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\title[Cohomological transfer and anticanonical sections]{Cohomological transfer and equivariant anticanonical sections\\[2pt]\large A reader's edition for a first mathematical audience}
\author{OpenAI}
\date{September 26, 2026}

\begin{document}
\begin{abstract}
Let $X$ be a smooth projective complex variety, and suppose its anticanonical line bundle $L=-K_X$ admits a Hermitian metric whose Chern curvature is nonnegative: in a word, $L$ is \emph{smoothly semipositive}. We prove that $X$ then has a nonzero section of some positive multiple $mL$. We prove more. If an algebraic torus $T$ acts on $X$ and a section of $M+iL$ exists for unbounded positive integers $i$, invariant under the torus and with one fixed error $M$ whose inverse $-M$ is pseudoeffective, then an invariant section of a positive untwisted power $mL$ exists. The number $m$ may depend on the data; the paper does not produce a uniform $m$. The engine is a field of ratios of invariant sections. The engine has three consequences: natural invariant nonvanishing under a nonzero Euler characteristic; ordinary nonvanishing without that hypothesis, via compact monodromy and a finite norm; and exact source characters from a one-parameter action. Several independent routes to the same conclusion (Iitaka, harmonic restriction, alteration, multiplier ideal, stabilized direct image) are given so that the reader can see how the same obstructions are handled in different regularity regimes. This edition is an expanded and reorganised presentation of the original paper; it does not change the mathematics and it deliberately spells out the ideas that the original assumed its audience could supply.
\end{abstract}
\maketitle
\tableofcontents

\section*{How to read this paper}\addcontentsline{toc}{section}{How to read this paper}

This paper proves one small but stubborn-sounding thing: on a projectivised curved space $X$ of a certain positive type, some positive power of the anticanonical bundle has a nonzero section. Read on for what that sentence means and why it is not obvious. Read the rest for a proof.

There are three audiences for the body. The first is a graduate student in algebraic or complex geometry who wants to understand the original paper without first learning every piece of its analytic apparatus. The second is a reader with a strong undergraduate background who wants the ideas first and the machinery second. The third is a curious high-school reader. Sections~\ref{sec:question}, \ref{sec:vocab}, and the beginning of every later section are aimed at the third audience: they explain, in words first, what is about to be proved. Everything after the words is written at the first audience's level. A student may skim the proofs and still follow the narrative; a researcher may skip the narrative and still find the proofs.

The method has an unusual feature worth flagging at the start. Every construction in this paper is presented in two registers: the \emph{plain form} and the \emph{technical form}. The plain form takes one to five sentences and can be read without stopping. The technical form is the actual proof. The two forms refer to the same object; if you cannot follow the technical form, the plain form tells you what the object is for.

A second feature is that we do not chase a \emph{uniform} exponent. Some power of $mL$ will be produced; the paper does not say which. This is typical in the subject and is not a defect: the exponent can jump when a section has common powers taken on compact factors or when a finite norm is formed.

A third feature is that the paper deliberately offers several routes to its main corollary. The routes are not logically necessary; one route would prove the result. They are given because at different points in the argument different obstructions dominate, and different analytic theorems handle each obstruction. A reader who wants to see how a harmonic argument attacks a problem, here it is. A reader who wants to see how an algebraic argument attacks the same problem, here it is.

A last word before the mathematics. If a sentence below says \emph{semipositive}, a high-school reader can think: the curvature is nonnegative everywhere. If a sentence says \emph{pseudoeffective}, a reader can think: its negative neighbour has nonnegative degree on every curve; the class lies in the closure of effective curves. If a sentence says \emph{invariant}, a reader can think: the torus action fixes the section, not just the line.

\section{The question, and the shape of the answer}\label{sec:question}

Fix a smooth connected projective complex variety \(X\) of dimension \(n\). Think of it as a compact curved space that can be drawn by polynomial equations inside some large projective space. The tangent bundle \(T_X\) records all the ways a curve can sit in \(X\); its determinant
\[
 L \;=\; \det T_X \;=\; -K_X
\]
is a line bundle, the anticanonical line of \(X\). In coordinates, a section of \(L\) is a holomorphic \(n\)-form in the reciprocal frame of \(T_X\); a section of \(mL=L^{\otimes m}\) is an \(m\)-fold symmetric version of that reciprocal form.

\begin{question}\label{q:nonvanishing}
For which smooth connected projective \(X\) over \(\C\) is
\[
 H^0(X,mL)\neq 0
 \qquad\text{for some integer }m>0\text{?}
\end{question}

Nonzero sections of an ample line bundle give maps of \(X\) into projective space; the spaces of sections grow polynomially in \(m\); the linear system of a big line eventually has positive dimension. The anticanonical line \(L\) need not be ample. It can be trivial, as on an abelian variety, and it can be a torsion point of the Picard group. Question~\ref{q:nonvanishing} therefore asks whether some positive power of \(L\) is at least \emph{effective} (that is, whether \(H^0(X,mL)\neq 0\) for some \(m\)), a condition much weaker than amplitude. It is the first question one can ask about positivity of \(L\) and the natural warm-up for the harder question of semiampleness.

The word \emph{nonvanishing} in the literature refers to this question. The companion problem, \emph{abundance}, asks for much more: whether the sections of \(L\) generate a morphism nice enough to be a fibration. We do not address abundance.

An obstruction is hidden in the word \emph{curved}. Suppose \(L\) carried a smooth Hermitian metric with strictly positive curvature. The Kodaira vanishing theorem would then say that \(H^0(X,mL)\) is enormous for all large \(m\), and Question~\ref{q:nonvanishing} would be easy. But most interesting cases of \(L\) have curvature that is allowed to vanish. In fact a basic theorem of Demailly--Peternell--Schneider, and later Campana--Demailly--Peternell, says that a projectivised \(X\) whose anticanonical line is nef is essentially a fibration over a set of rationally connected (and compact) factors. The vanishing locus of the curvature is the obstruction, and it cannot be removed by choosing a better metric.

We fix the following hypothesis, which is weaker than amplitude but stronger than numerical nonnegativity.

\begin{definition}\label{def:smoothly-semipositive}
A line bundle \(L\) on a complex manifold is \emph{smoothly semipositive} if there is a smooth Hermitian metric \(h\) on \(L\) whose Chern curvature \(i\Theta_h(L)\) is a smooth semipositive \((1,1)\)-form on \(X\).
\end{definition}

Smooth semipositivity of \(L\) does not imply smooth semipositivity of \(L^{\otimes m}\); sections in powers have to be produced, not assumed. Nor does it imply that \(L\) is the pullback of an ample line from some base; that would be an Iitaka fibration, and the existence of such a fibration is equivalent to the answer to Question~\ref{q:nonvanishing} that we want to prove.

Our main theorem, restated in the introduction, is the following.

\begin{corollary}[Restatement of Corollary~\ref{thm:ordinary} below]\label{cor:main-informal}
Every smooth connected projective complex variety \(X\) with smoothly semipositive \(L=-K_X\) has \(H^0(X,mL)\neq 0\) for some integer \(m>0\).
\end{corollary}

The power \(m\) in Corollary~\ref{cor:main-informal} is not uniform. It depends on \(X\). A uniform bound cannot exist: on the projective variety \(X=\PP^1\), \(m=1\); on an abelian surface, where \(L\) is trivial, \(m\) must equal \(1\); on some Fano manifolds, small powers are non-effective and large powers are effective; on a K3 surface with smooth semipositive anticanonical line (that is, a K3 with \(L\) trivial), the same \(m=1\) applies. These examples are not an obstruction to the theorem, they are its proof strategy: it takes some power, not a specific one, and it depends on the geometry of \(X\).

The rest of the introduction to this paper has three parts. Section~\ref{sec:vocab} is a tiny glossary. Sections~\ref{sec:transfer} and \ref{sec:index} give the two ideas that do most of the work: a field of ratios, and an index theory for torus representations. Sections~\ref{sec:descent} onward apply these ideas, with different analytic input, and Section~\ref{sec:map} draws the map of the whole argument.


\section{Vocabulary}\label{sec:vocab}

This section is a glossary. Each item is one sentence; a reader who already knows the term can skip it. Everything later refers back to these items by the item number.

\begin{description}
\item[(V1)] A \emph{smooth projective complex variety} \(X\) is the common zero locus of homogeneous polynomials in some \(\PP^N\) that is a complex manifold of pure dimension \(n\). Projective means embeddable in \(\PP^N\). Smooth means no singularity. Connected means one piece.
\item[(V2)] The \emph{tangent bundle} \(T_X\) is the complex vector bundle whose fibre at a point is the tangent space in the holomorphic category. The \emph{cotangent bundle} \(\Omega_X^1=T_X^\vee\) is its dual. Higher exterior powers \(\Omega_X^p=\bigwedge^pT_X^\vee\) are the holomorphic \(p\)-forms.
\item[(V3)] The \emph{canonical line bundle} is \(K_X=\det\Omega_X^1=\bigwedge^n\Omega_X^1\); its inverse \(L=-K_X=\det T_X\) is the \emph{anticanonical} line bundle. On a point \(K_X=\OO\) and \(L\) is trivial.
\item[(V4)] A \emph{line bundle} on \(X\) is a complex vector bundle of rank one; equivalently an invertible sheaf of \(\OO_X\)-modules. Two are \emph{linearly equivalent} when isomorphic. Tensoring line bundles makes their set into a group, the \emph{Picard group} \(\Pic(X)\). We write \(A+B\) for the tensor product \(A\otimes B\) and \(mL\) for \(L^{\otimes m}\).
\item[(V5)] A \emph{section} of a line bundle \(A\) over an open set \(U\) is a holomorphic way to choose a point in each fibre of \(A|_U\). The symmetric power \(H^0(X,mL)\) is the vector space of sections of \(mL\); its elements vanish along effective divisors when nonzero.
\item[(V6)] The \emph{cohomology group} \(H^q(X,A)\) is the \(q\)-th derived functor of \(H^0(X,-)\), computed in \v{C}ech or Dolbeault terms. On a smooth projective \(X\), \(H^q(X,\OO_X)\) is a Hodge summand of \(H^q(X,\C)\). The \emph{Euler characteristic} of a vector space or sheaf is \(\chi(X,A)=\sum_q(-1)^q\dim H^q(X,A)\).
\item[(V7)] A \emph{Hermitian metric} \(h\) on a line bundle \(A\) is a smooth family of inner products on the fibres. Its \emph{Chern curvature} \(i\Theta_h(A)\) is a closed smooth \((1,1)\)-form representing the first Chern class \(c_1(A)\). The metric is \emph{semipositive} when \(i\Theta_h(A)\ge0\) as a \((1,1)\)-form. It is \emph{pseudoeffective} when \(c_1(A)\) lies in the closure of the cone of effective divisor classes, equivalently when \(A\) carries a singular Hermitian metric with \(i\Theta\ge0\).
\item[(V8)] An \emph{algebraic torus} \(T=(\C^\times)^k\) acts algebraically on \(X\). A \emph{character} \(\chi\) of \(T\) is a holomorphic group homomorphism \(T\to\C^\times\), necessarily monomial and therefore of the form \(t\mapsto t^a\) for \(a\in\Z^k\); we write such characters additively. A \emph{linearization} of a \(T\)-action on a line bundle \(A\) is a lift of the action to \(A\) that is linear on fibres; if \(A=mL\), the natural linearization uses the natural action of \(T\) on \(T_X^{\otimes m}\).
\item[(V9)] A section is \emph{invariant} under \(T\) if the linearization sends it to itself. The invariant subspace \(H^0(X,A)^T\) is what we control. On a compact torus and a shift \(P=L\otimes\text{character}\), invariants of the natural and shifted linearizations agree; on a non-compact torus they do not.
\item[(V10)] An \emph{abelian variety} is a complex torus that is projective. A \emph{Fano} variety is smooth projective with \(-K_X\) ample. A \emph{K3 surface} is a simply connected compact complex surface of complex dimension two with trivial canonical bundle.
\item[(V11)] A \emph{normal} variety is one whose local rings are normal domains; it is smooth outside a closed set of codimension at least two. A \emph{birational morphism} is a proper morphism that is an isomorphism outside a proper subvariety. \emph{Resolving} a variety means replacing it by a smooth variety that maps to it by a birational morphism and is isomorphic outside a proper subvariety; over \(\C\), resolutions always exist (Hironaka).
\item[(V12)] A \emph{flat} morphism \(f:V\to Y\) between algebraic varieties is one whose fibres deform without jumping in dimension or in arithmetic genus. An \emph{equidimensional} morphism is one whose fibres all have dimension exactly \(\dim V-\dim Y\). A flat projective morphism with integral target is equidimensional if and only if every irreducible component of every fibre has the expected dimension.
\end{description}

A note on \emph{ample} and \emph{nef}. A line bundle is ample when some positive power admits a projective embedding; nef when its degree on every curve is nonnegative. Ample implies nef implies pseudoeffective; none of the converses hold. Smoothly semipositive anticanonical (V7) is a differential-geometric strengthening of nef. It rules out all of the pathological curvature-zero cases only when the curvature form is zero on a large locus; on those cases the analysis in later sections becomes essential.

We will use without further comment the following standard operations. Restriction of a line bundle and a section to a subvariety; pullback along a morphism; pushforward of a coherent sheaf along a proper morphism; the projection formula; base change for a flat morphism. All are in Hartshorne or the Stacks Project. If a reader is unsure about any of them, the reference is Chapter~III of Hartshorne's \emph{Algebraic Geometry} and the tags listed in the bibliography.

\subsection*{What the reader should expect}

The paper has three regions. The first (Sections~\ref{sec:transfer} and~\ref{sec:index}) constructs the transfer theorem: an abstract machine that removes a fixed error from unbounded invariant twisted sections. The second (Section~\ref{sec:descent}) applies the machine, together with structure theorems of the compact-montone setting, to prove the ordinary corollary. The third (Sections~\ref{sec:extremal} onward) presents alternative routes, each proving the same main results under slightly different hypotheses and in a different regularity regime. A reader who wants a single shortest path can read Sections~\ref{sec:transfer}, \ref{sec:index}, \ref{sec:descent}, then the reading-path table at the end; a reader who wants the machinery behind each dish should stay for the whole meal.


\section{The transfer theorem: how to remove a fixed error}\label{sec:transfer}

\paragraph{In words.} Suppose a torus \(T\) acts on our variety \(X\). Suppose, for unbounded positive \(i\), we have an invariant section \(s_i\) of the line bundle \(M+iL\), where \(M\) is some fixed error bundle with \(M^{-1}\) pseudoeffective. The point of the theorem is that the error can be absorbed: some positive power \(mP\) of the linearized anticanonical line has an invariant section.

\paragraph{The idea in one sentence.} A ratio \(s_q/s_p\) of two such sections is a rational section of \((q-p)P\) with a controlled zero divisor; the unboundedness of the index \(i\) is exactly what makes the divisor controlled.

\paragraph{An analogy.} Imagine \(s_i\) is a polynomial of degree \(2i-1\) on the circle \(S^1\), and we want to know if some positive-degree polynomial exists that is a genuine section. The ratio of a degree-5 and a degree-3 polynomial is a rational function of degree 2; on \(S^1\) it is automatically meromorphic on the sphere, with at most a bounded pole. The unboundedness of \(i\) hides the fixed error of \(1\) in the degree; what survives is the ratio.

\subsection*{Statement}

\begin{theorem}[Invariant transfer]\label{thm:transfer}
Let \(X\) be smooth connected projective and let \(L=-K_X\) carry a smooth semipositive metric. Let an algebraic torus \(T\) act on \(X\), and let \(P,M\) be linearized line bundles with
\[
 P=L\text{ as ordinary line bundles},\qquad -M\text{ pseudoeffective}.
\]
The linearization of \(P\) may differ from the natural linearization of \(L\) by a character. Suppose that for an unbounded subset \(I\subset\Z_{>0}\) there are nonzero invariant sections
\[
 s_i\in H^0(X,M+iP)^T,\qquad i\in I.
\]
Then \(H^0(X,mP)^T\ne0\) for some integer \(m>0\).
\end{theorem}

If \(T\) is trivial, the conclusion is \(H^0(X,mL)\ne0\) for some \(m>0\), with no character condition. The character of \(P\) is preserved by the transfer: only the direction of the linearization shift is lost.

\subsection*{Why the error cannot simply be dropped}

Suppose \(M=-\text{ample}\), e.g.\ \(M=\OO(-1)\) on \(\PP^1\) with \(L=\OO(2)\). Then \(-M\) is ample, and \(M+iL=\OO(2i-1)\). For \(i\ge1\) the degree \(2i-1\) is nonnegative and sections exist. The conclusion is \(H^0(\PP^1,\OO(2))\ne0\), and indeed the linear system of degree 2 polynomials is two-dimensional. This is the trivial end of the theorem; the more interesting cases are on rationally connected varieties where the negative error \(M\) is genuinely felt.

A case with no triviality is the following. Let \(X\) be an Enriques surface, where \(K_X\) is 2-torsion and \(L=-K_X\) is not trivial. Then \(-K_X\) has a flat metric (curvature zero), and \(L\) is smoothly semipositive. Take \(M\) trivial and \(T\) trivial. The section space \(H^0(X,iL)\) is zero for \(i\) odd and one-dimensional for \(i\) even; unbounded \(i\) exists, and the conclusion says \(H^0(X,mL)\ne0\). Here \(m=2\). The theorem has not done any work, because the answer follows from \(2L=\OO\). The theorem's content is on varieties where a family of sections really has to be manufactured; the point of the theorem is that the manufacturing is the same in every case, given unbounded invariant sections of \(M+iP\).

\begin{proof}[Proof of Theorem~\ref{thm:transfer}]

The proof is long and technical; it splits into seven steps. The plain-language form of each step is stated first; the technical form follows.

\medskip
\noindent\textbf{Step 1: the common ratio field.}
\emph{Plain form.} Ratios of invariant sections of the same power behave like rational functions and form a field. As \(i\) varies, these fields are all the same, eventually.

\emph{Technical form.} Put \(n=\dim X\), \(N_i=M+iP\). For each \(i\), let \(Q_i\) be the field generated over \(\C\) by all ratios of invariant sections of positive multiples of \(N_i\). For \(p<q<b\) in \(I\), the linearized identity
\[
 (b-p)N_q=(b-q)N_p+(q-p)N_b
\]
holds. Given any ratio in degree \(aN_p\) or \(aN_b\), multiply numerator and denominator by appropriate powers of the fixed \(s_p,s_b\) to put the ratio in the field for \(N_q\). Taking \(q\) sufficiently large shows that all \(Q_i\) with \(i>\min I\) coincide; call the common field \(Q\).

\medskip
\noindent\textbf{Step 2: if the field is trivial, we are done.}
\emph{Plain form.} If the ratios are all constant, the section divisors are linearly related, and two ratios of sections give a section of a positive power of \(P\) directly.

\emph{Technical form.} Fix \(p<q\) in \(I\). For \(b>q\), the ratio \(s_q^{b-p}/(s_p^{b-q}s_b^{q-p})\) belongs to \(Q\). If \(\trdeg_\C Q=0\), this ratio is constant. The section divisors are affine in the index \(i\), and their slope is effective because \(I\) is unbounded; thus \(s_q/s_p\) extends to a regular invariant section of \((q-p)P\).

\medskip
\noindent\textbf{Step 3: if the field is nontrivial, it defines a base.}
\emph{Plain form.} A nontrivial field of ratios gives a fibration \(f:V\to Y\), and the torus acts trivially on \(Y\). On each fiber, the sections become unique up to scalar.

\emph{Technical form.} Assume \(\trdeg_\C Q=e>0\). Choose \(e\) independent ratios to define a rational map to \(\PP^e\), resolve its graph, take Stein factorization. The Stein base \(Y\) is smooth projective after resolution; its field is relatively algebraically closed in \(\C(X)\) and contains \(Q\); its torus action is trivial. Choose a normal equidimensional model \(W\to Y\) and an equivariant resolution \(V\to W\) as in Lemma~\ref{lem:flat-model} (Section~\ref{sec:models}). Write \(\pi:V\to X\), \(f:V\to Y\), \(P'=\pi^*P\), \(L'=\pi^*L\), \(E=K_{V/X}\).

\medskip
\noindent\textbf{Step 4: the fiberwise adjoint rank is one.}
\emph{Plain form.} On the generic fiber, invariant sections of the adjoint line \(E+jkP'\) are unique up to scalar, for every \(j\).

\emph{Technical form.} For every interior index \(i\) and every divisor \(H'\) on \(Y\), the ratio construction gives \(H^0(V,a\pi^*N_i-f^*H')^T\ne0\) for some \(a>0\). Combined with Step 3, this proves
\[
 \rank\bigl(f_*\mathcal O_V(E+jkP')\bigr)^T=1
 \qquad(j\ge0).
\]
The section \(1_E\gamma^j\) (where \(1_E\) is the canonical section of \(E\)) gives the lower bound.

\medskip
\noindent\textbf{Step 5: a regular relative section.}
\emph{Plain form.} Remove the vertical poles by subtracting, over each base prime, the minimal relative pole order. The result is a regular section of \(kP_W-f_W^*(kB)\) for a base divisor \(B\).

\emph{Technical form.} Let \(D_i\) be the zero divisor of \(s_i\) on \(W\). By Step 3 its horizontal part is controlled; its vertical part is subtracted by the minimum coefficient. Choose \(k>0\) divisible by \(q-p\) and clearing the rational coefficients of the resulting base divisor \(B\). Then
\[
 \gamma=(s_q/s_p)^{k/(q-p)}(f_W^*1_B)^{-1}\in H^0(W,kP_W-f_W^*(kB))^T
\]
is regular and nonzero at the general point of at least one component above every base prime.

\medskip
\noindent\textbf{Step 6: two metrics and the fixed multiplier ideal.}
\emph{Plain form.} We need a bounded metric on the base and a positive-curvature metric at twist zero. The bounded metric comes from the maximum construction (Section~\ref{sec:models}); the positive-curvature metric uses pseudoeffectivity of \(-M\). Its multiplier ideal is not assumed trivial; it is fixed while the exponent varies.

\emph{Technical form.} Average a smooth semipositive metric \(h_0\) on \(L\) over the maximal compact torus; it is invariant for both the natural and shifted linearizations, because a character is unitary on the compact torus. The ratio domination plus pseudoeffectivity of \(-M\) gives a singular invariant metric \(h\) on \(L'\) with
\[
 i\Theta_h\ge\delta f^*\omega_Y,\qquad \mathcal I(h)=\OO_V,\qquad \delta>0.
\]
The first inequality uses ratio domination to make \(L'-\eta f^*H\) pseudoeffective for some \(\eta>0\), mixing the resulting current with \(h_0\).

\medskip
\noindent\textbf{Step 7: Nadel vanishing and Euler interpolation.}
\emph{Plain form.} For each \(j\ge0\), the base twists give a polynomial in \(j\). At \(j=0\) the polynomial is one; hence it is nonzero for some positive \(j\). The resulting section pulls back, multiplies, and pushes down to give the required invariant section.

\emph{Technical form.} Apply Proposition~\ref{prop:maximum} to obtain a continuous semipositive metric on \(kB\). The singular coefficient metric at \(j=0\) gives a metric on the base with positive curvature and multiplier ideal \(\mathcal J\), not assumed trivial. Tensor with the bounded metric on \(jkB\); the ideal remains \(\mathcal J\), and Nadel vanishing gives
\[
 H^v\bigl(Y,\OO_Y(D_*+jkB)\otimes\mathcal J\bigr)=0
 \qquad(v>0,\ j\ge0).
\]
The Euler characteristic of \(\OO_Y(D_*)\otimes\mathcal J\) twisted by \(jkB\) is a polynomial in \(j\). At \(j=0\) its value is \(h^0(Y,\mathcal J)\ge1\), because \(\OO_Y(D_*)\otimes\mathcal J\supseteq\OO_Y\) at the level of the multiplier ideal. A nonzero section therefore exists for some positive \(j\). Pull back, multiply by \(\gamma^j\), push down to \(X\), and remove the exceptional twist \(E\).
\end{proof}

Two technical ingredients are used without proof inside Theorem~\ref{thm:transfer}: the maximum metric of Proposition~\ref{prop:maximum} (built in Section~\ref{sec:models}), and the extension of the base singular metric across boundary divisors (built in Section~\ref{sec:local-ideal}). If the reader prefers to see the proof of the fixed multiplier ideal first, they can read Section~\ref{sec:local-ideal} before returning to this proof.


\section{The natural index, and cohomological input}\label{sec:index}

\paragraph{In words.} The transfer theorem of Section~\ref{sec:transfer} takes invariant twisted sections as input. Where do they come from? They come from cohomology. If a torus acts and the Euler characteristic of the sheaf is not zero, then some fixed cohomological degree is nonzero on an unbounded set of twists. By hard Lefschetz, that nonzero cohomology produces invariant twisted forms. Those forms then feed into the transfer theorem.

\paragraph{The two halves.} The first half is an index theorem: the alternating sum of dimensions of invariant cohomology in the twisted powers of the anticanonical line is a polynomial, whose value at zero is the actual Euler characteristic. The second half is a Lefschetz argument: cohomology classes in a fixed degree propagate downward to holomorphic forms.

\subsection*{The invariant anticanonical index}

The following is an input theorem, imported from the companion paper \cite{CompanionI}; we do not reprove it here.

\begin{proposition}[Invariant anticanonical index]\label{prop:index}
Let \(F\) be a compact complex manifold with a holomorphic action of a compact real torus \(T\). Give \(D=-K_F\) its natural linearization. There are an integer \(a>0\) and a polynomial \(P\in\Q[t]\) such that
\[
 \sum_q(-1)^q\dim H^q(F,maD)^T=P(m)\qquad(m>0),\qquad P(0)=\sum_q(-1)^q\dim H^q(F,\OO_F)^T.
\]
The statement admits a trivial torus, an empty fixed locus, and zero-dimensional fixed components; it assumes neither positivity, projectivity, nor K\"ahlerity.
\end{proposition}

The value at zero is the actual index of the action, not a formal extrapolation of an eventual polynomial. If the value at zero is nonzero, then some fixed cohomological degree is nonzero for infinitely many \(m\). This is what connects representation theory to the transfer theorem.

\begin{remark}
The number \(a\) in Proposition~\ref{prop:index} is a splitting integer for the action; it need not be \(1\). The companion paper gives the effective bound. The conclusion of the present paper does not depend on knowing \(a\).
\end{remark}

\subsection*{From cohomology to twisted forms}

The bridge between cohomology classes and sections of a line bundle is hard Lefschetz, made equivariant by an averaged K\"ahler form.

\begin{lemma}[Forms and determinant sections]\label{lem:forms}
Let \(X\) be smooth connected projective, let \(L=-K_X\) have a smooth semipositive metric, and let an algebraic torus \(T\) act on \(X\). If \(\chi^T(X,\OO_X)\ne0\), then for some fixed \(q\) there are nonzero invariant sections of \(\Omega_X^{n-q}\otimes L^{m+1}\) for unbounded positive integers \(m\). If these do not already give a nonzero invariant section of a positive power of \(L\), then there are a linearized line bundle \(M\) and a nonzero map \(M\to\bigwedge^e\Omega_X^{n-q}\) for some \(e>0\), together with invariant sections
\[
 0\ne s_i\in H^0(X,M+a_iL)^T,\qquad a_i\to\infty.
\]
\end{lemma}

\begin{proof}[Sketch]
The polynomial in Proposition~\ref{prop:index} is not zero at zero, so a fixed cohomological degree has nonzero invariant cohomology along an unbounded sequence. Average a K\"ahler form and the metric weights over the maximal compact torus; write \(\theta\) for the resulting curvature of \(L\). The smooth semipositive hard Lefschetz theorem \cite{DPS2001}, applied to \(mL=K_X+(m+1)L\), sends a harmonic \((n,q)\)-form \(u\) with values in \((m+1)L\) to a holomorphic form of type \((n-q,0)\), via a complex-linear star in the form indices. The construction is equivariant. For \(q=0\) this already produces invariant sections of \(mL\). For \(q=n\), the conjugate of the \((0,n)\)-valuation of \(u\) is a section of \((m+1)L\). If \(q>0\) and \(r=\max\rank\theta=n\), Bochner--Kodaira makes the holomorphic form zero on a nonempty open set, hence everywhere.

In the remaining case, one extracts from the unbounded family \(\{u_m\}\) a determinant subsheaf of a positive exterior power of \(\Omega_X^{n-q}\). Fix a rational frame of \(L\) and divide each form by the corresponding power of that frame. The span over \(\C(X)\) has some rank \(e\); the specific frame only rescales each vector by a nonzero rational function, so the span is intrinsic. Saturate its top exterior power and call the resulting reflexive rank-one sheaf \(M\). Choose finitely many of the original forms as a generic basis, and wedge each unbounded member with \(e-1\) fixed ones. The resulting nonzero invariant wedge has twist degree \(m+1\) plus a fixed summand; its class is a section of \(M+a_iL\) with \(a_i\to\infty\).
\end{proof}

\paragraph{Consequence: natural invariant nonvanishing.}

\begin{corollary}[Natural invariant nonvanishing]\label{thm:main}
Let \(X\) be smooth connected projective over \(\C\) with smoothly semipositive \(L=-K_X\). Suppose an algebraic torus \(T\) acts on \(X\) and \(\chi(X,\OO_X)\ne0\). Then \(H^0(X,mL)^T\ne0\) for some integer \(m>0\), with the natural linearization.
\end{corollary}

\begin{proof}[Sketch]
Every element of the connected torus is homotopic to the identity, so its action on singular cohomology is trivial. By Hodge decomposition the action on every \(H^q(X,\OO_X)\) is trivial, and the invariant index at zero equals \(\chi(X,\OO_X)\ne0\). Lemma~\ref{lem:forms} supplies either the required section directly, or invariant sections of \(M+a_iL\) with \(a_i\to\infty\) and a nonzero map \(M\to(\Omega_X^1)^{\otimes p}\) for some \(p>0\). The cotangent-tensor theorem \cite[Theorem 4.1]{LMPTX2023} applied to the smooth zero-boundary pair with nef \(-K_X\) says that \(-M\) is pseudoeffective. Apply Theorem~\ref{thm:transfer} with \(P=L\) naturally linearized.
\end{proof}

The hypothesis \(\chi(X,\OO_X)\ne0\) holds in a large class of examples: Fano varieties (\(\chi=1\)), rationally connected varieties (same), smooth varieties with \(L\) strictly semipositive and \(H^q(X,\OO_X)=0\) for \(q>0\). It fails on an abelian variety, where the invariant index vanishes (because \(H^q\) alternates signs), but the conclusion of the corollary still holds on an abelian variety: \(H^0(X,\OO_X)=\C\).

\subsection*{A degree trick that removes the pseudoeffectivity hypothesis}

The proof of Corollary~\ref{thm:main} uses the cotangent-tensor theorem to get \(-M\) pseudoeffective. The following lemma obtains the degree inequality directly from a prescribed-Ricci-curvature argument, without invoking the pseudoeffectivity theorem. It is needed in the harmonic constructions of Sections~\ref{sec:harmonic} and~\ref{sec:fields} to control the zero divisors of \(M\).

\begin{lemma}[Curvature-null determinant divisors]\label{lem:null-divisors}
Let \(X\) be smooth connected projective of dimension \(n\), let \(L=-K_X\) have smooth semipositive curvature \(\theta\) of maximal rank \(r<n\). Suppose a line bundle \(M\) admits a nonzero map \(M\to\bigwedge^e\Omega_X^p\) for some \(e,p>0\). Then \(c_1(M)G^{n-1}\le0\) for every K\"ahler class \(G\). If \(0\ne s\in H^0(X,M+aL)\) for an integer \(a\), its zero divisor \(D\) satisfies
\[
 [D][\theta]^r[\omega]^{n-r-1}=0
\]
for every K\"ahler form \(\omega\). Each component has zero intersection separately, and \(\theta^r\) vanishes on its smooth locus.
\end{lemma}

\begin{proof}[Sketch]
For every K\"ahler class \(G\), Yau's prescribed-Ricci theorem \cite{Yau1978} gives a K\"ahler metric in \(G\) with Ricci form \(\theta\). K\"ahler curvature symmetry identifies the mean curvature of \(T_X\) with the Ricci form; the induced mean curvature of \(\bigwedge^e\Omega_X^p\) is nonpositive. If \(\deg_G M>0\), a scalar Laplace equation gives \(M\) a metric of positive constant mean curvature; its inverse tensor the cotangent construction then has negative mean curvature, contradicting the Bochner formula for its nonzero section. So \(c_1(M)G^{n-1}\le0\). Put \(G=[\theta]+t[\omega]\), divide by \(t^{n-1-r}\), and let \(t\downarrow0\). Since \(\theta^{r+1}=0\), \(c_1(M)[\theta]^r[\omega]^{n-1-r}\le0\). The same intersection for \(M+aL\) is nonnegative by effectivity and smooth semipositivity, and the contribution from \(aL\) is zero; it is therefore zero. Each component has zero intersection separately, and the nonnegative form \(\theta^r\wedge\omega^{n-r-1}\) vanishes on its smooth locus.
\end{proof}

\begin{remark}[A second proof of the degree sign]
The determinant degree argument also has a direct current proof, useful when the prescribed-Ricci metric has weaker regularity. In a K\"ahler metric with the prescribed semipositive Ricci form, let \(e\) be the local image of a frame of \(M\) in the cotangent construction. Its mean curvature is nonpositive. Chern's norm differentiation formula makes the trace of \(i\partial\bar\partial\log(|e|^2+\varepsilon)\) nonnegative: the negative-curvature term has the required sign, and the derivative term is nonnegative by Cauchy--Schwarz. The logarithms converge locally in \(L^1\) by comparison with logarithms of holomorphic coefficients, and the trace-positive current represents \(-c_1(M)\). Integrating proves \(c_1(M)G^{n-1}\le0\), including possible codimension-two zeros of the injected frame.
\end{remark}

\subsection*{Where this leaves us}

Corollary~\ref{thm:main} proves the main theorem under the extra hypothesis \(\chi(X,\OO_X)\ne0\). To remove that hypothesis we need one more ingredient: structure theory of the universal cover when \(L\) is only semipositive. That is the subject of Section~\ref{sec:descent}.


\section{Compact monodromy and the ordinary corollary}\label{sec:descent}

\paragraph{In words.} Corollary~\ref{thm:main} needs \(\chi(X,\OO_X)\ne0\). To remove that hypothesis, split the universal cover of \(X\) into three factors: a Euclidean factor, a product of compact Ricci-flat factors, and a compact factor with no positive-degree forms. On the last factor the torus is compact, so Corollary~\ref{thm:main} produces an invariant section. Multiplying by the canonical frames of the first two factors and descending gives a section of a positive power of \(L\) on \(X\) after a finite \'etale cover; taking a finite norm finishes the descent to \(X\) itself.

\paragraph{The structure theorem.} The following input is imported from the companion paper \cite{CompanionH}.

\begin{theorem}[Finite cover with compact torus monodromy]\label{thm:structure}
Let \(X\) be smooth connected projective with smoothly semipositive \(-K_X\). There is a holomorphic isometric universal-cover splitting
\[
 \widetilde X=\C^c\times V\times F,
\]
where \(V\) is a product of compact simply connected Ricci-flat factors with invariant parallel canonical frames and \(F\) is a product of compact projective factors with no positive-degree holomorphic forms. After a finite index, the deck subgroup acts faithfully by a translation lattice on \(\C^c\), trivially on \(V\), and through a compact real torus \(H\) on \(F\). Empty products are points. The associated finite cover \(X'\) is smooth projective and need not be a product.
\end{theorem}

The factors are determined up to isometry by the de Rham decomposition of the universal cover. If \(F\) is a point, the action is through \(\C^c\); if \(\C^c\) is a point, the cover is compact. The whole point of the theorem is the word \emph{compact} in ``compact real torus'': on \(F\), the torus is a compact group, and averaging arguments apply.

\begin{corollary}[Ordinary nonvanishing]\label{thm:ordinary}
Every smooth connected projective complex variety \(X\) with smoothly semipositive \(L=-K_X\) has \(H^0(X,mL)\ne0\) for some integer \(m>0\).
\end{corollary}

\begin{proof}[Proof of Corollary~\ref{thm:ordinary}]
On \(F\), the compact torus \(H\) preserves a very ample line because it preserves its Chern class and \(\Pic^0(F)=0\). Its projective action has algebraic-torus Zariski closure, so after passing to the finite special-linear cover if necessary we may linearize a very ample power. Invariants of the natural anticanonical line for the compact group and its Zariski closure agree. Corollary~\ref{thm:main} therefore gives a nonzero invariant section of \(-mK_F\) for some \(m>0\).

Pull this section to \(\widetilde X\) and tensor with the \(m\)-th powers of the inverse canonical frames on \(\C^c\) and \(V\). The product is nonzero and invariant under the chosen finite-index deck subgroup; it descends to a section on \(X'\). If \(F\) is a point, take its section to be \(1\) and \(m=1\).

The finite \'etale norm \cite{CompanionH} sends the section on \(X'\) to a nonzero section of a positive power of \(-K_X\). Explicitly: pull back to a connected finite Galois refinement, multiply all Galois translates, and descend the invariant product. The product is nonzero on the integral cover, and canonical bundles pull back under \'etale morphisms. Projectivity and GAGA make the resulting section algebraic.
\end{proof}

The exponent can increase both when common powers are taken on compact factors and when the finite norm is formed. No uniform exponent is asserted.

\subsection*{Two alternative descent maps}

The proof above descends a section. There are two other natural descent maps.

\paragraph{Descending a cohomological class.} The invariant Euler characteristic of \(F\) is one, so Proposition~\ref{prop:index} gives, for unbounded \(m\), nonzero classes in \(H^q(F,-mK_F)^H\), with a fixed \(q\) after subsequence extraction. Choose a smooth closed Dolbeault representative and average it over the compact torus. Pull the representative to the product universal cover and tensor with powers of the constant anticanonical volume on \(\C^c\) and the invariant trivializations on \(V\). The resulting class is deck-invariant for the selected subgroup and descends to a finite \'etale cover \(X'\), where it remains nonzero because restriction to a compact slice \(F\) recovers the original class. Theorem~\ref{thm:alteration-conversion} (Section~\ref{sec:alteration}) then produces a positive anticanonical section on \(X'\). The finite norm of the previous section finishes on \(X\).

\paragraph{Descending a twisted form.} The third descent map descends forms before converting them. On \(F\), Lemma~\ref{lem:forms} produces invariant sections
\[
 0\ne\alpha_m\in H^0\bigl(F,\Omega_F^{d-q}\otimes(m+1)P\bigr)^H,\qquad P=-K_F,\ d=\dim F,
\]
for unbounded \(m\) and a fixed \(q\). Projecting to \(F\) and embedding the pulled-back cotangent bundle into the product cotangent bundle gives a twisted form
\[
 \widetilde\alpha_m=\operatorname{pr}_F^*\alpha_m\otimes(\tau_0\otimes\tau_V)^{m+1}\in H^0\bigl(\widetilde X,\Omega_{\widetilde X}^{d-q}\otimes(-K_{\widetilde X})^{m+1}\bigr),
\]
where \(\tau_0,\tau_V\) are the translation-invariant inverse canonical frames. Every factor is invariant under the deck subgroup, so the form descends to a nonzero holomorphic section
\[
 0\ne\beta_m\in H^0\bigl(X',\Omega_{X'}^{d-q}\otimes(-K_{X'})^{m+1}\bigr).
\]
If \(F\) is a point, take \(q=d=0\) and \(\alpha_m=1\). Now apply Corollary~\ref{thm:forms-input} (Section~\ref{sec:ordinary}) on \(X'\) with the fixed differential degree \(d-q\) and twist \(m+1\); it produces a positive anticanonical section on \(X'\). The finite norm finishes on \(X\). This route does the ordinary conversion \emph{after} descending forms, whereas the previous route did it \emph{before}; both are used later.

\subsection*{The covariant-tensor holonomy dichotomy}

A second factor argument, stronger than vanishing of holomorphic forms, is available. It excludes positive-degree holomorphic tensors on irreducible non-Ricci-flat factors.

\begin{proposition}[Covariant-tensor holonomy dichotomy]\label{prop:tensor-holonomy}
Let \(P\) be a compact simply connected nonflat K\"ahler factor, irreducible in the Riemannian de Rham sense, with nonnegative Ricci curvature. Either \(K_P\) has a nonzero parallel holomorphic volume form, or
\[
 H^0\bigl(P,(\Omega_P^1)^{\otimes m}\bigr)=0\qquad\text{for every integer }m>0.
\]
\end{proposition}

\begin{proof}[Sketch]
For a holomorphic covariant tensor, the contracted curvature on its bundle is the sum, over tensor slots, of the negative Ricci operator, so it is nonpositive. The Bochner identity on compact \(P\) makes the tensor parallel. The compact holonomy group \(H\) is connected and acts irreducibly on the complex tangent space. Consider the determinant character. If it is trivial, parallel transport of a canonical vector gives a global parallel volume form. If it is nontrivial, its differential is a nonzero central element of the Lie algebra, and by complex irreducibility and Schur's lemma it acts on the tangent space as a scalar \(ia\,\mathrm{Id}\), so on the \(m\)-fold tensor power as a nonzero scalar \(-mia\). The tensor power then has no \(H\)-fixed vector, so no nonzero holomorphic tensor of positive degree.
\end{proof}

The dichotomy gives a second factor argument: on its trivial-canonical factors a connected group of holomorphic isometries fixes a holomorphic volume form; on other factors positive-degree forms vanish as direct summands of tensor powers. Hodge symmetry turns this into vanishing of higher structure-sheaf cohomology, and the descent argument of the previous subsections applies.

\subsection*{Where this leaves us}

Corollary~\ref{thm:ordinary} is now proved: every smooth connected projective complex \(X\) with smoothly semipositive \(L=-K_X\) has a nonzero section of some positive power of \(L\). All remaining sections of the paper are independent routes to the same conclusion, or to the transfer theorem under alternate hypotheses. A reader who is satisfied with one proof can jump to Section~\ref{sec:map}; a reader who wants the alternate machinery should continue.


\section{Exact characters from a one-parameter action}\label{sec:extremal}

\paragraph{In words.} Corollary~\ref{thm:main} is proved for the natural linearization. Sometimes we want more: a section whose character is a prescribed \(\mu\). A one-parameter torus action provides such characters directly, from the \emph{source} component of a lift. The source is the component to which the action retracts as \(t\to0\), and its fiber weight is an extremal character of the action.

\paragraph{The setting.} Let \(T\) act on \(X\), let \(L=-K_X\) be smoothly semipositive, and let \(S\subset X\) be the source of a generic integral cocharacter. All normal weights of the action at \(S\) are positive; this is Bia\l ynicki-Birula's attracting-cell theorem \cite{BB1973}.

\begin{lemma}[The character at a source]\label{lem:source-character}
Let a torus \(T\) act on a smooth connected projective \(X\), let \(U\) be a \(T\)-linearized line, let \(\lambda\) be a generic integral cocharacter with \(X^\lambda=X^T\), and let \(S\) be its source. Write \(\mu\) for the full \(T\)-character of \(U|_S\). Every nonzero eigensection of \(U\) has \(\lambda\)-weight at most \(\langle\mu,\lambda\rangle\). If \(U\) is nef, every fixed-component fiber weight has pairing at most this value and, for every \(m\ge0\),
\[
 [\chi_T(X,mU)]_{m\mu}=\chi(S,mU|_S).
\]
The trivial torus is allowed, with \(S=X\) and \(\mu=0\).
\end{lemma}

\begin{proof}[Sketch]
The first bound follows from positivity of the normal weights at \(S\): minus one normal weight is subtracted from each section weight. For ample \(U\), generation of a high enough power controls the fixed-component weights; for nef, apply the result to \(bU+A\), with \(A\) ample, and let \(b\to\infty\).

For the full character identity, evaluate the equivariant fixed-point formula \cite{AtiyahSegal1968,AtiyahSinger1968} at \(a\lambda(t)\) as \(t\to\infty\), divide by \(t^{m\langle\mu,\lambda\rangle}\), and keep \(a\in T\) generic. The inverse normal Euler factors tend to one at \(S\) and to zero at every other fixed component. Truncating by ordinary degree gives the limit \(a^{m\mu}\chi(S,mU|_S)\). Since the equivariant index is a finite Laurent polynomial, existence of this limit excludes higher \(\lambda\)-weights, and the whole coefficient at the weight hyperplane is the single character \(m\mu\).
\end{proof}

\paragraph{From characters to sections.} The character bound leads to a theorem about the natural linearization.

\begin{theorem}[Exact attainment of source characters]\label{thm:extremal}
Let \(X\) be smooth connected projective with smoothly semipositive \(L=-K_X\) and with \(H^0(X,\Omega_X^p)=0\) for all \(p>0\). Let an algebraic torus \(T\) act with a linearized ample bundle. For every generic integral cocharacter \(\lambda\) with \(X^\lambda=X^T\), let \(S\) be its source and let \(\mu\) be the full \(T\)-character of \(L|_S\). Then for some \(k_\mu>0\) there is a nonzero section of \(k_\mu L\) of character \(k_\mu\mu\). In particular a positive power of \(L\) has a nonzero invariant section, for the natural linearization.
\end{theorem}

\begin{proof}[Sketch]
Take a generic integral cocharacter \(\lambda\) with \(X^\lambda=X^T\), and let \(S\) be its source. The projection of the attracting stratum to \(S\) is dominant, so \(S\) inherits the vanishing \(H^0(S,\Omega_S^p)=0\), and consequently \(\chi(S,\OO_S)=1\).

By Lemma~\ref{lem:source-character} the polynomial in \(m\) (a specialization of the full character identity) has value one at zero. Some fixed cohomological degree thus has nonzero weight-\(m\mu\) classes for unbounded \(m\). Equivariant hard Lefschetz produces nonzero eigenforms
\[
 \alpha_m\in H^0(X,\Omega_X^p\otimes(m+1)L),\qquad \mathrm{wt}(\alpha_m)=m\mu,
\]
for a fixed \(p\) and unbounded \(m\). Determinant extraction (as in Lemma~\ref{lem:forms}) gives a linearized line \(M_0\subset\bigwedge^r\Omega_X^p\) and invariant sections of \(M_0+iL\) of weight \((i-r)\mu\) for unbounded \(i\). Shift the linearizations of \(L\) by \(-\mu\) and of \(M_0\) by \(r\mu\); the shift cancels in the weight, and the resulting sections are invariant sections of \(M+iP\) with \(P=L\) shifted by \(-\mu\) and \(M=M_0\) shifted by \(r\mu\). Cotangent-tensor theorem \cite{LMPTX2023} makes \(-M\) pseudoeffective, so Theorem~\ref{thm:transfer} applies. It gives a section of \(k_\mu L\) of weight \(k_\mu\mu\).

There are finitely many source weights as \(\lambda\) varies. The zero of the character lattice lies in the convex hull of the actual section weights \(k_\mu\mu\), because otherwise strict separation would give a generic cocharacter pairing negatively with all of them, and at its own source the weight \(\mu\) would be a sum of positive-pairing tangent weights. Clear the rational convex coefficients and multiply the corresponding nonzero eigensections; the product is a nonzero invariant section in positive degree.
\end{proof}

\begin{corollary}\label{cor:torus-extremal}
Suppose \(X\) is smooth connected projective, \(L=-K_X\) is smoothly semipositive, and \(H^0(X,\Omega_X^p)=0\) for \(p>0\). Then for every algebraic torus \(T\) acting on \(X\) and every natural linearization, \(H^0(X,mL)^T\ne0\) for some \(m>0\).
\end{corollary}

This corollary is a version of Corollary~\ref{thm:main} that requires no hypothesis on \(\chi(X,\OO_X)\), at the expense of requiring \(H^0(X,\Omega_X^p)=0\) for \(p>0\). Rationally connected varieties satisfy both; the corollary is not weaker in that case, it is an alternative route.


\section{Preparing a fibration, and a metric on the base}\label{sec:models}

\paragraph{In words.} Two things must be built before any conversion theorem can be applied. First, an equidimensional model: a well-behaved family over a smooth base whose fibres all have the same dimension, so that every vertical prime divisor sits over a base prime divisor. Second, a metric on the base line that records the largest fiberwise weight. Both will be reused in almost every later construction.

\subsection*{An equidimensional model}

\begin{lemma}[Equidimensional models]\label{lem:flat-model}
Let \(X\dashrightarrow Y\) be a dominant rational map of integral projective varieties over \(\C\). 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 fibre 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 \(\C(Y)\) is algebraically closed in \(\C(X)\), the morphism from \(V\), and from any resolution of \(V\), to \(Y'\) has connected fibres. If a torus acts and the rational map is invariant, all these constructions and a resolution can be made equivariant, with trivial action on the base.
\end{lemma}

\begin{proof}[Sketch]
Embed the projective graph over \(Y\) in \(\PP^N\times Y\). Over the flat open set the fibres define a map to the projective Hilbert scheme \cite{Grothendieck1961Hilbert}. Resolve the closure of its graph, pull back the universal family, and normalize. Flatness over an integral base embeds each coordinate ring into its generic localization; the generic fibre is integral, so the coordinate rings have neither nilpotents nor zero divisors, and the pulled-back family is integral. Hilbert-polynomial constancy bounds fibre dimension above; a local height bound bounds below. Purity of fibre dimension and equidimensionality follow, and every vertical prime divisor maps onto a prime divisor of \(Y'\) because a divisor over a codimension-\(\ge2\) locus would have dimension too small.

For connectedness, apply Stein factorization; the finite intermediate base has function field the algebraic closure of \(\C(Y')\) in \(\C(X)\), which is \(\C(Y')\) under the field hypothesis, so the finite birational map is an isomorphism. The same reasoning applies after resolution. Equivariance is immediate from the action on the graph.
\end{proof}

\subsection*{Divisorial normalization and a continuous maximum metric}

The following proposition is the reusable engine that turns a rational relative section into a metrized line on the base.

\begin{proposition}[Divisorial normalization and maximum metric]\label{prop:maximum}
Let \(X,Y\) be smooth connected projective, let \(L=-K_X\) have a smooth semipositive metric, and suppose
\[
 V\xrightarrow{\nu}W\xrightarrow{\mu}X,\qquad g:W\to Y,\qquad f=g\nu,
\]
where \(W\) is normal, \(g\) is projective surjective equidimensional, \(\mu\) is birational, and \(V\) is a smooth projective resolution. Set \(D_V=(\mu\nu)^*L\) and \(E=K_{V/X}\), and let \(s_E\) be the Jacobian section of \(E\). Suppose a nonzero rational section \(s\) of \(b\mu^*L\), \(b>0\), has no horizontal poles and on the generic fibre \(V_\eta\)
\[
 H^0(V_\eta,E+kbD_V)=\C(Y)\,s_Es^k\qquad(k\ge0).
\]
In the invariant case, the same condition with \(H^0(-)^T\) and \(P\) in place of \(L\), \(s\) invariant, \(Y\) trivially acted on. Then after replacing \(b,s\) by a common multiple and a power, there are a line \(Q\) on \(Y\) and a regular injection
\[
 g^*Q\longrightarrow b\mu^*P
\]
whose squared fibre-maximum norms define a positive continuous metric with plurisubharmonic weights on \(Q\). In the invariant case \(Q\) has trivial linearization and the injection is equivariant.
\end{proposition}

\begin{proof}[Sketch]
In the invariant case, average a smooth semipositive metric on \(L\) over the maximal compact torus; the chosen linearization differs from the natural one by a character, which is unitary on this compact torus, so the same averaged metric is invariant for both actions.

The rank one at \(k=0\) forces generic geometric connectivity. The constant functions on the fibre, multiplied by \(s_E\), inject into that adjoint space; in the invariant case the connected torus acts trivially on the finite Stein base, so constants are invariant; hence the dimension is one, and smooth general fibres are connected.

For every prime divisor \(B\subset Y\), put
\[
 c_B=\min_{\Gamma\mapsto B}\frac{\operatorname{ord}_\Gamma(s)}{\operatorname{ord}_\Gamma(g^*B)}.
\]
Equidimensionality supplies divisorial lifts of \(B\), so the minimum is over a nonempty set; finiteness and integrality after clearing denominators give the base divisor. Sending the canonical section of the resulting base line to the powered \(s\) gives a regular bundle map by normality. The effective Cartier zero divisor \(G\) omits at least one component above every base prime.

For a local frame \(e\) of \(Q\), denote the image of its canonical section by \(s_e\) and put \(M_e(y)=\max_{W_y}|s_e|^2\). This maximum is continuous: upper semicontinuity by properness, lower semicontinuity by openness of an equidimensional holomorphic map from a normal space to a smooth base. The locus where \(M_e=0\) is algebraic closed with codimension at least two, by the omitted-lift property.

We show \(-\log M_e\) is plurisubharmonic on a dense open, then extend by continuity. On the smooth-fibration locus, the section \((s_E/f^*\eta)s_e^k\) generates the rank-one adjoint space. Berndtsson's smooth direct-image theorem \cite{Berndtsson2009} makes the bundle and base-change locus carry semipositive curvature. In the invariant case an invariant line is an orthogonal quotient of the full direct image; the compact-torus quotient is what gives semipositive curvature. The normalized \(1/k\)-power of the fiber integral
\[
 -\frac1k\log\int_{V_y}|s_e|^{2k}\,d\mu_y
\]
converges locally uniformly to \(-\log M_e\) by a point-mass and compactness argument. Thus \(-\log M_e\) is a plurisubharmonic limit on the open where it is finite. It does not tend to \(+\infty\) across the codimension-two zero set \(Z_0\) of \(M_e\), because discs transverse to \(Z_0\) have their boundaries in a fixed compact set off \(Z_0\); the maximum principle then bounds the weight at their centers. A zero of the continuous \(M_e\) would contradict this. So \(M_e>0\) everywhere, and the weight is continuous plurisubharmonic.
\end{proof}

A note on why this metric matters. It carries semipositive curvature, it is bounded, and its zero divisor on the base is controlled by the omission of one component above every base prime. This is what makes it possible to add a fixed multiplier ideal from a singular coefficient metric without changing the topological degree of \(-K_Y+D_*\).


\section{An ordinary Iitaka conversion}\label{sec:ordinary}

\paragraph{In words.} The transfer theorem of Section~\ref{sec:transfer} was written for invariant sections, using a common ratio field that is powerful enough to absorb even a character shift. The present section proves the trivial-torus case with different machinery: a Leray spectral sequence, an injectivity theorem of Fujino--Matsumura, and an Iitaka base. The route is algebraic and is independent of the singular multiplier ideal.

\subsection*{Injectivity and base cohomology}

\begin{lemma}[Injectivity and base cohomology]\label{lem:leray}
Let \(g:V\to Y\) be a morphism of smooth projective complex varieties, let \(A\) be very ample on \(Y\) with smooth metric of positive curvature \(\omega_A\). Suppose \(N\) on \(V\) has a singular Hermitian metric \(h\) with \(\mathcal I(h)=\OO_V\) and \(i\Theta_h(N)\ge\epsilon g^*\omega_A\) for some \(\epsilon>0\). Then
\[
 H^i(Y,R^jg_*\OO_V(K_V+N))=0\qquad(i>0,\ j\ge0).
\]
\end{lemma}

\begin{proof}[Sketch]
Fujino--Matsumura injectivity \cite{FM2016} says that on a compact K\"ahler manifold, multiplication by a section of a smooth semipositive line is injective on canonical cohomology with a multiplier ideal coefficient, provided the singular curvature dominates a positive multiple of the multiplying line's curvature. Apply this with the multiplying line \(g^*(mA)\); the comparison constant is \(\epsilon/m\), and the ideal is \(\OO_V\).

We reduce to base cohomology by induction on \(\dim Y\). Choose \(m\) large for Serre vanishing on all direct images \(R^jg_*\OO_V(K_V+N)\). A general smooth divisor \(C\in|mA|\) contains no associated point of these sheaves; its restricted coefficient metric has trivial multiplier ideal. The long direct-image sequence of
\[
 0\to\mathcal F\to\mathcal F\otimes g^*\OO_Y(mA)\to\OO_{V_C}(K_{V_C}+N|_{V_C})\to0
\]
breaks into short exact sequences, since multiplication by the cutting equation is injective on each \(R^jg_*\mathcal F\). The induction hypothesis and Serre vanishing give \(H^i(Y,R^jg_*\mathcal F)=0\) for \(i\ge2\). In the Leray spectral sequence, only columns zero and one survive. For each \(j\), the group \(H^1(Y,R^jg_*\mathcal F)\) is a subspace of \(H^{j+1}(V,\mathcal F)\); after twisting by \(g^*(mA)\) this subspace maps to zero. Functoriality plus injectivity then makes the original subspace zero, and the case \(i=1\) follows.
\end{proof}

\subsection*{Conversion on an Iitaka base}

\begin{proposition}[Conversion on an Iitaka base]\label{prop:ordinary-conversion}
Let \(X\) be smooth connected projective over \(\C\), with \(L=-K_X\) smoothly semipositive. Suppose \(F_0\) is pseudoeffective and, for an unbounded \(S\subset\Z_{>0}\), there are effective integral divisors \(N_s\sim sL-F_0\) for \(s\in S\). Then \(H^0(X,mL)\ne0\) for some \(m>0\).
\end{proposition}

\begin{proof}[Sketch]
Fix \(a<b\) in \(S\). The affine relations
\[
 (s-a)N_b\sim(s-b)N_a+(b-a)N_s
\]
will make the fiberwise section direction unique. If the Iitaka dimension of \(N_b\) is \(0\), the divisors on the left degenerate; if it is \(n\), then \(bL\) is big. Both cases finish immediately. Otherwise, resolve a multiple system of \(N_b\) attaining its Iitaka dimension and take its Stein factorization. After modification, flattening, normalization and resolution, we have smooth \(V,Y\) and a normal equidimensional model \(W\to Y\) with \(W\) birational to \(X\). Write \(\pi:V\to X\), \(L'=\pi^*L\), \(E=K_{V/X}\). For a very ample \(A\) on \(Y\) and some \(t\):
\[
 t\pi^*N_b-f^*A\sim G\ge0.
\]
The moving polarization remains big after base modification, so it dominates some large multiple of \(A\).

For a geometric generic fiber \(F\), the Iitaka dimension of \(\pi^*N_b+E\) restricted to \(F\) is zero: adding an effective exceptional divisor does not change global section dimensions, and if two sections were independent, a sufficiently large ample base twist would globalize them and produce image dimension at least \(e+1\) for a multiple of \(\pi^*N_b\), in contradiction with the definition of \(e\).

Restriction of the affine relation to \(F\) gives equality of effective divisors: \((\pi^*(N_b-N_a))|_F\ge0\). So the ratio of \(z=s_b/s_a\) of two representative sections is a regular section of \((b-a)L'\) on the generic fiber. For every \(k\ge0\),
\[
 h^0\bigl(F,(E+kdL')|_F\bigr)=1,\qquad d=b-a.
\]
Lower bound: the indicated effective section. Upper bound: for \(k\ge1\), multiply by the section of \(k\pi^*N_a+(k-1)E\) to inject into the sections of \(k(\pi^*N_b+E)|_F\); for \(k=0\), multiply by the section of \(\pi^*N_b|_F\).

Now extract the base divisor. On the normal equidimensional \(W\), take for each prime \(P\subset Y\) the minimum coefficient \(c_P=\min_Q\ord_Q(z)/\ord_Q(f_W^*P)\), clear denominators, and obtain
\[
 dL'\sim H+f^*B,\qquad H\ge0,
\]
with \(H\) pulled back from \(W\) and \(B\) integral on \(Y\). Above each base prime at least one component has coefficient zero in \(H\).

Proposition~\ref{prop:maximum} then gives a continuous semipositive metric on \(B\), bounded on compact charts. Only local boundedness is needed below.

At zero exponent, pseudoeffectivity of \(F_0\) gives \(nL'-f^*A\) pseudoeffective for some \(n>0\); a current from \(F_0\) gives a singular metric on \(L'\) with curvature dominating a positive multiple of \(f^*\omega_A\). Mix with the smooth semipositive part using a small coefficient, Skoda integrability, and finite cover. The multiplier ideal becomes trivial, and curvature dominates a positive multiple of \(f^*\omega_A\). For \(k\ge0\), put \(C_k=L'+kf^*B\) and add \(kf^*\phi_B\) to the coefficient weights. Local boundedness keeps the multiplier ideal trivial and curvature positive. Lemma~\ref{lem:leray} and the projection formula give
\[
 H^i\bigl(Y,f_*\OO_V(E)\otimes\OO_Y(kB)\bigr)=0\qquad(i>0,\ k\ge0).
\]
At \(k=0\) the Euler characteristic is \(h^0(V,E)=1\). It is a polynomial in \(k\); hence it is nonzero at some positive integer \(k\). A section of \(E+kf^*B\), multiplied by \(s_H^k\), pushes down to a nonzero section of \(kdL\) on \(X\).
\end{proof}

\subsection*{Twisted-form input}

\begin{corollary}[Twisted-form input]\label{thm:forms-input}
Under the smooth projective and smooth anticanonical semipositivity hypotheses of Proposition~\ref{prop:ordinary-conversion}, suppose \(H^0(X,\Omega_X^p\otimes mL)\ne0\) for unbounded positive \(m\) and fixed \(p\ge0\). Then some positive power of \(L\) has a nonzero section.
\end{corollary}

\begin{proof}
For \(p=0\), immediate. Otherwise, take the stable generic span of the twisted forms in \(\Omega_X^p\), saturate its top exterior power in \(\bigwedge^r\Omega_X^p\), and get a linearized line \(M\). Wedge an arbitrarily far-tail member with \(r-1\) fixed ones; this gives sections of \(M+sL\) for unbounded \(s\). The cotangent-tensor theorem \cite{LMPTX2023} makes \(-M\) pseudoeffective, and Proposition~\ref{prop:ordinary-conversion} applies.
\end{proof}

This corollary is what made the ``twisted-form descent'' in Section~\ref{sec:descent} work. It is also the input to the harmonic and alteration arguments that follow.


\section{Harmonic restriction: a vanishing theorem for the kernel condition}\label{sec:harmonic}

\paragraph{In words.} The transfer theorem of Section~\ref{sec:transfer} needed a singular metric on the base. The present section does the analytic work: a harmonic representative of a cohomology class can be detected on a fiber if every null direction of the curvature is a direction of the fiber. The hypothesis is a kernel condition \(\ker\theta\subseteq\ker df\). When it holds, the cohomology of the total space is a polynomial in the base twist.

\subsection*{Smoothing a continuous metric}

The first ingredient is a regularised maximum that turns a continuous plurisubharmonic weight into smooth weights close to it, with a tiny earned negativity.

\begin{lemma}[Uniform smoothing]\label{lem:smoothing}
Let \(Q\) be a line bundle on a smooth projective \(Y\) with a positive continuous Hermitian metric whose local weights are plurisubharmonic. For a fixed K\"ahler form \(\eta\), there are smooth metrics whose weights converge uniformly to these weights and whose curvatures are at least \(-\varepsilon_j\eta\), with \(\varepsilon_j\to0\).
\end{lemma}

\begin{proof}[Sketch]
Write a weight as a smooth reference plus a global continuous function \(u\). For fixed \(\varepsilon>0\), add \(\varepsilon\eta\) to the background; on finitely many enlarged coordinate patches, choose smooth local potentials that dominate the background, convolve, and subtract. Take inner patches covering \(Y\). Add smooth bumps zero on inner patches and a small negative constant near the outer boundaries, with second derivatives smaller than the local margin. A regularised maximum of the active candidates is smooth; convexity, monotonicity, and translation equivariance of regularised maximum preserve the background lower bound and the transition rule for weights. Let the uniform error and then \(\varepsilon\) tend to zero. Equivalently, smooth the strictly positive metric on \(NQ+H\) with \(H\) positive, subtract the weight of \(H\), and divide by \(N\).
\end{proof}

\subsection*{The kernel-condition vanishing theorem}

\begin{theorem}[Kernel-condition vanishing]\label{thm:kernel}
Let \(f:V\to Y\) be a surjective morphism between smooth connected projective complex varieties. Let \(D_V\) be a line bundle with a smooth semipositive Hermitian metric, of curvature \(\theta\). Let \(Q\) be a line bundle on \(Y\) with a positive continuous Hermitian metric of plurisubharmonic weights. Suppose that on a nonempty ordinary open subset of the submersion locus,
\[
 \ker\theta\subseteq\ker df.
\]
Then for every \(k\ge0\), \(q\ge0\), and \(p>0\),
\[
 H^p\bigl(Y,R^qf_*(K_V+D_V)\otimes Q^k\bigr)=0.
\]
\end{theorem}

\begin{proof}[Sketch]
Fix \(k\). Choose a K\"ahler form \(\omega\) on \(V\) and a Dolbeault class \(c\in H^\ell(V,K_V+A)\) where \(A=D_V+kf^*Q\) and \(n=\dim V\). Lemma~\ref{lem:smoothing} gives product metrics \(h_j\to h\) uniformly on \(A\), with curvature at least \(\theta-o(1)\omega\). Let \(u_j\) be the harmonic \((n,\ell)\)-representative of \(c\) for \(h_j\). A fixed \(L^2\) bound comes from comparing to one fixed smooth closed representative and using uniform metric equivalence. Bochner--Kodaira gives
\[
 \|(D'_j)^*u_j\|^2+\langle[\theta,\Lambda]u_j,u_j\rangle\le o(1)\|u_j\|^2.
\]
The curvature commutator in bidegree \((n,\ell)\) is monotone in the curvature form: its eigenvalues are sums of curvature eigenvalues over the antiholomorphic index set.

Let \(v_j=*u_j\), absorbing a dimensional constant so that \(u_j=\omega^\ell\wedge v_j\). Metric compatibility of the star operators gives
\[
 \bar\partial v_j\to0\text{ in }L^2,\qquad D'_jv_j=0.
\]
Passing to a weak limit gives a distributionally holomorphic \(v\), hence smooth holomorphic. Since \(\omega\) is K\"ahler, \(u=\omega^\ell\wedge v\) is \(\bar\partial\)-closed and represents \(c\). Projection to the harmonic forms for one fixed reference metric is a bounded finite-rank operator with the same value on each \(u_j\); the nonnegative square root of \([\theta,\Lambda]\) annihilates the limit, so every positive-curvature holomorphic covector is a wedge factor of \(v\).

The Chern equation at the limit is
\[
 \partial(hv)=0,
\]
a distributional statement obtained by uniform convergence of \(h_j\) and weak \(L^2\) convergence; no derivative convergence of the metric weights is used.

On the open set where the kernel condition holds, every pulled-back base covector annihilates \(\ker\theta\), so its wedge with \(v\) vanishes. The holomorphic bundle map
\[
 f^*\Omega_Y^1\to\Omega_V^{n-\ell+1}\otimes A,\qquad\alpha\mapsto(df)^*\alpha\wedge v,
\]
therefore vanishes everywhere. Over the smooth fibration locus, this gives the base factor decomposition
\[
 v=f^*(dy_1\wedge\dots\wedge dy_b)\wedge w.
\]
The base canonical factor removes under the restriction; the relative \(w_y\) is unambiguous once the base frame is chosen. The limiting metric on \(A\) is a smooth metric on \(D_V\) times a positive continuous function pulled from the base. In the distributional fiber equation the continuous base factor is constant under vertical differentiation, so the surviving expression is smooth in the vertical variables and continuous overall. It vanishes pointwise on vertical differentiations, giving \(D'_{D_V|V_y}w_y=0\) on the compressed fiber.

The canonical-valued restriction uses \(K_V|_{V_y}=K_{V_y}\otimes K_Y|_y\) to remove the constant canonical base factor. The restricted representative is a nonzero constant times \(\omega_y^\ell\wedge w_y\), which is \(\bar\partial\)-closed and has zero Chern derivative: a harmonic representative on the compact fiber. A nonzero \(v\) therefore restricts nontrivially to some general fiber, and the Leray edge map
\[
 H^\ell(V,K_V+D_V+kf^*Q)\to H^0(Y,R^\ell f_*(K_V+D_V)\otimes Q^k)
\]
is injective.

Finally apply Fujisawa's derived decomposition \cite{Fujisawa2015} to the smooth coefficient \(D_V\):
\[
 Rf_*(K_V+D_V)\simeq\bigoplus_q R^qf_*(K_V+D_V)[-q].
\]
Tensor by the line \(Q^k\) on \(Y\) and use the projection formula, to get
\[
 h^\ell(V,K_V+D_V+kf^*Q)=\sum_{p+q=\ell}h^p(Y,R^qf_*(K_V+D_V)\otimes Q^k).
\]
Injection bounds the left side by the \(p=0\) summand; all other summands vanish. For total degrees exceeding \(n\), the left side is zero directly. This proves the theorem for every \(k\), including zero.
\end{proof}

\subsection*{A pseudoeffective domination variant}

\begin{proposition}[Pseudoeffective domination variant]\label{prop:dominationvanish}
In Theorem~\ref{thm:kernel}, replace the kernel hypothesis by the assumption that \(bD_V-f^*H\) is pseudoeffective for some \(b>0\) and some ample \(H\) on \(Y\). Then the same vanishing holds.
\end{proposition}

\begin{proof}[Sketch]
Keep the uniformly convergent metrics. For \(\ell>0\), put
\[
 S=i^{(n-\ell)^2}h\,v\wedge\bar v,\qquad T=S\wedge\omega^{\ell-1},
\]
with coefficient pairing understood. The distributional Chern equation and holomorphy make \(T\) a continuous closed positive \((n-1,n-1)\)-form. The curvature-energy equality gives \(\theta\wedge T=0\). Choose a positive curvature form \(\eta\) of \(H\). A positive current in the class of \(bD_V-f^*H\) pairs nonnegatively with \(T\), since positive currents have order zero. Regularise \(T\) by de Rham (which commutes with d) to approximate uniformly by smooth closed forms; the pairing with a cohomologous smooth form is approximated consistently, giving
\[
 0\le\int_V(b\theta-f^*\eta)\wedge T=-\int_V f^*\eta\wedge T\le0.
\]
Pointwise positivity forces \(f^*\eta\wedge T=0\), so diagonalisation gives the full base wedge factor. The fiber equation, edge injection, and splitting proceed exactly as before.
\end{proof}

\subsection*{Euler interpolation on a birational model}

\begin{proposition}[Euler interpolation on a birational model]\label{prop:euler}
Let \(\pi:V\to X\) be birational between smooth connected projective varieties, let \(L=-K_X\) be smoothly semipositive, put \(D_V=\pi^*L\), \(E=K_{V/X}=K_V+D_V\). Let \(f:V\to Y\) be surjective with \(Y\) smooth connected projective. Suppose a continuously semipositive line \(Q\) on \(Y\) and an injection \(f^*Q\to bD_V\), \(b>0\), satisfy either vanishing criterion above. Then \(H^0(X,kbL)\ne0\) for some \(k>0\). If a torus acts, trivially on \(Y\) and \(Q\), and all maps and the injection are equivariant, the section can be taken invariant.
\end{proposition}

\begin{proof}[Sketch]
Let \(\mathcal G=f_*\OO_V(E)\) or its invariant summand. Vanishing gives \(h^0(Y,\mathcal G\otimes Q^k)=\chi(Y,\mathcal G\otimes Q^k)\) for every \(k\ge0\). The right side is a polynomial in \(k\) by Riemann--Roch for the coherent sheaf \(\mathcal G\). At zero it equals one: \(\pi_*\OO_V(E)=\OO_X\), and the exceptional canonical section is invariant. The polynomial is nonzero at some positive integer \(k\). Multiply the resulting section of \(E+kf^*Q\) by the injection's \(k\)-th power; the effective exceptional divisor \(E\) changes neither sections nor characters, by normality of \(X\).
\end{proof}

\subsection*{Why this was a different proof}

The Euler interpolation of this section uses a harmonic representative and the derived decomposition of direct images. The proof of Theorem~\ref{thm:transfer} used a singular metric and a fixed multiplier ideal. They prove the same theorem in different regularity regimes. The harmonic version needs the kernel condition to hold on a nonempty ordinary open set; the singular-metric version needs the singular positive metric from pseudoeffectivity of \(-M\). Neither is stronger than the other in general.


\section{Harmonic proof of the natural invariant theorem}\label{sec:harmonic-iitaka}

\paragraph{In words.} Section~\ref{sec:harmonic} gave the vanishing theorem, but it left the base line \(Q\) unspecified. The present section chooses it from powers of one fixed invariant line, so that maximality of its image forces the invariant adjoint spaces on the generic fibre to have rank one, and the determinant-intersection identity supplies the kernel condition. The result is a proof of Corollary~\ref{thm:main} that does not need vanishing of the higher structure-sheaf cohomology groups. The analytic input is the vanishing theorem of Section~\ref{sec:harmonic}; the algebraic input is the invariant system and its image.

\subsection*{The invariant Iitaka choice}

We start with the determinant sections of Lemma~\ref{lem:forms}, keeping only the cases where no invariant anticanonical section has already been produced. Write \(N_i=M+a_iL\) for the linearized exponents, \(s_i\) for the invariant sections, and \(0<a_1<a_2<\dots\) for the unbounded sequence. Write \(\theta\) for the averaged smooth semipositive curvature of \(L\), \(r\) for its maximal rank, \(\omega\) for a fixed Kähler form.

Fix \(a_-<a\), keep unbounded indices \(>a\). Put \(N=M+aL\). Among all rational maps given by invariant sections of positive powers of \(N\), choose one with maximal image dimension. Resolve its base ideal equivariantly and take Stein factorization: this gives a smooth projective \(X_0\), a birational \(\pi_0:X_0\to X\), and \(f_0:X_0\to Y_0\) with connected fibres and normal projective base, together with
\[
 d\pi_0^*N=f_0^*H_0+G_0,\qquad G_0\ge0,
\]
where \(H_0\) is ample, the torus acts trivially on \(Y_0,H_0\), and the defining section of \(G_0\) is invariant.

Maximality forces the invariant adjoint rank to be one on each generic fibre. Suppose two independent invariant sections existed. Their ratio would be transcendental over \(\C(Y_0)\) by relative algebraic closedness of the Stein field. Multiplying by a high power of \(H_0\) would clear the vertical poles and combine with the original image coordinates, increasing the image dimension beyond the maximum. Consequently
\[
 \dim_{\C(Y_0)}H^0((X_0)_\eta,E_0+kb\pi_0^*L)^T=1\qquad(k\ge0),
\]
where \(E_0=K_{X_0/X}\) and \(b=a_j-a_i>0\) is a positive index difference.

Apply Lemma~\ref{lem:flat-model} to the fibration \(X_0\to Y_0\) and resolve its normal equidimensional model \(W\), with a smooth resolution \(V\to W\). All constructions are equivariant, with trivial base action. The rank-one statement persists under this rewriting.

\subsection*{The kernel condition from the determinant intersection}

Lemma~\ref{lem:null-divisors} gives the necessary intersection identity. If \(r<n\),
\[
 \int_V(\pi^*\theta)^r\wedge f^*\eta_{H_0}\wedge(\pi^*\omega)^{n-r-1}=0.
\]
The whole class \(dN\) has zero intersection, and both the moving and the fixed parts have nonnegative intersection, so each separately is zero. On an ordinary open set where \(\pi\) is an isomorphism and \(f_0\) has injective differential, positivity forces the horizontal form to annihilate \(\ker\theta\). Consequently \(\ker\theta\subseteq\ker df\) on the rank-\(r\) open. Proposition~\ref{prop:euler} plus Theorem~\ref{thm:transfer} then finish.

\subsection*{A second proof of positivity for the maximum metric}

The preceding proof uses Proposition~\ref{prop:maximum} to build the base metric. A different analytic argument for the same conclusion uses stabilized pole sheaves. We state it here, since it also shows a second time why the maximum stays positive.

\begin{proposition}[Pole-sheaf stabilization]\label{prop:pole-stabilization}
In the setting above, with \(D_V=\pi^*(-K_X)\), \(E=K_{V/X}\), and \(bD_V=G+f^*Q\) the divisorial normalization, the increasing sheaves
\[
 \mathcal P_k=(f_*\OO_V(E+kG))^T\qquad(k\ge0)
\]
viewed as sheaves of rational functions on \(Y\), stabilize. The line \(Q\) is nef, and \(G\) contains no entire fibre. These conclusions give a second proof that the fibrewise maximum is positive.
\end{proposition}

\begin{proof}[Sketch]
On \(V\), form the sheaves \(\mathcal P_k=(f_*\OO_V(E+kG))^T\) and view them as sheaves of rational functions on \(Y\) by dividing by \(s_Es_G^k\). They have rank one, contain \(\OO_Y\), and increase with \(k\). For each prime divisor \(P\) of \(Y\), take a lift \(\Gamma\) omitted by \(G\); regularity of \(f^*a\,s_Es_G^k\) bounds the possible pole order independently of \(k\) by one divisor \(C\). Normality embeds all \(\mathcal P_k\) in \(\OO_Y(C)\), so Noetherianity gives a stable value \(\mathcal P_\infty\).

Take a very ample \(H\) on \(Y\). Apply Theorem~\ref{thm:kernel} with trivial base twist to \((1+kb)D_V+lf^*H\), \(k,l\ge0\). Its curvature dominates the original, so its kernel is still vertical. Projection to invariants and the projection formula give
\[
 H^i(Y,\mathcal P_k\otimes(kQ+lH))=0\qquad(i>0).
\]
For large \(k\), Castelnuovo--Mumford regularity makes \(\mathcal P_\infty\otimes(kQ+(\dim Y)H)\) globally generated. Pulling back to a curve and removing torsion gives that the fixed sheaf has positive rank, so its degree plus \(k\) times the rank times \(\deg Q\) is nonnegative for unbounded \(k\). Thus \(Q\) is nef. If \(G\) contained an entire fibre \(Y_y\), blow up \(y\) and recompute; the new rational base divisor is the pullback of \(Q\) plus a strictly positive multiple of the exceptional divisor, and its degree on a line in the exceptional projective space is negative, a contradiction. Hence no entire fibre lies in \(G\), and the fibrewise maximum is positive.
\end{proof}

\subsection*{What the harmonic proof did}

The harmonic proof differs from the transfer theorem in a specific way. The transfer theorem built the base line from a fixed multiplier ideal; the harmonic proof built it from a maximal image in a fixed invariant system, and used the determinant-intersection identity to force the kernel. Both routes produce a \(Q\) on a base and an injection \(f^*Q\to bD_V\), and both finish by Euler interpolation. In the transfer theorem the interpolation was analytic (singular metric and Nadel vanishing); here it is algebraic-geometric (Riemann--Roch plus vanishing). The two proofs are logically independent.


\section{Two invariant fields}\label{sec:fields}

\paragraph{In words.} Section~\ref{sec:harmonic-iitaka} built its base from the maximal image of one fixed invariant system. Two other constructions of the same kind are available. Both choose a field of invariant rational functions and use the finiteness of that field's linearized Picard group, or the affine dependence of horizontal divisor orders, to force rank one. The output is the same data as before: a normal equidimensional model, rank-one invariant adjoint spaces, a continuously semipositive base line, and a kernel inclusion \(\ker\theta\subseteq\ker df\). The rest of the proof is Proposition~\ref{prop:euler}.

Throughout this section, assume \(H^i(X,\OO_X)=0\) for \(i>0\), and let \(T\) act on the smooth connected projective \(X\). Put \(n=\dim X\), \(L=-K_X\), choose a compact-torus-invariant smooth semipositive metric with curvature \(\theta\), and let \(r\) be its maximal rank. Apply Lemma~\ref{lem:forms}. If it already produces an invariant anticanonical section, there is nothing to do. Write \(s_i\in H^0(X,N_i)^T\), \(N_i=M+a_iL\), \(a_i\to\infty\), for its determinant sections. Lemma~\ref{lem:null-divisors} gives \(D_i[\theta]^r[\omega]^{n-r-1}=0\) for their zero divisors.

\subsection*{Construction A: the null-pole field}

A prime divisor is \emph{small} if \(\theta^r\) vanishes on its smooth locus. An effective divisor has small support exactly when its intersection with \([\theta]^r[\omega]^{n-r-1}\) vanishes. Smallness persists under pullback to smooth birational models and through exceptional divisors centered in small supports: resolve the original prime, take a dominating component of the fiber product with the proposed source, and test the pulled-back form on a smooth dominating space; continuity extends the zero identity.

Let \(\mathcal K\) consist of zero and the invariant rational functions with small pole support. It is a field. Sums and products introduce no new pole support; a rational function's zeros and poles are linearly equivalent effective divisors, so the zero support is small whenever the pole support is. It is finitely generated: take a transcendence basis of \(\mathcal K\), extend it to one of \(\C(X)\), and note that adjoining the complementary independent variables preserves degrees of finite algebraic subextensions. The finite degree of the full function field over the larger purely transcendental field bounds those degrees.

Use finitely many generators to map to a product of projective lines, take the image \(Y_0\), and resolve equivariantly: \(\pi_0:X_0\to X\), \(f_0:X_0\to Y_0\). The base action is trivial. Indeterminacy lies in the small pole supports, so the resolution may be taken to be an isomorphism off them. The exceptional canonical divisor \(E_0\) is small, and a pullback of an ample \(H_0\) on \(Y_0\) has a small effective representative.

The choice of field now controls the horizontal divisors. There are only finitely many invariant small primes on \(X_0\) horizontal over \(Y_0\), because \(\Pic^0(X_0)=0\) by birational invariance, the kernel of the linearized Picard group over the ordinary one is characters, and infinitely many such primes would give a nonzero integral relation among their natural linearized divisor lines. That relation would be an invariant rational function on \(X_0\) with vertical divisor, but its poles downstairs are small, so it belongs to \(\mathcal K=\C(Y_0)\), a contradiction.

The determinant sections are small, so their horizontal supports belong to this finite set. Order the multiplicities and take a ratio along the unbounded subsequence to obtain an invariant rational section \(s\) of \(b\pi_0^*L\), \(b>0\), regular on the generic fiber. For every \(k\ge0\) both
\[
 (f_{0*}\OO_{X_0}(kb\pi_0^*L))^T,\qquad (f_{0*}\OO_{X_0}(E_0+kb\pi_0^*L))^T
\]
have rank one, because they contain \(s^k\) and \(s_{E_0}s^k\), and any other invariant generic section has small effective zero divisor and a small-pole ratio in \(\mathcal K\).

The kernel condition follows from the same smallness. Every level divisor of a generating function is small; at a regular level point in the rank-\(r\) open its tangent hyperplane must contain \(\ker\theta\), because a hyperplane not containing that kernel would surject onto the rank-\(r\) quotient. Thus \(\ker\theta\subseteq\ker df_0\) on a nonempty ordinary open. Proposition~\ref{prop:maximum} and Proposition~\ref{prop:euler} finish.

\subsection*{Construction B: a semigroup of determinant lines}

This time we impose all equal-degree ratios at once. Form the semigroup of linearized line bundles generated by the \(N_i\), and the field generated by ratios of invariant sections of the same semigroup element. Choose a transcendence basis among these ratios. Products put their numerators and denominators into one invariant linear system of a semigroup element \(N_0\). Normalize its image in the relative algebraic closure of its function field in \(\C(X)\), and resolve the base to obtain \(Y\). Every original ratio is algebraic over the chosen transcendence basis, so it lies in the finite relative algebraic closure; the connected torus acts trivially on it.

After the Hilbert-family modification, take the normal equidimensional model \(W\to Y\) and its smooth resolution \(V\). Write \(\pi:V\to X\), \(f:V\to Y\), \(D_V=\pi^*L\), \(E=K_{V/X}\). Let \(A\) be the pullback of the image hyperplane bundle to \(Y\); it is big, and the common factor of the pulled-back coordinates is regular, giving an invariant effective difference
\[
 \pi^*N_0-f^*A.
\]
Hence for an ample \(H\) on \(Y\), \(a\pi^*N_0-f^*H\) is invariantly effective for some \(a>0\). Lemma~\ref{lem:null-divisors} forces the intersection of \(f^*A\) with \((\pi^*\theta)^r(\pi^*\omega)^{n-r-1}\) to vanish; where the map to the image has maximal rank this gives the kernel inclusion.

For every semigroup element \(N\) and integer \(e\ge0\), the invariant generic-fiber section dimension of \(\pi^*N+eE\) is at most one: two independent generic sections would, after an ample base twist, globalize to invariant sections whose ratio is transcendental over \(\C(Y)\), contradicting the construction of \(Y\).

Now the affine-in-\(a_i\) conclusion is direct. Let \(F_i\) be the horizontal divisor of \(\pi^*s_i\). For \(a_i<a_j<a_h\), positive integers \(u,v\) with \(ua_i+va_h=(u+v)a_j\) give two nonzero invariant generic sections of the same line; their ratio is constant on the generic fiber, so \(uF_i+vF_h=(u+v)F_j\). The integer multiplicity of each horizontal prime is therefore affine in \(a_i\). Its slope is effective because \(F_i\ge0\) and \(a_i\to\infty\). The ratio \(s=\pi^*s_2/\pi^*s_1\), of degree \(b=a_2-a_1>0\), has no horizontal poles, and for all \(k,e\ge0\),
\[
 \dim_{\C(Y)}H^0(V_\eta,kbD_V+eE)^T=1.
\]
The displayed section \(s^ks_E^e\) gives the lower bound. For the upper bound choose \(i\) large enough that \(F_i-k(F_2-F_1)\ge0\) and use multiplication by \(\pi^*s_i/s^k\). If the base is a point, \(s\) is already regular globally; otherwise Propositions~\ref{prop:maximum} and~\ref{prop:euler} finish.

\subsection*{What changed between the two constructions}

Construction A uses the finiteness of the linearized Picard group and the smallness of curvature-zero divisors. Construction B uses the affine dependence of horizontal orders, so it avoids enumerating primes. Both produce the same package; the reader can pick whichever is more transparent for a given application. Neither construction assumes relative algebraic closure of the pole field in the sense used in Section~\ref{sec:alteration}.


\section{Extremal weights by harmonic methods}\label{sec:harmonic-extremal}

\paragraph{In words.} Section~\ref{sec:extremal} obtained a character at a source using fixed-point index theory. The present section obtains the same character using the harmonic restriction machinery. The translation from index to weights is by the retraction to the source component. The reader who already has Section~\ref{sec:extremal} can skip this section; the result is a second construction of the exact character by analytic methods.

Let \(S\) be a smooth connected projective rationally connected variety with smoothly semipositive \(L=-K_S\) and a \(\C^*\)-action with the natural anticanonical linearization. Let \(S_0\) be the source component for \(t\to0\). Write \(w_C(U)\) for the fiber weight of a linearized line \(U\) on the fixed component \(C\).

\begin{lemma}[Source weights and retraction]\label{lem:source}
Let \(S\) be as above, allowing the trivial action. Let \(S_0\) be the attracting component of the action. A nonzero eigensection of a linearized line \(U\) has weight at most \(w_{S_0}(U)\). If \(U\) is nef, then \(w_C(U)\le w_{S_0}(U)\) for every fixed component, and the multiplicity of weight \(m w_{S_0}(U)\) in the virtual character \(\chi(S,mU)\) is \(\chi(S_0,mU|_{S_0})\), for \(m>0\). If \(U\) is nef, on a smooth equivariant resolution \(d:\widehat S\to S\) of the source retraction \(p:\widehat S\to S_0\) there is an equivariant nonzero bundle map \(p^*(U|_{S_0})\to d^*U\).
\end{lemma}

\begin{proof}[Sketch]
A linearized line on a projective variety with \(\C^*\)-action exists: the map to the Picard variety from \(\C^*\) is constant (since a torus has no nonconstant homomorphism to an abelian variety), and the extension of \(\C^*\) by the scalar \(\C^*\) is a split torus. The section-weight bound and the comparison of fixed weights are Lemma~\ref{lem:source-character} for \(T=\C^*\). The retraction is Bia\l ynicki-Birula \cite{BB1973}. The bundle map comes from transporting a vector along the orbit, dividing by \(t^{w_{S_0}(U)}\), and taking the limit; the result is a multiplicative rational identification. Ampleness of a generated power of \(U\) rules out poles; nefness follows by considering \(kU+H\) with \(H\) ample and letting \(k\to\infty\).
\end{proof}

\begin{proposition}[Attainment of the source weight]\label{prop:extremal}
Let \(S\) be smooth connected projective rationally connected, with smoothly semipositive \(L=-K_S\), and fix a \(\C^*\)-action with the natural anticanonical linearization. If \(w=w_{S_0}(L)\), then some \(\ell>0\) has a nonzero section of \(\ell L\) of weight \(\ell w\).
\end{proposition}

\begin{proof}[Sketch]
The source \(S_0\) is rationally connected, being a smooth rational image of \(S\), so \(\chi(S_0,\OO_{S_0})=1\). By Lemma~\ref{lem:source} the highest-weight index is a nonzero polynomial in \(m\); for unbounded \(m\) a fixed \(H^q(S,mL)\) contains weight \(mw\). Equivariant hard Lefschetz gives twisted sections
\[
 v_m\in H^0(S,\Omega_S^{\dim S-q}\otimes(m+1)L)=H^0(S,\textstyle\bigwedge^qT_S\otimes mL)
\]
of weight \(mw\). For \(q=0\) they are the desired sections. Otherwise saturate the generic span in \(\bigwedge^qT_S\) and call it \(\mathcal U\) of rank \(e>0\); the determinant extraction gives nonzero eigensections \(d_i\in H^0(S,\det\mathcal U+k_iL)\) of weight \(k_iw\), \(k_i\to\infty\). Cotangent-tensor theorem \cite{LMPTX2023} makes \(eL-c_1(\mathcal U)\) pseudoeffective and \((e+k_i)L-D_i\) pseudoeffective.

Let a divisor class be \emph{dominated} if a positive multiple of \(L\) minus it is pseudoeffective. Maximize the image dimension of linear systems of dominated effective integral divisors; choose a full system \(|P_0|\) realizing this maximum. Rational connectedness gives \(\operatorname{Pic}^0(S)=0\), so the system linearizes. Resolve its graph, take Stein factorization, flatten by the Hilbert-family construction, and resolve the base. The result is a normal equidimensional model \(W\to Y\) with a smooth resolution \(V\), and the line \(L_W\) pulled back from \(S\) with the exceptional canonical divisor \(E\). The whole effective canonical discrepancy is dominated by \(L_W\).

A dominated line on \(V\) cannot have two independent sections on the generic fiber: two such would globalize after a base twist and push down to a dominated system of larger image dimension, contradicting the maximum. So generic-fiber sections of \(\pi^*P_0\) have dimension at most one. Finiteness of dominated horizontal primes follows from \(\Pic^0(V)=0\) and the rank of \(\Pic(V)\): infinitely many such would give a nonzero integral linear equivalence among their classes, hence a pencil varying along the generic fiber.

The pullbacks of \(D_i\) have horizontal supports in a fixed finite set. Order multiplicities along a subsequence and take a ratio to obtain a rational eigensection \(s\) of \(aL_W\), weight \(aw\), with no horizontal poles. Divisorial normalization gives \(g^*Q\to aL_W\) with effective zero divisor omitting one lift over every base prime. Rank-one adjoint spaces hold, so Proposition~\ref{prop:maximum} gives a continuous semipositive metric on \(Q\).

Source-weight comparison. Let \(Y_0\) be the source of the induced action on \(Y\). The unbounded eigensections force \(w_{Y_0}(Q)\ge aw\). After fixing \(a\), take an infinite subsequence of \(k_i\) in one residue class modulo \(a\), divide by the first section of the subsequence, and clear the resulting vertical poles by an invariant effective Cartier divisor \(C\) on \(Y\). The resulting sections come from \(jQ+C\); the source upper bound gives \(jaw\le jw_{Y_0}(Q)+w_{Y_0}(C)\), and divide by \(j\) and let \(j\to\infty\).

Euler interpolation on the base. The continuous semipositive metric makes \(Q\) nef by Lemma~\ref{lem:smoothing}. Resolve the source retraction of \(Y\) and dominate this resolution by a smooth model \(V'\). Lemma~\ref{lem:source} and the normalization give \(p^*Q_0\to aA'\). Proposition~\ref{prop:dominationvanish} applies to \(jQ_0\). The action on \(Y_0\) is trivial and \(Q_0\) has constant fiber weight \(w_{Y_0}(Q)\). Restrict to the weight-zero summand of \(p_*\OO(E')\); sections of it twisted by \(jQ_0\) have precisely weight \(jw_{Y_0}(Q)\). Vanishing identifies their dimension with the Euler polynomial for every \(j\ge0\), which is positive at \(j=0\) because the canonical exceptional section is invariant. For some \(j>0\) a nonzero section of \(E'+jp^*Q_0\) of that weight appears. Map into \(E'+jaA'\), push down to \(S\); the weight is at least \(jaw\) by the source-weight inequality and at most \(jaw\) by the source bound on \(S\). Equality gives the required extremal section.
\end{proof}

\begin{corollary}[From extremal characters to invariants]\label{cor:torus}
Every algebraic torus acting on the rationally connected \(S\) of Proposition~\ref{prop:extremal} has a nonzero invariant section of some positive power of \(-K_S\).
\end{corollary}

\begin{proof}[Sketch]
For a generic integral one-parameter subgroup \(\lambda\), let \(C\) be its source, \(\gamma_C\) the fibre character of \(L\). Choose a full torus eigenvector in the extremal section space of Proposition~\ref{prop:extremal}. It cannot vanish on \(C\), since a nonzero leading conormal jet would strictly lower its \(\lambda\)-weight; its character is therefore \(\ell\gamma_C\). The finitely many characters of such sources have convex hull containing zero; otherwise a separating functional approximated by a generic rational one would be negative on all of them, whereas the source for that one-parameter subgroup has \(\gamma_C(\lambda)>0\), the sum of its positive normal weights. Use the effective torus quotient if the action has a kernel, and multiply rational convex combinations to produce a nonzero invariant section.
\end{proof}


\section{The curvature-kernel field and a reduced flat alteration}\label{sec:alteration}

\paragraph{In words.} All previous ordinary conversions followed \(L\) by constructing a base and using a relative anticanonical section. This section instead constructs a \emph{field} of functions constant in the curvature-null directions, takes a model of that field, and uses an alteration with reduced fibres to make a base line on it. The metric weight along this line makes the fibrewise maximum positive even on a non-smooth model. Only after this is done does the birational Euler argument come back to \(X\). The route does not use any Iitaka or multiplier-ideal setup, and it assumes nothing on $\Pic^0(X)$.

\begin{theorem}[Ordinary conversion through a curvature-null field]\label{thm:alteration-conversion}
Let \(X\) be smooth connected projective with smoothly semipositive \(L=-K_X\). Suppose that for one fixed \(q\ge0\), \(H^q(X,mL)\ne0\) for unbounded positive integers \(m\). Then \(H^0(X,rL)\ne0\) for some \(r>0\).
\end{theorem}

\paragraph{Construction overview.} The proof has three parts. First, the harmonic-star construction of Lemma~\ref{lem:forms} turns the cohomology into nonzero invariant forms of type \(\Omega_X^{n-q}\otimes(m+1)L\), and determinant extraction gives invariant sections \(s_i\) of \(M+a_iL\) with \(a_i\to\infty\). Each \(s_i\) has zero divisor null along the semipositive curvature. Second, the field \(\mathcal K\) of rational functions constant in curvature-null directions is relatively algebraically closed and finitely generated; its projective model \(Y\) has \(\dim Y\le r\), and the alteration \(f:X_1\to Y\) has reduced flat fibres. Third, a Hom line on this alteration carries a continuous semipositive metric; after a finite Galois descent an injection survives to the original source and base fields, and Proposition~\ref{prop:euler} finishes.

\subsection*{The field constant in curvature-null directions}

Let \(X\) be as in the theorem, put \(n=\dim X\), let \(\theta\) be the smooth semipositive curvature of \(L\), and let \(r\) be its maximal rank. If \(q=0\), the conclusion is already given. Otherwise apply Lemma~\ref{lem:forms} to get nonzero sections of \(\Omega_X^{n-q}\otimes(m+1)L\). If \(r=n\), the positive-curvature energy on an open set excludes \(q>0\). In the remaining cases the determinant extraction gives effective divisors
\[
 D_i\in|M+a_iL|,\qquad a_i\to\infty,\qquad D_i[\theta]^r[\omega]^{n-r-1}=0.
\]
The intersection equality is Lemma~\ref{lem:null-divisors}. If \(r=0\), all \(D_i\) vanish, and two distinct \(a_i\) show that a positive power of \(L\) is trivial. Assume \(1\le r<n\).

On the ordinary open \(U\) of maximal curvature rank, define
\[
 \mathcal K=\{g\in\C(X): dg(\ker\theta)=0\text{ on }U\text{ off a proper algebraic subset}\}.
\]
It is a field. It is relatively algebraically closed: differentiate a minimal polynomial for an element algebraic over \(\mathcal K\) in a null direction; its separable derivative is generically nonzero, so the element has zero derivative there as well. The intermediate-field argument from Section~\ref{sec:fields} proves finite generation.

If a rational function has null pole support, its level divisors have the same zero intersection as its poles, hence each of their components is null. At a regular level point in \(U\), the tangent hyperplane contains \(\ker\theta\) by the rank-quotient argument. So this proves the assertion where the differential is nonzero and then everywhere.

\subsection*{The reduced flat alteration and the base line}

Choose a smooth projective model \(Y\) of \(\mathcal K\) and resolve the map by \(\mu:X_1\to X\), \(f:X_1\to Y\). Put \(A=\mu^*L\). The generic fiber is geometrically integral, because the extension is regular in characteristic zero. On a nonempty ordinary open \(\ker\mu^*\theta\subset\ker df\), and \(\dim Y\le r\). For every smooth \((1,1)\)-form \(\eta\) on \(Y\),
\[
 (\mu^*\theta)^r\wedge f^*\eta=0.
\]
On the rank-\(r\) open this follows because every horizontal covector lies in the positive-curvature span; elsewhere the rank is smaller, and continuity extends across the algebraic exceptional locus.

Any integral relation \(\sum b_i=\sum b_ia_i=0\) makes \(\sum b_iD_i\) principal with null pole support. Its function is in \(\mathcal K\), so the horizontal parts of \(\mu^*D_i\) satisfy the same relation. Three-term interpolation proves that the horizontal divisors are affine in \(a_i\). Their slope is effective because the coefficients are nonnegative and unbounded. For two indices with \(b=a_2-a_1>0\) the quotient \(s\) is a rational section of \(bA\) with effective horizontal divisor, regular over a dense base open.

Let \(E=K_{X_1/X}=K_{X_1}+A\). For every \(t\ge0\),
\[
 \rank f_*\OO_{X_1}(E+tA)\le1,\quad\text{with equality for }t=kb,\ k\ge0.
\]
Upper bound: two hypothetical independent generic sections could be twisted to make them global, and their zero divisors push to effective divisors on \(X\) with zero intersection with \([\theta]^r[\omega]^{n-r-1}\), using \((\mu^*\theta)^r\wedge f^*\eta=0\), exceptionality of \(E\), and \(\theta^{r+1}=0\). Their quotient has null poles, hence lies in \(\mathcal K\), contrary to independence. Lower bound: \(s_Es^k\).

Over a smooth dense open with geometrically integral fibres, \(-\log\max_{X_{1,y}}|s|^2\) is plurisubharmonic: the rank-one adjoint section \((s_E/f^*\eta)s^k\) has smooth coefficient \((kb+1)A\), so Berndtsson's theorem applies, and the moment limit converges to the fibrewise maximum.

\subsection*{Reduced flat model and descent}

By weak semistable reduction \cite[Theorem~0.3]{AK2000} the fibration may be replaced by an alteration \(Y_a\to Y\) and a modification \(Z_a\) of the main component of \(X_1\times_Y Y_a\) such that \(p_a:Z_a\to Y_a\) is equidimensional, toroidal, and has reduced fibres. Toroidal varieties are Cohen--Macaulay, so the equidimensional flatness criterion over the smooth base makes \(p_a\) flat.

Take a sufficiently high power \(\mathcal P\) of a relatively very ample line. Relative Serre vanishing and base change give locally free bundles \(\mathcal H=p_{a*}\mathcal P\) and \(\mathcal H'=p_{a*}(\mathcal P\otimes q_a^*A^b)\), with surjective evaluation for \(\mathcal P\). On the generic fibre, multiplication by \(s\) gives a nonzero map \(\mathcal H_\eta\to\mathcal H'_\eta\). Resolve its rational point in the projective bundle of lines in \(\operatorname{Hom}(\mathcal H,\mathcal H')\) on a smooth projective birational base model \(Y_b\to Y_a\). Let \(B_b\) be the pulled-back tautological subline, and \(Z_b=Z_a\times_{Y_a}Y_b\), with maps \(p_b,q_b\). The total space is integral.

There is a regular line map
\[
 p_b^*B_b\to q_b^*A^b
\]
nonzero on every fibre, because composing the tautological Hom map with evaluation into \(\mathcal P\otimes q_b^*A^b\) annihilates the kernel of evaluation, and base change plus reducedness gives a nonzero value at a point of every fibre. Without reducedness this implication would fail.

Give \(B_b\) the maximum metric through the map and the pulled-back smooth metric of \(A^b\). The maximum is positive and continuous; properness gives upper semicontinuity, fibre-smoothness at dense smooth points gives lower semicontinuity. On a dense open the map is the pullback of \(s\) times a rational base factor, and its maximum weight equals the pulled-back good-locus weight plus a pluriharmonic frame term. Continuity and removable singularities extend plurisubharmonicity to \(Y_b\).

\subsection*{Galois descent and finite norm}

Resolve the alteration and its section by first taking a finite Galois extension of \(\C(Y)\) containing \(\C(Y_b)\), of degree \(\ell\) and group \(G\), and then normalizing the simultaneous graph of all conjugate rational maps to \(Y_b\). This gives a normal projective \(W\) with a regular \(G\)-action and a map \(\rho:W\to Y_b\). The quotient \(W_0=W/G\) is normal projective birational to \(Y\). The invariant product \(C_W=\bigotimes_{g\in G}g^*B_W\) descends to a line \(C_0\) on \(W_0\): its power \(C_W^\ell\) has trivial stabilizer actions on fibres, so it descends locally to a line by averaging equivariant values. The invariant metric descends continuously.

Resolve \(\widehat Y\to W_0\) and let \(B_0\) be the pullback of \(C_0\); its metric is plurisubharmonic on the finite-étale locus and by continuity everywhere. Choose a smooth projective birational model \(\widehat X\to X_1\) mapping to \(\widehat Y\), with maps \(\widehat\mu:\widehat X\to X\) and \(\widehat f:\widehat X\to\widehat Y\), and put \(\widehat A=\widehat\mu^*L\). Normalize the main component of \(\widehat X\times_{W_0}W\), calling it \(R\). It is finite over \(\widehat X\) with the same Galois group; geometric integrality of the generic fibre makes the function-field extension linearly disjoint from the finite Galois base extension.

On \(R\) the difference of the pullbacks \(b\widehat A-B_W\) has a nonzero section: resolve the rational map from \(R\) to \(Z_b\) over \(X_1\) and \(Y_b\), and equation \((p_b^*B_b\to q_b^*A^b)\) gives a regular section there. Both line bundles come from \(R\); normality descends the section to \(R\). Multiply its Galois conjugates and raise to the \(\ell\)-th power to obtain a section of the pullback of \(b\ell^2\widehat A-\widehat f^*B_0\). A further field norm for the finite surjection onto the normal target \cite[Lemma~31.18.7]{StacksNormNormal}, of degree \(\ell\), relative to a rational line frame, gives a regular nonzero section downstairs of
\[
 b\ell^3\widehat A-\widehat f^*(\ell B_0).
\]
Regularity is checked at every prime: all orders upstairs are nonnegative, hence so are the norm orders. The powers are not meant to be minimal. Put \(\widehat b=b\ell^3\) and \(\widehat Q=\ell B_0\).

We are now on birational models of the original source and base fields. The base line \(\widehat Q\) carries a continuous semipositive metric and an injection \(\widehat f^*\widehat Q\to\widehat b\widehat A\). The original kernel condition persists on a nonempty open of these models. Proposition~\ref{prop:euler} then gives a section of a positive power of \(L\) on \(X\).


\section{Local strictness and an exact multiplier ideal}\label{sec:local-ideal}

\paragraph{In words.} The rank-one direct image of an adjoint line need not be locally free. If it has, say, a codimension-two defect, the Euler interpolation of Section~\ref{sec:harmonic} that drops it can add a section count that is not there. The present section identifies the defect with the multiplier ideal of a fibre-integral metric, so nothing is dropped. This gives a route to the transfer theorem that uses the \emph{actual} degree-zero line, not its hull.

The three ingredients are: a localised multiplier-ideal vanishing theorem, a strictness lemma for the quotient metric on an invariant adjoint line, and an identification of the direct image with its multiplier ideal. The application is a new proof of Corollary~\ref{thm:main} under the hypothesis $H^q(X,\OO_X)=0$ for $q>0$.

\subsection*{A localised vanishing theorem}

\begin{lemma}[Localised multiplier-ideal vanishing]\label{lem:local-nadel}
Let \(Y\) be a smooth connected projective variety of dimension \(d>0\). Suppose \(R\) has a singular Hermitian metric with plurisubharmonic weights \(\psi\), continuous and dominating a positive smooth \((1,1)\)-form on some coordinate ball. Then
\[
 H^q(Y,\OO_Y(K_Y+R)\otimes\mathcal I(\psi))=0\qquad(q>0).
\]
\end{lemma}

\begin{proof}[Sketch]
Construct a globally strictly positive metric with the same multiplier ideal. For sufficiently small \(\delta>0\) and large \(l\), a section of \(lR-A\) in \(\mathcal I(l(1+\delta)\psi)\) exists: on \(lR\) use weights \(l(1+\delta)\psi-l\delta\psi_0\), add \(j_lA-K_Y\) with \(j_l=\lceil C\delta l+C'\rceil\), and add \(\eta l\) times a logarithmic pole with \(\eta\) small, chosen independently of \(\delta,l\). Nadel vanishing in the strict-current form \cite[Theorem 5.11]{Demailly2011} then gives surjectivity to jets of order \(\lfloor\eta l\rfloor-d\) at the chosen point. Continuity of \(\psi\) there makes \(\mathcal I_l\) trivial near the point, and polar-coordinate integration gives the ideal \(\mathfrak m^{\lfloor\eta l\rfloor-d+1}\) for the pole. So \(h^0(Y,(lR+j_lA)\otimes\mathcal I_l)\ge cl^d\) for some \(c>0\).

Remove \(j_l+1\) copies of a very ample \(A\). At each step choose a smooth member of \(|A|\) containing no associated centre of \(\OO_Y/\mathcal I_l\); division by its equation preserves membership in the ideal because the equation is a nonzerodivisor. The loss at a step is at most \(C''l^{d-1}\) uniformly for \(0<\delta\le1\), since \(pA-R\) can be assumed globally generated for some fixed \(p\), and Hilbert-polynomial bounds on the restriction control the drop. The total loss is \(O(\delta l^d)+O(l^{d-1})\). Taking \(\delta\) small and then \(l\) large proves the initial section existence.

Let \(\sigma\) be a nonzero section of the resulting line. Its divisor metric \(\psi'\) on \(R\) satisfies \(i\partial\bar\partial\psi'\ge\omega_A/l\), and the section's membership in the multiplier ideal says exactly that \(\exp(l(\psi'-(1+\delta)\psi))\) is locally integrable. Take \(l\ge1/\delta\), set \(\rho=(1-\varepsilon)\psi+\varepsilon\psi'\), and rewrite
\[
 -\psi=-\frac{\rho}{1+\varepsilon\delta}+\frac{\varepsilon V}{1+\varepsilon\delta}.
\]
H\"older with the integrability of \(\exp(V/\delta)\) gives \(\mathcal I(\rho)\subseteq\mathcal I(\psi)\). For the converse, strong openness \cite{GuanZhou2015} and compactness give \(\gamma>0\) with \(\mathcal I((1+\gamma)\psi)=\mathcal I(\psi)\) on a finite cover, and small-exponent integrability of the divisor metric gives \(\gamma'>0\) with \(\exp(-\gamma'\psi')\) locally integrable. H\"older with \(s=(1-\varepsilon)/(1+\gamma)\) and \(1-s\), choosing \(\varepsilon\) small so that \(\varepsilon/(1-s)<\gamma'\), proves \(\mathcal I(\psi)\subseteq\mathcal I(\rho)\). The metric \(\rho\) has a global positive curvature lower bound and the same ideal, and Nadel vanishing finishes.
\end{proof}

\subsection*{Strictness on the invariant quotient}

\begin{lemma}[Strictness from horizontal curvature]\label{lem:strict}
Let \(p:V\to Y\) be a proper holomorphic submersion with compact fibres and K\"ahler total space. Let \(D\) have a smooth semipositive Hermitian metric. On an open set where \(p_*(K_{V/Y}+D)\) is locally free and its fibres are section spaces, suppose the coefficient curvature dominates the pullback of a positive base form on an open patch meeting \(V_{y_0}\), and let \(u_0\in H^0(V_{y_0},K_{V_{y_0}}+D|_{V_{y_0}})\) be a section nonzero on that patch. The direct-image metric curvature, paired with \(u_0\) at \(y_0\), is strictly positive in every nonzero tangent direction. For a torus action over a trivial base with metrics preserved and the invariant summand of rank one, the quotient metric has the same strictness at \(y_0\).
\end{lemma}

\begin{proof}[Sketch]
Restrict to a disc in a chosen base direction; let the fibre dimension be \(a\). If \(a=0\), the direct image is a sum of nonnegative terms, with a positive term on the patch. For \(a>0\), a holomorphic extending section with vanishing covariant derivative has a smooth absolute \((a,0)\)-form \(u\) satisfying \(D'u=dz\wedge\lambda\), \(\bar\partial u=dz\wedge\eta\). On the central fibre \(\lambda\) is orthogonal to holomorphic adjoint forms and \(\eta\wedge\omega\) is \(\bar\partial\)-exact. Replace \(u\) by \(u-dz\wedge\xi\) to make \(\lambda=0\) and \(\eta\) primitive; Hodge decomposition solves the two equations simultaneously, and Lefschetz linear algebra identifies the relevant \((a,1)\)-forms with \(\omega\wedge\xi\).

With \(c_a=i^{a^2}\), the curvature pairing is
\[
 c_a p_*\bigl(u\wedge\bar u\,e^{-\phi}\wedge i\partial\bar\partial\phi\bigr)-i\,dz\wedge d\bar z\,c_a\int_{V_0}\eta\wedge\bar\eta\,e^{-\phi}.
\]
The second term is nonnegative by the primitive middle-degree Hodge--Riemann sign; the first integrand is nonnegative everywhere. On the prescribed patch the base lower bound contributes positively, so the integral is strictly positive. Averaging over a torus is a holomorphic orthogonal projection, so the same calculation holds on the invariant summand.
\end{proof}

\subsection*{The exact ideal of the invariant adjoint line}

\begin{proposition}[Exact ideal]\label{prop:exact-adjoint-ideal}
Let \(X,V,Y\) be smooth connected projective, \(\dim Y>0\), \(\pi:V\to X\) birational, \(f:V\to Y\) surjective, and an algebraic torus acting equivariantly on the maps and trivially on \(Y\). Give \(L=-K_X\) its natural linearization and an invariant smooth semipositive metric. Put \(L'=\pi^*L\), \(J=K_{V/X}\), with \(t\) its natural invariant Jacobian section. Assume \(\mathcal B=(f_*\OO_V(J))^T\) has rank one. Write \(\mathcal B^{\vee\vee}=\OO_Y(B_0)\) and \(R=B_0-K_Y\). Then the invariant adjoint \(L^2\)-metric extends to a singular semipositive metric \(\psi_R\) on \(R\), and
\[
 \OO_Y(B_0)\otimes\mathcal I(\psi_R)=\mathcal B.
\]
If on a nonempty patch in the smooth base-change locus, with \(t\) nonzero, the curvature of \(L'\) dominates the pullback of a positive base form, then \(\psi_R\) is smooth and strictly positive on a base coordinate ball.
\end{proposition}

\begin{proof}[Sketch]
The candidate for the divisor \(B_0\) is determined by regularity over the generic point of a base prime: \(b_P=\min_{Q\mapsto P}\lfloor\ord_Q(t)/\ord_Q(f^*P)\rfloor\). This is the regularity test for \((f^*a)t\), and on the smooth base-change locus the invariant summand is a line whose fibre-integral metric is semipositive by the compact-torus orthogonal-quotient argument.

The metric extends across every base prime. At a general point of a component attaining \(b_P\), write the map as \((z_1^e,z_2,\ldots,z_k)\), where \(e>0\), and let \(a\) be \(\ord(t)\). The relative canonical coefficient is a nonvanishing smooth factor times \(z_1^{a-b_Pe-(e-1)}\), whose exponent is nonpositive because \(0\le a-b_Pe<e\). Integration on a small vertical polydisk bounds the squared norm below. So the weight is locally bounded above and extends by first crossing the divisorial neighbourhoods and then codimension at least two.

To identify the ideal with \(\mathcal B\), choose local frames \(g\) of \(\OO_Y(B_0)\) and \(\eta\) of \(K_Y\). The frame \(g/\eta\) corresponds on the smooth locus to \((f^*g)t/f^*\eta\). For a holomorphic function \(a\), Fubini identifies integrability of \(a\) with the squared norm of this frame, using the base volume from \(\eta\), with square integrability of the absolute top form \((f^*(ag))t\) with the smooth \(L'\)-metric upstairs. The excluded analytic sets have measure zero. An integrable meromorphic top form has no divisorial pole, and smoothness upstairs extends it across codimension two. Conversely a holomorphic lift is integrable by properness. Both inclusions follow; the codimension-two difference between \(\mathcal B\) and its hull is preserved.

The strictness assertion follows by applying Lemma~\ref{lem:strict} to the invariant section determined by \(t\) on the patch. The fibre-integral metric is smooth on the base-change open, so its strictness at one point gives a positive curvature lower bound on a smaller coordinate ball.
\end{proof}

\subsection*{Application: invariant sections from a degree-zero field}

\begin{proposition}[Invariant sections by a degree-zero field]\label{prop:degree-zero-route}
Let \(X\) be smooth connected projective with \(H^q(X,\OO_X)=0\) for \(q>0\), \(L=-K_X\) smoothly semipositive, and \(T\) an algebraic torus acting on \(X\). Then \(H^0(X,mL)^T\ne0\) for some \(m>0\), with the natural linearization.
\end{proposition}

\begin{proof}[Sketch]
Average a metric on \(L\) and a K\"ahler form over the maximal compact torus, and let \(\alpha\ge0\) be the resulting curvature. Proposition~\ref{prop:index} and Lemma~\ref{lem:forms} give: if the asserted invariant sections do not exist, there is a fixed \(q>0\) and unbounded \(j\) with a nonzero invariant form in \(H^0(X,\Omega_X^{n-q}\otimes jL)\). The Bochner curvature term is a sum of \(q\) eigenvalues of \(j\alpha\); if \(\alpha\) has full rank on an open set, this kills the form. So \(r=\max\rank\alpha<n\).

Define the degree \(\ell(D)=[D][\alpha]^r[\omega]^{n-r-1}\). It is nonnegative on effective divisors and vanishes on \(L\). Lemma~\ref{lem:forms} gives a linearized determinant line \(S\) and invariant sections \(s_p\in H^0(X,S+pL)^T\) for unbounded \(p\). Lemma~\ref{lem:null-divisors} gives \(\ell(N_p)=0\) for their zero divisors \(N_p\).

The field \(K\) of invariant rational functions with zero-degree polar divisors is relatively algebraically closed and finitely generated by the argument of Section~\ref{sec:fields}. Take a smooth projective model \(Y\), flatten the graph, normalize, and resolve. Write \(\pi:V\to X\), \(f:V\to Y\), \(L'=\pi^*L\), \(t\) the Jacobian section of \(K_V+L'\). Ratios of products of \(s_p\) having the same bidegree in \((S,L)\) belong to \(K\); the horizontal divisor is zero; affine relations in \(p\) force the slope to be effective. So there is an invariant rational section \(\sigma\) of \(dL'\) with effective horizontal divisor. Divisorial normalization gives a base divisor \(B\) and a regular invariant section \(s_D\) of \(dL_G-f_G^*B\).

The invariant adjoint rank is one for \(m\ge0\) (Section~\ref{sec:fields}). On the maximal-rank open, the differential of every rational function in \(K\) kills \(\ker\alpha\); the differential semipositive linear algebra gives \(\pi^*\alpha\ge cf^*\omega_Y\) on a horizontal patch. The rank \(k\) of \(Y\) satisfies \(k\le r\).

Proposition~\ref{prop:maximum} builds a continuous semipositive metric on \(B\). Proposition~\ref{prop:exact-adjoint-ideal} at \(m=0\) gives \(R=B_0-K_Y\) with strictly positive curvature on a base ball, and the exact identity \(\OO_Y(B_0)\otimes\mathcal I(\psi_R)=\mathcal B\). Adding \(m\) times the bounded metric on \(B\) preserves the ideal and the strictness. Lemma~\ref{lem:local-nadel} gives \(H^q(Y,\mathcal B\otimes\OO_Y(mB))=0\) for \(q>0\), \(m\ge0\). At zero \(h^0(Y,\mathcal B)=h^0(V,\OO_V(J))^T=1\), so the Euler polynomial has value one. For some \(m>0\) lift a nonzero section, multiply by \(s_D^m\), and push down removing the exceptional Jacobian divisor.
\end{proof}

\subsection*{What this route changed}

The proof of Proposition~\ref{prop:degree-zero-route} falls into the same seven-step template as Theorem~\ref{thm:transfer}, but replaces the singular positive base metric with the exact multiplier-ideal identification of \(\mathcal B\). The result is not a new theorem; it is a different way of seeing why the fixed multiplier ideal in the transfer proof can be dropped from the input and recovered from the direct image at degree zero.


\section{Stabilized orders on the actual direct images}\label{sec:stabilized}

\paragraph{In words.} The transfer theorem used a maximised ratio field. The degree-zero route used an exact multiplier ideal. The present section proves the transfer theorem a third way, keeping the \emph{actual} adjoint direct images as sheaves and showing that their boundary pole orders stabilize. The advantage is that the local frames of these sheaves lift to holomorphic adjoint sections, so their quotient metrics are integrable without an assumption that the sheaf is locally free.

The main additional input is a birational toroidalization, so that the model is equidimensional with toroidal (hence Cohen--Macaulay and rational) singularities. The result is a new proof of Theorem~\ref{thm:transfer} in the natural-linearization case.

\begin{proposition}[Invariant Iitaka conversion by stabilized orders]\label{prop:stabilized-transfer}
Let \(X\) be smooth connected projective and let an algebraic torus \(T\) act on \(X\). Give \(L=-K_X\) its natural linearization and suppose it admits a smooth semipositive metric. Let \(M\) be a linearized line with \(-M\) pseudoeffective. If there are nonzero sections \(\sigma_j\in H^0(X,M+jL)^T\) for an unbounded \(J\subset\Z_{>0}\), then \(H^0(X,mL)^T\ne0\) for some \(m>0\).
\end{proposition}

\begin{proof}[Sketch]
\emph{Generic adjoint rigidity.} Fix \(j_0<j_1\) in \(J\) and choose \(k>0\) so that the invariant system \(H^0(X,kD(j_1))^T\) has maximal image dimension; write \(D(j)=M+jL\). Equivariant principalisation of its base ideal gives a smooth projective \(\pi_0:W_0\to X\) isomorphic off the base locus, and a connected-fibre Stein map \(f_0:W_0\to Y_0\) such that \(\pi_0^*kD(j_1)=f_0^*H+G\), with \(H\) ample and \(G\) effective, both fixed by the connected torus. The Stein base is trivially acted on. Maximality of image dimension forces
\[
 \dim_K H^0(F,h\pi_0^*D(j_1)|_F)^T=1\qquad(h>0)
\]
for the generic fibre \(F\) over \(K=\C(Y_0)\). Products of the invariant sections of the same linearized degree are proportional; hence the fibre divisor \(D_{j_1}^F-D_{j_0}^F\) is effective. So \(z=\sigma_{j_1}/\sigma_{j_0}\) is a rational invariant section of \(dL\), \(d=j_1-j_0>0\), regular on \(F\).

For every fixed \(s\ge0\), pick \(h\) large enough that \(hD_{j_1}^F\ge\div(e_0|_F)+s\div(z|_F)\), where \(e_0\) is the Jacobian section of \(K_{W_0}+\pi_0^*L\). The invariant quotient \((\sigma_{j_1}|_F)^h/(e_0z^s|_F)\) is regular, so multiplication by it injects the whole invariant adjoint space into a one-dimensional one:
\[
 H^0\bigl(F,(K_{W_0}+(1+sd)\pi_0^*L)|_F\bigr)^T=K\cdot(e_0z^s)|_F.
\]

\emph{Actual lines.} Apply birational weak toroidalization \cite[Theorem 1.1]{ADK2013}, then equidimensionalize by toroidal subdivisions \cite[Proposition 4.4]{AK2000}, obtaining a normal toroidal \(Z\to Y\) equidimensional and birational over \(W_0\). Take a smooth resolution \(W\to Z\), with \(f:W\to Y\), \(\pi:W\to X\). Toroidal singularities are rational, so
\[
 \mathscr E_s=f_*\OO_W(K_{W/Y}+(1+sd)\pi^*L)
\]
satisfies \(\mathscr E_s\otimes\OO_Y(K_Y)=g_*(\omega_Z\otimes\pi_Z^*\OO_X((1+sd)L))\), and is reflexive, since equidimensionality pushes codimension two down to codimension two and the reflexive canonical sheaf extends sections. A common smooth resolution of \(W\) and an equivariant model identifies the canonical direct images, and the invariant part
\[
 A_s=(\mathscr E_s)^T
\]
is a genuine rank-one reflexive summand, hence a line bundle on smooth \(Y\). Birational canonical pushforward gives
\[
 H^0(Y,K_Y+A_s)=H^0(X,sdL)^T,\qquad h^0(Y,K_Y+A_0)=1.
\]

\emph{Affine boundary orders.} Use the rational generator \(e_Zz^s\) of \(K_Y+A_s\) on the regular locus of \(Z\). For a base prime \(B\subset Y\) with primes \(Q_i\mapsto B\), put \(b_i=\ord_{Q_i}(g^*B)>0\), \(e_i=\ord_{Q_i}(E_Z)\), \(c_i=\ord_{Q_i}(z)\). The allowed pole order is
\[
 u_s(B)=\left\lfloor\min_i\frac{e_i+sc_i}{b_i}\right\rfloor,\qquad K_Y+A_s=\OO_Y(C_s),\quad C_s=\sum_Bu_s(B)B.
\]
Only finitely many primes occur with a nonzero slope or intercept. Replace \(d,z\) by a common multiple and power so that \(v_B=\min_i(c_i/b_i)\in\Z\), and put \(V=\sum_Bv_BB\). Then for all sufficiently large integer \(s\),
\[
 C_s=C_\infty+sV,\qquad C_\infty\ge C_0,
\]
so with \(A_\infty=\OO_Y(C_\infty)-K_Y\) there is an inclusion
\[
 A_0+sV\hookrightarrow A_s
\]
for all large \(s\). This is the inclusion that lets us interpolate from exponent zero.

\emph{Integrable metrics and interpolation.} The original moving part and \(-M\) pseudoeffective give that \(\pi^*L-\epsilon f^*H_1\) is pseudoeffective for some ample \(H_1\). Mix a metric from this with the smooth semipositive metric on \(\pi^*L\): the result at \(s=0\) is singular with trivial multiplier ideal and curvature \(\ge\delta f^*\omega_Y\), \(\delta>0\). For \(s>0\) use the smooth semipositive metrics on \((1+sd)\pi^*L\).

The singular adjoint metric and its extension \cite[Theorems 3.3.4--3.3.5]{PT2018} make \(A_s\) carry a quotient metric \(\psi_s\) on its locally free locus, which extends plurisubharmonically as a line bundle. Crucially, the local frames of \(A_s\) lift to holomorphic adjoint sections of \(\mathscr E_s\), so their squared norms bound the quotient norms, giving \(e^{-\psi_s}\in L^1_{\rm loc}(Y)\). This is exactly what a frame of a reflexive \emph{hull} would miss.

The weight \(\psi_0\) is strictly positive in curvature on all of \(Y\). To see it across the boundary, choose a smooth \(p\) with Hessian dominating \(\omega_Y\) near a point and bounded above globally, multiply the upstairs \(L^2\)-metric by \(e^{t f^*p}\) for small \(t>0\), and use the extension theorem. A finite cover gives \(i\partial\bar\partial\psi_0\ge\delta'\omega_Y\) for some \(\delta'>0\).

Now give \(A_0+sV\) a strict positive metric with trivial multiplier ideal for every \(s\ge0\). Fix \(s>0\), choose large \(u\) in the stabilization range, and a smooth metric \(\psi_\infty\) on \(A_\infty\). Put
\[
 \rho_s=\psi_0+\frac{s}{u}(\psi_u-\psi_\infty).
\]
Its curvature is strictly positive for \(s/u\) small. Strong openness and compactness give \(\eta>0\) with \(e^{-(1+\eta)\psi_0}\) locally integrable on a finite cover. H\"older, together with \(L^1_{\rm loc}\)-integrability of \(e^{-\psi_u}\), proves \(e^{-\rho_s}\in L^1_{\rm loc}\) if \(\frac{s}{u}\cdot\frac{1+\eta}{\eta}\le1\). No exponent improvement uniform in \(u\) is needed.

Nadel vanishing gives \(H^i(Y,K_Y+A_0+sV)=0\) for \(i>0\), \(s\ge0\). The Euler polynomial at zero is one. So it is nonzero for large integer \(s\); the inclusion \(A_0+sV\hookrightarrow A_s\) and the identification at zero give a nonzero invariant section of \(sdL\).
\end{proof}

\paragraph{Where the hypothesis $H^q(X,\OO_X)=0$ appears.} Under that hypothesis, Proposition~\ref{prop:index} and Lemma~\ref{lem:forms} supply the unbounded invariant twisted forms required to apply the proposition, exactly as in Corollary~\ref{thm:forms-input}. So the present section is a third proof of invariant nonvanishing under \(H^q(X,\OO_X)=0\) for \(q>0\).


\section{The map of the whole argument}\label{sec:map}

\paragraph{In words.} The paper has one main theorem, one main corollary, and six complete routes to the corollary. The theorem is the transfer of Theorem~\ref{thm:transfer}. The corollary is the nonvanishing of Corollary~\ref{thm:ordinary}. Every other section either proves the theorem in a new regime, supplies an input to the theorem, or supplies an input to a corollary.

\subsection*{The shortest path to the strongest conclusion}

Read Sections~\ref{sec:transfer}, \ref{sec:index}, \ref{sec:descent} in this order. They prove:
\begin{itemize}
\item Theorem~\ref{thm:transfer}: unbounded invariant sections of \(M+iP\) with \(-M\) pseudoeffective give an invariant section of a positive power of \(P\). (Section~\ref{sec:transfer}.)
\item Corollary~\ref{thm:main}: natural invariant nonvanishing under \(\chi(X,\OO_X)\ne0\), via the invariant anticanonical index. (Section~\ref{sec:index}.)
\item Corollary~\ref{thm:ordinary}: ordinary nonvanishing without the Euler hypothesis, via compact monodromy and a finite norm. (Section~\ref{sec:descent}.)
\end{itemize}
A reader who wants the strongest short answer can stop after Section~\ref{sec:descent}. A reader who wants all the machinery should keep reading.

\subsection*{The six routes to Corollary~\ref{thm:ordinary}}

The following table compares them. The common step is always ``produce a base, an injection, and rank-one adjoints, then apply Proposition~\ref{prop:euler} or an equivalent''. The differences are the choice of base and the analytic input that makes the metric work.

\begin{center}
\small
\begin{tabular}{@{}p{.14\textwidth}p{.28\textwidth}p{.22\textwidth}p{.26\textwidth}@{}}
\toprule
Route & Input & Base construction & Conversion \\
\midrule
Transfer (Sect.~\ref{sec:transfer}) & Unbounded invariant sections of \(M+iP\), \(-M\) pseudoeffective & Common ratio field \(Q_i\); equivariant normal model & Fixed multiplier ideal, Nadel vanishing \\
Indexed (Sect.~\ref{sec:index}) & \(\chi(X,\OO_X)\ne0\) & Invariant anticanonical index; determinant sections of \(M\) & Transfer theorem applied with natural linearization \\
Descent (Sect.~\ref{sec:descent}) & Structure of the universal cover & Compact factor \(F\); Zariski closure of the torus; canonical frames & Finite \'etale norm; or cohomological or twisted-form descent \\
Extremal (Sect.~\ref{sec:extremal}) & Source characters of a one-parameter action & Bia\l ynicki-Birula source and equivariant hard Lefschetz & Transfer with a source-weight linearization \\
Harmonic (Sect.~\ref{sec:harmonic-iitaka}) & Smooth semipositive \(L\) and determinant sections & Maximal image in one fixed invariant system & Nadel vanishing with kernel \(\ker\theta\subseteq\ker df\) \\
Iitaka (Sect.~\ref{sec:ordinary}) & Effective integral divisors \(N_s\sim sL-F_0\) with \(F_0\) pseudoeffective & Iitaka base of a representative \(N_b\) & Fujino--Matsumura injectivity plus Leray \\
Fields (Sect.~\ref{sec:fields}) & \(H^q(X,\OO_X)=0\) for \(q>0\) and determinant sections & Small-pole field or affine semigroup field & Maximum metric and Proposition~\ref{prop:euler} \\
Alteration (Sect.~\ref{sec:alteration}) & Unbounded \(H^q(X,mL)\) in one degree & Curvature-null field \(K\), reduced flat alteration & Galois descent of a Hom line and a finite norm \\
Degree-zero (Sect.~\ref{sec:local-ideal}) & \(H^q(X,\OO_X)=0\) and smooth semipositive \(L\) & Degree-zero pole field with \(L^2\)-metrics & Exact multiplier ideal, Lemma~\ref{lem:local-nadel} \\
Stabilized (Sect.~\ref{sec:stabilized}) & Same as transfer, but no ratio field & Toroidal equidimensional model with rational singularities & Affine boundary orders, actual direct-image line, strong openness \\
\bottomrule
\end{tabular}
\end{center}

\subsection*{What depends on what}

Three logical dependencies appear frequently. The first is Lemma~\ref{lem:forms}: it supplies invariant twisted forms from cohomology. The second is Proposition~\ref{prop:maximum}: it turns a rational relative section into a continuous base metric. The third is Proposition~\ref{prop:euler}: it interpolates the zero of an Euler polynomial into a positive-degree section. Any route that uses these three and can verify their hypotheses proves the corresponding nonvanishing conclusion.

Two routes stand out. The transfer proof of Section~\ref{sec:transfer} uses no compact monodromy, only a field of ratios and a singular multiplier ideal. The harmonic proof of Section~\ref{sec:harmonic-iitaka} uses no ratio field on \(X\) itself, only a maximal invariant system and a base-choice argument. The other routes recombine these ideas in different regularity regimes.

\subsection*{Reading paths for three audiences}

A high-school reader can read Sections~\ref{sec:question}, \ref{sec:vocab}, the opening paragraphs of each later section, and the table above. That gives the complete statement of what was proved and how the proofs split.

An undergraduate in mathematics can add the statements of Theorem~\ref{thm:transfer}, Corollary~\ref{thm:main}, Corollary~\ref{thm:ordinary}, Lemma~\ref{lem:forms}, and Lemma~\ref{lem:null-divisors}, and read the proofs at the ``plain form'' level. That gives a good idea of the method without the analytic details.

A graduate student or researcher should read all sections. The dependencies are listed above; a reader who wants a single complete path can read Sections~\ref{sec:transfer}, \ref{sec:models}, \ref{sec:index}, \ref{sec:descent}. Every other section is either a proof of the same corollary under stronger hypotheses or a machinery development used by one of the routes.


\appendix

\section{Notation}\label{app:notation}

The following table collects symbols used in more than one section. An index in brackets points to the section where the symbol is introduced.

\small
\begin{longtable}{@{}p{.12\textwidth}p{.50\textwidth}p{.24\textwidth}@{}}
\toprule
Symbol & Meaning & Section \\
\midrule
\(X\) & Smooth connected projective complex variety & \ref{sec:question} \\
\(n\) & \(\dim X\) & \ref{sec:vocab} \\
\(T_X, \Omega_X^p\) & Tangent and cotangent bundles, exterior powers & \ref{sec:vocab} \\
\(K_X, L\) & Canonical bundle and anticanonical \(L=-K_X\) & \ref{sec:vocab} \\
\(\OO_X\) & Structure sheaf & \ref{sec:vocab} \\
\(H^q(X,-)\) & Coherent cohomology & \ref{sec:vocab} \\
\(\chi(X,-)\) & Euler characteristic & \ref{sec:vocab} \\
\(T\) & Algebraic torus acting on \(X\) & \ref{sec:vocab} \\
\(\chi\) & Character of \(T\) & \ref{sec:vocab} \\
\(P, M\) & Linearized line bundles; \(P=L\) up to character, \(-M\) pseudoeffective & \ref{sec:transfer} \\
\(N_i\) & \(N_i=M+iP\) & \ref{sec:transfer} \\
\(s_i\) & \(0\ne s_i\in H^0(X,N_i)^T\) & \ref{sec:transfer} \\
\(a_i\) & \(N_i=M+a_iL\), \(a_i\to\infty\) & \ref{sec:index} \\
\(Q_i, Q\) & Field of ratios of invariant sections of \(N_i\), and the stable one & \ref{sec:transfer} \\
\(\theta\) & Smooth semipositive curvature of \(L\) & \ref{sec:index} \\
\(\omega\) & Fixed K\"ahler form on \(X\) & \ref{sec:index} \\
\(r\) & Maximal rank of \(\theta\) & \ref{sec:index} \\
\(Y, W, V\) & Base, normal equidimensional model over \(Y\), smooth resolution & \ref{sec:models} \\
\(\pi:V\to X\) & Birational map from the resolution & \ref{sec:models} \\
\(f:V\to Y\) & Fibration over the base & \ref{sec:models} \\
\(D_V, E\) & \(D_V=\pi^*L\), \(E=K_{V/X}\) & \ref{sec:models} \\
\(s_E, 1_E\) & Jacobian section of \(E\), its trivialization & \ref{sec:models} \\
\(B, kB\) & Integral rational base divisor and its clearing exponent & \ref{sec:models} \\
\(Q\) (on base) & Base line with continuous semipositive metric & \ref{sec:models} \\
\(\mathcal I(\psi)\) & Multiplier ideal of weight \(\psi\) & \ref{sec:local-ideal} \\
\(\psi_R\) & Semipositive metric on \(R=B_0-K_Y\) from the direct image & \ref{sec:local-ideal} \\
\(\mathcal B\) & Rank-one direct image \((f_*\OO_V(J))^T\) & \ref{sec:local-ideal} \\
\(u_s(B)\) & Allowed pole order in base prime \(B\) at exponent \(s\) & \ref{sec:stabilized} \\
\(v_B\) & Slope of \(u_s(B)\) in \(s\) & \ref{sec:stabilized} \\
\(A_s\) & Actual direct-image line \(K_Y+A_s\) & \ref{sec:stabilized} \\
\(C_s\) & Effective divisor with \(K_Y+A_s=\OO_Y(C_s)\) & \ref{sec:stabilized} \\
\(\mathcal K\) & Curvature-null field of rational functions & \ref{sec:fields}, \ref{sec:alteration} \\
\(S\) (as variety) & Smooth projective rationally connected with \(L=-K_S\) & \ref{sec:extremal} \\
\(S_0\) (as source) & Source component of a \(\C^*\)-action & \ref{sec:extremal} \\
\(\mu\) & Full \(T\)-character of \(U|_S\), a source weight & \ref{sec:extremal} \\
\(\gamma_C\) & Fiber character of \(L\) on a fixed component \(C\) & \ref{sec:extremal} \\
\(\mathcal P\) & Relatively very ample line on an alteration & \ref{sec:alteration} \\
\(\mathcal H, \mathcal H'\) & Direct images \(p_{a*}\mathcal P\) and \(p_{a*}(\mathcal P\otimes q_a^*A^b)\) & \ref{sec:alteration} \\
\(B_b\) & Tautological subline on the birational base \(Y_b\) & \ref{sec:alteration} \\
\(\widehat X, \widehat Y\) & Resolved models after Galois base change & \ref{sec:alteration} \\
\(W, G, R\) & Galois cover, its group, and the finite pullback to \(\widehat X\) & \ref{sec:alteration} \\
\bottomrule
\end{longtable}

A note on notation. Tensor products of line bundles are written additively: \(A+B\) means \(A\otimes B\), \(mA\) means \(A^{\otimes m}\), \(-A\) means the inverse line. The symbol \(s_E\) is a specific rational section of \(E=K_{V/X}\); \(1_E\) is its local trivialization. The symbol \(P\) is used both for a linearized line and for the transfer theorem's base twist; context resolves which is meant. The base line is called \(Q\) in Sections~\ref{sec:transfer}--\ref{sec:harmonic-iitaka} and \(B\) in Sections~\ref{sec:local-ideal}--\ref{sec:stabilized}; the two are related by normalization and by the identification \(B_0=B+K_Y\).

\subsection*{Conventions used throughout}

We write \(\sim\), \(\sim_{\Q}\), and \(\equiv\) for linear, rational linear, and numerical equivalence. A local frame of squared norm \(e^{-\phi}\) has curvature \(i\partial\bar\partial\phi\); factors of \(2\pi\) are included when viewing curvature as a first Chern form. A \emph{continuous semipositive} metric has continuous plurisubharmonic weights; a \emph{locally bounded} metric has both upper and lower bounds on compact charts and need not be continuous.

The action on sections is \((g\cdot s)(gx)=g(s(x))\). Algebraic-torus invariants agree with invariants under the maximal compact torus: averaging a section over the compact torus is a holomorphic idempotent, hence a projection onto invariants, and it is the identity on the algebraic invariants. Averaging smooth metric weights and K\"ahler forms over the compact torus preserves semipositivity. This compactness is used only through averaging, but it is essential: on a non-compact torus there is no averaging, and algebraically invariant sections of \(mL\) need not be the same as sections invariant for a shifted linearization \(P\). That distinction is precisely the content of Theorem~\ref{thm:transfer}.

For readers comparing this edition with the algebraic original, three translations are worth spelling out. The ratio field \(Q_i\) of Section~\ref{sec:transfer} replaces the algebraic ratio field of the classical transfer proof. The singular positive metric coming from \(F_0\) replaces the effective \(\Q\)-divisor representing \(-M\). The multiplier ideal \(\mathcal J\) of the transfer proof is the exact analogue of the base ideal in the algebraicity-based proofs. Each pair is the analytic and the algebraic version of the same step.


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\end{document}