==== TABLE OF CONTENTS ====
A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//branched.tex
A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//differentials.tex
A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//inertia-descent.tex
A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//inertia-foundations.tex
A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//inertia-inputs.tex
A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//inertia-minimality.tex
A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//inertia.tex
A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//introduction.tex
A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//main.tex
A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//preamble.tex
A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//quadrics.tex
A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//transfer.tex
A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//references.bib
===========================

==== BEGIN A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//branched.tex ====
\section{An alternative construction using auxiliary double covers}
\label{app:branched}

We give an alternative construction of a manifold satisfying
Proposition~\ref{prop:transfer}.  Throughout this appendix, $X$ is a
connected smooth projective complex fourfold, $D\subset X$ is a nonempty
smooth connected reduced divisor, and $\mathcal X$ is its order-two root
orbifold. An affine action on a product of elliptic curves makes the
involution associated with $D$ act freely while retaining selected auxiliary fixed
loci. Blowing up these loci produces a smooth quotient with the ordinary
submersions needed along the auxiliary branch divisors.

\begin{lemma}[An auxiliary branched cover]
\label{lem:branched-cover}
There are distinct smooth prime divisors $B_1,\ldots,B_5$, with
$D+\sum_i B_i$ simple normal crossing, and a connected smooth projective
Galois cover
\[
 \pi:S\longrightarrow X,\qquad
 A=(\mathbb Z/2\mathbb Z)^5=\langle e_1,\ldots,e_5\rangle,
\]
with the following local descriptions.  Put $\tau=e_1+\cdots+e_5$ and
$I(x)=\{i:x\in B_i\}$.
\begin{enumerate}
\item If $x\notin D$, a chart over $x$ takes independent square roots of
the coordinates defining $B_i$, $i\in I(x)$.  Its stabilizer is generated
by the corresponding $e_i$, each changing the sign of its own root
coordinate.
\item If $x\in D$, choose coordinates $z_0,z_i$ defining $D,B_i$ for
$i\in I(x)$.  The charts over $x$ take independent roots
\[
 w_0^2=z_0,\qquad w_i^2=z_i\quad(i\in I(x)),
\]
and leave the remaining coordinates unchanged.  Their stabilizer is
$\langle\tau,e_i:i\in I(x)\rangle$; here $\tau$ changes the sign of
$w_0$ alone, and $e_i$ changes the sign of $w_i$ alone.
\end{enumerate}
In particular, $\pi^*D$ is twice a reduced divisor, and
$S^{e_i}$ is precisely the reduced inverse image of $B_i$, a smooth
divisor all of whose components dominate $B_i$.
\end{lemma}

\begin{proof}
Choose a line bundle $H$ such that $H^{\otimes2}(-D)$ is very ample.
Successive general members $B_i$ of this system are smooth and
irreducible and can be chosen transverse to all the preceding strata,
including those in $D$.  Thus the asserted simple normal crossing
condition follows from Bertini's theorem.  In particular, at a point of
$D$ at most three of the $B_i$ pass through the point.

Let $s_D$ and $s_i$ be the divisor sections for $D$ and $B_i$.  Form the
cover by adjoining roots
\[
 y_i^2=s_Ds_i\quad(1\leq i\leq5),
\]
where $y_i$ takes values in $H$, and normalize.  In a rational frame of
$H$, the five right-hand sides have independent classes modulo squares
in $\mathbb C(X)^*$.  Indeed, the order along $B_i$ of a product of these
functions is odd exactly when its $i$th factor occurs an odd number of
times.  A product representing a square must therefore have every
exponent even.  The resulting field extension has degree $2^5$, with
the independent sign changes as its Galois group.  Its normalization
$S$ is connected and finite over $X$, hence projective.

We compute this normalization locally analytically.  Units in the
equations have holomorphic square roots after shrinking the
neighborhood.  Off $D$, the equations give independent roots of the
passing $B_i$ coordinates; the other roots separate the local sheets.
At $x\in D$, the equations become
\[
 y_i^2=z_0z_i\quad(i\in I(x)),\qquad
 y_j^2=z_0\quad(j\notin I(x)).
\]
There is an index $j_0\notin I(x)$.  On a component set $w_0=y_{j_0}$
and, in its meromorphic function field, set $w_i=y_i/w_0$ for
$i\in I(x)$.  These functions satisfy the displayed independent root
equations.  The other $y_j$ equal $\pm w_0$, and their relative signs
separate the components.  The independent-root charts are finite and
normal over the base chart and have exactly the prescribed generic
sheets, so they are its normalization.  They are smooth.  The sign
changes give the stabilizers and their actions stated in the lemma.

For completeness, every element of the latter stabilizer with
nonzero $\tau$-coefficient has at least
\[
 5-|I(x)|\geq2
\]
nonzero coordinates in the basis $e_1,\ldots,e_5$.  Consequently $e_i$
belongs to a point stabilizer precisely over $B_i$.  In the local
charts its fixed locus is the root-coordinate hyperplane for $B_i$.
This proves the assertion about $S^{e_i}$.  Each of its components has
dimension three, so its finite image in the irreducible divisor $B_i$
has dimension three and equals $B_i$.  The equation $z_0=w_0^2$ also
proves the assertion about $\pi^*D$.
\end{proof}

\begin{lemma}[An action on a product of elliptic curves]
\label{lem:branched-action}
There is an action of $A$ on an abelian variety $F$ of dimension ten
such that every element with at least two nonzero coordinates in the
basis $e_i$ is fixed-point-free.  Each $e_i$ has a nonempty smooth fixed
locus of codimension four, and these five fixed loci are pairwise
disjoint.
\end{lemma}

\begin{proof}
Fix an elliptic curve $E$ and a nonzero point $c\in E[2]$.  Set
\[
 F=\prod_{1\leq i<j\leq5}E_{ij},\qquad E_{ij}=E.
\]
On the factor $E_{ij}$ let $e_i$ act by $v\mapsto-v$, let $e_j$ act by
$v\mapsto c-v$, and let the other generators act trivially.  The two
displayed involutions commute because $2c=0$, and their product is
translation by $c$.  If an element has both $i$ and $j$ in its support,
it therefore acts without fixed points on this factor and hence on
$F$.

For a single $e_i$, its four moving factors each carry an involution
of the form $v\mapsto a-v$.  The equation $2v=a$ has solutions and a
finite reduced solution set.  Thus $F^{e_i}$ is a nonempty disjoint
union of translates of products of the six remaining elliptic
factors.  It is smooth of codimension four.  Two different generators
cannot fix the same point, since their product is fixed-point-free.
\end{proof}

\begin{proposition}[A manifold model of dimension fourteen]
\label{prop:branched-model}
There is a connected smooth projective variety $Y$ of dimension
fourteen and a surjective morphism $f:Y\to X$ with the following
properties.
\begin{enumerate}
\item Over
\[
 U=X\setminus\bigl(D\cup B_1\cup\cdots\cup B_5\bigr),
\]
the map is a holomorphically locally trivial proper bundle with
connected fibre $F$, the abelian variety of
Lemma~\ref{lem:branched-action}.
\item The divisor $f^*D$ is twice a reduced divisor.  At every point
over $D$, the map admits a local holomorphic lift to its root chart.
At points over a general point of $D$, such a lift can be chosen
submersive.
\item Above a general point of every prime divisor of $X$ other than
$D$, there is a point at which $f$ is a submersion to $X$.
\end{enumerate}
\end{proposition}

\begin{proof}
Use the diagonal action of $A$ on $V=S\times F$.  By
Lemma~\ref{lem:branched-action}, a nontrivial point stabilizer can only
be generated by a single $e_i$.  The entire nonfree locus is the union
of the pairwise disjoint smooth subvarieties
\[
 Z_i=S^{e_i}\times F^{e_i}\qquad(1\leq i\leq5).
\]
They have codimension five.  They are invariant under $A$, since $A$
is abelian.  At a point of $Z_i$ the stabilizer is exactly
$\langle e_i\rangle$.  In local coordinates it fixes the tangent
coordinates to $Z_i$ and negates all five normal coordinates: one
from $S$, and four from the moving elliptic factors.

Blow up their disjoint union and take the quotient:
\[
 b:\widehat V=\operatorname{Bl}_{\cup_i Z_i}(V)\longrightarrow V,
 \qquad q:\widehat V\longrightarrow Y=\widehat V/A.
\]
The action lifts to the blowup.  We verify that $Y$ is smooth.  If
$(v_0,\ldots,v_4)$ are the normal coordinates just described, then in
the blowup chart with distinguished coordinate $v_j$ we have
coordinates
\[
 t=v_j,\qquad a_k=v_k/v_j\quad(k\ne j),
\]
besides the tangent coordinates.  The stabilizer acts by
\[
 (t,(a_k),\text{tangent coordinates})
 \longmapsto
 (-t,(a_k),\text{tangent coordinates}).
\]
Its quotient chart has smooth coordinates $t^2,(a_k)$ and the tangent
coordinates.  Every stabilizer on $\widehat V$ is contained in the
stabilizer of its image in $V$; away from the exceptional divisors it
is trivial.  These charts therefore establish smoothness everywhere.

The blowup is connected, smooth, and projective.  Its finite-group
quotient is projective: tensoring the translates of an ample line
bundle gives an invariant linearized ample line bundle, whose
invariant section ring constructs the projective quotient.  The
quotient is connected and has the same dimension as $V$, namely
$4+10=14$.  The morphism
$\pi\circ\operatorname{pr}_S\circ b$ is invariant and descends to a
surjective morphism $f:Y\to X$.

Over $U$, the map $S\to X$ is an \'etale $A$-torsor and the blowup has
no centre.  Hence
\[
 f^{-1}(U)=(S|_U\times F)/A.
\]
Analytically trivializing the finite \'etale torsor identifies this
map locally with projection from a product with $F$.  This proves the
first assertion, including properness, smoothness, and connectedness
of these fibres.  Their cotangent bundles are trivial.

We next check the divisor $D$ at all points, not only at generic
points away from the auxiliary divisors.  Let $x\in D$ and use the
coordinates of Lemma~\ref{lem:branched-cover}.  A stabilizer at a point
of $S\times F$ over $x$ is either trivial or $\langle e_i\rangle$ for
some $i\in I(x)$.  Indeed an inertia element involving $\tau$ has
support of size at least two and has no fixed point on $F$.  In
particular, every possible product stabilizer fixes the function
$w_0$, whose square is the pullback of $z_0$.

The pullback of $w_0$ to a blowup chart is still holomorphic and is
fixed by the stabilizer of every point in that chart.  A neighborhood
of the corresponding point of the finite quotient is the quotient
of a sufficiently small neighborhood by this stabilizer.  Thus
$w_0$ descends there to a holomorphic function whose square is
$f^*z_0$.  Together with the unchanged base coordinates it gives a
local holomorphic lift to the root chart of $D$.  This argument
includes points on exceptional divisors and all intersections with
the auxiliary branch divisors.

We can also determine the multiplicities exactly.  Write
$\pi^*D=2R$ with $R$ reduced.  No component of a centre $Z_i$ is
contained in $R\times F$, because every component of $S^{e_i}$
dominates $B_i\ne D$.  Hence blowing up introduces no exceptional
component in the pullback of $R\times F$, and
\[
 (f q)^*D=2\widehat R,
\]
where $\widehat R$ is its reduced strict transform.  At a general
point of every component of $\widehat R$ the image in $D$ avoids all
$B_i$.  There the only nontrivial inertia of $S\to X$ is generated
by $\tau$, which acts freely on $F$, so $q$ is unramified.  Every
prime component of $f^*D$ is dominated by a prime component of
$\widehat R$, since $q$ is finite.  The ramification index there is
one, and the displayed equality proves that its coefficient in
$f^*D$ is exactly two.

Over a general point of $D$, all the auxiliary divisors are absent.
There are no blowup centres and the diagonal action is free.
Consequently the quotient is locally isomorphic to $S\times F$, and
projection to $(w_0,z_2,z_3,z_4)$ is a submersive local lift to the
root chart.  This completes the second assertion.

Finally fix $i$ and take a general point of $B_i$ outside $D$ and the
other $B_j$.  Choose a point of $S^{e_i}$ above it and a point of
$F^{e_i}$.  Near the resulting point of $Z_i$, let $w$ be the cover
coordinate with $w^2=z_i$, where $z_i$ defines $B_i$.  The five normal
coordinates to the centre are $w,u_1,\ldots,u_4$, with the $u_j$
coming from the moving elliptic factors.  The other three base
coordinates are tangent to the centre and remain unchanged.

In the blowup chart distinguished by $w$, use
\[
 w,\quad u_1/w,\ldots,u_4/w
\]
as its normal-direction coordinates.  Passing to the local quotient
replaces $w$ by $\xi=w^2$ and leaves the ratios and all tangent
coordinates unchanged.  The map to $X$ then has coordinates
\[
 z_i=\xi,
 \qquad\text{the other three base coordinates unchanged}.
\]
It is a submersion, including at $\xi=0$ on the exceptional divisor.
This provides the required point above each general point of $B_i$.
Every other prime divisor distinct from $D$ has its general point in
$U$, where $f$ is smooth.  The third assertion follows.
\end{proof}

If $\mathcal X$ is special, Proposition~\ref{prop:transfer} now shows
that this $Y$ is special and gives an epimorphism
\[
 \pi_1(Y)\longrightarrow
 \pi_1(X\setminus D)/\langle\!\langle\gamma^2\rangle\!\rangle.
\]
Assumption~\ref{ass:manifold}, applied to this special smooth projective
fourteenfold, makes $\pi_1(Y)$ virtually abelian. Its finite-index
abelian subgroup has finite-index abelian image under the displayed
surjection. This gives a second proof of Theorem~\ref{thm:main}.
==== END A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//branched.tex ====

==== BEGIN A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//differentials.tex ====
\section{Differential lines and section ratios}

We record the elementary differential facts needed to transfer specialness. They apply on ordinary smooth projective manifolds; their root-chart use will be justified explicitly.

\begin{lemma}[Closed-form argument]\label{lem:ratios}
Let $V$ be a connected smooth projective manifold and let $i:L\to\Omega_V^p$ be a nonzero line-bundle map, where $1\le p\le\dim V$. If $s_0,\ldots,s_r$ are sections of the same positive power $L^{\otimes m}$, with $s_0\ne0$, then
\begin{equation}\label{eq:wedge-ratio}
 d(s_a/s_0)\wedge i(L)=0
\end{equation}
at the generic point. In particular, $\kappa(L)\le p$. If $\kappa(L)=p$, there are such sections $s_0,\ldots,s_p$ for which the image line of $i$ is generically spanned by
\begin{equation}\label{eq:wedge-description}
 d(s_1/s_0)\wedge\cdots\wedge d(s_p/s_0)\ne0.
\end{equation}
\end{lemma}

\begin{proof}
Adjoin simultaneous $m$th roots of the finitely many sections in copies of the total space of $L$. The equations $t_a^m=s_a$ are monic and define a finite space over $V$. Choose a reduced irreducible component dominating $V$ and take a smooth projective resolution $q:V'\to V$ of that component \cite{Hironaka}. The tautological roots are holomorphic sections of $q^*L$. Their differential images
\[
 \alpha_a\in H^0(V',\Omega_{V'}^p)
\]
are holomorphic, and $\alpha_0$ is generically nonzero because $q$ is generically finite in characteristic zero.

Holomorphic forms on a compact K\"ahler manifold are closed \cite{Demailly}. On the open set where $t_0\ne0$, we have $\alpha_a=(t_a/t_0)\alpha_0$, so
\[
 d(t_a/t_0)\wedge\alpha_0=0.
\]
Taking the $m$th power of the ratio gives \eqref{eq:wedge-ratio} upstairs; generic injectivity of differential pullback gives it downstairs.

If $r$ independent covectors each wedge to zero with a nonzero $p$-form, expansion in a basis containing those covectors shows that every nonzero term of the form contains all $r$ of them. Hence $r\le p$. In characteristic zero the image dimension of a section map is the generic rank of its ratio differentials, proving the bound. If that dimension is $p$, choose $p$ algebraically independent ratios from one section map. The same exterior-algebra argument gives \eqref{eq:wedge-description}.
\end{proof}

\begin{lemma}[Saturated meromorphic directions]\label{lem:saturation}
Let $E$ be a holomorphic vector bundle on a smooth complex manifold. The intersection with $E$ of a rank-one meromorphic subspace of its meromorphic sections is an invertible sheaf $K$ with a nonzero map $K\to E$. Moreover, a meromorphic tensor in the corresponding line inside $E^{\otimes m}$ is holomorphic if and only if it is a holomorphic section of $K^{\otimes m}$. Both assertions hold equivariantly in smooth root charts.
\end{lemma}

\begin{proof}
In a local frame clear the denominators of a meromorphic generator and divide out all common irreducible factors of its entries. Smooth local rings are factorial, so the resulting vector $v$ has holomorphic entries with no common height-one factor. If $av$ is holomorphic for a meromorphic scalar $a$, a pole of $a$ at a prime divisor would force that prime to divide every entry of $v$. Thus $a$ has no divisorial pole and is holomorphic. This proves that the saturated module is freely generated by $v$.

At every prime divisor some entry of $v$ is a unit at its generic point. The corresponding entry of $v^{\otimes m}$ has the same property. The identical pole test proves the tensor-power assertion. Holomorphic extension across codimension two gives the statements on the whole chart. The construction is intrinsic to the meromorphic direction and hence respects equivariance and changes of chart. The form-valued generator is allowed to vanish in codimension two; the assertion is about an invertible sheaf with a nonzero map, not about a subbundle everywhere.
\end{proof}

\begin{remark}[Comparison of differential conventions]\label{rem:conventions}
Under $z=w^m$, a coefficient $z^{-a}$ in a tensor containing $t$ occurrences of $dz$ pulls back with order
\[
 t(m-1)-ma.
\]
Thus the allowed pole is $a\le\lfloor t(1-1/m)\rfloor$, the usual orbifold symmetric-differential rule. For a fixed generic differential line, saturation of its powers has exactly the invariant sections of the corresponding saturated root line. The primitive-generator argument in Lemma~\ref{lem:saturation} verifies this assertion in codimension one and extends it across codimension two. This compares the section lattices; our specialness argument uses the line-bundle convention of Definition~\ref{def:special} directly.
\end{remark}

\begin{lemma}[Rational descent]\label{lem:rational-descent}
Let $g:Y\to X$ be a dominant morphism of connected smooth projective varieties with connected general fibre. A rational function on $Y$ which is constant on general fibres, wherever defined, is the pullback of a rational function on $X$.
\end{lemma}

\begin{proof}
Let $Z\subset X\times\PP^1$ be the closure of the joint image of $g$ and the function, on its regular locus. The actual image is constructible and dense in $Z$, so it contains a dense open of $Z$. Its fibres over general points of $X$ have at most one point. The fibre-dimension theorem therefore makes $Z\to X$ generically finite. After shrinking $X$, its generic degree is realized by distinct points, since the characteristic is zero, and all those points belong to the actual-image open. The degree is consequently one. Hence $\CC(Z)=\CC(X)$, giving the descent. Connectedness of the general fibre ensures that the constant is a single value on that fibre.
\end{proof}
==== END A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//differentials.tex ====

==== BEGIN A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//inertia-descent.tex ====
\section{Descent of positivity through the Iitaka fibration}
\label{in:descent}

We now derive a contradiction from Proposition~\ref{in:fibre-big}.
The proof keeps the boundary of actual meridian orders on every
prepared model.  An auxiliary boundary will be used only to study
Iitaka fibres; the final bigness statement will again concern the
actual boundary.

\subsection{A minimum over models and isogenies}

For every prepared model, including those after torus isogenies,
Assumption~\ref{in:input-tori} and Proposition~\ref{in:fibre-big} give
\begin{equation}\label{in:adjoint-lower-bound}
 \kappa(Z,K_Z+\Delta_Z)\ge\ell>0.
\end{equation}
Its hypotheses hold because $Z$ is smooth projective, $g:Z\to A$
is a fibration onto a torus, and $\Delta_Z$ is effective SNC with
coefficients strictly less than one.  Minimize the integer on the
left over \emph{all} these prepared models and all the indicated
isogenies.  The minimum is attained; denote it by $b$ and choose an
attaining model $Z_0$.  Relabel the torus, source, and quotients after
the isogeny producing that model.  Then
\begin{equation}\label{in:adjoint-minimum}
 b=\kappa(Z_0,K_{Z_0}+\Delta_{Z_0})\ge\ell,
\end{equation}
and the lower bound $b$ remains valid on every further prepared model
and after every further torus isogeny.  The source root orbifold
remains special.

Resolve the Iitaka fibration and all relevant supports on a smooth
projective modification $\mu:Z_1\to Z_0$, obtaining
\[
 h:Z_1\longrightarrow W,
 \qquad \dim W=b,
 \qquad g_1:Z_1\longrightarrow A,
\]
where $W$ is smooth projective and both maps are fibrations.  The
model may also resolve a plurisection system realizing the Iitaka
dimension.  Choose a rational SNC boundary $\Theta$ on $Z_1$ such
that
\begin{equation}\label{in:auxiliary-boundary}
 \Theta\ge\Delta_{Z_1},\qquad
 L:=K_{Z_1}+\Theta
 =\mu^*(K_{Z_0}+\Delta_{Z_0})+E,
 \qquad E\ge0\text{ exceptional},
\end{equation}
and all coefficients of $\Theta$ belong to $[0,1)$.

Here are the coefficient choices explicitly.  Keep the coefficients
of the strict transform of $\Delta_{Z_0}$, which equal the
corresponding actual coefficients. In the discrepancy formula
\[
 K_{Z_1}+\mu^{-1}_*\Delta_{Z_0}
 =\mu^*(K_{Z_0}+\Delta_{Z_0})+\sum_F a_F F,
\]
every exceptional $a_F$ is greater than $-1$ because the original
pair is klt.  The actual boundary coefficient $d_F$ is also less
than one.  Choose a rational $t_F<1$ satisfying
$t_F\ge\max(0,d_F,-a_F)$ and put that coefficient on $F$.
Then $a_F+t_F\ge0$, giving \eqref{in:auxiliary-boundary}.

Adding this effective exceptional divisor does not change the
plurisection spaces in divisible degrees.  A section upstairs pushes
to a meromorphic section downstairs with no poles at any prime
divisor, and extends across codimension two; the reverse inclusion is
pullback followed by multiplication by the exceptional section.
Thus $\kappa(Z_1,L)=b$, and $h$ is still its Iitaka fibration.
Moreover $L$ is big on a very general $g_1$-fibre: on such a fibre
\eqref{in:auxiliary-boundary} is a birational pullback of the big
adjoint on the corresponding $Z_0$-fibre, plus an effective divisor.

We give the two restriction arguments needed next.

\begin{lemma}\label{in:iitaka-fibre}
For a very general smooth fibre $H$ of $h$,
\[
 \kappa(H,L|_H)=0,
 \qquad L|_H=K_H+\Theta|_H.
\]
The restricted pair is smooth projective and klt.
\end{lemma}

\begin{proof}
Adjunction and the boundary assertion hold on very general smooth
fibres, using generic smoothness also on the boundary strata.
Some positive multiple of $L$ has a nonzero section which is not
identically zero on a very general fibre, so the restricted Iitaka
dimension is nonnegative.

There are an ample divisor $B$ on $W$, a positive integer $N$, and
an effective divisor $F$ with
\begin{equation}\label{in:ample-remainder}
 NL\sim_{\mathbb Q}h^*B+F,
\end{equation}
after clearing denominators.  Indeed, resolve a complete system of
maximal image dimension.  Its moving part comes from the pullback of
a hyperplane divisor on its image, which gives a big divisor on the
Stein base and on a smooth model of that base.  A sufficiently high
multiple of this big divisor contains an ample divisor.  The fixed
part and the effective remainder give \eqref{in:ample-remainder}.

Suppose that a positive multiple of $L|_H$ had ratios varying along
a very general $H$.  Choose this fibre also in the base-change loci
of all the countably many direct images $h_*\mathcal O_{Z_1}(mL)$
in divisible degrees.  For the relevant degree, twisting that
coherent direct image by a sufficiently high power $\mathcal O_W(kB)$
makes it globally generated.  The resulting sections of
$mL+h^*(kB)$ have ratios varying on general fibres.  Multiplying
them by a section of $k(NL-h^*B)$ from
\eqref{in:ample-remainder} puts them in $(m+kN)L$ and leaves their
ratios unchanged on general fibres.  Combining these sections with a
system having $b$ independent base ratios gives a plurisection map
of dimension at least $b+1$: products place the two systems in a
common degree, retaining both their base ratios and a nonconstant
fibre ratio.  This contradicts $\kappa(Z_1,L)=b$.
\end{proof}

\begin{lemma}\label{in:common-fibres}
Let $g:V\to A_0$ and $h_0:V\to W_0$ be surjective holomorphic maps
of smooth projective varieties with connected general fibres, and let
$M$ be a rational line bundle on $V$ which is big on a very general
smooth $g$-fibre.  Then $M$ is big on every component of a general
fibre of
\[
 (h_0,g):V\longrightarrow J\subset W_0\times A_0.
\]
Equivalently, one may take a very general $h_0$-fibre first and then
a general fibre of its map to its image in $A_0$.
\end{lemma}

\begin{proof}
Take a very general smooth $g$-fibre where base change holds for the
direct images of all divisible multiples of $M$.  Bigness supplies
one degree for which its section map has full generic differential
rank.  On an open of $A_0$ where the corresponding direct image is
locally free, relative evaluation therefore has injective vertical
differential on a dense open $U\subset V$.  The rank condition is
intrinsic and independent of a local frame of that direct image.

Every component of a general fibre of $V\to J$ meets $U$.  To see
this, apply the fibre-dimension theorem to the finitely many
components of the proper closed complement $V\setminus U$.  A
component dominating $J$ has strictly smaller general fibre dimension
than $V\to J$, and a component not dominating $J$ is avoided over
a general point.  The fibres themselves are equidimensional after
shrinking $J$.

At a general smooth point of each such fibre component, its tangent
space is contained in the $g$-vertical tangent space.  The relative
sections restrict to sections of $M$ on the component and still have
injective differential there.  They give a generically finite map,
which proves bigness.  The same open in $J$ meets the slices over
very general points of $W_0$ in dense opens, proving the last
formulation.
\end{proof}

\subsection{The Iitaka fibres are birational to torus translates}

\begin{lemma}\label{in:torus-fibres}
For a very general $h$-fibre $H$, its Albanese morphism
$\beta_H:H\to\operatorname{Alb}(H)$ is birational, and
$g_1|_H$ is the composition of $\beta_H$ with a translated isogeny
onto a subtorus translate in $A$.  Every component of $\Theta|_H$
has image of codimension at least two in that translate.
\end{lemma}

\begin{proof}
Lemma~\ref{in:iitaka-fibre} and
Assumption~\ref{in:input-tori}, applied with the point torus if
necessary, make $\beta_H$ surjective with connected fibres.
The map to $A$ factors through it by a translated homomorphism
$\operatorname{Alb}(H)\to A$.  Quotient $\operatorname{Alb}(H)$
by the connected component of that homomorphism's kernel.  We obtain
a fibration $H\to U_H$ onto a torus, followed by a translated
isogeny from $U_H$ onto the image of $H$ in $A$.

Apply Lemma~\ref{in:common-fibres} to $g_1,h$ and $L$.  It says
that $L|_H$ is big on the components of general fibres of $g_1|_H$,
and hence on a general fibre of $H\to U_H$.  On that smooth fibre
its restriction is the adjoint of the restricted klt pair.
Assumption~\ref{in:input-tori} on $H\to U_H$ now gives
\[
 0=\kappa(H,K_H+\Theta|_H)
 \ge\dim H-\dim U_H.
\]
The fibre dimension is therefore zero.  Since $H$ surjects onto
$\operatorname{Alb}(H)$ and that torus surjects onto $U_H$, all
three dimensions are equal.  Connected general fibres make
$\beta_H$ birational, and the homomorphism from its Albanese torus
to its image in $A$ has finite kernel.

Choose the effective relative canonical divisor for the birational
morphism $\beta_H$; its coefficient is positive on every exceptional
prime, by the vanishing of the Jacobian over a smooth target.  If a
component of $\Theta|_H$ mapped to a divisor $B_H$ in the Albanese
torus, a sufficiently small positive rational multiple of
$\beta_H^*B_H$ would be bounded by $K_H+\Theta|_H$.  At the strict
transform this uses its positive boundary coefficient, and at the
finitely many exceptional primes it uses the positive relative
canonical coefficients.  A nonzero effective divisor on an abelian
variety has positive Iitaka dimension, by the effective-divisor
structure used in Proposition~\ref{in:kernel}.  This contradicts
Lemma~\ref{in:iitaka-fibre}.  Thus the boundary is exceptional over
the Albanese torus.  The remaining map is finite, so its images also
have codimension at least two in the image translate.
\end{proof}

The subtorus occurring here is fixed as $H$ varies over very general
fibres.  More precisely, over a connected smooth-fibration open of
$W$, the image
\[
 M_0=\operatorname{im}\bigl(H_1(H,\mathbb Z)\longrightarrow
 H_1(A,\mathbb Z)\bigr)
\]
is constant by smooth proper transport and homotopy of the maps to
$A$.  Its real span determines a fixed subtorus $T_0\subseteq A$.
By Lemma~\ref{in:torus-fibres}, $M_0$ has finite index in
$\Lambda_{T_0}=\pi_1(T_0)$ and records the isogeny on a very general
fibre.  Choose $k>0$ with
\[
 k\Lambda_{T_0}\subseteq M_0.
\]
Make the multiplication isogeny $A^+\to A$ given by $[k]$, and let
$T^+\subset A^+$ be the identity component over $T_0$.  The
subtorus lattice is primitive in the ambient lattice, so the image
of $\pi_1(T^+)$ is $k\Lambda_{T_0}$.

Each connected component of the pullback of a very general $H$
maps birationally onto a translate of $T^+$.  Indeed, the covering
of that translate factors through the isogeny from
$\operatorname{Alb}(H)$ because of the lattice inclusion just
chosen.  Its fibre product with that isogeny is a disjoint union of
degree-one covers of the lifted translate.  The birational Albanese
morphism pulls back birationally to each of them.  There are no extra
components supported over its exceptional locus, since the map to
$H$ is finite \'etale and every component meets the dense open where
the birational morphism is an isomorphism.  The pulled-back boundary
on these fibres still has image of codimension at least two in the
lifted translate.

\subsection{A torus-bundle model and its actual boundary}

Put
\[
 \widetilde Z_1=Z_1\times_A A^+,
 \qquad X^+=X\times_A A^+.
\]
Both spaces are connected smooth projective by
\eqref{in:isogeny-groups}.  The root orbifold on $X^+$ is special.
The actual boundary on $\widetilde Z_1$ is the pullback of
$\Delta_{Z_1}$ and is dominated by the pullback of $\Theta$.

Let $W'$ be the Stein base of $\widetilde Z_1\to W$.  It is finite
over $W$.  The map to $A^+/T^+$ is constant on very general connected
Stein fibres, so it factors rationally through $W'$.  Resolve to a
smooth projective $W_1$ with a morphism to $A^+/T^+$, and form
\begin{equation}\label{in:torus-bundle}
 Z_*=W_1\times_{A^+/T^+}A^+,
 \qquad p:Z_*\longrightarrow W_1.
\end{equation}
The natural rational comparison from $\widetilde Z_1$ is birational:
on a very general connected Stein fibre it is the degree-one map
onto the lifted torus translate just established.  The map $p$ is
smooth with connected irreducible torus fibres.  Its relative canonical
bundle is trivial, since it is the pullback of a quotient map of
tori and the invariant relative volume form is nowhere zero.  Hence
\begin{equation}\label{in:canonical-bundle}
 K_{Z_*}=p^*K_{W_1}
\end{equation}
with compatible canonical divisors.  The construction works after
any further smooth projective modification of $W_1$.

Let $R^+=G^+/B_0\le R$ be the quotient for the pulled-back source.
Transfer its open monodromy to $Z_*$ and read the actual boundary
$\Delta_*$.  The same quotient is used on common open subsets of
all comparison models.

\begin{lemma}\label{in:vertical-boundary}
The support of $\Delta_*$ is vertical over $W_1$.  Consequently
there is an effective rational divisor $\Delta_{W_1}$, with finite
integral root coefficients, such that
\begin{equation}\label{in:boundary-pullback}
 \Delta_*=p^*\Delta_{W_1}.
\end{equation}
The coefficients on corresponding primes record the same actual
orders.  These assertions also hold after any further smooth
projective modification of $W_1$, with the orders read anew.
\end{lemma}

\begin{proof}
Compare $Z_*$ and $\widetilde Z_1$ on a common smooth projective
resolution, with compatible maps to $A^+$ and to $W'$.  Suppose a
prime of nontrivial order on $Z_*$ were horizontal over $W_1$.
Its strict transform maps, over a very general parameter, onto a
divisor in the corresponding torus translate in $A^+$.  If the
strict transform is exceptional over $\widetilde Z_1$, its centre
has codimension at least two there.  On very general fibres over
$W'$ that centre has codimension at least two or is absent, by the
fibre-dimension theorem.  Its image in the lifted translate cannot
be the indicated divisor.

If it is not exceptional, it corresponds to a divisor of nontrivial
actual order on $\widetilde Z_1$.  This divisor belongs to the
pullback of $\Theta$, because that pullback dominates the actual
boundary.  By Lemma~\ref{in:torus-fibres} and the \'etale pullback
description, its restriction to very general connected Stein fibres
has image of codimension at least two in the lifted torus translate.
Again it cannot supply a divisor there.  This contradiction excludes
horizontal support.  When $T^+$ has dimension zero, horizontal
prime divisors do not exist and the conclusion is immediate.

A vertical prime of the smooth map $p$ must dominate a prime of
$W_1$: the fibre dimension bound gives codimension at most one for
its image.  Over that prime the full inverse image is irreducible,
since its general torus fibre is geometrically irreducible, and it
is reduced in codimension one by smoothness.  Our vertical prime is
therefore exactly that inverse image, with pullback multiplicity
one.  Assign its actual coefficient to the corresponding prime on
$W_1$ to obtain \eqref{in:boundary-pullback}.  The same comparison
argument uses very general fibres over $W'$ after any further
modification of $W_1$, and proves the last assertion.
\end{proof}

\begin{proposition}\label{in:base-big}
For the actual boundary in \eqref{in:boundary-pullback},
\[
 K_{W_1}+\Delta_{W_1}\text{ is big}.
\]
This remains true for the version constructed on any further smooth
projective modification of $W_1$.
\end{proposition}

\begin{proof}
Dominate $Z_*$ by a prepared smooth projective model over $A^+$.
Its adjoint, defined by the actual boundary, has Iitaka dimension
at least $b$ by the minimum
\eqref{in:adjoint-minimum}, which included all further isogenies.
For every sufficiently divisible degree, its plurisections inject by
birational pushdown into the plurisections of $K_{Z_*}+\Delta_*$.
At every prime on $Z_*$, its strict transform has the same actual
meridian order.  Thus a meromorphic pluricanonical form pushed
downward has the prescribed pole bound at every prime.  It extends
as a section of the indicated line bundle across codimension two.
This proves the injection and preserves all section ratios.

Equations \eqref{in:canonical-bundle} and
\eqref{in:boundary-pullback}, together with
$p_*\mathcal O_{Z_*}=\mathcal O_{W_1}$, now give
\[
 b\le\kappa(Z_*,K_{Z_*}+\Delta_*)
   =\kappa(W_1,K_{W_1}+\Delta_{W_1}).
\]
Since $\dim W_1=\dim W=b$, the adjoint on $W_1$ is big.  For a
further modification of $W_1$, use its corresponding torus-bundle
model, Lemma~\ref{in:vertical-boundary}, and a new prepared model
dominating it.  The same minimum and pushdown argument applies.
No birational invariance of this actual adjoint is asserted or needed.
\end{proof}

\subsection{Preparing the base and constructing the differential line}

We finish by returning to the fixed smooth root source on $X^+$.
We shall only test primes on that source, and prepare the base to
ensure that the unresolved higher-codimension images do not contribute
additional prime-divisor conditions.

\begin{lemma}\label{in:neatness}
For a rational fibration $u_0:P\dashrightarrow W_0$ of smooth
projective varieties, one can modify the smooth projective base and
resolve the source to obtain
\[
 P\xleftarrow{\tau}\widehat P\overset{u}{\longrightarrow}W_1
\]
such that every prime divisor of $\widehat P$ whose image under $u$
has codimension at least two is exceptional over $P$.
\end{lemma}

\begin{proof}
First resolve the graph to a projective morphism over $W_0$.  Apply
projective flattening by a modification of the integral base, taking
the main transform of that graph; it is flat of the expected relative
dimension over the modified base.  Resolve the base.  The main
transform after this further base change is contained in the pullback
of the flat family, so its fibres have dimension at most the same
expected relative dimension.  It still maps properly and birationally
to $P$.

Resolve this main transform to $\widehat P$.  A prime mapping to
base codimension at least two cannot have a divisorial image in the
main transform, by the fibre dimension bound.  Its image there has
codimension at least two, and its image in $P$ consequently has
dimension at most $\dim P-2$.  It is exceptional over $P$, as
required.  All steps are projective in characteristic zero; the
flattening and base-change form used here is standard
\cite{Stacks,Hironaka}.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{in:main}]
Apply Lemma~\ref{in:neatness} to the induced rational fibration
$X^+\dashrightarrow W_1$, allowing a further modification of $W_1$.
Proposition~\ref{in:base-big} is valid for its new actual boundary.
Write
\[
 X^+\xleftarrow{\tau}\widehat X\overset{u}{\longrightarrow}W_1.
\]
The maps to $W_1$ and to $A^+$ give a holomorphic map
$\widehat X\to Z_*$ by the fibre product
\eqref{in:torus-bundle}: their composites to $A^+/T^+$ agree on a
dense open and hence everywhere.  It is compatible with the open
monodromy to $R^+$.

Let $E$ be the strict transform of a prime of $X^+$, let $r_E=2$
when that prime belongs to its root boundary and $r_E=1$ otherwise.
If $E$ dominates a boundary prime $B$ of $W_1$, write $m_B$ for
its order and $e=\operatorname{ord}_E(u^*B)$.  Lemma~\ref{in:local-divisibility},
applied to the smooth torus bundle and
\eqref{in:boundary-pullback}, gives
\begin{equation}\label{in:final-divisibility}
 m_B\mid e r_E.
\end{equation}
This also covers a source divisor whose image in $Z_*$ is a proper
subset of $p^{-1}(B)$.  Around that image over a general point of
$B$, filling the other deleted loci kills only meridians of trivial
image.  The remaining local group is the complement group of the
one smooth hypersurface $p^{-1}(B)$.  Its generator has order $m_B$;
a transverse source disc winds $e$ times and its boundary has order
dividing $r_E$ in $G^+$.  This explains precisely why the weighted
divisibility applies at these possibly exceptional intermediate
images.

Take the saturated line $\mathcal L$ in
$\Omega^b_{\mathcal X^+}$ determined generically by the top
differential of the rational map to $W_1$.  It is an orbifold line
bundle by the saturation argument in Section~\ref{in:foundations}.
We claim that, for all sufficiently divisible $j$, pullback gives
\begin{equation}\label{in:final-sections}
 H^0\bigl(W_1,\mathcal O_{W_1}(j(K_{W_1}+\Delta_{W_1}))\bigr)
 \longrightarrow H^0(\mathcal X^+,\mathcal L^{\otimes j}).
\end{equation}
It suffices to check regularity at every prime in the smooth root
charts.  At the general point corresponding to $E$, the modification
$\tau$ is an isomorphism.  If $E$ dominates a boundary prime as
above, a local equation of that prime has order $e r_E$ in the
source root coordinate.  The pulled-back local top differential has
common vanishing at least $e r_E-1$.  By
\eqref{in:final-divisibility},
\[
 e r_E-1\ \ge\ e r_E(1-m_B^{-1}),
\]
which cancels the pole permitted by the coefficient of $B$ in each
tensor factor.  Saturation makes this common divisorial factor the
relevant test for a section of the line power.

At primes dominating $W_1$, or mapping generically outside its
boundary, there is no boundary pole.  A strict source prime cannot
map into base codimension at least two by Lemma~\ref{in:neatness}.
These exhaust the primes of the original $X^+$.  Extension across
codimension two on its smooth root charts proves
\eqref{in:final-sections}.  Thus no specialness assertion is needed
on the resolved source $\widehat X$.

Pullback along a dominant map preserves ratios.  Bigness from
Proposition~\ref{in:base-big} therefore implies
$\kappa(\mathcal X^+,\mathcal L)\ge b$.  The root differential
bound of Lemma~\ref{in:root-bound} gives the reverse inequality.
Since $1\le\ell\le b\le\dim X^+=4$, this is a forbidden
differential line on the special orbifold $\mathcal X^+$.

The contradiction excludes the infinite-kernel case in
Proposition~\ref{in:group-reduction}.  The reductions in
Section~\ref{in:finite-cover-cases} handle a trivial meridian and a
meridian detected in a finite quotient by applications of
Assumption~\ref{in:input-manifold} to
the appropriate special smooth projective manifolds.  A nontrivial
meridian invisible in all finite quotients would give exactly the
infinite kernel excluded above.  Hence the original entire orbifold
group $G$ has an abelian subgroup of finite index.
\end{proof}
==== END A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//inertia-descent.tex ====

==== BEGIN A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//inertia-foundations.tex ====
\section{Root differentials and finite covers}\label{in:foundations}

We allow, throughout the preparatory arguments, that the reduced divisor
\(D\) is empty or is a disjoint union of smooth divisors.  Its nonempty
components all have order two.  Write \(\mathcal X\) for the resulting
root orbifold and \(\pi:\mathcal X\to X\) for its coarse map.  At a point
of \(D\), a chart has coordinates
\[
 z_1=w_1^2,\qquad z_j=w_j\quad(j>1),
\]
with the sign action on \(w_1\).  For a prime divisor \(E\) of \(X\), set
\(r_E=2\) when \(E\subset D\), and \(r_E=1\) otherwise.  Saturation of a
meromorphic line direction in a locally free sheaf is understood on
these charts, equivariantly.

\begin{lemma}[Auxiliary projective source]\label{in:auxiliary}
Let \(X\) be a connected smooth projective variety with the root datum
just described.  There is a connected smooth projective variety \(T\)
and a surjective generically finite morphism \(d:T\to X\), Galois on
function fields and equivariant for its finite Galois group, such that
\(d^*D\) is twice an integral divisor.  This includes every exceptional
prime of \(d\).  The morphism lifts to \(T\to\mathcal X\), and induces a
homomorphism
\[
 \chi_d:\pi_1(T)\longrightarrow \pi_1^{\mathrm{orb}}(\mathcal X)
\]
with finite-index image, compatible with the map on divisor
complements.  A deck transformation changes this homomorphism only by
the base-point conjugation coming from the target orbifold group.
\end{lemma}

\begin{proof}
Choose a line bundle \(L\) such that \(L^{\otimes2}(-D)\) is very ample,
and choose finitely many sections \(b_i\) of this bundle having no
common zero on \(D\).  If \(s_D\) is the canonical section of
\(\mathcal O_X(D)\), adjoin square roots of the sections \(s_Db_i\) of
\(L^{\otimes2}\), interpreted after a rational trivialization of \(L\).
Their compositum is a finite Galois field extension.  Normalize \(X\)
in this field and take a projective resolution equivariant for the
finite group, unchanged over the smooth locus; denote the composite
by \(d\).  Canonical characteristic-zero resolution is functorial
for smooth morphisms and equivariant for group actions
\cite[Theorems~1.0.2--1.0.3]{Wlodarczyk}.  For projectivity, embed the
normalization equivariantly in projective space by taking the product
of the translates of a projective embedding and then the Segre
embedding.  The embedded resolution consists of blowups in invariant
smooth centres; its ambient space and final strict transform are
projective.  The normalization is integral, so its resolution is
connected.

Let \(F\) be any prime of \(T\) whose image is contained in \(D\).  At
the generic point of its image some \(b_i\) is a unit.  In local frames,
the square-root equation gives
\[
 2\operatorname{ord}_F(t_i)
   =\operatorname{ord}_F(d^*s_D).
\]
The roots \(t_i\) are regular on the normalization, since they are
integral over its local rings, and remain regular after resolution.
The equality proves evenness also when \(F\) is exceptional.  As the
components of \(D\) are disjoint, each \(d^*D_j\) is separately twice
an integral effective divisor.  The line bundle and canonical section
of its half divisor are a square root of the pulled-back divisor
bundle and section.  These data give the asserted map to the root
orbifold.  For empty \(D\) one may take the identity.

On \(T\setminus d^{-1}D\) the map to \(X\setminus D\) induces a map
to the orbifold group.  Every meridian filled in on passing to \(T\)
maps either trivially or to an even power of a meridian about one
smooth branch of \(D\).  The normal-meridian description proved in
Lemma~\ref{in:topology} therefore gives \(\chi_d\).  On suitable dense
opens \(d\) is a finite \'{e}tale cover.  Its group image has finite
index in the complement group, hence also in its orbifold quotient.
This image is contained in \(\operatorname{im}\chi_d\).  Finally,
transport between two sheets projects to a loop in \(X\setminus D\);
compatibility there gives the stated conjugation formula, and the
surjection from the open group to \(\pi_1(T)\) extends it to the
compact source.
\end{proof}

\begin{lemma}[Saturated lines and the differential bound]
\label{in:root-bound}
A meromorphic line direction in \(\Omega^p_{\mathcal X}\) has a
saturated holomorphic line bundle with a nonzero map to
\(\Omega^p_{\mathcal X}\).  If a holomorphic line bundle
\(\mathcal L\) has such a map, then
\(\kappa(\mathcal X,\mathcal L)\le p\).  Saturating the image does not
decrease its Iitaka dimension.  In particular, specialness of
\(\mathcal X\) implies specialness of \(X\).
\end{lemma}

\begin{proof}
Intersect the meromorphic line with the locally free differential
sheaf.  This saturated rank-one subsheaf is reflexive: it is the
kernel of a morphism from a locally free sheaf to a torsion-free
sheaf, and the depth criterion gives the reflexive condition on a
smooth chart.  A rank-one reflexive sheaf on a smooth chart is
invertible.  Uniqueness of saturation supplies equivariant gluing.
Replacing a line map by its saturation permits an effective
divisorial enlargement and therefore cannot decrease the section-map
dimension.

Pull \(\mathcal L\) and its map to the source \(T\) of
Lemma~\ref{in:auxiliary}.  The induced map to \(\Omega_T^p\) is
nonzero, since \(d\) is generically \'{e}tale.  Ratios of sections keep
their transcendence degree under this dominant map.  The manifold
differential bound in Assumption~\ref{in:input-special}, applied to
the smooth projective \(T\), consequently implies
\(\kappa(\mathcal X,\mathcal L)\le p\).  For the last assertion,
ordinary differential line maps pull back to root charts, and
\(H^0(\mathcal X,\pi^*M^{\otimes j})=H^0(X,M^{\otimes j})\) for an
ordinary line bundle \(M\).  An ordinary witnessing line would
therefore witness nonspecialness on \(\mathcal X\).
\end{proof}

We make precise the specialness convention used when applying the
orbifold literature.  For a simple normal crossing root datum of
finite orders \(m_i\), write locally \(z_i=w_i^{m_i}\).  A monomial
in \(\operatorname{Sym}^j\Omega^p_X\) in which \(dz_i\) occurs
\(t_i\) times has an invariant holomorphic lift exactly when its
coefficient has pole order at most
\begin{equation}\label{in:rounding}
 \left\lfloor t_i(1-m_i^{-1})\right\rfloor
 \quad\text{along }z_i=0.
\end{equation}
Indeed a coefficient of order \(-a_i\) pulls back with exponent
\(-m_i a_i+(m_i-1)t_i\) in \(w_i\).  These are precisely the local
generators of Campana's orbifold symmetric differential sheaves.

For a coarse generic differential line, saturate upstairs to obtain
\(\mathcal L\).  At every chart prime its generator is primitive,
as is each tensor power.  Thus the invariant sections of
\(\mathcal L^{\otimes j}\) are exactly the saturation of the
corresponding coarse generic power in the sheaf given by
\eqref{in:rounding}; equality across codimension two follows by
reflexivity.  Conversely an orbifold differential line determines
such a coarse generic line.  The invariant sheaves are coherent and
algebraic on the projective coarse space \cite{GAGA}.  A power
divisible by all stabilizer orders has trivial stabilizer character
and descends to an ordinary line bundle: locally average a frame
whose value at the point is invariant.  Such a frame remains
nonvanishing after shrinking.  Divisible powers compute the same
Iitaka dimension, since ratios can be raised to common powers.
The usual section-growth definition agrees as well; any nonzero
residue degree can be shifted into divisible degrees by multiplying
by a fixed nonzero section in the complementary residue degree,
obtained as a power of an existing section.  This identifies the
line-bundle definition used here with the saturated Bogomolov-sheaf
criterion of \cite[Theorems~4.14--4.15]{CampanaSurvey}.

\begin{proposition}[Finite stability]\label{in:finite-stability}
Suppose a finite orbifold \'{e}tale morphism has smooth connected
projective coarse source and target, and that both orbifold
boundaries are empty or disjoint unions of smooth order-two
divisors.  If the target is special in the line-bundle convention,
so is the source.  This applies both to ordinary finite \'{e}tale
pullbacks of the pair and to finite manifold covers of
\(\mathcal X\) in the orbifold sense.
\end{proposition}

\begin{proof}
Campana's \cite[Proposition~9.11]{CampanaOrbifolds}, with orbifold
\'{e}taleness as in Definition~2.23 and Proposition~2.24 there,
preserves specialness for smooth connected geometric orbifolds in
Fujiki's class \(\mathcal C\) of the same dimension, under a
morphism \'{e}tale in codimension one in the orbifold sense.  In our
application these orbifolds are projective, integral, and smooth
with simple normal crossing support.  The local condition at a
source prime is
\[
 e_E\,m_{\mathrm{source}}(E)
       =m_{\mathrm{target}}(v(E)).
\]
Over a target order-two divisor the possibilities are
\((e_E,m_{\mathrm{source}}(E))=(1,2)\) or \((2,1)\);
off the boundary they are \((1,1)\).  Thus every hypothesis of that
proposition holds.  The preceding identification of symmetric
differentials identifies its specialness criterion with ours.
\end{proof}

\section{Open fibre images, inertia, and the Albanese kernel}
\label{in:inertia}

All group images below include a choice of path to the fixed base
point.  Changing that path conjugates the image.  We use actual
images of groups of open fibres, rather than abstract groups of
compact fibres.

\begin{lemma}[Complement groups and dominant maps]\label{in:topology}
If \(U\) is a dense Zariski open in a smooth connected complex
variety \(M\), then \(\pi_1(U)\to\pi_1(M)\) is surjective and its
kernel is normally generated by meridians of the divisorial
components of \(M\setminus U\).  A dominant morphism between smooth
connected algebraic varieties has finite-index image on fundamental
groups.  The same conclusion holds for a rational map on any dense
open where it is defined.
\end{lemma}

\begin{proof}
Perturb loops and null-homotopy discs transversely to smooth strata
of the deleted set.  Loops miss it; discs miss the strata of complex
codimension at least two and meet divisors in finitely many
transverse points.  Removing small discs about these points proves
normal generation.  This argument also applies in analytic balls.
For the second assertion, take the closure of a closed point of the
generic fibre of the dominant map.  It is generically finite over
the target.  After shrinking it and the target, separability and
generic finiteness give a finite \'{e}tale cover of a target open.
The resulting finite-index subgroup maps to a finite-index subgroup
of the full target group, by the first assertion, and its image is
contained in the image under consideration.
\end{proof}

In particular, local root charts and van Kampen give
\begin{equation}\label{in:orbifold-presentation}
 \pi_1^{\mathrm{orb}}(\mathcal X)
 =\pi_1(X\setminus D)
     /\langle\!\langle\gamma_j^2\rangle\!\rangle,
\end{equation}
where one takes the meridians of all components.  Equivalently,
ordinary covers of the complement extend in root charts precisely
when the squared meridians act trivially.  This description also
identifies subgroups with connected orbifold covers.

\begin{lemma}[Open fibre images]\label{in:open-images}
Let \(f:X\dashrightarrow Z\) be a rational fibration onto a smooth
projective model.  Resolve its graph, remove the inverse image of
\(D\), and take suitable dense opens of source and base.  The image
\(B_f\) of the general open fibre group in \(G\) is well defined up
to conjugacy, is normal in \(G\), and is unchanged by further
shrinking or resolution.  There is a surjection
\begin{equation}\label{in:base-monodromy}
 \psi_f:\pi_1(Z^\circ)\longrightarrow G/B_f
\end{equation}
compatible with the map from the open total space.
The same assertions hold for any fixed homomorphic image of the
open source group.
\end{lemma}

\begin{proof}
Resolve the graph and its deleted locus.  By generic smoothness,
after shrinking the base the compact family is smooth proper and
its deleted divisor is a relative simple normal crossing divisor;
include the exceptional loci among the deletions.  The family of
pairs is differentiably locally trivial.  One can see this directly
by lifting base vector fields tangent to all boundary components
in relative crossing coordinates, patching with a partition of
unity, and using properness to integrate the lifts.  The complement
therefore is a locally trivial fibration with connected fibres.
Its homotopy sequence ends in
\[
 \pi_1(F^\circ)\longrightarrow\pi_1(X^\circ)
       \longrightarrow\pi_1(Z^\circ)\longrightarrow1.
\]
The open total space maps onto \(G\), since it contains an open
identified with a dense open of \(X\setminus D\).  Exactness gives
normality and \eqref{in:base-monodromy}.  To compare models, pass
to common dense opens of their smooth general fibres.
Lemma~\ref{in:topology} says that deleting further analytic subsets
does not change their images in the fixed target group.  Transport
in the family handles the base-point choices.
\end{proof}

\begin{lemma}[Actual inertia]\label{in:actual-inertia}
For the quotient \(R=G/B_f\) and the monodromy
\eqref{in:base-monodromy}, the image of a meridian about every prime
\(J\) on any smooth projective model of \(Z\) has finite order
\(m_J\).  Set \(m_J=1\) on primes meeting the chosen open.  The divisor
\begin{equation}\label{in:boundary}
 \Delta_Z=\sum_J(1-m_J^{-1})J
\end{equation}
has finite support on each model, and orders agree on strict
transforms.  Smooth projective models whose divisorial deleted
locus has simple normal crossings can be chosen above any given
model, and over a fixed torus whenever the map to it is already
holomorphic.
\end{lemma}

\begin{proof}
Transfer the homomorphism through common base opens and resolve the
source map to the chosen model.  A source prime dominating \(J\)
has positive pullback multiplicity \(e\).  A transverse punctured
disc at its general point lies in the compatibility open and winds
\(e\) times about \(J\).  Its loop has finite image in \(G\): the
resolved source maps holomorphically to \(X\), and a small ball in
\(X\) meets at most one smooth boundary branch.  The local
complement group has image of order at most two in \(G\).  Thus
the \(2e\)-th power of the meridian of \(J\) dies in \(R\).
This reasoning includes exceptional primes on either model.
Only deleted divisors can have nontrivial inertia, giving finite
support.  Strict transforms have the same transverse loops.
Log resolution of the complement, together with the graph of the
map to the fixed torus when needed, gives the final assertion.
\end{proof}

\begin{lemma}[Weighted local divisibility]\label{in:local-divisibility}
In the setting of Lemma~\ref{in:actual-inertia}, suppose
\(p:Z\to W\) is smooth proper with connected fibres, both spaces
are smooth, and \(\Delta_Z=p^*\Delta_W\).  Let a resolution of
\(f\) give \(u:\widehat X\to W\).  If the strict transform \(E\)
of a prime of \(X\) dominates a boundary prime \(J\) of \(W\), then
\begin{equation}\label{in:divisibility}
 m_J\mid r_E\operatorname{ord}_E(u^*J).
\end{equation}
The image of \(E\) in \(Z\) need not dominate \(p^{-1}J\).
\end{lemma}

\begin{proof}
Work over a general point of \(J\), where it is smooth and no
other boundary component of \(W\) occurs.  In a small ball about
the relevant image point in \(Z\), \(p^{-1}J\) is smooth and is
the only divisor with potentially nontrivial meridian image.
Adding back all other deleted divisors kills normally generating
meridians which already map trivially; codimension-two deletions
do not affect this calculation.  Thus the local monodromy factors
through the complement of this single smooth hypersurface.
Its generator has order \(m_J\) in \(R\).  A transverse disc to
\(E\) winds \(\operatorname{ord}_E(u^*J)\) times about it.  Since
\(E\) is nonexceptional over \(X\), its source meridian has
\(r_E\)-th power trivial in \(G\).  Compatibility proves
\eqref{in:divisibility}.
\end{proof}

\begin{lemma}[Vanishing in root coordinates]\label{in:weighted-vanishing}
Let \(u:X\to W\) be a dominant holomorphic map with
\(b=\dim W>0\).  If \(E\) dominates a smooth divisor \(J\) and
\(e=\operatorname{ord}_E(u^*J)\), the pullback of a local volume
form on \(W\) vanishes to order at least \(r_Ee-1\) at the chart
prime over \(E\).  Consequently it absorbs the pole of coefficient
\(1-1/m_J\) whenever \(m_J\le r_Ee\).  If \(u(E)\) has
codimension at least two, the pulled-back volume form vanishes at
least once at that chart prime.
\end{lemma}

\begin{proof}
The normal equation of \(J\) pulls back to a unit times
\(w_1^{r_Ee}\).  Differentiation and wedging give the first bound,
and
\(r_Ee-1\ge r_Ee(1-1/m_J)\) gives the cancellation in each
divisible pluridegree.  In the second case the tangential rank on
the chart divisor is at most \(b-2\).  Adding the transverse
direction still leaves rank at most \(b-1\) along that divisor,
so every coefficient of the pulled-back top form vanishes there.
\end{proof}

\subsection{Removing the finite-cover cases}
\label{in:finite-cover-cases}
Initially \(D\) is smooth and connected.  By
Lemma~\ref{in:root-bound}, \(X\) is special.  The manifold theorem
in Assumption~\ref{in:input-manifold} therefore makes
\(\pi_1(X)\) virtually abelian.  If the meridian \(\gamma\) is
trivial in \(G\), then \eqref{in:orbifold-presentation} and
Lemma~\ref{in:topology} identify \(G\) with \(\pi_1(X)\), settling
this case.

If \(\gamma\) survives in a finite quotient of \(G\), take the
associated normal cover of \(X\setminus D\).  Its extension across
\(D\) has the smooth local form \(z_1=w_1^2\): normality makes
every inertia orbit have length two, and connectedness of \(D\)
makes all its meridians conjugate.  The resulting connected cover
\(Y\to X\) is finite and smooth projective, by normalization and
Riemann existence.  It is orbifold \'{e}tale with empty source
boundary.  The subgroup identification for
\eqref{in:orbifold-presentation} gives
\(\pi_1(Y)=\ker(G\to F)\), where \(F\) is the chosen finite image:
the meridians filled in upstairs are exactly the lifted squares,
with all their sheet transports.  Proposition~\ref{in:finite-stability}
and Assumption~\ref{in:input-manifold} now settle this case as well.

For the remainder assume \(\gamma\ne1\) and that it dies in every
finite quotient.  Put
\[
 K=\ker\bigl(G\longrightarrow\pi_1(X)\bigr).
\]
It is the normal closure of \(\gamma\).  Every finite-index subgroup
of \(G\) contains \(K\), since its finite-index normal core does.
Choose a connected finite \'{e}tale cover of \(X\) whose ordinary
fundamental group is free abelian; such a subgroup exists in any
finitely generated virtually abelian group.  Pull up the root
datum.  The new group is the preimage of this subgroup in \(G\),
and its coarse kernel is the same \(K\).  This follows either from
orbifold covering theory or by transporting all meridian-square
relations through the ordinary cover.  The pair stays special by
Proposition~\ref{in:finite-stability}.  Relabel these data by
\(X,D,\mathcal X,G\).  The support of \(D\) is now allowed to be
disconnected but remains a disjoint union of smooth divisors.

Assumption~\ref{in:input-special} gives a surjective Albanese map
\(\alpha:X\to A\) with connected fibres.  The abelian variety
\(A=\operatorname{Alb}(X)\) has lattice
\(\Lambda=\pi_1(A)=H_1(X,\mathbb Z)/\mathrm{torsion}\).
Our ordinary group is free abelian, so the Albanese map identifies
\(\pi_1(X)\) with \(\Lambda\).  Hence
\begin{equation}\label{in:lattice-extension}
 1\longrightarrow K\longrightarrow G
       \longrightarrow\Lambda\longrightarrow1.
\end{equation}

\begin{proposition}[The Albanese kernel]\label{in:kernel}
For a general point \(a\in A\), put \(P_a=\alpha^{-1}(a)\).
Then
\begin{equation}\label{in:kernel-image}
 K=\operatorname{im}\bigl(\pi_1(P_a\setminus D)\longrightarrow G\bigr).
\end{equation}
In particular \(K\) is finitely generated.
\end{proposition}

\begin{proof}
Let \(B\) be the open fibre image in
Lemma~\ref{in:open-images}, applied to \(\alpha\).  Then \(B\subset K\),
and the induced monodromy \(\pi_1(A^\circ)\twoheadrightarrow G/B\)
has the finite actual inertia orders of
Lemma~\ref{in:actual-inertia}.  Denote its boundary by \(\Delta_A\).
If \(A\) is a point, \eqref{in:kernel-image} is immediate, so assume
otherwise.

We recall the needed effective-divisor fact on abelian varieties.
Any nonzero effective rational divisor \(\Theta\) on \(A\) is
\(p^*\Theta_U\) for a quotient \(p:A\to U\) with connected fibres
and an ample effective rational divisor \(\Theta_U\), with
\(\dim U>0\).  To see this, clear denominators and let \(L\) be
the corresponding effective line bundle.  The connected kernel
of \(x\mapsto t_x^*L\otimes L^{-1}\) acts projectively on
\(H^0(A,L)\).  Its action is trivial, since a morphism from a
connected complete group to the affine group \(\operatorname{PGL}\)
is constant.  Thus it fixes the divisor, which descends to the
quotient by this connected kernel.  The descended effective line
bundle is nef (translate its divisor to test intersection with
any curve), and its translation kernel is finite.  The line-bundle
criterion on abelian varieties makes it ample: the associated
semipositive Hermitian form has radical the connected translation
kernel, so here is positive definite.  The quotient has positive
dimension because the divisor is nonzero.  Prime components and
their rational coefficients pull back unchanged under the smooth
quotient.

Suppose now \(\Delta_A\ne0\), and write
\(\Delta_A=p^*\Delta_U\) in this way.  Put \(b=\dim U\) and
\(u=p\alpha\).  Let \(\mathcal L_u\subset\Omega^b_{\mathcal X}\)
be the saturated line of pulled-back top forms.  For every prime
\(E\) dominating a boundary prime \(J\) of \(U\),
Lemma~\ref{in:local-divisibility} gives
\(m_J\mid r_E\operatorname{ord}_E(u^*J)\).  Thus
Lemma~\ref{in:weighted-vanishing} cancels the full allowed pole
along \(E\).  For a prime mapping into codimension at least two
in \(U\), that lemma gives positive vanishing.  There are only
finitely many primes in \(u^*\Delta_U\); choose a single rational
\(0<\epsilon\le1\) small enough that this positive vanishing
cancels the coefficient of \(u^*(\epsilon\Delta_U)\) at each of
these remaining primes, in root coordinates.

For every sufficiently divisible \(j\), pullback consequently gives
\[
 H^0\bigl(U,j(K_U+\epsilon\Delta_U)\bigr)
       \longrightarrow H^0(\mathcal X,\mathcal L_u^{\otimes j}).
\]
This is checked at chart primes and extends across codimension
two.  Dominance preserves section ratios.  Since \(K_U\) is
trivial and \(\epsilon\Delta_U\) is ample, the right-hand line
has Iitaka dimension at least \(b\), hence exactly \(b\) by
Lemma~\ref{in:root-bound}.  This contradicts specialness.

It follows that \(\Delta_A=0\).  All deleted divisorial meridians
are therefore killed, and Lemma~\ref{in:topology} factors the base
monodromy through \(\pi_1(A)\).  Its induced map to \(G/B\) is
inverse to the natural map \(G/B\to\Lambda\): both composites
are determined on the surjecting open groups.  Hence \(B=K\).
The smooth quasi-projective variety \(P_a\setminus D\) has finite
CW type, so its fundamental group, and its image \(K\), are
finitely generated.
\end{proof}

\begin{lemma}[Finite kernels over a lattice]\label{in:finite-by-lattice}
If \(1\to F\to E\to\mathbb Z^r\to1\) is exact with \(F\) finite,
then \(E\) has a finite-index free abelian subgroup disjoint from
\(F\).  If \(K\) is a finitely generated normal subgroup of a group
\(G\) with lattice quotient, every finite-index \(K_0\le K\)
contains \(K\cap G_0\) for some finite-index \(G_0\le G\).
\end{lemma}

\begin{proof}
Centralize \(F\) by passing to a finite-index subgroup \(E_1\).
Its commutator subgroup lies in the finite central group
\(F\cap E_1\).  If \(x_1,\ldots,x_t\) generate \(E_1\), and
\(e\) annihilates \(F\), then
\([x_i^e,x_j^e]=[x_i,x_j]^{e^2}=1\).  The subgroup generated by
these powers is abelian and has finite index: its image has finite
index in the abelianization, whose kernel here is finite.  A
torsion-free subgroup of finite index in this finitely generated
abelian group proves the first claim.

For the second, a finitely generated group has only finitely many
subgroups of each specified index.  Intersect all subgroups of
\(K\) having index \([K:K_0]\).  The result \(K_1\subset K_0\)
is characteristic and finite-index in \(K\), hence normal in
\(G\).  Apply the first claim to \(G/K_1\), whose finite kernel
is \(K/K_1\), and pull back a subgroup disjoint from this kernel.
\end{proof}

\begin{proposition}[The remaining group and its covers]
\label{in:group-reduction}
In the remaining case of \eqref{in:lattice-extension}, the group
\(K\) is infinite, finitely generated, and has no proper
finite-index subgroup.  All finite-dimensional complex linear
representations of \(K\), or of any quotient of \(K\), are trivial.
Every finite-index subgroup of \(G\) is the preimage of a
finite-index sublattice of \(\Lambda\), and is geometrically
realized by pulling the pair back along the corresponding
isogeny of \(A\).  These pullbacks are connected and special;
their kernels are the same \(K\), and the open fibre images in
\eqref{in:kernel-image} identify with the original ones.
\end{proposition}

\begin{proof}
We already know that every finite-index subgroup of \(G\)
contains \(K\).  Proposition~\ref{in:kernel} and
Lemma~\ref{in:finite-by-lattice} therefore rule out every proper
finite-index subgroup of \(K\).  The group \(K\) is nontrivial
because it contains the surviving meridian; it must be infinite,
since a nontrivial finite group has a proper finite-index trivial
subgroup.  A complex linear image is finitely generated and
residually finite by Malcev's theorem; see \cite{Nica}.  Any
nonidentity element in such an image would survive in a finite
quotient of \(K\), which is impossible.  The same reasoning
applies to its quotients.

If \(G_0\le G\) has finite index, then \(K\subset G_0\), so
\(G_0\) is exactly the preimage of its lattice image
\(\Lambda_0\).  The inclusion \(\Lambda_0\subset\Lambda\)
defines an isogeny \(A_0\to A\).  The pullback
\(X_0=X\times_A A_0\) is connected: on ordinary fundamental
groups the map \(\pi_1(X)\to\Lambda\) is an isomorphism, so
the monodromy on the isogeny's fibre is transitive.  Its pulled-back
root orbifold has group \(G_0\), by the complement presentation,
and is special by Proposition~\ref{in:finite-stability}.  A fibre
over a specified point of \(A_0\) is isomorphic to the original
fibre over its image, including the boundary.  These identifications
give the asserted equality of open images inside \(K\); images
of resolved subvarieties are compared on common dense opens in
the same way.
\end{proof}
==== END A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//inertia-foundations.tex ====

==== BEGIN A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//inertia-inputs.tex ====
\section{Statement, conventions, and conditional inputs}
\label{in:statement}

Campana's theory of special varieties and geometric orbifolds
\cite{Campana2004,CampanaOrbifolds} leads to the orbifold Abelianity
Conjecture \cite[Conjecture~12.10(1)]{CampanaOrbifolds}. This note
gives a conditional proof for a special smooth projective fourfold pair
with one smooth divisor of multiplicity two. It accompanies \emph{A
conditional abelianity theorem for special fourfold pairs with a
half-weight divisor}. The main article uses Theorem~1.1 of the companion
paper \cite{Supplied} as its sole result from that paper. The present proof uses its
Theorems~1.1, 2.1, 2.3, 2.5 and 4.1, stated below, and develops the
descent through inertia and quotient minimality in detail.

All varieties in the proof are over $\mathbb C$.  A fibration is a
dominant map with connected general fibre.  For a rational or meromorphic
fibration, this convention is applied on smooth models after resolution.
All fundamental groups are entire discrete topological fundamental
groups.  Base points and transporting paths are chosen compatibly; a
change of these choices conjugates the subgroup images under discussion.

Let $X$ be smooth and let $D$ be a smooth reduced divisor.  The
\emph{order-two root orbifold} $\mathcal X$ of $D$ agrees with $X$ off
$D$, and along $D=\{z_1=0\}$ it has the chart
\[
 z_1=w_1^2,\qquad z_j=w_j\quad (j>1),
\]
with the group $\mu_2$ acting by sign on $w_1$.  Orbifold line bundles,
forms, and their maps are equivariant holomorphic objects on these
charts, with compatible gluing.  We also use finite integral root
orbifolds on smooth spaces with simple normal crossing divisors.  An
order $m_i$ along $D_i$ corresponds to the coefficient $1-1/m_i$;
order one contributes no boundary.

For an orbifold line bundle $\mathcal L$, its Iitaka dimension is
$-\infty$ if every positive tensor power has zero global section space,
and otherwise is the largest dimension of the images of its complete
plurisection maps.  A root orbifold $\mathcal X$ of dimension $n$ is
\emph{special} if there is no nonzero sheaf map
\[
 \mathcal L\longrightarrow\Omega^p_{\mathcal X},
 \qquad 1\le p\le n,
 \qquad \kappa(\mathcal X,\mathcal L)=p.
\]
For a manifold the same definition uses ordinary line bundles and
ordinary differential forms.  A point is special.

\begin{theorem}\label{in:main}
Assume the five results in
Assumptions~\ref{in:input-manifold}--\ref{in:input-positive}.
Let $X$ be a connected
smooth projective variety of dimension four, and let $D$ be a nonempty
smooth connected reduced effective divisor.  If the order-two root
orbifold $\mathcal X$ of $D$ is special, then
\[
 G=\pi_1(X\setminus D,x)\big/\langle\!\langle\gamma^2\rangle\!\rangle
\]
is virtually abelian.  Here $\gamma$ is a positively oriented meridian
of $D$, transported to $x\in X\setminus D$, and virtual abelianity
means the existence of an abelian subgroup of finite index.
\end{theorem}

After the cases of a trivial meridian or one detected by a finite
quotient, the remaining obstruction is the kernel $K$ of
$G\to\pi_1(X)$. We identify it with the image of
the group of an open Albanese fibre. A first minimum, denoted by
$\ell$, ranges over rational fibrations of suitable compact Albanese
fibres whose open fibre images leave an infinite quotient of $K$.
It provides the dimension bound required for every infinite quotient
on one fixed auxiliary manifold in Theorem~4.1 of the companion paper.

This gives bigness on the general fibres of an intermediate fibration
over the Albanese torus. A second minimum, denoted by $b$, ranges over
the Iitaka dimensions of the adjoints defined by actual meridian
orders, on all prepared models and after all torus isogenies.
It provides a lower bound throughout the subsequent changes of model.
Descent through the Iitaka fibration then gives a big adjoint on a
base of dimension $b$. Its plurisections pull back to a differential
line of Iitaka dimension $b$ on a special root orbifold, a contradiction.
The auxiliary manifolds supply the ordinary compact groups required
by the reduction and positivity inputs. Specialness and the final
contradiction remain on the original root orbifold and its finite
coarse \'etale covers.

We use the following five results from the companion paper \cite{Supplied}.

\begin{assumption}[Companion Theorem 1.1]\label{in:input-manifold}
Every connected smooth compact K\"ahler manifold which is special in
the differential-line sense above has virtually abelian fundamental
group, in every complex dimension.
\end{assumption}

\begin{assumption}[Companion Theorem 2.1]\label{in:input-special}
For smooth compact K\"ahler manifolds, specialness is bimeromorphically
invariant and is preserved by connected finite \'etale covers.  The
Albanese morphism of a special manifold is surjective with connected
fibres.  If a holomorphic line bundle $L$ on a smooth compact K\"ahler
manifold $M$ admits a nonzero map $L\to\Omega_M^p$, then
$\kappa(M,L)\le p$.
\end{assumption}

\begin{assumption}[Companion Theorem 2.3]\label{in:input-reduction}
For a smooth compact K\"ahler manifold $M$ and a homomorphism
$\rho:\pi_1(M)\to P$, there is a meromorphic fibration
$\gamma_\rho:M\dashrightarrow B_\rho$ whose general fibre group has
finite image under $\rho$, and which contracts every compact
irreducible subvariety through a very general point whose fundamental
group has finite image under $\rho$.  These subvariety groups may be
computed on smooth resolutions.  The base has a compact K\"ahler
model.  Write
\[
 \operatorname{gdim}(M,\rho)=\dim B_\rho.
\]
\end{assumption}

Here and below, ``very general'' permits the exclusion of a countable
union of proper analytic subvarieties.  No linearity hypothesis is
imposed on $P$.

\begin{assumption}[Companion Theorem 2.5]\label{in:input-tori}
Let $Z$ be a smooth compact K\"ahler manifold, let $\Theta$ be an
effective rational divisor for which $(Z,\Theta)$ is Kawamata log
terminal (klt), and let
$g:Z\to A$ be a fibration onto a complex torus.  For a general smooth
fibre $F$,
\[
 \kappa(Z,K_Z+\Theta)\ge
 \kappa(F,K_F+\Theta|_F).
\]
If $\kappa(Z,K_Z+\Theta)=0$, the Albanese morphism of $Z$ is
surjective with connected fibres.
\end{assumption}

The torus in this statement may be a point.  In all our uses the spaces
are projective, and the boundaries have simple normal crossing support
and coefficients strictly less than one, which verifies the klt
hypothesis.  We use this formulation from the companion paper directly; see also the
results over tori in \cite{Wang}.

\begin{assumption}[Companion Theorem 4.1]\label{in:input-positive}
Let $Y$ be a connected smooth compact K\"ahler manifold of dimension
$d>0$, let
\[
 \Delta=\sum_i(1-m_i^{-1})D_i,
 \qquad m_i\in\mathbb Z,\quad m_i\ge2,
\]
have simple normal crossing support, and let $\mathcal Y$ be its root
orbifold.  Suppose that
\[
 q:\pi_1^{\mathrm{orb}}(\mathcal Y)\twoheadrightarrow Q
\]
is a surjection, that $Q$ is infinite, and that every finite-dimensional
complex linear representation of $Q$ has finite image.

Suppose 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.  Require compatibility
with $q$ on a dense analytic Zariski-open subset $S^\circ\subset S$
where $a$ is holomorphic with values in $Y\setminus\operatorname{Supp}
\Delta$: the restriction of $\nu$ is the composition of $a_*$ with
the natural map to $\pi_1^{\mathrm{orb}}(\mathcal Y)$ and then $q$.

Finally, for every finite-index subgroup $Q_0\le Q$, suppose there is a
connected finite \'etale cover $S'\to S$ such that, writing $\nu'$ for
the induced homomorphism and $Q'=\operatorname{im}\nu'$, one has
$Q'\subseteq Q_0$ and
\[
 \operatorname{gdim}(S',\psi\circ\nu')\ge d
 \quad\text{for every surjection }\psi:Q'\twoheadrightarrow H
 \text{ with }H\text{ infinite}.
\]
Then $K_Y+\Delta$ is big.
\end{assumption}

The descent below follows the strategy of Theorem~5.8 of the companion paper,
and the first dimension minimum follows the strategy of its Section~6.
Sections~\ref{in:minimality}--\ref{in:descent} establish the required
steps for the present orbifold source, including the compatible
dominating manifold and the full quantified quotient condition of
Assumption~\ref{in:input-positive}. Actual meridian orders determine
$\Delta$ on every model. The established orbifold specialness results
used separately are stated and checked in the next section.
==== END A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//inertia-inputs.tex ====

==== BEGIN A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//inertia-minimality.tex ====
\section{A minimum on open fibres and a compatible positivity source}
\label{in:minimality}

We work in the remaining case of Proposition~\ref{in:group-reduction}.
Thus $X$ is smooth connected projective, the support of $D$ is a
possibly empty disjoint union of smooth order-two divisors, and its
root orbifold is special.  There is a morphism
\begin{equation}\label{in:reduced-sequence}
 \alpha:X\longrightarrow A,
 \qquad
 1\longrightarrow K\longrightarrow G\longrightarrow
 \Lambda=\pi_1(A)\longrightarrow1,
\end{equation}
where $\alpha$ is surjective with connected fibres, the ordinary map
$\pi_1(X)\to\Lambda$ is an isomorphism, and $K$ is infinite, finitely
generated, and has no proper finite-index subgroup.  Every quotient of
$K$ has only trivial finite-dimensional complex linear images.  For a
general smooth fibre $P_a=\alpha^{-1}(a)$, the image of
$\pi_1(P_a\setminus D)$ in $G$ is exactly $K$.

\subsection{Descent of a reduction from an auxiliary manifold}

For clarity we first prove the descent statement used twice below.  If
$P$ is smooth projective, $P^\circ\subset P$ is a dense open, and
$\theta:\pi_1(P^\circ)\to E$ is a homomorphism, the image assigned
to a general fibre of a rational fibration of $P$ is computed on a
dense smooth open of a resolution which maps to $P^\circ$.  Removing
further proper algebraic subsets does not change that image, by
Lemma~\ref{in:topology}.  We call it the \emph{open fibre image}.

\begin{lemma}\label{in:descended-reduction}
Let $d:T\to P$ be a projective generically finite surjective map of
connected smooth projective varieties.  Suppose that the function-field
extension is Galois with finite group $J$, acting on $T$ over $P$.
Suppose $\theta:\pi_1(P^\circ)\to E$ induces a homomorphism
$\rho:\pi_1(T)\to E$, compatible on a dense open of $T$ mapping
to $P^\circ$.

There is a rational fibration $c:P\dashrightarrow Z$, with a smooth
projective base model, such that
\[
 \dim Z=\operatorname{gdim}(T,\rho)
\]
and the open group of its general fibre has finite image under
$\theta$.  The reduction of $(T,\rho)$ maps generically finitely to
$Z$, compatibly with $d$ and $c$.
\end{lemma}

\begin{proof}
Apply Assumption~\ref{in:input-reduction} to $(T,\rho)$ and write
$v:T\dashrightarrow Z_T$ for its reduction.  The base can be taken
projective.  Here is the usual justification in this setting.  Resolve
the graph of the meromorphic map from the projective source to a
compact K\"ahler model of the base.  Cutting the source by general
hyperplanes gives a projective subvariety of the base dimension mapping
generically finitely onto that base.  Under a proper generically finite
map the fields of meromorphic functions are a finite extension: after
Stein factorization, symmetric functions over the finite sheets give
the coefficients of the algebraic equations.  The base therefore has
full algebraic dimension.  It is Moishezon and admits a projective
birational model; resolving gives the asserted smooth projective model.

The finite-image test defining $v$ is invariant under $J$.  Indeed,
take a common $J$-stable dense open on which the compatibility with
$\theta$ holds.  The maps of this open to $P^\circ$ agree after deck
transformation.  With paths from a chosen point to its translates,
the induced homomorphisms differ by conjugation in $E$.  Images of
compact subvariety groups can be computed on dense opens of their
resolutions, because $\rho$ extends to $\pi_1(T)$.  Thus a deck
translate of a general finite-image fibre is again a finite-image
subvariety.  The contraction property makes it a fibre for the same
reduction, up to birational base equivalence.  Applying the argument
to inverse deck transformations gives invariance in both directions.

Consequently the relatively algebraically closed field
$F=\mathbb C(Z_T)\subset\mathbb C(T)$ is $J$-stable.  Its invariant
subfield
\[
 F^J\subset\mathbb C(T)^J=\mathbb C(P)
\]
is relatively algebraically closed in $\mathbb C(P)$.  To check this,
an element of $\mathbb C(P)$ algebraic over $F^J$ is algebraic over
$F$, hence belongs to $F$, and is $J$-invariant.  The field $F^J$
therefore defines a rational fibration $c:P\dashrightarrow Z$ with
connected general fibres.  The extension $F/F^J$ is finite, so the
base dimensions agree and $Z_T\dashrightarrow Z$ is generically
finite.

For general base points, a general fibre of $v$ maps dominantly and
generically finitely onto a general fibre of $c$: the source and base
maps in the commutative diagram are generically finite, and the two
fibre dimensions are equal.  On a smooth compact resolution the
upstairs fibre group has finite $\rho$-image.  The same is true on
any dense smooth open of that resolution, since its map to $T$ is
holomorphic and $\rho$ is defined on the compact group.  On suitable
opens the generically finite map has finite-index image on the
fundamental group of the downstairs fibre.  A group with a finite-index
subgroup of finite image itself has finite image.  Compatibility with
$\theta$ proves the assertion about the open fibre downstairs.
\end{proof}

For $P=X$, or $P=P_a$ with its smooth restricted boundary, the
auxiliary construction of Lemma~\ref{in:auxiliary} meets this lemma's
hypotheses for every quotient of the relevant open group image in
$G$.  Even pullback multiplicities, including on exceptional divisors,
are what extend the open group homomorphism to the compact manifold
$T$.

\subsection{Good fibres and a covering family}

Set $n=\dim X$ and $s=n-\dim A$.  We require a fixed notion of good
Albanese parameter which will work for all the fibrations subsequently
considered.

\begin{lemma}\label{in:cycle-families}
There is a countable collection of diagrams
\[
 h_i:V_i\longrightarrow T_i,
 \qquad j_i:V_i\longrightarrow X,
\]
such that $T_i$ is smooth connected quasi-projective, $h_i$ is smooth
proper with irreducible fibres, $j_i^{-1}D$ has relative simple normal
crossing support or is empty, and every compact irreducible subvariety
of $X$ not contained in $D$ is the image, birationally, of a fibre in
one of the families.

Outside a countable union of proper closed subsets of $A$, the fibre
$P_a$ is smooth, its boundary is smooth and disjoint, and no closure
$\overline{j_i(V_i)}\ne X$ contains $P_a$.
\end{lemma}

\begin{proof}
Use the countably many projective Hilbert schemes of $X$ and their
universal families.  The loci parametrizing reduced irreducible
subvarieties not contained in $D$ are constructible.  On a dense smooth
open of each irreducible parameter piece, resolve the reduced incidence
component dominating that piece, together with the inverse image of
$D$.  Characteristic-zero resolution and generic smoothness, applied
also to the finitely many boundary strata, give a smooth proper family
with relative normal crossings after further shrinking.  Its fibres
map birationally to the members: this holds on the generic fibre, and
the locus where the resolution is an isomorphism meets the members in
dense opens after shrinking.  The resolved fibres are geometrically
integral on that open.  Apply the same procedure to the complementary
lower-dimensional parameter pieces.  Noetherian induction, for each
Hilbert scheme, gives the required countable collection and loses no
member.  These are the usual projective parameter and resolution
constructions; see \cite{Hironaka,Stacks}.

Shrink $A$ so that $P_a$ and its boundary have the asserted smoothness
properties.  For any proper closed $Z_i=\overline{j_i(V_i)}$ dominating
$A$, the general fibre of $Z_i\to A$ has dimension at most $s-1$.
If $Z_i$ does not dominate $A$, its image is proper since $X\to A$
is proper.  In either case, outside a proper closed subset of $A$,
$Z_i$ cannot contain the irreducible smooth fibre $P_a$.  Take all
these countably many exclusions.  When $A$ is a point, a proper
$Z_i$ simply does not contain $X$.
\end{proof}

Call parameters satisfying Lemma~\ref{in:cycle-families} and the
open-fibre image conclusion $\operatorname{im}\pi_1(P_a\setminus D)=K$
\emph{good}.  They exist.  Moreover a good $P_a$ has points outside
all the nondominating evaluation closures: a countable union of
proper complex algebraic subsets does not cover a connected smooth
complex projective variety.

For every rational fibration $b:P_a\dashrightarrow V$ on a good
fibre, let $C_b\le K$ be its general open fibre image, computed in
$X\setminus D$.  Lemma~\ref{in:open-images} makes $C_b$ normal in
$K$.  Define
\begin{equation}\label{in:dimension-minimum}
 \ell=\min\{\dim V: a\text{ good},\quad
 b:P_a\dashrightarrow V\text{ a fibration},\quad K/C_b\text{ infinite}\}.
\end{equation}
The identity fibration is admissible, while a map to a point has
$C_b=K$.  Since $K$ is infinite and is an open fibre image, $s>0$,
and
\begin{equation}\label{in:ell-range}
 1\le\ell\le s.
\end{equation}
The minimum is over integers and is attained.  After any torus
isogeny pullback, a fibre over a lift of a good parameter is identified
with the old fibre, including its complement and its group images in
the same $K$.  Thus \eqref{in:dimension-minimum} remains a lower bound
for every such fibre and every further isogeny.  No countability of
the set of fibrations is needed for this definition.

Choose an attaining $b:P_a\dashrightarrow V$ and put $C=C_b$.
A general fibre image is a compact irreducible subvariety of dimension
$s-\ell$.  Choose it through a point outside all nondominating
evaluation closures, outside $D$, and in the good locus of the
resolved fibration.  It occurs in a family of
Lemma~\ref{in:cycle-families} whose evaluation is dominant.  Its open
group image in that family is $C$, since the two smooth birational
models have a common dense open.

The image of $\pi_1(V_i\setminus j_i^{-1}D)$ in $G$ normalizes $C$.
Indeed $h_i$, with its relative normal crossing boundary removed, is
a differentiably locally trivial family of connected opens; transport
identifies its fibre group images up to conjugation.  That total-space
image has finite index in $G$, by dominance of the evaluation and
Lemma~\ref{in:topology}.  The normalizer $G_1=N_G(C)$ therefore has
finite index and contains $K$.  It is the inverse image of a sublattice
of $\Lambda$.  Pass to the corresponding torus isogeny and its
connected coarse \'etale pullback of $X$.

The entire incidence family lifts to this pullback.  The open lifting
criterion holds by the preceding normalizer condition; composing to
$\pi_1(X)=\Lambda$ and using surjectivity from the incidence open to
$\pi_1(V_i)$ proves the lifting criterion for $V_i$ itself.  A
one-sheeted component of the finite \'etale pullback gives an algebraic
lift.  The lifted evaluation is dominant and its fibres still map
birationally to their images.  Transport now takes place in $G_1$,
where $C$ is normal, so all their open group images are $C$ with
compatible choices.  Relabel the resulting data as $X,A,G$.
We have obtained
\begin{equation}\label{in:normal-C}
 C\triangleleft G,\qquad C\subset K,\qquad K/C\text{ infinite},
\end{equation}
and a covering family of compact subvarieties of dimension $s-\ell$
whose open group images lie in $C$.

\subsection{The intermediate fibration and its quotients}

\begin{proposition}\label{in:intermediate}
After the preceding isogeny there is a factorization by fibrations
\[
 X\overset{c}{\dashrightarrow}Z\overset{g}{\longrightarrow}A,
 \qquad \alpha=g\circ c,
\]
with $Z$ smooth projective, such that the general open $c$-fibre image
$B_0$ is normal in $G$, is contained in $K$, and satisfies
\[
 \dim Z-\dim A=\ell>0,
 \qquad Q=K/B_0\text{ infinite}.
\]
Writing $R=G/B_0$, one has an exact sequence
\begin{equation}\label{in:RQ-sequence}
 1\longrightarrow Q\longrightarrow R\longrightarrow\Lambda
 \longrightarrow1.
\end{equation}
Every quotient of $Q$ has no proper finite-index subgroup and has only
trivial finite-dimensional complex linear images.
\end{proposition}

\begin{proof}
Choose an auxiliary $d:T\to X$ from Lemma~\ref{in:auxiliary}, and
apply Lemma~\ref{in:descended-reduction} to the quotient $G/C$.
Let $v:T\dashrightarrow Z_T$ be the manifold reduction and
$c:X\dashrightarrow Z$ its descended fibration.

The lifted covering family forces
$\dim Z_T\le n-(s-\ell)=\dim A+\ell$.  To verify this directly,
work over a dense open where $d$ is finite \'etale and the evaluation
family covers.  A member through the projection of a point of $T$
has a component of its inverse image through that point, of the same
dimension $s-\ell$, dominating the member.  On a resolution of this
compact component, its group image under $\pi_1(T)\to G/C$ is
trivial: compute on a common dense open mapping to the resolved member,
whose open image lies in $C$.  Surjectivity from that open to the
compact resolution group is applicable because the homomorphism
extends to $\pi_1(T)$.  These compact subvarieties pass through points
in a dense open.  At very general such points the contraction property
of Assumption~\ref{in:input-reduction} applies.  Its general fibres
therefore have dimension at least $s-\ell$.

The map $T\to A$ is constant on a general resolved fibre of $v$.
That fibre has finite group image in $G/C$ and thus finite image in
the torsion-free lattice $\Lambda$.  Its map to $A$ lifts to the
vector-space universal cover and is constant by compactness.  It
follows that $T\to A$ factors rationally through $Z_T$.  The factor
is invariant under the deck group, so it descends to $Z$.  Resolve
to make $g:Z\to A$ holomorphic.  Its general fibres are connected:
$\mathbb C(A)$ is relatively algebraically closed in $\mathbb C(X)$
by connectedness of the general $\alpha$-fibre, and hence also in
$\mathbb C(Z)$.

Let $B_0$ be the general open $c$-fibre image.  It is normal in $G$
by Lemma~\ref{in:open-images}, and lies in $K$ because $c$ is over
$A$.  Lemma~\ref{in:descended-reduction} says that its image in
$G/C$ is finite.  If $K/B_0$ were finite, the image of $K$ in $G/C$
would consequently be finite, contradicting \eqref{in:normal-C}.
Thus $Q$ is infinite.

For a very general good $a$, restrict the diagram to $P_a$.  The
result is a rational fibration onto the corresponding smooth connected
$g$-fibre, with general open fibre image $B_0$ in $K$.  This follows
on the common resolved fibration locus: the fibres are the same
general $c$-fibres, and deleting additional algebraic subsets preserves
their images.  Normality of $B_0$ in $G$ removes the ambiguity from
transporting paths.  The quotient $K/B_0$ is infinite, so the minimum
\eqref{in:dimension-minimum} gives $\dim Z-\dim A\ge\ell$.
The opposite inequality was already proved.  The exact sequence and
the remaining group properties follow from $B_0\subset K$ and from
the corresponding properties of $K$.
\end{proof}

We record carefully the effect of further isogenies.  If $A'\to A$
is one, the pullbacks of $X$ and a smooth model of $Z$ are connected.
For $X$ this follows from $\pi_1(X)\simeq\Lambda$; for $Z$ it
follows because a proper fibration with connected general fibres
induces a surjection on $\pi_1(A)$, as seen first over the smooth
fibration open.  The pulled-back $c$-fibres identify with the old
ones over chosen lifts of base points.  Thus their open group image,
under the inclusion $G'\le G$, is the same $B_0$, and
\begin{equation}\label{in:isogeny-groups}
 R'=G'/B_0\hookrightarrow R,
 \qquad Q=K/B_0\text{ is unchanged}.
\end{equation}
The relative dimension, the good-fibre minimum, and the source
specialness are unchanged as well.  We shall use these facts for
arbitrary further torus isogenies.

\subsection{Actual inertia and fibrewise bigness}

The open-fibration construction gives a compatible surjection
\[
 \psi:\pi_1(Z^\circ)\twoheadrightarrow R.
\]
On a smooth projective model over $A$, set $m_L$ equal to the order
of the image of a divisorial meridian about a prime $L$, and set it
equal to one for primes meeting $Z^\circ$.  Lemma~\ref{in:actual-inertia}
shows that these orders are finite.  Put
\begin{equation}\label{in:true-boundary}
 \Delta_Z=\sum_L(1-m_L^{-1})L.
\end{equation}
Monodromy is transported between models on common opens.  Strict
transforms retain their orders.  Under an isogeny pullback the
meridians lift with winding number one and
\eqref{in:isogeny-groups} preserves their orders.

Call a model \emph{prepared} if it is smooth projective over $A$ and
the divisorial part of a deleted locus defining this monodromy has
simple normal crossings. A prepared model can be chosen above every
smooth projective model in use: take common resolutions, resolve the
complement of a common monodromy open on the base, and resolve the
source diagram and its deleted sets.  Generic smoothness for all
boundary strata provides the required smooth family of complements
over a smaller base open.  The construction works after every torus
isogeny.  The coefficients on newly appearing primes are always read
from their actual meridians.

\begin{proposition}\label{in:fibre-big}
On every prepared model just described, including those after any
further torus isogeny, a very general smooth fibre $Y$ of $g:Z\to A$
satisfies
\[
 \dim Y=\ell,
 \qquad K_Y+\Delta_Z|_Y\text{ is big}.
\]
\end{proposition}

\begin{proof}
Choose a very general parameter $a$ lying above a good parameter of
the original torus, and put $Y=g^{-1}(a)$. Smoothness, connectedness, and dimension
$\ell$ follow from Proposition~\ref{in:intermediate}.  Generic
smoothness for the finitely many deleted strata ensures that
$\Delta_Y=\Delta_Z|_Y$ has simple normal crossing support and finite
integral root coefficients.  Higher-codimension deletions remain of
higher codimension on a very general fibre or miss it.

The restriction of $\psi$ to the open part $Y^\circ$ has image in
$Q$, since its composite with the lattice quotient is induced by the
constant map $Y\to A$.  It is onto $Q$: a dense open of $P_a$ maps
compatibly to $Y^\circ$, and its group image in $G$ is $K$.
The meridians of deleted divisors on $Y$ are transverse restrictions
of the corresponding meridians on $Z$, with the same images up to
transport.  Their orders in $Q$ equal their orders in $R$ because
$Q\le R$.  Normal meridian generation therefore gives a surjection
\begin{equation}\label{in:q-positive}
 q:\pi_1^{\mathrm{orb}}(Y,\Delta_Y)\twoheadrightarrow Q.
\end{equation}

Apply Lemma~\ref{in:auxiliary} to the smooth projective fibre $P_a$
with its smooth disjoint or empty boundary.  It gives a connected
smooth projective Galois auxiliary map $d:S\to P_a$ and a compatible
homomorphism $\pi_1(S)\to K$.  This homomorphism has finite-index
image: on dense opens the generically finite map supplies a
finite-index subgroup of the open group of $P_a$, whose image is
$K$.  Since $K$ has no proper finite-index subgroup, its image is
all of $K$.  Composition with $K\twoheadrightarrow Q$ gives
$\nu:\pi_1(S)\twoheadrightarrow Q$.  The dominant meromorphic map
$S\dashrightarrow Y$ and $\nu$ are compatible with
\eqref{in:q-positive} on a dense open by the construction of
monodromy.

It remains to verify the last, quantified hypothesis of
Assumption~\ref{in:input-positive}.  There is only one finite-index
subgroup of $Q$, namely $Q$ itself, so use the identity cover of this
fixed $S$.  Let $\psi_0:Q\twoheadrightarrow H$ be any infinite
quotient.  Apply Lemma~\ref{in:descended-reduction} on $P_a$ to
$S\to P_a$ and the compatible composition to $H$.  It gives a
rational fibration of $P_a$ whose base dimension is
$\operatorname{gdim}(S,\psi_0\nu)$ and whose general open fibre
image $C'\triangleleft K$ has finite image in $H$.  The quotient
$K/C'$ is infinite, since $K$ surjects onto the infinite group $H$
and the image of $C'$ there is finite.  This is one of the fibrations
tested by \eqref{in:dimension-minimum}.  Hence
\[
 \operatorname{gdim}(S,\psi_0\nu)\ge\ell.
\]
This argument holds for every infinite quotient of $Q$ on the
\emph{same} $S$.

We have now checked all the hypotheses of
Assumption~\ref{in:input-positive}: $Y$ and $S$ are connected smooth
projective manifolds, the source dominates compatibly and surjects on
the required group, the boundary is finite integral SNC, $Q$ is
infinite with finite linear images, and the full quotient condition
holds.  It proves the desired bigness.  The proof applies separately
to every prepared model and every further isogeny, using
\eqref{in:isogeny-groups} and the unchanged lower bound
\eqref{in:dimension-minimum}.  No single parameter is required to be
very general simultaneously for all possible models.
\end{proof}
==== END A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//inertia-minimality.tex ====

==== BEGIN A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//inertia.tex ====
\input{preamble.tex}

\title[Inertia and quotient minimality]{Conditional orbifold abelianity by inertia and quotient minimality}
\author{OpenAI}
\date{October 5, 2026}
\hypersetup{pdftitle={Conditional orbifold abelianity by inertia and quotient minimality},pdfauthor={OpenAI}}

\begin{document}

\begin{abstract}
Using five stated results from the companion paper on special compact
K\"ahler manifolds, we prove virtual abelianity of the entire orbifold
fundamental group of the special order-two root orbifold associated with
a pair $(X,\tfrac12D)$, where $X$ is a smooth projective fourfold and $D$
is a nonempty smooth connected divisor. Compatible smooth projective
sources and a dimension minimum give fibrewise adjoint bigness.
Tracking actual meridian orders through the Iitaka fibration then
produces a differential line excluded by specialness. The argument
includes a nontrivial meridian invisible in every finite quotient.
\end{abstract}

\maketitle
\tableofcontents

\input{sections/inertia-inputs.tex}
\input{sections/inertia-foundations.tex}
\input{sections/inertia-minimality.tex}
\input{sections/inertia-descent.tex}

\bibliographystyle{amsplain}
\bibliography{references}

\end{document}
==== END A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//inertia.tex ====

==== BEGIN A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//introduction.tex ====
\section{Statement and context}

Campana's theory of special varieties, developed in \cite{Campana2004}, describes the part of birational geometry with no positive-dimensional general-type orbifold base. The orbifold formulation records the multiplicities of fibres and divisors. The orbifold Abelianity Conjecture predicts virtual abelianity for the fundamental group of a smooth integral special geometric orbifold whose coarse space is bimeromorphic to a compact K\"ahler manifold; see \cite[Conjecture~12.10(1)]{CampanaOrbifolds} and its projective formulation in \cite[Conjecture~11.2]{CampanaSurvey}. Here, as there, virtually abelian means containing an abelian subgroup of finite index.

We give a conditional resolution of the case consisting of a smooth projective fourfold and one smooth divisor of multiplicity two. The argument uses the manifold abelianity theorem in every dimension and constructs a special manifold whose fundamental group surjects onto the orbifold group.

We first fix the differential definition of specialness used throughout. Let $X$ be a connected smooth projective complex variety and $D$ a smooth reduced divisor. Its order-two root orbifold is denoted by
\[
 \pi:\cX=\sqrt[2]{(X,D)}\longrightarrow X.
\]
Near $D=\{z_1=0\}$ it is presented by
\begin{equation}\label{eq:root-chart}
 z_1=w_1^2,\qquad z_i=w_i\ (i>1),
 \qquad w_1\longmapsto-w_1.
\end{equation}
Bundles, forms and their maps on $\cX$ mean equivariant holomorphic objects in these smooth charts, with the compatible identifications on overlaps. Write $\cD$ for the reduced root divisor. Its divisor line has the canonical relation
\begin{equation}\label{eq:root-line}
 T:=\OO_{\cX}(\cD),\qquad T^{\otimes2}=\pi^*\OO_X(D).
\end{equation}
The stabilizer acts by sign on $T$ along $\cD$. This line exists on $\cX$ even when $\OO_X(D)$ has no square root on $X$.

For an ordinary or orbifold line bundle $L$, its Iitaka dimension $\kappa(L)$ is $-\infty$ if every positive power has no nonzero section, and otherwise is the largest dimension of the images of its complete positive-power section maps. On an orbifold these are the global equivariant sections.

\begin{definition}\label{def:special}
A smooth compact complex manifold or a smooth root orbifold $V$ is \emph{special} if there is no integer $1\le p\le\dim V$ and no holomorphic line bundle $L$ admitting a nonzero sheaf map
\[
 L\longrightarrow\Omega_V^p,\qquad \kappa(L)=p.
\]
\end{definition}

This is the finite-integral root form of the Bogomolov-sheaf criterion in \cite[Theorem~8.9]{CampanaOrbifolds}. We retain Definition~\ref{def:special} throughout the proof. Remark~\ref{rem:conventions} gives the local comparison with orbifold symmetric differentials.

\begin{assumption}[Manifold abelianity]\label{ass:manifold}
Every connected smooth compact K\"ahler manifold which is special in the sense of Definition~\ref{def:special} has virtually abelian entire discrete topological fundamental group, in every complex dimension.
\end{assumption}

This is Theorem~1.1 of the companion paper \cite{Supplied}. No orbifold extension of that theorem is assumed. The following theorem, including the appendix proof, uses no other result of that paper.

\begin{theorem}\label{thm:main}
Assume Assumption~\ref{ass:manifold}. Let $X$ be a connected smooth projective complex fourfold and let $D\subset X$ be a nonempty smooth connected reduced divisor. If $\sqrt[2]{(X,D)}$ is special, then
\begin{equation}\label{eq:target-group}
 G=\pi_1(X\setminus D,x)\big/\langle\!\langle\gamma^2\rangle\!\rangle
\end{equation}
contains an abelian subgroup of finite index. Here $\gamma$ is a positively oriented meridian about $D$, transported to $x\in X\setminus D$.
\end{theorem}

The assertion concerns the entire discrete group in \eqref{eq:target-group}, not just its abelianization, finite quotients or linear images. No uniform index is claimed, and torsion is allowed. We impose no residual-finiteness or linearity assumption on $G$, no positivity condition on $K_X+\tfrac12D$, and no finite manifold-cover hypothesis.

The main construction is independent of specialness. It produces a smooth projective eightfold $Y$ over $\cX$ with general fibre a product of four genus-one curves. It also supplies a submersive lift at some point over the general point of \emph{every} base divisor. These points detect all poles in the descent of differential tensors. A direct argument then proves that specialness of $\cX$ forces specialness of $Y$, while even pullback multiplicities give an epimorphism $\pi_1(Y)\twoheadrightarrow G$. These two conclusions permit the application of Assumption~\ref{ass:manifold}.

Proposition~\ref{prop:transfer} isolates this passage in arbitrary base dimension for an order-two smooth boundary. Its hypotheses concern general fibres and divisorial submersions, so it does not require a globally smooth family. In the fourfold construction, four independent genus-one factors eliminate stabilizers over the three-dimensional divisor. Appendix~\ref{app:branched} gives an alternative construction using auxiliary branched covers, affine elliptic actions and a quotient resolution. Both constructions avoid a finite uniformization requirement.

The alternative proof \emph{Conditional orbifold abelianity by inertia and quotient minimality} retains the longer descent method and states its additional inputs from the companion paper. It is not needed for the proof of Theorem~\ref{thm:main} below.

\paragraph{Conventions.}
All varieties are over $\CC$. Projective spaces of vector bundles parametrize lines. A general fibre is taken over a sufficiently small nonempty Zariski open. Fundamental-group maps use compatible base points and paths; subgroup images are understood up to conjugacy. We use the standard complex-projective comparison between algebraic and analytic bundles, sections, and finite coherent algebras \cite{GAGA}.
==== END A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//introduction.tex ====

==== BEGIN A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//main.tex ====
\input{preamble.tex}
\title[A conditional orbifold abelianity theorem]{A conditional abelianity theorem for special fourfold pairs with a half-weight divisor}
\author{OpenAI}
\date{October 5, 2026}
\hypersetup{pdftitle={A conditional abelianity theorem for special fourfold pairs with a half-weight divisor},pdfauthor={OpenAI}}
\subjclass[2020]{14E20, 14F35, 32Q15, 32Q55}
\keywords{Special orbifold, fundamental group, abelianity conjecture, genus-one fibration}
\begin{document}
\begin{abstract}
Using the abelianity theorem for special compact K\"ahler manifolds in every dimension, we prove virtual abelianity of the entire orbifold fundamental group of the special order-two root orbifold associated with a pair $(X,\tfrac12D)$, where $X$ is a smooth projective fourfold and $D$ is a nonempty smooth connected divisor. We construct a special smooth projective eightfold whose fundamental group surjects onto the required group.
\end{abstract}
\maketitle
\input{sections/introduction.tex}
\input{sections/differentials.tex}
\input{sections/transfer.tex}
\input{sections/quadrics.tex}
\appendix
\input{sections/branched.tex}
\clearpage
\bibliographystyle{alpha}
\bibliography{references}
\end{document}
==== END A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//main.tex ====

==== BEGIN A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//preamble.tex ====
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\setlist[itemize]{leftmargin=*}
\allowdisplaybreaks[2]
==== END A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//preamble.tex ====

==== BEGIN A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//quadrics.tex ====
\section{Four genus-one factors over the root orbifold}\label{sec:quadrics}

We construct the manifold in Proposition~\ref{prop:transfer}. No specialness assumption is used in the construction itself.

\begin{proposition}\label{prop:eightfold}
Let $X$ be a connected smooth projective fourfold and $D$ a nonempty smooth connected reduced divisor. There is a connected smooth projective eightfold $Y$ with a map to $\sqrt[2]{(X,D)}$ whose coarse map $g:Y\to X$ satisfies all the hypotheses of Proposition~\ref{prop:transfer}. Its smooth general fibre is a product of four genus-one curves.
\end{proposition}

The parameter choice has two roles. Four independent factors eliminate stabilizers over the three-dimensional divisor, while a separate rank estimate supplies a submersive point over the general point of every base divisor. Both are needed in Proposition~\ref{prop:transfer}; generic smoothness alone would not justify the descent of all differential tensors.

Choose a line bundle $M$ such that $M$ and $M(-D)$ are globally generated. This is possible by taking a sufficiently high power of an ample line bundle. On $\cX$, with $T$ as in \eqref{eq:root-line}, put
\[
 V=\OO_{\cX}^{\oplus2}\oplus T^{\oplus2},\qquad
 \mathcal P=\underbrace{\PP_{\cX}(V)\times_{\cX}\cdots\times_{\cX}\PP_{\cX}(V)}_{4\text{ factors}}.
\]
All four factors are over the \emph{same} root orbifold. In a root chart there is one diagonal sign action on their product.

In factor $j$, use vector coordinates $z_{j1},\ldots,z_{j4}$ and choose two diagonal quadratic equations
\begin{equation}\label{eq:quadrics}
 q_{j\nu}=\sum_{i=1}^4a_{j\nu i}z_{ji}^2=0,
 \qquad j=1,\ldots,4,\quad \nu=1,2,
\end{equation}
where, independently for all $j,\nu,i$,
\[
 a_{j\nu i}\in
 \begin{cases}
 H^0(X,M),&i=1,2,\\
 H^0(X,M(-D)),&i=3,4.
 \end{cases}
\]
The squares of the last two coordinates take values in $T^2=\pi^*\OO_X(D)$, so each equation is a section of the appropriate $\OO(2)\otimes\pi^*M$. Let $\mathsf A$ denote the affine space of all these coefficients, and let $\mathcal S_a\subset\mathcal P$ be the simultaneous zero locus for $a\in\mathsf A$.

In compatible frames at $x\in X$, write the two rows in factor $j$ as a $2\times4$ matrix $C_j(x)$. The ranks of its specified column submatrices are independent of the choice of frames. Evaluation onto the four matrices is surjective at every $x$, by global generation and independence of the coefficient choices. This includes $x\in D$: a coefficient in $M(-D)$ is evaluated in the fibre of that line, and is paired with a square in $\OO_X(D)$. Its numerical value is not required to vanish on $D$.

\begin{lemma}[Simultaneous parameter choice]\label{lem:parameters}
There is a choice $a\in\mathsf A$ with the following properties.
\begin{enumerate}
\item\label{item:total-smooth} The zero locus $\mathcal S_a$ is smooth in its root charts.
\item\label{item:free} It has no nontrivial stabilizers.
\item\label{item:triples} There is a closed subset $Z\subset X$ of codimension at least two such that every matrix obtained from any $C_j(x)$ by deleting one column has rank two for $x\notin Z$.
\item\label{item:pairs} On a nonempty open $B\subset X\setminus D$, every $2\times2$ minor of every $C_j(x)$ is nonzero.
\end{enumerate}
\end{lemma}

\begin{proof}
Take a finite algebraic trivializing cover of $X$ for the bundles involved, with smooth root charts
\[
 W=\{(x,w):w^2=t(x)\}\longrightarrow U,
\]
where $t$ is the local divisor equation. These charts are smooth since $D$ is smooth and reduced; away from $D$ they are \emph{\'{e}tale}. Their products with the four projective spaces are smooth. At any point of such a product, each equation in \eqref{eq:quadrics} has nonzero evaluation as a linear functional on its own parameter block: at least one coordinate in that projective factor is nonzero. The eight evaluations are independent. Thus the universal incidence over the chart product is the kernel of a surjective vector-bundle map from the parameter bundle to the eight equation lines, and is smooth.

Generic smoothness in characteristic zero makes its fibres over a dense parameter open smooth. Indeed, each component of the critical locus has image of dimension smaller than $\dim\mathsf A$, since a dominating component would have generically surjective differential on its smooth locus, contradicting criticality. Images of nondominating components may likewise be discarded. There are finitely many algebraic charts, so one common dense open of parameters proves~\ref{item:total-smooth}.

Only points over $D$ can have nontrivial stabilizer. There the involution on a factor sends
\[
 [z_1:z_2:z_3:z_4]\longmapsto[z_1:z_2:-z_3:-z_4].
\]
Its fixed locus is the union of the two eigenspace lines. If a fixed point satisfies both equations, either the first pair of columns or the last pair of columns of $C_j(x)$ is dependent. Each of these determinant conditions is a proper hypersurface in matrix space. A fixed point in the fourfold product therefore requires one such condition in \emph{each} of the four independent matrices. At fixed $x\in D$ its parameter codimension is four. The bad incidence over $D$ has dimension at most
\begin{equation}\label{eq:fixed-dimension}
 \dim\mathsf A+\dim D-4=\dim\mathsf A-1.
\end{equation}
General parameters avoid its image. This excludes stabilizers above every point of $D$, including degenerate fibres, and proves~\ref{item:free}.

For one specified three-column submatrix, rank at most one has codimension two in the space of $2\times3$ matrices. Surjective evaluation implies that its incidence over $X$ has dimension at most $\dim\mathsf A+2$. For a general parameter its bad base locus has dimension at most two, by the fibre-dimension theorem. The finitely many choices of factor and deleted column can be treated simultaneously. Their union is closed and gives~\ref{item:triples}.

Finally choose a point $x_0\in X\setminus D$. By evaluation surjectivity, all pair minors can be made nonzero at $x_0$; for example the columns $(1,\lambda_i)^{\mathsf t}$ with distinct $\lambda_i$ have this property. This is a nonempty Zariski-open parameter condition and can be intersected with the preceding dense opens. Nonvanishing then holds on an open neighbourhood $B$ of $x_0$, proving~\ref{item:pairs}.
\end{proof}

Fix such a parameter and write $\mathcal S=\mathcal S_a$. Freeness and smoothness mean that its chart quotients glue to an ordinary smooth complex space $S$ with a map $S\to\cX$. In particular the quotient of each source chart is covered \emph{unramifiedly} by that chart; its map to the base root chart gives the local lifts on $S$.

\begin{lemma}[Projectivity]\label{lem:projectivity}
The space $S$ is projective.
\end{lemma}

\begin{proof}
Square the four coordinates in each factor. The identification $T^2=\pi^*\OO_X(D)$ gives a globally defined invariant map to the ordinary projective variety
\begin{equation}\label{eq:square-target}
 Q=\underbrace{\PP_X(E_0)\times_X\cdots\times_X\PP_X(E_0)}_{4\text{ factors}},
 \qquad E_0=\OO_X^{\oplus2}\oplus\OO_X(D)^{\oplus2}.
\end{equation}
Its map on a root chart is the finite root map $W\to U$ together with the finite projective coordinate-square maps. Restriction to the closed zero locus remains finite. Passing to the finite-group quotient preserves finiteness: over an affine target this takes an invariant subalgebra of a finite algebra, which is again finite. These invariant algebras glue under the bundle transitions, since the coordinate-square construction is invariant and global. Hence $S\to Q$ is a finite analytic map.

For completeness, a finite analytic space over a complex projective variety algebraizes by projective GAGA \cite{GAGA}. Its finite coherent direct-image algebra, including multiplication and unit, algebraizes, and relative spectrum recovers the space after analytification. A finite algebraic morphism is projective \cite[Tag~01WG]{Stacks}. Thus $S$ is projective and in particular compact.
\end{proof}

\begin{lemma}[Fibres and submersions over base divisors]\label{lem:fibres-witnesses}
The map $S\to X$ is smooth over $B$ with connected fibres that are products of four genus-one curves. Above every root-chart point with image in $X\setminus Z$, it has a point at which the lifted map to that chart is a submersion.
\end{lemma}

\begin{proof}
At $x\in B$, consider one matrix $C=C_j(x)$. It has rank two. Coordinate squaring on $\PP^3$ is finite and surjective; the common quadric zero set is its inverse image of $\PP(\ker C)\simeq\PP^1$. Thus it is nonempty and has dimension one. At a solution, put $b_i=z_i^2$. A nonzero vector $b\in\ker C$ cannot have support one or two, since every pair of columns is independent. The vertical derivative
\begin{equation}\label{eq:vertical-jacobian}
 2C\operatorname{diag}(z_1,z_2,z_3,z_4)
\end{equation}
therefore has rank two. It annihilates the radial direction at a solution, so it still has rank two on the projective tangent space. This proves both smoothness of the curve and submersivity of the family over $B$.

The curve is a smooth complete intersection of two quadrics in $\PP^3$. Its Koszul resolution is
\[
 0\longrightarrow\OO_{\PP^3}(-4)
 \longrightarrow\OO_{\PP^3}(-2)^{\oplus2}
 \longrightarrow\OO_{\PP^3}
 \longrightarrow\OO_C\longrightarrow0.
\]
The intermediate cohomology vanishings for line bundles on $\PP^3$ give $H^0(C,\OO_C)=\CC$, hence connectedness \cite[Tag~01XS]{Stacks}. Adjunction gives $K_C=\OO_C$. Thus $C$ is a genus-one curve and its cotangent line is trivial. Taking four factors proves the first assertion and triviality of the cotangent bundle of each product fibre.

Now let $x\notin Z$, possibly on $D$. In each matrix $C_j(x)$, the three columns remaining after any deletion span $\CC^2$. Its kernel is consequently contained in no coordinate hyperplane: one can prescribe a chosen coordinate to be one and solve for the other three. A vector space over $\CC$ is not the union of finitely many proper linear subspaces, so its kernel contains a vector $b$ with all four coordinates nonzero. Choose square roots $z_i$ of its entries. Formula~\eqref{eq:vertical-jacobian} then has rank two. Choosing such a point in each factor makes the full vertical derivative of the eight equations surjective. The implicit function theorem gives a submersion from their zero set to the base root chart at this tuple. The argument works for either chosen chart point over $x$ and in particular for $w=0$.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:eightfold}]
The smooth projective $S$ has disjoint connected components. Its fibres over $B$ are connected and nonempty, so there is exactly one component $Y$ dominating $X$. More explicitly, the proper image of a dominating component is all of $X$; two such components would disconnect the fibres over $B$. Every submersive point constructed in Lemma~\ref{lem:fibres-witnesses} lies on a dominating component, since its local image has full dimension. Therefore all these points lie on $Y$.

The variety $Y$ is connected smooth projective, of dimension $4+4=8$. Its map $g:Y\to X$ is surjective and has the asserted fibres over $B$. Its map to $\cX$ gives
\[
 g^*D=2f^*\cD.
\]
The pullback of the root-divisor section is generically nonzero, so $f^*\cD$ is an effective integral Cartier divisor. Thus every divisorial multiplicity over $D$ is even. Since $\codim_X Z\ge2$, every base prime divisor has general point outside $Z$. Lemma~\ref{lem:fibres-witnesses} supplies the ordinary or root-chart submersions there. All four conditions of Proposition~\ref{prop:transfer} follow.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Apply Proposition~\ref{prop:eightfold} to $(X,D)$. Specialness of $\cX$ and Proposition~\ref{prop:transfer} make the resulting eightfold $Y$ special and give $\pi_1(Y)\twoheadrightarrow G$. Assumption~\ref{ass:manifold}, in dimension eight, makes $\pi_1(Y)$ virtually abelian. Its quotient $G$ is virtually abelian as claimed.
\end{proof}
==== END A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//quadrics.tex ====

==== BEGIN A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//transfer.tex ====
\section{A manifold criterion for specialness and fundamental groups}

Here $X$ may have any positive dimension $n$, and $D$ is still a nonempty smooth connected reduced divisor. Write $\cX=\sqrt[2]{(X,D)}$.

If a map $g:Y\to X$ from a smooth manifold satisfies $g^*D=2E$, then the divisor line $\OO_Y(E)$ and its canonical section, with their squared identification with $g^*(\OO_X(D),s_D)$, define a map
\[
 f:Y\longrightarrow\cX,\qquad \pi f=g.
\]
Equivalently, a local defining equation of $D$ pulls back to a square after choosing a local square root of a unit. Choices differ by sign and give precisely the root-chart transition rule. Orbifold bundles and differential maps therefore pull back to ordinary ones on $Y$.

\begin{proposition}[Transfer criterion]\label{prop:transfer}
Let $g:Y\to X$ be a surjective morphism with $Y$ connected smooth projective. Assume:
\begin{enumerate}
\item\label{item:even} $g^*D=2E$ for an effective integral divisor $E$ on $Y$;
\item\label{item:fibres} over a nonempty Zariski open $B\subset X\setminus D$, the map $g$ is smooth with connected fibres $F$, and $\Omega_F^1$ is trivial;
\item\label{item:ordinary} above a general point of each prime divisor $P\ne D$ of $X$, there is a point at which $g$ is a submersion;
\item\label{item:root} above a general point of $D$, there is a point at which a local lift to the root chart is a submersion.
\end{enumerate}
If $\cX$ is special, then $Y$ is special. There is also a surjection
\begin{equation}\label{eq:group-surjection}
 \pi_1(Y)\twoheadrightarrow
 \pi_1(X\setminus D)/\langle\!\langle\gamma^2\rangle\!\rangle.
\end{equation}
Consequently Assumption~\ref{ass:manifold} implies virtual abelianity of the group on the right.
\end{proposition}

\begin{proof}[Specialness]
The map to the root orbifold described above will be denoted by $f$. Suppose that $i:L\to\Omega_Y^p$ contradicts specialness of $Y$. We first descend the plurisection ratios, then the associated differential tensors, and finally test their poles over base divisors. Choose sections $s_0,\ldots,s_p$ of $L^{\otimes m}$ as in Lemma~\ref{lem:ratios}, and put $u_a=s_a/s_0$.

For a smooth general fibre $F$ over $B$, the exact cotangent sequence
\[
 0\longrightarrow\OO_F^{\oplus n}
 \longrightarrow\Omega_Y^1|_F
 \longrightarrow\Omega_F^1\longrightarrow0
\]
gives a filtration of $\Omega_Y^p|_F$ by subbundles with trivial graded pieces. Choose $F$ for which both $i|_F$ and $s_0|_F$ are nonzero. The first nonzero projection of $i|_F$ to a graded piece, followed by a suitable coordinate projection, gives a nonzero section $t$ of $L^{-1}|_F$. Then
\[
 t^m s_0|_F\in H^0(F,\OO_F)
\]
is nonzero. It is a nonzero constant because $F$ is compact and connected. In particular $t$ and $s_0|_F$ have no zeros, and every $u_a$ is constant on $F$. Lemma~\ref{lem:rational-descent} gives
\[
 u_a=g^*v_a,\qquad v_a\in\CC(X).
\]
Their independent differentials imply $p\le n$ and
\[
 \sigma=dv_1\wedge\cdots\wedge dv_p\ne0.
\]
By Lemma~\ref{lem:ratios}, the generic image of $L$ is the line of $g^*\sigma$.

We next descend the plurisections through their images as differential tensors. Write
\begin{equation}\label{eq:tensor-coefficients}
 i^{\otimes m}(s_a)=h_a(g^*\sigma)^{\otimes m},
 \qquad h_a\in\CC(Y).
\end{equation}
Shrink the smooth base open so that $\sigma$ is holomorphic and nowhere zero. Because $g$ is a submersion there, $g^*\sigma$ is nowhere zero at every point of the full inverse image. Equation~\eqref{eq:tensor-coefficients} makes each $h_a$ holomorphic there: locally one divides by a nonvanishing coefficient of the reference tensor. Compact connected fibres force $h_a$ to be constant on each such fibre. Therefore
\[
 h_a=g^*b_a\quad(b_a\in\CC(X)),\qquad b_a/b_0=v_a.
\]

Let $\tau=\pi^*\sigma$ be the meromorphic form on the smooth root charts. Lemma~\ref{lem:saturation} gives an orbifold line bundle $K\to\Omega_{\cX}^p$ with generic direction $\tau$. We claim that every tensor
\begin{equation}\label{eq:descended-tensors}
 (\pi^*b_a)\tau^{\otimes m}
\end{equation}
is holomorphic. Its differential pullback to $Y$ is holomorphic by \eqref{eq:tensor-coefficients}. At the general point of any ordinary base prime, condition~\ref{item:ordinary} supplies a local section of a submersive lift. At $D$, condition~\ref{item:root} does the same in the root chart. Pulling back along such a local section proves that \eqref{eq:descended-tensors} has no pole at that divisorial point. Any polar chart divisor belongs to the invariant global polar locus of this rational tensor. Its finite image is a base divisor, so equivariance and the preceding tests exclude every polar component. Extension across codimension two proves the claim everywhere.

By Lemma~\ref{lem:saturation}, the tensors in \eqref{eq:descended-tensors} are sections of $K^{\otimes m}$. Their ratios are the independent $v_a$, so $\kappa(K)\ge p$. Conversely, $f^*K\to\Omega_Y^p$ is a nonzero line map: on the dense ordinary open it is the differential pullback by a dominant map. Pullback preserves independence of section ratios, and Lemma~\ref{lem:ratios} gives
\[
 \kappa(K)\le\kappa(f^*K)\le p.
\]
Thus $K$ contradicts specialness of $\cX$. This proves specialness of $Y$.
\end{proof}

\begin{proof}[Fundamental groups and conclusion]
Set $Y^\circ=Y\setminus g^{-1}(D)$ and $X^\circ=X\setminus D$. For a smooth manifold and a divisor, the inclusion of the complement surjects on fundamental groups, with kernel normally generated by meridians of the irreducible divisor components. Indeed, perturb loops off the divisor and perturb null-homotopy discs transversely to its smooth strata, avoiding the strata of complex codimension at least two. Puncturing a disc at its transverse intersections gives the normal generators. This applies even if the reduced support of $g^*D$ is singular.

Consider $\pi_1(Y^\circ)\to\pi_1(X^\circ)\to G$. At the general point of a component of $g^*D$, its meridian maps to a conjugate of $\gamma^{2e}$, where $2e$ is that component's multiplicity. The winding calculation uses a smooth local equation for $D$ and remains valid when the component maps into a smaller subset of $D$. Positive meridians of $D$ are conjugate because $D$ is smooth and connected. Thus all the meridians killed on filling $Y^\circ$ die in $G$, and the map factors through $\pi_1(Y)$.

Over $B$ the map is proper and smooth, hence a differentiable fibre bundle \cite{Ehresmann}. Connectedness of its fibres gives $\pi_1(g^{-1}B)\twoheadrightarrow\pi_1(B)$. General position gives $\pi_1(B)\twoheadrightarrow\pi_1(X^\circ)$. The map just factored is therefore surjective, proving \eqref{eq:group-surjection}. Finally, $Y$ is a connected smooth projective special manifold. Assumption~\ref{ass:manifold} applies, and the image of a finite-index abelian subgroup of $\pi_1(Y)$ is finite-index abelian in $G$.
\end{proof}
==== END A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//transfer.tex ====

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  doi = {10.4171/RLM/1088},
  eprint = {2512.18919},
  archivePrefix = {arXiv},
  primaryClass = {math.AG},
  url = {https://ems.press/journals/rlm/articles/14299533},
  note = {Published online 17 February 2026}
}

@article{Wang,
  author = {Wang, Juanyong},
  title = {On the {Iitaka} Conjecture {$C_{n,m}$} for {K{\"a}hler} Fibre Spaces},
  journal = {Annales de la Facult{\'e} des sciences de Toulouse : Math{\'e}matiques},
  series = {6},
  volume = {30},
  number = {4},
  year = {2021},
  pages = {813--897},
  doi = {10.5802/afst.1690},
  eprint = {1907.06705},
  archivePrefix = {arXiv},
  primaryClass = {math.AG},
  url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1690/}
}
==== END A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor//references.bib ====