==== TABLE OF CONTENTS ====
The-abelianity-conjecture-for-special-compact-Kahler-manifolds//figures/proof-overview.tex
The-abelianity-conjecture-for-special-compact-Kahler-manifolds//macros.tex
The-abelianity-conjecture-for-special-compact-Kahler-manifolds//paper.tex
The-abelianity-conjecture-for-special-compact-Kahler-manifolds//sections/completion.tex
The-abelianity-conjecture-for-special-compact-Kahler-manifolds//sections/descent.tex
The-abelianity-conjecture-for-special-compact-Kahler-manifolds//sections/hilbert.tex
The-abelianity-conjecture-for-special-compact-Kahler-manifolds//sections/introduction.tex
The-abelianity-conjecture-for-special-compact-Kahler-manifolds//sections/preliminaries.tex
The-abelianity-conjecture-for-special-compact-Kahler-manifolds//sections/spectral.tex
The-abelianity-conjecture-for-special-compact-Kahler-manifolds//references.bib
===========================

==== BEGIN The-abelianity-conjecture-for-special-compact-Kahler-manifolds//figures/proof-overview.tex ====
\begingroup
\small
\setlength{\fboxsep}{7pt}
\fbox{\begin{minipage}{0.9\linewidth}\centering
Infinite FAb group \(N\): minimize a base dimension \(\ell>0\).\\
Obtain \(Q=N/B_0\) and \((Y,\Delta_Y)\) over the Albanese torus.
\end{minipage}}
\par\smallskip
\(\Downarrow\)
\par\smallskip
\begin{tabular}{@{}c@{\hspace{0.025\linewidth}}c@{}}
\fbox{\begin{minipage}[t]{0.405\linewidth}\centering
\(Q\) has an infinite complex\\linear image in finite dimension.\par\smallskip
FAb, minimality, and the\\semisimple general-type criterion\par\smallskip
\(\Longrightarrow K_Y\) big
\end{minipage}}
&
\fbox{\begin{minipage}[t]{0.405\linewidth}\centering
Every finite-dimensional complex\\linear image of \(Q\) is finite.\par\smallskip
Hilbert tensors, discrete monodromy,\\and minimality\par\smallskip
\(\Longrightarrow K_Y+\Delta_Y\) big
\end{minipage}}
\end{tabular}
\par\smallskip
\(\Downarrow\)
\par\smallskip
\fbox{\begin{minipage}{0.9\linewidth}\centering
Torus descent with actual meridian orders\par\smallskip
A Bogomolov sheaf on a finite cover of \(X\): contradiction.
\end{minipage}}
\endgroup
==== END The-abelianity-conjecture-for-special-compact-Kahler-manifolds//figures/proof-overview.tex ====

==== BEGIN The-abelianity-conjecture-for-special-compact-Kahler-manifolds//macros.tex ====
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\newtheorem{remark}[theorem]{Remark}
==== END The-abelianity-conjecture-for-special-compact-Kahler-manifolds//macros.tex ====

==== BEGIN The-abelianity-conjecture-for-special-compact-Kahler-manifolds//paper.tex ====
\providecommand{\DoNotLoadEpstopdf}{}
\documentclass[11pt]{article}
\usepackage[letterpaper,margin=1.175in]{geometry}
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\title{The abelianity conjecture for\\special compact K\"ahler manifolds}
\author{OpenAI}
\date{September 23, 2026}
\hypersetup{
  pdftitle={The abelianity conjecture for special compact Kähler manifolds},
  pdfauthor={OpenAI}
}
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\makeatother
\begin{document}
\maketitle
\begin{abstract}
We prove that the fundamental group of every special compact K\"ahler
manifold is virtually abelian, resolving Campana's abelianity conjecture
in all dimensions. In particular, every smooth compact connected
K\"ahler manifold of Kodaira dimension zero has virtually abelian
ordinary fundamental group.
\end{abstract}
\begingroup
\small
\renewcommand{\baselinestretch}{0.9}\selectfont
\tableofcontents
\endgroup
\input{sections/introduction}
\input{sections/preliminaries}
\input{sections/spectral}
\input{sections/hilbert}
\input{sections/descent}
\input{sections/completion}
\bibliographystyle{plain}
\begingroup
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\bibliography{references}
\endgroup
\end{document}
==== END The-abelianity-conjecture-for-special-compact-Kahler-manifolds//paper.tex ====

==== BEGIN The-abelianity-conjecture-for-special-compact-Kahler-manifolds//sections/completion.tex ====
\section{Albanese fibre groups and completion of the proof}\label{sec:completion}

The remaining task is group-theoretic and geometric assembly: isolate
the Albanese fibre-group image, establish the minimality hypotheses
of the positivity theorem, and apply torus descent in both
representation cases.

Let \(X\) be special. If \(\dim X=0\), its fundamental group is trivial,
so assume \(\dim X>0\). Every connected finite \'etale cover is special
and has surjective Albanese morphism by Theorem~\ref{thm:specialness-inputs}.
Its first Betti number is therefore at most \(2\dim X\). Pass to a cover
on which this number is maximal, and continue to denote the cover by
\(X\). Pullback on \(H^1(-,\mathbb Q)\) is injective by transfer, so
any further connected finite \'etale cover has the same first Betti
number. Write
\[
 \alpha:X\longrightarrow A=\Alb(X),\qquad
 G=\pi_1(X),\qquad N=\im\bigl(\pi_1(S)\longrightarrow G\bigr),
\]
where \(S\) is a smooth general Albanese fibre. The group \(N\) is
finitely generated and normal. When \(A\) is a point, these statements
mean \(S=X\) and \(N=G\).

\subsection{Exactness over an arbitrary Albanese torus}

\begin{lemma}\label{lem:albanese-exactness}
The natural sequence is exact:
\[
 1\longrightarrow N\longrightarrow G
 \xrightarrow{\alpha_*}\pi_1(A)\longrightarrow1.
\]
\end{lemma}
\begin{proof}
Set \(R=G/N\). Over the smooth fibration locus in \(A\), fibrewise
exactness gives a monodromy homomorphism to \(R\), compatible with the
quotient map on the total space. A meridian around a base divisor has
finite order in \(R\), since a component of its inverse image kills a
positive power of that meridian. Form the actual inertial divisor
\(D_A\) using these orders, as in Section~\ref{sec:descent}.

Suppose \(D_A\ne0\). Lemma~\ref{lem:torus-divisor} gives a quotient
torus \(p:A\to B\) with connected fibres, where \(B\) is an abelian
variety of dimension \(b>0\), and an ample effective rational divisor
\(D_B\) such that \(D_A=p^*D_B\). The smooth quotient map pulls back
each prime divisor with multiplicity one, so the components of \(D_B\)
retain the meridian orders of their inverse images.

Lemma~\ref{de:inertia-divisibility}, applied to the smooth torus
bundle \(A\to B\), shows that each such order divides the
multiplicity of \emph{every} prime divisor of \(X\) dominating the
corresponding divisor of \(B\) under \(p\alpha\). This includes
components whose images in \(A\) have larger codimension. For prime
divisors whose images in \(B\) have codimension at least two, the
top differential of \(p\alpha\) vanishes: at their general points
its restriction to the divisor has rank at most \(b-2\), so the
total rank is at most \(b-1\). There are only finitely many relevant
components over the fixed support of \(D_B\). Thus a sufficiently
small rational \(\varepsilon>0\) makes their differential vanishing
orders compensate the poles allowed by \(\varepsilon D_B\).
This is the fixed-map argument of Lemma~\ref{de:small-boundary}.

Let \(L\subset\Omega_X^b\) be the saturated line generated by
the top differentials from \(B\). For sufficiently divisible \(m\),
pullback of pluriforms gives
\[
 H^0\bigl(B,m(K_B+\varepsilon D_B)\bigr)
 \longrightarrow H^0(X,L^{\otimes m}).
\]
The pullback is injective, and ratios of sections retain their
independence because \(X\to B\) is surjective. Since \(K_B\) is
trivial and \(\varepsilon D_B\) is ample, \(\kappa(X,L)\ge b\).
The opposite inequality follows from
Theorem~\ref{thm:specialness-inputs}. This contradicts specialness.

All meridians therefore have trivial image in \(R\). The monodromy
homomorphism factors through \(\pi_1(A)\), by normal meridian
generation for the complement of the discriminant. Its composition
with \(G\to\pi_1(A)\) is the quotient \(G\to R\). Conversely,
\(\alpha_*\) kills \(N\) and factors through \(R\).
These two factorizations are inverse because the maps from \(G\)
are surjective. This proves \(R\cong\pi_1(A)\).
\end{proof}

For a further connected finite cover \(X'\to X\) corresponding to
\(G'\le G\), write \(A'=\Alb(X')\). The induced homomorphism
\(A'\to A\) is surjective and, by maximality of the first Betti
number, is an isogeny. Lemma~\ref{lem:albanese-exactness} on both
manifolds shows that the new Albanese fibre-group image is
\begin{equation}\label{eq:fibre-intersection}
 N'=N\cap G'.
\end{equation}
Indeed the map on torus fundamental groups induced by an isogeny
is injective. Lemma~\ref{lem:realize-finite-index} consequently
realizes every prescribed finite-index condition on \(N\) after
shrinking the total group by finite index.

\subsection{Finite abelianization on all finite covers}

\begin{proposition}\label{prop:fibre-fab}
The group \(N\) is FAb.
\end{proposition}
\begin{proof}
We first show that every group \(N'=N\cap G'\) arising from a
finite cover has \(\Hom(N',\C)=0\). Set
\(\Lambda=\pi_1(A')\) and
\(V=\Hom(N',\C)\). The vector space \(V\) is finite dimensional
because \(N'\) is finitely generated. Conjugation defines an action
of the abelian group \(\Lambda\) on it. Write \(T_\lambda\) for the
operator
\[
 (T_\lambda v)(n)=v(\widetilde\lambda^{-1}n\widetilde\lambda),
\]
where \(\widetilde\lambda\) is any lift to \(G'\). Inner conjugation
acts trivially on additive homomorphisms, so the operator is
independent of this lift.

For a nontrivial character \(\beta:\Lambda\to\C^*\), the
cohomology of \(\Lambda\) with coefficients in \(\C_\beta\)
vanishes in every degree. This follows, for example, from the
Koszul resolution: one generator acts by a scalar different from
one, making that factor contractible. Inflation--restriction gives
\begin{equation}\label{eq:twisted-inflation}
 H^1(G',\C_\beta)
 \cong (V\otimes\C_\beta)^\Lambda.
\end{equation}
Choose a basis \(\lambda_1,\ldots,\lambda_r\) of \(\Lambda\) and put
\(T_i=T_{\lambda_i}\). A nonzero invariant on the right of
\eqref{eq:twisted-inflation} corresponds to a vector \(w\ne0\)
satisfying
\[
 T_iw=\beta(\lambda_i)^{-1}w\qquad(1\le i\le r).
\]
Each \(T_i\) has finitely many eigenvalues, and the values on this
basis determine a character. Hence only finitely many nontrivial
\(\beta\) have \(H^1(G',\C_\beta)\ne0\).

Inflation identifies the character group of \(\Lambda\) with the
open and closed identity component of \(\operatorname{Hom}(G',\C^*)\):
the Albanese lattice is \(H_1(X',\mathbb Z)\) modulo torsion, and the
character group has finitely many components. Group and manifold
cohomology agree in degree one for these local systems. The nontrivial
support points just found are therefore isolated in the full
degree-one rank-one cohomology jump locus. B.~Wang's torsion-translate
theorem for compact K\"ahler manifolds
\cite[author's version, Theorem~1.3]{BWang2016} makes them torsion.

Suppose \(V\ne0\). The commuting complex operators \(T_i\) have a
common eigenvector \(v\ne0\): successively take an eigenspace of the
next operator inside the nonzero common eigenspace already chosen,
which remains stable by commutativity. Their invertibility makes the
eigenvalues nonzero, and they define a character
\(\chi:\Lambda\to\C^*\). If \(\chi\ne1\), the vector \(v\) makes
\(\beta=\chi^{-1}\) one of the nontrivial support points above, so
\(\chi\) is torsion. If \(\chi=1\), it is already torsion. For
\(r=0\), take any nonzero \(v\) and the unique trivial character.

Let \(A_0\to A'\) be the isogeny with lattice
\(\Lambda_0=\ker\chi\), and pull it back to \(X'\). The total-space
cover \(X_0\) is connected because \(G'\to\Lambda\) is surjective;
its group is \(G_0=(\alpha'_*)^{-1}(\Lambda_0)\), with kernel still
exactly \(N'\). Thus the vector space \(V=\Hom(N',\C)\) has not
changed, and the same \(v\) is fixed by the restricted lattice.
The pulled-back fibration has connected fibres. By the Albanese
universal property it factors through a surjective homomorphism
\(\Alb(X_0)\to A_0\), which maximality makes an isogeny.
Connectedness of the pulled-back fibres forces its degree to be one.
We may therefore relabel \(X_0,A_0,G_0,\Lambda_0\) as
\(X',A',G',\Lambda\), with the same \(V\) and \(V^\Lambda\ne0\).

The untwisted inflation--restriction sequence contains
\begin{equation}\label{eq:transgression}
 \begin{aligned}
 0&\longrightarrow H^1(\Lambda,\C)
 \longrightarrow H^1(G',\C)
 \longrightarrow V^\Lambda\\
 &\longrightarrow H^2(\Lambda,\C)
 \longrightarrow H^2(G',\C).
 \end{aligned}
\end{equation}
The last arrow is injective. To see this, its composition with
the natural map to \(H^2(X',\C)\) is the pullback
\(H^2(A',\C)\to H^2(X',\C)\), since a torus is a
\(K(\Lambda,1)\). That pullback is injective for a surjective
map from a compact K\"ahler manifold. In fact, if
\(a=\dim A'\) and \(d=\dim X'-a\), fibre integration of a
K\"ahler class \([\omega]^d\) is a positive constant. For
\(\eta\in H^2(A',\C)\) and \(\theta\in H^{2a-2}(A',\C)\),
\[
 \int_{X'}\alpha'^*(\eta\smile\theta)\smile[\omega]^d
 =c\int_{A'}\eta\smile\theta,\qquad c>0.
\]
Poincar\'e duality proves the claim. For \(a=0\) there is no
degree-two group to consider.

The transgression in \eqref{eq:transgression} is zero.
Hence a nonzero invariant vector would give
\(b_1(X')>2\dim A'\), contrary to the defining equality for
the Albanese torus. The isogeny kept \(N'\) and \(V\) unchanged, so
this contradiction proves \(V=0\) for the original cover as well.
Maximality ensured that all intervening Albanese tori were isogenous
and that the pulled-back torus was itself the new Albanese torus.

Finally, let \(N_0\le N\) have finite index. By
Lemma~\ref{lem:realize-finite-index}, it contains some
\(N\cap G'\). The latter has no nonzero homomorphism to \(\C\).
A homomorphism from \(N_0\) to \(\C\) restricting to zero on
a finite-index subgroup is zero. Thus \(\Hom(N_0,\C)=0\).
Since \(N_0\) is finitely generated, its abelianization is finite.
\end{proof}

If \(N\) is finite, Lemmas~\ref{lem:albanese-exactness}
and~\ref{lem:finite-by-abelian} prove the theorem. For the rest
of the proof suppose that \(N\) is infinite.

\subsection{A simultaneous choice of general Albanese fibres}

We specify the generality needed to globalize a fibration from a
single Albanese fibre. A finitely generated group has countably
many finite-index subgroups. Thus there are countably many connected
finite covers of \(X\), up to the choices relevant here. Their
Albanese tori are isogenous to \(A\).

On each cover, Barlet's space of compact analytic cycles
\cite{Barlet1975} has countably many irreducible components, each
compact in the K\"ahler case
\cite[Section~3, Convention~3.1]{CampanaGamma}.
We will use a countable list of families with holomorphic maps
\[
 p_j:W_j\longrightarrow T_j,\qquad e_j:W_j\longrightarrow X,\qquad
 W_j^\circ=p_j^{-1}(T_j^\circ)\subset W_j.
\]
Here \(W_j\) is a connected smooth compact complex manifold,
\(T_j^\circ\) is a connected smooth dense analytic Zariski open
subset of a compact parameter space \(T_j\), and
\(p_j:W_j^\circ\to T_j^\circ\) is proper and smooth with connected
fibres. Every irreducible reduced compact cycle \(V\subset X\)
occurs as \(e_j((W_j)_t)=V\) for some \(t\in T_j^\circ\), with the
fibre map proper and bimeromorphic. In particular,
\(W_j^\circ\) is the complement of a proper analytic subset in
the same compact \(W_j\).

Here is a construction of this list. On a fixed compact irreducible
component \(P\) of the space of \(k\)-cycles, K\"ahler volume is
constant, say \(v\). A cycle is reducible or nonreduced precisely
when it is a sum of two nonzero cycles. Both summands have volume at
most \(v\). Bounded-volume compactness and local finiteness of the
components of the Barlet space leave only finitely many possible
component pairs for these summands. Addition of cycles is holomorphic,
and its images from these compact pairs are analytic by proper mapping.
Thus the decomposable locus in \(P\) is a closed analytic subset.
It is proper whenever \(P\) contains an irreducible reduced cycle;
components containing no such cycle can be omitted.

Resolve \(P\) and restrict to the locus where this resolution is an
isomorphism. Pull back the reduced universal incidence, take the
closure of its component dominating the irreducible-cycle locus,
and resolve that compact incidence. Delete the critical parameter
values, the
parameters whose whole support fibre lies in the exceptional image,
and those for which a component of the full non-isomorphism locus
has full fibre dimension.
These are proper analytic subsets by proper generic smoothness and
the proper fibre-dimension theorem. Over the remaining open, each
resolved fibre is smooth, connected, and bimeromorphic to its cycle.
Repeat on the irreducible components of the omitted parameter subsets,
including the image omitted in resolving \(P\). Each repetition lowers
the parameter dimension, and a compact analytic subset has finitely
many irreducible components. The countably many initial components
therefore give the claimed countable list. This construction requires
no K\"ahler metric on \(W_j\) or \(T_j\); their role is properness
and fundamental-group transport.

Each nondominant evaluation image \(e_j(W_j)\) is a proper analytic
subset of the cover. On the smooth Albanese locus let \(s\) be the
fibre dimension. An evaluation image dominating the torus has
general fibre dimension at most \(s-1\); the locus where this
dimension is at least \(s\) is proper analytic by the proper
fibre-dimension theorem. An image not dominating the torus already
projects to a proper analytic subset. Exclude these loci and the
singular-fibre locus. A smooth Albanese fibre contained in an evaluation image
has been excluded. Make these exclusions for every family and finite
cover, and project them to \(A\) through the finite isogenies.
The resulting union is countable. Call a point outside it \emph{good},
and use the same term for any of its lifts to a later Albanese torus.

If the torus has dimension zero, its fibre is the whole cover,
which no proper evaluation image contains; no exclusion is needed
for these images. Thus this convention includes irregularity zero.
For any fibre above a good point, every nondominant evaluation
image meets that fibre in a proper analytic subset. A countable
union of such subsets cannot cover an open part of that fibre.
This will allow us to choose a fibration cycle in a family
dominating the total space.

\subsection{The minimal relative base}

Consider all finite covers just described and their good Albanese
fibres \(S\). For each connected meromorphic fibration
\(S\dashrightarrow T\), let \(C\) be the image in the corresponding
Albanese fibre group \(N\) of a resolved general fibration fibre.
Then \(C\triangleleft N\). Among these fibrations for which
\(N/C\) is infinite, minimize \(\dim T\), and denote the minimum
by \(\ell\). The identity fibration is admissible because \(N\)
is infinite; the fibration to a point is not. Hence
\[
 1\le\ell\le\dim S.
\]
All covers used below remain within this simultaneous minimum.

\begin{proposition}\label{prop:minimal-base}
After a further connected finite \'etale cover, there is a
fibration \(c:X\dashrightarrow Z\) over the Albanese torus
\(A\), with connected general fibres and relative base dimension
\(\ell\). If \(B_0\) is the actual image of a general
\(c\)-fibre group, then \(B_0\subseteq N\), \(B_0\triangleleft G\),
and \(Q=N/B_0\) is infinite.

For each prepared smooth model of \(Z\), a very general good point
of \(A\) gives an Albanese fibre \(S\) and a torus fibre \(Y\).
With the restricted actual inertial boundary \(\Delta_Y\) for
\(R=G/B_0\), there is a surjection
\[
 \pi_1^\orb(Y,\Delta_Y)\twoheadrightarrow Q
\]
compatible with \(S\dashrightarrow Y\) and the natural
\(\nu:\pi_1(S)\twoheadrightarrow Q\). For every finite-index
subgroup \(Q_0\le Q\), there is a connected finite \'etale cover
\(S'\to S\) with induced homomorphism \(\nu'\) whose image
\(Q'=\im\nu'\) lies in \(Q_0\).
This one cover satisfies
\[
 \operatorname{gdim}(S',\psi\circ\nu')\ge\ell
\]
for every surjection \(\psi:Q'\twoheadrightarrow P\) with \(P\) infinite.
\end{proposition}
\begin{proof}
Choose a cover, a good fibre \(S\), and a fibration attaining
\(\ell\). Write \(C\triangleleft N\) for its fibre-group image.
The general fibration fibres give compact irreducible cycles of
dimension \(\dim S-\ell\), after taking their images in \(X\).
Choose one through a point of the fibration's general locus in
\(S\) outside all the nondominant evaluation images prepared
above. Choose a family from the list, with \(p:W\to T\) and
\(e:W\to X\), containing this member. Its compact evaluation must dominate \(X\),
and hence is surjective.

The homotopy sequence of the smooth proper family
\(W^\circ=p^{-1}(T^\circ)\to T^\circ\) makes its fibre-group image
normal in \(\pi_1(W^\circ)\). The selected fibre is bimeromorphic
to the selected cycle, so its image under \(e_*\) is \(C\).
Consequently \(H=e_*\pi_1(W^\circ)\) normalizes \(C\).
The complement of \(W^\circ\) in the smooth \(W\) is analytic,
so \(\pi_1(W^\circ)\to\pi_1(W)\) is surjective and
\(H=e_*\pi_1(W)\).

Put \(\Lambda=\pi_1(A)\) and
\(\Lambda_0=(\alpha e)_*\pi_1(W)\). Lift the surjective holomorphic
map \(\alpha e:W\to A\) to the connected complex covering
\(A_{\Lambda_0}\to A\). Its lifted image is compact analytic by
proper mapping. Surjectivity downstairs forces this image to have
dimension \(\dim A\), so it is all of the connected manifold
\(A_{\Lambda_0}\). The covering is therefore compact and has finite
degree. Thus \(\alpha_*(H)=\Lambda_0\) has finite index in \(\Lambda\).

Let \(L=N_G(C)\). It contains \(N\) and \(H\), so Albanese exactness
gives
\[
 [G:L]=[\Lambda:\alpha_*(L)]
 \le[\Lambda:\alpha_*(H)]<\infty.
\]
Pass to the cover corresponding to \(L\). Its fibre-group image is
still \(N\), by \eqref{eq:fibre-intersection}, and \(C\) is now normal
in the total group. Continue to write \(X,G,A,\Lambda\) for this
cover, its group, its Albanese torus and its lattice. The whole
compact evaluation \(e:W\to X\) lifts,
since \(e_*\pi_1(W)=H\subseteq L\). Its lift has full rank somewhere,
and its compact analytic image is therefore the entire connected cover.
The proper fibre-dimension theorem, applied to
\(W\setminus W^\circ\subsetneq W\), shows that a general evaluation
fibre meets \(W^\circ\). Hence members with parameters in \(T^\circ\)
cover general total points. Smooth proper transport gives their fibre-group image
\(C\) after compatible base-point transport.

Apply Theorem~\ref{thm:gamma-reduction} to \(G\to G/C\), giving
\(c:X\dashrightarrow Z\). The covering cycles just constructed
have dimension \(\dim S-\ell\) and are contracted through very
general points. Thus
\[
 \dim Z\le\dim X-(\dim S-\ell)=\dim A+\ell.
\]
This map is over \(A\): a compact
subvariety with finite image in the quotient lattice
\(G/N\) maps constantly to \(A\). To verify the last assertion,
resolve the subvariety. A finite image in the torsion-free lattice
is trivial, so its map lifts to the universal cover of \(A\),
a complex vector space; compactness makes that lift constant.
Consequently the Albanese map factors through \(c\), after
resolving. The induced map \(g:Z\to A\) has connected general
fibres: a nontrivial finite Stein factor of \(g\) would split a
general fibre of the connected Albanese map.

Let \(B_0\) be the actual group image of a general \(c\)-fibre.
It is normal in \(G\), lies in \(N\), and has finite image
\(B_0C/C\) in \(G/C\). If \(N/B_0\) were finite, the image of
\(N\) in \(G/C\) would be a finite union of translates of this
finite image, contradicting \(N/C\) infinite. Thus \(N/B_0\)
is infinite. For the dimension comparison, restrict \(c\) to a
very general good Albanese fibre. On simultaneous smooth loci,
its fibres are the same general \(c\)-fibres, so they are connected
and have image \(B_0\) in \(N\). This restricted fibration is
admissible in the minimum, giving \(\dim Z-\dim A\ge\ell\).
Equality follows. No equality or inclusion between \(B_0\) and
\(C\) has been used.

For inertia use \(R=G/B_0\), so
\[
 1\longrightarrow Q=N/B_0\longrightarrow R=G/B_0
 \longrightarrow\Lambda\longrightarrow1.
\]
Fix any prepared smooth model of \(Z\), with actual inertial
boundary \(D_Z\) for \(R\), and write \(\mu:\widetilde X\to X\)
for its resolved source, with \(c:\widetilde X\to Z\) holomorphic.
Choose afresh a very general good point
\(a\in A\) so that
\[
 S=\alpha^{-1}(a),\qquad
 \widetilde S=(\alpha\mu)^{-1}(a),\qquad Y=g^{-1}(a)
\]
are smooth, \(\widetilde S\to S\) is a modification, and \(Y\)
is transverse to every dominating stratum of the deleted divisor
and avoids the images of the other strata. This choice may depend
on the prepared model. The induced \(S\dashrightarrow Y\) is
meromorphic, and \(\dim Y=\ell\).

Let \(Z^\circ\) be the prepared smooth fibration locus for \(c\), put
\(Y^\circ=Y\cap Z^\circ\), and put
\(\widetilde S^\circ=c^{-1}(Y^\circ)\). The monodromy
\(\psi_Z:\pi_1(Z^\circ)\twoheadrightarrow R\) has composite
\(g_*\) in \(\Lambda\), by compatibility after precomposing with
the surjection from the open source group. Its restriction to
\(\pi_1(Y^\circ)\) therefore lands in \(Q\), giving the
commutative diagram
\[
\begin{array}{ccc}
 \pi_1(\widetilde S^\circ)&\longrightarrow&\pi_1(Y^\circ)\\
 \big\downarrow&&\big\downarrow\psi_Y\\
 \pi_1(S)&\xrightarrow{\nu}&N/B_0\ \subset\ G/B_0 .
\end{array}
\]
The left map is surjective, by removal of an analytic subset and
bimeromorphic invariance of the smooth fundamental group.
The map \(\nu\) is surjective by the definition of \(N\).
Thus \(\psi_Y\) is surjective. The upper map is the map of a
smooth proper fibration with connected fibres, so this is the
monodromy compatible with \(S\dashrightarrow Y\).

Transversality gives \(\Delta_Y=D_Z|_Y\) with simple normal
crossing support. For a deleted prime divisor \(D\), if \(E\) is a
component of \(D\cap Y\), its
meridian \(\mu_E\) maps to a conjugate of the meridian \(\mu_D\)
in \(Z^\circ\). The injection \(Q\subset R\) therefore gives
\[
 \operatorname{ord}_Q\bigl(\psi_Y(\mu_E)\bigr)
 =\operatorname{ord}_R\bigl(\psi_Z(\mu_D)\bigr)=m_D.
\]
The proper fibre-dimension theorem ensures that the deleted subsets
of codimension at least two remain of codimension at least two on
this very general restriction, and
filling them does not change the fundamental group. Filling
divisors of order one kills already trivial meridians. The
\(m_D\)-th powers of the remaining meridians are also killed,
so \(\psi_Y\) factors as a surjection
\[
 \pi_1^\orb(Y,\Delta_Y)\twoheadrightarrow Q.
\]

Finally fix a finite-index subgroup \(Q_0\le Q\), and let \(N_0\)
be its inverse image in \(N\). Lemma~\ref{lem:realize-finite-index}
gives a further global finite cover with group \(G'\) such that
\[
 N'=N\cap G'\subseteq N_0,\qquad
 Q'=\im(N'\to Q)\cong N'/(N'\cap B_0)\subseteq Q_0.
\]
The image \(Q'\) has finite index in \(Q\). Choose a point of the
new Albanese torus above the current \(a\). Its fibre \(S'\) is a
connected component of the pullback of \(S\), because the new
Albanese fibres are connected and the map of tori is finite.
Thus \(S'\to S\) is a connected finite \'etale cover, and its
induced homomorphism \(\nu'\) has image \(Q'\). The good-point
convention makes \(S'\) eligible for the same minimum.

Choose this cover before choosing a further quotient. For every
surjection \(\psi:Q'\twoheadrightarrow P\) with \(P\) infinite,
the \(\Gamma\)-reduction of \(\psi\circ\nu'\) on \(S'\) has
resolved general fibre image \(C'\triangleleft N'\) with finite
image in \(P\). If \(N'/C'\) were finite, the whole image in
\(P\) would be a finite union of translates of that finite image,
which is impossible. The reduction is therefore an admissible
fibration in the minimum, and its dimension is at least \(\ell\).
This one cover works for every infinite quotient of \(Q'\), as
required by \eqref{hi:minimality}. The prepared model was arbitrary;
repeating the choice of a very general good \(S\) on each model
proves the assertion on every prepared model.
\end{proof}

\subsection{The two representation cases}

Apply Proposition~\ref{prop:minimal-base}. The quotient \(Q\)
is FAb by Proposition~\ref{prop:fibre-fab}. Suppose first that
all finite-dimensional complex linear images of \(Q\) are
finite. On each prepared model of \(Z\), choose the very general
good fibre and the data in Proposition~\ref{prop:minimal-base}.
They satisfy the hypotheses of
Theorem~\ref{thm:hilbert-positivity}. Hence
\[
 K_Y+\Delta_Y\quad\text{is big}
\]
on the very general torus fibre, of positive dimension \(\ell\).
The same minimum applies after the fresh fibre choice on each
prepared model. Theorem~\ref{thm:torus-descent} contradicts the
specialness of \(X\).

Suppose instead that \(Q\) has an infinite complex linear image.
An infinite virtually solvable image is impossible for an FAb
group, as observed in Section~\ref{sec:preliminaries}. Pass to
finite index so that the Zariski closure \(\mathbf L\) is connected.
Its semisimple quotient is nontrivial. Choose a simple factor
\(\mathbf H\) of the adjoint semisimple quotient
\[
 \bigl(\mathbf L/\operatorname{Rad}(\mathbf L)\bigr)_{\mathrm{ad}},
\]
where \(\operatorname{Rad}(\mathbf L)\) is the solvable radical.
This quotient is a product of connected simple adjoint groups. Projection
gives a Zariski-dense image in \(\mathbf H\). In its faithful
adjoint representation this image is finitely generated and linear,
so Selberg's lemma \cite{Selberg1960} gives a finite-index subgroup
with torsion-free image. Pull this condition back to \(N\) and
realize it on a global finite cover by
Lemma~\ref{lem:realize-finite-index}. Write \(X',G',N',A'\) for
the resulting cover and its groups and Albanese torus. The induced
\(\rho_N:N'\to\mathbf H(\C)\) has torsion-free image. That image
is still Zariski dense, since it has finite index in a dense subgroup
of the connected group \(\mathbf H\).

Take the Stein factorization \(c':X'\dashrightarrow Z'\) of the
old map \(c\) after this cover. A connected \(c'\)-fibre maps under
the new Albanese map into a finite fibre of \(A'\to A\), and hence
to one point. Thus \(c'\) lies over a map \(g':Z'\to A'\);
connected Albanese fibres again make \(g'\) a fibration.
Use a smooth model for \(Z'\). Its relative dimension is still \(\ell\):
the new Stein base is finite over the old base on the general locus,
and \(A'\to A\) is an isogeny. Let \(B_0'\triangleleft G'\) be the
actual image of a general \(c'\)-fibre group. It lies in \(N'\).
A general \(c'\)-fibre is a connected component of a finite cover
of an old \(c\)-fibre, so its image in \(G\) lies in \(B_0\).
The representation \(\rho_N\), defined through the old \(Q=N/B_0\),
therefore kills \(B_0'\). Set
\[
 R'=G'/B_0',\qquad
 \overline\rho:N'/B_0'\longrightarrow\mathbf H(\C).
\]
Albanese exactness gives
\[
 1\longrightarrow N'/B_0'\longrightarrow R'
 \longrightarrow\pi_1(A')\longrightarrow1.
\]

On each prepared model of \(Z'\), use the actual inertial boundary
for \(R'\) and choose a fresh very general good Albanese fibre \(S'\)
and corresponding torus fibre \(Y\). Denote the restricted boundary
by \(\Delta_Y\), and let \(Y^\circ\) be the restricted smooth
fibration locus. The monodromy argument in
Proposition~\ref{prop:minimal-base}, applied to this restricted
fibration, gives
\[
 \psi'_Y:\pi_1(Y^\circ)\twoheadrightarrow N'/B_0'\subset R'.
\]
Compose it with \(\overline\rho\). A meridian has finite order in
\(R'\); its image under \(\overline\rho\circ\psi'_Y\) has order
dividing that actual inertial order. Since the representation image
is torsion free, every meridian is killed. Filling the deleted locus
therefore gives a representation
\(\rho_Y:\pi_1(Y)\to\mathbf H(\C)\) of the ordinary fundamental
group, with the same torsion-free Zariski-dense image.

This representation is generically large. Otherwise its
\(\Gamma\)-reduction has base dimension less than
\(\dim Y=\ell\). Compose it with \(S'\dashrightarrow Y\) and take
connected fibres by Stein factorization. A resolved general fibre
maps to a resolved \(\Gamma\)-fibre, so its group image
\(C'\triangleleft N'\) has finite image under \(\rho_N\).
The total image is infinite; if \(N'/C'\) were finite, the total
representation image would be a finite union of translates of that
finite fibre image.
Thus \(N'/C'\) is infinite, and this composed fibration would be
admissible in the minimum with base dimension less than \(\ell\),
a contradiction.

By Theorem~\ref{thm:linear-general-type}, \(K_Y\) is big, and hence
so is \(K_Y+\Delta_Y\). The argument applies after the fresh good
fibre choice on every prepared model, using the same minimum and
the boundary defined by the fixed actual quotient \(R'\) on each model.
Theorem~\ref{thm:torus-descent} yields the same contradiction.

Both cases exclude infinite \(N\). Thus \(N\) is finite and
the exact sequence of Lemma~\ref{lem:albanese-exactness}, followed
by Lemma~\ref{lem:finite-by-abelian}, makes \(G\) virtually
abelian. The group of the finite cover used at the start has
finite index in the original fundamental group, so the same
conclusion holds for the original \(X\). A finite-index
subgroup corresponds to a connected finite topological cover;
the complex structure lifts uniquely, making it an unramified
holomorphic cover. Base-point changes only conjugate the
subgroups. This completes the proof of
Theorem~\ref{thm:abelianity}.\qed
==== END The-abelianity-conjecture-for-special-compact-Kahler-manifolds//sections/completion.tex ====

==== BEGIN The-abelianity-conjecture-for-special-compact-Kahler-manifolds//sections/descent.tex ====
\section{Inertia and descent over a torus}\label{sec:descent}

We attach boundary coefficients to the orders of meridians in a fixed
group quotient. These orders also control components whose images are
exceptional for an intermediate map. Keeping them on every model will
allow us to descend positivity from the fibres over a torus.

\subsection{Meridian orders and differential pullback}

Let \(c:M\dashrightarrow Z\) be a fibration from a smooth compact
K\"ahler manifold, and let
\(\rho:\pi_1(M)\twoheadrightarrow R\) kill the image of the group
of a smooth general \(c\)-fibre. Replace the source and target by
smooth models so that \(c\) is holomorphic, and choose an analytic
Zariski open \(Z^\circ\) over which it is smooth. The homotopy
sequence of the smooth fibration gives a homomorphism
\[
 \psi:\pi_1(Z^\circ)\longrightarrow R
\]
compatible with \(\rho\). It is surjective: the open source maps
surjectively on the fundamental group of the compact source. The
homomorphism is independent of a further shrinking, in this
compatibility sense.

For a prime divisor \(D\) on the smooth base, let \(m_D\) be the
order in \(R\) of a meridian about its general point. Meridians are
defined up to conjugacy, which does not affect their orders. Set
\(m_D=1\) for divisors outside the deleted locus. Every \(m_D\)
is finite. Indeed, if a component \(E\) of \(c^*D\) dominates
\(D\) with multiplicity \(e>0\), a small transverse disc at a
general point of \(E\) has boundary mapping to a conjugate of the
\(e\)-th power of the meridian. This boundary contracts in the
compact source, so its image under \(\rho\) is trivial.

\begin{definition}\label{de:inertial-boundary}
The \emph{inertial boundary} associated with \(\rho\) on this model
of \(Z\) is
\[
 \Delta_Z=\sum_D(1-m_D^{-1})D.
\]
A \emph{prepared model} is a smooth compact K\"ahler base model,
together with a resolved source, on which the divisorial deleted locus
has simple normal crossings. If a map to a fixed torus is present,
the model is required to map holomorphically to that torus.
\end{definition}

There are only finitely many nonzero coefficients on a fixed model.
The pair \((Z,\Delta_Z)\) on a prepared model is klt. Preparation
uses resolution of the graph and discriminant; further smooth
modifications can again be prepared. The orders on all these models
refer to the same quotient \(R\). In particular, the strict transform
of a divisor has the same order. No invariance of
\(\kappa(Z,K_Z+\Delta_Z)\) under modification is assumed.
A collection of prepared models is \emph{cofinal} if every prepared
model is dominated by a member through a holomorphic modification.

\begin{lemma}[Inertia over a smooth intermediate map]
\label{de:inertia-divisibility}
Let \(f:X\to Z\) be a dominant holomorphic map of connected smooth
complex manifolds. Suppose that a homomorphism
\(\rho:\pi_1(X)\to R\) is compatible, over a dense analytic
Zariski open \(Z^\circ\), with a homomorphism
\(\psi:\pi_1(Z^\circ)\to R\). Let \(p:Z\to W\) be a smooth
proper surjective map with connected fibres, with \(W\) smooth.
Assume that the nontrivial divisorial meridian orders of \(\psi\)
give the boundary
\[
 \Delta_Z=p^*\Delta_W,
 \qquad
 \Delta_W=\sum_D(1-m_D^{-1})D,
 \qquad m_D\in\Z_{\ge2}.
\]
Thus every other divisorial meridian has trivial image. If a prime
divisor \(E\subset X\) dominates \(D\subset W\) under
\(p\circ f\), then
\begin{equation}\label{de:all-component-divisibility}
 m_D\mid \operatorname{ord}_E\bigl((p\circ f)^*D\bigr).
\end{equation}
This includes the case in which \(f(E)\) is a proper subvariety of
\(p^{-1}(D)\).
\end{lemma}

\begin{proof}
Take a general point \(w\in D\), avoiding its singularities and the
other components of \(\Delta_W\). The divisor \(p^{-1}(D)\) is
smooth near every point over \(w\). Fix such a point \(z\), and a
small coordinate ball \(U\) around it. In this ball the only
nontrivial inertial support is the smooth hypersurface
\(p^{-1}(D)\cap U\).

The map from the fundamental group of the original deleted-locus
complement in \(U\) to
\(\pi_1(U\setminus p^{-1}(D))\) is surjective. Its kernel is
normally generated by meridians around the other deleted divisors:
a transverse perturbation of a null homotopy gives precisely these
meridians. Their \(R\)-images are trivial. Removing the remaining
analytic subsets of complex codimension at least two does not change
this fundamental group, by the same transverse perturbation applied
to loops and homotopies. Consequently \(\psi\) factors locally
through
\[
 \pi_1(U\setminus p^{-1}(D))\cong\Z.
\]

At a general point of \(E\) over \(w\), choose a transverse disc
whose punctured part avoids all deleted inverse images. Its image
winds
\(e=\operatorname{ord}_E((p\circ f)^*D)\) times around the one
remaining hypersurface. Its boundary contracts in \(X\), so
compatibility with \(\rho\) gives \(\psi(\mu)^e=1\). The
order of \(\psi(\mu)\) is \(m_D\), proving
\eqref{de:all-component-divisibility}.
\end{proof}

The next consequence will also be used directly for the Albanese map.
It handles the source divisors whose images have codimension at least
two by decreasing the boundary on a fixed torus.

\begin{lemma}[A small ample boundary on a torus]
\label{de:small-boundary}
Let \(u:X\to B\) be a surjective holomorphic map from a smooth
compact K\"ahler manifold to a positive-dimensional complex torus.
Let
\(\Delta=\sum_D(1-m_D^{-1})D\) be an ample rational divisor on
\(B\), with integers \(m_D\ge2\). Suppose that
\[
 m_D\mid\operatorname{ord}_E(u^*D)
\]
for every prime divisor \(E\) dominating \(D\) under \(u\).
Then \(X\) is not special.
\end{lemma}

\begin{proof}
Put \(r=\dim B\), and let \(\cL\subset\Omega_X^r\) be the
saturated differential line of \(u\). For a local nonvanishing
generator \(\eta\) of \(K_B\), write \(a_E\) for the common
divisorial vanishing order of \(u^*\eta\) along \(E\), measured
in \(\cL\), and put \(v_E=\operatorname{ord}_E(u^*\Delta)\).
If \(E\) dominates a component \(D\), with multiplicity \(e\),
the local differential of a defining function of \(D\) gives
\[
 a_E\ge e-1\ge e(1-m_D^{-1})=v_E.
\]
If \(u(E)\) has codimension at least two, the tangent rank along
\(E\) is at most \(r-2\), and the normal direction increases
it by at most one. Thus every maximal minor of \(\dd u\)
vanishes along \(E\), and \(a_E\ge1\).

The pullback of the fixed divisor \(\Delta\) has finitely many
prime components on \(X\). We may therefore choose a rational
\(\epsilon>0\) such that \(\epsilon\le1\) and
\[
 \epsilon v_E\le a_E
 \quad\text{for every prime divisor }E\subset X.
\]
Only components with \(v_E>0\) impose a condition. This choice also
handles components whose images lie in intersections of boundary
divisors, since their \(v_E\) includes the sum of all pullback
coefficients.

The torus has trivial canonical bundle. For divisible \(m\), pullback
of a section of \(m\epsilon\Delta\), multiplied by the \(m\)-th
power of its top differential, is thus a holomorphic section of
\(\cL^{\otimes m}\). The inequality above checks every divisorial
pole; extension across codimension two finishes the check. The ratios
of these sections retain the \(r\) parameters of the ample linear
systems on \(B\). Hence \(\kappa(X,\cL)\ge r\), and
Theorem~\ref{thm:specialness-inputs} gives equality. This contradicts
specialness.
\end{proof}

\begin{remark}\label{de:fixed-map-vanishing}
The vanishing just used concerns the top differential of the fixed map
\(u:X\to B\). Saturation records its common vanishing in the map
\(u^*K_B\to\cL\). It does not assert that arbitrary lower-degree
forms vanish on exceptional divisors of a modification; that distinction
is relevant to \cite[Remark~2.2]{CampanaCorrection}.
\end{remark}

We will use the corresponding pullback statement on a neat model. A
prime divisor is \(u\)-exceptional when its image has codimension at
least two. The needed preparation is a smooth fibration model
\(u:\widehat X\to W\), with a modification
\(\nu:\widehat X\to X\) to a smooth source model, such that every
\(u\)-exceptional divisor is \(\nu\)-exceptional. Such models
exist after modifying the base; see
\cite[Lemma~1.4]{JWang2021}, whose base and source modifications
are projective morphisms. A projective initial base therefore remains
projective.
Concretely, flatten the strict transform over a modification of the
base and then resolve it. The flat strict transform has no divisors
mapping into codimension at least two. Any such divisor introduced by
the final resolution is exceptional over the original source.

\begin{lemma}[Pullback on a neat model]\label{de:neat-pullback}
Let \(u:\widehat X\to W\) and \(\nu:\widehat X\to X\)
be as above, with \(X\) smooth compact K\"ahler and \(W\) smooth
of dimension \(r>0\). Let
\(\Delta_W=\sum_D(1-m_D^{-1})D\), with \(m_D\ge2\).
Suppose that \(K_W+\Delta_W\) is big and that \(m_D\) divides
\(\operatorname{ord}_E(u^*D)\) for every component \(E\)
dominating \(D\). Then \(X\) is not special.
\end{lemma}

\begin{proof}
Let \(\cL\subset\Omega_X^r\) be the saturated differential
line of the meromorphic map induced by \(u\). A section of a
divisible multiple of \(K_W+\Delta_W\) pulls back as a meromorphic
section of \(\cL^{\otimes m}\). Along a component of multiplicity
\(e\) over \(D\), its allowed pole is cancelled because
\begin{equation}\label{de:differential-cancellation}
 e-1\ge e(1-m_D^{-1}).
\end{equation}
Divisors dominating \(W\) produce no boundary pole. The remaining
source divisors are \(u\)-exceptional and hence disappear in
codimension at least two on \(X\). Every prime divisor of \(X\)
therefore satisfies the holomorphicity condition. Extension across
codimension two gives
\[
 H^0\bigl(W,m(K_W+\Delta_W)\bigr)
 \hookrightarrow H^0\bigl(X,\cL^{\otimes m}\bigr).
\]
Pullback preserves the ratios of sections because the map is dominant.
Bigness gives \(\kappa(X,\cL)\ge r\), and
Theorem~\ref{thm:specialness-inputs} again gives the forbidden equality.
\end{proof}

\subsection{Two comparisons of models}

\begin{lemma}\label{de:birational-pushdown}
Let \(\mu:Z'\to Z\) be a modification between smooth base models
of the same fibration and quotient. For sufficiently divisible \(m\),
there is a natural injection
\[
 H^0\bigl(Z',m(K_{Z'}+\Delta_{Z'})\bigr)
 \hookrightarrow H^0\bigl(Z,m(K_Z+\Delta_Z)\bigr).
\]
\end{lemma}

\begin{proof}
View a section upstairs as a meromorphic pluricanonical form in the
common function field. Every prime divisor of \(Z\) has a strict
transform with the same meridian order. Its divisorial pole bound is
therefore the required bound on \(Z\). Extension across codimension
two gives the section downstairs, and the function-field identification
makes this map injective.
\end{proof}

Now suppose \(c:M\dashrightarrow Z\) lies over a torus \(A\),
and let \(B_c\) be the image of its general fibre group. Use
\(R=\pi_1(M)/B_c\). An isogeny \(A'\to A\) induces a subgroup
\(G'\le G=\pi_1(M)\) containing the kernel of
\(G\to\pi_1(A)\), hence containing \(B_c\). General
\(c\)-fibres lift unchanged to the source cover. The induced
quotient is consequently
\begin{equation}\label{de:quotient-inclusion}
 R'=G'/B_c\ \hookrightarrow\ G/B_c=R.
\end{equation}
On corresponding pulled-back base models, meridians lift with winding
number one. Their lattice image is zero, so their images lie in
\(R'\), and \eqref{de:quotient-inclusion} preserves their orders.
Thus both the canonical divisor and the true inertial boundary pull
back under this finite unramified base change.

\begin{lemma}[Restriction to common general fibres]
\label{de:common-fibres}
Let \(g:Z\to A\) and \(h:Z\to W\) be surjective holomorphic
maps with connected general fibres between compact complex manifolds,
with \(Z\) smooth. Let \(L\)
be a rational line bundle whose restriction to a very general smooth
\(g\)-fibre is big. Then \(L\) restricts big to every component
of a general fibre of the map
\[
 (h,g):Z\longrightarrow J\subseteq W\times A,
\]
where \(J\) denotes its image. Equivalently, one may first take a
very general \(h\)-fibre \(H\), and then a general fibre of
\(g|_H\) over its image. No equality \(g(H)=A\) is required.
\end{lemma}

\begin{proof}
For each divisible \(m\), properness gives the coherent direct image
\(g_*\cO_Z(mL)\). Choose a very general smooth fibre where base
change holds for all these countably many sheaves. Bigness on that
fibre supplies one \(m\) for which its section map has full generic
rank. On the open set where the direct image is locally free and
base change holds, evaluation defines the relative meromorphic map
to its projective bundle. The locus where its vertical differential
has full rank contains a dense analytic Zariski open \(U\subset Z\).
This condition is intrinsic to the relative evaluation and jet maps,
so it is defined over the base open, independently of a local frame
of the direct image.

A general fibre component of \(Z\to J\) meets \(U\). Otherwise
the proper analytic subset \(Z\setminus U\) would contain a
general component of a family of fibres covering \(Z\), contrary
to the fibre-dimension theorem. On such a component the relative
section map still has injective differential at a general point:
its tangent space is contained in the vertical tangent space of
\(g\). The restrictions of these sections therefore define a
generically finite meromorphic map, proving bigness.
General choices in \(J\), followed by general choices in its
projection to \(W\), give the last formulation. All assertions can
be restricted to the smooth loci supplied by generic smoothness.
\end{proof}

\Needspace{22\baselineskip}
\begin{samepage}
\subsection{Descent of positivity}

\begin{theorem}[Descent over a torus]\label{thm:torus-descent}
Let \(M\) be a special smooth compact K\"ahler manifold, and let
\(\alpha:M\to A\) be a surjective map to a complex torus with
connected fibres. Suppose that
\[
 M\dashrightarrow Z\overset{g}{\longrightarrow}A
\]
is a factorization by fibrations, and that
\(d=\dim Z-\dim A>0\). On every prepared model of \(Z\),
take the inertial boundary for
\[
 R=\pi_1(M)/\im\bigl(\pi_1(F_c)\to\pi_1(M)\bigr),
\]
where \(F_c\) is a smooth general fibre of \(M\dashrightarrow Z\).
It is impossible that, on a cofinal collection of prepared models,
\(K_Y+\Delta_Z|_Y\) is big for a very general \(g\)-fibre \(Y\).
The torus \(A\) is allowed to be a point.
\end{theorem}
\end{samepage}

\begin{proof}
Assume the stated bigness on the cofinal collection. We may further
prepare a model whenever necessary. By
Theorem~\ref{thm:wang-torus}, every model in this collection satisfies
\begin{equation}\label{de:minimum}
 \kappa(Z,K_Z+\Delta_Z)\ge d.
\end{equation}
Choose a model \(Z_0\) for which this integer is minimal, and put
\(b=\kappa(Z_0,K_{Z_0}+\Delta_{Z_0})\). Then \(b\ge d>0\).

\medskip\noindent
\emph{The Iitaka fibres.}
Resolve the Iitaka map on a modification \(\mu:Z_1\to Z_0\),
obtaining a fibration \(h:Z_1\to W\) with \(W\) smooth
projective of dimension \(b\). For this step introduce an auxiliary
klt boundary \(\Delta_1\). It has the strict-transform coefficients
of \(\Delta_{Z_0}\) and exceptional coefficients sufficiently close
to one to dominate the true inertial coefficients on \(Z_1\).
They may also be chosen so that
\begin{equation}\label{de:auxiliary-boundary}
 K_{Z_1}+\Delta_1
 =\mu^*(K_{Z_0}+\Delta_{Z_0})+E,
 \qquad E\ge0\text{ is }\mu\text{-exceptional}.
\end{equation}
Indeed all discrepancies of the log-smooth klt pair are greater than
\(-1\); an exceptional coefficient less than one can be chosen above
the negatives of these discrepancies and above the corresponding
inertial coefficient. A simultaneous log resolution makes all the
supports simple normal crossing.

Equation~\eqref{de:auxiliary-boundary} preserves sections in divisible
degrees, hence the Iitaka dimension and map. A very general smooth
\(h\)-fibre \(H\) consequently satisfies
\begin{equation}\label{de:iitaka-fibre}
 \kappa\bigl(H,K_H+\Delta_1|_H\bigr)=0.
\end{equation}
Here we use the ordinary Iitaka-fibre property, as in
\cite[Section~6.3]{JWang2021}. It can also be seen directly in this
setting. Write \(L=K_{Z_1}+\Delta_1\). By Kodaira's lemma on the projective
Iitaka base, after taking a further multiple and absorbing the
pullback of the effective remainder, the resolved system gives \(aL\sim h^*B+E\) for some divisible
\(a>0\), with \(B\) ample on \(W\) and \(E\ge0\). If a multiple \(jL|_H\) had a
positive-dimensional section map on a very general fibre, coherent
direct image and base change, followed by a sufficiently large ample
twist \(h^*(kB)\), would extend sections whose ratios vary on that
fibre. Multiplication by the canonical section of \(kE\) turns
them into sections of \((j+ka)L\). This fibrewise variation,
together with the \(b\) base parameters supplied by \(B\),
contradicts \(\kappa(Z_1,L)=b\). The restriction has a nonzero
section because a general fibre is not contained in \(E\), so its
Iitaka dimension is zero. Adjunction identifies \(L|_H\) with the
divisor appearing in \eqref{de:iitaka-fibre}.

We can arrange that a very general fibre of \(g:Z_1\to A\) has big
restriction of \(K_{Z_1}+\Delta_1\). For example, dominate
\(Z_1\) by a prepared member of the cofinal collection, perform
the same auxiliary-boundary construction relative to \(Z_0\), and
replace \(Z_1\) by this model. Its map to \(W\) is already
holomorphic. The auxiliary boundary dominates its true boundary,
whose fibre adjoint is big by assumption. Any further log resolution
preserves this fibre bigness by pullback and an effective exceptional
remainder, and \eqref{de:auxiliary-boundary} still holds.
Lemma~\ref{de:common-fibres} now makes the log canonical divisor big
on general components of common fibres of \(h\) and \(g\).
Generic smoothness, also for the finitely many boundary strata,
justifies the adjunction restrictions to these common fibres.

Apply Theorem~\ref{thm:wang-torus} to the klt pair in
\eqref{de:iitaka-fibre}. Its Albanese map is surjective with connected
fibres. The map \(g|_H\) factors through a homomorphism from
\(\Alb(H)\) to \(A\), followed by a translation. Quotienting by
the connected kernel shows that its Stein base is a torus finite
\'etale over its image subtorus translate. Torus subadditivity
on \(H\), using the bigness of the common fibres, forces their
dimension to be zero. The Albanese map of \(H\) is therefore
birational, and \(g|_H\) is, birationally, an isogeny onto a
subtorus translate.

Every component of \(\Delta_1|_H\) is exceptional for this
birational Albanese map. To prove this, suppose that a component maps
onto a divisor \(D\) of the torus. The relative canonical divisor
of the birational map to the smooth torus is effective, with positive
coefficients on all exceptional primes. For sufficiently small
rational \(\epsilon>0\), it absorbs the exceptional components of
the pullback of \(\epsilon D\), while the chosen boundary
component supplies its strict transform. Thus
\(K_H+\Delta_1|_H\) contains that pullback up to rational linear
equivalence. By Lemma~\ref{lem:torus-divisor}, a nonzero effective
divisor on a complex torus has positive Iitaka dimension. This contradicts
\eqref{de:iitaka-fibre}.

\medskip\noindent
\emph{A torus-bundle model.}
On the connected smooth parameter open of \(h\), the image of
\(H_1(H,\Z)\) in \(H_1(A,\Z)\) is locally constant. Transport
in the proper smooth family and the total map to \(A\) identify
these images. Their real span determines a fixed subtorus
\(T\subset A\), so the image of a general \(H\) is a translate
of \(T\). Write \(\Lambda_T\) for its lattice and \(L_H\)
for the finite-index image lattice. Choose \(k>0\) with
\(k\Lambda_T\subseteq L_H\), and use the isogeny
\(A_1\to A\) given by multiplication by \(k\).

Let \(T_1\) be the identity component of the inverse image of
\(T\). The lattice of a subtorus is primitive in the ambient
lattice, so the image of the lattice of \(T_1\) is
\(k\Lambda_T\). The choice of \(k\) makes each connected lifted
general \(H\)-fibre map with degree one onto a translate of
\(T_1\). This is a lattice construction; no splitting of the torus
extension is required.

Pull back both the source and \(Z_1\) to \(A_1\). These covers
are connected, since the original maps to \(A\) are surjective
with connected fibres and therefore surject on \(\pi_1(A)\).
Write \(M_1\) for the source cover. The Stein factorization of
the pulled-back map to \(W\) has a finite base over \(W\); take
a smooth projective model \(W_1\) of that base. The map to
\(A_1/T_1\) is constant on its general fibres and hence descends
meromorphically to \(W_1\). Resolve this map as well. The pair of
projection maps then gives a birational model
\begin{equation}\label{de:torus-bundle}
 Z_*=W_1\times_{A_1/T_1}A_1,
 \qquad p:Z_*\longrightarrow W_1.
\end{equation}
Indeed it has degree one on the general fibres just described.
The space \(Z_*\) is smooth, and it is a closed submanifold of the
compact K\"ahler product \(W_1\times A_1\). The map \(p\) is
a principal holomorphic bundle with connected torus fibre \(T_1\).
Its transition functions are translations, so an invariant top form
on \(T_1\) trivializes the relative canonical bundle. In particular,
\begin{equation}\label{de:canonical-pullback}
 K_{Z_*}=p^*K_{W_1}.
\end{equation}
We are free to replace \(W_1\) by any further smooth projective modification
and use its corresponding fibre product in
\eqref{de:torus-bundle}.

The true inertia on \(Z_*\) has no horizontal divisor over
\(W_1\). To check this on a morphism of models, resolve the maps
from the pulled-back \(Z_1\) to \(W_1\) and \(Z_*\), and
carry the auxiliary boundary by the rule
\eqref{de:auxiliary-boundary}. Its restriction to a general fibre
still has only Albanese-exceptional components. Indeed finite
\'etale covers of a birational torus are birational to torus
covers. An exceptional prime on a further model either misses the
general fibre or restricts to a divisor exceptional over the old
general fibre, so it remains Albanese-exceptional. The strict transform
of a horizontal prime of
\(Z_*\) maps onto a divisor of its general torus fibre. It cannot
occur in this auxiliary boundary, which dominates true inertia.
Its meridian order is therefore one.

Every vertical prime divisor of the smooth bundle \(p\) is the
full inverse image of a prime divisor of \(W_1\). A divisor cannot
map into codimension at least two by the fibre-dimension formula,
and connected smooth fibres are irreducible. We have consequently
obtained a rational boundary \(\Delta_{W_1}\) with
\begin{equation}\label{de:boundary-pullback}
 \Delta_{Z_*}=p^*\Delta_{W_1},
\end{equation}
with the same finite integral meridian orders on corresponding
divisors.

\medskip\noindent
\emph{The true boundary is big downstairs.}
Let \(F\) denote the finite deck group of \(A_1\to A\).
Write \(\widetilde Z_1=Z_1\times_A A_1\) and let
\(\phi:\widetilde Z_1\dashrightarrow Z_*\) be the birational
comparison. Take the closure of the simultaneous graph of
\(\phi\circ f\), for \(f\in F\), in
\(\widetilde Z_1\times (Z_*)^F\). The group acts on this graph
by its action on \(\widetilde Z_1\) and permutation of the other
factors. Resolve equivariantly and take a further equivariant
K\"ahler modification using
\cite[Theorem~1.1 in the author's version]{JiaMeng2024}; the
finite group preserves the average of a big class. This gives a
smooth K\"ahler \(V\) mapping to \(\widetilde Z_1\), to
\(Z_*\), and to \(A_1\).
The action is free because its map to \(A_1\) is equivariant for
the free translation action. The quotient \(V/F\) is thus smooth
K\"ahler and is a modification of \(Z_1\), hence of \(Z_0\),
over the original torus \(A\). Moreover, \(V\to V/F\) is the
pullback of \(A_1\to A\): each deck orbit maps bijectively onto
the corresponding isogeny fibre.

Further prepare this quotient, and, if necessary, dominate it by
a member of the cofinal collection. Pull this modification back
along \(A_1\to A\); all comparison morphisms persist. We continue
to call the resulting cover \(V\). Its true inertial adjoint is
the unramified pullback of that on its downstairs quotient, by
\eqref{de:quotient-inclusion}. The minimum in
\eqref{de:minimum} therefore gives
\[
 \kappa(V,K_V+\Delta_V)\ge b.
\]
All boundaries in this inequality are true inertial boundaries;
the auxiliary exceptional coefficients of \(\Delta_1\) are not
used here. Lemma~\ref{de:birational-pushdown} applied to
\(V\to Z_*\) gives the same lower bound on \(Z_*\).
Equations~\eqref{de:canonical-pullback} and
\eqref{de:boundary-pullback}, together with \(p_*\cO_{Z_*}=
\cO_{W_1}\), now yield
\[
 \kappa(W_1,K_{W_1}+\Delta_{W_1})\ge b=\dim W_1.
\]
Thus \(K_{W_1}+\Delta_{W_1}\) is big.

This argument applies after any chosen further smooth projective modification
of \(W_1\): form its fibre product, repeat the equivariant
comparison, and use the same minimum on the prepared quotient over
\(A\). The newly appearing true coefficients are read on the new
model. No comparison of exceptional coefficients on unrelated models
is needed.

\medskip\noindent
\emph{The differential contradiction.}
Choose \(W_1\) sufficiently modified that the induced fibration
from a resolution of \(M_1\) to \(W_1\) is neat, as described
before Lemma~\ref{de:neat-pullback}. The preceding argument still
gives bigness on this choice. The source resolution maps holomorphically
to \(Z_*\), using its maps to \(W_1\) and \(A_1\). Its
fundamental-group quotient is the one used to define the true inertia
on \(Z_*\). Lemma~\ref{de:inertia-divisibility}, with the smooth
map \(p\), shows that every boundary order on \(W_1\) divides
the multiplicity of every source component dominating that divisor.
Lemma~\ref{de:neat-pullback} therefore gives a Bogomolov sheaf on a
smooth source model of \(M_1\). This contradicts
Theorem~\ref{thm:specialness-inputs}, since \(M_1\) is a connected
finite \'etale cover of the special manifold \(M\).
\end{proof}
==== END The-abelianity-conjecture-for-special-compact-Kahler-manifolds//sections/descent.tex ====

==== BEGIN The-abelianity-conjecture-for-special-compact-Kahler-manifolds//sections/hilbert.tex ====
\section{Hilbert tensors and minimal quotients}\label{sec:hilbert}

The purpose of this section is to turn an infinite quotient with no
infinite linear image into positivity on a base of minimal dimension.
The auxiliary compact manifold in the following statement allows the
minimality condition to be imposed before passing to an orbifold base.
For a homomorphism \(\rho\) on the fundamental group of a compact
K\"ahler manifold \(V\), write
\(\operatorname{gdim}(V,\rho)\) for the dimension of its
\(\Gamma\)-reduction.

\begin{theorem}\label{thm:hilbert-positivity}
Let \(Y\) be a connected smooth compact K\"ahler manifold of dimension
\(\ell>0\), and let
\[
 \Delta=\sum_i(1-m_i^{-1})D_i,\qquad m_i\in\N,\quad m_i\ge2,
\]
have simple normal crossing support. Denote its root orbifold by
\(\cY\). Suppose that
\[
 q:\pi_1^\orb(\cY)\longrightarrow Q
\]
is surjective, that \(Q\) is infinite, and that every finite-dimensional
complex linear representation of \(Q\) has finite image.

Suppose that a connected smooth compact K\"ahler manifold \(S\) admits
a dominant meromorphic map \(a:S\dashrightarrow Y\) and a homomorphism
\(\nu:\pi_1(S)\to Q\) with finite-index image. Assume that \(a\) and
\(\nu\) are compatible with \(q\) on a dense analytic Zariski-open
subset where \(a\) is holomorphic and takes values in
\(Y\setminus\supp\Delta\).

Finally, suppose that for every finite-index subgroup \(Q_0\le Q\)
there is a connected finite \'etale cover \(S'\to S\) such that, writing
\(\nu'\) for the induced homomorphism and \(Q'=\im\nu'\), one has
\(Q'\subset Q_0\) and
\[
 \operatorname{gdim}(S',\psi\circ\nu')\ge\ell
 \quad\text{whenever }\psi:Q'\twoheadrightarrow H,\quad
 H\text{ is infinite}.
 \tag{M}\label{hi:minimality}
\]
Then \(K_Y+\Delta\) is big.
\end{theorem}

In \eqref{hi:minimality}, the cover may depend on \(Q_0\); after it has
been chosen, the lower bound is required for every infinite quotient
of its image. Each \(Q'\) is automatically of finite index in \(Q\).
No linearity assumption is imposed on the groups \(H\).

\subsection{Holomorphic forms with Hilbert coefficients}

We use flat unitary Hilbert bundles in the usual local sense: their
transition functions are constant unitary operators. Differential
operators act on the finite-dimensional form indices and on the
Hilbert coefficients componentwise. The local elliptic and Sobolev
estimates used in Section~\ref{sec:spectral} extend to these
coefficients, as explained below.

\begin{lemma}\label{hi:finite-images}
Every finite-index subgroup \(Q_0\) of \(Q\) has only finite
finite-dimensional complex linear images. In particular,
\(H^1(Q_0,\C)=0\).
\end{lemma}

\begin{proof}
Inducing a finite-dimensional representation of \(Q_0\) to \(Q\)
produces a finite-dimensional representation of \(Q\). Its restriction
to \(Q_0\) contains the original representation, whose image is
therefore finite. The group \(Q\), and hence \(Q_0\), is finitely
generated. A nonzero homomorphism \(Q_0\to(\C,+)\) would give an
infinite unipotent two-dimensional representation, proving the last
assertion.
\end{proof}

\begin{proposition}\label{hi:coefficients}
There exist an integer \(p>0\), a separable unitary representation
\(\rho:Q\to U(\mathcal H)\) with no nonzero finite-dimensional
invariant subspace, and a nonzero holomorphic section
\[
 \eta\in H^0\bigl(\cY,\Omega_{\cY}^p\otimes\mathcal H_\rho\bigr).
\]
\end{proposition}

\begin{proof}
We treat the nonamenable and amenable cases separately.

\medskip\noindent\emph{The nonamenable case.}
Let \(U\to\cY\) be the normal orbifold covering associated with
\(\ker q\). By Theorem~\ref{thm:zero-spectrum}, in some fixed degree
\((p,0)\) there are unit vectors with spectral support in
\([0,j^{-1}]\), for \(j\to\infty\). Folding forms from \(U\) to the
compact base identifies them with sections \(s_j\) having coefficients
in the regular representation \(\lambda_Q\) on \(\ell^2(Q)\).
This is the regular-coefficient bundle interpretation of covering-space
analysis \cite[Section~6.1]{Atiyah1976}. To include the orbifold
normalization, work on a chart \([V/G]\) and put
\(H=\ker(G\to Q)\). The cover has components \([V/H]\) indexed by
the cosets of \(\im(G\to Q)\) in \(Q\). A \(G\)-invariant form with
regular coefficients is exactly the corresponding family of scalar
forms on these components. In the norm, the \(|G/H|\) regular
coordinates cancel the downstairs factor \(1/|G|\), leaving the
upstairs factor \(1/|H|\). The differential operators agree
componentwise. Thus the identification preserves the \(L^2\) norms
and operators even when \(U\) has nontrivial isotropy.

Here is the compactness statement needed for these coefficients.
For every integer \(k\ge0\), elliptic estimates give
\[
 \|s_j\|_{H^{2k}}
 \le C_k\bigl(\|(\Delta'')^k s_j\|_2+\|s_j\|_2\bigr)
 \le 2C_k.
\]
The constants do not depend on the coefficient dimension. Indeed, in
a flat chart the coefficients of the operator are finite matrices
tensored with the identity on the Hilbert space. Summing the scalar
elliptic estimates over an orthonormal expansion gives the Hilbert
estimate. The same conclusion holds on invariant orbifold charts,
whose local group orders are uniformly bounded. A finite atlas then
gives the displayed global estimate. Hilbert-valued Sobolev embedding,
obtained from Fourier inversion and Cauchy--Schwarz, bounds every
derivative uniformly on smaller charts.

Fix a nonprincipal ultrafilter. Form the Hilbert ultrapower
\(\mathcal H^\omega\) and define the values and derivatives of \(s\)
by the corresponding pointwise classes of those of \(s_j\).
Uniform bounds for the next derivative give uniform Taylor
remainders. Thus these classes are the derivatives of a smooth
section, and the exact transition identities make it a section of
the ultrapower flat bundle.

This section retains its norm. The functions
\(x\mapsto\|s_j(x)\|^2\) are uniformly bounded and uniformly Lipschitz
on the finite atlas. A finite \(\varepsilon\)-net shows that their
pointwise ultralimit is approached uniformly along the ultrafilter:
first control the finitely many net values, then use the common
Lipschitz bound. Consequently
\[
 \|s\|_2^2=\lim_\omega\|s_j\|_2^2=1.
\]
Applying the same elliptic estimates to \(\Delta''s_j\) shows
convergence to zero in every fixed \(C^r\) norm, since
\(\|(\Delta'')^{k+1}s_j\|_2\le j^{-k-1}\).
Hence \(\Delta''s=0\). Integration by parts on the compact base,
in antiholomorphic degree zero, gives \(\dbar s=0\).

The ultrapower representation has no nonzero finite-dimensional
invariant subspace. Otherwise such a subspace would have finite
image, by the hypothesis on \(Q\), and would be fixed by a
finite-index subgroup \(Q_0\). This subgroup is nonamenable.
For a finite generating set \(F\subset Q_0\) and some
\(\varepsilon>0\),
\[
 \sum_{g\in F}\|\lambda(g)v-v\|^2
 \ge\varepsilon\|v\|^2
\]
on its regular representation. Indeed, failure would give unit
almost invariant vectors; their squared absolute values would be
almost invariant probability measures in \(\ell^1(Q_0)\), yielding
an invariant mean. Summing over copies gives the same
inequality on the restriction of \(\lambda_Q\) to \(Q_0\).
A nonzero \(Q_0\)-fixed ultrapower vector, represented by a bounded
sequence of limiting norm one, contradicts this inequality after
taking the ultralimit. There are only finitely many terms on its
left side.

Finally, restrict to the closed span of all coefficient values of
the section and their \(Q\)-translates. Countable charts, continuity,
and countability of \(Q\) make this subspace separable. It still
contains the nonzero section and has no finite-dimensional invariant
subspace.

\medskip\noindent\emph{The amenable case.}
An infinite amenable group does not have property~\((T)\): its regular
representation has almost invariant vectors and no invariant vector.
Shalom's reduced-cohomology theorem therefore gives a unitary
representation with nonzero reduced first cohomology
\cite[Theorem~4.2]{Shalom2006}.
We may restrict to a separable invariant subspace generated by the
values of a cocycle on the countable group.

The closed span of the finite-dimensional invariant subspaces
contributes no reduced first cohomology. To see this, decompose it
orthogonally into finite-dimensional irreducible summands. On any
finite sum the representation has finite image. Its kernel \(Q_0\)
has \(H^1(Q_0,\C)=0\), by Lemma~\ref{hi:finite-images}; restriction of
a cocycle to that kernel is zero, and averaging over the finite
quotient makes the cocycle a coboundary. Finite partial sums
approximate a cocycle on a finite generating set, so the same
vanishing holds for reduced cohomology of the closed sum.
The orthogonal complement therefore has a cocycle \(b\) that is not
a limit of coboundaries and has no finite-dimensional subrepresentation.

Pullback by the surjection \(q\) preserves this nonvanishing:
approximating the pullback cocycle on lifts of finitely many
generators of \(Q\) would approximate \(b\) itself. The associated
flat affine Hilbert bundle on \(\cY\) has a smooth section, constructed
by local sections and a partition of unity, averaged in the finite
orbifold charts. Its covariant differential is a smooth closed
Hilbert-valued one-form \(\alpha\) representing the pulled-back
cocycle.

The harmonic projection of \(\alpha\) is nonzero. Otherwise, the
orthogonal decomposition of the closed de Rham Hilbert complex
would put the closed form \(\alpha\) in the \(L^2\) closure of
exact forms. For fixed \(t>0\), the heat operator
\(e^{-t\Delta}\) commutes with the
differential and maps \(L^2\) convergence to \(C^0\) convergence,
by the dimension-independent elliptic estimates just used.
Thus \(e^{-t\Delta}\alpha\) would be a uniform limit of smooth exact
forms. Periods along finitely many fixed generator loops, with flat
parallel transport of coefficients, would approximate its cocycle
by coboundaries. Letting \(t\downarrow0\) gives the same conclusion
for \(\alpha\), contrary to the choice of \(b\).
The orthogonal decomposition of the closed de Rham Hilbert complex
now gives a nonzero harmonic one-form; local elliptic regularity
makes it smooth.

The K\"ahler identities for flat unitary coefficients split this
harmonic form into types. A nonzero \((1,0)\)-part is holomorphic.
If only the \((0,1)\)-part is nonzero, complex conjugation gives a
holomorphic \((1,0)\)-form with conjugate coefficients. The conjugate
representation again has no finite-dimensional invariant subspace.

In either case, degree zero is impossible: a holomorphic section of
a flat unitary Hilbert bundle on a compact K\"ahler orbifold is
parallel, by integration of the Laplacian of its squared norm.
A nonzero parallel section would give an invariant line.
\end{proof}

\subsection{The line associated with a tensor}

Proposition~\ref{hi:coefficients} supplies a nonzero tensor, but
nonvanishing alone does not contradict specialness. We next isolate
its coefficient line and measure the variation of that line by a
positive curvature current. This connects the group to cotangent
positivity, as in the finite-dimensional setting of
\cite{BrunebarbeKlinglerTotaro2013}; the Hilbert coefficient space
and the subsequent minimal-rank argument require the constructions
proved here.

For a finite-index subgroup \(\Gamma\le Q\), let
\(\cY_\Gamma\to\cY\) be the cover associated with \(q^{-1}(\Gamma)\).
All finite orbifold covers considered in this subsection and the
next are of this form.

Call a holomorphic tensor
\[
 \tau\in H^0\bigl(\cY_0,
       (\Omega_{\cY_0}^1)^{\otimes m}\otimes\mathcal B_\rho\bigr)
\]
a \emph{line tensor} if the map from the dual of its cotangent
tensor factor to its Hilbert coefficient space has generic rank one.
Here \(\cY_0\) is a connected finite orbifold cover of \(\cY\),
and \(\rho\) is a unitary representation of its corresponding
finite-index subgroup of \(Q\). On the \(Q\)-cover, the coefficient
lines define a holomorphic map to \(\PP(\mathcal B)\) away from an
analytic degeneracy locus. We always replace \(\mathcal B\) by the
closed span of these lines.

\begin{lemma}\label{hi:plucker}
There is a line tensor whose coefficient lines have infinite-dimensional
closed span.
\end{lemma}

\begin{proof}
Let \(\eta\) be given by Proposition~\ref{hi:coefficients}, and let
\(r\) be the generic rank of
\((\Omega_{\cY}^p)^*\to\mathcal H\).
Taking its \(r\)-th exterior power gives a nonzero rank-one tensor
with coefficient space \(\bigwedge^r\mathcal H\).
Antisymmetrization embeds its finite-dimensional tensor factor into
\((\Omega_{\cY}^1)^{\otimes pr}\).

If the resulting decomposable lines spanned a finite-dimensional
space, \(Q\) would act on that span through a finite group. The orbit
of one of these lines would be finite. A decomposable exterior line
determines its \(r\)-dimensional original coefficient plane, so the
corresponding orbit of planes would also be finite. Their sum would
be a nonzero finite-dimensional invariant subspace of \(\mathcal H\),
a contradiction.
\end{proof}

\begin{lemma}\label{hi:saturated-line}
A line tensor determines a saturated orbifold line subsheaf
\[
 \cL\subset(\Omega_{\cY_0}^1)^{\otimes m}
\]
and a positive curvature current on \(\cL\). On the regular locus
this current is the pullback of the Fubini--Study form of its
coefficient-line map.
\end{lemma}

\begin{proof}
Work in a uniformizing chart with a flat coefficient trivialization.
Any nonzero scalar coefficient of \(\tau\) determines its
meromorphic direction in the finite-rank tensor bundle. All scalar
coefficients have this direction, since the coefficient rank is one.
These local meromorphic directions agree on overlaps. Their
saturations are coherent rank-one subsheaves of the tensor bundle.
A saturated subsheaf of a locally free sheaf is reflexive in rank
one, and on a smooth chart a rank-one reflexive sheaf is a line
bundle. The invariant constructions give the asserted orbifold line.

Write a local line generator as \(e\). Off its codimension-two
degeneracy set the tensor has the form \(e\otimes u\), with \(u\)
Hilbert-valued and holomorphic. There are no divisorial poles, by
saturation. The Hilbert-valued Hartogs extension, obtained from the
Cauchy integral formula on transverse polydiscs, extends \(u\)
across the remaining set.

The weights \(\log\|u\|^2\) are plurisubharmonic. If
\(e_\beta=g_{\alpha\beta}e_\alpha\), then
\(u_\beta=g_{\alpha\beta}^{-1}u_\alpha\), so
\[
 \log\|u_\beta\|^2
 =\log\|u_\alpha\|^2-\log|g_{\alpha\beta}|^2.
\]
They therefore define the metric
\(\|e_\alpha\|^2=\|u_\alpha\|^{-2}\) on \(\cL\), with positive
curvature. Where \(u\ne0\), its curvature is
\(\dd\dc\log\|u\|^2\), the pulled-back Fubini--Study form.
A positive power of \(\cL\) kills the finitely many chart
stabilizer actions and descends to an ordinary line bundle on the
coarse space; its weights descend as psh weights.
\end{proof}

The degeneracy loci used here and below are analytic even though the
coefficient space may be infinite-dimensional. Locally their rank
conditions are the vanishing of holomorphic scalar minors.
The ideal of their germs at a point is generated by finitely many
of those minors. On sufficiently small representatives of the
finitely many local components, every remaining minor vanishes by
the identity theorem. This proves local analyticity of the common
zero set; it does not require a global finite list of coefficients.

\subsection{Minimal rank and discrete monodromy}

Among all line tensors with infinite-dimensional span on finite
orbifold covers, choose one for which the generic complex rank \(s\)
of the coefficient-line map is minimal. This is a minimum in the
finite set \(\{1,\ldots,\ell\}\): rank zero would make the map
constant on its connected covering, giving a one-dimensional span.
Let \(\Gamma\le Q\) be the subgroup defining the selected cover,
let \(\mathcal B\) be its separable coefficient span, and write
\[
 F:U^\circ\longrightarrow\PP(\mathcal B)
\]
on the connected regular open subset of the \(Q\)-cover. We may
delete the chart isotropy locus as well. These deletions are proper
analytic sets and do not disconnect the covering.

\begin{proposition}\label{hi:discrete-monodromy}
After replacing \(\Gamma\) by a finite-index subgroup, the action on
the regular image germs of \(F\) has infinite discrete image in a
Lie group acting properly on a connected complex manifold of
dimension \(s\).
\end{proposition}

\begin{proof}
\medskip\noindent\emph{The manifold of image germs.}
At each point of \(U^\circ\), the holomorphic constant-rank theorem
writes \(F\) as a submersion onto an embedded \(s\)-dimensional
plaque. This theorem also holds for the present Hilbert target:
choose a finite-dimensional projection of rank \(s\), use it for
source coordinates, and observe that all derivatives in the
remaining source directions vanish.

Let \(Z\) consist of the image points together with the germs of
these embedded plaques at those points. A plaque supplies its own
coordinate chart on \(Z\). The map \(U^\circ\to Z\) is an open
holomorphic submersion, so \(Z\) is connected and second countable.
It is Hausdorff. Indeed, two germs over the same image point can
both be written as graphs over one finite-dimensional transverse
projection and one small connected ball. If their plaque
neighborhoods shared a germ, the graph functions would agree on an
open set and hence everywhere on that ball. The original germs
would be equal. Points lying over different image points are
separated by the Hilbert projective space.

The natural immersion \(\iota:Z\to\PP(\mathcal B)\) gives \(Z\)
a K\"ahler metric. We use its intrinsic Riemannian length distance,
which induces the manifold topology and is locally compact even when
the metric is incomplete. Its isometry group acts properly
\cite[Corollary~4]{Manoussos2010} and is a Lie group
\cite[author's version, Theorem~6.3]{MatveevTroyanov2017}.
Let \(I\) be the closure of the \(\Gamma\)-image in this isometry
group with its compact-open topology. Its elements are holomorphic.
Indeed, if holomorphic isometries \(g_j\) converge to \(g\), choose
a relatively compact neighborhood whose image under \(g\) lies
compactly inside one target complex chart. Eventually the \(g_j\)
also take its closure into that chart, and their coordinate maps
converge locally uniformly. The Weierstrass theorem makes \(g\)
holomorphic there. Inversion is continuous in the isometry group,
so the same argument applies to \(g^{-1}\).

\medskip\noindent\emph{A locally compact unitary extension.}
Let \(J\) be the closure of the simultaneous image of \(\Gamma\) in
\(I\times U(\mathcal B)\), with the strong group topology on the
unitary factor. Every \((g,A)\in J\) satisfies
\(A\iota(z)=\iota(gz)\) projectively, by continuity.
The projection \(J\to I\) is proper and surjective.
For properness, suppose that the isometry components of a sequence
converge. Choose countably many unit vectors in coefficient lines
whose span is dense in \(\mathcal B\). The images of each such
vector lie in convergent projective lines. Compactness of the
scalar circle gives a convergent subsequence of the unit vectors.
Diagonal extraction, performed also for inverse operators, gives
strong convergence to a unitary operator. This proves properness,
since the spaces involved are metrizable. The image is consequently
closed and contains the dense \(\Gamma\)-image, proving surjectivity.

The kernel is a closed subgroup of the scalar circle. A kernel
element preserves every coefficient line; a unitary operator has
the same eigenvalue on two nonorthogonal eigenlines. Nearby lines
are nonorthogonal, so connectedness of \(Z\) makes this scalar the
same everywhere. Their span is all of \(\mathcal B\).
It follows that \(J\) is locally compact. It has no small subgroups:
use such a neighborhood in the Lie quotient \(I\), then one in the
scalar kernel. The locally compact no-small-subgroups theorem
therefore makes \(J\) a Lie group
\cite[author's version, Corollary~1.5.8]{Tao2014}.

Its adjoint representation, complexified, has finite image on
\(\Gamma\), by Lemma~\ref{hi:finite-images}. Pass to its kernel in
\(\Gamma\), and replace both \(I\) and \(J\) by the corresponding
closures. They are open finite-index subgroups of the old closures,
with unchanged identity components; \(J\to I\) remains surjective
and proper. The \(Q\)-cover and the map \(F\) are unchanged.
Every element of the new \(J\) centralizes \(J^\circ\), since
its conjugation has identity differential on the connected group.
In particular, \(J^\circ\) is abelian.

\medskip\noindent\emph{Spectral projection of holomorphic data.}
The spectral theorem for the continuous unitary representation of
the connected abelian Lie group \(J^\circ\), in its commutative
\(C^*\)-algebra formulation \cite{Williams2007}, gives a direct integral
\[
 \mathcal B=\int^\oplus \mathcal B_\chi\,\dd\mu(\chi),\qquad
 t|_{\mathcal B_\chi}=\chi(t)\Id .
\]
We take \(\mu\) finite, as is possible because \(\mathcal B\) is separable.
Because \(\Gamma\) centralizes \(J^\circ\), its operators are
decomposable in this integral \cite{Williams2007}.
Write \(j_\gamma\) for the simultaneous image of \(\gamma\) in \(J\),
and put
\[
 D=\{j_\gamma:\gamma\in\Gamma\}\cap J^\circ.
\]
Countability of \(\Gamma\) allows the group identities to hold on
one common full-measure set, giving unitary representations
\(\rho_\chi\) of \(\Gamma\). On that set we also impose the countably
many identities
\[
 \rho_\chi(\gamma)=\chi(j_\gamma)\Id
 \qquad\text{when }j_\gamma\in D.
\]

We explain carefully how the holomorphic tensors pass to these
fibres. These are almost-everywhere decompositions of a countable
family of Taylor coefficients; no bounded point-evaluation projection
from the direct integral is asserted. Let
\(\varpi:U^\circ\to Z\) be the submersion. Choose a countable
uniformizing atlas for \(\cY_\Gamma\) and all its lifts to the
\(Q\)-cover, still a countable family. Choose also local nonvanishing
holomorphic vector representatives \(v_\alpha\) of \(\iota\) on a
countable cover of \(Z\). On a source polydisc mapping
into the domain of \(v_\alpha\), the finitely many tensor coefficients
satisfy vector identities
\[
 \tau_a(x)=c_a(x)v_\alpha(\varpi(x)),
\]
with holomorphic scalar functions \(c_a\), obtained after shrinking
by applying a bounded functional nonzero on \(v_\alpha\).
On suitable overlap
domains the equivariance identities have the form
\[
 v_\beta(\gamma z)
 =c_{\gamma,\alpha\beta}(z)\rho(\gamma)v_\alpha(z),
\]
where \(c_{\gamma,\alpha\beta}\) is a holomorphic unit.
Here \(\rho\) is the selected unitary representation of \(\Gamma\).
These identities are kept as vector identities before projectivizing.

Expand the tensor coefficients and the \(v_\alpha\) in
Hilbert-valued power series. On every strictly smaller polydisc the
sums of coefficient norms times the monomial radii are finite.
Cauchy--Schwarz in the spectral variable, followed by Tonelli,
gives the corresponding absolute sums in
\(\mathcal B_\chi\) for almost every \(\chi\). The projected series
are therefore holomorphic there. After restricting to countably
many smaller overlap polydiscs, normal convergence permits
composition with the fixed coordinate maps and multiplication by
the scalar functions above term by term. For each resulting coefficient
sum, its partial sums converge in the direct-integral norm, while
its fibre series converges absolutely almost everywhere. A subsequence
converging almost everywhere identifies the two limits. There are
only countably many such sums and Taylor-coefficient identities,
including the tensor transition and orbifold isotropy identities.
They therefore hold, together with the group identities above,
on one common full-measure set and give global
holomorphic tensors on
\(\cY_\Gamma\) of coefficient rank at most one, and holomorphic
projective maps
\[
 F_\chi:Z\setminus E_\chi\longrightarrow\PP(\mathcal B_\chi)
\]
of rank at most \(s\), where \(E_\chi\) is the common zero locus of
the projected representatives.

The countable family of Taylor coefficients of the tensor on the
lifted charts has closed span \(\mathcal B\): the series give all
coefficient values, while Cauchy integrals express their coefficients
as limits of linear combinations of those values. The orthogonal
projection onto the fibrewise complement of the projected coefficient
spans is a measurable field, obtained by countable Gram--Schmidt
orthogonalization. Its direct integral annihilates this dense family
and is therefore zero. Thus the projected Taylor coefficients span
\(\mathcal B_\chi\) almost everywhere. By the same series and Cauchy
formulas, so do the projected coefficient values. On every nonzero
such spectral fibre the projected tensor is therefore nonzero,
has generic coefficient rank one, and is a line tensor whose
coefficient lines span \(\mathcal B_\chi\). We henceforth restrict
to these nonzero fibres.

\medskip\noindent\emph{The identity component cannot move an image germ.}
The subgroup \(D\) is dense in \(J^\circ\), since that component is open.
It acts by the character \(\chi\) on each spectral fibre, so
\(F_\chi\) is invariant under its action on \(Z\).
The closed analytic set \(E_\chi\) is invariant as well.
Continuity extends both invariances to \(J^\circ\).

Suppose that \(J^\circ\) acts nontrivially on \(Z\). If some
\(F_\chi\) had rank \(s\) at \(z\notin E_\chi\), a finite-dimensional
target projection of full rank and the inverse function theorem
would make \(F_\chi\) locally injective at \(z\). The connected
\(J^\circ\)-orbit of \(z\) stays in the domain and in its fibre by
invariance. The point \(z\) is isolated in that fibre, and its
singleton is open there by isolation and closed because \(Z\) is Hausdorff.
The orbit is therefore \(\{z\}\). Applying this at every point of
the nonempty open full-rank locus, each element of \(J^\circ\)
fixes that open set. It is holomorphic, so the identity theorem on
connected \(Z\) makes it the identity everywhere, a contradiction.
Thus every \(F_\chi\) has rank strictly less than \(s\).
Minimality of \(s\) then forces
\(\mathcal B_\chi\) to be finite-dimensional almost everywhere.
By Lemma~\ref{hi:finite-images}, \(\rho_\chi(\Gamma)\) is finite.
The character \(\chi\) consequently has finite image on the dense
subgroup of \(J^\circ\) just used. Continuity and connectedness
force \(\chi\) to be trivial on \(J^\circ\).
Hence \(J^\circ\) acts trivially on \(\mathcal B\).
It then acts trivially on \(Z\): each orbit maps to a point under
the immersion \(\iota\), and is connected. This contradicts the
assumed nontrivial action.

We have proved that \(J^\circ\) acts trivially on \(Z\).
The surjective proper Lie-group homomorphism \(J\to I\) maps
identity component onto identity component; since \(I\) is an
effective isometry group, \(I^\circ=\{1\}\).
Thus \(I\) is discrete and acts properly.

Its \(\Gamma\)-image is infinite. Otherwise a finite-index subgroup
would fix every image germ and would act on \(\mathcal B\) by
scalars. Its scalar character has finite image; on a further finite
orbifold cover the tensor would be untwisted. An untwisted
Hilbert-valued tensor on a compact orbifold has finite-dimensional
coefficient span: its scalar tensor space is finite-dimensional,
and expansion in a basis writes the tensor as a finite sum of
scalar tensors times fixed Hilbert vectors. This contradicts our
choice of the line tensor.
\end{proof}

\subsection{Compact fibres from transverse volume}

Discrete monodromy gives a genuine quotient of the image-germ
manifold. To use minimality, its local fibres must determine compact
subvarieties with finite group image. The following argument obtains
these subvarieties from the mass of the positive current; only the
source is assumed compact.

\begin{lemma}\label{hi:compact-fibres}
Let \(V\) be a connected smooth compact K\"ahler manifold of
dimension \(d\), and let \(V^\circ\) be the complement of a proper
analytic subset. Let a discrete group \(I\) act properly and
effectively by holomorphic isometries on a connected K\"ahler
manifold \(Z\) of dimension \(s>0\).
Suppose that \(f:V^\circ\to Z/I\) has local holomorphic lifts to
\(Z\) of rank \(s\), whose continuation is compatible with a
homomorphism \(\rho:\pi_1(V)\to I\).
If a positive closed \((1,1)\)-current on \(V\) restricts to
a positive constant multiple of \(f^*\omega_Z\), then
\begin{equation}
 \operatorname{gdim}(V,\rho)\le s.
 \label{hi:gamma-upper-bound}
\end{equation}
\end{lemma}

\begin{proof}
Let \(T\) be the current and let \(\omega\) be a K\"ahler form on
\(V\). The mass of \(T^s\wedge\omega^{d-s}\) on \(V^\circ\) is
finite; this is the positive-current mass bound
\cite{Boucksom2002}. One can see the bound directly
by writing \(T=\alpha+\dd\dc\varphi\), choosing \(C\) with
\(\beta=\alpha+C\omega>0\), and truncating
\(\varphi_j=\max(\varphi,-j)\). The bounded-potential products satisfy
\[
 \int_V(\beta+\dd\dc\varphi_j)^s\wedge\omega^{d-s}
 =\int_V\beta^s\wedge\omega^{d-s}.
\]
On \(\{\varphi>-j\}\cap V^\circ\) they equal
\((T+C\omega)^s\wedge\omega^{d-s}\), which dominates
\(T^s\wedge\omega^{d-s}\). Exhaustion gives the bound.

Use the free locus of the orbifold \(Z/I\) for the coarea formula.
The discarded isotropy values, and their inverse images under the
local submersions, have measure zero. With normalized volumes,
\[
 f^*\frac{\omega_Z^s}{s!}\wedge
       \frac{\omega^{d-s}}{(d-s)!}
 =J_f\,\frac{\omega^d}{d!},
\]
where \(J_f\) is the real normal Jacobian. Its value is the product
of the squared nonzero complex singular values of the differential.
The coarea formula and the mass bound show that almost every
fibre has finite \((d-s)\)-dimensional complex volume in \(V^\circ\).
The formula is applied to a countable disjoint measurable
decomposition subordinate to target charts, so it counts whole
fibres and requires neither compactness nor bounded geometry of
the target. Moreover, this yields a full-measure set of source
points: the inverse image of the exceptional null set has integral
of \(J_f\) equal to zero, and \(J_f>0\) on the submersion locus.

For a good free value, its fibre is closed analytic and pure of
dimension \(d-s\) in \(V^\circ\). Its integration current has finite
mass. Extension by zero across \(V\setminus V^\circ\) is positive
and closed, by the extension theorem of El Mir
\cite{ElMir1984}.
Siu's analyticity theorem for positive Lelong superlevel sets
\cite{Siu1974} then makes its closure analytic of pure dimension
\(d-s\): the level set with Lelong number at least one contains
the fibre, has dimension at most \(d-s\), and agrees with the
smooth fibre on \(V^\circ\). Take its components meeting
\(V^\circ\). Equivalently, one may apply the finite-volume
closure theorem directly \cite{ElMir1984}.

Let \(\widetilde C\to C\) resolve one such closure component.
The preimage of \(C\cap V^\circ\) has proper analytic complement
in the smooth manifold \(\widetilde C\), and its inclusion induces
a surjection on fundamental groups: every loop may be perturbed
off an analytic set of real codimension at least two.
Along the open fibre the continued lift to \(Z\) is constant.
Thus its monodromy fixes one point of \(Z\), and lies in a finite
stabilizer. The same is true of the image of
\(\pi_1(\widetilde C)\), by that surjection.

The \(\Gamma\)-reduction contracts these compact subvarieties through
very general points, by Theorem~\ref{thm:gamma-reduction}.
The full-measure set just obtained meets the complement of its
countable union of proper analytic exceptional sets. Its general
fibres consequently have dimension at least \(d-s\), proving
\eqref{hi:gamma-upper-bound}.
\end{proof}

\subsection{Descent of the big line}

\begin{proof}[Proof of Theorem~\ref{thm:hilbert-positivity}]
Choose the minimal line tensor and the finite-index subgroup
\(\Gamma\) of Proposition~\ref{hi:discrete-monodromy}.
Let \(\cL\) and its positive curvature current be as in
Lemma~\ref{hi:saturated-line}. The current has rank \(s\) on the
regular open set.

Suppose that \(s<\ell\). Apply \eqref{hi:minimality} with
\(Q_0=\Gamma\), obtaining \(S'\) with image \(Q'\subset\Gamma\).
The map \(S'\dashrightarrow Y\) lifts meromorphically to the
coarse space of \(\cY_\Gamma\). Indeed, the monodromy condition
gives a lift on the complementary open set; the selected component
of the normalized finite pullback extends its graph.
Resolve this map on a smooth compact K\"ahler modification \(V\)
of \(S'\).

The descended power of \(\cL\) pulls back to an ordinary line
bundle on \(V\). Its psh weights pull back, since the map is
dominant, giving a positive current. On a regular analytic
Zariski-open subset the local coefficient-line maps give
\[
 V^\circ\longrightarrow Z/I
\]
of rank \(s\), with monodromy the composite
\(\pi_1(V)\to Q'\to I\). The pulled-back current is a positive
constant multiple of its transverse K\"ahler form.
Lemma~\ref{hi:compact-fibres} gives a \(\Gamma\)-dimension at most
\(s\). The image of \(Q'\) in \(I\) is infinite: it has finite
index in the infinite image of \(\Gamma\).
This image is an infinite quotient to which \eqref{hi:minimality}
applies. Bimeromorphic invariance of the \(\Gamma\)-reduction
therefore gives a dimension at least \(\ell\), a contradiction.
Hence \(s=\ell\).

Take a finite Galois orbifold cover
\(\widehat{\cY}\to\cY\) dominating \(\cY_\Gamma\), and pull
\(\cL\) to it. If \(G\) is its deck group, form
\[
 \mathcal M=\bigotimes_{\gamma\in G}\gamma^*\cL.
\]
The permutation action gives a canonical \(G\)-linearization.
All its factors inject into the pullback of the same cotangent
tensor bundle. Multiplication in the symmetric algebra gives a
nonzero morphism from \(\mathcal M\) to the \(|G|\)-th symmetric
power of that tensor bundle, and hence into a cotangent tensor power.
It is nonzero because a product of nonzero vectors in a symmetric algebra is
nonzero. Equivariant descent gives an orbifold line \(\mathcal M_0\)
and a nonzero cotangent-tensor morphism on \(\cY\).

The tensor product of the translated metrics is invariant.
Its curvature is positive and strictly positive on a nonempty
open set, since \(s=\ell\). A power of \(\mathcal M_0\) kills
the chart stabilizers and descends to an ordinary line bundle
\(A\) on \(Y\) with the same positivity property. The volume
criterion \cite[author's preprint, Theorem~1.2]{Boucksom2002} makes \(A\) big.
The Iitaka map of a big line bundle makes \(Y\) Moishezon.
The projectivity theorem for K\"ahler Moishezon manifolds
\cite{Moishezon1967} therefore makes \(Y\) projective.

It remains to interpret the tensor inclusion on an adapted
projective cover. Choose the smooth Galois Kawamata cover
\(\pi:Y'\to Y\) supplied by \cite[Lemma~5.2]{CampanaPaun2019}.
It is projective, being finite over \(Y\), and its ramification
order over \(D_i\) is exactly the denominator \(m_i\).
Near that component the root chart and the cover are
\(z_i=u_i^{m_i}=w_i^{m_i}\). Choosing the local root lift
\(u_i=w_i\), the cotangent generator pulls back as
\[
 \dd u_i=\dd w_i.
\]
Along an auxiliary ramification divisor \(z_j=w_j^{a_j}\), the
orbifold coefficient is zero and the generator is
\(\dd z_j=a_jw_j^{a_j-1}\dd w_j\).
These are precisely the local generators of the adapted orbifold
cotangent bundle in \cite[Definition~5.3]{CampanaPaun2019}.
Thus a suitable tensor power of the descended inclusion gives
a nonzero map
\[
 \pi^*A\longrightarrow
 \bigl(\pi^*\Omega^1(Y,\Delta)\bigr)^{\otimes M}
\]
on this projective adapted cover. The big-line criterion for orbifold
cotangent tensors \cite[Theorem~7.11]{CampanaPaun2019} now implies
that \(K_Y+\Delta\) is big. Its hypotheses are satisfied because
projectivity and the ordinary big line \(A\) have already been
established. No pseudoeffectivity assumption is needed in that
criterion.
\end{proof}
==== END The-abelianity-conjecture-for-special-compact-Kahler-manifolds//sections/hilbert.tex ====

==== BEGIN The-abelianity-conjecture-for-special-compact-Kahler-manifolds//sections/introduction.tex ====
\section{Introduction}\label{sec:introduction}

Campana's classification program separates compact K\"ahler geometry
into special varieties and bases of general type. Specialness records
the absence of differential forms that would detect such a base. One
of the program's central predictions is topological: the fundamental
group of a special compact K\"ahler manifold should become abelian
after passage to a finite unramified cover
\cite[Conjecture~A.1]{ClaudonHoring2013}.

We first specify the notion of specialness. For a holomorphic line
bundle \(L\) on a compact complex manifold \(X\), its Iitaka dimension
\(\kappa(X,L)\) is \(-\infty\) if no positive tensor power has a
nonzero section, and otherwise is the largest dimension of the images
of the meromorphic maps defined by its complete plurisection systems.
A manifold \(X\) of dimension \(n\) is \emph{special} if there is no
integer \(1\le p\le n\) and no nonzero sheaf morphism
\[
 L\longrightarrow\Omega_X^p
 \qquad\text{with}\qquad \kappa(X,L)=p.
\]
A line bundle with such a morphism and equality is a
\emph{Bogomolov sheaf}. A point is special. We prove the following
form of Campana's abelianity conjecture.

\begin{theorem}\label{thm:abelianity}
Let \(X\) be a connected smooth compact K\"ahler manifold of arbitrary
complex dimension. If \(X\) is special, then \(\pi_1(X)\) has an abelian
subgroup of finite index. Equivalently, \(X\) has a connected finite
unramified holomorphic cover with abelian fundamental group.
\end{theorem}

This is a positive resolution of the conjecture in the compact
K\"ahler setting. The conclusion concerns the entire discrete
topological fundamental group. It requires no linearity, residual
finiteness, projectivity, or prescribed Kodaira dimension. The finite
index may depend on \(X\), and torsion in \(\pi_1(X)\) is allowed.

\subsection{Kodaira dimension zero}

Campana's Theorem~6.7, applied to the zero-boundary orbifold
\((X|0)\), shows that a smooth connected compact K\"ahler manifold
of Kodaira dimension zero is special
\cite[Theorem~6.7, p.~92]{CampanaOrbifolds}. Combining this implication
with Theorem~\ref{thm:abelianity} gives the following consequence.

\begin{corollary}[Fundamental groups in Kodaira dimension zero]
\label{cor:kappa-zero-abelianity}
Let \(X\) be a smooth compact connected K\"ahler manifold with
\(\kappa(X)=\kappa(X,K_X)=0\). Then its ordinary topological fundamental
group \(\pi_1(X)\) is virtually abelian.
\end{corollary}

\begin{proof}
By Campana's established specialness implication, \(X\) is special.
Theorem~\ref{thm:abelianity} therefore applies.
\end{proof}

The specialness implication is also recorded in the orbifold core
formulation of \cite[Corollary \emph{Kodaira dimension along the core}]{OpenAIIitaka2026},
where its established provenance is noted. No projectivity,
vanishing of \(c_1(X)\), linearity, or residual-finiteness assumption
is required for Corollary~\ref{cor:kappa-zero-abelianity}.

\subsection{Conjecture S and affine-space uniformization}

We also record two geometric consequences identified by Campana.
For a connected compact K\"ahler manifold \(X\), let
\(\gamma_X:X\dashrightarrow\Gamma(X)\) be its ordinary
\(\Gamma\)-reduction. An irreducible compact analytic subspace \(Z\)
through a very general point is contained in the corresponding fibre
precisely when \(\pi_1(\widehat Z)\to\pi_1(X)\) has finite image,
where \(\widehat Z\) is the normalization of \(Z\). Set
\(\gamma d(X)=\dim\Gamma(X)\). This is the reduction for the identity
quotient of \(\pi_1(X)\), and maximal ordinary \(\Gamma\)-dimension
means \(\gamma d(X)=\dim X\)
\cite[Definition~2.2 and the paragraph preceding Conjecture~A.3]{ClaudonHoring2013}.
The two cases below are, respectively, Campana's Conjecture~S and the
bimeromorphic formulation of Iitaka's conjecture in his appendix
\cite[Conjecture~A.3 and Remark~A.4(2)]{ClaudonHoring2013}.

\begin{corollary}[Conjecture S and bimeromorphic Iitaka uniformization]
\label{cor:conjecture-s-iitaka}
Let \(X\) be a connected compact K\"ahler manifold of complex dimension
\(n\). Suppose that at least one of the following holds:
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(X\) is special and \(\gamma d(X)=n\);
\item the ordinary universal cover of \(X\) is biholomorphic to \(\C^n\).
\end{enumerate}
Then \(X\) has a connected finite \'etale cover \(X'\to X\) such that
\(X'\) is bimeromorphic to a complex torus.
\end{corollary}

\begin{proof}
Theorem~\ref{thm:abelianity} supplies the Abelianity Conjecture.
Campana proves that the Abelianity Conjecture implies Conjecture~S in
\cite[Remark~A.4(3)]{ClaudonHoring2013}. We spell out the
finite-cover step. A finitely generated virtually abelian group has
a torsion-free abelian subgroup of finite index, so take the
corresponding connected finite \'etale cover \(X'\).
Specialness persists on \(X'\). Maximal ordinary
\(\Gamma\)-dimension persists as well. A positive-dimensional compact
subvariety through a very general point with finite group image
upstairs would have an image of the same dimension downstairs.
The induced map between the normalizations is finite \'etale, so
their fundamental-group images differ by finite index. The image
downstairs would therefore have finite group image, contrary to the
\(\Gamma\)-reduction criterion.

The Albanese map of \(X'\) is surjective with connected fibres by
Theorem~\ref{thm:specialness-inputs}. Since \(\pi_1(X')\) is
torsion-free abelian, its map to the Albanese lattice
\(H_1(X',\Z)/\mathrm{torsion}\) is an isomorphism. The group of a
smooth general Albanese fibre therefore has trivial image in
\(\pi_1(X')\). Maximal ordinary \(\Gamma\)-dimension forces that
fibre to have dimension zero. Connectedness makes the general fibre
one point, so the Albanese map is bimeromorphic to its torus target.
This proves~\textup{(i)}. Campana's second implication,
Conjecture~S to the stated bimeromorphic Iitaka conclusion, is
\cite[Remark~A.4(2)]{ClaudonHoring2013}; it gives~\textup{(ii)}.
\end{proof}

\subsection{Holomorphic convexity of the universal cover}

The compact K\"ahler linear Shafarevich theorem gives a further
geometric consequence. Recall that a complex manifold is
\emph{holomorphically convex} if the holomorphic hull of every nonempty
compact subset is compact.

\begin{corollary}[Holomorphic convexity for special manifolds]
\label{cor:special-shafarevich}
Let \(X\) be a connected smooth compact K\"ahler manifold. If \(X\) is
special, then its ordinary universal cover is holomorphically convex.
\end{corollary}

\begin{proof}
The group \(G=\pi_1(X)\) is finitely generated because \(X\) is a compact
smooth manifold. By Theorem~\ref{thm:abelianity}, it has an abelian
subgroup \(H\) of finite index, which is also finitely generated.
Write \(H\cong\Z^r\oplus T\) with \(T\) finite. Decomposing \(T\) into
cyclic factors gives a faithful finite-dimensional complex
representation \(\sigma:H\to\operatorname{GL}(V)\): represent each
infinite cyclic factor by powers of \(2\) on its own coordinate and
each finite cyclic factor by a primitive root of the corresponding
order on its own coordinate.
For the trivial group, use the one-dimensional trivial representation.

The induced representation \(\operatorname{Ind}_H^G\sigma\) is finite
dimensional because \([G:H]<\infty\), and it is faithful. Indeed, in
its direct-sum realization indexed by the left cosets \(G/H\), an
element of its kernel must preserve the summand indexed by \(H\), so
it belongs to \(H\); its action on that summand is given by the faithful
representation \(\sigma\). Thus \(G\) admits a faithful finite-dimensional
complex linear representation. The compact K\"ahler linear Shafarevich theorem
of Campana--Claudon--Eyssidieux
\cite[author's version, Corollary~5.9]{CCE2015} now gives the claimed
holomorphic convexity of the ordinary universal cover.
\end{proof}

No projectivity or maximal ordinary \(\Gamma\)-dimension is required
here. The conclusion is holomorphic convexity, not Steinness: for
example, \(\PP^1\) is special and its universal cover is compact.
In contrast, the smooth projective surface constructed in
\cite{OpenAIShafarevich2026} has a universal cover that is not
holomorphically convex. That surface is therefore not special.

\subsection{Compactifiable covers}

Claudon--H\"oring's compactifiable-cover results give another
consequence. Here a \emph{Zariski-open} embedding is holomorphic and
has closed analytic complement. The two parts require different
compactifications.

\begin{corollary}[Analytically compactifiable covers]
\label{cor:compactifiable-covers}
\begin{enumerate}[label=\textup{(\roman*)}]
\item Let \(X\) be a connected normal compact K\"ahler space and
\(p:V\to X\) a connected infinite \'etale Galois covering of complex
spaces, with deck group \(\Gamma\). If \(V\) embeds as a Zariski-open
subset of a normal compact complex space, then \(\Gamma\) is virtually
abelian.
\item Let \(X\) be a connected compact K\"ahler manifold and \(U\) its
ordinary universal cover. If \(U\) embeds as a Zariski-open subset of a
normal compact K\"ahler space, then there is a connected finite \'etale
cover \(X'\to X\) whose Albanese map
\[
 \alpha_{X'}:X'\longrightarrow A=\Alb(X')
\]
is a locally trivial holomorphic fibre bundle with simply connected
compact K\"ahler fibre \(F\). Writing \(q=\dim_{\C}A\), there is a
biholomorphism
\[
 U\simeq F\times\C^q.
\]
A point is allowed as \(A\) or \(F\).
\end{enumerate}
\end{corollary}

\begin{proof}
Suppose (i) has a counterexample and choose one of minimal base
dimension. The cover is normal because it is \'etale over normal
\(X\); normalizing an analytic compactification leaves this open set
unchanged. Claudon--H\"oring's minimal-counterexample reduction
\cite[Proposition~1.4]{ClaudonHoring2013} says that this minimal base is
smooth and special. Theorem~\ref{thm:abelianity} makes its fundamental
group virtually abelian. Its quotient \(\Gamma\) is then virtually
abelian, a contradiction.

For (ii), if \(\pi_1(X)\) is infinite, apply (i) to the ordinary
universal cover; if it is finite, it is already virtually abelian.
The K\"ahler compactification hypothesis is the one in
\cite[Definition~2.3]{ClaudonHoring2013}, so
\cite[Theorem~1.5]{ClaudonHoring2013} gives the asserted Albanese bundle
on a connected finite \'etale cover \(X'\). Pull this bundle back along
the universal cover \(\C^q\to A\). The automorphism group of the
compact K\"ahler fibre \(F\) is a complex Lie group, as recalled in
\cite[Section~2.C]{ClaudonHoring2013}. The associated principal
\(\operatorname{Aut}(F)\)-bundle is topologically trivial over the
contractible base \(\C^q\), and the Oka--Grauert principle makes it
holomorphically trivial over this Stein base. Thus the pullback is
\(F\times\C^q\), the product consequence also recorded in
\cite[Conjecture~1.1]{ClaudonHoring2013}. Since \(F\) is simply
connected, the pullback is the
ordinary universal cover of \(X'\), and hence also of \(X\).
\end{proof}

In (i), the compactifying space need not be K\"ahler and the deck action
need not extend to it. For a nonuniversal cover, only the deck group
\(\Gamma\), not the entire fundamental group of \(X\), is controlled.
The stronger K\"ahler compactification is used in (ii): the finite
cover \(X'\) need not be a product, and its fibre \(F\) need not be
projective.

\subsection{Prior work and the remaining group-theoretic obstacle}

Campana introduced specialness together with its orbifold framework
and the special/general-type decomposition in \cite{Campana2004}.
The same work established restrictions on solvable and linear
quotients of special fundamental groups. An important advance for the
full group was the proof of the abelianity conjecture for compact
K\"ahler threefolds by Campana--Claudon
\cite[Theorem~1.1]{CampanaClaudonThreefolds}, using the geometry of
orbifold surfaces. Their later work proves specialness of general
Albanese fibres in the smooth projective setting and reduces the
projective conjecture to a finiteness question when all finite covers
have irregularity zero
\cite[author's version, Theorem~2.4 and Proposition~3.2]{CampanaClaudon}.
These results explain the role of finite covers and Albanese fibre
groups in the argument below. We do not assume specialness of the
general Albanese fibre in the arbitrary compact K\"ahler case.

A classical predecessor is the case \(c_1(X)=0\) in real cohomology:
Beauville's decomposition gives a finite \'etale cover that is a
product of a complex torus and simply connected factors, and therefore
gives virtual abelianity in that case
\cite[Section~5, Theorem~2]{Beauville1983}.

The linear case has a separate geometric structure theory.
Building on work of Zuo and Eyssidieux, Campana--Claudon--Eyssidieux
describe linear Shafarevich reductions
by torus fibrations over bases of general type after finite cover and
modification \cite[author's version, Theorem~6.5]{CCE2015}.
Their semisimple theorem gives a general-type base for the torsion-free
representations used here
\cite[author's version, Theorems~1 and~6.3]{CCE2015}.
This supplies one branch of our proof. An infinite group, however,
need not have an infinite finite-dimensional complex linear image.
Controlling these remaining quotients is essential for a statement
about the full fundamental group.

The relation between fundamental groups and holomorphic tensors has
several relevant predecessors. Brunebarbe--Klingler--Totaro prove that
an infinite finite-dimensional linear image over any field forces a
nonzero symmetric differential on a compact K\"ahler manifold
\cite[Theorem~0.1]{BrunebarbeKlinglerTotaro2013}.
Brunebarbe--Campana prove that vanishing of all positive symmetric
powers of all exterior cotangent powers forces simple connectedness;
their argument develops the use of \(L^2\) forms and the
Poincar\'e--Atiyah--Gromov method
\cite[Theorem~1.1]{BrunebarbeCampana2016}.
Specialness permits many nonzero tensors, so their vanishing
hypothesis differs from ours. The task here is to obtain sufficient
\emph{positivity} from tensors with unitary Hilbert space coefficients.
Only after constructing a big ordinary line bundle, whose Iitaka
dimension equals the dimension of the base, and establishing projectivity do we invoke the orbifold cotangent positivity theorem
of Campana--P\u aun \cite[Theorem~7.11]{CampanaPaun2019}.

The spectral step also belongs to the study of \(L^2\) invariants of
infinite coverings. Atiyah's index theorem and regular-coefficient
bundle interpretation \cite[Theorem~3.8 and Section~6.1]{Atiyah1976},
and Dodziuk's reduced \(L^2\) de Rham--Hodge theory
\cite{Dodziuk1977}, provide foundational models for working on a
cover through Hilbert coefficients downstairs. Lott surveys the
zero-in-the-spectrum question and several geometric cases
\cite{Lott1996}. The unrestricted Riemannian assertion has
counterexamples \cite{FarberWeinberger2001}; the K\"ahler spectral
statement needed here is proved in Section~\ref{sec:spectral}.

A second issue is geometric descent over a possibly nonprojective
torus. Campana's correction to the Albanese theorem preserves
surjectivity and connectedness for special compact K\"ahler
manifolds, but its repaired assertion excluding multiple fibres
requires a projective target \cite[Theorem~1.2]{CampanaCorrection}.
Our proof therefore keeps the actual orders of meridians in the
chosen group quotient, including on exceptional divisors of new
models. Juanyong Wang's subadditivity and Albanese theorems provide
the required geometric input over arbitrary complex tori
\cite[Theorems~A(II) and~C]{JWang2021}. They apply to the smooth klt
pairs arising here: the boundary has simple normal crossings and
rational coefficients strictly less than one.

\subsection{Overview of the proof}

\paragraph{The group to be excluded.}
Finite \'etale covers preserve specialness. Since the
Albanese map of each cover is surjective, its first Betti number is
bounded by twice the dimension of \(X\). Pass to a finite cover on
which that number is maximal. Write
\(\alpha:X\to A=\Alb(X)\), and let \(N\) be the image in
\(G=\pi_1(X)\) of the fundamental group of a smooth general
Albanese fibre. We prove
\[
 1\longrightarrow N\longrightarrow G\longrightarrow
 \pi_1(A)\longrightarrow1,
\]
and show that \(N\) is finitely generated and \emph{FAb}: every
finite-index subgroup of \(N\) has finite abelianization.
The latter step uses Botong Wang's torsion theorem for isolated
characters in cohomology jump loci \cite[Theorem~1.3]{BWang2016}.
It remains to exclude infinite \(N\); a finite kernel of a lattice
quotient gives virtual abelianity by elementary group theory.

\paragraph{A minimal base carrying an infinite quotient.}
Assuming \(N\) infinite, consider fibrations of Albanese fibres on
all finite covers for which the quotient by the fibre-group image
is still infinite. Minimize the dimension \(\ell>0\) of their
bases. Compact analytic cycle spaces and Campana's meromorphic
\(\Gamma\)-reduction \cite{Barlet1975,CampanaGamma} globalize a
minimizer to a factorization
\[
 X\dashrightarrow Z\longrightarrow A.
\]
If \(B_0\) is the actual group image of a general fibre of the first
map, then \(Q=N/B_0\) is an infinite subgroup of \(R=G/B_0\), and
\[
 1\longrightarrow Q\longrightarrow R\longrightarrow\pi_1(A)
 \longrightarrow1.
\]
A very general fibre \(Y\) of \(Z\to A\) has dimension \(\ell\).
The actual meridian orders in \(R\) restrict to the same orders in
\(Q\) on \(Y\). For each boundary divisor \(D\subset Y\), denote
this finite order by \(m_D\), and put
\(\Delta_Y=\sum_D(1-m_D^{-1})D\).
The resulting orbifold \((Y,\Delta_Y)\) maps onto \(Q\) on
fundamental groups. Minimality holds simultaneously after the
compatible finite covers and for every further infinite quotient.
This is the input to the two branches in Figure~\ref{fig:proof}.

\paragraph{Positivity in the absence of an infinite linear image.}
Theorem~\ref{thm:zero-spectrum} first shows that an infinite normal
cover of a compact K\"ahler orbifold has zero in the spectrum of
its scalar Dolbeault Laplacian, with ordinary complex-valued
coefficients, in \emph{some} holomorphic degree.
The argument places the compact diagonal on
\((U\times U)/P\), where \(U\) is the cover and its deck
group \(P\) acts diagonally. The diagonal \(\{(u,u):u\in U\}\)
therefore has compact quotient. Under a contrary spectral gap, weighted solutions and
finite-propagation smoothing preserve the nonzero cohomology class
of the diagonal after push-down, while making its pairing tend to
zero. The contradiction produces the required low spectrum.

When every finite-dimensional complex linear image of \(Q\) is
finite, we construct nonzero holomorphic tensors with unitary Hilbert
coefficients. We use the spectral construction in the nonamenable
case and Shalom's reduced-cohomology theorem in the amenable case
\cite[Theorem~4.2]{Shalom2006}. Taking exterior powers gives tensors with rank-one coefficient
image; these lines define maps to Hilbert projective space. We minimize the rank among such
maps, prove discreteness of the monodromy on their regular image
germs, and compactify the fibres using transverse volume bounds.
A rank smaller than \(\ell\) would contradict the minimal-base
condition. Full rank yields a big cotangent line, and hence
\(K_Y+\Delta_Y\) is big by Theorem~\ref{thm:hilbert-positivity}.
When \(Q\) instead has an infinite linear image, its FAb property
produces the required semisimple image and minimality gives generic
largeness. The general-type criterion of
Campana--Claudon--Eyssidieux then gives \(K_Y\) big.

\paragraph{Descent and the contradiction.}
Both branches give fibre positivity on every sufficiently prepared
model. Theorem~\ref{thm:torus-descent} turns it into a
positive-dimensional orbifold base of general type whose orbifold
pluriforms pull back
to a Bogomolov sheaf on a finite cover of \(X\). The key point is
to retain actual quotient meridian orders on each model; no
birational invariance of the associated log Kodaira dimension is
assumed. Differential vanishing cancels the boundary poles on a
neat model. Specialness excludes the resulting sheaf, so \(N\)
is finite and Theorem~\ref{thm:abelianity} follows.

\begin{figure}[htbp]
\centering
\input{figures/proof-overview}
\caption{The contradiction when the Albanese fibre-group image
\(N\) is infinite. The minimal quotient \(Q\) and the fibre
\((Y,\Delta_Y)\) are constructed together. Each branch must give
positivity on all sufficiently prepared models before torus descent
applies. The arrows show logical implications.}
\label{fig:proof}
\end{figure}

\paragraph{Organization.}
Section~\ref{sec:preliminaries} fixes specialness, group reductions,
and the precise external geometric inputs.
Section~\ref{sec:spectral} proves the scalar spectral theorem.
Section~\ref{sec:hilbert} constructs the Hilbert tensors and proves
their positivity theorem under the minimality condition.
Section~\ref{sec:descent} develops meridian bookkeeping and torus
descent. Section~\ref{sec:completion} establishes the Albanese exact
sequence and FAb property, constructs the minimal relative base,
and assembles the two representation cases to finish the proof.
==== END The-abelianity-conjecture-for-special-compact-Kahler-manifolds//sections/introduction.tex ====

==== BEGIN The-abelianity-conjecture-for-special-compact-Kahler-manifolds//sections/preliminaries.tex ====
\section{Geometric and group-theoretic preliminaries}\label{sec:preliminaries}

We collect the geometric input that detects a forbidden Bogomolov
sheaf and the group lemmas that will convert a finite Albanese fibre
image into virtual abelianity.

All manifolds are connected unless stated otherwise. A fibration is a
dominant meromorphic map with connected general fibre. We resolve its
graph when making differential or fundamental-group arguments. A smooth
compact space in Fujiki's class \(\mathcal C\) is replaced, when needed,
by a compact K\"ahler modification. Smooth compact bimeromorphic models
have the same fundamental group. Fibre groups always mean groups of
smooth models of general fibres; their images are understood up to
base-point transport. The image of a general fibre group is normal in
the total fundamental group: over a smooth fibration locus this follows
from the homotopy exact sequence, and the fundamental group of the open
total space surjects onto that of the total space.

\subsection{Specialness and quotient reductions}

We recall the precise established inputs used in the proof.

\begin{theorem}[Bogomolov--Campana]\label{thm:specialness-inputs}
For smooth compact K\"ahler manifolds, specialness in the definition of
Section~\ref{sec:introduction} is bimeromorphically invariant and is
preserved by connected finite \'etale covers. The Albanese morphism
of a special manifold is surjective with connected fibres. Moreover, if
a line bundle \(L\) has a nonzero morphism to \(\Omega_X^p\), then
\(\kappa(X,L)\le p\).
\end{theorem}

The Bogomolov-sheaf characterization and stability under finite
\'etale covers are given in
\cite[author's version, Theorems~2.26 and~5.12]{Campana2004};
the remaining specialness properties are developed there. The Iitaka
bound originates in Bogomolov's projective work \cite{Bogomolov1979};
for its compact K\"ahler formulation, see \cite{Campana2015Survey}.
The Albanese statement is recorded with its correction in
\cite[Theorem~1.2]{CampanaCorrection}. Related projective
stability results are developed in \cite{CampanaClaudon}. We only use
surjectivity and connectedness from that correction. The meridian
statement needed for arbitrary, possibly nonprojective Albanese tori
will be proved in Lemma~\ref{lem:albanese-exactness}.
Saturating a rank-one subsheaf and taking its double dual does not
decrease its Iitaka dimension. Thus the line-bundle definition agrees
with the saturated Bogomolov-sheaf formulation.

\begin{lemma}[Effective divisors on a torus]\label{lem:torus-divisor}
Let \(D\ne0\) be an effective rational divisor on a compact complex
torus \(A\). There are a quotient homomorphism \(p:A\to B\) with
connected fibres, where \(B\) is a positive-dimensional abelian
variety, and an ample effective rational divisor \(D_B\) on \(B\)
such that \(D=p^*D_B\). In particular, \(\kappa(A,D)>0\).
\end{lemma}
\begin{proof}
We give the standard torus argument; see also \cite{Debarre1999}.
After multiplying \(D\) by a positive integer, assume it is integral.
Average its positive integration current \([D]\) over translations of
\(A\). The result is a smooth translation-invariant semipositive
\((1,1)\)-form \(\theta\) representing
\(c_1(\mathcal O_A(D))\). Write \(A=V/\Lambda\). The alternating
form of \(\theta\) is integral on \(\Lambda\), so its radical \(W\)
is defined over \(\mathbb Q\). It is a complex subspace because
\(\theta\) has type \((1,1)\). Thus
\(K=W/(W\cap\Lambda)\) is a subtorus. Let \(p:A\to B=A/K\) be
the quotient.

On every \(K\)-coset, \(\mathcal O_A(D)\) has zero real first Chern
class. If its defining section is not identically zero on that
coset, its zero divisor must be empty: a nonzero effective divisor
on the positive-dimensional K\"ahler torus \(K\) has strictly
positive pairing with the \((\dim K-1)\)-st power of a K\"ahler class.
For \(K=0\) the same conclusion is immediate. Each coset is
therefore either contained in \(\operatorname{Supp}D\) or disjoint
from it. If \(B\) were a point, this argument with \(K=A\) would
make the defining section of \(D\) nowhere zero, contrary to
\(D\ne0\). Hence \(\dim B>0\).

Translations by the connected group \(K\) consequently preserve
each prime component \(D_i\) of \(D\), since a connected group
cannot permute their finite set.
The image \(\overline D_i=p(D_i)\) is a prime divisor on \(B\):
properness gives an analytic image, and invariance gives
\(\dim\overline D_i=\dim D_i-\dim K=\dim B-1\).
Moreover \(p^{-1}(\overline D_i)=D_i\), and smoothness of \(p\)
gives \(p^*\overline D_i=D_i\) with multiplicity one. Keeping the
coefficients of the \(D_i\) therefore defines an effective integral
divisor \(D_B\) with \(D=p^*D_B\).

Since \(p^*D_B=D\), uniqueness of translation-invariant cohomology
representatives identifies \(c_1(\mathcal O_B(D_B))\) with the
positive definite form induced by \(\theta\) on \(B\). The positivity
criterion for line bundles on a complex torus makes \(D_B\) ample
and \(B\) an abelian variety. Dividing \(D_B\) by the initial integer
proves the rational statement. Ratios of
sections of its ample multiples remain independent after pullback,
so \(\kappa(A,D)\ge\dim B>0\).
\end{proof}

\begin{theorem}[Meromorphic \(\Gamma\)-reduction]\label{thm:gamma-reduction}
Let \(M\) be a smooth compact K\"ahler manifold and let
\(\rho:\pi_1(M)\to P\) be a homomorphism. There is a fibration
\(\gamma_\rho:M\dashrightarrow B_\rho\) whose general fibre has finite
fundamental-group image under \(\rho\), and which contracts every
compact irreducible subvariety through a very general point whose
fundamental group has finite image under \(\rho\). The groups of
subvarieties may be computed on their smooth resolutions. The base has
a compact K\"ahler model.
\end{theorem}

The contraction assertion is Campana's \(\Gamma\)-reduction
\cite[Theorem~3.5 and Remark~3.6]{CampanaGamma}; the arbitrary-quotient
formulation is also stated in
\cite[author's version, Theorem~2.1 and Corollary~2.2]{CCE2015}.
The K\"ahler-model assertion uses separately the standard stability
of Fujiki's class \(\mathcal C\) under meromorphic images: the base
belongs to \(\mathcal C\), and a smooth model admits a compact
K\"ahler modification.
We call \(\dim B_\rho\) the \emph{\(\Gamma\)-dimension} for \(\rho\),
and call \(\rho\) \emph{generically large} when this dimension is
\(\dim M\). The target group need not be linear. The qualifier
``very general'' means outside a countable union of proper analytic
subsets. The contraction criterion also applies when the relevant
subvarieties exist through a full-measure set: intersect this set with
the very general locus in the theorem.

\begin{theorem}[Linear general-type criterion]\label{thm:linear-general-type}
Suppose a compact K\"ahler manifold has a generically large complex
linear representation with torsion-free image and connected simple
semisimple Zariski closure. Then it is of general type.
\end{theorem}

\begin{proof}
Let \(M\) be the manifold, and regard \(\rho\) as a Zariski-dense
representation into its semisimple closure. By
\cite[author's version, Theorem~1 and Theorem~6.3]{CCE2015}, it has
a Shafarevich morphism
\[
 \operatorname{sh}_\rho:M\longrightarrow\operatorname{Sh}_\rho(M)
\]
with normal projective target of general type. This morphism is a
model of the connected \(\Gamma\)-reduction
\cite[author's version, Definition~2.13]{CCE2015}. Generic largeness
gives \(\dim\operatorname{Sh}_\rho(M)=\dim M\). The connected general fibre
is then a point, so \(\operatorname{sh}_\rho\) is bimeromorphic.
Bimeromorphic invariance of Kodaira dimension proves that \(M\) is
of general type.
\end{proof}

\subsection{Root orbifolds and adjoint positivity}

For a smooth compact K\"ahler manifold \(Y\) and a simple normal
crossing divisor \(\sum D_i\) with integers \(m_i\ge1\), let
\(\cY\) be the analytic root orbifold of orders \(m_i\), and write
\[
 \Delta=\sum_i(1-m_i^{-1})D_i.
\]
Locally, if the divisor coordinates are \(z_i\), its charts have
\(z_i=w_i^{m_i}\) with the corresponding product of cyclic groups.
Tensors and smooth metrics are invariant tensors and smooth metrics
upstairs. Its fundamental group is the fundamental group of the
divisor complement modulo the normal subgroup generated by the
\(m_i\)-th powers of meridians. Orders equal to one add no boundary.
Orbifold covering space theory supplies a connected cover for each
subgroup; the cover need not be a manifold.

These orbifolds are K\"ahler. Add to a sufficiently large pullback
of a coarse K\"ahler form the \(\ii\partial\bar\partial\) of local
squared root norms, using smooth metrics on the divisor bundles and
cutoffs. Their root-direction terms are positive near the divisor and
their negative parts away from it are bounded by the coarse metric.
On a cover of a compact orbifold, finitely many uniformizing charts
give uniform local elliptic estimates, bounded-overlap partitions and
cutoffs. All complete-metric differential operators below are their
closed \(L^2\) extensions. We use
\(\dd\dc=\ii\partial\bar\partial\).

\begin{theorem}[Klt K\"ahler results over tori]\label{thm:wang-torus}
Let \(Z\) be a smooth compact K\"ahler manifold, let \(D\) be an
effective rational divisor such that \((Z,D)\) is klt, and let
\(g:Z\to A\) be a fibration onto a complex torus. For a general
smooth fibre \(F\),
\[
 \kappa(Z,K_Z+D)\ge\kappa(F,K_F+D|_F).
\]
If \(\kappa(Z,K_Z+D)=0\), the Albanese morphism of \(Z\) is
surjective with connected fibres.
\end{theorem}

These are Theorems~A(II) and~C of \cite{JWang2021}; the torus is
not required to be projective. We apply them to smooth pairs with
simple normal crossing boundaries and coefficients strictly less
than one.

For a projective smooth pair, we also use the bigness consequence of
orbifold cotangent positivity in \cite[Theorem~7.11]{CampanaPaun2019}:
a big ordinary line bundle injecting into a tensor power of the
orbifold cotangent, on adapted covers, forces the log canonical
divisor to be big. The projectivity hypothesis will be established
before this result is applied in Section~\ref{sec:hilbert}.

\subsection{Finite kernels and finite covers}

\begin{lemma}\label{lem:finite-by-abelian}
Let \(E\) be a finitely generated group in an exact sequence
\[
 1\longrightarrow F\longrightarrow E\longrightarrow\Z^r
 \longrightarrow1
\]
with \(F\) finite. Then \(E\) is virtually abelian and residually
finite. In particular it has a finite-index subgroup disjoint from
\(F\).
\end{lemma}
\begin{proof}
The centralizer \(E_0\) of \(F\) has finite index. Its commutator
subgroup is contained in the finite central group \(F\cap E_0\).
Choose generators \(x_1,\ldots,x_t\) of \(E_0\) and an integer
\(e>0\) annihilating \(F\). Commutators are central, so
\([x_i^e,x_j^e]=[x_i,x_j]^{e^2}=1\).
The subgroup generated by the \(x_i^e\) is abelian and has finite
index: its image has finite index in the finitely generated
abelianization, and the commutator subgroup is finite.
This abelian subgroup has a torsion-free subgroup of finite index.
Its core in \(E\) is a normal free abelian subgroup of finite index.
Multiples of this core have finite index in \(E\) and separate its
nonzero elements; the finite quotient by the core separates elements
outside it. Hence \(E\) is residually finite. Since \(F\) is finite,
intersecting finitely many finite-index subgroups separates all its
nonidentity elements at once.
\end{proof}

\begin{lemma}\label{lem:realize-finite-index}
Suppose \(G\) has a finitely generated normal subgroup \(N\) with
\(G/N\cong\Z^r\). For every finite-index subgroup \(N_0\le N\),
there is a finite-index subgroup \(G_0\le G\) satisfying
\(G_0\cap N\subseteq N_0\).
\end{lemma}
\begin{proof}
A finitely generated group has only finitely many subgroups of any
given finite index. Intersect all automorphic images of the core of
\(N_0\) in \(N\). The result is a characteristic finite-index
subgroup \(K\le N\) contained in \(N_0\), and hence is normal in
\(G\). The group \(G/K\) is finite-by-\(\Z^r\).
By Lemma~\ref{lem:finite-by-abelian}, it has a finite-index subgroup
disjoint from \(N/K\). Its inverse image is a suitable \(G_0\).
\end{proof}

A group is \emph{FAb} if every finite-index subgroup has finite
abelianization. Quotients of FAb groups and their finite-index
subgroups are FAb. A finitely generated complex linear FAb group
with virtually solvable image has finite image: after passage to
finite index its Zariski closure is connected and solvable and hence
triangularizable. The image is then solvable, and applying finite
abelianization successively to the finite-index derived subgroups
shows that the image is finite. This observation is used only for
linear images; it imposes no linearity hypothesis on the original
fundamental group.
==== END The-abelianity-conjecture-for-special-compact-Kahler-manifolds//sections/preliminaries.tex ====

==== BEGIN The-abelianity-conjecture-for-special-compact-Kahler-manifolds//sections/spectral.tex ====
\section{Zero in the Dolbeault spectrum}
\label{sec:spectral}

We first establish the analytic input needed for representations on Hilbert
spaces.  All metrics and differential forms on an orbifold are smooth in its
finite uniformizing charts.  In particular, the covering in the following
theorem need not be a manifold.

\begin{theorem}\label{thm:zero-spectrum}
Let $(M,\omega)$ be a connected compact K\"ahler orbifold of complex
dimension $n$, and let $U\to M$ be a connected normal orbifold covering with
infinite deck group $P$.  For some $p\in\{0,\ldots,n\}$, zero belongs to the
spectrum of the $L^2$ Dolbeault Laplacian on scalar $(p,0)$-forms on $U$,
with the lifted metric.
\end{theorem}

We use the nonnegative Laplacian
$\Delta''=\dbar\dbar^*+\dbar^*\dbar$.
For functions, the operator
$\tr_\omega\dd\dc$ has the opposite sign, up to a fixed positive
normalization.  Norms without a bundle subscript use the fixed bundle
metrics introduced below.

The invariant throughout the proof is the cohomology class of the
compact diagonal after push-down to the compact product. The estimates
below construct representatives with decreasing norm while controlling
the volume of their supports. We first build negative potentials and
a weighted inverse, then localize and smooth repeatedly. The final
pairing with the diagonal class makes these estimates incompatible
with a spectral gap in every degree.

\subsection{Complete operators and uniform local estimates}
\label{sp:operators}

Lifting a finite collection of nested uniformizing charts on $M$ gives
uniform coordinate radii, uniformly equivalent Euclidean metrics, bounds
for all metric derivatives, and partitions of unity with bounded overlap
and bounded derivatives.  A lifted chart has a subgroup of the original
finite uniformizing group.  The orders of all such groups are consequently
bounded.  Euclidean estimates applied to invariant lifts descend with
uniform constants, since the chart integrals differ from the orbifold
integrals by these bounded orders.  The same assertions hold on coverings
of $M\times M$ and for vector bundles with metrics pulled back from the
compact base.

We write $H^k$ for the real $L^2$ Sobolev spaces defined with these charts.
The local interior elliptic estimates, followed by summation over bounded
overlap, have the following consequence.  If $A$ is a scalar uniformly
elliptic second-order operator, or a system with scalar uniformly elliptic
principal symbol, whose coefficients have bounded derivatives of all
orders, then
\begin{equation}\label{sp:global-elliptic}
 \|v\|_{H^{k+2}}
 \le C_k\bigl(\|Av\|_{H^k}+\|v\|_2\bigr).
\end{equation}
Here the estimate applies whenever the right side is finite, initially in
the distributional sense.  One obtains it first on a smaller chart from
the corresponding larger chart and then sums the squared estimates.
The constants depend only on the ellipticity and the indicated coefficient
bounds; see the interior estimates in
\cite[Theorems~6.2 and~9.11]{GilbargTrudinger2001}, followed by
differentiation for higher Sobolev orders.
Sobolev embedding is uniform for the same reason.  In particular, an
$H^k$ function, with $k$ sufficiently large, tends to zero at infinity
together with any prescribed fixed number of derivatives: the local
Sobolev norms on fixed-radius charts outside a compact set are bounded by
the tail of its global Sobolev norm.

The lifted metric is complete.  Smooth compactly supported cutoffs
$\chi_R$, tending to one, can be chosen with
$|\dd\chi_R|\le C/R$ by smoothing truncated distance functions in the
uniform charts.  They show that the minimal and maximal closed extensions
of $\dbar$ agree.  They also justify the identities involving formal
adjoints: first approximate locally by smooth invariant sections and then
use
\[
 [\dbar,\chi_R]=\dbar\chi_R\wedge\ ,
 \qquad
 \|[\dbar,\chi_R]\|\le C/R.
\]
The same commutator bound holds for its adjoint and for the Chern
differential.  Thus compactly supported smooth sections form a core for
the associated quadratic forms.  Equivalently, the symmetric Dolbeault
Dirac operator $\sqrt2(\dbar+\dbar^*)$ is essentially self-adjoint.  For
completeness, its deficiency equation $D^*v=\pm\ii v$, tested against
$\chi_R^2v$, gives
$\|\chi_Rv\|_2^2\le C R^{-1}\|v\|_2^2$; local elliptic regularity
makes this test legitimate.  Letting $R$ tend to infinity eliminates both
deficiency spaces.  These arguments remain valid with each of the smooth
bundle multipliers used below, which have positive upper and lower bounds
for each fixed value of the parameters.  The K\"ahler identities
and the Bochner--Kodaira identity therefore extend from compact supports
to their complete-metric form domains; their local identities are those
of \cite[Chapter~VI, Section~6, and Chapter~VII, Theorems~1.1--1.2]{DemaillyCADG}.

Suppose, towards a contradiction to Theorem~\ref{thm:zero-spectrum}, that
all the scalar $(p,0)$ Laplacians have positive lower bounds.  Since there
are only finitely many degrees, choose a common lower bound $\delta>0$,
decreasing it whenever necessary by a fixed normalization constant.
The pointwise Lefschetz isomorphisms
\[
 L^{n-p}:\Lambda^{p,0}\longrightarrow\Lambda^{n,n-p},
 \qquad L=\omega\wedge\ ,
\]
commute with the scalar Laplacian and, after normalization, are isometries.
Thus the same type of positive lower bound holds on every $(n,q)$ degree.
In degree zero it gives
\begin{equation}\label{sp:poincare}
 \|v\|_2^2\le C\|\dd v\|_2^2,
 \qquad v\in H^1(U).
\end{equation}
The group $P$ is infinite, so $U$ is noncompact and $n>0$.

\subsection{Negative potentials with controlled mass}
\label{sp:potentials}

We need negative potentials with increasing depth on each fixed compact
set and controlled global \(L^2\) mass. The depth will make the later
weights large near that set, while the mass bound limits the volume of
regions where the potential is very negative and hence the growth of
supports during localization.

\begin{lemma}\label{sp:potential-lemma}
Under \eqref{sp:poincare}, there are smooth functions $u_N$ on $U$, for
all sufficiently large integers $N$, such that
\begin{equation}\label{sp:potential-bounds}
 u_N\le0,\qquad \dd\dc u_N\ge-\omega,
 \qquad \|u_N\|_2\le C N^n.
\end{equation}
For every compact set $K\subset U$, there are $c_K>0$ and $N_K$ with
\begin{equation}\label{sp:potential-depth}
 \sup_Ku_N\le-c_KN\qquad(N\ge N_K).
\end{equation}
For each fixed $N$, the function $u_N$ belongs to every $H^k$ and has
bounded derivatives of all orders.
\end{lemma}

\begin{proof}
Fix a nonnegative $f\in C_c^\infty(U)$ which is strictly positive on a
chart ball, and put
\[
 F_N=\log(1+N^nf),\qquad \lambda=N^{-2n}.
\]
We solve
\begin{equation}\label{sp:ma-equation}
 \omega_u^n=e^{F_N+\lambda u}\omega^n,
 \qquad \omega_u=\omega+\dd\dc u>0.
\end{equation}
We give the complete continuity argument, including the global estimates
which distinguish this problem from a bounded-solution existence result.

Fix an integer $k>n+6$ and consider
\[
 \mathcal B=\{u\in H^{k+2}(U,\R):
                 \omega_u\ge c\omega\text{ for some }c>0\}.
\]
This is open in $H^{k+2}$.  The map
$u\mapsto\log(\omega_u^n/\omega^n)-\lambda u$ is smooth from
$\mathcal B$ to $H^k$, by Sobolev multiplication and composition in the
uniform charts.  Its linearization at $u$ is
\begin{equation}\label{sp:ma-linearization}
 v\longmapsto\tr_{\omega_u}\dd\dc v-\lambda v.
\end{equation}
At a smooth solution with bounded derivatives this is an isomorphism
$H^{k+2}\to H^k$.  Indeed its negative is coercive on $H^1$ when
integrated with $\omega_u^n$: its quadratic form is a positive multiple
of the gradient norm plus $\lambda\|v\|_2^2$.  Lax--Milgram gives an
$H^1$ inverse, and \eqref{sp:global-elliptic} gives the asserted higher
regularity and bounded inverse.  The metric $\omega_u$ is complete and
uniformly equivalent to $\omega$ at this individual solution.  A solution
in $\mathcal B$ is smooth with bounded derivatives by local elliptic
bootstrapping, since initially $k>n+6$.  The implicit function theorem
therefore gives openness for
\begin{equation}\label{sp:ma-path}
 \log(\omega_u^n/\omega^n)-\lambda u=sF_N,
 \qquad 0\le s\le1.
\end{equation}
At $s=0$ the solution is $u=0$.

We next obtain estimates uniform in $s$, with $N$ fixed.  Every solution
in $\mathcal B$ tends to zero at infinity with its first several
derivatives.  At an interior maximum of $u$, the determinant ratio is at
most one; at an interior minimum it is at least one.  The same bounds
hold when an extremum is approached at infinity, where $u$ tends to zero.
Consequently
\begin{equation}\label{sp:ma-czero}
 -\lambda^{-1}\|F_N\|_\infty\le u\le0.
\end{equation}
In particular the determinant ratio in \eqref{sp:ma-path} has positive
upper and lower bounds depending only on $N$.

Here are the second-order and higher local estimates in the order in
which they are needed.  Write $\Delta_u=\tr_{\omega_u}\dd\dc$.
Since $\tr_\omega\dd\dc u\ge-n$,
\[
 \Delta_\omega(sF_N+\lambda u)\ge-C_N-\lambda n.
\]
The differential trace estimate
\cite[Lemma~2.2]{CampanaGuenanciaPaun2013} is
\[
 \Delta_u\log\tr_\omega\omega_u
 \ge\frac{\Delta_\omega(sF_N+\lambda u)}
            {\tr_\omega\omega_u}
       -B\tr_{\omega_u}\omega.
\]
It gives
\begin{equation}\label{sp:yau-trace}
 \Delta_u\log\tr_\omega\omega_u
 \ge-B\tr_{\omega_u}\omega-C_N.
\end{equation}
In this estimate $B$ depends on a lower bound for the background
holomorphic bisectional curvature.  The possible negative term involving
$\Delta_\omega(sF_N+\lambda u)$ is divided by
$\tr_\omega\omega_u$, which is bounded below by the determinant lower
bound.
Apply \eqref{sp:yau-trace} to
$Q=\log\tr_\omega\omega_u-Au$, where $A>B+1$ is fixed.  Since
$\Delta_u u=n-\tr_{\omega_u}\omega$, at a maximum of $Q$ one obtains
\[
 (A-B)\tr_{\omega_u}\omega\le An+C_N.
\]
If the supremum is not attained, it is bounded by the value $\log n$ at
infinity.  At an attained maximum, the determinant upper bound and the
inverse-trace bound control $\tr_\omega\omega_u$ there.  The oscillation
bound in \eqref{sp:ma-czero} then controls it everywhere.  It follows that
\begin{equation}\label{sp:ma-equivalence}
 c_N\omega\le\omega_u\le C_N\omega
\end{equation}
uniformly along the path.

This also bounds the background real Laplacian of $u$.  Local Poisson
$W^{2,p}$ estimates, for $p$ larger than the real dimension, give a
uniform $C^{1,\alpha}$ bound on smaller charts.  The equation now has a
uniformly elliptic complex Hessian and a controlled H\"older right side.
Fix $0<\beta<\alpha$.  The local complex Monge--Amp\`ere
$C^{2,\beta}$ estimate
\cite[arXiv:1111.0902v1, Theorem~1.1]{YuWang2012} applies after adding
a local potential for $\omega$: the resulting function is strictly
plurisubharmonic with local sup norm controlled by \eqref{sp:ma-czero},
its Euclidean Laplacian is bounded by \eqref{sp:ma-equivalence}, and
the positive density is bounded below with its $n$-th root uniformly
$C^\alpha$.  Differentiation of the equation and local Schauder
estimates then give bounds for every derivative.  These are local bounds
uniform over all lifted charts and
all $s$, with constants depending on $N$ and the derivative order; no
global higher Sobolev bound has been assumed in obtaining them.

There is a global estimate with a better dependence on $N$.  Integration
by parts gives
\begin{align}
 \int_U(-u)(\omega_u^n-\omega^n)
 &=\sum_{a=0}^{n-1}\int_U
   \ii\partial u\wedge\dbar u\wedge
   \omega_u^a\wedge\omega^{n-1-a},\label{sp:ma-energy}\\
 c\|\dd u\|_2^2
 &\le\int_U(-u)(\omega_u^n-\omega^n)
 \le N^n\|f\|_2\|u\|_2.\label{sp:ma-energy-bound}
\end{align}
The first mixed term is the fixed background metric, so the lower
constant is independent of $N$.  For the upper bound use $u\le0$ and
\[
 e^{sF_N+\lambda u}-1\le (1+N^nf)^s-1\le N^nf.
\]
The integrand in \eqref{sp:ma-energy} is absolutely integrable: away from
$\supp f$, its absolute value is at most $\lambda u^2$, and on
$\supp f$ it is integrable.  To justify the integration by parts, insert
the complete-metric cutoffs.  The mixed metrics are bounded by
\eqref{sp:ma-equivalence}, and the cutoff errors tend to zero by
$u,\dd u\in L^2$.  Combining \eqref{sp:ma-energy-bound} with
\eqref{sp:poincare} gives
\begin{equation}\label{sp:ma-ltwo}
 \|u\|_2\le C N^n
\end{equation}
uniformly in $s$ and $N$.

It remains to establish closedness in the global Banach space.  In local
coordinates, \eqref{sp:ma-path} can be written
\[
 a_u^{i\bar j}u_{i\bar j}-\lambda u=sF_N,
 \qquad
 a_u^{i\bar j}=\int_0^1(g_{a\bar b}+t u_{a\bar b})^{i\bar j}\,dt.
\]
The superscript $i\bar j$ denotes the corresponding entry of the inverse
Hermitian matrix.
The coefficients have uniform ellipticity and all required derivative
bounds by the preceding local estimates.  Thus
\eqref{sp:global-elliptic} and \eqref{sp:ma-ltwo} give global bounds in
every fixed Sobolev order.  If $s_\nu\to s$, a subsequence of the
corresponding solutions converges smoothly on compact sets.  The limit
solves \eqref{sp:ma-path}, satisfies the uniform positive lower bound in
\eqref{sp:ma-equivalence}, and belongs to $H^{k+2}$ by lower
semicontinuity of the global norms.  It is therefore still in
$\mathcal B$.  This proves closedness and hence solvability at $s=1$.
Denote that solution by $u_N$.  The first three assertions of
\eqref{sp:potential-bounds} and the stated regularity have been proved.

Suppose \eqref{sp:potential-depth} fails for a fixed nonempty compact set
$K$.  Along a subsequence,
$\sup_K(u_N/N)\to0$.  Local plurisubharmonic compactness, after adding
a local potential for $\omega/N$, gives an $L^1_{\mathrm{loc}}$ limit
$v$ which is plurisubharmonic and nonpositive.  The alternative of
uniform collapse to $-\infty$ is excluded by the supremum on $K$.
Hartogs' lemma implies that $v$ attains zero on $K$.  Since $U$ is
connected, the maximum principle gives $v=0$ everywhere.

The trace--determinant inequality applied to \eqref{sp:ma-equation}
gives
\begin{equation}\label{sp:ma-trace-limit}
 \tr_\omega\dd\dc(u_N/N)
 \ge n f^{1/n}e^{\lambda u_N/n}-n/N.
\end{equation}
Pass to a further almost-everywhere convergent subsequence of $u_N/N$.
As $n\ge1$,
$\lambda u_N=N^{1-2n}(u_N/N)\to0$ almost everywhere.  The right side of
\eqref{sp:ma-trace-limit} converges in $L^1$ on a ball where $f>0$ to
$nf^{1/n}>0$, by dominated convergence and $u_N\le0$.  The left side
converges to zero as a distribution because $u_N/N\to0$ in local
$L^1$.  Testing against a nonzero nonnegative function supported in that
ball is a contradiction.  This proves \eqref{sp:potential-depth}.
\end{proof}

Choose once and for all a compact region $L\subset U$, of positive
volume, whose translates have interiors covering $U$.  It may be
enlarged to be connected.  Properness and cocompactness of the deck action
give bounded overlap and a bounded number of neighbors within any fixed
distance.  For $a,g\in P$ put
\begin{equation}\label{sp:cell-depth}
 A_a(g)=-\sup_{x\in gL}u_N(a^{-1}x).
\end{equation}

\begin{lemma}\label{sp:cell-lemma}
There are $C,\epsilon>0$, independent of $N,a,g$, such that
\begin{equation}\label{sp:cell-count}
 \#\{g\in P:A_a(g)\ge1\}\le C N^{2n},
\end{equation}
and
\begin{equation}\label{sp:cell-exponential}
 \int_{gL}\exp\!\left(
       \frac{-\epsilon u_N(a^{-1}x)}{A_a(g)+1}\right)\omega^n\le C.
\end{equation}
For each $R<\infty$ there is $C_R$ such that
\begin{equation}\label{sp:cell-neighbors}
 C_R^{-1}(A_a(g)+1)\le A_a(h)+1\le C_R(A_a(g)+1)
\end{equation}
whenever the distance between $gL$ and $hL$ is at most $R$.
\end{lemma}

\begin{proof}
Every cell counted in \eqref{sp:cell-count} contributes at least
$\vol(L)$ to the integral of $|u_N(a^{-1}x)|^2$.  Bounded overlap and
\eqref{sp:potential-bounds} prove the count.

Translate $gL$ to $L$ and normalize the potential by $A_a(g)+1$.
The resulting functions are nonpositive, have $\dd\dc$ bounded below
by $-\omega$, and have supremum in $[-1,0]$ on $L$.  Plurisubharmonic
compactness on a connected neighborhood of any fixed enlargement of
$L$ makes this a relatively compact family in local $L^1$.  In
particular its supremum cannot tend to $-\infty$ on another fixed
cell.  There are only finitely many possible relative neighbors; this
gives the upper comparison in \eqref{sp:cell-neighbors}, and exchanging
the two cells gives the lower comparison.

On finitely many smaller charts covering $L$, add fixed local potentials
to make the normalized functions plurisubharmonic.  Each limit has a
positive local exponential integrability exponent.  Compactness and the
semicontinuity and integrability statement of
\cite[Main Theorem~0.2]{DemaillyKollar2001} give a common sufficiently
small exponent and a uniform integral bound.  The added local potentials
are bounded, so they can be removed at the cost of a constant.  Summing
the chart estimates and translating back gives
\eqref{sp:cell-exponential}.
\end{proof}

\subsection{A weighted inverse on the diagonal covering}
\label{sp:weighted}

The potentials now have both a depth estimate and a uniform exponential
integrability bound on cells. The next step is an inverse estimate
that makes a closed form inexpensive to solve where the potential is
deep. Its dependence on the weight, not merely existence of an inverse,
will be needed when the solution is localized.

Set
\[
 D=(U\times U)/P,\qquad \Omega_S=\omega_x+S\omega_y,
 \qquad E=p_y^*K_M,
\]
where $P$ acts diagonally and $S\ge1$.  The maps $p_x,p_y$ here are to
$M$.  We give $E$ the metric from the original, unscaled metric
$\omega$.  The space $D$ is an orbifold covering of $M\times M$;
write $m=2n$ for its complex dimension.  The subgroup defining this
covering is the inverse image of the diagonal in $P\times P$.
In particular the diagonal map $M\to M\times M$ lifts to $D$.

Unweighted norms for $\Omega_S$ are denoted by $\|\cdot\|_S$; an
additional bundle metric multiplier $h$ is indicated by
$\|\cdot\|_{S,h}$.  On the $(m,*)$ complex with coefficients $E^*$,
the unweighted Dolbeault Laplacian has a positive lower bound $\delta$
independent of $S$.  To verify this on a compactly supported section of
$D$, lift it to $U\times U$ and integrate first in $x$, with $y$ in a
fundamental region.  Product splitting expresses the energy as its
nonnegative $x$ and $y$ parts.  The $x$ part uses scalar $(n,q_x)$-forms,
so the previously obtained gaps apply.  The same argument on
$E^*$-valued $(p,0)$-forms uses the original scalar $(p_x,0)$ gaps and
gives
\begin{equation}\label{sp:x-gap}
 \|\dbar\xi\|_S^2\ge\delta\|\xi\|_S^2
 \quad\text{for $\xi$ of type $(p,0)$.}
\end{equation}
These calculations are local product calculations and hence descend to
the diagonal quotient.  Bounded finite chart stabilizers do not change
them.

\begin{lemma}\label{sp:weighted-inverse}
Suppose a smooth positive multiplier $h$ on $E^*$ satisfies
\begin{equation}\label{sp:weight-hypotheses}
 1\le h\le e^b,
 \qquad \ii\Theta_{E^*,h}\ge-C_0t\Omega_S,
 \qquad b,t\ge0.
\end{equation}
Then the complete $\dbar$ complex of $E$-valued $(0,*)$-forms with
multiplier $k=h^{-1}$ is exact with closed ranges.  Every closed form
$F$ in this complex has a solution $\dbar v=F$ satisfying
\begin{equation}\label{sp:weighted-solution}
 \|v\|_{S,k}\le
 C\bigl(e^{b/4}+\sqrt t\,e^{b/2}\bigr)\|F\|_{S,k}.
\end{equation}
The constant depends on $M$, $C_0$ and $\delta$, but not on $S,b,t$.
\end{lemma}

\begin{proof}
First work on the $(m,*)$ complex with coefficients $E^*$.  Its
unweighted gap gives exactness and closed ranges: the bounded Green
operator yields the decomposition into the images of $\dbar$ and
$\dbar^*$.  Weighted and unweighted differential graph norms are
equivalent for fixed $b$.  Thus the domains, kernels and ranges of the
closed differential $d=\dbar$ are the same, and weighted exactness and
closed ranges follow.  This argument does not assert equality of the
weighted and unweighted adjoint domains.

Let $D'_h$ be the $(1,0)$ part of the Chern connection and let $*_S$ be
the complex-linear Hodge star on form indices.  On $(m,q)$-forms it
takes values in $(m-q,0)$, and
\[
 (D'_h)^*=\pm *_S\dbar *_S.
\]
This formula acts trivially on the coefficient index.  The derivative of
the bundle metric in the adjoint cancels the corresponding Chern
connection term.  The Bochner--Kodaira identity and
\eqref{sp:weight-hypotheses} imply
\begin{equation}\label{sp:bochner-weighted}
 \mathcal E_{S,h}(\eta)+C_1t\|\eta\|_{S,h}^2
 \ge\|(D'_h)^*\eta\|_{S,h}^2
 \ge\delta\|\eta\|_S^2.
\end{equation}
Here
$\mathcal E_{S,h}(\eta)=
\|\dbar\eta\|_{S,h}^2+\|\dbar_h^*\eta\|_{S,h}^2$
is the weighted Dolbeault energy.
Indeed $D'_h\eta=0$ in top holomorphic degree, and the curvature
commutator there is a sum of $q$ curvature eigenvalues.  For the last
inequality drop $h\ge1$ from the differential norm and apply
\eqref{sp:x-gap} to $*_S\eta$.  The star is an isometry.  Compact
support approximation from Section~\ref{sp:operators} proves the same
inequality on the form domain.

Fix a degree $q$ and put $A=dd_h^*$ on the closed subspace
$\im d$.  For a nonzero vector $\eta$ in a spectral band of $A$ up to
$\mu>0$, its minimal weighted lift has squared norm
$\langle A^{-1}\eta,\eta\rangle_{S,h}$, hence at least
$\mu^{-1}\|\eta\|_{S,h}^2$.  The minimal unweighted lift is also an
admissible weighted lift, with norm at most
$e^{b/2}\delta^{-1/2}\|\eta\|_S$.  Consequently
\[
 \|\eta\|_S^2
 \ge\delta e^{-b}\mu^{-1}\|\eta\|_{S,h}^2.
\]
On this spectral band $d\eta=0$ and
$\mathcal E_{S,h}(\eta)\le\mu\|\eta\|_{S,h}^2$.
Substitution in \eqref{sp:bochner-weighted} gives
\begin{equation}\label{sp:spectral-quadratic}
 \mu(\mu+C_1t)\ge\delta^2e^{-b}.
\end{equation}
This holds for every nonzero lower spectral band.  Solving the quadratic
inequality shows that the minimal inverses of $d$ have norm at most
$C(e^{b/4}+\sqrt t\,e^{b/2})$ in every degree.

Finally use the conjugate-linear bundle-metric star for Serre duality,
which is distinct from the linear star used above.  It identifies the
$(m,q)$ complex for $E^*$ with multiplier $h$ with the reversed
$(0,m-q)$ complex for $E$ with multiplier $h^{-1}$, interchanging the
differential and its adjoint.  It is an isometry for these norms.  The
complete-domain identities therefore transfer exactness, closed ranges,
and the inverse bounds to the latter complex.  This proves
\eqref{sp:weighted-solution}.
\end{proof}

\subsection{A smoothing operator with fixed propagation}
\label{sp:smoothing}

For the rest of the proof all distances, cell neighborhoods, and support
volumes refer to the standard metric $\Omega_1$.  This convention is
independent of the parameter $S$ in the preceding estimates.
Write $\pi:D\to M\times M$ for the covering and
$E_0=p_y^*K_M$ for the bundle downstairs, so that $E=\pi^*E_0$.

\begin{lemma}\label{sp:smoothing-lemma}
There is a smoothing operator $K$ on the complete Dolbeault complex of
$E$-valued $(0,*)$-forms on $D$ with the following properties:
\begin{enumerate}[label=\textup{(\roman*)}]
\item $K$ commutes with $\dbar$ and preserves separately the
antiholomorphic degrees in $x$ and $y$.
\item Its propagation is at most a fixed $R$, and its kernel and all
kernel derivatives are uniformly bounded in the lifted charts.
\item For compactly supported currents, $K$ commutes with trace to
$M\times M$.  On closed such currents it preserves the cohomology class
of their trace.
\end{enumerate}
If $B$ is an $E$-valued $(0,n)$ current with cellwise $L^1$ coefficients,
then
\begin{equation}\label{sp:smoothing-norm}
 \|KB\|_S\le C S^{n/2}
 \left(\sum_C\left(\int_C|B|_S\,\dd V_1\right)^2\right)^{1/2},
\end{equation}
where $C$ runs through the product cells modulo the diagonal action.
\end{lemma}

\begin{proof}
Choose an even Schwartz function $f_0$ whose Fourier transform is smooth
and compactly supported, with $f_0(0)=1$, and set
$K=f_0(\sqrt{\Delta''_1})$, using the standard unweighted product
metric and the fixed metric on $E$.  The product splitting of
$\Delta''_1$ proves the degree assertions.  Commutation with $\dbar$
follows first on its domain from the Dolbeault identities and then on
currents by duality.

The wave equation has finite propagation with a speed bounded by the
principal symbol.  In the present setting this can be seen from the
symmetric first-order operator $D_1=\sqrt2(\dbar+\dbar^*)$:
the energy identity for a moving cutoff contains only its commutator,
whose norm is bounded by a fixed multiple of the cutoff gradient.
It gives the domain-of-dependence estimate for $e^{\ii tD_1}$.
Since $f_0$ is even, the Fourier expression for
$f_0(\sqrt{\Delta''_1})$ uses only these finite-time wave operators.
Thus $K$ has fixed propagation.  The coefficient curvature contributes
only lower-order terms and does not alter this argument.  The identical
energy calculation in invariant charts proves it on the orbifold.

To see smoothing with uniform kernel bounds, use
\eqref{sp:global-elliptic} and local Sobolev embedding to bound point
evaluation, and derivative evaluation, after $(1+\Delta''_1)^{-a}$
for $a$ sufficiently large.  The factorization
\[
 K=(1+\Delta''_1)^{-a}
 \bigl((1+\Delta''_1)^{2a}K\bigr)
 (1+\Delta''_1)^{-a}
\]
has bounded middle factor by the Schwartz decay of $f_0$.  Evaluating
the outside factors and their adjoints bounds the kernel at any two
points, uniformly in the charts.  Increasing $a$ gives every derivative
bound.  This argument also defines $K$ on distributions.

For a compactly supported $E$-valued current $A$, define its orbifold
trace by
\[
 \langle\operatorname{Tr}A,\beta\rangle
 =\langle A,\pi^*\beta\rangle
\]
for smooth $E_0^*$-valued test forms $\beta$ of complementary bidegree.
On a covering chart $[V/H]\to[V/G]$, this is the sum over cosets
$G/H$, with the natural action on form and coefficient indices, summed
also over the local sheets met by the support.  The $|G:H|$ terms
convert the downstairs integration factor $1/|G|$ into the upstairs
factor $1/|H|$.  This description applies even when $H$ is not normal;
it includes the stabilizer multiplicities at singular strata.

For a compactly supported current, the wave-evolved support stays in a
fixed compact neighborhood for bounded time, by completeness and finite
propagation.  Its local trace is therefore a finite sum.  Since the
operator and coefficient metric are pulled back from the base, the
local transfer commutes with the differential operator.  The traced
wave is thus the downstairs wave with traced initial data, by
uniqueness.  Integrating in time proves that trace commutes with $K$.
On the compact base, a spectral function with value one at zero is the
identity on harmonic representatives and therefore on Dolbeault
cohomology, also computed by currents.  This proves~\textup{(iii)}.

It remains to check the dependence on $S$.  For an $E$-valued $(0,n)$
block of $y$ antiholomorphic degree $q$, the unscaled metric on $E$ gives
\begin{equation}\label{sp:scaling-block}
 |B|_S=S^{-q/2}|B|_1,\qquad
 \dd V_S=S^n\dd V_1,\qquad
 \|B\|_S=S^{(n-q)/2}\|B\|_1.
\end{equation}
Put $a_C^S(B)=\int_C|B|_S\dd V_1$.
The kernel bound and fixed propagation imply
$\|KB\|_1\le C\|(a_C^1(B))_C\|_{\ell^2}$: each output cell sees
only boundedly many input neighbors, and the neighbor matrix has
uniformly bounded row and column sums.  Since $K$ preserves $q$,
\[
 \|KB\|_S
 \le C S^{(n-q)/2}S^{q/2}
             \|(a_C^S(B))_C\|_{\ell^2}
 = C S^{n/2}\|(a_C^S(B))_C\|_{\ell^2}.
\]
The finitely many orthogonal blocks give
\eqref{sp:smoothing-norm}.  Product cells may overlap and have bounded
finite chart multiplicities; these change only the fixed constant.
\end{proof}

\subsection{Cell weights and one localization step}
\label{sp:localization}

We now combine the weighted inverse with the fixed-propagation
smoother. One step replaces a compactly supported closed form by a
smaller representative of the same traced class. The thresholds are
nested so that the output support of one step is an admissible input
for the next.

Fix an integer $r>30n+6$ and choose $0<\zeta<1/(8r)$.  These choices
are made before $N$ tends to infinity.  At step $j\in\{1,\ldots,r\}$ set
\begin{equation}\label{sp:thresholds}
 S=N^2,\quad H_j=N^{3/4-2(j-1)\zeta},\quad
 H_j^-=H_jN^{-\zeta},\quad t_j=\frac{2\log N}{H_j}.
\end{equation}
In particular $H_j\ge N^{1/2}$, $H_j\le S$, and $t_jS\to\infty$.
Constants in the following construction may depend on the fixed $r$.

For each step choose a bump $w^{(j)}\in C_c^\infty(U,[0,1])$, equal
to one on a neighborhood of $L$, and let
$w_a^{(j)}(y)=w^{(j)}(a^{-1}y)$.  The bumps can be chosen as squares so
that
\[
 |\dd w_a^{(j)}|^2\le C_jw_a^{(j)}.
\]
Their translates have bounded overlap and bounded derivatives.  Choose
the support at step $j+1$ sufficiently large that $w^{(j+1)}$ equals
one on every fixed-radius neighborhood of $\supp w^{(j)}$ required
below.  Only finitely many enlargements are made, all independent of
$N$.

We suppress $j$ temporarily and write $H,H^-,t,w_a$ for the data at
that step.  Let $\rho$ be a smooth convex regularization of
$\max(0,s)$, equal to zero for $s\le-1$ and to $s$ for $s\ge1$,
with $0\le\rho'\le1$, $\rho''\ge0$, and $\rho\ge\max(0,s)$.
Define
\begin{align*}
 \ell_a(x)&=-2H+\rho\bigl(u_N(a^{-1}x)+2H\bigr),\\
 p_H(\ell)&=\ell+\frac{\ell^2}{8H},\\
 T(x,y)&=\sum_{a\in P}w_a(y)p_H(\ell_a(x)),\\
 \varphi(x,y)&=-H/2+\rho\bigl(T(x,y)+H/2\bigr).
\end{align*}
For large $N$, the clipped functions satisfy
$-2H\le\ell_a\le0$ and $-H/2\le\varphi\le0$.
The sum is locally finite and invariant under the diagonal action.
Moreover
\begin{equation}\label{sp:weight-curvature}
 \dd\dc\varphi\ge-C_j(\omega_x+H\omega_y).
\end{equation}
Here is the estimate including mixed terms.  On $[-2H,0]$,
$1/2\le p_H'\le1$ and $p_H''=1/(4H)$.  The $x$ Hessian of each
summand contains
$w_ap_H''\ii\partial\ell_a\wedge\dbar\ell_a$ as a positive
term.  The mixed derivative is bounded by half this term plus a
constant times
$H|\dd w_a|^2/w_a$ in the $y$ direction, using Cauchy--Schwarz;
the expression is interpreted as zero where $w_a=0$.  The bump
inequality bounds the latter term by $C_jH\omega_y$.  The pure $y$
Hessian has the same bound because $|p_H(\ell_a)|\le2H$.
The remaining $x$ Hessian is bounded below by $-w_a\omega_x$.
Bounded overlap and the final convex clipping give
\eqref{sp:weight-curvature}.

Put
\begin{equation}\label{sp:actual-weights}
 h_j=e^{-t_j\varphi},\qquad k_j=h_j^{-1},\qquad b=\log N.
\end{equation}
These are smooth multipliers on $D$.  They satisfy
\eqref{sp:weight-hypotheses}, with a constant independent of $N$ and
$S$.  Indeed $1\le h_j\le N$, and
\[
 \ii\Theta_{E^*,h_j}
 =\ii\Theta_{E^*}+t_j\dd\dc\varphi.
\]
The first term is bounded below by $-C\omega_y$, which is absorbed
by $-C't_j\Omega_S$ because $t_jS\to\infty$; the second is controlled
by \eqref{sp:weight-curvature} and $H_j\le S$.

Call the following set the deep region at step $j$:
\begin{equation}\label{sp:deep-region}
 \mathcal D_j=\{(x,y): x\in gL,\ A_a(g)\ge H_j,
                       \ w_a^{(j)}(y)=1
                       \text{ for some }a,g\}/P.
\end{equation}
On this region $\ell_a\le-H_j$ for a witnessing index and
$p_{H_j}(\ell_a)\le-7H_j/8$.  The final clipping is consequently
constant, $\varphi=-H_j/2$, and
\begin{equation}\label{sp:deep-weight}
 h_j=N,\qquad k_j=N^{-1}\quad\text{on }\mathcal D_j.
\end{equation}

A product cell $gL\times g'L$, modulo the diagonal action, is marked
at step $j$ when some $a$ satisfies
\begin{equation}\label{sp:marked-cells}
 A_a(g)\ge H_j^-,\qquad
 g'L\cap\supp w_a^{(j)}\ne\varnothing.
\end{equation}
There are at most $C_jN^{2n}$ marked cells.  In fact, translating by
$a^{-1}$ leaves only a bounded number of possible second cells meeting
$\supp w^{(j)}$, and \eqref{sp:cell-count} bounds the number of first
cells.  Any fixed enlargement of their union has the same bound on its
number of cells and on its standard volume.

On every unmarked cell $C$ one has
\begin{equation}\label{sp:unmarked-weight}
 \int_C h_j\,\dd V_1\le C_j.
\end{equation}
Indeed $\ell_a\ge u_N(a^{-1}x)$, $p_H(\ell_a)\ge\ell_a$, and
$\varphi\ge T$.  Since all the potentials are nonpositive and
$0\le w_a\le1$, this bounds $h_j$ on a cell by the product of
$\exp(-t_ju_N(a^{-1}x))$ over the bounded number of indices whose
$y$ supports meet that cell.  Each such index has $A_a(g)<H_j^-$
when the cell is unmarked.  H\"older's inequality and
\eqref{sp:cell-exponential} apply, because
\[
 t_j(H_j^-+1)\le 2(\log N)(N^{-\zeta}+N^{-1/2})\longrightarrow0.
\]
Integration in the $y$ cell contributes only a fixed volume factor.

Choose a smooth compactly supported cutoff $\chi_j$ equal to one on
a neighborhood of all marked cells, with support in a fixed enlargement
of their union and with $|\dd\chi_j|_1\le C_j$.  Such cutoffs are
obtained by smoothing the distance cutoff in the uniform charts.
Every cell meeting $\supp\dd\chi_j$ is unmarked.
Enlarge the supports of $w^{(j+1)}$ as specified above so that
\begin{equation}\label{sp:nested-supports}
 \text{the $R$-neighborhood of $\supp\chi_j$ lies in }
 \mathcal D_{j+1}\qquad(j<r).
\end{equation}
To verify this, a point in that neighborhood has an $x$ cell within a
fixed distance of a cell witnessing \eqref{sp:marked-cells}.
By \eqref{sp:cell-neighbors}, its depth is at least
$c_jH_j^--1$, which exceeds $H_{j+1}$ for large $N$, since
$H_j^-/H_{j+1}=N^\zeta$.  Its $y$ coordinate lies within a fixed
distance of $\supp w_a^{(j)}$, where the next bump is one.  This
proves \eqref{sp:nested-supports}.  Similarly,
\eqref{sp:potential-depth} shows that $\mathcal D_1$ contains every
prescribed fixed-radius neighborhood of the lifted diagonal for large
$N$.

\begin{lemma}\label{sp:one-step}
Let $F$ be a smooth compactly supported closed $E$-valued $(0,n)$-form
supported in $\mathcal D_j$.  There is a smooth compactly supported
closed form $F_{\mathrm{new}}$ of the same type whose traced
cohomology class is that of $F$, with
\begin{equation}\label{sp:one-step-norm}
 \|F_{\mathrm{new}}\|_S\le C_jN^{-1/6}\|F\|_S.
\end{equation}
Its support is in the fixed $R$-neighborhood of $\supp\chi_j$, has
standard volume at most $C_jN^{2n}$, and lies in $\mathcal D_{j+1}$
when $j<r$.
\end{lemma}

\begin{proof}
By Lemma~\ref{sp:weighted-inverse} and
\eqref{sp:deep-weight}, solve $\dbar v=F$ with
\begin{align*}
 \|v\|_{S,k_j}
 &\le C_j\bigl(N^{1/4}+\sqrt{t_j}\,N^{1/2}\bigr)
                              N^{-1/2}\|F\|_S\\
 &\le C_jN^{-1/6}\|F\|_S.
\end{align*}
The last inequality uses
$\sqrt{t_j}\le C(\log N)^{1/2}N^{-1/4}$.
The compactly supported closed current
\[
 B=F-\dbar(\chi_jv)=-\dbar\chi_j\wedge v
\]
has the same traced cohomology class as $F$.  The equality uses
$\chi_j=1$ near $\supp F$.  On any cell meeting $\supp B$, which is
unmarked, Cauchy--Schwarz and \eqref{sp:unmarked-weight} give
\[
 \int_C|B|_S\,\dd V_1
 \le C_j\left(\int_C|v|_S^2 k_j\,\dd V_1\right)^{1/2}.
\]
Here wedging with $\dbar\chi_j$ has bounded operator norm for
$\Omega_S$, because $S\ge1$.  After summing squares and using bounded
overlap and $\dd V_S=S^n\dd V_1$, we obtain
\[
 \left(\sum_C\left(\int_C|B|_S\,\dd V_1\right)^2\right)^{1/2}
 \le C_jS^{-n/2}N^{-1/6}\|F\|_S.
\]
Set $F_{\mathrm{new}}=KB$ with the fixed operator of
Lemma~\ref{sp:smoothing-lemma}.  Its norm estimate is
\eqref{sp:one-step-norm}; its closedness and traced class follow from
the same lemma.  Fixed propagation, the marked-cell count and
\eqref{sp:nested-supports} give all support assertions.  Only the
compactly supported current $\chi_jv$ is traced; no sum of the global
solution $v$ is used.
\end{proof}

\subsection{The compact diagonal class}
\label{sp:diagonal-class}

\begin{proof}[Proof of Theorem~\ref{thm:zero-spectrum}]
Continue under the assumption of positive lower bounds in all $(p,0)$
degrees.  Let $\iota:M\to D$ be the lifted diagonal and let
$[\iota(M)]$ be its integration current of type $(n,n)$.  Orbifold
integration defines this current directly by pullback of test forms to
$M$, regardless of singularities of the coarse spaces.
Project it to the component whose holomorphic degree lies entirely in
the $y$ factor.  Under the natural identification with $E$-valued
$(0,n)$ currents, call the result $T_0$.  This projection commutes with
$\dbar$, so $T_0$ is closed and compactly supported.

Its trace to $M\times M$ has a nonzero cohomology class.  More
explicitly, the canonical isomorphism
$K_{M\times M}\otimes E_0^*\simeq p_x^*K_M$ identifies
$p_x^*\omega^n$ with a closed $E_0^*$-valued $(m,n)$ test form $\beta$.
The holomorphic-degree projection used to define $T_0$ does not change
the pairing with this form, since all other components vanish on
wedging with $p_x^*\omega^n$.  Hence
\begin{equation}\label{sp:diagonal-pairing}
 \langle\operatorname{Tr}T_0,\beta\rangle
 =\int_M\omega^n>0.
\end{equation}

Let $F_0=KT_0$.  It is smooth, compactly supported, and closed; its
traced class has the same pairing \eqref{sp:diagonal-pairing}.  It is
independent of $N$, and its support lies in a fixed neighborhood of the
lifted diagonal.  Thus it lies in $\mathcal D_1$ for all sufficiently
large $N$.  Apply Lemma~\ref{sp:one-step} successively for the fixed
number $r$ of steps to obtain $F_r$.  Its standard support volume is
at most $C_rN^{2n}$, while
\[
 \|F_r\|_S\le C_rN^{-r/6}\|F_0\|_S.
\]
For degree $(0,n)$, \eqref{sp:scaling-block} and $0\le q\le n$ show
that $\|F_r\|_1\le\|F_r\|_S$ and
$\|F_0\|_S\le S^{n/2}\|F_0\|_1$.  The constant norm conversions
are used only at these endpoints.  Since $S=N^2$,
\[
 \|F_r\|_1\le C_rN^{n-r/6}.
\]
The pullback of the fixed test form $\beta$ is uniformly bounded for
the standard metric.  Therefore Cauchy--Schwarz gives
\[
 \bigl|\langle\operatorname{Tr}F_r,\beta\rangle\bigr|
 \le C\vol_1(\supp F_r)^{1/2}\|F_r\|_1
 \le C_rN^{2n-r/6}\longrightarrow0.
\]
Our fixed choice $r>30n+6$ more than suffices for this decay.  Every
step preserves the traced cohomology class, so its pairing is the
strictly positive constant in \eqref{sp:diagonal-pairing}.  This is a
contradiction.  At least one scalar $(p,0)$ Dolbeault Laplacian has no
positive lower bound, which proves the theorem.
\end{proof}
==== END The-abelianity-conjecture-for-special-compact-Kahler-manifolds//sections/spectral.tex ====

==== BEGIN The-abelianity-conjecture-for-special-compact-Kahler-manifolds//references.bib ====
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  note = {\href{https://doi.org/10.1007/s12220-024-01796-2}{doi:10.1007/s12220-024-01796-2}; author's version \href{https://arxiv.org/abs/2201.06748}{arXiv:2201.06748}}
}

@article{Kawamata1981,
  author = {Kawamata, Yujiro},
  title = {Characterization of abelian varieties},
  journal = {Compositio Math.}, volume = {43}, number = {2},
  year = {1981}, pages = {253--276},
  url = {https://www.numdam.org/article/CM_1981__43_2_253_0.pdf}
}

@book{Ueno1975,
  author = {Ueno, Kenji},
  title = {Classification Theory of Algebraic Varieties and Compact Complex Spaces},
  series = {Lecture Notes in Mathematics}, volume = {439},
  publisher = {Springer-Verlag}, address = {Berlin--Heidelberg},
  year = {1975}, doi = {10.1007/BFb0070570},
  url = {https://link.springer.com/book/10.1007/BFb0070570},
  note = {\href{https://doi.org/10.1007/BFb0070570}{doi:10.1007/BFb0070570}}
}

@article{CampanaClaudonThreefolds,
  author = {Campana, Fr{\'e}d{\'e}ric and Claudon, Beno{\^i}t},
  title = {Abelianity conjecture for special compact {K{\"a}hler} 3-folds},
  journal = {Proc. Edinb. Math. Soc. (2)}, volume = {57}, number = {1},
  year = {2014}, pages = {55--78}, doi = {10.1017/S0013091513000849},
  url = {https://arxiv.org/abs/1107.0168},
  note = {\href{https://doi.org/10.1017/S0013091513000849}{doi:10.1017/S0013091513000849}}
}

@book{Debarre1999,
  author = {Debarre, Olivier},
  title = {Tores et vari{\'e}t{\'e}s ab{\'e}liennes complexes},
  series = {Cours Sp{\'e}cialis{\'e}s}, volume = {6},
  publisher = {Soci{\'e}t{\'e} Math{\'e}matique de France and EDP Sciences},
  address = {Paris and Les Ulis}, year = {1999}, isbn = {2-86883-427-2},
  url = {https://smf.emath.fr/publications/tores-et-varietes-abeliennes-complexes-0}
}

@book{Demailly2012,
  author = {Demailly, Jean-Pierre},
  title = {Analytic Methods in Algebraic Geometry},
  series = {Surveys of Modern Mathematics}, volume = {1},
  publisher = {International Press and Higher Education Press},
  address = {Somerville, MA, and Beijing}, year = {2012},
  isbn = {978-1-57146-234-3},
  url = {https://intlpress.com/site/pub/files/preview/bookpubs/00000371.pdf}
}

@article{Chernoff1973,
  author = {Chernoff, Paul R.},
  title = {Essential self-adjointness of powers of generators of hyperbolic equations},
  journal = {J. Funct. Anal.}, volume = {12}, number = {4},
  year = {1973}, pages = {401--414}, doi = {10.1016/0022-1236(73)90003-7},
  url = {https://www.sciencedirect.com/science/article/pii/0022123673900037},
  note = {\href{https://doi.org/10.1016/0022-1236(73)90003-7}{doi:10.1016/0022-1236(73)90003-7}}
}

@article{YuWang2012,
  author = {Wang, Yu},
  title = {On the {$C^{2,\alpha}$}-regularity of the complex {Monge--Amp{\`e}re} equation},
  journal = {Math. Res. Lett.}, volume = {19}, number = {4},
  year = {2012}, pages = {939--946}, doi = {10.4310/MRL.2012.v19.n4.a16},
  url = {https://intlpress.com/site/pub/files/_fulltext/journals/mrl/2012/0019/0004/MRL-2012-0019-0004-a016.pdf},
  note = {\href{https://doi.org/10.4310/MRL.2012.v19.n4.a16}{doi:10.4310/MRL.2012.v19.n4.a16}; author's version \href{https://arxiv.org/abs/1111.0902v1}{arXiv:1111.0902v1}, November 3, 2011. The cited regularity result is Theorem~1.1 in that version}
}

@book{GilbargTrudinger2001,
  author = {Gilbarg, David and Trudinger, Neil S.},
  title = {Elliptic Partial Differential Equations of Second Order},
  edition = {Second}, series = {Classics in Mathematics},
  publisher = {Springer-Verlag}, address = {Berlin}, year = {2001},
  isbn = {978-3-540-41160-4}, doi = {10.1007/978-3-642-61798-0},
  url = {https://link.springer.com/book/10.1007/978-3-642-61798-0},
  note = {Reprint of the 1998 edition. \href{https://doi.org/10.1007/978-3-642-61798-0}{doi:10.1007/978-3-642-61798-0}}
}

@article{Yau1978,
  author = {Yau, Shing-Tung},
  title = {On the {Ricci} curvature of a compact {K{\"a}hler} manifold and the complex {Monge--Amp{\`e}re} equation, {I}},
  journal = {Comm. Pure Appl. Math.}, volume = {31}, number = {3},
  year = {1978}, pages = {339--411}, doi = {10.1002/cpa.3160310304},
  url = {https://onlinelibrary.wiley.com/doi/10.1002/cpa.3160310304},
  note = {\href{https://doi.org/10.1002/cpa.3160310304}{doi:10.1002/cpa.3160310304}}
}

@article{CampanaGuenanciaPaun2013,
  author = {Campana, Fr{\'e}d{\'e}ric and Guenancia, Henri and P{\u a}un, Mihai},
  title = {Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields},
  journal = {Ann. Sci. {\'E}c. Norm. Sup{\'e}r. (4)},
  volume = {46}, number = {6}, year = {2013}, pages = {879--916},
  doi = {10.24033/asens.2205},
  url = {https://www.numdam.org/item/10.24033/asens.2205.pdf},
  note = {\href{https://doi.org/10.24033/asens.2205}{doi:10.24033/asens.2205}}
}

@book{Tao2014,
  author = {Tao, Terence},
  title = {{Hilbert}'s Fifth Problem and Related Topics},
  series = {Graduate Studies in Mathematics}, volume = {153},
  publisher = {American Mathematical Society}, address = {Providence, RI},
  year = {2014}, isbn = {978-1-4704-1564-8},
  url = {https://terrytao.wordpress.com/wp-content/uploads/2014/11/gsm-153.pdf},
  note = {\href{https://terrytao.wordpress.com/wp-content/uploads/2014/11/gsm-153.pdf}{Author's version}; numbering there is used when specified}
}

@book{Williams2007,
  author = {Williams, Dana P.},
  title = {Crossed Products of {$C^*$}-Algebras},
  series = {Mathematical Surveys and Monographs}, volume = {134},
  publisher = {American Mathematical Society}, address = {Providence, RI},
  year = {2007}, doi = {10.1090/surv/134},
  url = {https://www.ams.org/books/surv/134/surv134-endmatter.pdf},
  note = {\href{https://doi.org/10.1090/surv/134}{doi:10.1090/surv/134}}
}

@article{Moishezon1967,
  author = {Moishezon, B. G.},
  title = {On {$n$}-dimensional compact varieties with {$n$} algebraically independent meromorphic functions},
  journal = {Amer. Math. Soc. Transl. (2)}, volume = {63},
  year = {1967}, pages = {51--177},
  note = {English translation of Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966), 133--174, 345--386, 621--656},
  url = {https://www.mathnet.ru/eng/im2823}
}

@article{Fujiki1982,
  author = {Fujiki, Akira},
  title = {On the {Douady} space of a compact complex space in the category {$\mathcal{C}$}},
  journal = {Nagoya Math. J.}, volume = {85},
  year = {1982}, pages = {189--211}, doi = {10.1017/S002776300001970X},
  url = {https://doi.org/10.1017/S002776300001970X},
  note = {\href{https://doi.org/10.1017/S002776300001970X}{doi:10.1017/S002776300001970X}}
}

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

@article{BrunebarbeKlinglerTotaro2013,
  author = {Brunebarbe, Yohan and Klingler, Bruno and Totaro, Burt},
  title = {Symmetric differentials and the fundamental group},
  journal = {Duke Mathematical Journal},
  volume = {162}, number = {14}, year = {2013}, pages = {2797--2813},
  doi = {10.1215/00127094-2381442},
  eprint = {1204.6443}, archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/1204.6443v3},
  note = {Theorem locations refer to arXiv:1204.6443v3}
}

@article{BrunebarbeCampana2016,
  author = {Brunebarbe, Yohan and Campana, Fr{\'e}d{\'e}ric},
  title = {Fundamental group and pluridifferentials on compact {K\"ahler} manifolds},
  journal = {Moscow Mathematical Journal},
  volume = {16}, number = {4}, year = {2016}, pages = {651--658},
  doi = {10.17323/1609-4514-2016-16-4-651-658},
  eprint = {1510.07922}, archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/1510.07922},
  note = {Author preprint used for Theorem 1.1 and Corollary 1}
}

@article{Bogomolov1979,
  author = {Bogomolov, F. A.},
  title = {Holomorphic tensors and vector bundles on projective varieties},
  journal = {Mathematics of the USSR-Izvestiya},
  volume = {13}, number = {3}, year = {1979}, pages = {499--555},
  doi = {10.1070/IM1979v013n03ABEH002076},
  url = {https://www.mathnet.ru/eng/im1965}
}

@incollection{Campana2015Survey,
  author = {Campana, Fr{\'e}d{\'e}ric},
  title = {Special manifolds, arithmetic and hyperbolic aspects: a short survey},
  booktitle = {Rational points, rational curves, and entire holomorphic curves on projective varieties},
  series = {Contemporary Mathematics}, volume = {654},
  publisher = {American Mathematical Society},
  year = {2015}, pages = {23--52},
  doi = {10.1090/conm/654/13214},
  url = {https://doi.org/10.1090/conm/654/13214}
}

@article{Beauville1983,
  author = {Beauville, Arnaud},
  title = {Vari{\'e}t{\'e}s k{\"a}hl{\'e}riennes dont la premi{\`e}re classe de {Chern} est nulle},
  journal = {Journal of Differential Geometry},
  volume = {18}, number = {4}, year = {1983}, pages = {755--782},
  doi = {10.4310/jdg/1214438181},
  url = {https://math.univ-cotedazur.fr/~beauvill/pubs/jdg.pdf}
}

@article{Lott1996,
  author = {Lott, John},
  title = {The zero-in-the-spectrum question},
  journal = {L'Enseignement Math{\'e}matique},
  volume = {42}, year = {1996}, pages = {341--376},
  url = {https://math.berkeley.edu/~lott/zero.pdf}
}

@article{FarberWeinberger2001,
  author = {Farber, Michael and Weinberger, Shmuel},
  title = {On the zero-in-the-spectrum conjecture},
  journal = {Annals of Mathematics},
  volume = {154}, year = {2001}, pages = {139--154},
  doi = {10.2307/3062113},
  url = {https://annals.math.princeton.edu/articles/10999}
}

@incollection{Atiyah1976,
  author = {Atiyah, Michael F.},
  title = {Elliptic operators, discrete groups and von {Neumann} algebras},
  booktitle = {Colloque ``Analyse et Topologie'' en l'Honneur de Henri Cartan},
  series = {Ast{\'e}risque}, volume = {32--33},
  publisher = {Soci{\'e}t{\'e} Math{\'e}matique de France},
  year = {1976}, pages = {43--72},
  url = {https://www.numdam.org/item/AST_1976__32-33__43_0.pdf}
}

@article{Dodziuk1977,
  author = {Dodziuk, Jozef},
  title = {De {Rham--Hodge} theory for {$L^2$}-cohomology of infinite coverings},
  journal = {Topology}, volume = {16}, year = {1977}, pages = {157--165},
  doi = {10.1016/0040-9383(77)90013-1},
  url = {https://doi.org/10.1016/0040-9383(77)90013-1}
}
==== END The-abelianity-conjecture-for-special-compact-Kahler-manifolds//references.bib ====