoplus_{a,b\in\N}
H^0\bigl(Y,\OO_Y(aB_1+bB_0)\bigr)x^ay^b
\subset\C(Y)[x,y].
\end{equation}
Projectivity and integrality give \(H^0(Y,\OO_Y)=\C\), as required
in Lemma~\ref{lem:endpoint}.

We also use rational powers of rational functions, interpreted as
elements of
\(\C(Y)^\times\otimes_{\Z}\Q\), written multiplicatively.
The divisor map and every valuation extend \(\Q\)-linearly to this
group. Each element becomes an ordinary rational function after a
positive integral power is taken. No extension of the function field
by actual roots is intended.

A valuation of \(\C(X)\) is centered on \(X\) if its valuation ring
dominates \(\OO_{X,\xi}\) for some point \(\xi\in X\). For a Cartier
divisor \(D\), define \(v(D)\) by applying \(v\) to a local defining
rational function at \(\xi\). This is independent of the equation,
because units have valuation zero, and extends \(\Q\)-linearly to
rational Cartier divisors. In particular, \(v(D)\geq0\) for every
effective rational Cartier divisor \(D\). The zero valuation is
centered at the generic point.

\begin{proposition}
\label{prop:geometric}
Let \(X,Y\) be smooth integral projective complex varieties, let \(W\)
be normal and projective, and let
\[
X\xleftarrow{\ p\ }W\xrightarrow{\ f\ }Y
\]
be morphisms, with \(p\) birational and \(f\) surjective and
equidimensional. Identify \(\C(Y)\) with its induced subfield of
\(\C(X)=\C(W)\). Fix a centered discrete valuation \(v\) of
\(\C(X)\), allowing also the zero valuation.

Let \(A\geq0\) be a rational divisor on \(Y\) such that
\(p_*f^*A=0\). Let \(D_0,D_1\) be rational Cartier divisors on \(X\)
and put
\[
D(t)=(1-t)D_0+tD_1\qquad(0\leq t\leq1).
\]
For each prime divisor \(Q\) of \(Y\), define
\begin{equation}
\label{eq:minimum-divisor}
P(t)=\sum_Q
\min_{E:\,f(E)=Q}
\frac{\coeff_E p^*D(t)}{\ord_E(f^*Q)}\,Q,
\end{equation}
where \(E\) ranges over the prime divisors of \(W\) dominating \(Q\).
Assume that \(P(t)=(1-t)P(0)+tP(1)\) throughout \([0,1]\).

Suppose that rational numbers \(t_j\in[0,1]\) tend to zero and that
\(h_j\in\C(Y)^\times\otimes_{\Z}\Q\) satisfy
\[
N_j=D(t_j)+\divisor_X(h_j)\geq0,\qquad v(N_j)\longrightarrow0.
\]
Finally, suppose that for some \(\epsilon\in\Q_{>0}\) and integer
\(k>0\), the divisors
\[
B_1=k\bigl(A+\epsilon P(1)\bigr),\qquad
B_0=k\bigl(A+\epsilon P(0)\bigr)
\]
are Cartier and the ring \(R(Y;B_1,B_0)\) is finitely generated.
Then there is an effective rational divisor
\[
N\sim_{\Q}D_0,\qquad v(N)=0.
\]
\end{proposition}

The minimum in \eqref{eq:minimum-divisor} is over a finite nonempty
set, since \(f\) is surjective and equidimensional. Only finitely many
base primes occur: the pullbacks of \(D_0,D_1\) have fixed finite
support. Each coefficient is a minimum of finitely many rational
affine functions, so \(P(t)\) is piecewise affine. In applications one
restricts to a sufficiently small rational segment ending at \(t=0\)
and rescales the parameter to obtain the stated affine hypothesis.
If \(Y\) is a point, the sum is empty and means \(P(t)=0\).

\begin{proof}
Write
\[
t_j=\frac{a_j}{d_j},\qquad d_j=a_j+b_j>0,
\qquad a_j,b_j\in\N.
\]
Scale the pair \((a_j,b_j)\) by a positive integer whenever necessary
to clear denominators. The ratio \(a_j/d_j=t_j\) is unchanged.
In particular we can arrange that
\[
s_j=h_j^{\epsilon k d_j}
\]
is an ordinary rational function on \(Y\).

The effective divisor \(N_j\) is rational Cartier because \(X\) is
smooth, so \(p^*N_j\geq0\). For a prime \(E\) of \(W\) dominating a
base prime \(Q\), let \(e_E=\ord_E(f^*Q)>0\). Taking its coefficient
in this pullback gives
\[
\frac{\coeff_E p^*D(t_j)}{e_E}
+\coeff_Q\divisor_Y(h_j)\geq0.
\]
Taking the minimum proves
\begin{equation}
\label{eq:base-effectivity}
P(t_j)+\divisor_Y(h_j)\geq0.
\end{equation}
The affine hypothesis now gives
\[
a_jB_1+b_jB_0=k d_j\bigl(A+\epsilon P(t_j)\bigr).
\]
Since \(A\geq0\), equation~\eqref{eq:base-effectivity} shows that
\(s_jx^{a_j}y^{b_j}\) is a nonzero homogeneous element of
\(R(Y;B_1,B_0)\).

The valuation of \(D(t)\) is affine in \(t\), since the two divisors
have fixed support and valuation is \(\Q\)-linear. Thus
\[
v(h_j)=v(N_j)-v(D(t_j))\longrightarrow-v(D_0),
\qquad
\frac{v(s_j)}{d_j}\longrightarrow-\epsilon k\,v(D_0).
\]
Apply Lemma~\ref{lem:endpoint} to this section ring and to the
restriction of \(v\) to \(\C(Y)\). The lemma only uses this restriction
as a real-valued valuation. We obtain a
generator represented by a rational function \(r\), of degree
\((0,b)\) with \(b>0\), such that
\[
\frac{v(r)}{\epsilon k b}\leq-v(D_0).
\]
Put \(h_*=r^{1/(\epsilon k b)}\) in the rational-power group.
The section inequality for \(r\) gives
\begin{equation}
\label{eq:endpoint-effectivity}
P(0)+A/\epsilon+\divisor_Y(h_*)\geq0.
\end{equation}

We have selected the endpoint section on \(Y\). It remains to show that
it gives an effective divisor on \(X\) without retaining the correction
\(A/\epsilon\).

A prime of \(W\) is horizontal if it dominates \(Y\), and vertical
otherwise. Principal divisors pulled back from \(Y\) have no horizontal
components. Consequently the horizontal coefficients of \(p^*D(t_j)\)
are nonnegative, by effectivity of \(p^*N_j\); the same is true for
\(p^*D_0\) by passage to the limit. Every vertical prime dominates a
base prime, by equidimensionality. For such a prime \(E\) over \(Q\),
the minimum defining \(P(0)\) and
\eqref{eq:endpoint-effectivity} show that its coefficient in
\[
p^*D_0+
f^*\bigl(\divisor_Y(h_*)+A/\epsilon\bigr)
\]
is nonnegative. This divisor is therefore effective.

Push it forward by \(p\). The assumption \(p_*f^*A=0\) removes the
correction, while pullback and pushforward of the principal divisor
give \(\divisor_X(h_*)\). Hence
\[
N=D_0+\divisor_X(h_*)\geq0,\qquad N\sim_{\Q}D_0.
\]
Centering gives \(v(N)\geq0\). The selected generator gives
\(v(N)=v(D_0)+v(h_*)\leq0\), proving equality.
\end{proof}

The proof never evaluates the correction \(f^*A\) at \(v\).
Its removal is a divisor identity under pushforward. This permits an
exceptional correction with positive valuation on a higher model.

==== END Exact-orders-and-invariant-anticanonical-linear-systems//endpoint.tex ====

==== BEGIN Exact-orders-and-invariant-anticanonical-linear-systems//exact-proof.tex ====
\section{Proof of exact attainment}
\label{V:exact-proof}

The preceding construction depends only on the curvature-null field.
We now place the given divisors in a two-dimensional section ring and
apply Proposition~\ref{prop:geometric}.

\begin{proof}[Proof of Theorem~\ref{V:exact}]
Choose two input divisors with distinct exponents, write them as
$N_1,N_2$, and denote their exponents by $u_1,u_2$. For a rational
vector $t=(t^1,t^2)$ put $D(t)=t^1N_1+t^2N_2$. Let $t_j$ solve
\[
 u_1t_j^1+u_2t_j^2=1,\qquad t_j^1+t_j^2=1/m_j.
\]
After discarding finitely many terms, these points approach from one
side the rational point $t_*$ given by the same first equation and
$t_*^1+t_*^2=0$. Hence $D(t_*)\sim_{\Q}L$, and there are rational
powers of rational functions $h_j$ such that
\begin{equation}\label{V:principal-corrections}
 N_j/m_j=D(t_j)+\divisor_X(h_j).
\end{equation}
A positive power clearing denominators is an ordinary rational function.
Its polar divisor is bounded by a positive linear combination of $N_1,N_2$,
because the left side of \eqref{V:principal-corrections} is effective.
That bound is null. Thus the power belongs to $k$, and we may regard
$h_j$ as an element of $k^\times\otimes_{\Z}\Q$.

If $k=\C$, the principal correction vanishes. The fixed-support limit
of the effective divisors $D(t_j)$ is effective, and linearity on that
finite support gives
\[
 v(D(t_*))=\lim_j v(N_j)/m_j=0.
\]
Then $N=D(t_*)$ proves the assertion.

Otherwise use the equidimensional model of Section~\ref{V:null-field}.
Define the divisor $P(t)$ on $Y$ by
\[
 P(t)=\sum_Q\min_{E:\,f(E)=Q}
 \frac{\coeff_E p^*D(t)}{\ord_E f^*Q}\,Q.
\]
Its support is finite and fixed, since the support of $p^*D(t)$ lies
in that of $p^*N_1+p^*N_2$. Each coefficient is a minimum of finitely
many rational linear functions. On a sufficiently short rational
segment $[t_*,t_0]$ containing a tail of $t_j$, all these minima are
linear. Reparameterize this segment by $u\in[0,1]$, so that $u=0$
corresponds to $t_*$ and $u=1$ to $t_0$.

Apply the correction-ring theorem to $P(t_0),P(t_*)$. It gives a
finitely generated bigraded section ring for
\[
 B_1=k_0(A_0+\epsilon P(t_0)),\qquad
 B_0=k_0(A_0+\epsilon P(t_*)).
\]
All the hypotheses of Proposition~\ref{prop:geometric} now hold:
$p_*f^*A_0=0$ by the toroidal-volume theorem; the normalized effective
divisors in \eqref{V:principal-corrections} approach $D(t_*)$ through
principal corrections from $Y$; their valuations tend to zero; and
the minimum divisor is affine on the chosen segment. That proposition
gives an effective rational divisor $N\sim_{\Q}D(t_*)\sim_{\Q}L$
with $v(N)=0$.
\end{proof}

In particular, no valuation of the exceptional correction is assumed to
vanish. The proof of Proposition~\ref{prop:geometric} first removes it
by pushforward and then uses centeredness on $X$. This is also why the
fixed valuation may have center of any codimension.
==== END Exact-orders-and-invariant-anticanonical-linear-systems//exact-proof.tex ====

==== BEGIN Exact-orders-and-invariant-anticanonical-linear-systems//exact-weights.tex ====
\section{Exact attainment of source weights}
\label{subsec:valuation-extremal-weights}

We prove Theorem~\ref{V:invariant-main} by constructing the small input
orders required by Theorem~\ref{V:exact}. Let $X$ be smooth connected
projective with smoothly semipositive $L=-K_X$, and assume
$H^i(X,\mathcal O_X)=0$ for all $i>0$. Let an algebraic torus $T$
act effectively on $X$. If $X$ is a point, its constant section proves
the theorem. Otherwise use $G_0$ and $\ell$ as in
\eqref{V:null-functional}. The anticanonical line $L=\det T_X$ has its
natural linearization.  The action on sections is the left action
induced by the action on the total space.

\subsection{Source weights and cohomology}

Fix a cocharacter $\lambda:\mathbb G_m\to T$.  Its unique source
component $S$ attracts a dense open subset as $t\to0$, by the
Bia\l ynicki-Birula decomposition \cite{BB1973}; see also the smooth attracting decomposition in \cite[Theorem~1.5 and Corollary~7.3]{JelisiejewSienkiewicz2019}.  Let $c$ be
the weight of $L$ on $S$.  This is the sum of the positive normal
tangent weights, so $c>0$ for a nonzero cocharacter.  For the zero
cocharacter take $S=X$ and $c=0$.  The rational attracting map to $S$
is dominant.  Resolve it and pull back holomorphic forms; birational
invariance and Hodge symmetry give
$H^i(S,\mathcal O_S)=0$ for $i>0$.

Average the logarithmic weights of the metric over the circle in
$\lambda(\mathbb G_m)$.  For a nonzero vector $u\in L_x$, put
$q_u(s)=\log|\lambda(e^s)u|$.  The logarithmic norm of this
nonvanishing holomorphic orbit section is superharmonic by semipositive
curvature.  Circle invariance therefore makes $q_u$ concave in $s$.
There is a smooth function $\theta:X\to\mathbb R$ such that
$q_u'(s)=\theta(\lambda(e^s)x)$; it is independent of the scaling
of $u$ and equals the fiber weight on each fixed component.  If $x$
is in the dense attracting set, continuity gives
$q_u'(s)\to c$ as $s\to-\infty$.  Concavity then gives
$\theta(x)\le c$.  Density and continuity extend this inequality
to all of $X$, so every fixed-component fiber weight is at most $c$.

For all sufficiently large integers $l$, the weight $(l-1)c$ occurs in
some $H^q(X,(l-1)L)$.  Indeed, expand the equivariant holomorphic
Lefschetz formula at infinity in its weight variable $z$.
For a normal tangent weight $w$, the inverse normal factor is
$(1-z^{-w}e^{-x})^{-1}$.  If $w>0$, it has constant term one and lower
powers; if $w<0$, all its powers are strictly negative.  Every component
other than the source has a negative normal weight.  Maximality of $c$
therefore makes the coefficient of $z^{(l-1)c}$ precisely
$\chi(S,(l-1)L|_S)$.  This polynomial in $l-1$ has value
$\chi(S,\mathcal O_S)=1$ at zero, and is nonzero for all large $l$.
See \cite{AtiyahSegal1968,AtiyahSinger1968} for the fixed-component formula.

Equivariant hard Lefschetz with the smooth semipositive coefficient
$lL$ \cite[Theorem~0.1]{DPS2001} gives a surjection
\[
 H^0(X,\Omega_X^{n-q}\otimes lL)\longrightarrow
 H^q(X,K_X+lL)=H^q(X,(l-1)L).
\]
A circle-invariant K\"ahler form makes this map equivariant.  On an
infinite subsequence we obtain nonzero eigensections $u_l$ of
$\Omega_X^a\otimes lL$, with one fixed $a$, of weight $(l-1)c$ for
$\lambda$.  Decompose each into $T$-weight components and select a
nonzero component; all the components have the same $\lambda$-weight.
Thus the $u_l$ can be chosen to be $T$-eigensections.

\subsection{Determinants with bounded orbit order}

The use of saturated determinants follows the method of
Lazi\'c--Peternell \cite[Lemma~4.1]{LP2018} and its anticanonical
adaptation in \cite[Lemma~5.1]{LMPTX2023}. Those statements include
a numerical-nontriviality hypothesis on the twisting line. Here the
positive unbounded twists are already available, and the generic-span
and wedge construction below proves exactly the assertion needed,
also for a numerically trivial $L$.

Each section $u_l$ defines a map $L^{-l}\to\Omega_X^a$. Take the
span of their image lines over $\C(X)$, of rank $e>0$, and saturate
its determinant line in $\bigwedge^e\Omega_X^a$. This construction is
independent of a choice of rational frame for $L$.  Denote the resulting
line bundle by $M$.  Rank-one saturation in a locally free sheaf is
reflexive and hence a line bundle on the smooth variety; its natural
linearization extends from the subbundle locus across codimension two.
The needed curvature-intersection sign is $\ell(M)\leq0$; it is
automatic when $\nu=n$.  Suppose $\nu<n$.  The smooth metric on
$L=-K_X$ determines a positive volume form whose Ricci form is
$\alpha$.  For every $b>0$, scale this volume to the volume of the
K\"ahler class $G_0+bL$ and apply the Calabi--Yau theorem
\cite{Yau1978}.  The resulting K\"ahler metric has Ricci form
$\alpha\ge0$.  K\"ahler curvature symmetries identify the mean
curvature of its tangent bundle with the Ricci endomorphism.  Duals
and exterior powers therefore make the mean curvature of
$\bigwedge^e\Omega_X^a$ nonpositive.

Apply the subbundle curvature inequality to the inclusion of $M$
where it is nonzero.  Its logarithmic norm extends as a locally
quasiplurisubharmonic function across the zero set.  Its Hessian has
a smooth lower bound there, so the singular trace measure is
nonnegative and the extension preserves the degree inequality.  Hence
\[
 M\cdot(G_0+bL)^{n-1}\leq0.
\]
Since $L^{\nu+1}=0$, all powers above $b^\nu$ vanish in this
intersection polynomial.  The coefficient of $b^\nu$ is a positive
multiple of $\ell(M)$, so letting $b\to\infty$ proves the required
sign.

Choose a fixed finite collection of the $u_l$ whose image lines form
a generic basis. Every further nonzero $u_l$ can be completed to a basis
by $e-1$ members of this collection. There are finitely many choices;
on an infinite subsequence one choice is fixed. Taking these wedges
gives nonzero eigensections of $M+m_jL$, where $m_j$ is the sum of the
$e$ twists and tends to infinity. Their weights are $(m_j-e)c$, since
each of the $e$ constituent forms contributes its twist minus one.  Their zero divisors $N_j$
satisfy $\ell(N_j)=0$, by effectivity and $\ell(M)\leq0$.
Let $v$ be the order obtained by substituting $\lambda(t)x$ for the
generic point $x$ of $X$.  Properness extends this orbit to a formal arc
at zero, with center at the generic point of $S$; its order is a
discrete valuation up to rescaling, or is zero.  If $c_M$ is the fiber
weight of $M$ at $S$, then
\[
 v(N_j)=m_jc+c_M-(m_j-e)c=c_M+ec.
\]
This is Lemma~\ref{V:order-weight} for the declared left action. The
constant $c_M+ec$ retains the determinant-rank shift.

\subsection{Exact source weights and an invariant product}

Theorem~\ref{V:exact} now gives an effective rational
representative of $L$ of order zero at the generic source.  After
clearing denominators, some section of $mL$ restricts nontrivially to
$S$.  Its weight decomposition consequently contains a normalized
weight whose value on $\lambda$ is exactly $c$.  If $T$ is trivial,
this is already a nonzero invariant section.  Assume from now on that
$T$ has positive dimension.

The function $c$ is the maximum of the finitely many $T$-fixed fiber
weights evaluated on $\lambda$: the source is $T$-stable and contains
a $T$-fixed point by the Borel fixed-point theorem.  It thus extends
to a continuous rational polyhedral homogeneous function on real
cocharacters.  It is strictly positive away from zero.  Indeed a
nontrivial zero or negative locus would, by rational polyhedrality,
contain a nonzero rational cocharacter, contrary to the source-weight
calculation.  Every normalized section weight is bounded above by $c$
on each cocharacter, by the same orbit-order test, and the preceding
argument attains this upper bound.  The comparison between fixed-fiber
and section-weight polytopes for semiample linearized line bundles,
and the resulting invariant-product criterion, are established in
M\"uller~\cite[Proposition~3.13 and Corollary~3.14]{Muller2025}.
Here exact order attainment supplies the supporting section weights.

Testing a lattice basis of cocharacters and its negatives bounds
all normalized section weights.  Their closed convex hull is therefore
compact; its support function equals $c$ on rational cocharacters by
exact attainment, and then on all real cocharacters by continuity.
Strict positivity of $c$ away from zero puts zero in its interior.
Choose a small simplex surrounding zero in that convex hull and
approximate its vertices by finite convex combinations of actual
normalized weights.  Sufficiently close approximations still surround
zero, so the finitely many actual weights in these combinations have
zero in their convex hull.  Lemma~\ref{V:convex-product} now multiplies finitely many corresponding
eigensections to produce a nonzero invariant section.

This proves Theorem~\ref{V:invariant-main}. For an arbitrary torus action, replace $T$ by its effective quotient
$T/\ker(T\to\operatorname{Aut}(X))$. The entire kernel, including
its finite part, acts trivially on the natural anticanonical line, so
a section invariant under the quotient is invariant under $T$.
==== END Exact-orders-and-invariant-anticanonical-linear-systems//exact-weights.tex ====

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==== BEGIN Exact-orders-and-invariant-anticanonical-linear-systems//introduction.tex ====
\section{Introduction}
\label{V:introduction}

An effective divisor can have normalized order tending to zero while
vanishing at the same center in every degree. For example, order one
becomes negligible after division by an unbounded sequence of integers.
Producing a section which is actually nonzero at that center is a stronger
problem. We prove an exact attainment theorem for divisors approaching
the anticanonical class of a smooth projective variety with a smoothly
semipositive anticanonical bundle. We then use generic orbit arcs to
translate this control of orders into statements about characters and
invariant sections.

The broader nonvanishing problem asks whether a nef anticanonical
bundle on a smooth projective variety has a nonzero section in some
positive power.  Numerical effectivity and effectivity in rational
linear equivalence are distinct here.  Lazi\'c, Matsumura, Peternell,
Tsakanikas and Xie prove numerical effectivity for smooth projective
threefolds with nef anticanonical class and develop nonvanishing methods based on twisted
differential forms \cite[Theorem~A and Section~5]{LMPTX2023}.
M\"uller proves anti-log-canonical nonvanishing for projective klt
threefold pairs with nef anti-log-canonical divisor \cite[Corollary~B]{Muller2025}.
Our positivity hypothesis is smooth semipositivity, and our first
result retains both rational linear equivalence and the order at a
prescribed valuation.

Let $X$ be a smooth connected projective complex variety, of dimension
$n>0$, and put $L=-K_X$. Fix a smooth Hermitian metric on $L$ with
semipositive curvature form $\alpha$, normalized to represent $c_1(L)$.
Let $\nu$ be the maximum complex rank of $\alpha$, and let $G_0$ be an
ample line bundle. Define
\begin{equation}\label{V:null-functional}
 \ell(D)=
 \begin{cases}
 D\cdot L^\nu\cdot G_0^{n-\nu-1},&\nu<n,\\
 0,&\nu=n.
 \end{cases}
\end{equation}
This functional is nonnegative on effective divisors. Moreover
$\ell(L)=0$, since $\alpha^{\nu+1}=0$ when $\nu<n$.
A discrete valuation $v$ of $\C(X)$ is \emph{centered on $X$} if its
valuation ring dominates the local ring of a point of $X$. We allow the
zero valuation as well. The value $v(D)$ of a Cartier divisor is the
valuation of a local defining rational function at the center. It is
well defined because units have value zero, and extends rationally to
rational divisors.

\begin{samepage}
\begin{theorem}[Exact attainment of zero order]\label{V:exact}
With $X,L,\alpha,\nu,G_0,\ell$ as above, let $v$ be either a discrete
valuation of $\C(X)$ centered on $X$ or the zero valuation. Let $M$
be any fixed line bundle. Suppose that
effective integral divisors $N_j$ and positive integers $m_j\to\infty$
satisfy
\[
 N_j\sim m_jL+M,\qquad \ell(N_j)=0,\qquad
 \frac{v(N_j)}{m_j}\longrightarrow0.
\]
There is an effective rational divisor $N\sim_{\Q}L$ such that
$v(N)=0$.
\end{theorem}
\end{samepage}

The null intersection condition is the geometric input; the auxiliary
line $M$ is arbitrary. No irregularity or Euler-characteristic hypothesis
is used. The theorem preserves rational linear equivalence, which is
necessary for constructing sections. A limit in the numerical effective
cone would not give its conclusion.

Here is a useful special case. A \emph{generic formal arc} on $X$ is a
morphism $\gamma:\Spec F[[t]]\to X$, for an extension field $F/\C$,
whose fraction point maps to the generic point of $X$. If $-M$ is
pseudoeffective and $N_j\sim m_jL+M$ is effective, nef intersection
gives $0\le\ell(N_j)=\ell(M)\le0$. Thus, if the orders of $N_j$
along $\gamma$ are $o(m_j)$, Theorem~\ref{V:exact} supplies a section
of some positive multiple of $L$ whose order along $\gamma$ is exactly
zero. Corollary~\ref{cor:arc} verifies the centered-valuation and
integrality details, including a point variety and the zero valuation.

\subsection{Why finite generation attains the limit}
The proof first places the principal corrections to the given divisors
in one function field, and then uses finite generation to attain the
limiting class.  Let $k\subset\C(X)$ be the field of rational functions
whose polar divisors have $\ell$-value zero.  It is relatively
algebraically closed.  Two fixed input divisors give representatives
of $L+M/m_j$ on a common finite support, and the principal corrections
between those representatives and $N_j/m_j$ belong to $k$.

The geometric task is to find a model $Y$ of $k$ on which these
corrections lie in a finitely generated divisor section ring.  On a smooth resolution $\pi:Z\to X$ of the rational map to $Y$,
the null-polar field makes the section space of the effective divisor
$K_{Z/X}$ one-dimensional on the generic fiber over $Y$.  The null
intersection also forces the curvature kernel to be vertical on a
nonempty open set.  Section~\ref{V:null-field} verifies these two
inputs to the volume-metric theorem of \emph{Metric descent and rank-preserving
contractions} \cite[Theorem \textup{(Toroidal volume metric)}]{CompanionG}.
It gives an effective rational correction $A_0$ on $Y$ whose pullback
is exceptional over $X$.  The semipositive metric it supplies on $-K_Y+A_0$ has
locally integrable squared frame norms and is smooth and strictly
positive on an ordinary open set.  The correction-ring theorem of
\emph{Integrable metrics and effectivity with controlled boundary}
\cite[Theorem \textup{(Finite generation near a correction divisor)}]{CompanionE}
then gives finite generation in the two rational divisor directions
selected by the input sequence.

The remaining argument uses the actual rational functions representing
sections.  In a finitely generated bigraded ring, a monomial whose
normalized degree approaches a boundary ray uses off-ray generators
in negligible degree.  For each input section, the valuation inequality
selects a monomial with no larger valuation.  One boundary-ray generator
therefore has sufficiently small normalized valuation.  Its divisor
is effective up to the correction $A_0$.  Birational pushforward removes
that correction; centeredness then turns the resulting upper bound on
the valuation into exact equality with zero.  This order of operations
allows the exceptional correction itself to have positive valuation.

The endpoint argument belongs to the study of asymptotic vanishing and
multigraded section rings.  Ein, Lazarsfeld, Musta\c{t}\u{a}, Nakamaye
and Popa explain how finite generation controls asymptotic orders and
base loci \cite[Section~4]{ELMNP2006}, with the corrected polyhedral
argument \cite[Proposition~1.1]{ELMNP2023Erratum}; Corti and Lazi\'c
give a multigraded formulation \cite[Theorem~3.5]{CortiLazic2013}.
Lemma~\ref{lem:endpoint} gives the direct one-ray, one-valuation
argument used here.  The geometric finite-generation input rests on
the adjoint-ring theory of Birkar, Cascini, Hacon and McKernan
\cite{BCHM2010} and Cascini--Lazi\'c \cite{CasciniLazic2012}, in the
form \cite[Theorem~3.2(1)]{CortiLazic2013}.

\subsection{From orders to invariant sections}
For a torus action, the differential of the action gives $-K_X$ its
natural linearization.  A nonzero section can transform by a nontrivial
character, so scalar nonvanishing does not by itself give invariance.
Along the generic orbit of a one-parameter subgroup, an eigensection
of weight $w$ has order $\mu-w$, where $\mu$ is the fiber weight at
the limit.  Exact order zero therefore attains that fiber weight.

M\"uller's equivariant nonvanishing theorem gives invariant
anti-log-canonical sections for projective sub-log-canonical pairs
with semiample anti-log-canonical divisor, under commutative linear
algebraic group actions \cite[Theorem~C]{Muller2025}.  For smooth
varieties with a torus action, we obtain the following result under
smooth semipositivity and a cohomological hypothesis.

\begin{theorem}\label{V:invariant-main}
Let $X$ be smooth connected projective over $\C$, with $-K_X$ smoothly
semipositive and $H^i(X,\OO_X)=0$ for every $i>0$. For every algebraic
torus $T$ acting on $X$, some positive anticanonical multiple has a
nonzero $T$-invariant section for its natural linearization.
\end{theorem}

The proof starts with the source component, which attracts the generic
orbit as the one-parameter subgroup tends to zero.  The attracting
decomposition of Bia\l ynicki-Birula identifies this component
\cite{BB1973}.  The holomorphic fixed-point formula of Atiyah--Bott,
Atiyah--Segal and Atiyah--Singer
\cite{AtiyahBott1968,AtiyahSegal1968,AtiyahSinger1968} supplies
cohomology classes of the corresponding extremal weight.  Semipositive
hard Lefschetz \cite[Theorem~0.1]{DPS2001} turns them into twisted
differential forms.  Saturated determinants, following the method of
Lazi\'c--Peternell \cite[Lemma~4.1]{LP2018} and its anticanonical
adaptation \cite[Lemma~5.1]{LMPTX2023}, give effective
divisors with bounded order along the generic orbit.  Their character
calculation retains the determinant-rank shift.  Theorem~\ref{V:exact}
then supplies actual normalized section weights whose convex hull
contains zero.  A rational convex combination becomes
an invariant product after clearing denominators.  The comparison of
fixed-fiber and section-weight polytopes in the semiample setting is
developed in \cite[Proposition~3.13 and Corollary~3.14]{Muller2025};
here exact attainment supplies the supporting section weights.

Together with the structural inputs described below, the invariant
sections yield the following scalar consequence: for every smooth
connected projective complex variety $X$ with smoothly semipositive
$-K_X$, there is an integer $m>0$ such that
\[
 H^0(X,-mK_X)\ne0.
\]
Sections~\ref{V:isometric-descent} and \ref{V:sec-mrc-descent} give
two descents.  The first uses the compact-monodromy structure theorem
of \emph{Anticanonical nonvanishing from smooth semipositivity}
\cite[Theorem \textup{(Finite cover with compact torus monodromy)}]{CompanionH},
based on the universal-cover structure of
Demailly--Peternell--Schneider and its rational-connectedness refinement
by Campana--Demailly--Peternell \cite{DPS1996,CDP2015}.
The second uses a locally constant maximal rationally connected
fibration, supplied by Matsumura--Wang \cite{MatsumuraWangV3},
following the earlier smooth structure theorem of Cao--H\"oring
\cite{CaoHoring}.  It follows M\"uller's invariant-section reduction
\cite[Theorem~2.2]{Muller2025} and includes the unipotent part of the
commutative algebraic monodromy.  The compact-monodromy argument
uses only the structural and finite-cover norm results of the
nonvanishing companion; its nonvanishing theorem is not an input.

\subsection{Independent methods and their additional outputs}
The exact-order theorem and its invariant application are followed by
two independent constructions.  Their purpose is to retain different
information during the passage from twisted sections to anticanonical
sections.

The first uses nef reduction and extension.  Given bounded orders in
$|m_jL+M|$, with $q(X)=0$ and $-M$ pseudoeffective, it separates a
fixed divisor of zero arc order from a line bundle on a base.  A
fiber-maximum metric makes that line nef; the nef reduction of Bauer
and collaborators \cite{BCEKPRSW2002} removes its numerical null
directions.  Extension of first jets by the theorem of
Cao--Demailly--Matsumura \cite{CDM2017} proves that the remaining base
line is big and yields arbitrarily small normalized arc orders.
Theorem~\ref{V:approximate-arc} records this approximate transfer,
which the exact theorem already strengthens under the same hypotheses.
The independent proof also gives the bigness and jet construction.
Its character application, Proposition~\ref{V:extremal-invariants},
uses $q(X)=0$ and $\chi(X,\OO_X)\ne0$ instead of vanishing of all
higher structure-sheaf cohomology, together with the stated equivariant
very ample linearization.

There is a second ending to this nef-reduction construction.  Starting
with invariant sections of $jL-G$ for unbounded $j$, where $G$ is a
fixed pseudoeffective linearized line bundle, it forms a base from
invariant rational functions with pseudoeffectively controlled poles.
It then extends one nonzero invariant class in a multiplier-ideal
quotient.  Theorem~\ref{V:invariant-conversion} thus removes the fixed
error while keeping invariance throughout.  This ending needs
$q(X)=0$ but does not require the first-jet or weight-polytope steps.
The common tools occupy Section~\ref{V:sec-nef}, the arc and character
applications Sections~\ref{V:sec-transfer} and \ref{V:sec-extremal},
and the invariant conversion Section~\ref{V:sec-invariant-field}.

The other finite-generation construction assumes rational
connectedness and controls an entire collection of divisor classes.
A prime divisor $D$ is controlled when $aL-D$ is pseudoeffective for
some positive integer $a$.  Proposition~\ref{V:controlled-cone} places
the naturally linearized classes of all invariant controlled primes
in one rational polyhedral cone generated by invariant effective
rational divisors.  Its generators need not themselves be controlled.
Here the effectivity companion's weighted-approximation theorem
produces an adjoint boundary on a base; its correction disappears
under birational pushforward.  Finite generation of the full
multigraded ring and then of its invariant subring gives the cone.
Character separation supplies another proof of invariant nonvanishing
in the rationally connected case, as well as the fiberwise input used
in the locally constant MRC descent.  Section~\ref{V:controlled-cone-section}
keeps this weighted metric construction distinct from the unweighted
integrability used for exact attainment.

The proof of exact attainment is organized as follows.
Section~\ref{sec:generator} proves the finite-generator lemma and its
birational application.  Sections~\ref{V:null-field} and
\ref{V:exact-proof} construct the null-polar base and prove
Theorem~\ref{V:exact}; Section~\ref{sec:arc} gives the formal-arc
consequence.  Section~\ref{V:sec-order-weight} establishes the
order/weight identity, and Section~\ref{subsec:valuation-extremal-weights}
proves Theorem~\ref{V:invariant-main}.  The independent methods then
supply the additional conclusions just described, before the paired
monodromy descents finish the scalar nonvanishing consequence.
==== END Exact-orders-and-invariant-anticanonical-linear-systems//introduction.tex ====

==== BEGIN Exact-orders-and-invariant-anticanonical-linear-systems//invariant-field.tex ====
\section{Invariant conversion by a controlled function field}
\label{V:sec-invariant-field}

The preceding method works one cocharacter at a time and then multiplies
sections.  We now give a second nef-reduction argument which keeps
invariance throughout.  Its starting point is a sequence of invariant
twisted sections.  The base is defined by invariant functions with
pseudoeffectively controlled poles, and the final extension produces
one nonzero invariant quotient class directly.  In particular this
argument does not need the jet-generation or weight-polytope steps.

\begin{theorem}\label{V:invariant-conversion}
Let $X$ be smooth connected projective with $q(X)=0$, and let
$A=-K_X$ carry a smooth semipositive Hermitian metric.  Let an
algebraic torus $T$ act on $X$, and use the natural linearization on
$A$.  Let $G$ be a pseudoeffective $T$-linearized line bundle.
Suppose that for an unbounded set $I$ of positive integers there are
nonzero sections
\[
 \delta_j\in H^0(X,jA-G)^T\qquad(j\in I).
\]
Then $H^0(X,mA)^T\ne0$ for some integer $m>0$.
\end{theorem}

\begin{proof}
We use the maximal compact torus $T_c$ for averaging.  Average the
local weights of the given smooth metric using the natural
linearization of $A$, and call the resulting invariant smooth
semipositive metric $h_A$.  Use its pullbacks on all subsequent models.
Define $F\subset\mathbb C(X)$ to consist of the invariant rational
functions $u$ for which
\begin{equation}\label{V:controlled-poles}
 bA-D_\infty(u)\quad\hbox{is pseudoeffective for some integer }b>0,
\end{equation}
and include zero.  Here $D_\infty(u)$ denotes the effective polar
divisor on $X$.  Sum and product preserve this condition by the
usual pole inequalities, and inversion preserves it because a
rational function's zero and pole divisors are linearly equivalent.
Thus $F$ is a field.

It is relatively algebraically closed in $\mathbb C(X)$.  For if
$u^d+\sum_{i<d}c_i u^i=0$ with $c_i\in F$, then at a pole of $u$
of order $a>0$ some term satisfies
$\operatorname{ord}(D_\infty(c_i))\ge(d-i)a$.  Consequently
$D_\infty(u)\le\sum_{c_i\ne0}D_\infty(c_i)$.
The root is invariant as well: the connected torus permutes the
finite set of roots trivially.  It therefore belongs to $F$.
The field $F$ is finitely generated over $\mathbb C$.  Indeed take
a transcendence basis $x$ for $F/\mathbb C$ and extend it by $z$ to
one for $\mathbb C(X)/\mathbb C$.  The finite degree of
$\mathbb C(X)/\mathbb C(x,z)$ bounds $[F:\mathbb C(x)]$, because
$z$ is algebraically independent over $F$.  Hence the latter degree
is finite.

Choose a projective model of $F$ and apply
Lemma~\ref{V:flat-model} equivariantly.  Denote the smooth base by
$Y$, the normal equidimensional model by $V$, and a smooth equivariant
resolution by $S$, with maps
\[
 p:V\to X,\quad f:V\to Y,
 \qquad p:S\to X,\quad f:S\to Y.
\]
The base action is trivial and $f$ has connected fibers.  Put
$A_S=p^*A$ and $E=K_S-p^*K_X\ge0$.  The natural section
$s_E$ is invariant; this follows directly from pullback of top forms.

We record a domination consequence of the definition of $F$.
For every smooth birational model of $X$ mapping to a smooth
projective base $Y_1$ with $\mathbb C(Y_1)\subset F$, and every ample
Cartier divisor $H$ on $Y_1$, there is an integer $b>0$ with
\begin{equation}\label{V:field-domination}
 b p^*A-f_1^*H\quad\hbox{pseudoeffective}.
\end{equation}
Choose sections $s_0,\ldots,s_N$ generating $hH$ with $s_0\ne0$.
The ratios $s_i/s_0$ are in $F$.  On every prime of the total space,
generation ensures that one pulled-back section has zero order, so
\[
 \operatorname{div}(f_1^*s_0)
 \le p^*\sum_iD_\infty(s_i/s_0).
\]
The pole inequality is valid also on exceptional primes, since the
pole divisor of the pullback of a rational function is bounded by
the pullback of its polar divisor.  Sum the pseudoeffective bounds
\eqref{V:controlled-poles} and then add $(h-1)f_1^*H$ to obtain
\eqref{V:field-domination}.

\subsection*{An invariant relative section with vertical minima}

Choose $r<s$ in $I$.  For $j\in I$ with $j>s$, the invariant function
\[
 u_j=\frac{\delta_j^{s-r}\delta_r^{j-s}}
             {\delta_s^{j-r}}
\]
belongs to $F$.  Its poles are bounded by
$(j-r)\operatorname{div}(\delta_s)$, and
$sA-\operatorname{div}(\delta_s)\sim G$ is pseudoeffective.
At each horizontal prime of $V\to Y$, the order of $u_j$ is zero.
If $v_j$ denotes the order there of $p^*\delta_j$, then
\[
 (s-r)v_j=(j-r)v_s-(j-s)v_r.
\]
Since $v_j\ge0$ for arbitrarily large $j$, we have $v_s\ge v_r$.
Thus the invariant rational section
$t=p^*(\delta_s/\delta_r)$ of $(s-r)p^*A$ has no horizontal pole.
If $Y$ is a point, $t$ is regular by normality and descends to $X$,
proving the assertion.

Otherwise, over each prime $Q$ of $Y$, take the minimum of
$\operatorname{ord}_\Gamma(t)/\operatorname{ord}_\Gamma(f^*Q)$
over components $\Gamma$ dominating $Q$.  Let $k_0$ clear their
finitely many nonzero denominators.  Subtracting the pullback of the
resulting Cartier divisor $B$ on $Y$ gives an invariant section
\begin{equation}\label{V:relative-invariant}
 0\ne\tau\in H^0(V,a p^*A-f^*B)^T,
 \qquad a=k_0(s-r)>0.
\end{equation}
It has zero order on at least one component over every base prime.
There are no vertical primes above higher codimension on $V$, so
nonnegative orders on these primes and on horizontal primes prove
regularity.  Pull $\tau$ to $S$, and write $P_\tau$ for its effective
divisor.  Every base line bundle in this construction has trivial
linearization.

\subsection*{Rank one for the invariant direct image}

For each $l\ge1$ consider
\[
 \mathcal E_l=f_*(K_{S/Y}+(al+1)A_S).
\]
Over a dense base-change open set its invariant part has rank one.
To prove the upper bound, twist by a sufficiently ample $H$ on $Y$
so that $H-K_Y$ is ample and the twisted direct image is generated.
Averaging shows that invariant global sections generate the invariant
fiber spaces there.  The ratio of any two invariant global sections
lies in $F$: if the denominator has divisor $N$ on $S$, then
\[
 N\sim alA_S+E+f^*(H-K_Y).
\]
Equation~\eqref{V:field-domination} bounds the last summand
pseudoeffectively by a multiple of $A_S$.  Pushforward to the smooth
$X$ preserves pseudoeffectivity and kills $E$, so the polar divisor
of the ratio on $X$ satisfies \eqref{V:controlled-poles}.
Since all these ratios come from $Y$, the invariant fiber space is
at most one-dimensional.  It is at least one-dimensional, because
in local frames $\beta$ of $B$ and $\tau_Y$ of $-K_Y$ the section
\[
 v_l=(\tau f^*\beta)^l s_E f^*\tau_Y
\]
is a nonzero invariant relative canonical section with coefficient
$(al+1)A_S$.

We can now apply the invariant version of
Lemma~\ref{V:bounded-maximum}.  The action on the base and its line
$B$ is trivial, $s_E$ is the natural invariant Jacobian section,
$\tau$ is invariant, and the coefficient metric is smooth and
compact-invariant.  The rank-one assertion has been proved for
\emph{every} $l\ge1$.  The lemma therefore gives locally bounded
semipositive weights on $B$, so $B$ is nef.  Notice that this step
uses the invariant adjoint rank and the orthogonal averaging
projection; ordinary rank one has not been assumed.

\subsection*{Nef reduction and a nonzero quotient class}

Since $Y$ is dominated rationally by $X$, pullback of holomorphic
one-forms gives $q(Y)=0$.  Apply Lemma~\ref{V:nef-descent}, lift $S$
further equivariantly, and clear the common denominators.  Retaining
$p:S\to X$, $A_S=p^*A$, $E=K_S-p^*K_X$, and writing $g:S\to Z$,
we have a nef Cartier divisor $L$ and an invariant section
\[
 \tau\in H^0(S,aA_S-g^*L)^T,
 \qquad P_\tau=\operatorname{div}(\tau)\ge0.
\]
The line-bundle identification comes from a base with trivial action,
so invariance is preserved.  A zero-dimensional $Z$ finishes the
proof by descending $\tau$.  Otherwise let $b=\dim Z>0$.

Fix very ample $H$ on $Z$.  Equation~\eqref{V:field-domination},
using $\mathbb C(Z)\subset F$, gives a pseudoeffective divisor
$R_0=b_0A_S-g^*H$ for some integer $b_0>0$.
Take a $T_c$-invariant semipositive singular metric $h_R$ on it.
Such a metric is obtained by averaging its local weights relative
to an invariant smooth reference metric.  The quasi-plurisubharmonic
potentials have uniformly bounded $L^1$ norm under the compact group,
so the average is well defined, has semipositive curvature
distributionally, and gives invariant plurisubharmonic weights.
Choose a small rational $\epsilon>0$ with
$\mathcal I(h_R^\epsilon)=\mathcal O_S$.

Apply Lemma~\ref{V:sesha} with $c=2b$.  For sufficiently large $k$,
the ample rational divisor $B_k=kL+\epsilon H$ has Seshadri constant
at least $4b$ at very general points.  Choose such a point $z$ with
some $x\in g^{-1}(z)$ where $g$ is submersive and $P_\tau\cup E$
is absent.  Put $\beta=3b$.  The pole construction in the same lemma,
applied to $B_k/2$ with $c=\beta/2$, gives a smooth positive metric
of curvature $\omega_k/2$ and a function $\psi$ with neat analytic
singularities, smooth away from $z$, such that
\[
 i\partial\bar\partial\psi\ge-\tfrac12\omega_k,
 \qquad \psi\le\tfrac{\beta}{2}\log|w|^2+O(1).
\]
Give $B_k$ the induced smooth metric of curvature $\omega_k$, and
put $\Psi=g^*\psi$.  As in the first-jet argument, surjectivity of
$g$ preserves the local analytic-singularity form and prevents the
pullback from being identically $-\infty$.  Thus $\Psi$ has neat
analytic singularities, is smooth away from $g^{-1}(z)$, and satisfies
\[
 i\partial\bar\partial\Psi\ge-\tfrac12 g^*\omega_k,
 \qquad
 \Psi\le\tfrac{\beta}{2}\log|w|^2+O(1)
\]
in transverse base coordinates at the chosen submersion point.

Choose an integer $p_0\ge k$ sufficiently large, set $m=ap_0$, and
use the decomposition
\begin{equation}\label{V:invariant-metric-decomposition}
 (m+1)A_S
 =k(aA_S-g^*L)+g^*B_k+\epsilon R_0
        +(m+1-ka-\epsilon b_0)A_S.
\end{equation}
The last coefficient is nonnegative.  Give the four terms respectively
the divisor metric of $\tau^k$, the indicated smooth base metric,
$h_R^\epsilon$, and the remaining smooth semipositive metric.
They define a $T_c$-invariant metric $h_J$ on $J=(m+1)A_S$ with
\[
 i\Theta_{h_J}+(1+s)i\partial\bar\partial\Psi\ge0
 \qquad(0\le s\le1).
\]
Its multiplier ideal is exactly
\begin{equation}\label{V:fixed-divisor-ideal}
 \mathcal I(h_J)=\mathcal O_S(-kP_\tau).
\end{equation}
Divisibility by the equation of $kP_\tau$ is sufficient because the
remaining singular factor $h_R^\epsilon$ is integrable.  It is
necessary at the generic smooth point of each prime divisor: the
remaining local weights are bounded above, and an uncancelled
integral pole already prevents square integrability there.  The
resulting codimension-one divisibility implies divisibility as a
holomorphic germ, since the local rings of the smooth variety are
factorial.  This proves \eqref{V:fixed-divisor-ideal}.

Apply \cite[Theorem~1.1]{CDM2017}.  It gives
\[
 H^0\!\left(S,\mathcal O_S(K_S+J)\otimes\mathcal I(h_J)\right)
 \longrightarrow
 H^0\left(S,\mathcal O_S(K_S+J)\otimes
       \frac{\mathcal I(h_J)}{\mathcal I(h_Je^{-\Psi})}\right)
 \quad\hbox{surjectively}.
\]
The quotient is supported over $z$.  In a local frame $e_L$ of $L$
near $z$, the section
\[
 s_E(\tau g^*e_L)^{p_0}
\]
is a section of the larger ideal twist near the whole fiber, because
$p_0\ge k$.  Its quotient class glues with zero away from that fiber
and is invariant, as the base action and its frame action are trivial.
The class is nonzero.  At $x$ its coefficient is a unit; the local
weight of $h_J$ is bounded above, while the transverse pole has
exponent $\beta/2=3b/2\ge b$.  Thus its squared norm for
$h_Je^{-\Psi}$ is not locally integrable in the $b$ transverse
variables.

Lift this nonzero class and average the lift over $T_c$.
The ideals are invariant, the map is equivariant, and averaging
preserves the chosen invariant quotient class.  We obtain a nonzero
invariant section of
\[
 K_S+J=mA_S+E.
\]
Remove the canonical exceptional factor and push to $X$.
It is a section of $mA$ off a codimension-two subset, hence extends
holomorphically over the smooth $X$.  It remains nonzero and
$T_c$-invariant, therefore $T$-invariant.  This proves the theorem.
\end{proof}

\begin{corollary}\label{V:invariant-differentials}
Let $X,T,A$ satisfy the geometric hypotheses of
Theorem~\ref{V:invariant-conversion}.  If for a fixed
$0\le p\le\dim X$ there are nonzero invariant sections of
$\Omega_X^p\otimes jA$ for an unbounded set of positive integers
$j$, then $H^0(X,mA)^T\ne0$ for some $m>0$.
\end{corollary}

\begin{proof}
For $p=0$ the conclusion is part of the assumption.  Otherwise use
the determinant construction of Section~\ref{V:sec-extremal}, now
with invariant forms throughout.  Form
the saturated invariant subsheaf $\mathcal S\subset\Omega_X^p$
spanned generically by the chosen forms, and put
$G=-(\bigwedge^{\operatorname{rank}\mathcal S}\mathcal S)^{**}$.
This is a line bundle on the smooth $X$, with its induced
linearization.  The inclusion into the tensor power of $\Omega_X^1$
and \cite[Theorem~4.1]{LMPTX2023} make $G$ pseudoeffective.
A form with arbitrarily large twist can be completed to a generic
basis using a fixed finite collection of forms.  Their wedges,
followed by the reflexive determinant map, give nonzero invariant
sections of $j'A-G$, with $j'$ unbounded.  Theorem~\ref{V:invariant-conversion}
applies.
\end{proof}

\begin{remark}\label{V:invariant-index-application}
Under the geometric hypotheses of Theorem~\ref{V:invariant-conversion},
in particular $q(X)=0$, this conversion also supplies the nef-reduction
route from invariant cohomology.  The invariant-index theorem
\cite[Theorem \textup{(Invariant anticanonical index)}]{CompanionI} applies to the compact
complex manifold $X$ and its maximal compact torus, with the natural
linearization of $A=-K_X$.  It says that the invariant index agrees
with a rational polynomial on every positive multiple of some
integer, and that this polynomial's value at zero is the actual
invariant index at zero.  The connected torus acts trivially on
ordinary cohomology by homotopy and therefore on its Hodge summands.
Consequently this zero value is $\chi(X,\mathcal O_X)$.  If it is nonzero,
some fixed $q$ has nonzero invariant $H^q(X,mA)$ for unbounded $m$.
The compact-averaged hard Lefschetz map with coefficient $(m+1)A$
produces the invariant twisted forms needed by
Corollary~\ref{V:invariant-differentials}.  Thus
$H^0(X,mA)^T\ne0$ for some $m>0$ whenever
$\chi(X,\mathcal O_X)\ne0$.  In particular the hypothesis $H^q(X,\mathcal O_X)=0$ for every $q>0$ gives the zero
index value $1$ and $q(X)=0$.  The index theorem and the conversion
play separate roles: the former retains the actual constant term,
whereas the latter produces an invariant section without presuming
an equivariant version of an ordinary nonvanishing criterion.
\end{remark}

The compact-factor application of these invariant conclusions is
given in the descent section.  Only the geometric structure and
finite-cover norm are needed there; neither of the proofs above
uses the headline nonvanishing theorem.
==== END Exact-orders-and-invariant-anticanonical-linear-systems//invariant-field.tex ====

==== BEGIN Exact-orders-and-invariant-anticanonical-linear-systems//isometric-descent.tex ====
\section{Descent with compact isometric monodromy}
\label{V:isometric-descent}

We first apply the invariant results when the residual monodromy lies
in a compact torus. The geometric structure is supplied by the
finite-cover theorem of \emph{Anticanonical nonvanishing from smooth semipositivity} \cite[Theorem~2.1]{CompanionH}, based
on Demailly--Peternell--Schneider and Campana--Demailly--Peternell
\cite{DPS1996,CDP2015}. Its nonvanishing theorem is not used in the
argument below.

Let $X$ be smooth connected projective with smoothly semipositive
$-K_X$. The cited structure theorem gives a connected finite \'etale
cover $\nu:S\to X$ and an equivariant product description of its
universal cover
\[
 \widetilde X=\C^a\times F\times Z,
 \qquad S=(\C^a\times F\times Z)/\Lambda.
\]
Here $\Lambda$ acts by translations on $\C^a$, trivially on the
compact simply connected Ricci-flat projective factor $F$, and through
a compact connected real torus on $Z$. The variety $Z$ is smooth
projective, its anticanonical line is smoothly semipositive, and
$H^i(Z,\OO_Z)=0$ for every $i>0$. The compact action has algebraic
torus closure $G\subset\operatorname{Aut}(Z)$, with its natural
anticanonical linearization. There are invariant canonical frames
$\eta_0,\eta_F$ on $\C^a,F$. Any factor may be a point.

Theorem~\ref{V:invariant-main} gives a nonzero
$s\in H^0(Z,K_Z^{-m})^G$ for some $m>0$. If $Z$ is a point, take
$m=1$ and $s=1$. Equally, the controlled-field conversion and its
index application can be used here: $Z$ has irregularity zero and
actual index one, and its natural compact action meets precisely those
hypotheses.

Under the determinant identity for the product tangent bundle, the
section
\[
 \operatorname{pr}_0^*(\eta_0^{-m})\otimes
 \operatorname{pr}_F^*(\eta_F^{-m})\otimes
 \operatorname{pr}_Z^*s
\]
belongs to $H^0(\widetilde X,K_{\widetilde X}^{-m})$. It is nonzero,
since the first two factors never vanish and $s$ is nonzero on a
nonempty open set of $Z$. Translations preserve $\eta_0$, the action
on $F$ preserves $\eta_F$, and the residual action on $Z$ fixes $s$.
Thus this product is $\Lambda$-invariant and descends to a nonzero
holomorphic section of $K_S^{-m}$. It is algebraic by projectivity.
The finite \'etale norm \cite[Lemma~2.2]{CompanionH}
then gives a nonzero section of $K_X^{-m\deg\nu}$.

The product is used on the universal cover; the finite cover $S$ need
not split as a product. The next argument starts from a locally constant
rationally connected fibration instead. Its algebraic monodromy can
have a nontrivial unipotent part, which requires an additional
fixed-vector argument.
==== END Exact-orders-and-invariant-anticanonical-linear-systems//isometric-descent.tex ====

==== BEGIN Exact-orders-and-invariant-anticanonical-linear-systems//main.tex ====
\documentclass[11pt,reqno]{amsart}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern,microtype,amsmath,amssymb,mathtools,mathrsfs}
\input{glyph-unicode}
\usepackage[margin=1.08in]{geometry}
\usepackage{enumitem}
\usepackage[hyphens]{url}
\usepackage{xcolor}
\usepackage[unicode,colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}\newtheorem{remark}[theorem]{Remark}
\newcommand{\C}{\mathbb C}\newcommand{\Q}{\mathbb Q}
\newcommand{\R}{\mathbb R}\newcommand{\N}{\mathbb N}\newcommand{\Z}{\mathbb Z}
\newcommand{\OO}{\mathcal O}\newcommand{\PP}{\mathbb P}
\newcommand{\PE}{\overline{\operatorname{Eff}}}
\newcommand{\ddc}{i\partial\bar\partial}
\DeclareMathOperator{\divisor}{div}\DeclareMathOperator{\ord}{ord}
\DeclareMathOperator{\Spec}{Spec}\DeclareMathOperator{\coeff}{coeff}
\DeclareMathOperator{\rank}{rank}\DeclareMathOperator{\rk}{rk}
\DeclareMathOperator{\Pic}{Pic}\DeclareMathOperator{\Supp}{Supp}
\DeclareMathOperator{\Sym}{Sym}\DeclareMathOperator{\trdeg}{trdeg}
\DeclareMathOperator{\im}{im}\DeclareMathOperator{\codim}{codim}
\title[Exact orders and invariant anticanonical systems]{Exact orders and invariant anticanonical linear systems}
\author{OpenAI}\date{September 26, 2026}
\hypersetup{pdfauthor={OpenAI},pdftitle={Exact orders and invariant anticanonical linear systems}}
\begin{document}
\raggedbottom
\begin{abstract}
On a smooth connected projective complex variety with smoothly
semipositive anticanonical bundle, we prove that asymptotically zero valuation on the curvature-null face is attained by
an effective rational anticanonical divisor. The fixed auxiliary line
bundle is arbitrary.
\end{abstract}
\maketitle
\tableofcontents
\input{sections/introduction}
\input{sections/endpoint}
\input{sections/null-field}
\input{sections/exact-proof}
\input{sections/arc-corollary}
\input{sections/order-weight}
\input{sections/exact-weights}
\input{sections/nef-tools}
\input{sections/approximate-arc}
\input{sections/approximate-weights}
\input{sections/invariant-field}
\input{sections/controlled-cone}
\input{sections/isometric-descent}
\input{sections/mrc-descent}
\bibliographystyle{amsplain}
\bibliography{references}
\end{document}
==== END Exact-orders-and-invariant-anticanonical-linear-systems//main.tex ====

==== BEGIN Exact-orders-and-invariant-anticanonical-linear-systems//mrc-descent.tex ====
\section{Descent through a locally constant MRC fibration}
\label{V:sec-mrc-descent}

We apply Theorem~\ref{V:controlled-invariant-nonvanishing} to the
rationally connected fibers of a locally constant fibration.  The constant extension of a section
on one fiber gives a relative anticanonical section precisely when
the section is fixed by monodromy.  The argument below first reduces monodromy to
a connected commutative algebraic group, then uses its unipotent part
to pass from torus invariance to monodromy invariance.  The final two
steps remove the base canonical bundle and descend through a finite
cover.  This is the invariant-section reduction of
M\"uller~\cite[Theorem~2.2 and its proof]{Muller2025}, with the
fiberwise section supplied here by the controlled-cone theorem.

Let $X$ be a smooth connected projective complex variety, and assume
that $-K_X$ has a smooth Hermitian metric with semipositive curvature.
The structure input is Matsumura--Wang's theorem for projective klt
pairs with nef anti-log-canonical divisor
\cite[Theorem~1.1, Definition~2.3, Theorem~4.12, and the conclusion
of~\S4.4]{MatsumuraWangV3}.  We use its zero-boundary case.  The
hypotheses hold because smoothness implies the klt condition and
smooth semipositivity implies nefness.  The theorem gives a finite
quasi-\'etale cover $p_0:X_0\to X$ and a holomorphic maximal
rationally connected fibration
\[
 f_0:X_0\longrightarrow B_0,
 \qquad K_{B_0}\sim_{\mathbb Q}0,
\]
where $B_0$ is projective and the fibration is locally constant.
Concretely, for a fixed connected rationally connected fiber $F$, there
are a representation
\[
 \rho_0:\pi_1(B_0)\longrightarrow\operatorname{Aut}(F)
 \quad\text{and an isomorphism}\quad
 X_0\simeq(\widetilde B_0\times F)/\pi_1(B_0)
\]
over $B_0$; the group acts by deck transformations on $\widetilde B_0$
and through $\rho_0$ on $F$.  The rational linear equivalence of
$K_{B_0}$ is the conclusion of Theorem~4.12, not merely the numerical
triviality in the introductory statement of Theorem~1.1.  The same
formulation appears in M\"uller~\cite[Theorem~2.1]{Muller2025}.
Cao--H\"oring~\cite{CaoHoring}
provides the earlier smooth locally trivial structure theorem; the
constant transition maps used here come from Matsumura--Wang.

Purity of the branch locus makes $p_0$ \'etale, since its target $X$
is smooth.  Hence $X_0$ is smooth and projective.  Local product charts
for $f_0$ imply that $B_0$ and $F$ are smooth as well: the embedding
dimension and the dimension of a product of analytic germs are sums,
so a smooth product has smooth factors.  The fiber $F$ is projective
because it is closed in $X_0$.  On the fiber over $b\in B_0$ the tangent
sequence gives
\[
 K_{X_0}^{-1}|_F\simeq K_F^{-1}\otimes
 \bigl(\det T_bB_0\bigr)\otimes\mathcal O_F.
\]
A choice of nonzero vector in the constant determinant line therefore
transfers the restricted smooth semipositive metric to $-K_F$.
Rational connectedness and Hodge symmetry give
$H^i(F,\mathcal O_F)=0$ for every $i>0$.  Thus $F$ has all the
nonequivariant hypotheses of
Theorem~\ref{V:controlled-invariant-nonvanishing}.

\subsection{Making the monodromy group connected and commutative}

The structure theorem of Demailly--Peternell--Schneider
\cite[Structure Theorem~(iii), p.~218]{DPS1996} shows that
$\pi_1(X_0)$ is virtually abelian.  Since $f_0$ is a fiber bundle with
connected fiber, its homotopy sequence makes
$\pi_1(X_0)\to\pi_1(B_0)$ surjective.  The image of an abelian
finite-index subgroup is again abelian of finite index.  Consequently
$\pi_1(B_0)$ is virtually abelian.

To place monodromy in a fixed algebraic group, choose an ample line
bundle $A$ on $X_0$ and put $A_F=A|_F$.  The restrictions of $c_1(A)$
form a flat section of $R^2(f_0)_*\mathbb Z$, so monodromy fixes the
integral class $c_1(A_F)$.  Since $H^1(F,\mathcal O_F)=0$, the exponential
sequence makes the Chern-class map
$\operatorname{Pic}(F)\to H^2(F,\mathbb Z)$ injective.  It follows that
$\rho_0(\gamma)^*A_F\simeq A_F$ for every $\gamma\in\pi_1(B_0)$.
Choose $a>0$ such that $A_F^a$ is very ample.  The projective embedding
by $A_F^a$ identifies $\operatorname{Aut}(F,A_F^a)$ with the closed stabilizer
of the embedded $F$ in $\operatorname{PGL}(H^0(F,A_F^a)^*)$.
Monodromy therefore lies in a linear algebraic group.  This argument
uses a projective action and requires no choice of linearization on
$A$ or $A_F$.

Pass to the connected finite \'etale cover of $B_0$ corresponding to
an abelian subgroup of finite index in $\pi_1(B_0)$, and pull back
$X_0$.  The resulting monodromy has commutative Zariski closure $H$:
the commutator morphism is trivial on the dense product of the
monodromy image with itself.  Pass next to the cover corresponding to
the inverse image of the identity component $H^\circ$.  The new
monodromy is dense in $H^\circ$, since each component of an algebraic
group is open and closed.  Write
\[
 f:Y\longrightarrow B,\qquad p:Y\longrightarrow X,
 \qquad G=\overline{\rho(\pi_1(B))}^{\mathrm{Zar}}=H^\circ
\]
for the resulting fibration, composite finite \'etale cover, and
monodromy closure.  These finite covers of projective varieties are
projective, and $K_B$ is still torsion.  In particular $G$ is a
connected commutative linear algebraic group acting on the same
fiber $F$.

\subsection{From a torus-fixed section to a section on \texorpdfstring{$X$}{X}}

We use the natural action on anticanonical bundles: an automorphism
acts on $K_F^{-1}=\det T_F$ by the determinant of its differential.
It induces an algebraic representation on every finite-dimensional
space $H^0(F,K_F^{-m})$.  This specified action is also the action in
the relative canonical transition functions of the locally constant
fibration.

Let $T\subset G$ be a maximal algebraic torus.  Theorem~\ref{V:controlled-invariant-nonvanishing}
gives an integer $m>0$ for which
\[
 W:=H^0(F,K_F^{-m})^T\ne0.
\]
If $F$ is a point, take $m=1$ and the constant section; the rest of the
argument is unchanged.  The trivial torus is allowed.  The inclusion
$G\subset\operatorname{Aut}(F)$ makes the action faithful, so there
is no ineffective subgroup that could introduce a scalar character
on the natural anticanonical line.

For completeness, the following elementary argument explains why
$W$ contains a vector fixed by all of $G$.  Jordan decomposition in a
connected commutative linear algebraic group over $\mathbb C$ gives
\[
 G=T\times U,
\]
where $U$ is connected and unipotent.  Commutativity makes $W$ stable
under $U$.  Its Lie algebra acts on $W$ by commuting nilpotent
endomorphisms $N_1,\ldots,N_e$, after choosing a basis of the Lie
algebra.  Their kernels have nonzero intersection: successively
restrict $N_j$ to the intersection of the preceding kernels, which
it preserves by commutativity; a nilpotent endomorphism of a nonzero
finite-dimensional space has nonzero kernel.  A nonzero vector in
this intersection is fixed by $U$, because $U$ is connected and is
generated by the exponentials of its Lie algebra.  We obtain
\[
 0\ne s\in W^U=H^0(F,K_F^{-m})^G.
\]
This is the connected-group step in
\cite[Lemma~2.3 and the proof of Theorem~2.2]{Muller2025}; we have
retained the unipotent argument because $G$ need not be a torus.
The input here is a fixed vector for the natural linearization, not
a section of an unspecified character.

Since $s$ is fixed by monodromy, its constant extension to
$\widetilde B\times F$ descends to a nonzero section
\[
 \sigma\in H^0(Y,K_{Y/B}^{-m}).
\]
The canonical bundle formula reads
$K_Y=K_{Y/B}\otimes f^*K_B$.  Choose $r>0$ with
$K_B^r\simeq\mathcal O_B$ and a nowhere-zero section
$\tau\in H^0(B,K_B^{-r})$.  Then
\[
 \sigma^r\otimes f^*\tau^m
 \in H^0(Y,K_Y^{-mr})\setminus\{0\}.
\]
All these holomorphic sections are algebraic by projectivity.  Finally,
let $d=\deg p$.  Since $p$ is finite \'etale,
$K_Y^{-mr}\simeq p^*K_X^{-mr}$, and the norm of this section is a
section of $K_X^{-mrd}$.  It is nonzero: over the generic point the
norm is a product of nonzero conjugates in a finite field extension.
Thus $X$ has a nonzero section of a positive anticanonical power.
The construction uses the locally constant product only after passing
to the universal cover of the base; no product decomposition of a
finite cover of $X$ is required.
==== END Exact-orders-and-invariant-anticanonical-linear-systems//mrc-descent.tex ====

==== BEGIN Exact-orders-and-invariant-anticanonical-linear-systems//nef-tools.tex ====
\section{Nef bases and extension of sections}
\label{V:sec-nef}

The nef-reduction arguments in Sections~\ref{V:sec-transfer} and
\ref{V:sec-invariant-field} start from twisted anticanonical sections.
Their common construction is a diagram
\[
 X\xleftarrow{p}W\xrightarrow{f}Y,
 \qquad ap^*L\sim D+f^*B,
\]
where $L=-K_X$, $a$ is a positive integer, $W$ is normal and
equidimensional over the smooth base $Y$, and $p$ is birational.
The divisor $D$ is effective Cartier, and above each prime of $Y$ it has
coefficient zero on at least one component.  The line bundle $B$ is
the part whose positivity must be established.

On a smooth resolution of $W$, a rank-one hypothesis for every relative
adjoint twist identifies the relevant sections with powers of the section
defining $D$, multiplied by the Jacobian section.  The bounded maximum
metric lemma below states this hypothesis in ordinary and invariant
forms.  It shows that the fiber maximum of the norm of the section defining $D$
gives a locally bounded semipositive metric on $B$, hence makes $B$
nef.  Nef reduction then identifies its numerical null directions, and
irregularity zero turns the resulting numerical descent into rational
linear descent.

The two applications finish differently.  For a prescribed arc,
Section~\ref{V:sec-transfer} proves $v(D)=0$ and extends first jets at
one point of the reduced base; this makes its descended line big and
yields arbitrarily small normalized arc orders.  Starting instead with
invariant twisted sections, Section~\ref{V:sec-invariant-field} extends
one nonzero invariant class in a quotient of multiplier ideals and
obtains an invariant anticanonical section.  We now prove the model,
metric, descent and curvature-concentration statements shared by these
constructions.

\subsection{Models and connected fibers}

The relative section must be normalized over every base divisor.
For that purpose we use a normal equidimensional model.  Its smooth
resolution is used for integration, and is not required to remain
equidimensional.

\begin{lemma}[Equidimensional models]\label{V:flat-model}
Let $X\dashrightarrow Y$ be a dominant rational map of integral projective
varieties over $\mathbb C$.  There are a smooth projective birational model
$Y'$ of $Y$ and an integral projective flat family $X_0\to Y'$ with a
birational morphism to the graph of the original map.  Every irreducible
component of every fiber has dimension $\dim X-\dim Y$.  The normalization
$V$ of $X_0$ is equidimensional over $Y'$; in particular, every vertical
prime divisor of $V$ maps onto a prime divisor of $Y'$.  If
$\mathbb C(Y)$ is algebraically closed in $\mathbb C(X)$, the morphism
from $V$, and from any smooth resolution of $V$, to $Y'$ has connected
fibers.  If a torus acts on $X$ and the rational map is invariant, these
constructions and a projective resolution of $V$ can be made equivariant,
with trivial action on the base.
\end{lemma}

\begin{proof}
Embed the projective graph over $Y$ in $\mathbb P^N\times Y$.
Over the flat open set the fibers define a map to the projective
Hilbert scheme \cite[Theorem~3.2 and the Hilbert-scheme specialization
after Proposition~3.8]{Grothendieck1961Hilbert}.
Resolve the closure of its graph, also resolving the base, and pull back the
universal family.  Flatness over an integral base embeds each affine
coordinate ring in its generic localization.  The generic fiber is
integral, so these coordinate rings have neither nilpotents nor zero
divisors.  Thus the pulled-back family is integral and is the closure of
the original family over the flat open set.  Its map to the graph is
birational.

Constancy of the Hilbert polynomial bounds the dimension of every fiber
above by the generic relative dimension.  At any fiber component the
height bound for the ideal generated by local parameters of the smooth
base gives the opposite bound.  This proves purity of fiber dimension.
Normalization is finite, so fiber dimensions and dimensions of images of
components are preserved.  A prime divisor mapping into codimension at
least two in $Y'$ would have dimension at most
$\dim Y'-2+(\dim X-\dim Y')=\dim X-2$, a contradiction.

For connectedness apply Stein factorization to the normal total space.
Its finite intermediate base has function field the algebraic closure
of $\mathbb C(Y')$ in $\mathbb C(X)$; the asserted field condition and
normality of $Y'$ make the finite birational morphism an isomorphism.
The same reasoning applies after resolution.  In the equivariant case,
the family over the original open set is stable under the action.
Its closure and normalization are therefore stable, and equivariant
resolution in characteristic zero supplies the last assertion.
\end{proof}

\subsection{A bounded metric from all relative moments}
\label{V:sec-maximum}

The two nef-reduction arguments below construct a section of a
relative line bundle and normalize its orders over base divisors.
The purpose of this normalization is analytic: it prevents the
maximum of the section's norm on nearby fibers from tending to zero.
Together with rank one for every relative adjoint twist, it produces
a nef line bundle on the base.  Ordinary and invariant rank one
are separate hypotheses in the statement.

\begin{lemma}[Bounded maximum metric]\label{V:bounded-maximum}
Let $X,Y$ be smooth connected projective varieties, let
$L=-K_X$ have a smooth semipositive metric $h$, and consider
\[
 W\xrightarrow{p}X,\qquad W\xrightarrow{f}Y,
 \qquad Z\xrightarrow{r}W,
\]
where $W$ is normal integral projective, $p$ is a birational morphism,
$f$ is surjective and
equidimensional with connected general fiber, and $r$ is a smooth
projective resolution.  Write $\pi=pr$, $g=fr$ and
$E=K_Z-\pi^*K_X\ge0$.  Let $B$ be a line bundle on $Y$, let $a\in\mathbb Z_{>0}$,
and suppose
\[
 0\ne\tau\in H^0(W,ap^*L-f^*B)
\]
has order zero on at least one component above every prime divisor
of $Y$.  Assume that, for every integer $l\ge1$, the ordinary
generic adjoint space for
$K_{Z/Y}+(al+1)\pi^*L$ has dimension one and is generated, in
base frames, by $s_E(r^*\tau)^l$.
Then $B$ has a semipositive metric with locally bounded weights.
On a smooth connected-fiber open set its weight in a frame $\beta$
is the continuous plurisubharmonic function
\begin{equation}\label{V:sup-weight}
 \varphi_\beta(y)=-\log\max_{Z_y}
       |r^*\tau\,g^*\beta|^2_{(\pi^*h)^a}.
\end{equation}
In particular, $B$ is nef.

The same conclusion holds with invariant generic rank one in place
of ordinary rank one, provided an algebraic torus acts compatibly
on the diagram with trivial action on $Y$, $L$ has its natural
linearization, $B$ has trivial linearization, $\tau$ is invariant,
and $h$ is invariant under the maximal compact torus.
\end{lemma}

\begin{proof}
The point-base case is immediate, so assume $\dim Y>0$.
The generic fiber is connected; the fibers over a sufficiently
small smooth-submersion open set $U$ are therefore connected and
smooth, hence irreducible.  Shrink $U$ so that neither $s_E$ nor
$r^*\tau$ vanishes identically on any fiber.

Choose frames $\beta$ of $B$ and $\kappa$ of $-K_Y$ on a base
chart.  For each $l\ge1$, the relative adjoint section
\[
 u_l=s_Eg^*\kappa\,(r^*\tau g^*\beta)^l
\]
generates the relevant generic space.  On its locally free
base-change locus, Berndtsson's smooth direct-image theorem
\cite[Theorem~1.2]{Berndtsson2009} applies to the smooth
semipositive coefficient $(al+1)\pi^*L$.  In the invariant case,
compact averaging is a holomorphic projection on the direct image.
It is orthogonal for the fiber integral metric, by change of
variables and invariance of the coefficient metric.  Its invariant
line is consequently also a metric quotient and inherits
semipositive curvature.  This is the only extra positivity step in
the invariant case.  If the relative dimension is zero, the
connected smooth fiber is one point and the same assertion follows
directly from the coefficient metric.

Put $b=|r^*\tau g^*\beta|^2_{(\pi^*h)^a}$, and let $d\mu_y$
be the smooth nonnegative density of $s_Eg^*\kappa$ with
coefficient $\pi^*L$.  The preceding positivity gives
plurisubharmonicity of
\[
 -\frac1l\log\int_{Z_y}b^l\,d\mu_y.
\]
The density has full support on each fiber in $U$.  Smooth proper
integration extends the displayed function across the possible
base-change exceptions for this particular $l$, preserving
plurisubharmonicity.  Thus no common base-change locus for infinitely
many $l$ is required.

Normalize each density to mass one.  Its $l$-th power mean of $b$
increases to the maximum.  The maximum is positive and continuous
under a proper smooth submersion, so this convergence is uniform
on compact subsets by monotonicity.  The logarithm of the mass,
divided by $l$, tends uniformly to zero there.  Passing to the
limit proves that \eqref{V:sup-weight} is plurisubharmonic on $U$.

It remains to extend the metric over $Y\setminus U$.  Regularity
of $\tau$ and properness bound $b$ above over compact base charts;
hence $\varphi_\beta$ is locally bounded below.  At a general point
of every boundary prime $Q$, choose a component $\Gamma$ above
$Q$ on which $\tau$ has order zero.  Normality makes $W$ smooth
at a general point of $\Gamma$.  We can choose this point so that
$\tau$ is nonzero and $\Gamma\to Q$ is submersive.  In local
coordinates a parameter for $Q$ pulls back to a unit times a
positive power of a parameter for $\Gamma$.  Taking a local root
of the unit, and completing with the tangential coordinates, shows
that $f$ is open there.  A neighborhood on which the norm of
$\tau$ is bounded below by a positive constant therefore covers
a base neighborhood.  Surjectivity of $Z\to W$ gives the same
lower bound for the fiber maximum.  Thus $\varphi_\beta$ is
locally bounded above near a general point of $Q$.

The removable-singularities theorem for plurisubharmonic functions
now extends the weights across the divisors, except over a closed
analytic set of codimension at least two.  The Hartogs extension
for plurisubharmonic functions removes that set as well.  More
explicitly, choose a small transverse disc whose boundary avoids
the missing set and parallel nearby discs with boundaries in a
fixed compact subset of the complement.  Outside a proper set of
parameters these discs miss the analytic set.  Their boundary
maximum gives a common upper bound, and the submean inequality
extends it to the other points of the domain.  The original lower
bounds persist under extension.  Finally,
\[
 \varphi_{u\beta}=\varphi_\beta-\log|u|^2
\]
for a nowhere-zero holomorphic function $u$, so the local weights
glue to a metric on $B$.  Restricting the bounded plurisubharmonic
weights to the normalization of any curve gives nonnegative
degree.  This proves nefness.
\end{proof}

The conclusion over the entire base is local boundedness.  Continuity
on the smooth locus is used for the power-mean limit; the extension
argument does not require continuity at every boundary point.

\subsection{Descending through the numerical null directions}

A nef divisor has degree zero on the general fibers of its nef
reduction.  The next argument upgrades this to every contracted
curve, which is the hypothesis needed for numerical pullback.
Irregularity zero is then used to turn that numerical identity into
a rational linear identity of line bundles.

\begin{lemma}[Descent of a nef divisor]\label{V:nef-descent}
Let $Y$ be smooth connected projective with $q(Y)=0$, and let $B$ be a
nef Cartier divisor on $Y$.  There are smooth connected projective
varieties $Y'$ and $Z$, a birational morphism $\rho:Y'\to Y$, a
surjective morphism $h:Y'\to Z$ with connected fibers, and a nef rational
Cartier divisor $L$ on $Z$ such that
\[
 \rho^*B\sim_{\mathbb Q}h^*L.
\]
If $\dim Z>0$, every irreducible curve through a very general point of
$Z$ has positive $L$-degree.  After multiplying $B$ and $L$ by one
positive integer, $L$ is Cartier and the displayed equivalence is linear.
\end{lemma}

\begin{proof}
Apply the nef reduction theorem of Bauer, Campana, Eckl, Kebekus,
Peternell, Rams, Szemberg and Wotzlaw
\cite[Theorem~2.1]{BCEKPRSW2002}.  It gives an almost holomorphic
dominant map $Y\dashrightarrow Z_0$ with connected general fibers,
numerical triviality of $B$ on its compact general fibers, and positive
$B$-degree for every noncontracted curve through a very general point
of $Y$.  Apply Lemma~\ref{V:flat-model} to its graph, over a smooth
projective birational model $Z$ of $Z_0$.  Write $T\to Z$ for the flat
family and $B_T$ for the pullback of $B$.

We claim that $B_T$ is numerically trivial on every curve in every fiber.
Almost holomorphicity identifies sufficiently general graph fibers with
compact fibers in the domain of the original map.  Indeed the complement
of that domain has smaller relative dimension over a general base point;
the compact fiber in the domain is already closed, and so acquires no
additional points in the graph.  Thus $B_T$ is numerically trivial on
a general fiber.

Let $d$ be the relative dimension.  If $d=0$ there are no contracted
curves on $T$.  Suppose $d>0$ and choose an ample Cartier divisor
$H$ on $T$.
Both $H$ and $H+B_T$ are relatively ample.  Their top degrees on the
fundamental fiber cycles are constant, by flatness and the leading
coefficients of their Hilbert polynomials, and agree on the general
fiber.  On any fiber the binomial expansion of their difference is a
sum of nonnegative mixed intersections.  Since the fiber has pure
dimension $d$, every component $R$ occurs with positive multiplicity;
we obtain
\[
 B_T\cdot H^{d-1}\cdot R=0
 \qquad\hbox{for every fiber component }R.
\]
If $d=1$, every integral curve $C\subset R$ equals $R$, so its degree
is already zero. For $d\ge2$, choose $d-1$ sufficiently high ample
hypersurfaces through a prescribed curve $C\subset R$. Global generation
of its twisted ideal permits successive general choices avoiding all
existing components not contained in $C$, so the final intersection is
proper and one-dimensional. At the generic point of $C$, their local
equations form a system of parameters; their intersection multiplicity
there is positive. Thus the resulting effective curve cycle contains
$C$ with positive coefficient, also when $R$ is singular.
Its $B_T$-degree is
zero, and nefness makes every component degree nonnegative.  Hence
$B_T\cdot C=0$, as claimed.

Normalize $T$ and resolve it to $Y'$, with maps $\rho:Y'\to Y$ and
$h:Y'\to Z$.  A curve contracted by $h$ either maps to a curve in a
fiber of $T$ or maps to a point, so $\rho^*B$ has degree zero on every
such curve.  The fibers of $h$ are connected, by Stein factorization
and the connected general fibers of the nef reduction.  Lehmann's
numerical pullback theorem \cite[Theorem~1.2]{Lehmann2015} applies:
the divisor $\rho^*B$ is pseudoeffective, is numerically trivial on
all contracted curves, and the smooth base $Z$ is $\mathbb Q$-factorial.
It gives $\rho^*B\equiv h^*L$ for a real numerical divisor class $L$.
The pullback map on numerical divisor spaces is defined over
$\mathbb Q$, so the linear equation with rational right-hand side
has a rational solution.  Choose such an $L$.  It is nef: for each
curve in $Z$, a projective slice of its inverse image gives a curve
dominating it, and the projection formula gives nonnegative degree.

Birational invariance gives $q(Y')=0$.  Numerically trivial integral
line bundles on a smooth projective variety belong to $\operatorname{Pic}^0$
up to torsion; see \cite[Definition~6.1 and Theorem~6.3]{Kleiman2005}.  Therefore the rational numerical equivalence just
obtained is a rational linear equivalence.  Clearing its torsion and
denominators gives the asserted integral scaling.

Finally, choose a very general $z\in Z$ and a point $y'\in h^{-1}(z)$
where $h$ is smooth and $\rho$ is an isomorphism, such that $\rho(y')$
lies outside the countable exceptional union in the nef reduction
theorem.  Such a point exists: for each proper closed exception, the
image of its complement contains a dense open subset of $Z$; on a very
general irreducible fiber, the resulting countably many proper closed
subsets cannot cover the fiber.  Given any curve through $z$, normalize
it and slice a component of its inverse image through $y'$ to obtain
a curve through $y'$ dominating it.  Smoothness at $y'$ permits a
nonvertical curve germ, so the slicing may be chosen to retain that
property.  Its image in $Y$ is not contracted by the original nef
reduction.  The positivity part of that theorem, followed by the
projection formula, proves that the given curve has positive
$L$-degree.
\end{proof}

The reduced base now has positive degree on every curve through a
very general point.  We next turn this qualitative condition into
an arbitrarily large Seshadri bound after adding a fixed small ample
term.  This will give an isolated logarithmic pole strong enough to
detect either a first jet or a nonzero quotient class.

We use the following metric conventions in this extension argument.
If a singular Hermitian metric has squared frame norm $h=e^{-\varphi}$,
its multiplier ideal $\mathcal I(h)$, also denoted $\mathcal J(h)$,
consists of holomorphic germs $u$ for which $|u|^2e^{-\varphi}$ is
locally integrable. The metric $he^{-\psi}$ has weight
$\varphi+\psi$ and curvature $i\Theta_h+i\partial\bar\partial\psi$.
A quasi-plurisubharmonic function has \emph{neat analytic singularities}
if locally it has the form $c\log\sum_j|f_j|^2+u$, where $c>0$,
the $f_j$ are holomorphic and $u$ is smooth.

\begin{lemma}[Concentrating curvature at one base point]
\label{V:sesha}
Let $Z$ be smooth projective of dimension $b>0$, let $L$ be a nef
Cartier divisor, and suppose every curve through a very general point
of $Z$ has positive $L$-degree.  Fix a very ample Cartier divisor $H$
and rational $\epsilon>0$.  For every $c>0$, all sufficiently large
integers $k$ make
\[
 J_k=\tfrac12(kL+\epsilon H)
\]
an ample rational divisor with Seshadri constant greater than $c$ at
a very general point.  At any such point $z$ one can choose a smooth
positive metric of curvature $\eta$ on $J_k$ and a quasi-plurisubharmonic
function $\psi$ with neat analytic singularities, smooth away from $z$,
such that
\[
 \eta+i\partial\bar\partial\psi\ge0,
 \qquad \psi(w)\le c\log|w|^2+O(1)
\]
in coordinates centered at $z$.
\end{lemma}

\begin{proof}
Fix the countable union of proper closed subsets outside which the
curve positivity holds.  If an integral $d$-dimensional subvariety
$U$ is not contained in this union, then
$L\cdot H^{d-1}\cdot U$ is a positive integer.  Otherwise nefness
and the cutting-through-a-curve argument in the proof of
Lemma~\ref{V:nef-descent} would make $L$ numerically trivial on
$U$, contrary to a curve in $U$ through a point outside the union.
The mixed term in the binomial expansion consequently gives
\[
 J_k^d\cdot U\ge 2^{-d}d k\epsilon^{d-1}.
\]
Choose $\alpha>c$ and then $k$ large enough that this is at least
$(d\alpha)^d$ for every $1\le d\le b$.  The theorem of
Ein--K\"uchle--Lazarsfeld \cite[Theorem~3.1]{EKL1995}, applied after
clearing rational denominators, gives the required Seshadri bound.

Choose a rational number $c_0$ strictly between $c$ and the Seshadri
constant.  On the blowup of $z$, the pullback of $J_k$ minus $c_0$
times the exceptional divisor is ample.  To see ampleness rather than
mere nefness, choose $c_0<c'<\epsilon(J_k;z)$; a sufficiently small positive
exceptional subtraction is ample, and the desired divisor is a
positive convex combination of that ample divisor and the nef divisor
with coefficient $c'$.  For a sufficiently divisible large integer
$p$, a basis of sections of $pJ_k$ vanishing to order at least $pc_0$
at $z$ therefore has no common zero away from $z$.  If these sections
are $s_j$, put
\[
 \psi=\frac1p\log\sum_j|s_j|^2_{h_{J_k}^{p}}.
\]
The curvature inequality follows directly, and the pole with coefficient
$c_0>c$ gives the asserted upper bound after shrinking the coordinate
neighborhood.  For $\dim Z=1$ use $J_k-c_0z$ on the curve.
\end{proof}

==== END Exact-orders-and-invariant-anticanonical-linear-systems//nef-tools.tex ====

==== BEGIN Exact-orders-and-invariant-anticanonical-linear-systems//null-field.tex ====
\section{The field selected by the curvature intersection}
\label{V:null-field}

We now construct the base needed for Theorem~\ref{V:exact}. The data
$X,L,\alpha,\nu,G_0,\ell$ remain as in the introduction. A divisor
is called \emph{null} in this section if it is effective and has
$\ell$-value zero. Nonnegativity of $\ell$ implies that every effective
divisor bounded by a null divisor is again null.

\begin{lemma}\label{V:polar-field}
The set
\[
 k=\{0\}\cup\{h\in\C(X)^\times:\ell((h)_\infty)=0\}
\]
is a finitely generated subfield of $\C(X)$, relatively algebraically
closed in $\C(X)$.
\end{lemma}
\begin{proof}
The polar divisors of $h_1+h_2$ and $h_1h_2$ are bounded by the sum
of the polar divisors of $h_1,h_2$. The zero and polar divisors of a
nonzero rational function are linearly equivalent, and therefore have
the same $\ell$-value. These observations prove closure under addition,
multiplication and inversion.

Suppose that $b\in\C(X)$ is algebraic over $k$. In a monic equation
$b^r+a_1b^{r-1}+\cdots+a_r=0$, each coefficient belongs to $k$.
At a prime divisor $P$ where $b$ has a pole, the term $b^r$ cannot
have strictly smaller order than every other term. Thus, for some $i$,
$i\ord_P(b)\ge\ord_P(a_i)$. In particular the pole order of $b$
is bounded by the sum of the pole orders of the coefficients. Its polar
divisor is null, so $b\in k$.

Choose a transcendence basis $t_1,\ldots,t_d$ of $k/\C$.
The algebraic closure of $\C(t_1,\ldots,t_d)$ inside the finitely
generated field $\C(X)$ is a finite extension of that rational function
field. The intermediate field $k$ is algebraic over the same field
and is therefore finite over it. This proves finite generation.
\end{proof}

Choose smooth projective models and morphisms
\[
 X\xleftarrow{\ \pi\ } Z\xrightarrow{\ g\ }Y,
 \qquad \C(Y)=k,
\]
with $\pi$ birational and $g$ surjective. Characteristic zero and relative
algebraic closedness imply that its generic fiber is geometrically
integral. We first verify two properties which persist under every
birational change of these models.

\begin{lemma}\label{V:null-rank}
For every very ample divisor $H$ on $Y$,
\begin{equation}\label{V:ample-null}
 \ell(\pi_*g^*H)=0.
\end{equation}
If $E_Z=K_{Z/X}$ is the effective Jacobian divisor, then
\begin{equation}\label{V:zero-rank}
 \dim_{\C(Y)}H^0(Z_\eta,\OO_Z(E_Z)|_{Z_\eta})=1.
\end{equation}
\end{lemma}
\begin{proof}
Choose a basis $s_0,\ldots,s_N$ of the very ample system. The functions
$s_i/s_0$ belong to $k$, so their polar divisors on $X$ are bounded by
one null divisor $D$. Their poles on $Z$ are then bounded by $\pi^*D$.
Base point freeness says that the componentwise maximum of those polar
divisors on $Z$ is $g^*\divisor(s_0)$. After pushforward, effectivity
and monotonicity give
$0\le\ell(\pi_*g^*H)\le\ell(D)=0$.

The Jacobian section $s_{E_Z}$ gives one nonzero section in
\eqref{V:zero-rank}. Divide any other generic-fiber section by it.
The resulting rational function $a\in\C(Z)=\C(X)$ has horizontal
poles bounded by $E_Z$. There are finitely many remaining polar primes.
Each maps into a proper closed subset of $Y$. A sufficiently high very
ample system has an effective member $B$ vanishing to prescribed orders
on their images; increasing these orders ensures
\[
 (a)_\infty\le E_Z+g^*B.
\]
This also works when a vertical prime maps to codimension greater than
one. Since $E_Z$ is exceptional over $X$, pushforward and
\eqref{V:ample-null} show that the polar divisor of $a$ on $X$ is null.
Thus $a\in k$. Every generic section is a scalar multiple of the
Jacobian section, as claimed.
\end{proof}

The equality in \eqref{V:zero-rank} concerns the full adjoint space in
degree zero. It has not been deduced from an eventual rank statement.
It identifies the integrated Jacobian volume with a line metric on the
base, rather than with a selected subspace of a larger direct image.

\subsection{An equidimensional model without a field extension}
Assume $d=\dim Y>0$. We need a normal projective variety $W$ and a
diagram
\begin{equation}\label{V:model}
 X\xleftarrow{\ p\ }W\xrightarrow{\ f\ }Y
\end{equation}
with $p$ birational, $Y$ smooth projective, and $f$ equidimensional
and toroidal for strict toroidal embeddings. All modifications here
are birational; in particular the field of the base remains $k$.
A toroidal embedding is locally modeled on a toric variety with its
dense torus; the complement is its boundary. Strictness means that the
irreducible boundary components are normal. Toroidal morphisms are
locally toric morphisms, so boundary strata determine rational cones and
the morphism determines their lattice-linear maps.

Weak toroidalization \cite[Theorem~1.1]{ADK2013} first gives a
birational toroidal diagram with smooth projective target. For a
toroidal morphism, equidimensionality is equivalent to the condition
that every source cone maps onto a target cone
\cite[Lemma~4.1 and Remark~4.2]{AK2000}. Refine the target cone
complex so that the image of every source cone is a union of cones.
Projective refinements exist by \cite[Lemma~4.3]{AK2000}; further
projective toroidal resolution makes the target nonsingular. Intersecting
the source cones with inverse images of target cones gives projective
source subdivisions and a compatible toroidal morphism. In a target
cone of the new subdivision, an old cone image meets it in a union of
faces which is convex, hence in a face. Thus the new source cones map
onto target cones. This is the equidimensionalization of
\cite[Proposition~4.4]{AK2000}.

In local toric models, a source stratum maps submersively onto its
target stratum. Cone surjectivity bounds the dimension of the fiber in
each stratum above by $n-d$; the opposite bound for full fibers follows
because the target is smooth. This gives equidimensionality. The normal
source may remain singular, and no covering to reduce multiplicities
is needed. Resolve it by $Z\to W$ when using smooth differential forms;
that resolution need not itself be equidimensional. We continue to write
$\pi:Z\to X$ and $g:Z\to Y$ for the induced maps.

\subsection{The horizontal curvature patch}
The next calculation verifies the remaining analytic hypothesis of the
toroidal-volume theorem. Let $\omega_0$ be a positive form representing
$c_1(G_0)$, and let $\omega_H$ represent a very ample class on $Y$.
When $\nu<n$, equation~\eqref{V:ample-null} implies
\begin{equation}\label{V:vanishing-form}
 (\pi^*\alpha)^\nu\wedge g^*\omega_H
       \wedge\pi^*\omega_0^{n-\nu-1}=0.
\end{equation}
Indeed the left side is a semipositive top-degree form whose integral
is the displayed intersection number. It therefore vanishes pointwise.

At a maximum-rank point away from the Jacobian divisor, diagonalize
$\pi^*\alpha$ in coordinates orthonormal for $\pi^*\omega_0$.
Its first $\nu$ eigenvalues are positive and the rest are zero.
The left side of \eqref{V:vanishing-form} is a positive multiple of
the sum of the diagonal entries of $g^*\omega_H$ in the remaining
$n-\nu$ directions. These entries vanish. A semipositive Hermitian
form vanishing on a vector has zero mixed terms with that vector, so
\[
 \ker\pi^*\alpha\subset\ker dg.
\]
For $\nu=n$ this inclusion is automatic. The maximum-rank locus is a
nonempty ordinary open and intersects the dense smooth base-change
locus, away from the Jacobian divisor. Near a point in their intersection
the rank is constant. On a smaller relatively compact neighborhood the
positive eigenvalues are bounded below, and the kernel inclusion gives
\begin{equation}\label{V:strict-patch}
 \pi^*\alpha\ge c\,g^*\omega_H\qquad(c>0).
\end{equation}
The Jacobian section is nonzero on this patch. If $\nu=0$, the same
calculation would force $dg=0$ on a nonempty open; thus a positive-
dimensional base cannot occur in that case.

We have now proved both analytic inputs from the null-polar field:
full rank one in \eqref{V:zero-rank} and the patch
\eqref{V:strict-patch}. Apply the toroidal-volume theorem
\cite[Theorem \textup{(Toroidal volume metric)}]{CompanionG} to \eqref{V:model}.
For a prime $E$ of $W$ put $a_E=\ord_E K_{W/X}$. Its conclusion uses
\begin{equation}\label{V:correction}
 A_0=\sum_Q\max\left\{0,
       \min_{E\mapsto Q}\left(\frac{1+a_E}{\ord_E f^*Q}-1\right)
       \right\}Q.
\end{equation}
It supplies $A_0\ge0$, $p_*f^*A_0=0$, and a semipositive metric on
$-K_Y+A_0$ with locally integrable squared frame norms, smooth and
strictly positive on a nonempty ordinary open. The cited theorem proves
the all-valuation discrepancy calculation and extension across boundary
divisors and codimension two. Its strictness requirement is exactly
\eqref{V:strict-patch}; no global strict curvature bound is being
inferred directly from that patch.

The correction-ring theorem \cite[Theorem \textup{(Finite generation near a correction divisor)}]{CompanionE}
now applies. For every finite list of rational divisors $P_i$ on $Y$,
there exist $\epsilon\in\Q_{>0}$ and $k_0\in\Z_{>0}$ such that
all $k_0(A_0+\epsilon P_i)$ are Cartier and their multigraded section
ring is finitely generated. This statement retains the actual rational
linear equivalences and the common value of $\epsilon$. We next choose
the two directions dictated by the divisors in Theorem~\ref{V:exact}.
==== END Exact-orders-and-invariant-anticanonical-linear-systems//null-field.tex ====

==== BEGIN Exact-orders-and-invariant-anticanonical-linear-systems//order-weight.tex ====
\section{Orbit orders and characters}
\label{V:sec-order-weight}

We use the left action on sections of a linearized line bundle:
$(g\cdot s)(x)=g\bigl(s(g^{-1}x)\bigr)$.  Thus a character
$\alpha$ means $g\cdot s=\alpha(g)s$.  Fixing this convention is
necessary when an order estimate is converted into a weight estimate.
The next two elementary lemmas will be used both for approximate
weights and for exact attainment of a supporting weight.

\begin{lemma}[The order on a generic orbit]\label{V:order-weight}
Let an algebraic torus $T$ act on a smooth integral projective variety
$X$, and let $L$ be a $T$-linearized line bundle.  Let
$\lambda:\mathbb G_m\to T$ be a cocharacter, and let $S$ be its
source component: a dense open subset of $X$ tends to $S$ as the
parameter tends to zero.  Let $\mu$ be the $\lambda$-weight on
$L|_S$.  If $0\ne s\in H^0(X,L)$ has $\lambda$-weight $w$, then
\[
 v_\lambda(\operatorname{div}s)=\mu-w,
\]
where $v_\lambda$ is the order on the arc
$t\mapsto\lambda(t)\xi$ and $\xi$ is the generic point, viewed as
a $\mathbb C(X)$-valued point.  For the zero cocharacter the order
and both weights are zero.
\end{lemma}

\begin{proof}
The orbit map is defined over $\mathbb C(X)(t)$ and remains generic
on $X$.  By properness it extends to the formal disc over
$\mathbb C(X)$.  Its center belongs to $S$.  Pull back $L$ to this disc.
If the pulled-back section has order $d$, its first nonzero class
belongs to the fiber of $L$ tensored with the degree-$d$ parameter
ideal quotient.  Parameter scaling acts on this quotient, under
the declared action on sections, with weight $-d$.  The first
nonzero class therefore has weight $\mu-d$.  Equivariance gives
$w=\mu-d$.  The zero-cocharacter arc is constant at the generic
point; every nonzero rational function is a unit there, which
proves the last assertion directly.
\end{proof}

\begin{lemma}[Multiplying characters]\label{V:convex-product}
Let $T$ act on an integral projective variety $X$, and let $L$ be
a $T$-linearized line bundle.  Suppose, for $1\le i\le N$, that $s_i\in H^0(X,r_iL)$ are
nonzero eigensections with characters $\alpha_i$, where
$r_i\in\mathbb Z_{>0}$.
If zero lies in the convex hull of the rational points
$\alpha_i/r_i$, then $H^0(X,rL)^T\ne0$ for some $r>0$.
\end{lemma}

\begin{proof}
The defining linear equations and inequalities have rational
coefficients, so there are rational numbers $c_i\ge0$ satisfying
$\sum_i c_i=1$ and $\sum_i c_i\alpha_i/r_i=0$.  Clear the
denominators of $c_i/r_i$ to obtain integers $n_i\ge0$, not all
zero, with $\sum_i n_i\alpha_i=0$.  The product
$\prod_i s_i^{n_i}$ has weight zero and belongs to
$H^0(X,rL)$, where $r=\sum_i n_i r_i>0$.  Its factors are
nonzero at the generic point of the integral variety, so the
product is nonzero.
\end{proof}
==== END Exact-orders-and-invariant-anticanonical-linear-systems//order-weight.tex ====

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@article{CasciniLazic2012,
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@article{CDM2017,
  author = {Cao, Junyan and Demailly, Jean-Pierre and Matsumura, Shin-ichi},
  title = {\href{https://arxiv.org/abs/1703.00292v2}{A general extension theorem for cohomology classes on non reduced analytic subspaces}},
  journal = {Science China Mathematics},
  volume = {60},
  number = {6},
  year = {2017},
  pages = {949--962},
  doi = {10.1007/s11425-017-9066-0},
  eprint = {1703.00292},
  archivePrefix = {arXiv},
  primaryClass = {math.AG},
  url = {https://arxiv.org/abs/1703.00292v2},
  note = {Preprint version: arXiv:1703.00292v2}
}

@misc{Demailly2011,
  author = {Demailly, Jean-Pierre},
  title = {\href{https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/analmeth_book.pdf}{Analytic Methods in Algebraic Geometry}},
  year = {2011},
  month = jul,
  howpublished = {Author manuscript},
  note = {Manuscript dated July 2011; Lemma 5.6(a), pp. 35--36, and Theorem 5.11, p. 39},
  url = {https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/analmeth_book.pdf}
}

@misc{Demailly2012,
  author        = {Demailly, Jean-Pierre},
  title         = {\href{https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/agbook.pdf}{Complex Analytic and Differential Geometry}},
  year          = {2012},
  howpublished  = {Author's online book},
  url           = {https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/agbook.pdf},
  note          = {Version of 21 June 2012}
}

@article{EKL1995,
  author = {Ein, Lawrence and K{\"u}chle, Oliver and Lazarsfeld, Robert},
  title = {\href{https://arxiv.org/abs/alg-geom/9408003v1}{Local positivity of ample line bundles}},
  journal = {Journal of Differential Geometry},
  volume = {42},
  number = {2},
  year = {1995},
  pages = {193--219},
  doi = {10.4310/jdg/1214457231},
  eprint = {alg-geom/9408003},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/alg-geom/9408003v1},
  note = {Preprint version: arXiv:alg-geom/9408003v1}
}

@article{FM2016,
  author = {Fujino, Osamu and Matsumura, Shin-ichi},
  title = {\href{https://arxiv.org/abs/1605.02284v3}{Injectivity theorem for pseudo-effective line bundles and its applications}},
  journal = {Transactions of the American Mathematical Society, Series B},
  volume = {8},
  year = {2021},
  pages = {849--884},
  doi = {10.1090/btran/86},
  eprint = {1605.02284v3},
  archivePrefix = {arXiv},
  primaryClass = {math.CV},
  url = {https://arxiv.org/abs/1605.02284v3},
  note = {Author version: arXiv:1605.02284v3, 21 May 2022}
}

@misc{Fujisawa2015,
  author        = {Fujisawa, Taro},
  title         = {\href{https://arxiv.org/abs/1512.03887v1}{A remark on the decomposition theorem for direct images of canonical sheaves tensorized with semipositive vector bundles}},
  year          = {2015},
  eprint        = {1512.03887v1},
  archivePrefix = {arXiv},
  primaryClass  = {math.AG},
  doi           = {10.48550/arXiv.1512.03887},
  url           = {https://arxiv.org/abs/1512.03887v1},
  note          = {Version 1, 12 December 2015}
}

@misc{Kleiman2005,
  author = {Kleiman, Steven L.},
  title = {\href{https://arxiv.org/abs/math/0504020v1}{The {Picard} scheme}},
  year = {2005},
  eprint = {math/0504020},
  archivePrefix = {arXiv},
  primaryClass = {math.AG},
  url = {https://arxiv.org/abs/math/0504020v1},
  note = {Version: arXiv:math/0504020v1, 1 April 2005}
}

@article{Lehmann2015,
  author = {Lehmann, Brian},
  title = {\href{https://arxiv.org/abs/1109.4382v3}{Numerical triviality and pullbacks}},
  journal = {Journal of Pure and Applied Algebra},
  volume = {219},
  number = {12},
  year = {2015},
  pages = {5637--5649},
  doi = {10.1016/j.jpaa.2015.05.037},
  eprint = {1109.4382},
  archivePrefix = {arXiv},
  primaryClass = {math.AG},
  url = {https://arxiv.org/abs/1109.4382v3},
  note = {Preprint version: arXiv:1109.4382v3}
}

@misc{Matsumura2018,
  author        = {Matsumura, Shin-ichi},
  title         = {\href{https://arxiv.org/abs/1511.04222v3}{A vanishing theorem of {Koll\'ar--Ohsawa} type}},
  year          = {2018},
  eprint        = {1511.04222v3},
  archivePrefix = {arXiv},
  primaryClass  = {math.CV},
  doi           = {10.48550/arXiv.1511.04222},
  url           = {https://arxiv.org/abs/1511.04222v3},
  note          = {Version 3, 26 January 2018; Theorem 1.3}
}

@article{Richberg1968,
  author        = {Richberg, Rolf},
  title         = {\href{https://link.springer.com/article/10.1007/BF02063212}{Stetige streng pseudokonvexe {Funktionen}}},
  journal       = {Mathematische Annalen},
  volume        = {175},
  year          = {1967},
  pages         = {257--286},
  doi           = {10.1007/BF02063212},
  url           = {https://link.springer.com/article/10.1007/BF02063212},
  note          = {Also commonly cited with the year 1968}
}

@misc{StacksNormNormal,
  author = {{The Stacks Project Authors}},
  title = {\href{https://stacks.math.columbia.edu/tag/0BD3}{The {Stacks Project}, Lemma 31.18.7}},
  year = {2026},
  url = {https://stacks.math.columbia.edu/tag/0BD3},
  note = {Tag 0BD3}
}

@article{Kodaira1954,
 author = {Kodaira, Kunihiko},
 doi = {10.1073/pnas.40.5.313},
 journal = {Proceedings of the National Academy of Sciences of the United States of America},
 number = {5},
 pages = {313--316},
 title = {\href{https://doi.org/10.1073/pnas.40.5.313}{On {K\"ahler} varieties of restricted type}},
 url = {https://doi.org/10.1073/pnas.40.5.313},
 volume = {40},
 year = {1954}
}

@article{Zassenhaus1948,
 author = {Zassenhaus, Hans},
 journal = {Commentarii Mathematici Helvetici},
 pages = {117--141},
 title = {\href{https://www.e-periodica.ch/cntmng?pid=com-001\%3A1948\%3A21\%3A\%3A13}{{\"U}ber einen Algorithmus zur Bestimmung der Raumgruppen}},
 url = {https://www.e-periodica.ch/cntmng?pid=com-001\%3A1948\%3A21\%3A\%3A13},
 volume = {21},
 year = {1948}
}

@article{ELMNP2006,
  author = {Ein, Lawrence and Lazarsfeld, Robert and Musta{\c t}{\u a}, Mircea and Nakamaye, Michael and Popa, Mihnea},
  title = {Asymptotic invariants of base loci},
  journal = {Annales de l'Institut Fourier},
  volume = {56}, number = {6}, year = {2006}, pages = {1701--1734},
  doi = {10.5802/aif.2225},
  url = {https://www.numdam.org/item/10.5802/aif.2225.pdf}
}

@misc{ELMNP2023Erratum,
 author={Ein, Lawrence and Lazarsfeld, Robert and Musta{\c t}{\u a}, Mircea and Nakamaye, Michael and Popa, Mihnea},
 title={Erratum to the paper: Asymptotic Invariants of Base Loci},
 year={2023},
 note={arXiv:2309.16722v1, September 20, 2023},
 url={https://arxiv.org/abs/2309.16722v1}
}
@misc{CompanionG,
  author = {{OpenAI}},
  title = {{Metric descent and rank-preserving contractions}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Metric-descent-and-rank-preserving-contractions-September-26-2026/main.pdf}{OAI:Metric-descent-and-rank-preserving-contractions-September-26-2026}},
  year = {2026}
}

@misc{CompanionE,
  author = {{OpenAI}},
  title = {{Integrable metrics and effectivity with controlled boundary}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Integrable-metrics-and-effectivity-with-controlled-boundary-September-26-2026/main.pdf}{OAI:Integrable-metrics-and-effectivity-with-controlled-boundary-September-26-2026}},
  year = {2026}
}

@misc{CompanionI,
  author = {{OpenAI}},
  title = {{Invariant anticanonical indices and conversion of twisted differentials}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Invariant-anticanonical-indices-and-conversion-of-twisted-differentials-September-26-2026/main.pdf}{OAI:Invariant-anticanonical-indices-and-conversion-of-twisted-differentials-September-26-2026}},
  year = {2026}
}

@misc{CompanionH,
  author = {{OpenAI}},
  title = {{Anticanonical nonvanishing from smooth semipositivity}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026/main.pdf}{OAI:Anticanonical-nonvanishing-from-smooth-semipositivity-September-26-2026}},
  year = {2026}
}

@article{AtiyahBott1968,
  author = {Atiyah, Michael F. and Bott, Raoul},
  title = {A {Lefschetz} fixed point formula for elliptic complexes. {II}. Applications},
  journal = {Annals of Mathematics},
  series = {2},
  volume = {88},
  number = {3},
  pages = {451--491},
  year = {1968},
  doi = {10.2307/1970721}
}
==== END Exact-orders-and-invariant-anticanonical-linear-systems//references.bib ====