64c3814
==== TABLE OF CONTENTS ====
B-semiampleness-for-compact-log-smooth-Kahler-fibrations//comparison.tex
B-semiampleness-for-compact-log-smooth-Kahler-fibrations//conclusion.tex
B-semiampleness-for-compact-log-smooth-Kahler-fibrations//descent.tex
B-semiampleness-for-compact-log-smooth-Kahler-fibrations//hodge.tex
B-semiampleness-for-compact-log-smooth-Kahler-fibrations//introduction.tex
B-semiampleness-for-compact-log-smooth-Kahler-fibrations//orders.tex
B-semiampleness-for-compact-log-smooth-Kahler-fibrations//paper.tex
B-semiampleness-for-compact-log-smooth-Kahler-fibrations//periods.tex
B-semiampleness-for-compact-log-smooth-Kahler-fibrations//preliminaries.tex
B-semiampleness-for-compact-log-smooth-Kahler-fibrations//spreading.tex
B-semiampleness-for-compact-log-smooth-Kahler-fibrations//references.bib
===========================
==== BEGIN B-semiampleness-for-compact-log-smooth-Kahler-fibrations//comparison.tex ====
\section{The projective comparison factor}
\label{compare:section}
We now extend the projective factor supplied by
Proposition~\ref{spread:product}. The relevant positivity input is
algebraic b-semiampleness. We apply it to a projective comparison
family, retaining the specified identification of its plurivolumes.
\begin{proposition}
\label{compare:semiample}
Use the notation of Proposition~\ref{spread:product}, and write
\(P^\circ\subset P\) for its good open set. After a proper modification
\(\pi:P'\to P\), with \(P'\) smooth compact, and a further shrinking of
\(P^\circ\), there are a positive integer \(a\), a globally generated
holomorphic line bundle \(\mathcal H_Q\) on \(P'\), and a specified
isomorphism
\begin{equation}
\label{compare:open-identification}
\mathcal H_Q|_{\pi^{-1}(P^\circ)}
\simeq
\bigl(\pi^*\mathcal G_{m,Q}\bigr)^{\otimes a}
|_{\pi^{-1}(P^\circ)} .
\end{equation}
Transport to the left the \(a\)-th power of the volume norm
\[
N_{m,Q}(s)=
\left(\int_{Q_x}|s|^{2/m}\right)^{m/2}.
\]
For any holomorphic disk in \(P'\) whose punctured disk lies in the
good open, a local generator of the pulled-back \(\mathcal H_Q\)
and its dual have norm bounded by powers of
\(1+|\log|t||\), after a finite power substitution if necessary.
\end{proposition}
\begin{proof}
Let \(H\) be the product of the fixed Hilbert schemes used to record
the embedded projective factor and its boundary components.
Proposition~\ref{spread:product} gives a holomorphic map
\(\eta:P\to H\).
Its reduced image \(H_0\) is a compact irreducible analytic subspace.
Since \(H\) is projective, the proper mapping theorem and Chow's
theorem make \(H_0\) an irreducible projective variety.
We first identify an algebraic open of \(H_0\) carrying the desired
family. Smoothness, connectedness of the fibers, the condition that
the marked subschemes are divisors, and their relative simple normal
crossing condition are algebraically constructible conditions on the
universal marked family. The coefficients of the marks are the fixed
rational coefficients of \(C\), all in \([0,1)\).
On its smooth locus the bundle
\[
H_m=\omega_{\mathrm{rel}}^{\otimes m}\otimes\cO(mC)
\]
is defined after the fixed choice of \(m\). On a smooth connected
projective fiber, it is trivial precisely when
\[
h^0(H_m)\geq1
\quad\hbox{and}\quad
h^0(H_m^{-1})\geq1.
\]
Indeed, the product of nonzero sections is a nonzero holomorphic
function and is therefore a nonzero constant.
These cohomology conditions are constructible by semicontinuity.
All the conditions hold on \(\eta(P^\circ)\), which is dense in
\(H_0\). A constructible subset containing a dense subset of an
irreducible variety contains a nonempty algebraic open.
We may therefore choose an algebraic open \(H_0^\circ\) on which
the universal marked family is smooth projective, its fibers are
connected, and \(H_m\) is trivial on every fiber.
After shrinking the good open of \(P\), its projective factor is the
pullback of this family.
Cohomology and base change gives a line bundle
\(\mathcal V_m\) on \(H_0^\circ\).
Its evaluation map on the universal family is an isomorphism:
on every fiber it evaluates the one-dimensional space of sections
of a trivial line bundle.
Take a smooth projective model \(R\to H_0\), and resolve the
dominant component of the pulled-back universal family. We obtain
a projective morphism
\[
q:Z_Q\longrightarrow R
\]
with \(Z_Q\) smooth projective, unchanged over a smaller smooth
open \(R^\circ\subset R\).
The generic fiber is smooth and geometrically connected.
Consequently the finite map in the Stein factorization of \(q\)
is birational; it is an isomorphism because \(R\) is normal.
Thus \(q\) has connected fibers.
Choose a rational frame of \(\mathcal V_m\) over \(R\).
On \(Z_Q\) it defines a rational relative \(m\)-pluriform
\(\sigma\). Shrink \(R^\circ\) so that this frame is regular and
nonvanishing there. As a meromorphic section of
\(\omega_{Z_Q/R}^{\otimes m}\), its divisor has horizontal part
\(-mC\). Define the rational divisor
\begin{equation}
\label{compare:volume-boundary}
D_Q=-\frac1m\operatorname{div}(\sigma).
\end{equation}
The form \(\sigma\) then supplies an actual trivialization
\[
\omega_{Z_Q/R}^{\otimes m}\otimes\cO(mD_Q)\simeq\cO_{Z_Q},
\]
and hence an adjoint identity
\begin{equation}
\label{compare:adjoint}
K_{Z_Q}+D_Q\sim_{\Q}q^*K_R.
\end{equation}
Over \(R^\circ\), the divisor \(D_Q\) is exactly \(C\).
Its remaining coefficients are vertical and may have either sign.
Resolve their support, taking the boundary crepantly; this preserves
\eqref{compare:adjoint} and does not change the smooth family over
\(R^\circ\).
We check the algebraic b-semiampleness input explicitly.
Theorem~1.5 of \cite{BFMT25}, with Definition~6.18, applies to a
projective lc-trivial fibration whose boundary is effective over the
generic point. The required conditions are generic sub-log-canonicity,
the discrepancy rank-one condition, and a rational adjoint pullback
identity. Here the generic pair is the smooth effective klt pair
\((Q,C)\), and \eqref{compare:adjoint} is the required identity.
For the rank condition, on the generic fiber the rounded
discrepancy divisor is \(\lceil-C\rceil=0\); connectedness gives
\(h^0(Q,\cO_Q)=1\). A resolution of the vertical locus does not
affect this generic computation. Thus all these hypotheses hold.
For completeness, vertical adjustments do not change the moduli
object. If a rational divisor \(E\) on \(R\) is added to the
adjoint base bundle and \(q^*E\) is added to \(D_Q\), then
\[
t_P\longmapsto t_P-\operatorname{coeff}_P(E),
\qquad
B_R\longmapsto B_R+E,
\qquad
M_R\longmapsto M_R.
\]
The same equalities hold on every higher base model.
Thus one may, if desired, arrange upper bounds on the vertical
coefficients by subtracting sufficiently large pullbacks.
Alternatively one may arrange global effectivity by adding such
pullbacks. Each operation is possible because the finitely many
vertical components have proper algebraic images contained in
divisors on the projective base. Neither operation is needed:
the theorem imposes effectivity and log canonicity over the generic
point.
It follows from Theorem~1.5 of \cite{BFMT25} that the moduli
b-divisor is b-semiample. Its identification with the moduli object
defined by the log canonical thresholds is supplied by
Theorem~6.28 of \cite{BFMT25}: the Hodge-theoretic and threshold
moduli divisors agree up to a fixed rational linear equivalence.
Consequently, on a sufficiently high smooth projective model
\(\mu_R:R'\to R\), the actual line bundle
\[
\mathcal L_Q=\cO_{R'}(amM_{R'})
\]
is globally generated for some positive integer \(a\).
Increase \(a\) to clear all denominators.
Further projective modifications preserve global generation of the
pullback, so we may also arrange a simple normal crossing complement
to the good open.
We must retain the particular open identification with plurivolumes.
On a smooth source model over \(R'\), take the crepant transform
\(D'_Q\) and the base bundle \(L_{R'}=\mu_R^*K_R\).
Then
\[
K_{Z'_Q/R'}+D'_Q
\sim_{\Q}
(q')^*(L_{R'}-K_{R'}).
\]
Over the common good open this identity identifies the direct image
of the log \(m\)-pluricanonical bundle with the line of the original
volumes. Theorem~\ref{orders:trace} applies to this projective
family: its generic boundary is effective klt, and the theorem
permits arbitrary vertical coefficients.
It identifies the extension specified by the threshold trace
\(mM_{R'}\) under precisely this open-volume identification.
The base Jacobian on \(R'\) is included in that theorem's relative
order calculation. In particular, the \(k=0\) case of
Corollary~\ref{orders:residuenorm} gives the two-sided logarithmic
norm bounds for local generators and their duals, including after
pullback to a disk.
Finally resolve the graph of the meromorphic lift
\(P\dashrightarrow R'\) of \(\eta\), together with the complement
of the good open, obtaining \(\pi:P'\to P\) and
\(\rho:P'\to R'\), with \(P'\) smooth compact and its boundary
simple normal crossing.
Set
\[
\mathcal H_Q=\rho^*\mathcal L_Q.
\]
This line bundle is globally generated. The preceding specified
identification, pulled back along \(\rho\), is
\eqref{compare:open-identification}, and the same pullback gives the
asserted norm bounds.
\end{proof}
\begin{remark}
The rational form in \eqref{compare:volume-boundary} is taken on the
projective comparison space \(Z_Q\). The construction makes no
assumption that the canonical bundle, or an arbitrary holomorphic
line bundle, on the original nonprojective manifold has a global
meromorphic section. It also specifies an isomorphism of actual
bundles over the open set; a numerical identification would not
suffice for the final comparison.
\end{remark}
==== END B-semiampleness-for-compact-log-smooth-Kahler-fibrations//comparison.tex ====
==== BEGIN B-semiampleness-for-compact-log-smooth-Kahler-fibrations//conclusion.tex ====
\section{Proof of the main theorem}\label{sec:conclusion}
We assemble the preceding constructions, preserving the specified
isomorphisms of volume lines.
Start with the smooth compact K\"ahler model \(S\) prepared in
Section~\ref{sec:prelim}. Corollary~\ref{orders:bdescent} proves that
\[
M_{S_1}=\nu^*M_S
\]
for every smooth compact K\"ahler modification \(\nu:S_1\to S\).
This proves part~\textup{(a)} of Theorem~\ref{thm:main}.
For part~\textup{(b)}, Proposition~\ref{spread:product} supplies a
proper surjection \(r:P\to S\) from a smooth compact connected
auxiliary base, and an actual product decomposition of log-plurivolume
lines on a dense open \(P^\circ\). The connected deepest-stratum
selection in that proposition is compatible with the original
top root line by Proposition~\ref{hodge:residue}.
Theorem~\ref{orders:trace} therefore identifies the extension of
the original factor on the left with \(r^*(mM_S)\), after taking
the necessary common multiples and testing on local unipotent
covers.
By Proposition~\ref{compare:semiample}, the projective-factor line
has a semiample extension, with the required logarithmic norm
bounds. By Proposition~\ref{periods:semiample}, so do the remaining
highest-form factor lines after a further finite cover and
resolution of \(P\). Replace \(P\) by that compact smooth model.
It is still proper and surjective over \(S\).
All powers and all finitely many finite covers can be chosen once
for this family. Their composite and a common further exponent
suffice simultaneously for every factor.
The product isomorphism on \(P^\circ\) is induced by the actual
volume maps, and its norms agree up to the fixed covering-degree
constant. Corollary~\ref{orders:residuenorm} and the two factor
propositions supply upper and lower powers-of-logarithm estimates
for local extended frames on general transverse disks.
Lemma~\ref{descent:unit}, applied also to the inverse isomorphism,
extends the product identity to all of \(P\).
Thus, for some positive integer \(a\), the actual line
\(r^*\cO_S(aM_S)\) is a tensor product of semiample lines.
A common power is globally generated.
Lemma~\ref{descent:semiample} now gives a positive multiple of
\(aM_S\) generated at every point of \(S\).
The exponent is independent of the point, and the conclusion
holds for the holomorphic line bundle itself.
This proves part~\textup{(b)} and completes
Theorem~\ref{thm:main}.
\qed
\begin{remark}
The compact auxiliary base is used only to establish generation.
It is not asserted to be the modification in
Theorem~\ref{thm:main}. The latter is the compact K\"ahler model
\(S\) on which the threshold calculation already proves pullback
compatibility. Nor is a choice of global divisor representatives
for canonical or moduli line bundles on the original nonprojective
manifolds needed anywhere in the argument.
\end{remark}
==== END B-semiampleness-for-compact-log-smooth-Kahler-fibrations//conclusion.tex ====
==== BEGIN B-semiampleness-for-compact-log-smooth-Kahler-fibrations//descent.tex ====
\section{Extending the comparison and descending sections}
\label{descent:section}
Two analytic facts complete the passage from the auxiliary parameter
space to the original base model. The first extends a specified
isomorphism of volume lines; the second descends global generation.
\begin{lemma}
\label{descent:unit}
Let \(P\) be a complex manifold, \(D\) a simple normal crossing
divisor, and \(P^\circ=P\setminus D\).
Let \(E_1,E_2\) be holomorphic line bundles on \(P\), equipped on
\(P^\circ\) with Hermitian norms, and let
\[
\phi:E_1|_{P^\circ}\xrightarrow{\ \sim\ }E_2|_{P^\circ}
\]
be a specified holomorphic isomorphism.
Assume the following near the smooth locus of each component of
\(D\).
In a coordinate polydisk \((t,z)\) with \(D=(t=0)\), choose
nonvanishing holomorphic frames \(e_1,e_2\).
For a dense set of centers \(z\), their norms and dual norms along
the parallel punctured disks satisfy bounds by powers of
\(1+|\log|t||\). The constants and powers may depend on the disk.
Assume also that \(\phi\) compares the two norms by factors bounded
above and below on each such disk.
It is enough for these bounds to hold after a finite power
substitution on each disk.
Then \(\phi\) extends uniquely to a holomorphic isomorphism
\(E_1\simeq E_2\) on \(P\).
\end{lemma}
\begin{proof}
Write \(\phi(e_1)=g e_2\), where \(g\) is holomorphic and
nowhere zero on the punctured polydisk.
The bounds on the frames and their duals give, on every disk under
consideration,
\begin{equation}
\label{descent:scalar-bounds}
|g(t,z)|+|g(t,z)|^{-1}
\leq C_z\bigl(1+|\log|t||\bigr)^{b_z}
\end{equation}
for suitable positive constants.
If the estimate is initially given after \(t=u^N\), it implies
an estimate of the same form downstairs, since every nonzero \(t\)
has an \(N\)-th root and
\(|\log|u||=|\log|t||/N\).
Fix one such \(z\).
For \(j\geq1\), the coefficient of \(t^{-j}\) in the Laurent
expansion of \(g(t,z)\) is
\[
a_{-j}(z)=\frac{1}{2\pi i}
\int_{|t|=\varepsilon}g(t,z)t^{j-1}\,dt.
\]
It is independent of \(\varepsilon\), and
\eqref{descent:scalar-bounds} yields
\[
|a_{-j}(z)|
\leq C_z\varepsilon^j
\bigl(1+|\log\varepsilon|\bigr)^{b_z}
\longrightarrow0.
\]
Thus every negative Laurent coefficient vanishes.
The same argument applies to \(g^{-1}\), so both functions extend
holomorphically on this disk, and their extensions multiply to one.
For a fixed integration radius, \(a_{-j}(z)\) is holomorphic in
the remaining coordinates. It vanishes on the dense set of
centers from the hypothesis and therefore vanishes identically.
The Laurent expansion in the polydisk consequently has no negative
powers. Equivalently, Cauchy's integral formula constructs a
jointly holomorphic extension across \(t=0\).
Applying this also to \(g^{-1}\) gives a holomorphic unit throughout
the polydisk. This reasoning does not require uniform logarithmic
constants as the center varies.
We have extended \(\phi\) and its inverse across each component of
\(D\) away from its intersections with the others.
The remaining subset has complex codimension at least two.
In local line-bundle frames, Hartogs' extension theorem extends the
coefficient functions of both maps across that subset.
Their compositions remain the identity by analytic continuation.
Local extensions agree because they agree on the dense open
\(P^\circ\), proving existence and uniqueness globally.
\end{proof}
\begin{remark}
\label{descent:tangential-disks}
Local power charts also handle disks tangent to the boundary.
Indeed, write such a chart as
\[
(u_1,\ldots,u_\ell,z)\longmapsto
(u_1^{N_1},\ldots,u_\ell^{N_\ell},z).
\]
For a holomorphic disk \(\gamma\) whose punctured disk avoids the
boundary, each boundary coordinate has the form
\[
t_i\circ\gamma(t)=t^{b_i}a_i(t),
\qquad b_i\geq0,\quad a_i(0)\ne0.
\]
After a substitution \(t=v^N\), with every \(N_i\) dividing \(N\),
the functions \(a_i(v^N)\) have holomorphic \(N_i\)-th roots on a
smaller disk, so \(\gamma\) lifts to the power chart.
Thus the disk bounds for generators supplied by
Theorem~\ref{orders:trace} and the factor-extension results can be
used after arbitrary auxiliary pullbacks, including at tangential
disks. Lemma~\ref{descent:unit} itself needs only the general
parallel transverse disks.
\end{remark}
We next give the descent statement in a form that does not require
either a generically finite auxiliary map or a finite flat
intermediate map.
\begin{lemma}
\label{descent:semiample}
Let \(r:P\to S\) be a proper surjective holomorphic map of
connected compact complex manifolds, and let \(A\) be a
holomorphic line bundle on \(S\).
If \(r^*A\) is semiample, then \(A\) is semiample.
More precisely, let
\[
P\xrightarrow{h}T\xrightarrow{q}S
\]
be the Stein factorization, and let \(d\) be the generic degree of
\(q\).
If \(r^*(A^{\otimes n})\) is globally generated, then
\(A^{\otimes nd}\) is globally generated.
The same descent conclusion holds for rational holomorphic line
bundles after clearing denominators.
\end{lemma}
\begin{proof}
The intermediate space \(T\) is normal and irreducible,
\(q\) is finite surjective, and \(h_*\cO_P=\cO_T\).
The projection formula gives an identification of section spaces
\begin{equation}
\label{descent:stein-sections}
H^0\bigl(P,r^*(A^{\otimes n})\bigr)
\simeq
H^0\bigl(T,q^*(A^{\otimes n})\bigr).
\end{equation}
For \(t\in T\), choose \(p\in h^{-1}(t)\).
A section on the left that is nonzero at \(p\) descends to a
section on the right that is nonzero at \(t\).
Thus \(q^*(A^{\otimes n})\) is globally generated.
We recall directly the analytic norm needed for \(q\), without
assuming flatness.
Outside a proper analytic subset of \(S\), the map \(q\) is a
covering with \(d\) distinct sheets.
If \(b\) is a holomorphic function on \(q^{-1}(V)\), its product
over these sheets is a single-valued holomorphic function on the
covering locus in \(V\):
\[
\Nm_q(b)(s)=\prod_{t\in q^{-1}(s)} b(t).
\]
This function is locally bounded near the omitted analytic subset.
To see this near \(s_0\), choose neighborhoods of the finitely many
points of \(q^{-1}(s_0)\) on which \(b\) is bounded.
Properness allows us to shrink \(V\) so that \(q^{-1}(V)\) is
contained in their union.
The product is then bounded by the \(d\)-th power of a common
bound. Since \(S\) is normal, the removable singularities theorem
extends it uniquely to a holomorphic function on \(V\).
The extended norm is multiplicative and satisfies
\[
\Nm_q(q^*c)=c^d.
\]
Both identities hold on the covering locus and hence everywhere.
If \(b\) is nonzero at every point above \(s_0\), the same
neighborhoods can be chosen so that \(b\) and \(b^{-1}\) are
bounded. The preceding construction then extends both
\(\Nm_q(b)\) and \(\Nm_q(b^{-1})\), whose product is one.
In particular, \(\Nm_q(b)(s_0)\ne0\).
This argument uses local sheets and normality, rather than a
determinant of a locally free pushforward.
For any section
\[
v\in H^0\bigl(T,q^*(A^{\otimes n})\bigr),
\]
apply the function norm to its coefficients in local frames of
\(A^{\otimes n}\).
The identity \(\Nm_q(q^*c)=c^d\) shows that these coefficients glue
to a section
\[
\Nm_q(v)\in H^0\bigl(S,A^{\otimes nd}\bigr).
\]
Fix \(s_0\in S\).
For each point \(t\) of the finite set \(q^{-1}(s_0)\), sections
vanishing at \(t\) form a proper linear hyperplane in the section
space in \eqref{descent:stein-sections}.
Over \(\C\), finitely many proper hyperplanes cannot cover a
vector space. We may therefore choose \(v\) nonzero at every
point of \(q^{-1}(s_0)\).
Its norm is nonzero at \(s_0\).
The exponent \(nd\) is independent of \(s_0\), so these norm
sections prove global generation of \(A^{\otimes nd}\).
Finally, for a rational line bundle, first choose an actual line
bundle representing a positive integral multiple.
Any torsion ambiguity in such a representative disappears after
a further positive tensor power. The preceding argument then
applies to that actual line bundle.
\end{proof}
\begin{remark}
No Kähler assumption on the auxiliary manifold \(P\) is used in
Lemma~\ref{descent:semiample}. Positive-dimensional fibers are
handled by \eqref{descent:stein-sections}; branching and possible
nonflatness of the finite map are handled by the bounded norm.
In particular, this descent gives global generation on the whole
compact base, with a single exponent.
\end{remark}
==== END B-semiampleness-for-compact-log-smooth-Kahler-fibrations//descent.tex ====
==== BEGIN B-semiampleness-for-compact-log-smooth-Kahler-fibrations//hodge.tex ====
\section{The root eigenline and its deepest residue}
\label{hodge:section}
The coefficient-one part of the boundary naturally produces a mixed
variation. We first identify the line of volumes inside that variation
and then identify the same line in a pure variation associated with a
deepest boundary stratum. The comparison will retain the specified
volume, which is necessary for the extension calculation in
Section~\ref{orders:section}.
\subsection{A local root construction}
\label{hodge:rootconstruction}
Use the notation of Section~\ref{sec:prelim}. Thus \(S\) is a smooth
compact K\"ahler modification with good open \(U\), the complement
\(S\setminus U\) is an SNC divisor, and
\[
p:W\longrightarrow S,\qquad
D_W=h^*\Delta-K_{W/Y},\qquad
K_{W/S}+D_W\sim_{\Q}p^*Q_S.
\]
The equivalence is equipped with the actual bundle identification in a
fixed degree \(m\). Over \(U\), the map and the boundary strata are
smooth, the boundary is relatively SNC with coefficients in \([0,1]\),
and
\[
\mathcal G_m
=p_*\cO_W\bigl(m(K_{W/S}+D_W)\bigr)|_U
\simeq \cO_S(mQ_S)|_U.
\]
Only this good-open condition will be used in the present section.
Choose a sufficiently small coordinate neighborhood \(V\subset S\)
and a frame \(s\) of \(\cO_S(mQ_S)|_V\). Its image on \(W_V\) is a
meromorphic section of \(\omega_{W/S}^{\otimes m}\), with divisor
\(-mD_W\). Form its normalized \(m\)-th-root cover. This construction
does not require a meromorphic trivialization of \(\omega_{W/S}\):
choose an effective integral divisor \(J\) that clears the poles, form
the finite root cover of the holomorphic section in
\((\omega_{W/S}(J))^{\otimes m}\), and normalize. The resulting cover
with its meromorphic root is independent of the pole clearing. In a
local canonical frame \(\alpha\), if \(s=a\alpha^{\otimes m}\), it is
given by adjoining \(z\) with \(z^m=a\), and the root form is
\(z\alpha\), pulled back as a differential form.
Keep every component of this cover. The group \(\mu_m\) acts on it,
and the tautological root has a fixed character \(\chi\). Take a
projective equivariant resolution and write
\[
\rho:Z\longrightarrow W_V,\qquad
\tau=\text{the pulled-back relative root form}.
\]
The space \(Z\) and its fibers are allowed to be disconnected. The
finite-cover and resolution facts from Section~\ref{sec:prelim}
give K\"ahler total spaces after shrinking \(V\), with the resolution
performed over a slightly larger neighborhood.
Put \(V^\circ=V\cap U\). On this open the construction can be made
smooth over the base, with a relative reduced SNC divisor \(H\)
consisting of the inverse images of the coefficient-one components.
Here is the local reason for this assertion. In relative SNC
coordinates the tensor has the form
\begin{equation}
s=
a\,\prod_{\nu=1}^{r}x_\nu^{-m\delta_\nu}
(dx_1\wedge\cdots\wedge dx_d)^{\otimes m},
\qquad a\in\cO^*,\quad 0\leq\delta_\nu\leq1.
\label{hodge:localtensor}
\end{equation}
After taking a local root of the unit and removing full \(m\)-th
powers, the normalized cover is a monomial cover in the coordinates
with \(0<\delta_\nu<1\). The coefficient-one coordinates, the other
fiber coordinates, and the base coordinates are smooth product
parameters. Functorial resolution, which commutes with smooth
morphisms, resolves the monomial part while preserving these product
parameters; one can equally use a toroidal resolution in the fractional
directions. Thus the inverse images of the coefficient-one
components are smooth reduced divisors, their intersections are the
corresponding resolved covers of the original intersections, and all
these strata are smooth over \(V^\circ\). This use of functorial
resolution is as in \cite[Theorems~1.1 and~1.3]{BM08}.
\begin{lemma}
\label{hodge:rootpoles}
Over \(V^\circ\), the form \(\tau\) is a relative top form with at most
logarithmic poles on \(H\) and no other poles. At a strict covering
prime over a horizontal divisor of coefficient \(\delta\), if \(j\)
is the ramification index, its order is
\[
j(1-\delta)-1.
\]
\end{lemma}
\begin{proof}
At the generic point of the divisor, substitute \(x=z^j\) in
\(x^{-\delta}dx\). The order is \(j-1-j\delta\). It is integral;
for \(\delta<1\) it lies between \(0\) and \(j-1\), while for
\(\delta=1\) it equals \(-1\).
For exceptional divisors in a monomial resolution in the fractional
directions, the shifted order is the sum of the nonnegative coordinate
orders multiplied by \(1-\delta_\nu\). At least one fractional
coordinate has positive order, so this sum is strictly positive.
The actual order is integral, hence nonnegative. Equivalently, write
the root of \eqref{hodge:localtensor} in a logarithmic differential
basis. The coefficient has a strictly positive gain in every
exceptional fractional direction, whereas the logarithmic Jacobian
has at most a simple pole. The coefficient-one coordinates remain
transverse product parameters. Consequently the only poles are the
asserted simple logarithmic poles along \(H\).
\end{proof}
\subsection{Extension facts and the mixed top step}
\label{hodge:extensionfacts}
We record precisely the extension input needed below. Let
\(\Delta_t\) be a disk and \(\Delta_t^*=\Delta_t\setminus\{0\}\).
Suppose
\[
g:(Z,H)\longrightarrow\Delta_t
\]
is proper, \(Z\) is smooth K\"ahler, \(H\) is a reduced SNC divisor,
every stratum of \((Z,H)\) dominates the disk, and all strata are
smooth over \(\Delta_t^*\). Write \(d=\dim Z-1\).
Fujino--Fujisawa's Theorem~1.1 supplies the graded-polarizable real
variation on \(H_c^d(Z_t\setminus H_t)\) and its lower canonical
extension, with locally free double Hodge/weight grades
\cite[Theorem~1.1]{FF25}. Its dual description gives
\begin{equation}
g_*\omega_{Z/\Delta_t}(H)
\simeq
\left(\Gr_F^0\,
\overline{H_c^d(Z_t\setminus H_t)}^{\,\mathrm{lower}}\right)^*.
\label{hodge:FF}
\end{equation}
Poincar\'e duality identifies \(H_c^d{}^*\) with \(H^d(d)\).
Therefore \eqref{hodge:FF} is the extended \(F^d\) step of ordinary
cohomology in the \emph{upper} canonical extension. All its double
grades remain locally free. Upper residues lie in \((-1,0]\), and
lower residues in \([0,1)\); in the unipotent case the conventions
coincide \cite[Section~2.14]{FF25}.
For pure variations we will use the following consequences of the
nilpotent orbit theorem. We include the functorial consequences in
the statement to specify exactly what is required of an extension.
\begin{lemma}[Pure extension rules]
\label{hodge:purerules}
Let a real-polarizable variation of pure Hodge structure be given on
the complement of an SNC divisor in a polydisk, with quasi-unipotent
local monodromies.
\begin{enumerate}[label=\textup{(\roman*)}]
\item
After coordinate power substitutions making all local monodromies
unipotent, its Hodge filtrations extend by subbundles of the canonical
flat bundle. On a disk, the filtration also extends by subbundles
of either the upper or the lower canonical bundle before
unipotentization.
\item
After coordinate power substitutions making the monodromies
unipotent, the canonical extension and its Hodge filtration commute
with further powers and with pullback to any disk whose punctured
image lies in the given open.
\item
A real pure Hodge morphism has flat Hodge kernels and images, with flat
Hodge complements. Its split maps and finite-group eigenspace
projectors are compatible with the canonical extensions in
\textup{(i)}. After unipotentization, tensor constructions are
compatible as well.
\item
On a unipotent disk, a nonvanishing frame \(e\) of an extended highest
Hodge line satisfies, for some constants \(C,A>0\),
\begin{equation}
C^{-1}(1+|\log|t||)^{-A}
\leq \|e(t)\|
\leq C(1+|\log|t||)^A.
\label{hodge:logbounds}
\end{equation}
The constants may depend on the disk. The same assertion applies to
Hodge determinant lines.
\end{enumerate}
\end{lemma}
\begin{proof}
In the unipotent case, extension by subbundles is the nilpotent orbit theorem
\cite[Theorem~4.12]{Schmid73}; a formulation with real polarization
and its canonical-frame interpretation is given in
\cite[Section~2, \((2.1)\)--\((2.2)\)]{CK89}.
For a quasi-unipotent disk, make the cyclic substitution \(t=u^N\)
that removes the semisimple monodromy. Its finite deck group acts
on the extended filtered bundle. Choose a character basis at the
origin adapted to the Hodge flag. Lift these vectors to local
sections in the corresponding Hodge subbundles and average with
the character projectors. The resulting equivariant sections
form an adapted frame near the origin. Multiplying its vectors
by the integral powers of \(u\) prescribed by their characters
gives a descending frame with residue eigenvalues in either chosen
interval \((-1,0]\) or \([0,1)\). The same subsets of this frame
span the respective extended Hodge steps. This proves the disk
assertion for both canonical conventions.
For unipotent monodromies with commuting logarithms \(N_i\), the
canonical flat bundle is described in logarithmic frames. Under a
disk map, each boundary coordinate is \(t^{a_i}v_i(t)\), with \(v_i\)
a unit and \(a_i\geq0\). The pulled-back logarithmic connection has
nilpotent residue \(\sum_i a_iN_i\). Thus the pulled-back flat bundle
is the canonical one. The pulled-back Hodge subbundles are the
canonical filtration as well, by its characterization inside this
bundle.
For a pure Hodge map, its kernel and image are flat real Hodge
subbundles. The restriction of a polarization to a real Hodge
subspace is nondegenerate. Orthogonal complements therefore split
the map in the category of real variations. These projectors are
flat, so in logarithmic frames their extensions have the same
constant ranks. A polarization can be averaged over a finite group;
one then complexifies and applies the character projectors.
This uses no semisimplicity assertion for arbitrary local systems.
The abstract norm estimate is
\cite[Theorem~\(6.6'\)]{Schmid73}. Applied to a flat frame and its
dual, it bounds the metric and its inverse by powers of
\(1+|\log|t||\). Logarithmic frames differ from flat frames by finite
polynomials in the logarithm. A nonvanishing extended line frame
has bounded coefficients in a logarithmic frame and one coefficient
bounded away from zero locally. The metric and inverse-metric bounds
give \eqref{hodge:logbounds}. Tensor and determinant constructions
preserve this conclusion.
\end{proof}
The variations occurring here have integral lattices up to isogeny.
Indeed they arise from the cohomology of compact SNC strata and the
rational weight filtration. A K\"ahler class on the total space
restricts to flat real K\"ahler classes on these strata, so their pure
cohomologies, and the relevant kernels and images, are real-polarizable.
Their local monodromies are quasi-unipotent by the integral monodromy
theorem. One may use its real-polarized formulation; alternatively,
the elementary transport in Lemma~\ref{periods:rationalization}
reduces it to the rationally polarized statement without changing
the integral monodromy. Applying the theorem to weight grades also
gives quasi-unipotence for the mixed variation.
\begin{lemma}[The top step and its weight grade]
\label{hodge:doublegrades}
Let a mixed variation on a punctured disk be extended in its upper
canonical flat bundle, with the canonical weight filtration and
locally free double Hodge/weight grades, as in \eqref{hodge:FF}.
Suppose a finite-group character summand satisfies
\[
\dim F^d=1,\qquad F^{d+1}=0,
\]
and the sole nonzero \(F^d\) weight grade has weight \(w\).
Then projection to \(\Gr^W_w\) identifies the extended top line with
the canonically extended top line of that pure grade.
\end{lemma}
\begin{proof}
The induced filtration of \(F^d\) by the weights has just one
nonzero successive quotient. Local freeness of the double grades
implies that the same ranks and exact quotient sequences hold at the
origin. Its terms below weight \(w\) vanish and its terms from
weight \(w\) onward equal the whole line. Projection is therefore
an isomorphism there as well. The upper canonical extension is
exact on the weight filtration, and the induced filtration on a
pure grade is its canonical Hodge filtration. These statements also
follow by dualizing the lower extension in \eqref{hodge:FF}.
The character projector is an idempotent on all these locally free
objects, so taking its image preserves the argument.
\end{proof}
\subsection{The pure line detected by residues}
\label{hodge:pureresidue}
Over \(V^\circ\), put
\[
\mathbb V=R^d(g|_{Z\setminus H})_*\C,
\qquad g:Z\longrightarrow V,
\]
where the notation denotes ordinary cohomology on the smooth open
fibers. We use \(W_\bullet\) for its weight filtration. The
logarithmic mixed Hodge construction identifies
\[
F^d\mathbb V_x
\simeq H^0(Z_x,\omega_{Z_x}(H_x)).
\tag{\(\ast\)}
\]
It is the ordinary-cohomology version of the compact-support SNC
construction above: compact supports are computed by the simple
complex of the compactification and its successive intersections;
duality gives logarithmic forms and residues. These constructions
work for compact K\"ahler strata, using their pure Hodge structures
and Hodge degeneration, as in \cite[Section~4]{FF25}; compare
\cite[Section~3]{Deligne71}.
\Needspace{9\baselineskip}
\begin{proposition}[Root eigenline and deepest residue]
\label{hodge:residue}
Let \(k\) be the maximal number of horizontal coefficient-one
components meeting in a good fiber, constant after shrinking \(U\).
For the local root family above:
\begin{enumerate}[label=\textup{(\roman*)}]
\item
\[
F^d\mathbb V_\chi=\cO_{V^\circ}\tau.
\]
This line lies in the single pure weight grade
\(\Gr^W_{d+k}\mathbb V_\chi\).
\item
Choose a depth-\(k\) intersection component downstairs dominating
the good open. After a good-open base change selecting one
connected component \(A_x^*\) of its fibers, take every root-cover
sheet above that component. Iterated residue identifies the line
in \textup{(i)}, through its pure grade and with the Tate twist
removed, with the highest eigenline of the resolved root cover of
the induced klt volume on \(A_x^*\). In degree \(m\) this is the
specified plurivolume identification from SNC adjunction.
\item
These highest-line identifications preserve generators in
canonical extensions on unipotent disks. In a disk model satisfying
the hypotheses of \eqref{hodge:FF}, its extended mixed top line
projects isomorphically to the same pure line.
\end{enumerate}
If \(k=0\), the intersection is the whole original fiber, there is
no removed divisor, and the pure root-cover cohomology itself
provides the line.
\end{proposition}
\begin{proof}
By Lemma~\ref{hodge:rootpoles}, \(\tau_x\) belongs to the top
logarithmic-form space. If \(\eta\) is another such eigenform with
character \(\chi\), the ratio \(\eta/\tau_x\) is invariant under
\(\mu_m\). It descends to a meromorphic function on the original
connected compact smooth fiber, also when the normalized cover is
disconnected: the invariant meromorphic algebra of the full root
cover is the downstairs algebra. At a strict covering divisor, a
downstairs pole of order \(n\geq1\) would lower the order in
Lemma~\ref{hodge:rootpoles} by \(jn\). For \(\delta<1\) this gives a
pole where no pole is allowed; for \(\delta=1\) it gives an order
below \(-1\). The ratio has no pole along any downstairs prime.
It extends across the remaining codimension-two set, and compactness
makes it constant. This proves the first displayed equality.
There is no Hodge step above \(F^d\). Exactness of the induced
weight filtration on this one-dimensional space shows that exactly
one weight grade carries it. To determine that grade, let
\(H_x^{[r]}\) be the disjoint union of the \(r\)-fold intersections,
with \(H_x^{[0]}=Z_x\). The residue spectral sequence has terms
\begin{equation}
E_1^{-r,q}
=H^{q-2r}(H_x^{[r]},\C)(-r)
\ \Longrightarrow\ H^{q-r}(Z_x\setminus H_x,\C).
\label{hodge:weightspectral}
\end{equation}
The differentials on the first page are pure Hodge maps. Taking
their kernels modulo images leaves pure objects, and subsequent
differentials vanish by their different weights. Since
\(H_x^{[k+1]}\) is empty, there is no incoming first differential at
\((-k,d+k)\). Consequently \eqref{hodge:weightspectral} gives a
natural pure map
\begin{equation}
\Gr^W_{d+k}H^d(Z_x\setminus H_x,\C)
\lhook\joinrel\longrightarrow
H^{d-k}(H_x^{[k]},\C)(-k).
\label{hodge:residuemap}
\end{equation}
On the highest Hodge step it is iterated residue. In local SNC
coordinates this extracts the coefficient of
\(dx_1/x_1\wedge\cdots\wedge dx_k/x_k\); integration over the
corresponding normal circles gives the same cohomological map,
up to fixed conventional constants. Thus it is horizontal in
families.
At a point of a selected deepest intersection away from fractional
boundary, the root cover is unramified and \(\tau_x\) has exact
logarithmic poles in the \(k\) transverse coordinates. Its residue
is nonzero. It extends as a holomorphic top form on the compact
resolved intersection upstairs and hence has a nonzero cohomology
class. The unique weight carrying \(F^d\mathbb V_\chi\) is therefore
\(d+k\).
Consider now the selected connected component \(A_x^*\) downstairs.
By maximality of \(k\), no further coefficient-one component meets
it. The remaining boundary is effective fractional SNC, so the
induced pair is klt. In the product coordinates of the root
construction, taking residue simply deletes the \(k\) logarithmic
parameters. The union of all sheets above \(A_x^*\) is thus the
resolved root cover of its induced log plurivolume. Its top
eigenline has dimension one by the same ratio argument. The
projection of \eqref{hodge:residuemap} to these sheets is nonzero on
the source line, and therefore identifies the two top lines.
With an ordering of the boundary components, tensor power of
ordinary residue gives
\[
(\operatorname{Res}\tau_x)^{\otimes m}
=\rho^*(\operatorname{Res}_m s_x).
\]
Here \(\operatorname{Res}_m\) denotes the bundle map supplied by
SNC adjunction; fixed sign or normalization choices do not affect
the argument.
The maps on pure grades and compact intersection cohomology are
real Hodge maps before taking the character summand. Their
kernels and images split by Lemma~\ref{hodge:purerules}, including
after projection to all the selected sheets. They hence preserve
the extended highest-line generators. On a disk to which
\eqref{hodge:FF} applies, Lemma~\ref{hodge:doublegrades} identifies
the mixed top line with that on its unique pure grade. This proves
\textup{(iii)}.
For \(k=0\) there is no logarithmic boundary, so ordinary compact
root-cover cohomology is pure of weight \(d\), and the same
eigenform argument applies directly.
\end{proof}
\subsection{Changing the resolved root model}
\label{hodge:modelchanges}
The order calculation will use resolved models adapted to a chosen
disk. Such a model can modify a generic root fiber, so one needs
the following invariance statement rather than a claim that every
cohomology group is birationally invariant.
\begin{lemma}
\label{hodge:birational}
Over a smooth good open, let
\(\varphi:(Z',H')\to(Z,H)\) be a proper equivariant modification
between smooth families of SNC pairs, with \(H'\) the reduced
inverse image of \(H\). Pullback is an isomorphism on top
logarithmic-form spaces. In the character summand of
Proposition~\ref{hodge:residue}, it identifies the top line and
the top line of its unique pure weight grade.
Suppose further that, over a disk, the new compactification
satisfies the hypotheses of \eqref{hodge:FF}. If the original
variation has been made unipotent, testing extension of the
pulled-back volume in \(g'_*\omega_{Z'/\Delta}(H')\) tests the
canonical extension of the original top line. It is not necessary
to assume unipotence of every summand newly appearing in the
cohomology of \(Z'\setminus H'\).
\end{lemma}
\begin{proof}
Pullback of an SNC logarithmic top form has at most logarithmic
poles along the reduced inverse image of the boundary. To descend
a top logarithmic form, use the isomorphism away from the
codimension-two exceptional image on each smooth original fiber.
The resulting section of \(\omega_Z(H)\) extends across that image
because this is a line bundle. Thus pullback and descent are
inverse on top forms.
Pullback of logarithmic complexes gives the cohomological morphism
of mixed variations. It respects residues and both filtrations;
one can see the logarithmic pullback in SNC coordinates, and the
compact-stratum construction gives the same rational map.
Morphisms of mixed Hodge structures are strict for the weight
filtration on each Hodge step. Since the top line maps
isomorphically, its unique carrying weight is unchanged, and the
induced pure map is an isomorphism on these highest lines.
Use the upper canonical extension for the new model. Its extended
mixed top eigenspace projects isomorphically to its pure grade by
Lemma~\ref{hodge:doublegrades}. The comparing pure map splits onto
its image and is compatible with upper extension by
Lemma~\ref{hodge:purerules}. That image is the unipotent summand
pulled from the original variation, so upper extension on it is
the unipotent canonical extension. All other weight grades have
zero top step of this character. Formula~\eqref{hodge:FF} therefore
tests precisely the original extended line, as claimed.
\end{proof}
==== END B-semiampleness-for-compact-log-smooth-Kahler-fibrations//hodge.tex ====
==== BEGIN B-semiampleness-for-compact-log-smooth-Kahler-fibrations//introduction.tex ====
\section{Introduction}\label{sec:intro}
The canonical bundle formula separates the canonical geometry of a
fibration into a discriminant, which measures singularities over divisors,
and a moduli part, which measures variation.
The b-semiampleness conjecture predicts that a suitable multiple of the
moduli part defines a morphism on a birational model of the base.
Hodge-theoretic semipositivity underlies this formula
\cite{Kawamata81}. Ambro established birational stabilization and
positivity of the moduli part in the generic-klt setting
\cite{Ambro04,Ambro05}; Fujino--Gongyo proved b-nefness and abundance
for projective lc-trivial fibrations \cite{FujinoGongyo14}.
In the algebraic category, the conjecture is proved by
Bakker--Filipazzi--Mauri--Tsimerman \cite[Theorem~1.5]{BFMT25}.
Their discussion of the analytic category retains projectivity of the
morphism; the extension to K\"ahler morphisms is explicitly distinguished
in \cite[Remarks~6.31 and~7.5]{BFMT25}.
We resolve the compact log-smooth case of this K\"ahler b-semiampleness
question. Both the source and the base may be nonprojective.
The boundary may have coefficient one, so the relevant Hodge structure
can be mixed. The assertion concerns actual holomorphic line bundles:
numerical semipositivity alone would not suffice.
\subsection{Statement}
All spaces and maps are complex analytic unless stated otherwise.
A \emph{modification} is a proper bimeromorphic holomorphic map.
Write \(\Pic(X)_{\Q}=\Pic(X)\otimes_{\Z}\Q\), using additive notation.
Equality in this group means an isomorphism of holomorphic line bundles
after a common positive integral multiple.
Let \(f:Y\to X\) be a surjective holomorphic map with connected fibers,
where \(Y\) and \(X\) are smooth compact connected K\"ahler manifolds.
Let \(\Delta\) be an effective rational divisor with simple normal
crossing support and coefficients in \([0,1]\), and assume that
\begin{equation}\label{eq:adjoint}
K_Y+\Delta\sim_{\Q}f^*L,\qquad L\in\Pic(X)_{\Q}.
\end{equation}
These assumptions give an lc-trivial fibration in the usual rank-one
sense; the rank condition is recalled in
Section~\ref{sec:prelim}.
Here is the moduli object in the statement.
For a smooth compact K\"ahler modification \(\mu:X'\to X\), resolve
the main component of \(Y\times_X X'\):
\[
\begin{tikzcd}[column sep=large,row sep=large]
Y' \arrow[r,"h"] \arrow[d,"f'"'] & Y\arrow[d,"f"]\\
X' \arrow[r,"\mu"'] & X .
\end{tikzcd}
\]
Take \(Y'\) smooth compact K\"ahler and set
\[
\Delta'=h^*\Delta-K_{Y'/Y}.
\]
Thus \(K_{Y'}+\Delta'=h^*(K_Y+\Delta)\) under the natural
canonical-bundle identification. Exceptional coefficients of
\(\Delta'\) can be negative.
For a prime divisor \(P\subset X'\), put
\begin{equation}\label{eq:threshold}
t_P=\inf_{E\mapsto P}
\frac{a(E;Y',\Delta')}{\ord_E((f')^*P)}.
\end{equation}
Here the infimum ranges over all prime divisors on smooth proper
bimeromorphic models of \(Y'\) that dominate \(P\), with positive
denominator; \(a(E;Y',\Delta')\) denotes log discrepancy.
Equivalently, \(t_P\) is the largest \(t\) for which
\((Y',\Delta'+t(f')^*P)\) is sub-log-canonical over a general point of
\(P\). Define
\begin{equation}\label{eq:moduli}
B_{X'}=\sum_P(1-t_P)P,\qquad
M_{X'}=\mu^*L-K_{X'}-B_{X'}\ \in\Pic(X')_{\Q}.
\end{equation}
These definitions are independent of the chosen resolution of the
source. The coefficients are rational and only finitely many are
nonzero; the elementary local calculation is recalled below.
The family \(\mathbf M=(M_{X'})\) is the \emph{moduli rational
b-line bundle}.
\Needspace{8\baselineskip}
\begin{theorem}\label{thm:main}
Under the assumptions above, \(\mathbf M\) is b-semiample.
More explicitly, there is a smooth compact K\"ahler modification
\(\mu_0:S\to X\) such that:
\begin{enumerate}[label=\textup{(\alph*)}]
\item for every smooth compact K\"ahler modification \(\nu:S_1\to S\),
\[
M_{S_1}=\nu^*M_S\quad\text{in }\Pic(S_1)_{\Q};
\]
\item for some positive integer \(m\), the actual line bundle
representing \(mM_S\) is generated by its global holomorphic sections.
\end{enumerate}
The theorem includes a point base and relative dimension zero.
No projectivity of \(f\), \(X\), or \(Y\) is assumed.
\end{theorem}
The discriminant in \eqref{eq:moduli} is the log-canonical-threshold
discriminant. It is not the weighted inf-multiplicity divisor of an
orbifold base. In particular, a Hodge line produced by a decomposition
of pluriforms must still be identified with \eqref{eq:moduli}.
That identification is one of the principal steps of the proof.
\subsection{Proof and technical contributions}
Choose a compact K\"ahler modification \(S\) whose boundary is simple
normal crossing. Locally on \(S\), take a cyclic root of a generator of
the relative log-plurivolume line. Its highest Hodge eigenspace is
one-dimensional. If horizontal coefficient-one components are present,
an iterated residue identifies that line, through a single pure weight
grade, with the volume line of a deepest boundary stratum.
The mixed Hodge extension theorem of Fujino--Fujisawa
\cite[Theorem~1.1]{FF25} makes this reduction compatible with extension
over a disk.
The local extension calculation keeps track of all divisorial
valuations and of the base Jacobian. For a local root volume \(\tau\),
it proves the precise normalized order
\[
\ord_P^{\mathrm{Hodge}}(\tau)=t_P-1.
\]
It follows that the canonical Hodge extension is the actual moduli
line and that it pulls back on every further smooth model. The same
calculation works with arbitrary crepant vertical coefficients.
It is therefore also available later on a projective comparison family.
Adjunction to a deepest stratum produces smooth compact K\"ahler klt
fibers with trivial log-pluricanonical bundle.
A theorem of Matsumura--Wang--Wu--Zhang gives finite product covers of
these fibers \cite[version~1, Corollary~1.3]{MWWWZ25}.
A pointwise product decomposition is insufficient for semiampleness.
We construct a marked family with a section, realize its finite fiber
covers after finite base changes, and parameterize their product graphs
inside fixed compact K\"ahler ambients.
Compact Douady components and a proper-constructibility argument give
a compact proper surjection \(P\to S\) carrying the required product
family on a dense open. The auxiliary base need not be projective or
generically finite over \(S\).
The product has a projective log Calabi--Yau factor and factors whose
highest forms are controlled by weight-one or weight-two periods.
The first is handled by the algebraic b-semiampleness theorem on a
projective comparison base. For the others, a flat change of the Hodge
filtration turns a real polarization into a rational one without
changing the underlying integral monodromy. Arithmetic period
compactifications then supply semiample extensions. This use of
period compactification is analytic on the auxiliary base
\cite[Theorems~5.2 and~5.5]{BFMT25}.
Finally, a product identity of volume norms and two-sided logarithmic
bounds extend a specified open isomorphism of line bundles across the
boundary. This step excludes an undetected flat twist.
Semiampleness descends from \(P\) through Stein factorization and finite
analytic norms. These arguments yield global generation at every point
of \(S\), with one exponent.
The proof does not use the Campana orbifold Iitaka theorem.
==== END B-semiampleness-for-compact-log-smooth-Kahler-fibrations//introduction.tex ====
==== BEGIN B-semiampleness-for-compact-log-smooth-Kahler-fibrations//orders.tex ====
\section{Threshold orders and descent on higher models}
\label{orders:section}
We now identify the extension of the specified volume line with the
moduli trace defined by log canonical thresholds. The calculation is
local on the base, but its compatibility with pullback will give the
assertion on every higher modification.
We use a slightly more general setup than the original fibration.
Let \(p:W\to S\) be a proper connected-fiber map between smooth compact
K\"ahler manifolds, let \(U\subset S\) have SNC complement, and suppose
that over \(U\) the family is smooth with effective relatively SNC
boundary whose coefficients lie in \([0,1]\). Let \(D_W\) be a
rational divisor extending that boundary, with
\begin{equation}
K_{W/S}+D_W\sim_{\Q}p^*Q_S
\label{orders:adjoint}
\end{equation}
and a specified bundle identification in degree \(m\).
The coefficients on vertical components of \(D_W\) are arbitrary
rational numbers; in particular, they need not be at most one.
We use the thresholds of the sub-pair \((W,D_W)\) to define
\[
B_S=\sum_P(1-t_P)P,\qquad M_S=Q_S-B_S.
\]
All source modifications below carry the crepant transform of
\(D_W\). The original setup satisfies these conditions. The
projective comparison in Section~\ref{compare:section} will use the
allowance of arbitrary vertical coefficients.
\subsection{Thresholds on a resolved transverse disk}
\label{orders:thresholds}
\begin{lemma}
\label{orders:threshold}
Fix a prime divisor \(P\subset S\). On a resolution over a general
point of \(P\), arrange joint SNC support for the crepant divisor
and \(p^*P\). Write \(\delta_E\) for the crepant coefficient and
\(a_E=\ord_E(p^*P)>0\) for the components dominating \(P\). Then
\begin{equation}
t_P=\min_{E\mapsto P}\frac{1-\delta_E}{a_E}.
\label{orders:thresholdformula}
\end{equation}
This minimum computes the threshold over all divisorial valuations,
including valuations exceptional over the chosen resolution.
\end{lemma}
\begin{proof}
For a rational SNC divisor, sub-log-canonicity is equivalent to all
coefficients being at most one. To recall why this tests further
valuations, in coordinates for its support write the logarithmic
top form
\[
\frac{dx_1}{x_1}\wedge\cdots\wedge\frac{dx_r}{x_r}
\wedge dx_{r+1}\wedge\cdots\wedge dx_n.
\]
At any prime on a smooth model its pullback has at most a simple
pole: all singular normal terms in the logarithmic differentials
are multiples of the same normal differential \(dz/z\), so at
most one can occur in the wedge. If \(b_i\) are the coordinate
orders there, this gives the logarithmic Jacobian inequality.
For coefficients \(c_i\leq1\), the resulting log discrepancy is at
least \(\sum_i b_i(1-c_i)\geq0\). Necessity is already detected
by a component of coefficient greater than one. Negative
coefficients cause no change in this criterion.
The horizontal coefficients on the resolved model are at most one,
because the sub-pair is log canonical over the good open and their
coefficients are constant. The coefficient of a vertical component
in \(D_W+t\,p^*P\) is \(\delta_E+ta_E\). The preceding criterion
therefore gives exactly \eqref{orders:thresholdformula}.
\end{proof}
At a general smooth point of \(P\), choose base coordinates
\((t,z_2,\ldots,z_b)\) with \(P=(t=0)\), and hold the last \(b-1\)
coordinates fixed. First resolve the joint divisor on the source.
Properness and generic smoothness of the finitely many relevant
strata allow the chosen point to avoid their bad images. The
resulting transverse slice has a smooth source, and restriction of
the joint divisor is SNC. Adjunction of the other base coordinates
changes none of the orders in
\eqref{orders:thresholdformula}. Components mapping to higher
codimension subsets of \(S\) are absent from this general slice.
We may therefore perform the order calculation on a disk.
Throughout that calculation, a relative canonical form means a
section of
\(\omega_W\otimes p^*\omega_{\Delta_t}^{-1}\).
Wedging with the chosen frame \(dt\) identifies it with a total
canonical form. This convention remains valid at the central
fiber; it does not presume that \(p\) is a submersion there.
\begin{lemma}[Disk models for the extension test]
\label{orders:diskmodels}
Fix the local root family of Section~\ref{hodge:rootconstruction}
and a resolved transverse disk downstairs. For every sufficiently
divisible power substitution \(t=u^N\), there is a smooth
equivariant K\"ahler model over \(\Delta_u\) with the following
properties:
\begin{enumerate}[label=\textup{(\roman*)}]
\item
It retains every component of the generic root cover and maps to
the downstairs order resolution. On the punctured disk it is a
modification of the pulled-back root family.
\item
The horizontal removed divisor \(H_N\), defined by the reduced full
inverse image of the original removed divisor on the punctured
disk and then closed up, is SNC. Its support and the full central
fiber have joint SNC support.
\item
Every stratum of \((Z_N,H_N)\) dominates the disk and is smooth
over a smaller punctured disk.
\item
For each vertical prime \(E\) on the downstairs order resolution,
some prime on \(Z_N\) dominates \(E\).
\end{enumerate}
Consequently \eqref{hodge:FF} applies to this model, and its top
eigenline tests the extension of the original line as in
Lemma~\ref{hodge:birational}.
\end{lemma}
\begin{proof}
Take the main parts of the fiber products of the root cover, the
downstairs order resolution, and \(\Delta_u\). Here main parts
means every component dominating the disk, so no generic root
sheet is discarded. Normalize the finite covers and take a common
projective equivariant resolution, including embedded resolution
of the horizontal boundary and central fiber. This gives the
required maps. Finite covers and projective resolutions preserve
the local K\"ahler models from Section~\ref{sec:prelim}. A prime
above a downstairs prime exists already on the normalized finite
cover; retaining its strict transform proves \textup{(iv)}.
Only horizontal components belong to \(H_N\). Joint SNC with the
central fiber shows that none of their intersection components can
be vertical. Indeed, if such a stratum were contained in the
central fiber, it would be contained in one of its components. In
SNC coordinates the intersection of the specified horizontal
coordinate hyperplanes cannot be contained in the additional
central-fiber coordinate hyperplane. Properness now makes the
image of every stratum the whole disk. Generic smoothness and
shrinking the disk give smoothness of all strata off its origin.
On the punctured disk the comparison is an equivariant
modification of smooth SNC families, with the reduced full
inverse image as boundary. Lemma~\ref{hodge:birational} identifies
the top eigenline and its extension. If the new ambient
cohomology is merely quasi-unipotent, use its upper canonical
extension; the summand coming from the unipotent original line
has the same canonical extension there.
\end{proof}
\subsection{The exact order calculation}
\label{orders:localcalculation}
Let \(\sigma\) be a meromorphic relative \(m\)-plurivolume on the
downstairs disk model whose good-fiber divisor is \(-m\) times the prescribed
effective SNC log Calabi--Yau boundary. It may have arbitrary
vertical orders. Let \(\Omega\) denote its multivalued total root,
obtained by multiplying the relative root by \(dt\), and put
\[
l_E=\frac1m\ord_E(\sigma\otimes dt^{\otimes m}),
\qquad a_E=\ord_E(t).
\]
Resolve the joint support of the divisor of this tensor and the
central fiber. The root construction gives a section \(\tau\)
of the pure highest eigenline on the punctured disk. We compare
it with the canonical extension after unipotentization.
\begin{lemma}[Order of a root volume]
\label{orders:order}
The section \(\tau\) is meromorphic in that extended line. Its
normalized order is
\begin{equation}
\ord^{\mathrm{Hdg}}_0(\tau)
=\min_E\frac{l_E+1-a_E}{a_E},
\label{orders:generalorder}
\end{equation}
where \(E\) runs through the central-fiber primes on the downstairs
resolution. The normalization divides an ordinary order after
\(t=u^N\) by \(N\). At the displayed order, the corrected section
generates the extension after clearing its denominator.
\end{lemma}
\begin{proof}
For a rational number \(b\), take \(N\) sufficiently divisible to
clear all exponents being used and to make the original local
monodromy unipotent. On the model of
Lemma~\ref{orders:diskmodels}, test \(u^{-Nb}\tau\), with \(\tau\)
now pulled back as a relative form. The total form being tested is
\begin{equation}
(u^{-Nb}\tau)\wedge du
=\frac{u}{N}\,q^*(t^{-1-b}\Omega),
\label{orders:changevariables}
\end{equation}
where \(q\) maps the resolved composite cover to the downstairs
model. This identity is simply \(dt=Nu^{N-1}du\), and includes
the base ramification Jacobian.
Set
\[
\alpha_E=l_E+1-a_E(1+b).
\]
Suppose first that every \(\alpha_E\) is nonnegative.
Along horizontal primes the divided order is at least \(-1\),
because the good-fiber coefficients lie in \([0,1]\).
Thus \(t^{-1-b}\Omega\) has logarithmic bound on the entire
downstairs SNC chart. More explicitly, in a logarithmic
differential basis its coefficient is a unit times a product of
nonnegative rational powers of boundary coordinates. On a cover
making the form single-valued those factors are bounded. Pullback
of the logarithmic basis has at most a simple pole along each
prime, by the wedge argument in
Lemma~\ref{orders:threshold}. Hence its pullback has at most
logarithmic poles.
The factor \(u\) in \eqref{orders:changevariables} removes every
vertical logarithmic pole, since \(u\) has positive integral order
on each central-fiber prime. Along horizontal primes, the tested
relative form has the required logarithmic behavior by
Lemma~\ref{hodge:birational}; the base factor is a unit on the
punctured disk. It follows that the tested form belongs to
\[
(g_N)_*\omega_{Z_N/\Delta_u}(H_N).
\]
By \eqref{hodge:FF} and Lemma~\ref{orders:diskmodels}, it extends
in the original canonical highest line.
Conversely, suppose \(\alpha_E<0\) for one downstairs prime \(E\).
Choose a prime above it that dominates it, and let \(e\) be the
ramification index. Shifted orders of a pulled-back total form
multiply by \(e\). Also
\(\ord(u)=ea_E/N\) there, since \(t=u^N\).
The shifted order of \eqref{orders:changevariables} is therefore
\begin{equation}
e\alpha_E+\frac{ea_E}{N}.
\label{orders:failure}
\end{equation}
This is \(e(\alpha_E+a_E/N)\), which is negative for every
\(N>a_E/(-\alpha_E)\), regardless of the dependence of \(e>0\)
on \(N\). The
tested form then has a vertical pole and fails the extension test.
Such a prime survives on a common resolution, so an unfavorable
valuation cannot be lost by choosing a different resolved model.
For completeness, these two tests give an exact order, not just
one inequality. First choose an integer \(b\) sufficiently negative
that all \(\alpha_E\) are nonnegative, and then choose a fixed
unipotent power \(N_0\) divisible enough for that first test.
That test shows that a power of \(u\) times the pulled-back \(\tau\) is
holomorphic. Thus \(\tau\) is meromorphic in the canonical line.
Let \(v\) be its ordinary order after that substitution, and put
\(\beta=v/N_0\). Under a further power \(N=N_0a\) the order is
\(av\), by Lemma~\ref{hodge:purerules}. For \(Nb\in\Z\), the
extension test is therefore equivalent to \(b\leq\beta\).
The first part proves extension for
\[
b\leq b_0:=\min_E\frac{l_E+1-a_E}{a_E},
\]
whereas \eqref{orders:failure} proves failure for every rational
\(b>b_0\) after a sufficiently large further power. Consequently
\(\beta=b_0\), which proves \eqref{orders:generalorder}. After
clearing its denominator, the section corrected at \(b=b_0\)
has order zero and hence generates the extended line.
\end{proof}
\subsection{The actual moduli extension}
\label{orders:actualextension}
\begin{theorem}[The threshold trace is the volume extension]
\label{orders:trace}
In the setup of \eqref{orders:adjoint}, choose locally a frame
\(s\) of \(\cO_S(mQ_S)\), and let \(\tau\) be its tautological
root as in Section~\ref{hodge:rootconstruction}. For every prime
\(P\subset S\),
\begin{equation}
\ord_P^{\mathrm{Hdg}}(\tau)=t_P-1.
\label{orders:traceorder}
\end{equation}
More precisely, let \(\pi:\widetilde V\to V\) be a coordinate
power chart making the local pure variation unipotent, and let
\(\overline{\mathcal E}_\chi\) be its extended highest eigenline.
For a sufficiently divisible positive integer \(a\), the open
identification \(s^{\otimes a}\leftrightarrow
\tau^{\otimes am}\) extends to an isomorphism of actual bundles
\begin{equation}
\pi^*\cO_S(amM_S)|_{\widetilde V}
\simeq \overline{\mathcal E}_\chi^{\otimes am}.
\label{orders:tracebundle}
\end{equation}
One integer \(a\) suffices on the compact base. The resulting
extension rule commutes with pullback to disks generically
meeting \(U\), after further power substitution on the disk.
All assertions allow arbitrary rational coefficients on the
vertical components of \(D_W\).
\end{theorem}
\begin{proof}
For the chosen frame \(s\), the divided order on every crepant
source resolution is \(l_E=-\delta_E\). Lemmas~\ref{orders:threshold}
and~\ref{orders:order} give
\[
\ord_P^{\mathrm{Hdg}}(\tau)
=\min_{E\mapsto P}\frac{1-\delta_E}{a_E}-1=t_P-1.
\]
Over a prime meeting \(U\), the family is smooth and relatively
SNC, so its threshold is one and the same equality is immediate.
The discriminant is therefore supported on the finite SNC bad
divisor and has rational coefficients.
We spell out the extension at intersections of its components.
Choose coordinates \(t_1,\ldots,t_b\) on \(V\) with bad divisor
\(t_1\cdots t_r=0\), and write
\[
B_S|_V=\sum_{i=1}^r b_i(t_i=0).
\]
Coefficients \(b_i\) may be negative or zero. Take the coordinate
power chart \(t_i=w_i^{N_i}\) with \(N_i\) divisible enough to
make monodromy unipotent and \(N_i b_i\) integral. On its good
open consider
\begin{equation}
\gamma=
\left(\prod_{i=1}^r w_i^{N_i b_i}\right)\pi^*\tau.
\label{orders:correctedroot}
\end{equation}
The just-proved disk calculation says that this section has order
zero in \(\overline{\mathcal E}_\chi\) at a general point of every
coordinate divisor. In fact, on general parallel transverse
disks it extends as a nonvanishing section.
This statement suffices without an a priori multivariable
meromorphicity assertion. In a local frame of
\(\overline{\mathcal E}_\chi\), let \(g\) be the coefficient of
\(\gamma\) on the good open. Near a general point of one boundary
component, expand \(g\) as a Laurent series in its normal
coordinate. The coefficients are holomorphic in the tangential
parameters. Every negative coefficient vanishes for the dense
set of centers where the disk test applies, hence vanishes
identically. The same argument applies to \(g^{-1}\), since
the corrected disk section generates. Thus \(\gamma\) and its
inverse extend across the divisors away from their intersections.
Hartogs extension then extends both across the remaining
codimension-two set. Their product is still one, so
\(\gamma\) is a frame everywhere.
Choose \(a\) so that \(amM_S\) is an actual integral power and
all \(am b_i\) are integers. A frame of that power on \(V\) is
\[
s^{\otimes a}\prod_{i=1}^r t_i^{am b_i}.
\]
Its pullback corresponds exactly to \(\gamma^{\otimes am}\).
This proves \eqref{orders:tracebundle}. Notice the sign:
as a meromorphic section of \(mM_S=mQ_S-mB_S\), the original
frame \(s\) has divided order \(-b_i=t_{P_i}-1\).
Changing \(s\) by a unit \(v\) multiplies the local root by a chosen
\(m\)-th root of \(v\) and gives a locally isomorphic root
family. Choose these roots on a refinement of each overlap by
small coordinate neighborhoods. They are holomorphic units, and
two choices differ by a locally constant element of \(\mu_m\).
On the \(am\)-th tensor power the change is exactly \(v^a\),
independent of that choice; the possible root-of-unity factors on
triple overlaps also become one. Thus no root on an entire
overlap is required. The identifications match the specified
volume identification and its actual bundle cocycle.
Compactness permits finitely many
charts, so one can choose a common positive \(a\).
Finally, a disk map generically meeting \(U\) lifts to each
needed power chart after a further power on the disk: its
boundary coordinates are powers of the parameter times units,
and the units have roots on a small disk. On that lifted disk,
\eqref{orders:tracebundle} and
Lemma~\ref{hodge:purerules} give the canonical pullback rule.
Nothing in the proof imposed a restriction on vertical
coefficients; only the horizontal good-fiber bounds were used.
\end{proof}
\begin{corollary}[Descent on every higher model]
\label{orders:bdescent}
For the original fibration and the smooth compact K\"ahler model
\(S\) fixed above, every smooth compact K\"ahler modification
\(\nu:S_1\to S\) satisfies
\[
M_{S_1}=\nu^*M_S
\quad\text{in }\Pic(S_1)\otimes\Q.
\]
No assumption that \(\nu\) be an isomorphism over \(U\) is needed.
\end{corollary}
\begin{proof}
Resolve the main-component family over \(S_1\), using its crepant
divisor \(D_1\), and put
\[
L_{S_1}=\nu^*L_S,\qquad
Q_{S_1}=L_{S_1}-K_{S_1}.
\]
The original bundle identification gives
\[
K_{W_1/S_1}+D_1
\sim_{\Q}p_1^*Q_{S_1}
\]
with its specified degree-\(m\) identification.
Fix a prime \(P\subset S_1\). Choose a local frame \(s\) of
\(mQ_S\) near a general point of its image. On the common
isomorphism open it is a section of the same relative volume
line. Regard it as a meromorphic section of \(mQ_{S_1}\), and
write \(c_P\) for its divided order there. This interpretation
uses
\begin{equation}
Q_{S_1}=\nu^*Q_S-K_{S_1/S}.
\label{orders:basejacobian}
\end{equation}
In particular, a pulled-back local frame has order
\(c_P=-\ord_P K_{S_1/S}\); the base Jacobian is divided out.
At a general point of \(P\), take a transverse disk whose
punctured part lies in the common good isomorphism open.
After resolving the source jointly with the disk divisor,
the divided order of the relative volume along a component
\(E\) dominating \(P\) is
\begin{equation}
l_E=-\delta_E+a_Ec_P,
\label{orders:higherorders}
\end{equation}
where \(\delta_E\) is the crepant coefficient and
\(a_E=\ord_E(p_1^*P)\). This is the divisor identity obtained
from the specified relative adjoint isomorphism. Adjunction of
the other base coordinates introduces no further order.
Use the pulled-back original local root family, and then the
common resolved disk models of
Lemma~\ref{orders:diskmodels}. Their punctured fibers differ
from that family only by the modifications covered by
Lemma~\ref{hodge:birational}. Thus
Lemma~\ref{orders:order} computes the order of the original
pulled-back Hodge line and gives
\[
\min_{E\mapsto P}\frac{l_E+1-a_E}{a_E}
=c_P+t_{P,S_1}-1.
\]
The right side is precisely the divided order of \(s\) in
\(M_{S_1}=Q_{S_1}-B_{S_1}\).
By Theorem~\ref{orders:trace} and its disk pullback rule, the
same Hodge order is the divided order in \(\nu^*M_S\).
The specified open identification therefore has neither a zero
nor a pole at \(P\).
This argument applies to every prime, including exceptional
primes with image of codimension at least two in the bad
divisor, and primes whose centers lie inside \(U\).
For the latter case one can also check the orders directly:
before further source resolution, smooth base change replaces
the boundary by its pullback minus \(p_1^*K_{S_1/S}\).
If \(j_P=\ord_P K_{S_1/S}\), the threshold is \(1+j_P\);
together with \(c_P=-j_P\) this gives order zero.
To conclude globally in the holomorphic Picard group, the
difference of the two rational lines has the canonical
divisorial description
\[
M_{S_1}-\nu^*M_S
=-K_{S_1/S}-B_{S_1}+\nu^*B_S.
\]
The Jacobian supplies the relative canonical divisor even
though the individual canonical bundles need not have global
meromorphic sections. The order comparison shows that this
rational divisor is zero. Equivalently, after clearing
denominators the fixed open isomorphism extends across every
divisor and then across codimension two. This proves the
asserted equality of actual rational line bundles.
\end{proof}
\subsection{The residue integral as an extension norm}
\label{orders:residuenormsection}
We finish with the form of the extension rule needed for the
auxiliary product families. Let \(r:P\to S\) be a holomorphic
map from a smooth compact complex manifold. On a dense open
\(P^\circ\) mapping into \(U\), suppose a connected fiber
component \(A_x^*\) of the deepest stratum has been selected
in family. Write \(s_A\) for the adjunction plurivolume
induced by \(s\in r^*\mathcal G_m\), and define
\begin{equation}
\|s\|_{\mathrm{res}}
=\left(\int_{A_x^*}|s_A|^{2/m}\right)^{m/2}.
\label{orders:residuenormformula}
\end{equation}
For \(k=0\) use the whole connected original fiber, and for a
zero-dimensional selected fiber interpret the integral as
evaluation at its point.
\begin{corollary}
\label{orders:residuenorm}
The expression \eqref{orders:residuenormformula} is a positive
finite norm, homogeneous of degree one, on the volume line.
Let \(a\) clear the indicated powers. Along any disk in \(P\)
whose punctured part lies in \(P^\circ\), after a sufficiently
divisible power substitution, a nonvanishing local frame of
\[
r^*\cO_S(amM_S)
\]
and its dual have at most powers-of-log growth in the \(a\)-th
tensor power of this norm and its dual norm. No K\"ahler
assumption on \(P\) is required.
\end{corollary}
\begin{proof}
The selected fiber has effective SNC coefficients less than
one. In local coordinates its measure has factors
\(|z_i|^{-2\delta_i}\), which are integrable for
\(\delta_i<1\). The integral is therefore finite and positive
for a nonzero volume. Multiplication by a scalar \(c\)
multiplies it before the outer exponent by \(|c|^{2/m}\),
so the norm has degree one.
Pull the local root family to the punctured disk and retain
all sheets above the selected connected component. Its
resolved residue cover has a fixed finite degree \(e\).
Proposition~\ref{hodge:residue} gives the specified top-line
identification and
\[
\int_{\widetilde A_x^*}|\operatorname{Res}\tau|^2
=e\int_{A_x^*}|s_A|^{2/m},
\]
up to fixed normalization constants. Exceptional subsets do
not affect either integral. On a compact K\"ahler
\((d-k)\)-fold a holomorphic top form is primitive, and its
Hodge norm is this volume integral, again up to a fixed
constant. Hence \eqref{orders:residuenormformula} is the
\(m\)-th power of the residue Hodge norm, multiplied by
\(e^{-m/2}\).
Theorem~\ref{orders:trace} identifies a pulled-back frame of
the asserted power of \(M_S\) with a canonical highest-line
frame after power substitution. The pure residue map
preserves such frames by Proposition~\ref{hodge:residue}.
The two-sided bounds in Lemma~\ref{hodge:purerules}, applied
to the compact residue cohomology and then raised to the
required power, give the conclusion. These are pullbacks
of polarized variations and comparisons of their specified
lines; the smooth compact parameter space itself need not
be K\"ahler.
\end{proof}
==== END B-semiampleness-for-compact-log-smooth-Kahler-fibrations//orders.tex ====
==== BEGIN B-semiampleness-for-compact-log-smooth-Kahler-fibrations//paper.tex ====
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\hypersetup{pdftitle={B-semiampleness for compact log-smooth K\"ahler fibrations},pdfsubject={The moduli rational line bundle of an lc-trivial fibration},pdfauthor={OpenAI}}
\title{B-semiampleness for compact log-smooth\\K\"ahler fibrations}
\author{OpenAI}
\date{September 10, 2026}
\begin{document}
\maketitle
\begin{abstract}
We prove the compact log-smooth K\"ahler case of b-semiampleness. Let
$f:Y\to X$ be a surjective holomorphic map with connected fibers between
smooth compact connected K\"ahler manifolds, and let $\Delta$ be an effective
rational divisor with simple normal crossing support and coefficients in
$[0,1]$, with $K_Y+\Delta\sim_{\Q}f^*L$ for $L\in\Pic(X)_{\Q}$. There is a
smooth compact K\"ahler modification $S\to X$ for which the threshold-moduli
line satisfies $M_{S_1}=\nu^*M_S$ in $\Pic(S_1)_{\Q}$ for every smooth compact
K\"ahler modification $\nu:S_1\to S$, and some positive multiple of $M_S$ is
represented by a holomorphic line bundle generated by global sections.
Horizontal components of coefficient one are allowed; neither projectivity
nor a Campana orbifold Iitaka hypothesis is assumed.
\end{abstract}
\setcounter{tocdepth}{1}
\tableofcontents
\input{sections/introduction}
\input{sections/preliminaries}
\input{sections/hodge}
\input{sections/orders}
\input{sections/spreading}
\input{sections/comparison}
\input{sections/periods}
\input{sections/descent}
\input{sections/conclusion}
\begingroup
\raggedright
\bibliographystyle{amsplain}
\bibliography{references}
\endgroup
\end{document}
==== END B-semiampleness-for-compact-log-smooth-Kahler-fibrations//paper.tex ====
==== BEGIN B-semiampleness-for-compact-log-smooth-Kahler-fibrations//periods.tex ====
\section{Semiample extensions of the torus and symplectic factors}
\label{periods:section}
We now treat the nonprojective factors in Proposition~\ref{spread:product}.
Their first or second cohomology has a classical period domain. Two points
require care: the available K\"ahler classes are real, and the compact auxiliary
base need not be algebraic. We first address the polarization and extension
questions independently of the factor construction.
\subsection{Real polarizations and the integral local system}
An integral variation in this section consists of a local system
$\mathbb V_{\Z}$ of finite free abelian groups and a holomorphic Hodge filtration
on $E=\mathbb V_{\Z}\otimes_{\Z}\cO$, satisfying opposedness and Griffiths
transversality. A real polarization is a flat nondegenerate real bilinear
form on $\mathbb V_{\R}$ satisfying the Hodge--Riemann relations. We use the
sign convention in which, on type $(p,q)$ of weight $w$, the associated
positive Hermitian form is
\begin{equation}\label{periods:sign}
(-1)^{w(w-1)/2}(\sqrt{-1})^{p-q}B(v,\overline v).
\end{equation}
Changing this convention by the usual weight-dependent sign changes none of
the arguments.
\begin{lemma}[Changing the polarization]\label{periods:rationalization}
Let $(\mathbb V_{\Z},F^\bullet)$ be an integral pure variation with a flat real
polarization $B$. There is a flat real automorphism $C$ of $\mathbb V_{\R}$
such that $(\mathbb V_{\Z},C^{-1}F^\bullet)$ is polarized by a rational form
$B'$, and
\[
B'(x,y)=B(Cx,Cy).
\]
Consequently the new and original variations are isomorphic as real
polarized variations, and their holomorphic Hodge bundles are isomorphic.
Their canonical Hodge extensions are also identified wherever local
monodromy is unipotent. After multiplying $B'$ by a positive integer, its
values on $\mathbb V_{\Z}$ are integral.
\end{lemma}
\begin{proof}
Work on a connected base and fix a lattice fiber $\Lambda$ and its monodromy
representation. In a basis of $\Lambda$, the invariant bilinear forms of the
required symmetry satisfy rational linear equations
\[
g^{\mathsf t}Bg=B\quad(g\text{ in the monodromy group}),
\qquad B^{\mathsf t}=(-1)^wB.
\]
Although the first collection may be infinite, the rows of these equations
span a finite-dimensional rational vector space. Finitely many suffice.
Thus rational invariant forms are dense in the space of real invariant
forms. Choose such a form $B'$ sufficiently close to $B$.
Put $J=B^{-1}B'$. Both invariance identities imply that $J$ commutes with
monodromy, and the symmetry identities give
$J^{\mathsf t}B=BJ$. If $B'$ is sufficiently close, the power series for the
square root at the identity defines $C=J^{1/2}$. This is a real invertible
matrix, commutes with monodromy, and satisfies $C^{\mathsf t}B=BC$.
Therefore
\[
C^{\mathsf t}BC=BC^2=BJ=B'.
\]
In particular $C$ defines a global flat real automorphism. The filtration
$F'^p=C^{-1}F^p$ is holomorphic and transverse because $C$ is flat; opposedness
holds because $C$ is real. The displayed identity transfers the
Hodge--Riemann relations exactly, at every base point. Hence $B'$ polarizes
$F'^\bullet$.
The lattice and its monodromy have not changed. We do not require $C$ to
preserve the lattice: it supplies an isomorphism of real variations and of
their holomorphic filtered bundles. On a unipotent normal-crossing chart,
$C$ commutes with every monodromy logarithm. In the canonical logarithmic
frames it is therefore the same invertible constant matrix. It identifies
the flat extensions and the extended Hodge filtrations. Finally, clear the
denominators of $B'$ on a lattice fiber; monodromy invariance clears them
everywhere. If this multiplies $B'$ by $a>0$, replace $C$ by $\sqrt a\,C$;
the transported filtration is unchanged and the polarization identity
remains exact.
\end{proof}
Thus Lemma~\ref{periods:rationalization} does not assert that a nonprojective
fiber has a rational polarization of its original Hodge structure. It
constructs a different Hodge filtration on the same integral local system,
with isomorphic holomorphic Hodge bundles. This is precisely what is needed
to use arithmetic period spaces for these bundles.
\begin{lemma}[Polarizations from a fixed K\"ahler class]
\label{periods:lefschetz}
Let $t:\mathcal T\to B^\circ$ be a smooth proper family of connected compact
complex manifolds of dimension $n$ over a complex manifold $B^\circ$,
and suppose a closed real $(1,1)$-form
on $\mathcal T$ restricts to a K\"ahler form on every fiber. Its restriction
class $\kappa$ is flat. The cohomological variations, with their integral
lattices modulo torsion, are real-polarizable. In particular:
\begin{enumerate}[label=\textup{(\roman*)}]
\item on $H^1$, a polarization is
\[
B_1(\alpha,\beta)=\int_{\mathcal T_b}
\alpha\wedge\beta\wedge\kappa^{n-1};
\]
\item if $n\geq2$, write
$H^2_{\R}=P^2_{\R}\oplus\R\kappa$, where
$P^2_{\R}=\Ker(\kappa^{n-1}\cup-)$.
For $\alpha=\alpha_0+a\kappa$ and $\beta=\beta_0+b\kappa$, a full
weight-two polarization is
\begin{equation}\label{periods:full-h2}
B_2(\alpha,\beta)=
\int_{\mathcal T_b}\alpha_0\wedge\beta_0\wedge\kappa^{n-2}
-ab\int_{\mathcal T_b}\kappa^n.
\end{equation}
If $h^{2,0}=1$, this real form has signature $(2,h^{1,1})$.
\end{enumerate}
\end{lemma}
\begin{proof}
Restriction of a closed total-space class is invariant under parallel
transport in a smooth proper family. For example, a differentiable
trivialization over a path and Stokes' theorem show that its integrals on
transported cycles are constant. Thus $\kappa$, its cup powers, and the
fiber integrals in the statement are flat. Locally on the base, properness
allows us to add a sufficiently large K\"ahler form pulled back from a
coordinate ball and obtain a K\"ahler form on the total space. The smooth
proper K\"ahler Hodge
theorem gives the Hodge filtrations and transversality; one may use the
smooth, boundary-free case of \cite[Theorem~1.1]{FF25}. The K\"ahler
Lefschetz decomposition and Hodge--Riemann relations then polarize its
primitive summands. Transporting their forms by Lefschetz and inserting
the signs for the resulting Tate twists polarizes full cohomology.
For clarity, $H^1$ is primitive and its Hodge--Riemann form is $B_1$.
In degree two the summands $P^2_{\R}$ and $\R\kappa$ are orthogonal for the
intersection form with $\kappa^{n-2}$. This form polarizes the primitive
summand with convention~\eqref{periods:sign}. The positive K\"ahler line
must have its sign reversed, exactly as in~\eqref{periods:full-h2}.
The resulting form is positive on the real plane underlying $H^{2,0}$
and negative on real $H^{1,1}$ when $h^{2,0}=1$. This proves the asserted
signature. The primitive projectors and these forms are flat, so all
constructions apply to variations.
\end{proof}
For the families in Proposition~\ref{spread:product}, the form in this
Lemma is obtained by pulling back a fixed K\"ahler form from the compact
ambient manifold $V$. Locally over a small polydisk in the auxiliary base,
the family embeds in the product of that polydisk with $V$, so its total
space is K\"ahler there. No K\"ahler metric on the entire auxiliary base
is required.
\subsection{Canonical extensions and volume norms}
For unipotent monodromy along a simple normal crossing divisor, we write
$\overline E$ for the Deligne extension with nilpotent residues, and
$\overline F^p$ for its Schmid Hodge subbundles. Their existence in the
analytic setting follows from \cite[Theorem~4.12]{Schmid73}; this theorem
concerns a product of punctured disks and disks. By
Lemma~\ref{periods:rationalization}, its usual polarized integral
formulation applies equally to the real-polarized variations used here.
\begin{lemma}[Functoriality and logarithmic bounds]
\label{periods:extensions}
Let a pure real-polarizable integral variation be defined on the complement
of an SNC divisor in a complex manifold, with unipotent local monodromy.
Canonical Hodge extensions commute with tensor products, exterior powers,
duals, and holomorphic pullbacks from smooth manifolds with SNC boundary,
whose boundary complements map into the original complement. This
includes disks whose punctured disks map into that complement.
A morphism of pure real variations induces an isomorphism
between its coimage and image on these extensions. In particular, an
isomorphism of highest Hodge lines induced by such a morphism remains an
isomorphism after extension.
If $\mathcal H$ is a highest Hodge line, $\gamma:\Delta\to B$ is a disk
with $\gamma(\Delta^*)$ in the open set, and $e$ is a nowhere-zero local
section of $\gamma^*\overline{\mathcal H}$, there are constants $c>0$ and
$a\geq0$ such that, for $0<|z|$ sufficiently small,
\begin{equation}\label{periods:logbounds}
c^{-1}(1+|\log|z||)^{-a}
\ \leq\ \|e(z)\|\ \leq\
c(1+|\log|z||)^a.
\end{equation}
The same assertion holds for its powers and their duals.
\end{lemma}
\begin{proof}
On a normal-crossing chart let $N_j$ be the commuting logarithms of the
local monodromies. Applying the exponential of
$-\sum_j(\log z_j)N_j/(2\pi\sqrt{-1})$ to multivalued flat frames gives
canonical logarithmic frames, with a compatible choice of monodromy
convention. Tensor and dual constructions have the corresponding sums
and duals of the nilpotent residues, so their flat extensions agree.
Their extended Hodge filtrations agree as well: the induced filtrations
are holomorphic subbundles, and coincide on the dense open with the given
ones.
After a holomorphic pullback the residue along a prime divisor is a sum
$\sum_j a_jN_j$, where the $a_j$ are the orders of the pulled-back boundary
coordinates. It is nilpotent because the $N_j$ commute. Thus the pulled-back
flat extension is again canonical; pulling back the Hodge subbundles gives
their canonical extension by the same uniqueness. Locally on a disk,
write the coordinates as $z^{a_j}u_j(z)$ with nonvanishing holomorphic
units $u_j$; choosing logarithms of the units gives the explicit
identification. This also explains the usual pullback compatibility
\cite[Lemma~5.6]{BFMT25} in the analytic situation needed here.
For a morphism of pure variations, Lemma~\ref{hodge:purerules} supplies
flat Hodge splittings of its kernel and image. In logarithmic frames
these splitting projectors extend as the same constant matrices, so the
splittings persist in the extended filtrations. This proves the
coimage-to-image assertion and its consequence for highest lines.
It remains to check the norm statement. Pull back to the disk, using the
compatibility just proved. Schmid's abstract norm estimate
\cite[Theorem~6.6$'$]{Schmid73}, applied to a flat basis and to the dual
variation, bounds both the Hodge metric and its inverse by powers of
$1+|\log|z||$ on sectors. The matrices relating a flat basis to a
canonical logarithmic basis are polynomials in $\log z$; hence the same
kind of estimates holds in canonical frames. Finitely many sectors
cover a punctured disk. The coefficients of $e$ in a holomorphic frame
of $\overline E$ are bounded. Since its highest line is a subbundle and
$e$ does not vanish, a local holomorphic section $\xi$ of
$\overline E^{\vee}$ can be chosen with $\xi(e)=1$. Its coefficients are
also bounded. The metric estimates and
$1\leq\|\xi\|\,\|e\|$ give both inequalities in
\eqref{periods:logbounds}. Powers and duals follow immediately.
\end{proof}
In degree $n$ on an $n$-dimensional compact K\"ahler manifold, a
holomorphic $n$-form is primitive. Its Hodge norm is, with a fixed
normalization,
\begin{equation}\label{periods:volume}
\|\eta\|_{\mathrm{vol}}^2
=(\sqrt{-1})^{n^2}\int_{\mathcal T_b}\eta\wedge\overline\eta.
\end{equation}
Thus Lemma~\ref{periods:extensions} gives bounds for the actual volume norms
appearing in the product formula, as well as for abstract Hodge norms.
\subsection{Arithmetic period maps on an analytic base}
For an effective pure variation of weight $w$, we use
the Griffiths line
\[
\lambda=\bigotimes_{p=1}^{w}\det F^pE.
\]
Including the full step $F^0E=E$ only adds a full-local-system determinant;
we will account for this harmless finite-order factor explicitly.
We use the following precise consequence of
\cite[Theorems~5.2 and~5.5]{BFMT25}. If $Z$ is an algebraic variety with
a quasifinite horizontal period map for a polarized integral variation,
then it has a projective period compactification $Z^{\mathrm{BB}}$.
For a sufficiently divisible positive integer $a$ an ample line bundle
$\mathcal A$ on this compactification restricts to $\lambda_Z^a$.
Every analytic map $(\Delta^*)^b\to Z^{\mathrm{an}}$ whose period map is
locally liftable extends to $\Delta^b\to Z^{\mathrm{BB},\mathrm{an}}$;
when the induced local monodromies are unipotent, the pullback of
$\mathcal A$ is canonically the Schmid extension of the corresponding
$a$-th Griffiths power. The last assertion is part of Theorem~5.5, not an
algebraicity assumption on the source.
\begin{lemma}[The Griffiths line on a compact analytic base]
\label{periods:griffiths}
Let $B$ be a smooth compact complex manifold, let $D$ be an SNC divisor,
and let $\mathbb V$ be a real-polarizable integral variation on
$B^\circ=B\setminus D$. Suppose its Hodge numbers are either
\[
w=1,\quad h^{1,0}=h^{0,1}=g,
\qquad\text{or}\qquad
w=2,\quad h^{2,0}=h^{0,2}=1.
\]
After a finite cover of $B^\circ$, extended and resolved over $B$, the
canonical extension of its Griffiths line is semiample. The resulting
base is smooth and compact and its boundary can be taken SNC.
\end{lemma}
\begin{proof}
Apply Lemma~\ref{periods:rationalization} and scale the rational
polarization to be integral. In weight one its domain $\mathcal D$ is
Siegel space. In weight two, use the sign convention giving signature
$(2,h^{1,1})$. The highest filtration step is a line $\ell$ satisfying
$B'(\ell,\ell)=0$ and $B'(v,\overline v)>0$ for $v\ne0$ in $\ell$;
the middle step is $\ell^\perp$. A connected component of this domain
is a type-IV domain. A zero-dimensional domain causes no difficulty and
can be treated as a point.
Let $\Gamma$ be the integral isometry group preserving a component, after
passing to the corresponding finite cover if necessary. It is an
arithmetic group and contains the monodromy. Choose a finite-index neat
subgroup $\Gamma'$, as in the level construction of
\cite[\S5.2]{BFMT25}. Here neat means that the multiplicative group
generated by the eigenvalues of each element has no nontrivial torsion.
The inverse image of $\Gamma'$ under monodromy gives a finite \emph{\'etale}
cover of $B^\circ$. This cover has a finite normal extension over $B$
without any K\"ahler assumption on $B$. Indeed, on an SNC chart
$\Delta^{*k}\times\Delta^{b-k}$ its permutation monodromy is finite.
A sufficiently divisible coordinate power map kills that monodromy.
The cover is then a quotient of finitely many copies of this power
cover, with its finite deck group also permuting the copies. Extend
each copy to the full polydisk and take the same finite-group quotient.
This is a finite normal analytic extension. Such extensions are unique,
being the normalization in the finite meromorphic algebra defined on the
open, so they glue. The resulting normal space is compact because it
is finite over $B$. Resolve it with SNC boundary, leaving the smooth
covering open unchanged. This is the finite-extension construction also
used in the proof of \cite[Lemma~2.21]{Villadsen24}; the additional
K\"ahler conclusion of that Lemma is not needed here.
Every boundary monodromy on this resolution lies in $\Gamma'$.
By the monodromy theorem, in its analytic formulation
\cite[Lemma~4.5]{Schmid73}, all its eigenvalues are roots of unity.
Neatness therefore makes them all equal to one. Hence all local
monodromies are unipotent, including those about exceptional boundary
divisors. We keep the notation $B$ for the resulting base.
The quotient $Z=\Gamma'\backslash\mathcal D$ is a smooth algebraic
quasiprojective variety by Baily--Borel
\cite[\S10, especially Theorem~10.11]{BailyBorel66}. The theorem applies
to the effective adjoint group of the classical domain, and the finite
central quotient does not change this conclusion. We keep $\Gamma'$
inside the original integral isometry group: neatness kills the finite
kernel of its action and the finite point stabilizers. Thus the local
system $(\mathcal D\times V_{\Z})/\Gamma'$, with $V_{\Z}$ the marked
lattice, and the tautological
filtration give an integral polarized variation on $Z$, retaining the
original representation. In the zero-dimensional weight-two case,
the isometry group of the positive-definite integral lattice is finite,
so its neat subgroup is trivial and this variation is constant on a
point. Its period map to this
arithmetic quotient is the identity, so it is quasifinite. It is
horizontal: for weight one transversality is automatic, and in weight
two differentiation of $B'(v,v)=0$ gives $dv\in\ell^\perp$.
Thus $Z$ satisfies the Griffiths-line hypotheses of
\cite[Theorems~5.2 and~5.5]{BFMT25}.
The conjugated variation on $B^\circ$ is the pullback of this tautological
variation by its analytic period map $\phi$. This assertion concerns
the local system as well as the filtration: the marked period map is
equivariant for its monodromy representation, and at neat level the
quotient has no stabilizers. In particular $\phi$ is locally liftable.
The quoted analytic extension theorem extends it to
\[
\overline\phi:B\longrightarrow Z^{\mathrm{BB},\mathrm{an}},
\qquad
\overline\phi^*\mathcal A\simeq\overline\lambda^{\,a}.
\]
To apply the theorem on a chart $\Delta^{*k}\times\Delta^{b-k}$,
restrict first to $(\Delta^*)^b$. Its extension agrees with the original
map on $\Delta^{*k}\times\Delta^{b-k}$ by uniqueness. The unused
coordinates have trivial monodromy, so the same statement holds for the
canonical bundle identification. Both maps and identifications glue
uniquely on overlapping charts.
A sufficiently high power of $\mathcal A$ is generated by global
sections on the projective variety $Z^{\mathrm{BB}}$. Pulling these
sections back gives global generation of a power of $\overline\lambda$
on the compact analytic base. Finally, the real isomorphism in
Lemma~\ref{periods:rationalization} identifies this extension with the
one for the original variation.
\end{proof}
\subsection{From Griffiths determinants to the required highest lines}
We record the determinant calculation as an identity of holomorphic
bundles. For any integral local system, $\det E$ has monodromy in
$\{1,-1\}$, since its monodromy matrices are invertible over $\Z$.
If the boundary monodromies are unipotent, its canonical extension has
trivial local monodromy and extends as a flat line across the boundary.
Its square is therefore holomorphically trivial on the compactification.
At neat level the determinant itself is trivial. These statements concern
the full determinant, and do not assert finite monodromy of individual
Hodge pieces.
In weight one, the Griffiths line is $\det F^1E$ up to this full
determinant. In weight two put $\ell=F^2E$. The polarization identifies
$E/F^1E$ with $\ell^\vee$, giving
\begin{equation}\label{periods:determinants}
\det F^1E\simeq\det E\otimes\ell,
\qquad
\lambda\simeq\det E\otimes\ell^{\otimes2}.
\end{equation}
The possible additional $F^0$ factor changes only the power of $\det E$.
These identities hold on canonical extensions. Indeed, the flat
polarization extends as a nondegenerate bilinear form in logarithmic
frames, and $\overline F^1=(\overline F^2)^\perp$ by continuity of the
subbundles. Taking determinants then gives the extended identities.
It follows that semiampleness of $\overline\lambda$ implies semiampleness
of $\overline\ell$: kill the determinant torsion and use a further even
power. Thus the square in~\eqref{periods:determinants} is retained.
\begin{proposition}[Semiample period-factor extensions]
\label{periods:semiample}
Let $r:P\to S$ and the smooth families $t_i:T_i\to P^\circ$ be as in
Proposition~\ref{spread:product}, with $n_i=\dim T_{i,u}>0$, and put
\[
\mathcal H_i=F^{n_i}\bigl(R^{n_i}(t_i)_*\C\otimes\cO_{P^\circ}\bigr).
\]
Thus $\mathcal H_i$ is the highest line denoted $\lambda_i$ in
Proposition~\ref{spread:product}; the symbol $\lambda$ above denotes
the Griffiths line.
After a proper generically finite surjection $\pi:P'\to P$, obtained by
extending a finite \emph{\'etale} cover of the good open and resolving,
there are semiample holomorphic line bundles $\overline{\mathcal H}_i$
on the smooth compact manifold $P'$ whose specified restrictions are
$\pi^*\mathcal H_i$.
The boundary may be taken SNC. Along any disk in $P'$ whose punctured
disk lies in the good open, a generator of
$\overline{\mathcal H}_i$ and its inverse have at most powers-of-logarithm
growth in the volume norm~\eqref{periods:volume} and its dual. The same
is true for all positive powers, including the powers used in the
product-volume identification of Proposition~\ref{spread:product}.
\end{proposition}
\begin{proof}
For a torus-type factor take
$\mathbb V_i=R^1(t_i)_*\Z/\mathrm{tors}$, and for a symplectic-type
factor take $\mathbb V_i=R^2(t_i)_*\Z/\mathrm{tors}$.
The cohomological conditions in Proposition~\ref{spread:product} give
the Hodge numbers in Lemma~\ref{periods:griffiths}. The fixed ambient
K\"ahler class supplies their flat real polarizations by
Lemma~\ref{periods:lefschetz}; in degree two we use the full
polarization~\eqref{periods:full-h2}.
Apply the finite-level construction simultaneously to these finitely many
variations and to $R^{n_i}(t_i)_*\Z/\mathrm{tors}$ for every $i$.
The latter variations also have flat real polarizations and can be
rationalized by Lemma~\ref{periods:rationalization} for this purpose.
Intersect the resulting finite-index subgroups in the base fundamental
group. Extending this one finite cover and resolving gives $P'$ with
unipotent local monodromies for all these variations. We may first
resolve the complement of $P^\circ$ and shrink the good open, so the
final complement is SNC. Subsequent cup products are taken in the
original real variations, whose monodromies are unchanged by
rationalization.
Lemma~\ref{periods:griffiths} makes the extended Griffiths lines of
$\mathbb V_i$ semiample on this common cover; the same proof uses the
chosen common neat level and requires no additional alteration.
For a torus-type factor, $h^{1,0}=n_i$, and cup product gives the
specified isomorphism
\[
\bigwedge^{n_i}F^1E_i=\det F^1E_i
\xrightarrow{\ \sim\ }\mathcal H_i.
\]
It is induced by the pure Hodge morphism
$\bigwedge^{n_i}\mathbb V_i\to R^{n_i}(t_i)_*\R$.
By Lemma~\ref{periods:extensions} it identifies the extended lines.
The determinant calculation above therefore makes
$\overline{\mathcal H}_i$ semiample.
For a symplectic-type factor, write $n_i=2k_i$ and $\ell_i=F^2E_i$.
Equation~\eqref{periods:determinants} makes $\overline\ell_i$ semiample,
and the specified top cup product is
\[
\ell_i^{\otimes k_i}\xrightarrow{\ \sim\ }\mathcal H_i.
\]
This is induced by the pure morphism
$\mathbb V_i^{\otimes k_i}\to R^{2k_i}(t_i)_*\R$.
Again Lemma~\ref{periods:extensions} identifies the extensions, so
$\overline{\mathcal H}_i\simeq\overline\ell_i^{\otimes k_i}$ is
semiample. In particular the extended cup maps take generators to
generators, rather than to sections vanishing at the boundary.
We choose $\overline{\mathcal H}_i$ throughout to be the Schmid highest
line in the original top cohomological variation. Its norm is the
geometric volume norm~\eqref{periods:volume}. The last assertion is
therefore exactly Lemma~\ref{periods:extensions}, including its pullback
statement for disks. This proves both the specified holomorphic
extension and the required metric control.
\end{proof}
An absent factor, or a zero-dimensional factor retained by convention,
contributes the trivial line with constant norm. Thus an empty collection
of period factors requires no modification and no extra hypothesis.
==== END B-semiampleness-for-compact-log-smooth-Kahler-fibrations//periods.tex ====
==== BEGIN B-semiampleness-for-compact-log-smooth-Kahler-fibrations//preliminaries.tex ====
\section{Analytic models and plurivolumes}\label{sec:prelim}
\subsection{Discrepancies and the rank condition}
For a smooth model \(h:V\to Y\), log discrepancy is normalized by
\[
a(E;Y,\Delta)
=1+\operatorname{coeff}_E(K_{V/Y}-h^*\Delta).
\]
A \emph{sub-pair} allows negative boundary coefficients; it is
sub-log-canonical when these numbers are nonnegative.
On a smooth simple normal crossing pair, this is equivalent to every
boundary coefficient being at most one.
For comparison with the standard definition of an lc-trivial
fibration, write
\[
A_V=K_{V/Y}-h^*\Delta,\qquad
A_V^*=A_V+\sum_{\operatorname{coeff}_E(A_V)=-1}E.
\]
The usual rank condition is
\(\rk(f\circ h)_*\cO_V(\lceil A_V^*\rceil)=1\).
In the effective log-smooth situation of
Theorem~\ref{thm:main}, the identity is a log resolution.
For a coefficient less than one, \(\lceil-\delta\rceil=0\);
for a coefficient equal to one, the correction in \(A_Y^*\) gives
zero. Hence \(\lceil A_Y^*\rceil=0\), and connected fibers give the
rank condition. Its resolution independence is the usual
discrepancy convention. All later applications to projective
comparison families will check the generic rank directly.
\begin{lemma}\label{prelim:threshold}
Fix a model \(f':(Y',\Delta')\to X'\) as in
\eqref{eq:moduli} and a prime \(P\subset X'\).
On a joint log resolution of \(\Delta'\) and \((f')^*P\), let
\(\delta_E\) be the crepant coefficient and
\(a_E=\ord_E((f')^*P)>0\). Over a general point of \(P\),
\[
t_P=\min_{E\mapsto P}\frac{1-\delta_E}{a_E}.
\]
In particular, the threshold is rational and unchanged by a further
resolution of the source. The divisor \(B_{X'}\) has finite support.
\end{lemma}
\begin{proof}
On the joint simple normal crossing model the vertical coefficients
of the tested sub-pair are \(\delta_E+ta_E\).
The horizontal coefficients are already at most one.
The sub-log-canonical condition is therefore precisely
\(\delta_E+ta_E\leq1\) for every vertical \(E\) dominating \(P\).
For a simple normal crossing sub-pair these bounds also test all
further exceptional valuations: the log discrepancy of a blowup
stratum is the sum of the corresponding nonnegative coordinate log
discrepancies, with any remaining smooth directions contributing
nonnegative terms. Equivalently, this is the standard local
simple normal crossing discrepancy criterion. Thus the minimum is
the infimum in \eqref{eq:threshold}.
Outside a proper analytic subset of \(X'\), the morphism and all
horizontal boundary strata are smooth and the boundary is relatively
simple normal crossing, with no vertical component. At each prime
meeting this open the same calculation gives \(t_P=1\).
Properness gives an analytic bad locus; on a compact base it has
finitely many irreducible components. Only its divisorial components
can contribute to \(B_{X'}\).
\end{proof}
\subsection{Model constructions}
We use analytic embedded and equivariant resolution, chosen projective
over the space being resolved. Functorial resolution is compatible
with smooth product directions; see
\cite[Theorems~1.1 and~1.3]{BM08}.
In particular, the resolution of a fractional monomial cover can
preserve additional smooth logarithmic coordinate directions.
Only compact or relatively compact neighborhoods are used, so no
global finiteness issue for a noncompact resolution arises.
We will also use the following standard analytic model facts.
\begin{lemma}\label{prelim:models}
\begin{enumerate}[label=\textup{(\roman*)}]
\item A closed analytic subspace of a compact K\"ahler manifold,
and a finite cover of such a space, admit smooth compact K\"ahler
models obtained by projective resolution. The corresponding
assertion holds locally over relatively compact base charts.
\item Let \(V\) be a smooth compact K\"ahler manifold and
\(V^\circ\subset V\) a dense analytic Zariski open. A finite
unramified holomorphic cover of \(V^\circ\) extends to a finite
normal cover of \(V\). After resolution it has a smooth compact
K\"ahler model unchanged over the smooth covering open.
\end{enumerate}
\end{lemma}
\begin{proof}
For \textup{(i)}, closed analytic subspaces inherit K\"ahler
structures as complex spaces. Finite maps preserve the K\"ahler
property on the source, and projective resolutions over a K\"ahler
space have K\"ahler total space. The same construction on a slightly
larger chart gives the local assertion. These are the analytic
K\"ahler model facts used in \cite[Section~4]{Fujiki78}.
For \textup{(ii)}, apply the analytic extension theorem for finite
covers, followed by normalization; a convenient formulation for
the present K\"ahler setting is
\cite[version~1, Lemma~2.21]{Villadsen24}.
Locally, after resolving the complement to normal crossings, the
cover has finite monodromy about the coordinate divisors. Sufficiently
divisible coordinate power substitutions trivialize that monodromy.
The resulting trivial finite covers extend over the polydisk, and
their finite quotients give the normal extension before the power
substitution. Normal extensions agree on overlaps by uniqueness.
Part~\textup{(i)} and projective resolution give the last assertion.
\end{proof}
A compact parameter space used below need not be K\"ahler.
Whenever a K\"ahler form on a family is needed, it will come from
one of the fixed ambient spaces of Lemma~\ref{prelim:models},
not from a presumed polarization of its parameter space.
\subsection{The prepared family}
Choose a dense analytic Zariski open \(U\subset X\) on which
\(f\), every relevant boundary stratum, and their incidence diagrams
are smooth over the base. Remove vertical boundary images.
The boundary over \(U\) is then relatively simple normal crossing.
Existence follows from properness and generic smoothness applied
to the finitely many strata.
Fix a sufficiently divisible \(m>0\) so that \(m\Delta\) is integral
and \eqref{eq:adjoint} is realized by an actual bundle isomorphism
in degree \(m\). It will be harmless to replace \(m\) by a further
multiple finitely often. Set \(d=\dim Y-\dim X\).
The line of relative log plurivolumes on \(U\) is
\begin{equation}\label{eq:open-volume-line}
\cG_m
=f_*\cO_{Y_U}\bigl(m(K_{Y_U/U}+\Delta|_{Y_U})\bigr)
\simeq \cO_U\bigl(m(L-K_X)\bigr).
\end{equation}
The last notation denotes a holomorphic line bundle, whether or not
it has a global meromorphic section. The isomorphism follows from
\eqref{eq:adjoint}, the projection formula and
\(f_*\cO_Y=\cO_X\).
The evaluation map identifies its pullback with the relative
log-pluri bundle. A local generator therefore has no zeros as a
log pluriform with the specified twist; as a meromorphic relative
pluriform it has divisor \(-m\Delta\).
Take a compact K\"ahler log resolution
\(\mu_0:S\to X\), isomorphic over \(U\), such that the complement of
\(U\) is supported on a simple normal crossing divisor.
Resolve the main component of the pulled-back family to
\(p:W\to S\), unchanged over \(U\), and put
\begin{equation}\label{eq:prepared-data}
D_W=h^*\Delta-K_{W/Y},\qquad
L_S=\mu_0^*L,\qquad Q_S=L_S-K_S .
\end{equation}
The natural canonical-bundle identifications give
\begin{equation}\label{eq:relative-adjoint}
K_{W/S}+D_W\sim_{\Q}p^*Q_S .
\end{equation}
We retain the actual degree-\(m\) identification coming from
\eqref{eq:adjoint}. Over \(U\), \(mQ_S\) is the line \(\cG_m\).
On \(S\), the desired moduli trace is \(M_S=Q_S-B_S\).
The local calculations below use only \eqref{eq:relative-adjoint}
and the smooth effective log-smooth generic pair.
Thus they continue to apply if the vertical coefficients of
\(D_W\) are arbitrary rational numbers.
This observation will be used for the projective comparison.
\subsection{Degenerate dimensions}
If \(X\) is a point, its rational Picard group is zero, and
Theorem~\ref{thm:main} is immediate.
If \(d=0\), connected general fibers make \(f\) bimeromorphic.
A proper bimeromorphic morphism to a smooth space is an isomorphism
outside an analytic subset of codimension at least two in the base.
At the general point of each base divisor,
\[
t_P=1-\operatorname{coeff}_P(f_*\Delta).
\]
The actual adjoint identity on this isomorphism locus extends, with
its inverse, across codimension two. Hence
\(L=K_X+f_*\Delta\) in \(\Pic(X)_{\Q}\), and \(M_X=0\).
The same argument with the crepant sub-boundary applies on every
higher model, so all moduli traces vanish.
Here a rational divisor on a smooth base is rational Cartier;
the extension assertion is the removable-singularities theorem for
local functions expressing a line-bundle isomorphism.
We may therefore assume \(\dim X>0\) and \(d>0\) in the rest of the
proof, while allowing zero-dimensional strata and empty products.
==== END B-semiampleness-for-compact-log-smooth-Kahler-fibrations//preliminaries.tex ====
==== BEGIN B-semiampleness-for-compact-log-smooth-Kahler-fibrations//spreading.tex ====
\section{A compact base carrying a product decomposition}\label{sec:spreading}
The single-fiber decomposition theorem does not, by itself, give a
decomposition of a family. We obtain one after a proper surjective base
change by parameterizing finite covers, factor submanifolds, and graphs.
The auxiliary base may have larger dimension than \(S\). Its compactness,
together with a fixed compact K\"ahler ambient for the factors, will be
the essential feature.
\subsection{A marked family of klt strata}
We retain the good open \(U\), the log-smooth family, and the integer \(m\)
of the preceding sections. In particular, the line
\(\mathcal G_m\) consists of relative log plurivolumes. The integer \(k\)
is the largest number of horizontal coefficient-one components that meet
over \(U\); when there are none, \(k=0\).
All opens in this section are analytic Zariski opens.
\begin{lemma}\label{spread:marked}
There are a smooth compact K\"ahler manifold \(\Sigma\), a proper
surjection \(\rho:\Sigma\to S\), and a dense open
\(\Sigma^\circ\subseteq\rho^{-1}(U)\) with the following data:
\begin{enumerate}[label=\textup{(\roman*)}]
\item a proper smooth family
\(a:\mathscr A^\circ\to\Sigma^\circ\) with connected fibers and a
holomorphic section;
\item an effective relative SNC rational boundary \(D_{\mathscr A}\)
with coefficients in \([0,1)\);
\item an identification
\[
a_*\cO_{\mathscr A^\circ}
\bigl(m(K_{\mathscr A^\circ/\Sigma^\circ}+D_{\mathscr A})\bigr)
\simeq \rho^*\mathcal G_m.
\]
The relative log-pluri bundle itself is the pullback of this line.
Moreover, \(\mathscr A^\circ\) and its boundary have extensions in a
smooth compact K\"ahler manifold mapping to \(\Sigma\).
\end{enumerate}
Each fiber pair is a connected component of a deepest coefficient-one
stratum of the original good family, with its adjunction boundary.
\end{lemma}
\begin{proof}
Suppose first that \(k>0\). Choose an irreducible component \(A\) of an
intersection of \(k\) coefficient-one components whose image contains
\(U\), after decreasing \(U\) if necessary. The original SNC hypothesis
makes \(A\) a smooth compact K\"ahler manifold. Maximality of \(k\) says
that no further coefficient-one component meets \(A\) over \(U\).
Adjunction there gives
\[
(K_Y+\Delta)|_A=K_A+D_A,
\]
where \(D_A\) is effective and has SNC support and coefficients less than
one. Indeed, the remaining boundary equations are transverse coordinate
equations on \(A\). Two distinct such equations cannot define the same
divisor on \(A\), by the SNC codimension condition. The preparation of
\(U\) makes \(A_U\to U\), and all its boundary strata, smooth.
For \(k=0\), set \(A=Y\) and use the original boundary over \(U\).
Let \(A\to B\to X\) be the Stein factorization. Over \(U\), the finite
map \(B\to X\) is \'etale, and \(A\to B\) has connected smooth fibers.
Resolve the graph of the map \(A\dashrightarrow S\), taking the
resolution to be unchanged over \(A_U\); call the resulting manifold
\(\Sigma\). It is compact K\"ahler and maps holomorphically both to
\(A\) and to \(S\). The first map also gives \(\Sigma\to B\). Consider
\[
A\times_B\Sigma\longrightarrow\Sigma.
\]
Over \(\Sigma^\circ=A_U\) this is the desired family: its fiber is the
connected component selected by the point of \(B\), and the map
\[
\sigma\longmapsto \bigl(\sigma,\sigma\bigr)
\]
under the identification with \(A_U\) gives its diagonal section.
Resolve its main component, without changing this open family.
The fiber product is a closed analytic subspace of \(A\times\Sigma\),
so a projective resolution is compact K\"ahler. Pullbacks and strict
transforms of the original boundary give analytic extensions of all
boundary data.
Ordered pluriresidue gives the indicated pullback of
\(\mathcal G_m\). On every fiber it is a nowhere-vanishing section of
the \(m\)-th log canonical bundle, interpreted with its boundary twist.
That bundle is therefore trivial on each connected fiber, and its
space of sections has dimension one. Grauert's base-change theorem
identifies its direct image with a line bundle. The evaluation map is
an isomorphism because its restriction to each fiber is an isomorphism.
This proves the assertion about the actual relative bundle as well.
\end{proof}
For a section \(s\) of \(\rho^*\mathcal G_m\), denote its adjunction
plurivolume on the selected fiber by \(s_A\). We use the norm
\begin{equation}\label{spread:residue-norm}
\|s\|_{\mathrm{res}}
=\left(\int_{A_\sigma}|s_A|^{2/m}\right)^{m/2}.
\end{equation}
The measure in this formula is the measure of a meromorphic
pluricanonical form with the specified boundary poles.
Since the boundary is SNC and every coefficient is less than one,
the integral is finite, and it is positive for \(s\ne0\).
The same definition will be used after any further base change.
\begin{lemma}\label{spread:residue-growth}
Let \(P\) be a smooth compact complex manifold and \(r:P\to S\) a
holomorphic map factoring through the marked-family construction
on a dense open \(P^\circ\).
Equip \(r^*\mathcal G_m\) there with
\eqref{spread:residue-norm}. Along a disk in \(P\) mapping generically
to \(P^\circ\), generators of any sufficiently divisible positive
power of \(r^*(mM_S)\), and the dual generators, have at most
powers-of-logarithm growth in the corresponding norms, after a
sufficiently divisible power substitution.
\end{lemma}
\begin{proof}
This is Corollary~\ref{orders:residuenorm}; we recall the norm
identification. By Theorem~\ref{orders:trace}, the corrected generator
is a power of a generator of the canonical extension of the pure Hodge line.
Proposition~\ref{hodge:residue} maps this line isomorphically to the
top residue eigenline over the chosen connected stratum component,
including all sheets over that component. This identification
preserves canonical generators after unipotentization.
The residue is a holomorphic top form on the resolved cyclic root
cover of the klt stratum. Its Hodge norm is its volume integral.
Its \(m\)-th power is the pullback of \(s_A\), so change of variables
identifies its squared norm with
\(\int_{A_\sigma}|s_A|^{2/m}\), multiplied by the fixed degree of the
root cover and a fixed normalization constant. Resolution exceptional
sets have measure zero. The canonical highest-line norm estimates
therefore give the assertion and its dual. They apply to the
pulled-back pure line on the stated disk; no extension of a geometric
product over the center of that disk is required.
\end{proof}
\subsection{Splitting a fiber and realizing its finite covers}
We first specify the single-fiber consequence that will be spread.
\begin{lemma}\label{spread:fiber-splitting}
Let \((F,D)\) be a smooth connected compact K\"ahler pair, with
effective SNC rational boundary of coefficients less than one, and
suppose that \(m(K_F+D)\) is trivial. There is a finite \'etale cover
\(\nu:J\to F\) and an isomorphism of pairs
\begin{equation}\label{spread:fiber-product}
(J,\nu^*D)\simeq (Q,C)\times T_1\times\cdots\times T_j,
\end{equation}
where \((Q,C)\) is a smooth projective klt SNC pair, and each \(T_i\)
is a positive-dimensional complex torus or an irreducible holomorphic
symplectic manifold. The boundary is entirely on \(Q\).
A point is allowed for \(Q\), and \(j=0\) is allowed. Moreover,
\(m(K_Q+C)\) and all \(mK_{T_i}\) are trivial.
\end{lemma}
\begin{proof}
Linear triviality implies numerical triviality, so the decomposition
theorem of Matsumura--Wang--Wu--Zhang applies to this smooth effective
klt pair
\cite[version~1, Corollary~1.3]{MWWWZ25}.
It gives a product of a rationally connected pair and boundary-free
torus, Calabi--Yau, and holomorphic symplectic factors after a finite
\'etale cover. The covered manifold and all its product factors are
smooth. Apply the ordinary
Beauville--Bogomolov decomposition to any boundary-free factors still
requiring decomposition, and compose the further finite \'etale
covers \cite[Theorem~1]{Beauville83}.
For completeness, a rationally connected compact K\"ahler manifold
\(R\) is projective. To see the relevant vanishing directly, graphs
in the Douady space of \(\PP^1\times R\) parameterize its holomorphic
rational curves. There are countably many finite-dimensional
parameter families \cite{Fujiki79}. Matching marked endpoints
parameterizes chains, again in countably many analytic families.
Stratify their parameter spaces into smooth pieces. General pairs
of points of \(R\) occur as endpoints, so Sard's theorem implies that
one two-endpoint evaluation
\[
e=(e_0,e_\infty):T\longrightarrow R\times R
\]
is submersive at some point. Otherwise all these countably many
images would have measure zero. Mark the ends of each chain segment
by \(0,\infty\), using reparameterization of \(\PP^1\).
For \(\alpha\in H^0(R,\Omega_R^2)\), its pullback by each segment
\(\PP^1\times T\to R\) comes from \(T\), because \(\PP^1\) has no
holomorphic forms of positive degree. Comparing successive endpoints
gives
\[
e^*(\operatorname{pr}_1^*\alpha-\operatorname{pr}_2^*\alpha)=0.
\]
Submersivity forces \(\alpha=0\) on an open set, hence everywhere.
Thus \(H^{2,0}(R)=0\). Hodge decomposition now makes every real
degree-two class of type \((1,1)\). Openness of the K\"ahler cone
gives a rational K\"ahler class, and an integral multiple is the first
Chern class of a positive line bundle. Kodaira's embedding theorem
makes \(R\) projective \cite{Kodaira54}.
The same criterion makes every remaining strict Calabi--Yau factor
with \(H^{2,0}=0\) projective. Dimension-two symplectic factors are
kept among the \(T_i\). Combine all projective factors into \(Q\).
The boundary on \(Q\) remains effective klt SNC under this product
description. Finally, restrict the trivial log-pluri bundle of
\(J\) to factor slices. The canonical lines of the complementary
factors contribute only constant one-dimensional vector spaces,
so these restrictions give \(m(K_Q+C)\sim0\) and \(mK_{T_i}\sim0\).
\end{proof}
\begin{lemma}\label{spread:finite-covers}
Let \(g:E^\circ\to B^\circ\) be a proper smooth holomorphic map of
connected complex manifolds, with connected compact fibers and a
section. Suppose \(B^\circ\) is a dense open in a smooth compact
K\"ahler manifold \(B\), and the family has a smooth compact
K\"ahler extension \(E\to B\).
Then all finite connected \'etale covers of individual fibers occur
in countably many families, each obtained after a finite \'etale base
cover of \(B^\circ\). Each resulting base and total space admit
smooth compact K\"ahler extensions preserving the open families.
The same holds with pulled-back relative SNC boundary data.
\end{lemma}
\begin{proof}
Fix a base point and write \(\Pi=\pi_1(E_b)\) and
\(\Gamma=\pi_1(B^\circ)\), with fiber base points supplied by the
section. Ehresmann's theorem makes \(g\) a differentiable fiber
bundle. Its homotopy sequence contains
\[
\pi_2(E^\circ)\longrightarrow\pi_2(B^\circ)
\longrightarrow\Pi\longrightarrow\pi_1(E^\circ)
\longrightarrow\Gamma\longrightarrow1.
\]
The section makes the first arrow surjective, so the connecting
homomorphism into \(\Pi\) is zero. It also splits the last
homomorphism. Thus there is a genuine semidirect product
\begin{equation}\label{spread:semidirect}
\pi_1(E^\circ)\simeq\Pi\rtimes\Gamma.
\end{equation}
The group \(\Pi\) is finitely generated, since the fiber is a compact
manifold. For each integer \(e>0\), it has only finitely many
subgroups of index \(e\): their coset actions are represented among
the homomorphisms from a fixed finite generating set to the finite
group of permutations of \(e\) elements. If \(H\subseteq\Pi\) is
one such subgroup, its stabilizer \(\Gamma_H\) under the action in
\eqref{spread:semidirect} has finite index in \(\Gamma\).
Pass to the corresponding finite base cover. The subgroup
\[
H\rtimes\Gamma_H\subseteq\Pi\rtimes\Gamma_H
\]
then defines a finite connected cover of the pulled-back total
space. Its restriction to a fiber is the connected cover
corresponding to \(H\). Topological covering charts lift the complex
structure, so this is a holomorphic \'etale cover. Transport to
other fibers, and then all indices \(e\), realizes every required
fiber cover among countably many such constructions.
Extend the finite base cover first across \(B\setminus B^\circ\)
by the analytic finite-cover extension theorem, and resolve the
normal extension. Pull back \(E\), resolve its main component, and
extend the finite total-space cover over that resolved compact
space in the same way. Projective resolutions give smooth compact
K\"ahler manifolds, unchanged on the open under consideration
\cite[Lemma~2.21]{Villadsen24}.
One can also describe the extension locally after making the
complement SNC. A punctured coordinate chart has commuting
meridians. Sufficiently divisible powers of its boundary coordinates
kill their finite permutation action. The cover becomes a disjoint
union of trivial covers on the punctured power chart; the finite
normal extensions descend and glue by uniqueness. Thus this step
uses finite-cover extension, for which essential singularities do
not arise. Finite maps over K\"ahler targets and projective
resolutions preserve the K\"ahler property; equivalently one may
use the compactification in the cited Lemma. Boundary pullbacks
and their strict transforms give the required analytic closures.
\end{proof}
\subsection{Compact parameter spaces}
We call a subset of a compact complex space \emph{constructible} if it
is a finite union of differences of closed analytic subsets.
Compactness is important in the following elementary form of analytic
constructibility.
\begin{lemma}\label{spread:constructible}
Let \(q:Z\to T\) be a proper holomorphic map of compact complex
spaces. The image of a constructible subset of \(Z\) is
constructible in \(T\).
Consequently, any finite combination of incidence, emptiness,
surjectivity, and injectivity conditions on fibers of compact
analytic diagrams is constructible.
\end{lemma}
\begin{proof}
It suffices to consider \(A\setminus C\), where \(C\subseteq A\)
are closed analytic subsets of \(Z\). There are finitely many
irreducible components of \(A\), so we may assume that \(A\) is
irreducible and \(C\subsetneq A\). Set \(T'=q(A)\), a compact
irreducible analytic subspace by the proper mapping theorem.
On a dense open of \(T'\), the fibers of \(A\to T'\) have dimension
\(\dim A-\dim T'\). Each component of \(C\) either has proper image
in \(T'\), or has smaller generic fiber dimension. The proper
fiber-dimension theorem therefore gives a dense open
\(T'_0\subseteq T'\) on which no fiber of \(A\to T'\) is contained
in \(C\). Hence
\[
T'_0\subseteq q(A\setminus C).
\]
The part of the image over \(T'\setminus T'_0\) is the image of
\[
\bigl(A\cap q^{-1}(T'\setminus T'_0)\bigr)\setminus C
\]
under a proper map whose image has smaller dimension. Induction
proves that this remainder is constructible. The case of a
zero-dimensional image is immediate. This proves the first
assertion.
For example, failure of containment of two fiber subspaces is the
image of their difference. Failure of surjectivity is detected by
points of the target outside the image, itself analytic under a
proper map. Failure of injectivity is detected on the fiber product
of the source with itself, after removing its diagonal. Every
ambient projection here is proper; the first assertion applies
to the indicated differences. Finite Boolean combinations remain
constructible.
\end{proof}
We also use smoothness and rank conditions in these diagrams.
A universal Douady or Hilbert family is proper and flat.
Its nonsmooth locus is analytic, and its proper image gives the
complement of the smooth-fiber locus. On this locus, fiber dimensions
are locally constant. Ranks of maps of relative tangent bundles
can be tested using relative differentials and their Fitting ideals.
Thus they too give constructible conditions, also after restricting
to a constructible incidence locus. A smooth compact fiber is
connected exactly when \(h^0(\cO)=1\), which is an analytic
semicontinuity condition in a proper flat family. For marked
divisors, smoothness and the ranks of their defining differentials
test the relative SNC condition, including the expected codimensions
of all intersections.
\begin{proposition}\label{spread:product}
After a proper surjection \(r:P\to S\), with \(P\) a smooth compact
connected manifold, there is a dense open \(P^\circ\subseteq r^{-1}(U)\)
on which the marked klt family has a finite \'etale cover and a product
decomposition
\begin{equation}\label{spread:family-product}
(\mathscr J,D_{\mathscr J})
\simeq
(\mathscr Q,C)\times_{P^\circ}
\mathscr T_1\times_{P^\circ}\cdots\times_{P^\circ}\mathscr T_j .
\end{equation}
Here all families are proper and smooth with connected fibers.
They have the following properties.
\begin{enumerate}[label=\textup{(\roman*)}]
\item The pair \((\mathscr Q,C)\) is a projective family of effective
klt SNC pairs. It is embedded in \(P^\circ\times\PP^N\) and is the
pullback of universal marked families by the restriction of a
holomorphic map from \(P\) to a fixed product of projective Hilbert
schemes.
\item Each \(\mathscr T_i\) is a family of submanifolds of a fixed
compact K\"ahler manifold \(V\), of positive dimension \(n_i\).
Its cohomological local systems have their integral lattices modulo
torsion and flat real polarizations. Its top Hodge line has rank one
and is given by
\[
\lambda_i
=(\pi_i)_*\omega_{\mathscr T_i/P^\circ}
=F^{n_i}R^{n_i}(\pi_i)_*\C
\]
Each index is of one of the following two types:
\[
\begin{array}{ll}
\text{torus type:}&
h^{1,0}(T_i)=n_i,\quad
\bigwedge^{n_i}F^1R^1(\pi_i)_*\C
\xrightarrow{\ \sim\ }\lambda_i;\\[2mm]
\text{symplectic type:}&
n_i\ \text{even},\quad h^{2,0}(T_i)=1,\quad
\bigl(F^2R^2(\pi_i)_*\C\bigr)^{\otimes n_i/2}
\xrightarrow{\ \sim\ }\lambda_i .
\end{array}
\]
The displayed maps are cup products on the underlying variations.
\item On every fiber, \(m(K_Q+C)\) and \(mK_{T_i}\) are trivial.
With
\[
\mathcal G_{m,Q}
=(\pi_Q)_*\cO_{\mathscr Q}
\bigl(m(K_{\mathscr Q/P^\circ}+C)\bigr),
\]
there is an identification of actual holomorphic line bundles
\begin{equation}\label{spread:volume-line}
r^*\mathcal G_m
\simeq \mathcal G_{m,Q}\otimes
\bigotimes_i\lambda_i^{\otimes m}.
\end{equation}
\item If the finite cover in \eqref{spread:family-product} has degree
\(e\), the norm in \eqref{spread:residue-norm} satisfies
\begin{equation}\label{spread:norm-product}
\|s\|_{\mathrm{res}}
=c\,\|s_Q\|_{Q,\mathrm{res}}\,
\prod_i\|\omega_i\|_{\mathrm{Hdg}}^m
\quad\text{when}\quad
s\longleftrightarrow
s_Q\otimes\bigotimes_i\omega_i^{\otimes m}.
\end{equation}
Here \(c>0\) is constant; with compatible volume conventions,
\(c=e^{-m/2}\). The norm on the left gives generators and their duals
of sufficiently divisible powers of \(r^*(mM_S)\) the two-sided
logarithmic bounds of Lemma~\ref{spread:residue-growth}.
\end{enumerate}
These assertions are preserved on the good open under further smooth
compact auxiliary base changes. Neither generic finiteness of \(r\)
nor a global K\"ahler form on \(P\) is required.
\end{proposition}
\begin{proof}
Apply Lemma~\ref{spread:finite-covers} to the marked family from
Lemma~\ref{spread:marked}. We obtain countably many compact K\"ahler
total spaces \(V_\alpha\to B_\alpha\), whose good fibers supply all
finite \'etale fiber covers that may occur in
Lemma~\ref{spread:fiber-splitting}. Each \(B_\alpha\) maps properly
to \(S\). Fix one such family temporarily, and write \(V\to B\).
The boundary on its good open is the restriction of a fixed rational
divisor on \(V\); only its restriction to good fibers will be tested.
\smallskip
\noindent\emph{The parameters and their geometric conditions.}
For an individual splitting, embed \(Q\) into some \(\PP^N\).
Choose points in complementary factors and realize each \(T_i\)
as a factor slice of \(J\subset V\). The product isomorphism is
represented by its graph in
\[
V\times\PP^N\times V^j.
\]
We parameterize the fiber \(J\) by \(B\), the embedded \(Q\) and its
marked boundary components by Hilbert schemes of \(\PP^N\), the
slices by Douady spaces of \(V\), and the graph by the Douady space
of the displayed ambient.
For each choice of \(N,j\), dimensions, boundary coefficients,
Hilbert polynomials, and Douady components, take the product
\[
K=B\times\mathscr H\times
\prod_{i=1}^j\mathscr D_i\times\mathscr D_\Gamma,
\]
where \(\mathscr H\) is the corresponding product of Hilbert schemes.
This is a compact complex space. Indeed, connected components of
the Douady space of a compact K\"ahler manifold are compact
\cite[Corollary~5.3]{Toma11}, and all ambients here are compact
K\"ahler. Fujiki's countability theorem \cite{Fujiki79} gives
countably many components. There are also only countably many
choices of the discrete data and of \(\alpha\). Thus all the
individual splittings have been included in countably many compact
parameter spaces.
In each \(K\), impose the following conditions on the universal data:
\begin{enumerate}[label=\textup{(\alph*)}]
\item the base parameter is good; the projective factor, the slices,
and the graph are nonempty smooth fibers of the prescribed
dimensions; the factors are connected;
\item every marked boundary component is a smooth divisor on the
projective factor, distinct marks have distinct supports, and all
their intersections have the prescribed SNC codimensions;
\item the slices lie in \(V_b\), and the graph lies in the product
of \(J=V_b\) with the indicated projective factor and slices;
\item the projections of the graph to \(J\) and to the factor
product are bijective, with invertible vertical differentials;
\item the graph identifies the weighted boundary on \(J\) with the
boundary pulled back from the marked projective factor.
\end{enumerate}
These conditions define a constructible subset of \(K\).
For (a)--(b), use proper flat smoothness, connectivity, and the
relative rank tests described above. Condition (c) is incidence.
For (d), all ambient coordinate projections are holomorphic on
the compact universal graph. Bijectivity is tested by
Lemma~\ref{spread:constructible}, and the vertical rank conditions
are imposed by relative differentials. For (e), group marks of
equal coefficient and test equality of the corresponding reduced
supports on the graph. The boundary extensions on \(V\) are
analytic, so these are again incidence conditions on compact
diagrams. On the SNC locus, equality by coefficient groups is
exactly equality of the rational divisors.
Decompose this constructible locus into finitely many irreducible
locally closed analytic pieces and then stratify their singular loci.
This yields countably many smooth connected locally closed analytic
strata, each with compact analytic closure. Over every such stratum,
the graphs give isomorphisms of smooth families: fiberwise they are
bijective and have invertible vertical differential, so the relative
inverse function theorem and properness give holomorphic inverses.
\smallskip
\noindent\emph{The cohomological types and domination.}
Retain the strata containing at least one splitting with a specified
list of torus and symplectic types. This is still a countable
collection; no constructibility of a Hodge locus is being assumed.
We check that the cohomological conditions in (ii) then hold
throughout the chosen stratum.
For a smooth factor family over such a stratum \(T\), pull back a
K\"ahler form \(\omega_V\) from \(V\). It is relatively K\"ahler,
and its fiberwise class is flat for the Gauss--Manin connection,
because it is represented by a closed form on the total family.
The smooth proper K\"ahler Hodge theorem therefore supplies the
usual pure variations and locally constant Hodge numbers; one may
apply \cite[Theorem~1.1(i)]{FF25} with empty removed boundary on
small smooth base charts, and use purity of compact K\"ahler cohomology.
Fiberwise Lefschetz decomposition for this flat class is a
decomposition of real variations. Poincar\'e duality on the
primitive summands, with the Hodge--Riemann signs, gives flat real
polarizations on the full cohomology. This construction requires
no rational K\"ahler class. Locally over \(T\), adding a K\"ahler
form from a coordinate neighborhood to the ambient form makes
the total space K\"ahler as well.
The cup products
\[
\bigwedge^{n_i}R^1(\pi_i)_*\C
\longrightarrow R^{n_i}(\pi_i)_*\C,
\qquad
\bigl(R^2(\pi_i)_*\C\bigr)^{\otimes n_i/2}
\longrightarrow R^{n_i}(\pi_i)_*\C
\]
are flat morphisms of pure variations of equal weight in each case.
Their kernels and images are subvariations; equivalently,
polarizations give flat Hodge complements and strictness gives
the induced Hodge filtrations. The ranks on the highest Hodge
pieces are consequently locally constant. At a torus fiber,
exterior multiplication of holomorphic one-forms gives its
nonzero top form. At an irreducible holomorphic symplectic fiber,
the top power of the symplectic form gives its nonzero top form.
Thus the Hodge numbers and cup-product isomorphisms in (ii)
hold throughout \(T\). Only these cohomological consequences of
the two types are needed.
Every point of the original good base has splitting data on at
least one of the retained strata: first select the connected
stratum fiber, then its finite cover, then its product and
embeddings. The compact closures of these strata have proper
analytic images in \(S\). If all their images were proper analytic
subsets, their countable union could not contain the nonempty open
\(U\), by Baire category. One closure therefore dominates \(S\).
Resolve that irreducible closure, choosing a projective resolution
unchanged on its smooth open stratum. The result is a smooth
compact connected \(P\), with a proper surjection \(r:P\to S\).
Let \(P^\circ\) be the inverse image of the chosen stratum.
The product of families on it is
\eqref{spread:family-product}. Projection to \(\mathscr H\) gives
the holomorphic Hilbert-scheme map on all of \(P\) required by (i).
The ambient \(V\) is fixed by the compact parameter space containing
this closure. This proves (i)--(ii).
\smallskip
\noindent\emph{Volumes and their norms.}
The finite cover of the selected klt fiber preserves the trivial
\(m\)-th log canonical bundle. Restriction of
\eqref{spread:family-product} to factor slices proves the fiberwise
trivialities in (iii), even for parameter points where only the
cohomological types have been retained.
Grauert's base-change theorem gives the direct image line
\(\mathcal G_{m,Q}\). For each \(T_i\), the Hodge-number calculation
gives \(\rk\lambda_i=1\). A nonzero holomorphic top form on \(T_i\)
is nowhere vanishing: its \(m\)-th power is a nonzero section of a
trivial line bundle on a connected compact manifold, hence has
no zeros. Consequently, relative multiplication induces an
isomorphism
\[
\lambda_i^{\otimes m}
\simeq(\pi_i)_*\omega_{\mathscr T_i/P^\circ}^{\otimes m}.
\]
Finite \'etale pullback of the selected adjunction volume, followed
by the product formula for relative canonical bundles and their
boundary twists, now gives \eqref{spread:volume-line}. Every map
in this construction identifies actual one-dimensional spaces
of sections on the fibers, so these are holomorphic line-bundle
isomorphisms, with their specified identification on the open.
Take a decomposable local section on the right side of
\eqref{spread:volume-line}. With compatible volume conventions,
change of variables under the \'etale cover and Fubini's theorem
give
\[
e\int_{A_\sigma}|s_A|^{2/m}
=
\left(\int_{Q_\sigma}|s_Q|^{2/m}\right)
\prod_i\left(\int_{T_{i,\sigma}}|\omega_i|^2\right).
\]
The integral on \(Q_\sigma\) is finite by the klt SNC condition.
Raising the equality to the power \(m/2\) proves
\eqref{spread:norm-product}. The degree \(e\) is fixed on the
connected open parameter stratum. Other fixed volume conventions
only change the constant \(c\).
Lemma~\ref{spread:residue-growth} gives the assertion for
\(r^*(mM_S)\) and its dual.
All constructions are compatible with pullback on their smooth
opens. The argument has used compactness of \(P\) for proper
images and later global arguments, and a K\"ahler form on \(V\)
for the relative Hodge theory. It has imposed no dimension
restriction on \(r\) and no global K\"ahler hypothesis on \(P\).
Empty products and zero-dimensional projective factors satisfy
the same formulas, with integration over a point taken with mass one.
\end{proof}
==== END B-semiampleness-for-compact-log-smooth-Kahler-fibrations//spreading.tex ====
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==== END B-semiampleness-for-compact-log-smooth-Kahler-fibrations//references.bib ====