# On semi-ampleness of the moduli part

Stefano Filipazzi  EPFL SB MATH CAG MA C3 625 (Bâtiment MA) Station 8 CH-1015
Lausanne, Switzerland<EMAIL_ADDRESS>and  Calum Spicer
Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK
<EMAIL_ADDRESS>
###### Abstract.

We discuss a conjecture of Shokurov on the semi-amplenes of the moduli part of
a general fibration.

###### 2020 Mathematics Subject Classification:

14E30, 37F75

SF and CS were partially supported by ERC starting grant #804334.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Property $(*)$ pairs and locally stable families
  4. 4 On semi-ampleness of the moduli part
  5. 5 $1$-dimensional bases

##  1\. Introduction

In [Sho21], the notion of the moduli and discriminant parts of a generically
log canonical (GLC) fibration $(X/Z,B)$ is defined. By a GLC fibration we mean
the datum of a log pair $(X,B)$ and a contraction $f\colon X\rightarrow Z$,
i.e., a projective surjective morphism with connected fibres, between normal
quasi-projective varieties, such that $(X,B)$ is log canonical above the
generic point of $Z$. We may define a discriminant divisor $B_{Z}$ on $Z$
which (roughly) measures the singularities of the fibres of $(X,B)$ over $Z$,
and the moduli part is then defined as
$M_{X}\coloneqq(K_{X}+B)-(K_{Z}+B_{Z})$. These definitions are straightforward
extensions of the corresponding notions for LC trivial fibrations.

For GLC fibrations, it is known that, after passing to a sufficiently high
model, the moduli part of the fibration is compatible with pull-backs.
Moreover, in the case of an LC trivial fibration, it is expected that the
moduli part becomes semi-ample on such a model. This conjecture is known as
the b-semi-ampleness conjecture, see [PS09]. Analogously, one might hope that
a similar statement holds for a general GLC fibration, namely, after replacing
$(X/Z,B)$ by a birationally equivalent pair
$(X^{\prime}/Z^{\prime},B^{\prime})$, the moduli part $M_{X^{\prime}}$ of
$(X^{\prime}/Z^{\prime},B^{\prime})$ becomes semi-ample. However, as examples
of Keel show, see [ACSS21, §5.4], this is not in general possible.

In light of these examples Shokurov conjectured ([Sho21, Conjecture 1]) after
replacing $(X/Z,B)$ by an appropriate model, that $M_{X}$ becomes semi-ample
after any small perturbation by an ample divisor coming from the moduli space
of fibres. We consider here a variant on Shokurov’s conjecture.

######  Conjecture 1.1.

Suppose that $(X,B)$ is log canonical, $B\geq 0$, $(X/Z,B)$ has maximal moduli
(see [ACSS21, Definition 2.20]), $M_{X}$ is $f$-nef and is BP stable. Then the
following hold

  1. (1)

for all $m\gg 0$ and any $\epsilon\in\mathbb{Q}_{>0}$, $M_{X}+\epsilon
f^{*}\operatorname{det}f_{*}\mathcal{O}(mM_{X})$ is semi-ample;

  2. (2)

$\kappa(M_{X})\geq\kappa(X/Z,B)+{\mathrm{v}ar}(X/Z,B)$.

We note that if $B=0$, $X$ has canonical singularities and $f\colon
X\rightarrow B$ is a family of good minimal models then item (2) was proven in
[Kaw85]. In [KP17], Kovács and Patakfalvi settled item (2) when $K_{X}+B$ is
$f$-big and the generic fiber has klt singularities. In the same work, en
route to proving the projectivity of the moduli space of stable pairs, they
settled item (1) when $f$ is a family of stable varieties.

In the case of an lc trivial fibration item (1) reduces to the b-semi-
ampleness conjecture. Our first theorem shows that in fact the general case of
item (1) of Conjecture 1.1 is implied by the b-semi-ampleness conjecture.

######  Theorem 1.2 ( = Theorem 4.7).

Assume the b-semi-ampleness conjecture holds. Let $f\colon(X,B)\rightarrow Z$
be a GLC fibration between projective varieties with $B\geq 0$ and
$\operatorname{dim}X=n$. Suppose that a general fibre of $f$ admits a good
minimal model.

Then we may find a birationally equivalent GLC fibration
$f^{\prime}\colon(X^{\prime},B^{\prime})\rightarrow Z^{\prime}$ fitting into
the following commutative diagram

${(X^{\prime},B^{\prime})}$${(X,B)}$${Z^{\prime}}$${Z}$$\scriptstyle{\alpha}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$$\scriptstyle{\beta}$

such that the following hold, where $M^{\prime}$ is the moduli part of
$(X^{\prime}/Z^{\prime},B^{\prime})$

  1. (1)

$(X^{\prime}/Z^{\prime},B^{\prime})$ is BP stable and has maximal moduli;

  2. (2)

$M^{\prime}+\epsilon f^{\prime*}\lambda_{m}^{\prime}$ is semi-ample for $m\gg
0$ and any $\epsilon\in\mathbb{Q}_{>0}$, where
$\lambda_{m}^{\prime}=\operatorname{det}f^{\prime}_{*}\mathcal{O}(mM^{\prime})$;
and

  3. (3)

$M^{\prime}$ and $\lambda_{m}^{\prime}$ are compatible with base change in an
appropriate sense.

We refer to Theorem 4.7 for a precise statement.

Over a curve base, by using ideas from foliation theory, from [CH19] and the
Simpson correspondence, we are able to remove our assumption on the b-semi-
ampleness conjecture and are therefore able to prove Conjecture 1.1. The
following is proven in Theorem 5.6.

######  Theorem 1.3 (Conjecture 1.1 holds if $\operatorname{dim}Z=1$).

Let $f\colon(X,B)\rightarrow Z$ be a GLC fibration between projective
varieties with $B\geq 0$, $\operatorname{dim}X=n$ and $\operatorname{dim}Z=1$.
Suppose that $(X,B)$ is klt over the generic point of $Z$ and the generic
fibre of $f$ admits a good minimal model.

Then we may find a birationally equivalent GLC fibration
$f^{\prime}\colon(X^{\prime},B^{\prime})\rightarrow Z$ fitting into the
following commutative diagram

${(X^{\prime},B^{\prime})}$${(X,B)}$${Z}$${Z.}$$\scriptstyle{\alpha}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$

such that $K_{X^{\prime}}+B^{\prime}$ is nef over $Z$ and

  1. (1)

$(X^{\prime}/Z^{\prime},B^{\prime})$ is BP stable and has maximal moduli;

  2. (2)

$M^{\prime}+\epsilon f^{\prime*}\lambda_{m}^{\prime}$ is semi-ample for $m\gg
0$ and any $\epsilon\in\mathbb{Q}_{>0}$ where
$\lambda_{m}^{\prime}=\operatorname{det}f^{\prime}_{*}\mathcal{O}(mM^{\prime})$;
and

  3. (3)

$\kappa(M^{\prime})\geq\kappa(X^{\prime}/Z,B^{\prime})+{\mathrm{v}ar}(X^{\prime}/Z,B^{\prime})$.

Lastly, as immediate consequence of Theorem 1.2, we have the following
statement.

######  Corollary 1.4.

Let $f\colon(X,B)\rightarrow Z$ be a GLC fibration between projective
varieties with $B\geq 0$, $\operatorname{dim}X=n$, and
$\operatorname{dim}Z=m$. Assume that a general fibre $(X_{z},B_{z})$ of $f$
satisfies $\kappa(X_{z},B_{z})\geq n-m-2$ and that $(X,B)$ is klt over the
generic point of $Z$. Then, Theorem 1.2 holds for $f$.

Corollary 1.4 holds unconditionally, since good minimal models exist for klt
pairs with Kodaira dimension in the given range (see, e.g., [Fil20b, Theorem
1.5]), and the b-semi-ampleness conjecture is known in relative dimension at
most 2, see [PS09, Fuj03, Fil20a, ABB+22].

### Acknowledgements

We thank Zsolt Patakfalvi for many insightful discussions. We also thank Paolo
Cascini, Giulio Codogni, Christopher D. Hacon, Giovanni Inchiostro, Jihao Liu,
Roberto Svaldi and Luca Tasin for many useful discussions. We thank Jihao Liu
for feedback on a draft version of this work.

##  2\. Preliminaries

We work over $\mathbb{C}$. We refer to [KM98] for the standard terminology in
birational geometry. For the language of generalised pairs and b-divisors, we
refer to [FS20].

Our varieties are connected and quasi-projective unless otherwise stated.
Given a normal variety $X$ and an open subset $U\subset X$, we say that $U$ is
big if $\mathrm{codim}_{X}(X\setminus U)\geq 2$.

All divisors have coefficients in $\mathbb{Q}$. Unless otherwise stated, a
divisor means a Weil $\mathbb{Q}$-divisor. Given a dominant morphism $f\colon
X\rightarrow Y$ and a divisor $D=\sum a_{i}P_{i}$, we define its horizontal
part $D^{h}$ and its vertical part $D^{v}$ as

$D^{h}\coloneqq\sum_{i|P_{i}\;\mathrm{dominates}\;Y}a_{i}P_{i},\quad
D^{v}\coloneqq D-D^{h}.$

A log pair $(X,B)$ consists in a normal variety $X$ and a divisor $B$ such
that $K_{X}+B$ is $\mathbb{Q}$-Cartier.

Given a coherent sheaf $\mathcal{F}$ of rank $r$ on a normal variety we define
$\operatorname{det}\mathcal{F}\coloneqq(\bigwedge^{r}\mathcal{F})^{**}$.

###  2.1. (Locally) stable families

We recall the definition of locally stable family of pairs over a reduced base
as in [Kol22, Definition-Theorem 4.7].

######  Definition 2.1.

Let $S$ be a reduced scheme, $f\colon X\rightarrow S$ a flat morphism of
finite type and $f\colon(X,\Delta)\rightarrow S$ a well defined family of
pairs (see [Kol22, Theorem-Definition 4.7]). Assume that $(X_{s},\Delta_{s})$
is slc for every $s\in S$. Then $f\colon(X,\Delta)\rightarrow S$ is locally
stable if the following equivalent conditions holds.

  * (1)

$K_{X/S}+\Delta$ is $\mathbb{R}$-Cartier.

  * (2)

$f_{T}\colon(X_{T},\Delta_{T})\rightarrow T$ is locally stable whenever $T$ is
the spectrum of a DVR and $q\colon T\rightarrow S$ is a morphism (see [Kol22,
Theorem-Definition 2.3] for the notion of a locally stable family over a DVR).

  * (3)

There is a closed subset $Z\subset X$ such that ${\mathrm{c}odim}(Z\cap
X_{s},X_{s})\geq 3$ for all $s\in S$ and $f_{X\setminus Z}\colon(X\setminus
Z)\rightarrow S$ satisfies the above (1-2).

Such a family is called stable if, in addition, $K_{X/S}+\Delta$ is $f$-ample.

By [Kol22, Theorem 4.8] if $f\colon(X,\Delta)\rightarrow S$ is a (locally)
stable family over a reduced base $S$ and $\phi\colon W\rightarrow S$ is a
morphism with $W$ reduced, then the family over $W$ obtained by fibre product
is again a (locally) stable family.

We recall the following criterion for local stability.

######  Proposition 2.2.

Let $S$ be a smooth variety and $p\colon(X,\Delta)\rightarrow S$ a projective
morphism with $\Delta\geq 0$. Then $p\colon(X,\Delta)\rightarrow S$ is locally
stable if and only if the pair $(X,\Delta+p^{*}D)$ is semi-log canonical for
every reduced simple normal crossing divisor $D\subset S$.

###### Proof.

This is a direct consequence of [ACSS21, Lemma 2.12] and [Kol22, Corollary
4.55]. ∎

###  2.2. Property $(*)$, BP stable and all that

We recall some definitions from [ACSS21].

Let $f\colon X\rightarrow Z$ be a projective morphism between normal varieties
and let $(X,B)$ be a log pair. We say that $f\colon(X,B)\rightarrow Z$ (or
$(X/Z,B)$ when the morphism $f$ is understood) is generically log canonical or
GLC, if $Z$ is irreducible and $(X,B)$ is log canonical over the generic point
of $Z$. In this case, we say that $(X/Z,B)$ is a GLC pair induced by the
morphism $f$. When $f\colon X\rightarrow Z$ is a contraction, i.e.,
$f_{*}\mathcal{O}_{X}=\mathcal{O}_{Z}$, we will refer to $f$ as a GLC
contraction. When $f$ is a GLC contraction and
$\operatorname{dim}X>\operatorname{dim}Z$, we will refer to $f$ as a GLC
fibration.

######  Definition 2.3.

Let $(X/Z,B)$ be a GLC pair induced by a morphism $f\colon X\to Z$. The pair
$(X/Z,B)$ is said to be BP stable (resp. BP semi-stable) over $Z$ if
$K_{Z}+B_{Z}$ is $\mathbb{R}$-Cartier and for any contraction
$f^{\prime}\colon X^{\prime}\to Z^{\prime}$ which is birationally equivalent
to $f$ and such that the induced maps $\alpha\colon Z^{\prime}\to Z$ and
$\beta\colon X^{\prime}\to X$ are projective birational morphisms, if
$B^{\prime}_{Z^{\prime}}$ is the discriminant of the induced pair
$(X^{\prime}/Z^{\prime},B^{\prime})$ and if we write

$K_{Z^{\prime}}+B^{\prime}_{Z}=\alpha^{*}(K_{Z}+B_{Z})$

then $B^{\prime}_{Z}=B^{\prime}_{Z^{\prime}}$ (resp. $B^{\prime}_{Z}\geq
B^{\prime}_{Z^{\prime}}$).

We say that a GLC pair $(X/Z,B)$ induced by a morphism $f\colon X\to Z$
satisfies the boundary property if there exist a surjective projective
morphism $f^{\prime}\colon X^{\prime}\to Z^{\prime}$ between normal varieties,
projective birational morphisms $\alpha\colon Z^{\prime}\to Z$ and
$\beta\colon X^{\prime}\to X$, and a commutative diagram

${X^{\prime}}$${X}$${Z^{\prime}}$${Z}$$\scriptstyle{\beta}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$$\scriptstyle{\alpha}$

such that the induced pair $(X^{\prime}/Z^{\prime},B^{\prime})$ is BP stable
over $Z^{\prime}$. Here, we define $B^{\prime}$ via the identity
$K_{X^{\prime}}+B^{\prime}=\beta^{*}(K_{X}+B)$.

######  Definition 2.4.

Let $(X/Z,B)$ be a GLC pair induced by a morphism $f\colon X\to Z$. We say
that $(X/Z,B)$ satisfies Property $(*)$ if $f$ is a projective contraction and
the following hold:

  1. (1)

there exists a reduced divisor $\Sigma_{Z}$ on $Z$ such that $(Z,\Sigma_{Z})$
is log smooth and the vertical part of $B$ is a reduced divisor that coincides
with $f^{-1}(\Sigma_{Z})$; and

  2. (2)

for any closed point $z\in Z$ and for any $\Sigma\geq\Sigma_{Z}$ reduced
divisor on $Z$ such that $(Z,\Sigma)$ is log smooth around $z$, we have that
$(X,B+f^{*}(\Sigma-\Sigma_{Z}))$ is log canonical around $f^{-1}(z)$.

Given a generalised pair $(X,B,\mathbf{M})$ and a contraction $f\colon
X\rightarrow Z$ such that $(X,B,\mathbf{M})$ is generalised log canonical
above the generic point of $Z$ we say that $(X/Z,B,\mathbf{M})$ is a
generalised GLC pair. If a generalised GLC pair satisfies the conditions of
Definition 2.4 for a generalised pair, mutatis mutandis, we will say that
$(X/Z,B,\mathbf{M})$ satisfies generalised Property $(*)$.

######  Lemma 2.5.

Let $f\colon(X,B,\mathbf{M})\rightarrow Z$ be a generalised GLC contraction
which satisfies generalised Property $(*)$ and let $(Z,\Sigma_{Z})$ be the
associated log smooth pair. Let $W$ a generalised log canonical centre of
$(X,B,\mathbf{M})$. Then $f(W)$ is a log canonical centre of $(Z,\Sigma_{Z})$.

###### Proof.

Follows directly from the definition of generalised Property $(*)$. ∎

######  Lemma 2.6.

Let $f\colon(X,B)\rightarrow Z$ be a GLC contraction where $B\geq 0$. Assume
that $(X,B)$ is log canonical and that every log canonical centre of $(X,B)$
dominates $Z$. Then, if a general fibre of $f$ has a good minimal model, then
$(X,B)$ has a good minimal model over $Z$.

###### Proof.

By [LT22, Theorem 2.9], we may show the claim up to extracting some log
canonical places of $(X,B)$. Furthermore, since all log canonical places are
horizontal over $Z$, by [HX13, Theorem 1.1], it suffices to show the claim
over a non-empty open subset of $Z$. Therefore, we may assume that $(X,B)$ is
dlt, and throughout the proof we are free to shrink $Z$ as needed.

Let $\pi\colon X^{\prime}\rightarrow X$ be a log resolution of $(X,B)$ that
only extracts valuations with positive log discrepancy, which exists by [KM98,
Theorem 2.44]. We denote the strict transform of $B$ by $B^{\prime}$, and we
let $E^{\prime}$ be the reduced $\pi$-exceptional divisor. Up to shrinking
$Z$, we may assume that $(X^{\prime},B^{\prime}+E^{\prime})\rightarrow Z$ is
log smooth. By construction, a good minimal model for $(X,B)$ over $Z$ is also
a good minimal model for $(X^{\prime},B^{\prime}+E^{\prime})$ over $Z$, and
vice versa.

By assumption, for $z\in Z$ general, $(X_{z},B_{z})$ has a good minimal model.
Since $\pi$ is a fibrewise log resolution that only extracts valuations of
positive log discrepancy, this good minimal models is also a good minimal
model for $(X^{\prime}_{z},B^{\prime}_{z}+E^{\prime}_{z})$. Then, by [HMX18,
Theorem 1.2], $(X^{\prime},B^{\prime}+E^{\prime})$ has a good minimal over
$Z$. In turn, this is a good minimal model for $(X,B)$ over $Z$, and the claim
follows. ∎

######  Lemma 2.7.

Let $f\colon(X,B)\rightarrow Z$ be a GLC contraction where $B\geq 0$ and which
satisfies Property $(*)$. Suppose that a general fibre of $(X,B)\rightarrow Z$
admits a good minimal model. Then $(X,B)$ has a good minimal model over $Z$.

###### Proof.

By Lemma 2.5, we see that for $0<t\ll 1$ every log canonical centre of
$(X,B-tf^{*}\Sigma)$ dominates $Z$ and $B-tf^{*}\Sigma\geq 0$. Then, by Lemma
2.6, $(X,B-tf^{*}\Sigma)$ admits a good minimal model over $Z$. Since
$K_{X}+B\sim_{\mathbb{Q}}K_{X}+B-tf^{*}\Sigma$ we see that this is a good
minimal model for $(X,B)$ over $Z$ as well. ∎

######  Lemma 2.8.

Let $f\colon(X,B)\rightarrow Z$ be a GLC contraction which satisfies Property
$(*)$ and such that $B=B^{h}\geq 0$. Then $f\colon(X,B)\rightarrow Z$ is a
locally stable family. Conversely, any locally stable family over a smooth
base satisfies Property $(*)$.

###### Proof.

From the equality $B=B^{h}$ we see that if $\Sigma_{Z}$ is the reduced divisor
associated to $(X/Z,B)$ then $\Sigma_{Z}=0$. The definition of Property $(*)$
together with Proposition 2.2 immediately implies our claim. ∎

###  2.3. Preliminaries on foliations

We will need some ideas from the theory of foliations during our analysis of
the variation of a family of varieties. For completeness, we provide a brief
summary of some relevant definitions and results from the study of foliations
on algebraic varieties.

Given a normal variety $X$, we define a foliation $\mathcal{F}$ to be the
datum of a saturated subsheaf $T_{\mathcal{F}}\subset T_{X}$, called the
tangent sheaf of the foliation, such that $T_{\mathcal{F}}$ is closed under
Lie bracket. The rank of $\mathcal{F}$ is defined to be the rank of
$T_{\mathcal{F}}$ as a sheaf. We define the normal sheaf to be
$N_{\mathcal{F}}\coloneqq(T_{X}/T_{\mathcal{F}})^{**}$ and the cotangent sheaf
to be $\Omega^{1}_{\mathcal{F}}\coloneqq T_{\mathcal{F}}^{*}$.

We refer to [Bru00] for basic notions and definitions regarding foliations. We
recall that given a smooth foliation $\mathcal{F}$ on a smooth variety $X$, at
any point $x\in X$ there is an analytic neighbourhood $U$ and a holomorphic
submersion $F\colon U\rightarrow V$ such that $T_{\mathcal{F}}|_{U}=T_{U/V}$.
The leaves of $\mathcal{F}$ are locally given by the fibres of this
submersion.

We refer to [CS21, §2.3] for the notion of the transform of a foliation under
a birational map.

Given a variety, a foliation $\mathcal{F}$ and a sheaf $E$, we say that $E$
admits a partial $\mathcal{F}$-connection provided there exists a
$\mathbb{C}$-linear map $\nabla\colon E\rightarrow
E\otimes\Omega^{1}_{\mathcal{F}}$ which satisfies the Leibniz rule. We recall
that the normal sheaf of a foliation $N_{\mathcal{F}}$ is always equipped with
a partial $\mathcal{F}$-connection, called the Bott connection.

######  Lemma 2.9.

Let $X$ be a normal variety, let $\mathcal{F}$ be a foliation on $X$ and let
$L$ be a line bundle equipped with a partial $\mathcal{F}$-connection,
$\nabla\colon L\rightarrow L\otimes\Omega^{1}_{\mathcal{F}}$. Let $s\in
H^{0}(X,L)$ and suppose that $\nabla(s)=0$. Then $D=(s=0)$ is
$\mathcal{F}$-invariant.

###### Proof.

The invariance of a divisor may be checked on a big open subset and so we may
freely assume that $X$ and $\mathcal{F}$ are smooth.

Let $\\{U_{i}\\}$ be a trivialising open cover for $L$ and let $t_{i}$ be
local generators for $L$. Write $s=f_{i}t_{i}$, where $f_{i}=0$ is a local
equation for $D$. We may compute

$0=\nabla(s)=\nabla(f_{i}t_{i})=df_{i}\otimes t_{i}+f_{i}\nabla(t_{i})\mod
L\otimes N^{*}_{\mathcal{F}}.$

Since $\nabla(t_{i})=\omega\otimes t_{i}$ for a holomorphic one form $\omega$,
we see that $df_{i}+f_{i}\omega=0\mod N^{*}_{\mathcal{F}}$, from which we may
conclude that $df_{i}=0\mod N^{*}_{\mathcal{F}}$. This implies that
$\\{f_{i}=0\\}$ is $\mathcal{F}$-invariant as required. ∎

Let $X$ be a smooth variety, let $X_{0}\subset X$ be a local complete
intersection subvariety and let $\mathcal{F}$ be a smooth foliation on $X$. We
say that $X_{0}$ is everywhere transverse to $\mathcal{F}$ provided the
natural map $T_{\mathcal{F}}|_{X_{0}}\rightarrow N_{X_{0}/X}$ is an
isomorphism.

######  Lemma 2.10.

Let $f\colon P\rightarrow Z$ be a proper fibration between complex manifolds,
let $\mathcal{F}$ be a smooth foliation on $P$ with
$\operatorname{rk}\mathcal{F}=\operatorname{dim}Z$ and let $Y$ be a
$\mathcal{F}$-invariant subvariety. Let $z\in Z$ be a closed point and let
$P_{z}$ and $Y_{z}$ be the fibre over $z$ of $P$ and $Y$ respectively.

If $P_{z}$ is everywhere transverse to $\mathcal{F}$ then there exists an open
Euclidean neighbourhood $U\subset Z$ of $z$ such that $(f^{-1}(U),Y\cap
f^{-1}(U))\cong(P_{z},Y_{z})\times U$ as analytic spaces and
$\mathcal{F}|_{f^{-1}(U)}$ is given by projection onto the first coordinate.

###### Proof.

Since $P_{z}$ is everywhere transverse to $\mathcal{F}$, by definition, the
natural map $T_{P}|_{P_{z}}\rightarrow N_{P_{z}/P}$ is surjective, in
particular, $P_{z}$ is smooth.

We recall the graphic neighbourhood construction of [BM16, §2.1]. We remark
that [BM16] only considers the graphical neighbourhood in the case where
$P_{z}$ is a curve, but the construction works for general smooth varieties
just as well. Let $\delta\colon P_{z}\rightarrow P_{z}\times P$ be the
diagonal embedding. We may find a germ $\Gamma$ of an analytic variety
containing $\delta(P_{z})$ so that for all $t\in P_{z}$ we have
$q(p^{-1}(t))\subset P$ is a germ of the leaf of $\mathcal{F}$ through $t$,
where $p\colon\Gamma\rightarrow P_{z}$ and $q\colon\Gamma\rightarrow P$ are
the projections.

Observe $\operatorname{dim}\Gamma=\operatorname{dim}P$ and so in fact $q$
gives an isomorphism between an open neighbourhood $V$ of $P_{z}$ and
$\Gamma$. By the properness of the fibration, perhaps shrinking this
neighbourhood, we may assume that $V=f^{-1}(U)$ for some open Euclidean
neighbourhood of $z$ and so we get a morphism $(p\circ q^{-1},f)\colon
V\rightarrow P_{z}\times U$ which is easily seen to be an isomorphism.

Since the fibres of $p\circ q^{-1}$ are exactly the leaves of $\mathcal{F}$ we
see that if $Y$ is an invariant subvariety, then $(p\circ q^{-1})(Y)=Y_{z}$,
and we may conclude. ∎

We also recall some well known properties relating holomorphic connections and
foliations.

######  Lemma 2.11.

Let $X$ be a smooth variety and let $E$ be a vector bundle on $X$ which admits
a flat holomorphic connection $\nabla\colon E\rightarrow
E\otimes\Omega^{1}_{X}$. Let $p\colon P\coloneqq\mathbb{P}(E)\rightarrow X$ be
the associated projective space bundle. Then

  1. (1)

$\nabla$ gives a splitting $p^{*}T_{X}\rightarrow T_{P}$ whose image defines a
smooth foliation $\mathcal{F}$ on $P$;

  2. (2)

$\mathcal{O}_{P}(1)$ admits a partial $\mathcal{F}$-connection; and

  3. (3)

if $E^{\prime}$ is a vector bundle on $P$ and admits a partial
$\mathcal{F}$-connection, then $p_{*}E^{\prime}$ admits a holomorphic
connection.

###### Proof.

Item (1) is a standard fact and we omit the proof.

To prove item (2) let $U\subset X$ be a small open subset such that $T_{X}$ is
generated by $\partial/\partial x_{1},\dots,\partial/\partial x_{n}$ and let
$s_{i}$ be a basis of flat sections of $E$. Let $\partial_{i}$ be the lift of
$\partial/\partial x_{i}$ to a vector field on $P$ given by $\nabla$.

Write $p^{-1}(U)=\cup_{i}U_{i}$ where $U_{i}=\\{\overline{s}_{i}\neq 0\\}$,
where we denote by $\overline{s}_{i}$ the section of $\mathcal{O}_{P}(1)$
corresponding to $s_{i}$.

We define a partial connection $\overline{\nabla}$ on $\mathcal{O}_{P}(1)$ by
requiring $\overline{\nabla}(\overline{s}_{i})=0$ for all $i$. This is well
defined because $dg_{ij}=0\mod N^{*}_{\mathcal{F}}$ where
$g_{ij}=\overline{s_{i}}/\overline{s_{j}}$.

Item (3) is straightforward consequence of the fact that
$p_{*}\Omega^{1}_{\mathcal{F}}=\Omega^{1}_{X}$. ∎

We remark that under the correspondence in item (1) of Lemma 2.11 the leaves
of $\mathcal{F}$ are exactly the flat sections of $\nabla$.

Given a normal variety $X$ we say a resolution of singularities $p\colon
X^{\prime}\rightarrow X$ is equivariant provided $p_{*}T_{X^{\prime}}=T_{X}$.
Such resolutions exist by [Kol07, Theorem 3.26]. We remark that if
$\mathcal{F}$ is a foliation on $X$ such that $T_{\mathcal{F}}$ is locally
free, then there exists a foliation on $X^{\prime}$ such that
$p_{*}T_{\mathcal{F}^{\prime}}=T_{\mathcal{F}}$.

###  2.4. Numerical flatness to isotriviality

We recall that a vector bundle $E$ on a smooth variety is said to be
numerically flat provided $E$ is nef and $-c_{1}(E)$ is nef.

######  Lemma 2.12.

Let $f\colon(X,B)\rightarrow Z$ be a projective fibration over a smooth
projective variety $Z$, where $B\geq 0$, $(X,B)$ is klt and $K_{X}+B$ is
$f$-semi-ample.

Suppose that for all $m\gg 0$ sufficiently divisible we have that the sheaf
$E_{m}\coloneqq f_{*}\mathcal{O}_{X}(ml(K_{X/Z}+B))$ is locally free and
numerically flat.

Let $h\colon X\rightarrow Y$ be the relatively ample model of $(X,B)$ over
$Z$, and let $(Y,B_{Y},\mathbf{M})$ be the generalized pair induced by the LC
trivial fibration $h$.

  1. (1)

There exists a foliation $\mathcal{F}_{Y}$ on $Y$ everywhere transverse to the
fibres of $Y\rightarrow Z$, such that if $0\leq D$ is any divisor such that
$D\sim_{\mathbb{Q}}\mathbf{M}_{Y}$ then $\mathcal{F}_{Y}$-leaves $B_{Y}+D$
invariant.

  2. (2)

In particular, $(Y,B_{Y}+D)\rightarrow Y$ is a locally trivial family.

  3. (3)

If $(Y_{0},(B_{Y}+D)_{0})$ denotes a general fibre of $(Y,B_{Y}+D)\rightarrow
Z$, then there exists a finite étale cover $Z^{\prime}\rightarrow Z$ such that
$(Y,B_{Y}+D)\times_{Z}Z^{\prime}\cong(Y_{0},(B_{Y}+D)_{0})\times Z^{\prime}$.

###### Proof.

Step 1. In this step we construct a foliation.

For $m\gg 0$ and sufficiently divisible, we have an embedding $Y\rightarrow
P\xrightarrow{p}Z$, where $P\coloneqq\mathbb{P}(E_{m})$. Set
$L=\mathcal{O}_{P}(1)|_{Y}$. The connection $\nabla_{m}$ on $E_{m}$ guaranteed
to exist by [Sim92, §3] is in fact a holomorphic connection (see [Den21] or
[Ou21, Theorem 3.5]) and therefore by Lemma 2.11 gives a smooth foliation
$\mathcal{H}$ on $P$ such that $T_{\mathcal{H}}\cong p^{*}T_{Z}$, in
particular $T_{\mathcal{H}}$ is locally free. Observe that
$K_{\mathcal{H}}\cong p^{*}K_{Z}$. Moreover, $\mathcal{H}$ is everywhere
transverse to the fibres of $P\rightarrow Z$.

It follows from [Ou21, Corollary 3.9] that for all $k>0$ the kernel of
$\operatorname{Sym}^{k}E_{m}\rightarrow L^{\otimes k}\rightarrow 0$ is a
$\operatorname{Sym}^{k}\nabla_{m}$-invariant subbundle, and therefore $Y$ is
invariant by $\mathcal{H}$. We may then restrict $\mathcal{H}$ to $Y$ to get a
foliation $\mathcal{F}_{Y}$ on $Y$.

Step 2.  In this step we show local triviality of $Y\rightarrow Z$.

Lemma 2.10 implies that $Y\rightarrow Z$ is locally trivial. It remains to
show that $(Y,B_{Y}+D)$ is locally trivial. This will follow by another
application of Lemma 2.10 if we can show ${\mathrm{S}upp}(B_{Y}+D)$ is
$\mathcal{F}_{Y}$-invariant.

Step 3. In this step we show local triviality of $(Y,B_{Y}+D)\rightarrow Z$.

Let $s\colon(Y^{\prime},B_{Y}^{\prime}+D^{\prime})\rightarrow(Y,B_{Y}+D)$ be a
small $\mathbb{Q}$-factorialisation, which exists by [BCHM10, Corollary
1.4.4], and let
$\sigma\colon(\overline{Y},\overline{B}_{Y}+\overline{D})\rightarrow(Y,B_{Y}+D)$
be the ample model of $K_{Y}+(1+\eta)(B^{\prime}_{Y}+D^{\prime})$ over $Y$ for
$0<\eta\ll 1$, which exists by [BCHM10, Theorem 1.2]. Note that
$\overline{B}_{Y}+\overline{D}$ is $\sigma$-ample by construction.

Let $\bar{p}\colon\overline{Y}\rightarrow Z$ be the composition of $p$ with
$\sigma$. We claim that $\mathcal{F}_{Y}$ lifts to a foliation
$\overline{\mathcal{F}}_{Y}$ on $\overline{Y}$. Indeed, let $W\rightarrow Y$
be an equivariant resolution of singularities, and let $\mathcal{F}_{W}$ be
the lift of $\mathcal{F}_{Y}$ to $W$. By construction of a small
$\mathbb{Q}$-factorialisation, there exists a birational contraction
$\rho\colon W\dashrightarrow\overline{Y}$. We may take
$\overline{\mathcal{F}}_{Y}=\rho_{*}\mathcal{F}_{W}$ to be of the transform of
a foliation under $\rho$. Note that
$K_{\overline{\mathcal{F}}_{Y}}=\sigma^{*}K_{\mathcal{F}_{Y}}$. By Lemma 2.11
$\mathcal{O}_{\overline{Y}}(m(K_{\overline{Y}/Z}+\overline{B}_{Y}+D))$ admits
a partial $\overline{\mathcal{F}}_{Y}$-connection.

By the Bott partial connection we know that
$\operatorname{det}N^{*}_{\overline{\mathcal{F}}_{Y}}\cong
K_{\overline{Y}}-K_{\overline{\mathcal{F}}_{Y}}=K_{\overline{Y}/Z}$ also
admits a partial connection. It follows that
$\mathcal{O}_{\overline{Y}}(m(\overline{B}_{Y}+\overline{D}))$ admits a
partial $\overline{\mathcal{F}}_{Y}$-connection, and so
$\mathcal{O}_{\overline{Y}}(m(K_{\overline{Y}/Z}+\overline{B}_{Y}+D)+k(\overline{B}_{Y}+\overline{D}))$
admits a partial $\overline{\mathcal{F}}_{Y}$-connection for all $m\gg k\gg
0$.

By Lemma 2.11, $\tilde{E}_{m}\coloneqq
p_{*}\mathcal{O}_{\overline{Y}}(m(K_{\overline{Y}/Z}+\overline{B}_{Y}+D)+k(\overline{B}_{Y}+\overline{D}))$
is a vector bundle which admits a holomorphic connection, and is therefore
numerically flat. Let $\tilde{\nabla}_{m}$ be the connection guaranteed to
exist by [Sim92, §3]. Let $s\in
H^{0}(\overline{Y},\mathcal{O}_{\overline{Y}}(k(\overline{B}_{Y}+\overline{D}))$
be a global section such that $\\{s=0\\}=k(\overline{B}_{Y}+\overline{D})$,
and let $\varphi\colon E_{m}\rightarrow\tilde{E}_{m}$ be the morphism induced
by multiplication by $s$. By the proof of [Ou21, Corollary 3.9] we see that
$\varphi$ is a morphism of vector bundles equipped with connections, i.e., if
$t$ is any flat local section of $E_{m}$, we see that $t\otimes s$ is a flat
local section of $\tilde{E}_{m}$.

Without loss of generality we may assume that $m\gg k\gg 0$ are sufficiently
large so that
$m(K_{\overline{Y}/Z}+\overline{B}_{Y}+D)+k(\overline{B}_{Y}+\overline{D})$ is
$\bar{p}$-very ample. Let $\tilde{P}\coloneqq\mathbb{P}(\tilde{E}_{m})$ and by
repeating the construction of Step 1, we produce a foliation
$\tilde{\mathcal{F}}$ on $\overline{Y}$ which is everywhere transverse to the
fibres of $\overline{Y}\rightarrow Z$. Let us continue to denote by
$\tilde{\nabla}_{m}$ the $\tilde{\mathcal{F}}$-partial connection on
$m(K_{\overline{Y}/Z}+\overline{B}_{Y}+\overline{D})+k(\overline{B}_{Y}+\overline{D})$
guaranteed by Lemma 2.11. As noted, if $t$ is any flat local section of
$E_{m}$, then $\tilde{\nabla}_{m}(t\otimes s)=0$, and so by Lemma 2.9 it
follows that $\\{t\otimes s=0\\}$ is invariant, from which we may conclude
that
${\mathrm{S}upp}(\overline{B}_{Y}+\overline{D})={\mathrm{S}upp}(\\{s=0\\})$ is
invariant.

Since $\varphi$ was a morphism of vector bundles equipped with connections it
follows that $\mathcal{F}_{Y}=\sigma_{*}\tilde{\mathcal{F}}$, and we so
$\mathcal{F}_{Y}$ leaves $B_{Y}+D$ invariant.

Our final claim follows by noting that the automorphism group of a variety of
log general type is finite, and so our locally trivial family is in fact
trivial after a finite étale base change. ∎

We remark that the use of [Sim92] in the above proof is closely related to a
similar application in [CH19]. The use of [Sim92] to produce a foliation
transverse to a fibration was also considered in [Ou21].

######  Proposition 2.13.

Let $f\colon(X,B)\rightarrow Z$ be a projective fibration over a smooth
projective variety $Z$, where $B\geq 0$, $(X,B)$ is klt and $K_{X}+B$ is
$f$-semi-ample.

Suppose that for all $m\gg 0$ sufficiently divisible we have that
$E_{m}\coloneqq f_{*}\mathcal{O}(ml(K_{X/Z}+B))$ is locally free and
numerically flat.

Let notation be as in Lemma 2.12.

  1. (1)

There exists a resolution of singularities
$(\overline{X},\overline{B})\rightarrow(X,B)$ such that the foliation
$\mathcal{F}_{Y}$ lifts to a foliation $\mathcal{F}_{\overline{X}}$ on
$\overline{X}$. Moreover, there is a Zariski open subset $U\subset Y$ such
that $\mathcal{F}_{\overline{X}}$ is everywhere transverse to the fibres of
$V\coloneqq\overline{X}\times_{Y}U\rightarrow U$ and
$\mathcal{F}_{\overline{X}}$ leaves $\overline{B}\cap V$ invariant.

  2. (2)

For general $t,t^{\prime}\in Z$, there exists an isomorphism $\psi\colon
Y_{t}\rightarrow Y_{t^{\prime}}$ such that if
$(\overline{X}_{s},\overline{B}_{s})$ denotes the fibre of
$(\overline{X},\overline{B})\rightarrow Y$ over $s$, then for general $s\in
Y_{t}$,
$(\overline{X}_{s},\overline{B}_{s})\cong(\overline{X}_{\psi(s)},\overline{B}_{\psi(s)})$.

  3. (3)

For general $t,t^{\prime}\in Z$ the geometric generic fibres of
$(X_{t},B_{t})\rightarrow Y_{0}$ and
$(X_{t^{\prime}},B_{t^{\prime}})\rightarrow Y_{0}$ are isomorphic, i.e. there
exists a finite cover $\tilde{Y}_{0}\rightarrow Y_{0}$ (perhaps depending on
$t$ and $t^{\prime}$) such that $(X_{t},B_{t})\times_{Y_{0}}\tilde{Y}_{0}$ and
$(X_{t^{\prime}},B_{t^{\prime}})\times_{Y_{0}}\tilde{Y}_{0}$ are birational.

  4. (4)

If $y\in Y$ is a general point and $L$ is the leaf of $\mathcal{F}_{Y}$
passing through $y$, then the family $(X,B)\times_{Y}L\rightarrow L$ is
isotrivial.

###### Proof.

Following [Amb05, §2], we may find a generically finite morphism
$\sigma\colon\widetilde{Y}\rightarrow Y$, a morphism
$\tau\colon\widetilde{Y}\rightarrow Y^{!}$ and a big and nef divisor
$\mathbf{M}_{Y^{!}}$ such that
$\overline{\tau^{*}\mathbf{M}_{Y^{!}}}=\sigma^{*}\mathbf{M}$ as equality of
b-divisors. Notice that we use b-divisors to express this identity, since
$\mathbf{M}$ may not descend onto $Y$, and so $\mathbf{M}$ may not agree with
the Cartier closure of $\mathbf{M}_{Y}$. It follows that for some $k\gg 0$
some sublinear system of $~{}|k\mathbf{M}_{Y}|~{}$ defines a rational map
$\phi\colon Y\dashrightarrow W$ such that at the generic point $\eta$ of $Y$
$~{}\operatorname{Ker}d\phi$ agrees with $\operatorname{Ker}\kappa$, where
$\kappa\colon T_{Y,\eta}\rightarrow
H^{1}(T_{\overline{X}_{\eta}}(-\log\overline{B}))$ is the Kodaira–Spencer map,
see [Amb05, Theorem 2.2], and $(\overline{X},\overline{B})\rightarrow(X,B)$ is
a log smooth model as guaranteed by [Amb05, Lemma 1.1]. Observe that we may
not take the full linear series $~{}|k\mathbf{M}_{Y}|~{}$, since $\mathbf{M}$
may not descend onto $Y$. Let $\bar{h}\colon\overline{X}\rightarrow Y$ be the
natural morphism.

The general fibre of $Y\dashrightarrow W$ is generically an intersection of
divisors $D\sim k\mathbf{M}_{Y}$. By Lemma 2.12, these divisors are all
$\mathcal{F}_{Y}$ invariant, and so the general fibre is $\mathcal{F}_{Y}$
invariant.It follows that at the generic point of $Y$, $T_{\mathcal{F}_{Y}}$
is contained in the kernel of the Kodaira–Spencer morphism $T_{Y}\rightarrow
R^{1}\bar{h}_{*}T_{\overline{X}/Y}(-\log\overline{B})$, as the latter
generically coincides with $\operatorname{Ker}d\phi$. Thus, at the generic
point of $Y$ we have a lift $\rho\colon h^{*}T_{\mathcal{F}_{Y}}\rightarrow
T_{\overline{X}}(-\log\overline{B})$, which induces a foliation
$\mathcal{F}_{\overline{X}}$ on $\overline{X}$. Let $U$ be the largest open
subset over which we have a lift $h^{*}T_{\mathcal{F}_{Y}}\rightarrow
T_{\overline{X}}(-\log\overline{B})$.

By construction $\mathcal{F}_{\overline{X}}$ is everywhere transverse to the
fibres of $\overline{X}\rightarrow Y$ over $U$, and since $\rho$ takes values
in $T_{\overline{X}}(-\log\overline{B})$ we see that
$\mathcal{F}_{\overline{X}}$ leaves the support of $\overline{B}$ invariant.
This proves item (1).

We now prove item (2). Fix a general point $t\in Z$, and let $t^{\prime}\in Z$
be a nearby point. By Lemma 2.12, we have isomorphisms $\psi\colon
Y_{t}\rightarrow Y_{t^{\prime}}$ given by the flows of the foliation
$\mathcal{F}_{Y}$. Perhaps shrinking $U$, we may assume this gives an
isomorphism $\psi\colon U_{t}\rightarrow U_{t^{\prime}}$.

For any closed point $s\in U_{t}$, let $L$ denote a germ of a leaf of
$\mathcal{F}_{Y}$ through $s$, which up to shrinking, we may assume is
contained in $U$. By construction, we have $L\cap U_{t^{\prime}}=\psi(s)$.
Consider the family
$(\overline{X}_{L},\overline{B}_{L})\coloneqq(\overline{X},\overline{B})\times_{Y}L$.
By construction, $\overline{X}_{L}$ is $\mathcal{F}_{\overline{X}}$-invariant,
and so we have a restricted foliation $\mathcal{F}_{\overline{X}_{L}}$ which
is everywhere transverse to the fibres of $\overline{X}\rightarrow L$ and
leaves $B$ invariant. Applying Lemma 2.10 to $\mathcal{F}_{\overline{X}_{L}}$
and $(\overline{X}_{L},\overline{B}_{L})\rightarrow L$, it follows that
$(\overline{X}_{s},\overline{B}_{s})\cong(\overline{X}_{\psi(s)},\overline{B}_{\psi(s)})$,
as required.

By considering the relative ${\mathrm{I}som}$ functor
${\mathrm{I}som}_{Y_{0}}((X_{t},B_{t}),(X_{t^{\prime}},B_{t^{\prime}}))$ for
general $t,t^{\prime}$, we see that standard arguments imply the geometric
generic fibres $(X_{t},B_{t})\rightarrow Y_{0}$ and
$(X_{t^{\prime}},B_{t^{\prime}})\rightarrow Y_{0}$ are isomorphic, from which
we may conclude item (3).

Item (4) is a direct consequence of the previous arguments. We remark that by
the last point in Lemma 2.12 the leaves of $\mathcal{F}_{Y}$ are all
algebraic. ∎

###  2.5. Harder–Narasimhan filtrations

Given a vector bundle $E$ on a smooth projective curve we define its slope to
be $\mu(E)\coloneqq\frac{\operatorname{deg}E}{\operatorname{rk}E}$. We define
$\mu_{\max}(E)$ to be the largest slope $\mu(F)$ among all non-zero subbundles
$0\rightarrow F\rightarrow E$, and we define $\mu_{\min}(E)$ to be the
smallest slope $\mu(Q)$ among non-zero quotient bundles $E\rightarrow
Q\rightarrow 0$.

######  Definition 2.14.

Let $f\colon(X,B)\rightarrow Z$ be a projective fibration to a smooth
projective curve, where $B\geq 0$, $(X,B)$ is log canonical. Let $l$ be a
positive integer such that $l(K_{X/Z}+B)$ is Cartier. Let $P\in Z$ be a closed
point. We define

$\lambda_{-}(X/Z,B)\coloneqq\sup\\{t:l(K_{X/Z}+B)-tf^{*}P\text{ is nef}\\}$

and

$\lambda_{+}(X/Z,B)\coloneqq\sup\\{t:l(K_{X/Z}+B)-tf^{*}P\text{ is pseudo-
effective}\\}.$

When clear from context we will write $\lambda_{-}=\lambda_{-}(X/Z,B)$ and
$\lambda_{+}=\lambda_{+}(X/Z,B)$.

We remark that $\lambda_{-}$ and $\lambda_{+}$ depend on the choice of $l$. We
feel this is unlikely to cause confusion and so do not include the choice of
$l$ in our notation.

######  Proposition 2.15.

Let $f\colon(X,B)\rightarrow Z$ be a GLC fibration where $B\geq 0$, $(X,B)$ is
log canonical and klt over the generic point of $Z$, $Z$ is a projective curve
and $B\geq 0$. Let $l>0$ be a sufficiently divisible integer such that
$l(K_{X/Z}+B)\eqqcolon L$ is Cartier and suppose that for all $m\gg 0$ we have
that $E_{m}\coloneqq f_{*}\mathcal{O}(ml(K_{X/Z}+B))$ is locally free. Suppose
in addition that $K_{X/Z}+B$ is relatively semi-ample. Then, for $m\gg 0$, we
have

$\lambda_{-}\leq\frac{1}{m}\mu_{{\mathrm{m}in}}(E_{m})\leq\frac{1}{m}\mu_{{\mathrm{m}ax}}(E_{m})\leq\lambda_{+}.$

###### Proof.

Suppose first that $K_{X/Z}+B$ is relatively ample. In this case the result
follows from [XZ20, Lemma-Definition 2.26] (see also [CP21, Proposition 5.4]
and [CP21, Proposition 6.4]) by taking $L=l(K_{X/Z}+B)$.

In general, let $g\colon X\rightarrow Y$ be the ample model of $K_{X/Z}+B$
over $Z$, and, perhaps replacing $l$ by a multiple we may find a Cartier
divisor $L^{\prime}$ be such that $g^{*}L^{\prime}\sim L$. By [Amb05, Theorem
0.2], we may write $L^{\prime}\sim_{\mathbb{Q}}K_{Y/Z}+B_{Y}$ where
$(Y,B_{Y})$ is klt over the generic point of $Z$. Perhaps replacing $l$ by
$2l$ we may assume that $L^{\prime}-(K_{Y/Z}+B_{Y})$ is $f$-ample. We have
equalities $\lambda_{+}(X/Z,B)=\lambda_{+}(Y/Z,B_{Y})$ and
$\lambda_{-}(X/Z,B)=\lambda_{-}(Y/Z,B_{Y})$ and so we may freely reduce to the
previous case in order to conclude. ∎

##  3\. Property $(*)$ pairs and locally stable families

######  Lemma 3.1.

Let $f\colon(X,B)\rightarrow Z$ be a GLC contraction which is also a locally
stable family of log varieties. The moduli part of $(X/Z,B)$ is $K_{X/Z}+B$.

###### Proof.

By Proposition 2.2 it follows that the discriminant $B_{Z}$ of $(X/Z,B)$ is
zero, and we may conclude. ∎

The following openness property of Property $(*)$ morphisms is implicit in
[ACSS21].

######  Proposition 3.2.

Let $f\colon(X,B)\rightarrow Z$ be a GLC contraction where $B\geq 0$. Suppose
that there exists a reduced divisor $\Sigma\subset Z$ such that $(Z,\Sigma)$
is log smooth and such that the vertical part of $B$ coincides with
$f^{-1}(\Sigma)$. Suppose moreover for all closed points $z\in Z$ there is a
reduced divisor $D\subset Z$ such that $(Z,\Sigma+D)$ is log smooth, $z$ is a
log canonical centre of $(Z,\Sigma+D)$, and $(X,B+f^{*}D)$ is log canonical in
a neighbourhood of $f^{-1}(z)$. Then $(X/Z,B)$ satisfies Property $(*)$.

###### Proof.

The problem is local on $Z$, so by replacing $Z$ by an open neighbourhood of
an arbitrary point we may assume that $Z$ is quasi-projective. Next, we reduce
to the case where $Z$ is projective. Indeed, we may first take a log smooth
compactification $(\overline{Z},\overline{\Sigma}+\overline{D})$ of
$(Z,\Sigma+D)$. Then, we may take a log canonical closure of $(X,B+f^{*}D)$
over $\overline{Z}$, call it $(\overline{X},\overline{B}+f^{*}\overline{D})$,
see [HX13]. By replacing $X$ (resp. $Z$) by $\overline{X}$ (resp.
$\overline{Z}$) we may freely assume that $X$ and $Z$ are projective.

The proof of [ACSS21, Proposition 2.19] implies the following fact. Let
$z^{\prime}\in Z$ be a point and $D_{z^{\prime}}$ a divisor such that

  1. (1)

$(Z,\Sigma+D_{z^{\prime}})$ is log smooth;

  2. (2)

$z^{\prime}$ is a log canonical centre of $(Z,\Sigma+D_{z^{\prime}})$; and

  3. (3)

$(X,B+f^{*}D_{z^{\prime}})$ is log canonical.

Then, if $D^{\prime}_{z^{\prime}}$ is any other divisor such that
$(Z,\Sigma+D^{\prime}_{z^{\prime}})$ is log smooth and $z^{\prime}$ is a log
canonical centre of $(Z,\Sigma+D^{\prime}_{z^{\prime}})$, it follows that
$(X,B+f^{*}D^{\prime}_{z^{\prime}})$ is log canonical. This in turn
immediately implies the proposition. ∎

######  Lemma 3.3.

Let $f\colon(X,B,\mathbf{M})\rightarrow Z$ be a generalised GLC contraction.
Consider a generalised dlt modification
$p\colon(X^{\prime},B^{\prime},\mathbf{M})\rightarrow(X,B,\mathbf{M})$.

If $(X/Z,B,\mathbf{M})$ satisfies generalised Property $(*)$, then
$(X^{\prime}/Z,B^{\prime},\mathbf{M})$ satisfies generalised Property $(*)$.

###### Proof.

Let $\Sigma\subset Z$ be the reduced divisor associated to
$(X/Z,B,\mathbf{M})$.

Note that $(X,B,\mathbf{M})$ is generalised log canonical. From the equality
$K_{X^{\prime}}+B^{\prime}+\mathbf{M}_{X^{\prime}}=p^{*}(K_{X}+B+\mathbf{M}_{X})$,
it is immediate that $(X,B+f^{*}D,\mathbf{M})$ is generalised log canonical if
and only if $(X^{\prime},B^{\prime}+p^{*}f^{*}D,\mathbf{M})$ is generalised
log canonical.

Thus to conclude it suffices to show that if $E$ is a vertical component of
$\lfloor B^{\prime}\rfloor$, then $p(E)$ is contained in a vertical component
of $\lfloor B\rfloor$. Generalised Property $(*)$ implies that, if $W\subset
X$ is any generalised log canonical centre of $(X,B,\mathbf{M})$, then by
Lemma 2.5 $f(W)$ is a log canonical centre of $(Z,\Sigma)$ and so $W$ is
contained in $\lfloor B\rfloor$. Since $p(E)$ is a log canonical centre, we
may therefore conclude. ∎

######  Lemma 3.4.

Let $f\colon(X,B)\rightarrow Z$ be a GLC contraction satisfying Property $(*)$
with $B\geq 0$. Suppose in addition that

  1. (1)

$f\colon X\rightarrow Z$ is equidimensional; and

  2. (2)

if $D\subset Z$ is any reduced divisor, then $f^{*}D$ is reduced.

Then $(X,B^{h})$ satisfies Property $(*)$.

###### Proof.

We first claim that $K_{X}+B^{h}$ is $\mathbb{Q}$-Cartier. Observe that
Property $(*)$ implies that, if $D$ is a vertical component of $B$, then in
fact $D$ is a component of $\lfloor B\rfloor$. This together with item (2)
implies that $B^{v}=f^{*}G$ where $G$ is a reduced smooth normal crossings
divisor on $Z$.

For any $z\in Z$, let $G^{\prime}\geq 0$ be reduced divisor such that
$(Z,G+G^{\prime})$ is log smooth and $z$ is a log canonical centre of
$(Z,G+G^{\prime})$. Since $(X,B)$ satisfies Property $(*)$, we see that
$(X,B^{h}+f^{*}(G+G^{\prime}))$ is log canonical. Since $z\in Z$ is arbitrary,
we may apply Proposition 3.2 to conclude $(X,B^{h})$ satisfies Property $(*)$.
∎

######  Proposition 3.5.

Let $f\colon(X,B)\rightarrow Z$ be an equidimensional GLC contraction which
satisfies Property $(*)$ and where $B\geq 0$.

Then, there exists a diagram

${Y}$${X}$${W}$${Z}$$\scriptstyle{g}$$\scriptstyle{s}$$\scriptstyle{f}$$\scriptstyle{r}$

where

  1. (1)

$r\colon W\rightarrow Z$ is a finite Galois morphism;

  2. (2)

$(Y,C)$ is the normalisation of $(X,B^{h})\times_{Z}W$; and

  3. (3)

$g\colon(Y,C)\rightarrow W$ is a locally stable family.

Moreover, if $M$ (resp. $N$) is the moduli part of $(X/Z,B)$ (resp.
$(Y/W,C)$), then $s^{*}M\sim N$, where $s\colon Y\rightarrow X$ is the natural
projection, and for all $k>0$ sufficiently divisible,
$r^{*}f_{*}\mathcal{O}(kM)\cong g_{*}\mathcal{O}(kN)$.

###### Proof.

Let $\Sigma_{Z}\subset Z$ be the reduced divisor associated to the Property
$(*)$ pair $(X/Z,B)$.

Because $X\rightarrow Z$ is equidimensional, then away from subsets of
codimension $\geq 2$ in $X$ and $Z$ we have that $f\colon X\rightarrow Z$ is
smooth, hence toroidal. Therefore, by [AK00, Proposition 5.1], we may find a
Kawamata cover $r\colon W\rightarrow Z$ which is finite and Galois with Galois
group $G$, see [KMM87, Theorem 1-1-1], such that if $Y$ is the normalisation
of $X\times_{W}Z$ then for any reduced divisor $D\subset W$ we have $g^{*}D$
is reduced, where $g\colon Y\rightarrow W$ is the natural projection.

Let $\Sigma\subset Z$ denote the branch locus of $r\colon W\rightarrow Z$. If
$C\subset Z$ is any divisor such that $f^{*}C$ is not reduced, then
$C\subset\Sigma_{Z}$ and $C\subset\Sigma$. Therefore, the prime divisors in
$Z$ over which $f$ does not have reduced fibres form a simple normal crossing
divisor. By the construction of a Kawamata cover, we may therefore choose
$r\colon W\rightarrow Z$ so that $\Sigma_{Z}\subset\Sigma$ and so that
$(Z,\Sigma)$ is log smooth. Replacing $B$ by $B+f^{*}(\Sigma-\Sigma_{Z})$ we
may assume that the branch locus of $s\colon Y\rightarrow X$ is contained in
$\lfloor B\rfloor$.

Define $\overline{C}$ by the identity $K_{Y}+\overline{C}=s^{*}(K_{X}+B)$. By
the Riemann–Hurwitz formula, we have that $\overline{C}\geq 0$. We claim that
$(Y/W,\overline{C})$ satisfies Property $(*)$. Indeed, let
$\Sigma_{W}=r^{-1}\Sigma_{Z}$ and let $w\in W$ be arbitrary. Let $H$ be a
divisor on $Z$ such that $(Z,\Sigma_{Z}+H)$ is log smooth and $r(w)$ is a log
canonical centre of $(Z,\Sigma_{Z}+H)$. Then $(W,\Sigma_{W}+r^{*}H)$ is log
smooth and $w$ is a log canonical centre of $(W,\Sigma_{W}+r^{*}H)$. By the
construction of Kawamata covers, $(W,\Sigma_{W})$ is log smooth, and the log
smoothness of $(W,\Sigma_{W}+r^{*}H)$ immediately follows from the fact that
$\Sigma+H$ is a simple normal crossing divisor. The claim regarding log
canonical centres follows from [KM98, Proposition 5.20]. Moreover,
$K_{Y}+\overline{C}+g^{*}r^{*}H=s^{*}(K_{X}+B+f^{*}H)$ and so
$(Y,\overline{C}+g^{*}r^{*}H)$ is log canonical. Letting $w$ vary over all
points of $W$ we may apply Proposition 3.2 to see that $(Y,\overline{C})$ has
Property $(*)$.

By Lemma 3.4, $(Y,C\coloneqq\overline{C}^{h})$ has Property $(*)$ and so we
may apply Lemma 2.8 to see that $(Y,C)$ is locally stable.

We have an equality $s^{*}M=N$ because $K_{Y}+\overline{C}=s^{*}(K_{X}+B)$ and
$K_{W}+\Sigma_{W}=r^{*}(K_{Z}+\Sigma_{Z})$. The final claim is a consequence
of the flatness of $r$ and [Har77, Proposition III.9.3]. ∎

######  Lemma 3.6.

Let $f\colon(X,B)\rightarrow Z$ and
$f^{\prime}\colon(X^{\prime},B^{\prime})\rightarrow Z^{\prime}$ be
birationally equivalent equidimensional GLC fibrations with $B,B^{\prime}\geq
0$ fitting into the following commutative diagram

${(X^{\prime},B^{\prime})}$${(X,B)}$${Z^{\prime}}$${Z.}$$\scriptstyle{\alpha}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$$\scriptstyle{\beta}$

Suppose moreover that the following hold

  1. (1)

$(X,B)$ and $(X^{\prime},B^{\prime})$ are crepant birational over the generic
point of $Z$;

  2. (2)

the moduli part $M$ (resp. $M^{\prime}$) of $(X/Z,B)$ (resp.
$(X^{\prime}/Z^{\prime},B^{\prime})$) is $f$-nef (resp. $f^{\prime}$-nef);

  3. (3)

$(X/Z,B)$ and $(X^{\prime}/Z^{\prime},B^{\prime})$ satisfy Property $(*)$ and
are BP stable;

  4. (4)

$f_{*}\mathcal{O}(nM)$ is locally free for $n\gg 0$ sufficiently divisible;
and

  5. (5)

$\operatorname{det}f^{\prime}_{*}\mathcal{O}(nM^{\prime})$ is nef for $n\gg 0$
sufficiently divisible.

Then for all $n\gg 0$ sufficiently divisible and $k>0$ we have isomorphisms
$\beta^{*}\operatorname{det}f_{*}\mathcal{O}(nM)\cong\operatorname{det}f^{\prime}_{*}\mathcal{O}(nM^{\prime})$.
Moreover, if $p\colon W\rightarrow X$ and $q\colon W\rightarrow X^{\prime}$
resolves the rational map $X^{\prime}\dashrightarrow X$ we have
$q^{*}(kM^{\prime}+(f^{\prime})^{*}\operatorname{det}f^{\prime}_{*}\mathcal{O}(nM^{\prime}))\sim_{\mathbb{Q}}p^{*}(kM+f^{*}\operatorname{det}f_{*}\mathcal{O}(nM))$.

###### Proof.

For ease of notation set $E=f_{*}\mathcal{O}(nM)$ and
$E^{\prime}=f^{\prime}_{*}\mathcal{O}(nM^{\prime})$. Let $q\colon R\rightarrow
X$ and $p\colon R\rightarrow X^{\prime}$ resolve the birational map
$\alpha\colon X^{\prime}\dashrightarrow X$. Let $g\colon R\rightarrow
Z^{\prime}$ be the composition of $f^{\prime}$ and $p$. By [ACSS21,
Proposition 2.21], we know that $q^{*}M\sim_{\mathbb{Q}}p^{*}M^{\prime}$.

By the projection formula, it follows that for $n\gg 0$ sufficiently divisible
we have $\beta_{*}E^{\prime}=E$. This gives us a morphism
$\beta^{*}E\rightarrow\beta^{*}\beta_{*}E^{\prime}\rightarrow E^{\prime}$,
which is an isomorphism away from ${\mathrm{e}xc}(\beta)$. Taking determinants
we get a non-zero morphism
$\beta^{*}\operatorname{det}E\rightarrow\operatorname{det}E^{\prime}$, which
is an isomorphism away from ${\mathrm{E}xc}(\beta)$ and so we can deduce that
$\operatorname{det}E^{\prime}-\beta^{*}\operatorname{det}E\sim_{\mathbb{Q}}\mathcal{O}(F)$
where $F\geq 0$ is $\beta$-exceptional.

By assumption $\operatorname{det}E^{\prime}$ is $\beta$-nef and so the same is
true of $F$, and so the negativity lemma, [KM98, Lemma 3.39], implies that
$F=0$. We deduce that
$\beta^{*}\operatorname{det}E\cong\operatorname{det}E^{\prime}$ and
$q^{*}(kM^{\prime}+(f^{\prime})^{*}\operatorname{det}E^{\prime})\sim_{\mathbb{Q}}p^{*}(kM+f^{*}\operatorname{det}E)$,
from which we may conclude. ∎

##  4\. On semi-ampleness of the moduli part

###  4.1. Stable pairs

######  Lemma 4.1.

Let $f\colon(X,B)\rightarrow Z$ be a family of stable log varieties where $Z$
is normal and projective. Let $M$ be the moduli part of $(X/Z,B)$.

  1. (1)

For $m\gg 0$ sufficiently divisible, $f_{*}\mathcal{O}(mM)$ is a vector bundle
whose formation commutes with base change.

  2. (2)

For $m\gg 0$ sufficiently divisible,
$\lambda_{m}\coloneqq\operatorname{det}f_{*}\mathcal{O}(mM)$ is semi-ample.

  3. (3)

For $m\gg 0$ sufficiently divisible and all $\epsilon\in\mathbb{Q}_{>0}$,
$M+\epsilon f^{*}\lambda_{m}$ is semi-ample.

###### Proof.

Part (1) is a direct consequence of [KP17, Lemma 7.7] (and its proof) and the
fact that $M=K_{X/Z}+B$.

To show (2) and (3) consider the following commutative diagram, which is
guaranteed to exist by [KP17, Corollary 6.20]:

(1)
${(X_{0},B_{0})}$${(\overline{X},\overline{B})}$${(X,B)}$${Z_{0}}$${\overline{Z}}$${Z}$$\scriptstyle{f_{0}}$$\scriptstyle{s}$$\scriptstyle{\sigma}$$\scriptstyle{\overline{f}}$$\scriptstyle{f}$$\scriptstyle{t}$$\scriptstyle{\tau}$

where all three of our vertical morphisms are families of stable log
varieties, $\tau$ is finite, and the moduli map associated to
$(X_{0},B_{0})\rightarrow Z_{0}$ is finite.

By [KP17, Corollary 7.3] and its proof for $m\gg 0$ sufficiently divisible,
$\lambda^{0}_{m}\coloneqq\operatorname{det}f_{0*}\mathcal{O}(mM_{0})$ is
ample, where $M_{0}$ is the moduli part of $(X_{0}/Z_{0},B_{0})$. This implies
that $M_{0}+\epsilon f_{0}^{*}\lambda^{0}_{m}$ is ample for all
$\epsilon\in\mathbb{Q}_{>0}$, since $M_{0}$ is globally nef and relatively
ample, see [KP17, Lemma 7.7 and proof of Theorem 7.1.1]. Next, note that we
have $\sigma^{*}M=s^{*}M_{0}$ and by (1) we know
$\tau^{*}\lambda_{m}=t^{*}\lambda^{0}_{m}$, which implies both (2) and (3). ∎

###  4.2. Intermediate Kodaira dimension

######  Lemma 4.2.

Let $(X,\Delta)$ be a log canonical pair with $\Delta\geq 0$, and let
$f\colon(X,\Delta)\rightarrow Z$ be a fibration with
$K_{X}+\Delta\sim_{\mathbb{Q}}0/Z$. Let $(Z,B_{Z},\mathbf{M})$ be the
generalized pair induced on $Z$ by the canonical bundle formula. Then, any
union of irreducible components of $\mathrm{Nklt}(Z,B_{Z},\mathbf{M})$ is
semi-normal.

###### Proof.

It follows immediately from [LX22, Theorem 1.6] and the fact that Du Bois
singularities are semi-normal. ∎

######  Lemma 4.3 ([FI22, Lemma 2.3]).

Let $p\colon X\to Y$ be a proper surjective morphism with connected fibres,
with $Y$ semi-normal and $X$ reduced. Then
$p_{*}\mathcal{O}_{X}=\mathcal{O}_{Y}$.

######  Proposition 4.4.

Let $f\colon(X,B)\rightarrow Z$ be a GLC fibration which is also a locally
stable family, where $Z$ is quasi-projective and smooth. Assume that
$K_{X/Z}+B$ is relatively semi-ample over $Z$. Let $c\colon X\rightarrow Y/Z$
denote the relative Iitaka fibration of $(X,B)$ over $Z$ and let
$(Y,B_{Y},\mathbf{M})$ be the generalised pair induced on $Y$.

Assume that the b-semi-ampleness conjecture holds.

Then, for every $z\in Z$, we may find
$0\leq\Delta_{Y}\sim_{\mathbb{Q}}B_{Y}+\mathbf{M}_{Y}$ such that
$g\colon(Y,\Delta_{Y})\rightarrow Z$ is a stable family of pairs over a
neighbourhood of $z\in Z$.

###### Proof.

By Lemma 2.8, $(X,B)\rightarrow Z$ satisfies Property $(*)$, and so by [Amb04,
Theorem 3.1] we see that $(Y,B_{Y},\mathbf{M})\rightarrow Z$ satisfies
generalised Property $(*)$. Observe also that if $D\subset Z$ is any reduced
divisor, then $g^{*}D$ is reduced as well.

Let $p\colon\overline{Y}\rightarrow Y$ be a generalised dlt modification, and
let $(\overline{Y},\overline{B}_{Y},\mathbf{M})$ be the transformed pair. By
Lemma 3.3, we see that
$\bar{g}\colon(\overline{Y},\overline{B}_{Y},\mathbf{M})\rightarrow Z$
satsifies generalised Property $(*)$, and so it follows that
$(\overline{Y}/Z,\overline{B}_{Y})$ satisfies Property $(*)$, and let
$\Sigma\subset Z$ be the associated boundary. Since $(X,B)\rightarrow Z$ is
locally stable we see that $\overline{B}_{Y}^{h}=\overline{B}_{Y}$, and so by
Lemma 2.8 $(Y,\overline{B}_{Y})\rightarrow Z$ is locally stable.

Since we are assuming the b-semi-ampleness conjecture, for a general choice of
$0\leq H\sim_{\mathbb{Q}}\mathbf{M}_{\overline{Y}}$, by Bertini’s theorem
(applied on a log resolution where $\mathbf{M}$ descends) we may guarantee
that $(\overline{Y}_{z},\overline{B}_{Y,z}+H_{z})$ is slc. By [Kol22,
Corollary 4.45] there is an open neighbourhood of $z$ such that
$(\overline{Y}/Z,\overline{B}_{Y}+H)$ is locally stable as required. ∎

######  Theorem 4.5.

Let $f\colon(X,\Delta)\rightarrow Z$ be a GLC fibration which is also a
locally stable family, where $Z$ is quasi-projective and normal.

Assume that the geometric generic fibers over the generic points of the
irreducible components of $Z$ are normal. Assume that, for every $z\in Z$,
$K_{X_{z}}+\Delta_{z}$ is semi-ample. Then, we may find $m_{0}\in\mathbb{N}$
depending on the family $f$ such that the following holds:

  * (1)

for every $k\in\mathbb{N}$,
$h^{0}(X_{z},\mathcal{O}_{X_{z}}(km_{0}(K_{X_{z}}+\Delta_{z})))$ is
independent of $z\in Z$;

  * (2)

for every $k\in\mathbb{N}$, $f_{*}\mathcal{O}_{X}(km_{0}(K_{X/Z}+\Delta))$ is
a vector bundle, whose formation commutes with base change;

  * (3)

the relative Iitaka fibration $c\colon(X,\Delta)\rightarrow Y$ of $f$ is a
morphism and computes the ample model of each fibre; and

  * (4)

the formation of the relative Iitaka fibration commutes with base change by
reduced schemes.

Furthermore, $g\colon Y\rightarrow Z$ is flat. In particular, the scheme
theoretic fibres of $g$ are pure-dimensional reduced and semi-normal.

###### Proof.

Item (2) immediately follows from item (1) by Grauert’s theorem and cohomology
and base change, see [Har77, Corollary III.12.9 and Theorem III.12.11].
Similarly, item (4) follows from item (3). Thus, we are left with showing
items (1) and (3).

Let us first suppose that $\operatorname{dim}Z=1$. Let
$c\colon(X,\Delta)\rightarrow Y$ be the relative Iitaka fibration of $f$,
which exists by the assumptions and [HX13]. Since $X$ is normal, so is $Y$.
Let $(Y/Z,B_{Y},\mathbf{M})$ be the generalized pair induced by the canonical
bundle formula. Since $Z$ is a curve, $g\colon Y\rightarrow Z$ is flat, see
[Har77, Proposition III.9.7]. By semi-continuity [Har77, Theorem III.12.8], it
suffices to show that, for a special closed point $0\in Z$, we have
$Y_{0}=\operatorname{Proj}(R(X_{0},\Delta_{0}))$, where we set
$R(X_{0},\Delta_{0})=\oplus_{m\in\mathbb{N}}\Gamma(X_{0},\mathcal{O}_{X_{0}}(m(K_{X_{0}}+\Delta_{0})))$.
Since $f$ is a locally stable family, $(X,\Delta+X_{0})$ is log canonical.
Then, by inversion of adjunction for LC-trivial fibrations [Amb04, Theorem
3.1], the induced generalized pair $(Y/Z,B_{Y}+g^{*}0,\mathbf{M})$ is
generalized log canonical. In particular, we have $Y_{0}=g^{*}0$ as divisors.
In particular, the scheme theoretic fibre $Y_{0}$ is $R_{0}$. Since $Y$ is
normal and $g^{*}0$ is Cartier, it follows that $\mathcal{O}_{Y}$ and
$\mathcal{O}_{Y}(-g^{*}0)$ are $S_{2}$. Then, by [Kol13, Corollary 2.61],
$Y_{0}$ is $S_{1}$. But then, by [Sta22, Tag 0344], $Y_{0}$ is reduced.
Furthermore, by Lemma 4.2, $Y_{0}$ is semi-normal.

Now, we have a tower of morphisms as follows

$X_{0}\xrightarrow{\alpha}\operatorname{Proj}(R(X_{0},\Delta_{0}))\xrightarrow{\beta}Y_{0}.$

For $m_{0}$ sufficiently large and divisible, we have the following:

  * •

the morphism $\alpha$ is given by the full linear series
$|\Gamma(X_{0},\mathcal{O}_{X_{0}}(m_{0}(K_{X_{0}}+\Delta_{0})))|$; and

  * •

the morphism $c_{0}\colon X_{0}\rightarrow Y_{0}$ is given by the restricted
linear series corresponding to

(2)
$\operatorname{Im}(\Gamma(X,\mathcal{O}_{X}(m_{0}(K_{X/Z}+\Delta)))\rightarrow\Gamma(X_{0},\mathcal{O}_{X_{0}}(m(K_{X_{0}}+\Delta_{0})))).$

Since $c_{0}$ is given by a sub-series of the one giving $\alpha$, and since
$c_{0}$ contracts exactly the same curves as $\alpha$, the induced morphism
$\beta$ is finite. By construction, we have
$\alpha_{*}\mathcal{O}_{X_{0}}=\mathcal{O}_{\operatorname{Proj}(R(X_{0},\Delta_{0}))}$.
Similarly, we have $c_{*}\mathcal{O}_{X}=\mathcal{O}_{Y}$ and so
$X_{0}\rightarrow Y_{0}$ has connected fibers. Then, by Lemma 4.3, we have
$(c_{0})_{*}\mathcal{O}_{X_{0}}=\mathcal{O}_{Y_{0}}$. In turn, this gives
$\beta_{*}\mathcal{O}_{\operatorname{Proj}(R(X_{0},\Delta_{0}))}=\mathcal{O}_{Y_{0}}$.
But then, as $\beta$ is finite, we have $R(X_{0},\Delta_{0})=Y_{0}$. Since $g$
is flat, its fibres are pure-dimensional. This settles items (1) and (3) when
$\operatorname{dim}Z=1$.

We now handle the general case. Let $l_{0}$ denote the Cartier index of
$K_{X/Z}+\Delta$. By generic flatness applied to $g\colon Y\rightarrow Z$ and
relative Serre vanishing, we may find a non-empty open subset $U\subset Z$ and
$m_{0}\in\mathbb{N}$ divisible by $l_{0}$ such that (1) holds over $U$. Then,
for every closed point $z^{\prime}\in Z$, by restricting to a general curve
passing through $z^{\prime}$ and which meets $U$, we may find
$m_{z^{\prime}}\in\mathbb{N}$ divisible by $m_{0}$ such that, for every
$k\in\mathbb{N}$,
$h^{0}(X_{z^{\prime}},\mathcal{O}_{X_{z}}(km_{z^{\prime}}(K_{X_{z}^{\prime}}+\Delta_{z^{\prime}})))$
attains the generic value achieved over $U$. Then, by semi-continuity and
Grauert’s theorem [Har77, Theorem III.12.8 and Corollary III.12.9],
$z^{\prime}$ is in the open subset where
$f_{*}\mathcal{O}_{X}(km_{z^{\prime}}(K_{X}+\Delta))$ is a vector bundle, for
every $k\in\mathbb{N}$. Thus, by replacing $m_{0}$ with a multiple, we may
assume that any fixed closed point $z^{\prime}$ belongs to the open set $U$.
Since the $\mathcal{O}_{Z}$-algebra
$\oplus_{k\in\mathbb{N}}f_{*}\mathcal{O}_{X}(kl_{0}(K_{X/Z}+\Delta))$ is
finitely generated, after finitely many iterations, the above procedure
extends the open subset $U$ to the whole $Z$.

Since (1) and (2) are settled and (3) is known over a curve, item (3) now
follows immediately. Similarly, the additional properties of the fibres of $g$
from (3) hold since they may be checked in the case the base is a curve.
Lastly, the flatness of $g$ follows from [Kol22, Theorem 3.20]. ∎

######  Theorem 4.6.

Let $f\colon(X,\Delta)\rightarrow Z$ be a GLC fibration which is also locally
stable family, where $Z$ is quasi-projective and normal. Assume
$K_{X/Z}+\Delta$ is $f$-nef and that a general fibre is a good minimal model.
Further assume that the b-semi-ampleness conjecture holds.

Then, for $m\in\mathbb{N}$ sufficiently divisible,

  1. (1)

$\lambda_{m}\coloneqq\operatorname{det}f_{*}\mathcal{O}_{X}(m(K_{X/Z}+\Delta))$
is basepoint-free; and

  2. (2)

for all $\epsilon\in\mathbb{Q}_{>0}$, $K_{X/Z}+\Delta+\epsilon
f^{*}\lambda_{m}$ is semi-ample.

###### Proof.

Since $(X,\Delta)\rightarrow Z$ is locally stable, we know that every log
canonical centre of $(X,\Delta)$ dominates $Z$. By applying [HX13, Theorem
1.1] to a dlt modification of $(X,\Delta)$, it follows that $K_{X/Z}+\Delta$
is relatively semi-ample over $Z$.

By Theorem 4.5, the coherent sheaf $f_{*}\mathcal{O}_{X}(m(K_{X/Z}+\Delta))$
is a vector bundle and its formation commutes with base change. In particular,
it follows that the line bundle
$\lambda_{m}\coloneqq\operatorname{det}f_{*}\mathcal{O}_{X}(m(K_{X/Z}+\Delta))$
is compatible with base change. Lastly, we recall that, given a line bundle
$\mathcal{L}$ on a normal variety $U$ and a surjective projective morphism
$\phi\colon V\rightarrow U$ where $V$ is normal, then $\mathcal{L}$ is semi-
ample if an only if so is $\phi^{*}\mathcal{L}$. Thus, when needed, we are
free to replace $Z$ with a generically finite cover and all the relevant
varieties and morphisms by base change. In particular, we may assume that $Z$
is smooth.

Let $c\colon X\rightarrow Y$ denote the relative Iitaka fibration of
$(X,\Delta)$, and let $(Y,B_{Y},\mathbf{M})$ be the generalized pair induced
on $Y$. By Noetherianity and Proposition 4.4, we may find finitely many
effective divisors $D_{1},\ldots,D_{k}\in|\mathbf{M}_{Y}|_{\mathbb{Q}}$ and
open sets $U_{i}\subset Z$ such that $Z=\cup_{i=1}^{k}U_{i}$ and
$g\colon(Y,B_{Y}+D_{i})\rightarrow Z$ is a stable family of pairs over
$U_{i}$.

Now, fix $i\in\\{1,\dots,k\\}$. We will show that for integers $m_{i}\gg 0$
and $\ell>0$ and $\epsilon\in\mathbb{Q}_{>0}$, the base locus of
$\lambda_{\ell m_{i}}$ (resp. $\mathbf{M}_{Y}+\epsilon f^{*}\lambda_{\ell
m_{i}}$) is contained in $Z\setminus U_{i}$ (resp. $f^{-1}(Z\setminus
U_{i})$). Supposing this, we see that by taking $m=m_{1}m_{2}\dots m_{k}$ the
claims of the theorem hold.

For ease of notation set $D\coloneqq D_{i}$ and $U\coloneqq U_{i}$.

By [KP17, Corollary 6.18], we may find a generically finite cover
$\sigma\colon\overline{U}\rightarrow U$ such that
$g\colon(Y,B_{Y}+D)\times_{U}\overline{U}\rightarrow\overline{U}$ can be
recompactified to a family of stable pairs over a normal projective variety
$\overline{Z}\supset\overline{U}$. Call this compactified family
$(\overline{Y},\overline{B}_{Y}+\overline{D})\rightarrow\overline{Z}$. Perhaps
replacing $\overline{Z}$ by a higher model, we may assume that $\sigma$
extends to a morphism $\sigma\colon\overline{Z}\rightarrow Z$.

Let $\tau\colon\overline{Z}\rightarrow\tilde{Z}$ be the Stein factorisation of
$\sigma$, and set $\tilde{Y}\coloneqq Y\times_{Z}\tilde{Z}$.

Let $m_{0}$ be the integer guaranteed to exist by Theorem 4.5. Up to replacing
$m_{0}$ with a multiple, we may further assume that $f_{*}\mathcal{O}_{X}(\ell
m_{0}(K_{X/Z}+\Delta))$ is a vector bundle that commutes with base change.
Note that
$\operatorname{det}h_{*}\mathcal{O}_{\overline{Y}}(km_{0}(K_{\overline{Y}/\overline{Z}}+\overline{B}+\overline{D}))|_{\overline{U}}$
coincides with
$\sigma^{*}\operatorname{det}f_{*}\mathcal{O}_{X}(km_{0}(K_{X/Z}+\Delta))|_{U}$.

Now, we claim that we have an inclusion of coherent sheaves

(3) $0\rightarrow
h_{*}\mathcal{O}_{\overline{Y}}(km_{0}(K_{\overline{Y}/\overline{Z}}+\overline{B}+\overline{D}))\rightarrow\sigma^{*}f_{*}\mathcal{O}_{X}(km_{0}(K_{X/Z}+\Delta))$

that is an isomorphism over $\overline{U}$. The latter claim is immediate by
the previous observations. Furthermore, as the two sheaves are torsion free,
to conclude it suffices to show there is a non-zero morphism between the two.

Now, let $\hat{Y}$ be the normalisation of the closure of the graph of
$Y\dashrightarrow\overline{Y}$ in $Y\times_{Z}\overline{Y}$, and let
$p\colon\hat{Y}\rightarrow Y$ and $q\colon\hat{Y}\rightarrow\overline{Y}$ be
the corresponding morphisms. As $Y$ and $\overline{Y}$ are normal and
isomorphic over the generic point of $Z$, $p$ and $q$ are isomorphisms over
the generic point of $Z$. To conclude, it is then sufficient to show that

(4)
$q^{*}(K_{\overline{Y}}+\overline{B}+\overline{D})+\Theta=p^{*}(K_{Y}+B_{Y}+D),$

where $\Theta\geq 0$. Note that, by construction, $p_{*}\Theta$ is contained
in $f^{-1}(Z\setminus U)$.

To this end, we follow the proof of [Kol22, Theorem 11.40]. Then, we may write

$p^{*}(K_{Y}+B_{Y}+D)=K_{\hat{Y}}+\hat{B}+\hat{D}+A-B,$

where $\hat{B}$ and $\hat{D}$ denote the strict transforms of $B_{Y}$ and $D$,
respectively, and $A$ and $B$ are effective divisors with no common
components. Since
$h\colon(\overline{Y},\overline{B}+\overline{D})\rightarrow\overline{Z}$ is a
stable family, we may write

$q^{*}(K_{\overline{Y}}+\overline{B}+\overline{D})=K_{\hat{Y}}+\hat{B}+\hat{D}-E,$

where $E$ is an effective divisor, cf. [Kol22, Proposition 2.15]. Then, it
follows that $-(E+A-B)$ is $p$-nef and $p_{*}(E+A-B)=p_{*}E\geq 0$. Thus, by
the negativity lemma [KM98, Lemma 3.39], we have $E+A-B\geq 0$. Thus, in
particular, $A-B\geq-E$, and (4) follows.

By taking determinants of (3), we get a morphism

(5) $\bar{\lambda}_{\ell
m_{0}}\coloneqq\operatorname{det}h_{*}\mathcal{O}_{\overline{Y}}(\ell
m_{0}(K_{\overline{Y}/\overline{Z}}+\overline{B}+\overline{D}))\rightarrow\sigma^{*}\operatorname{det}f_{*}\mathcal{O}_{X}(\ell
m_{0}(K_{X/Z}+\Delta))$

that is an isomorphism over $\overline{U}$, and so $\sigma^{*}\lambda_{\ell
m_{0}}=\bar{\lambda}_{\ell m_{0}}+G$ where $G\geq 0$ is supported on
$\overline{Z}\setminus\overline{U}$. Likewise, we see that
$q^{*}(\overline{M}+\epsilon\bar{f}^{*}\bar{\lambda}_{\ell
m_{0}})+\Theta+\epsilon q^{*}\bar{f}^{*}G=p^{*}(M+\epsilon f^{*}\lambda_{\ell
m_{0}})$.

By Lemma 4.1, we know that $\bar{\lambda}_{\ell m_{0}}$ (resp.
$(\overline{M}+\epsilon\bar{f}^{*}\bar{\lambda}_{\ell m_{0}})$) is semi-ample,
which from our above equalities implies that the base locus of $\lambda_{\ell
m_{0}}$ (resp. $M+\epsilon f^{*}\lambda_{\ell m_{0}})$) is contained in $G$
(resp. $p_{*}(\Theta+\epsilon q^{*}\bar{f}^{*}G)$) as required. ∎

######  Theorem 4.7.

Assume that the b-semi-ampleness conjecture holds.

Let $f\colon(X,B)\rightarrow Z$ be a GLC fibration with $B\geq 0$ and
$\operatorname{dim}X=n$. Suppose that a general fibre of $f$ admits a good
minimal model.

Then we may find a birationally equivalent GLC fibration
$f^{\prime}\colon(X^{\prime},B^{\prime})\rightarrow Z^{\prime}$ fitting into
the following commutative diagram

${(X^{\prime},B^{\prime})}$${(X,B)}$${Z^{\prime}}$${Z,}$$\scriptstyle{\alpha}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$$\scriptstyle{\beta}$

where $\alpha$ and $\beta$ are birational, such that the following holds

  1. (1)

$(X,B)$ and $(X^{\prime},B^{\prime})$ are crepant birational over the generic
point of $Z$;

  2. (2)

$(X^{\prime}/Z^{\prime},B^{\prime})$ satisfies Property $(*)$, is BP stable
and has maximal moduli;

  3. (3)

$f\colon X^{\prime}\rightarrow Z^{\prime}$ is equidimensional;

  4. (4)

the moduli part, $M^{\prime}$, of $(X^{\prime}/Z^{\prime},B^{\prime})$ is nef;

  5. (5)

$f^{\prime}_{*}\mathcal{O}(mM^{\prime})$ is locally free and
$\lambda_{m}^{\prime}\coloneqq\operatorname{det}f^{\prime}_{*}\mathcal{O}(mM^{\prime})$
is nef and semi-ample for $m\gg 0$ sufficiently divisible; and

  6. (6)

$M^{\prime}+\epsilon f^{\prime*}\lambda_{m}^{\prime}$ is semi-ample for any
$\epsilon\in\mathbb{Q}_{>0}$.

Moreover, $\lambda_{m}^{\prime}$ is compatible with base change in the
following sense. Consider another birationally equivalent GLC fibration
$f^{\prime\prime}\colon(X^{\prime\prime},B^{\prime\prime})\rightarrow
Z^{\prime\prime}$, where $\gamma\colon Z^{\prime\prime}\rightarrow Z^{\prime}$
is a birational morphism, such that
$(X^{\prime\prime}/Z^{\prime\prime},B^{\prime\prime})$ satisfies Property
$(*)$ and is BP stable, $f^{\prime\prime}$ is equidimensional and such that
the moduli part of $(X^{\prime\prime}/Z^{\prime\prime},B^{\prime\prime})$ is
nef. Then, we have
$\lambda_{m}^{\prime\prime}\sim\gamma^{*}\lambda_{m}^{\prime}$.

###### Proof.

By [ACSS21, Theorem 1.1] we may find a model
$(X^{\prime},B^{\prime})\rightarrow Z^{\prime}$ which satisfies (1)-(4). By
Lemma 2.7 we see that that the assumption on termination of flips in [ACSS21,
Theorem 1.1] is unneeded.

Let $(Y,C)\rightarrow W$ be the cover of $X$ guaranteed to exist by
Proposition 3.5. By Proposition 3.5, (5) and (6) hold by the corresponding
statement for $(Y/W,C)$, which follows from Theorem 4.6.

To prove our final claim, observe that
$(X^{\prime\prime},B^{\prime\prime})\rightarrow Z^{\prime\prime}$ also
satisfies (1)-(6) and so our final claim is a consequence of Lemma 3.6. ∎

######  Remark 4.8.

In the case where the moduli part is $f$-big, it is not necessary to assume
the b-semi-ampleness conjecture.

######  Corollary 4.9.

Assume that the b-semi-ampleness conjecture holds.

Let $f\colon(X,B)\rightarrow Z$ be an equidimensional GLC fibration with
$B\geq 0$ and $\operatorname{dim}X=n$ which satisfies Property $(*)$. Suppose
that a general fibre of $f$ admits a good minimal model.

For $m\gg 0$ sufficiently divisible, set
$\lambda_{m}\coloneqq\operatorname{det}f_{*}\mathcal{O}(mM)$. Then, for any
$\epsilon\in\mathbb{Q}_{>0}$, there exists a birational model
$b\colon\tilde{Z}\rightarrow Z$ (resp. $a\colon\tilde{X}\rightarrow X$) and a
semi-ample line bundle $L$ on $Z^{\prime}$ (resp. a semi-ample line bundle $N$
on $X^{\prime}$) such that $b_{*}c_{1}(L)=c_{1}(\lambda_{m})$ (resp.
$a_{*}N=M+\epsilon f^{*}\lambda_{m}$).

###### Proof.

Consider the model $(X^{\prime}/Z^{\prime},B^{\prime})$ with moduli part
$M^{\prime}$ guaranteed to exist by Theorem 4.7, and let
$p\colon\tilde{X}\rightarrow X$ and $q\colon\tilde{X}\rightarrow X^{\prime}$
resolve the rational map $\alpha\colon X^{\prime}\dashrightarrow X$.

By [ACSS21, Proposition 2.21], we see that $p^{*}M=q^{*}M^{\prime}$. It
follows that we have
$\beta_{*}f^{\prime}_{*}\mathcal{O}(mM^{\prime})=f_{*}\mathcal{O}(mM)$, from
which we may conclude that $\beta^{*}\lambda_{m}=\lambda^{\prime}+F$ where
$\lambda^{\prime}=\operatorname{det}f^{\prime}_{*}\mathcal{O}(mM^{\prime})$
and $F$ is $\beta$-exceptional. Since $\lambda^{\prime}$ is nef [KM98, Lemma
3.39] implies $F\geq 0$. Take $L=\lambda^{\prime}$ to conclude. The claim for
$M+\epsilon f^{*}\lambda_{m}$ follows similarly. ∎

##  5\. $1$-dimensional bases

######  Proposition 5.1.

Let $f\colon(X,B)\rightarrow Z$ be a locally stable family of good minimal
models over a smooth curve $Z$ and let $M$ be the moduli part associated to
$(X/Z,B)$. Further assume that $\lambda_{+}>0$.

Then

  1. (1)

$M+\epsilon f^{*}\lambda_{m}$ is semi-ample for $m\gg 0$ any $\epsilon>0$
where $\lambda_{m}=\operatorname{det}f_{*}\mathcal{O}(mM)$; and

  2. (2)

$\kappa(M)\geq\kappa(X/Z,B)+{\mathrm{v}ar}(X/Z,B)$.

###### Proof.

Let $(Y,B_{Y},\mathbf{M}_{Y})$ denote the relatively ample model of $(X,B)$
over $Z$, with morphisms $c\colon X\rightarrow Y$ and $g\colon Y\rightarrow
Z$. Then, we have that $\kappa(M_{X})=\kappa(K_{Y/Z}+B_{Y}+\mathbf{M}_{Y})$,
as we have $M_{X}=K_{X/Z}+B$.

Item (1) is a direct consequence of the fact that $\lambda_{m}$ is ample and
$K_{X/Z}+B$ is nef and $f$-semi-ample.

By dimension reasons, item (2) is implied by the bigness of
$K_{Y/Z}+B_{Y}+\mathbf{M}_{Y}$. By construction,
$K_{Y/Z}+B_{Y}+\mathbf{M}_{Y}$ is ample over $Z$. Furthermore, since
$\lambda_{+}>0$, we may find an ample divisor $H$ on $Z$ such that
$K_{Y/Z}+B_{Y}+\mathbf{M}_{Y}-g^{*}H$ is pseudo-effective. Now, since $H$ is
ample and $K_{Y/Z}+B_{Y}+\mathbf{M}_{Y}$ is $g$-ample, for $0<\epsilon\ll 1$,
we have that $\epsilon(K_{Y/Z}+B_{Y}+\mathbf{M}_{Y})+g^{*}H$ is ample. And so
we have we have that

$\epsilon(K_{Y/Z}+B_{Y}+\mathbf{M}_{Y})+g^{*}H+(K_{Y/Z}+B_{Y}+\mathbf{M}_{Y}-g^{*}H)=(1+\epsilon)(K_{Y/Z}+B_{Y}+\mathbf{M}_{Y})$

is big as required. ∎

######  Lemma 5.2.

Let $X$ be a normal projective variety, let $Z$ be normal quasi-projective
variety and let $Y$ be a normal variety which is projective over $Z$.

  1. (1)

Suppose that we have a surjective contraction $\phi\colon X\times Z\rightarrow
Y$ over $Z$. Then there is a morphism $\varphi\colon X\rightarrow X^{\prime}$
and an isomorphism $Y\cong X^{\prime}\times Z$.

  2. (2)

Let $D$ be a divisor on $Y=X\times Z$ and suppose that the $D$-flip (resp.
$D$-flop) over $Z$, $Y^{+}$, exists. Then there exists a divisor $D_{0}$ on
$X$ such that the $D_{0}$-flip (resp. $D_{0}$-flop), $X^{+}$, exists and
$Y^{+}\cong X^{+}\times Z$.

###### Proof.

To see (1) take $X^{\prime}=\phi(X\times\\{z\\})$ for any $z\in Z$ and set
$\varphi=\phi|_{X\times\\{z\\}}$. Let $A^{\prime}$ be an ample divisor on
$X^{\prime}$ and let $A_{Y}$ be an ample divisor on $Y$. We observe that
$(\phi^{*}A_{Y})^{\perp}\cap\overline{NE}(X\times
Z)=(\phi^{*}A_{Y})^{\perp}\cap\overline{NE}(X\times
Z/Z)\cong(\varphi^{*}A^{\prime})^{\perp}\cap\overline{NE}(X)$, where the first
equality holds because $\phi$ is a morphism over $Z$. It follows that the
morphisms over $Z$ defined by $\phi^{*}A_{Y}$ and $\varphi^{*}A^{\prime}$
contract the same curves, and so by the rigidity lemma, [Deb01, Lemma 1.15],
they are the same morphism.

To prove (2), let $W$ be the base of the $D$-flipping (resp. $D$-flopping)
contraction. Arguing as in the previous case if $D_{0}=D|_{X\times\\{z\\}}$
for a general point $z\in Z$ then $D_{0}\times Z\equiv D/W$. It follows that
the $D$-flip (resp. $D$-flop) is isomorphic to the $D_{0}\times Z$-flip (resp.
$D_{0}\times Z$-flop), which is easily seen to be $X^{+}\times Z$, as
required. ∎

######  Lemma 5.3.

Let $C$ be a smooth projective variety, and let $(X_{0},\Delta_{0})$ be a
projective klt pair with $\Delta_{0}\geq 0$ and such that
$K_{X_{0}}+\Delta_{0}$ is nef. Let $(X,\Delta)$ denote the product
$(X_{0},\Delta_{0})\times C$, and let $f\colon(X,\Delta)\rightarrow C$ denote
the induced morphism. Let
$f^{\prime}\colon(X^{\prime},\Delta^{\prime})\rightarrow C$ be a birational
model of $(X,\Delta)$, where $\Delta^{\prime}\geq 0$,
$(X^{\prime},\Delta^{\prime})$ is klt and $K_{X^{\prime}}+\Delta^{\prime}$ is
nef. Then, there exists a crepant model $(X^{\prime}_{0},\Delta^{\prime}_{0})$
of $(X_{0},\Delta_{0})$ such that
$(X^{\prime},\Delta^{\prime})=(X^{\prime}_{0},\Delta^{\prime}_{0})\times C$.

###### Proof.

By taking a $\mathbb{Q}$-factorial terminalisation of $(X_{0},\Delta_{0})$,
which exists by [BCHM10, Corollary 1.4.3], we may further assume that
$(X,\Delta)\dashrightarrow(X^{\prime},\Delta^{\prime})$ is a rational
contraction and $X$ and $X_{0}$ are $\mathbb{Q}$-factorial. Taking a
$\mathbb{Q}$-factorial terminalisation
$(X^{\prime\prime},\Delta^{\prime\prime})\rightarrow(X^{\prime},\Delta^{\prime})$,
which exists by [BCHM10, Corollary 1.4.3], we may assume $X\dashrightarrow
X^{\prime\prime}$ is an isomorphism in codimension 1.

By [Kaw08] $X\dashrightarrow X^{\prime\prime}$ is a sequence of
$(K_{X}+\Delta)$-flops. By Lemma 5.2 we see that if $X_{0}\dashrightarrow
X^{\prime\prime}_{0}$ is the induced birational contraction on a general
fibre, then in fact $X^{\prime\prime}\cong X^{\prime\prime}_{0}\times C$. We
conclude by another application of Lemma 5.2 to
$(X^{\prime\prime},\Delta^{\prime\prime})\rightarrow(X^{\prime},\Delta^{\prime})$.
∎

######  Definition 5.4.

Given a GLC fibration $(X,B)\rightarrow Z$ with $B\geq 0$ and
$\operatorname{dim}Z=1$ we define ${\mathrm{v}ar}(X/Z,B)$ to be $0$ if there
exists a pair $(X_{0},B_{0})$ and a generically finite morphism
$Z^{\prime}\rightarrow Z$ such that $(X_{0},B_{0})\times Z^{\prime}$ is
birational to $(X,B)\times_{Z}Z^{\prime}$. Otherwise we define
${\mathrm{v}ar}(X/Z,B)$ to be 1.

######  Remark 5.5.

  1. (1)

Note that ${\mathrm{v}ar}(X/Z,B)$ is invariant by crepant birational
transformations of $(X,B)$.

  2. (2)

If $X$ has canonical singularities and $B=0$ this agrees with the definition
of variation found in [Kaw85].

  3. (3)

A family of terminal minimal models with $B=0$ and ${\mathrm{v}ar}=0$ is
always (up to base change) crepant birational to a product, and so by Lemma
5.3 is in fact a product.

######  Theorem 5.6.

Let $f\colon(X,B)\rightarrow Z$ be a GLC fibration between projective
varieties with $B\geq 0$, $\operatorname{dim}X=n$ and $\operatorname{dim}Z=1$.
Suppose that $(X,B)$ is klt over the generic point of $Z$ and the generic
fibre of $f$ admits a good minimal model.

Then we may find a birationally equivalent GLC fibration
$f^{\prime}\colon(X^{\prime},B^{\prime})\rightarrow Z$ fitting into the
following commutative diagram

${(X^{\prime},B^{\prime})}$${(X,B)}$${Z}$${Z.}$$\scriptstyle{\alpha}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$

such that $K_{X^{\prime}}+B^{\prime}$ is nef over $Z$ and

  1. (1)

$(X^{\prime}/Z^{\prime},B^{\prime})$ is BP stable and has maximal moduli;

  2. (2)

$M^{\prime}+\epsilon f^{\prime*}\lambda_{m}^{\prime}$ is semi-ample for $m\gg
0$ and any $\epsilon\in\mathbb{Q}_{>0}$ where
$\lambda_{m}^{\prime}=\operatorname{det}f^{\prime}_{*}\mathcal{O}(mM^{\prime})$;
and

  3. (3)

$\kappa(M^{\prime})\geq\kappa(X^{\prime}/Z,B^{\prime})+{\mathrm{v}ar}(X^{\prime}/Z,B^{\prime})$.

######  Remark 5.7.

We remark that $M^{\prime}$ and $\lambda^{\prime}_{m}$ are compatible with any
further base change, as in the statement of Theorem 4.7.

###### Proof.

Take $(X^{\prime},B^{\prime})$ to be the model of $(X,B)$ guaranteed by
[ACSS21, Theorem 1.1]. By Lemma 2.7 we see that that the assumption on
termination of flips in [ACSS21, Theorem 1.1] is unneeded.

Taking a base change along the finite morphism guaranteed by Proposition 3.5
does not alter $\kappa(M^{\prime}),\kappa(X/Z,B),{\mathrm{v}ar}(X/Z,B)$ or the
semi-ampleness of $M^{\prime}+\epsilon f^{\prime*}\lambda_{m}^{\prime}$, so we
may freely replace $(X^{\prime},B^{\prime})$ by this cover, and therefore may
assume that $(X^{\prime},B^{\prime})$ is a locally stable family. For ease of
notation we replace $(X,B)$ by $(X^{\prime},B^{\prime})$. By [HX13, Theorem
1.1], we see that $K_{X}+B$ is semi-ample over $Z$. Next, observe that since
$(X,B)$ is locally stable and klt over the generic point of $Z$ that $(X,B)$
is in fact klt, see Lemma 2.5.

Let $(\tilde{X},\tilde{B})\rightarrow(X,B)$ be a $\mathbb{Q}$-factorial
terminalisation of $(X,B)$, which exists by [BCHM10, Corollary 1.4.3]. By
Proposition 2.2, $(\tilde{X},\tilde{B})$ is locally stable. So we may freely
replace $(X,B)$ by $~{}(\tilde{X},\tilde{B})$, and so we may assume that $X$
is $\mathbb{Q}$-factorial with canonical singularities.

By Proposition 5.1, we may assume that $\lambda_{+}=0$. By [ACSS21,
Proposition 4.1] we see that $K_{X/Z}+B$ is nef and so $\lambda_{-}\geq 0$.
For $m\gg 0$ Theorem 4.5 implies that $E_{m}\coloneqq
f^{\prime}_{*}\mathcal{O}(m(K_{X/Z}+B)$ is a vector bundle and by Proposition
2.15 $\mu_{\min}(E)=\mu_{\max}(E)=0$ and so $E_{m}$ is numerically flat.

Semi-ampleness of $M$ is a direct consequence of Lemma 2.12, and so item (1)
is proven.

To prove item (2), let $Y$ be the relatively ample model of $(X,B)$ over $Z$
and let $(Y,B_{Y},\mathbf{M}_{Y})$ be the generalised pair given by the
canonical bundle formula. By Lemma 2.12 after replacing $Z$ by an étale cover
we may assume
$(Y,B_{Y}+\mathbf{M}_{Y})=Z\times(Y_{0},B_{Y_{0}}+\mathbf{M}_{Y_{0}})$. As we
are assuming that $\lambda_{+}=0$ it suffices to show that
${\mathrm{v}ar}(X/Z,B)=0$ to conclude. So we will show that after a
generically finite base change, $(X,B)\rightarrow Z$ becomes isomorphic to a
product $(X_{0},B_{0})\times Z$. We will argue by induction on
$\operatorname{dim}(X/Z)$. The case where $\operatorname{dim}(X/Z)=0$ is
obvious.

We proceed with a case by case analysis.

Case 1: In this case we assume that ${\mathrm{S}upp}(B)$ is vertical over $Y$.

First, we run a $K_{X}$-MMP with scaling relatively over $Y$. By [HX13,
Theorem 1.1] and the assumption that $B$ is vertical over $Y$, the MMP
terminates with a good minimal model. We denote by $X^{\prime}$ the final
model of this MMP and $Y^{\prime}$ the relatively ample model, which is a
birational model of $Y$. Since $K_{X}+B\sim_{\mathbb{Q}}0/Y$,
$(X^{\prime},B^{\prime})\rightarrow Z$ is still a family of good minimal
models, where $B^{\prime}$ denotes the push-forward of $B$ along the MMP. Note
that with respect to $(X^{\prime},B^{\prime})\rightarrow Z$ we still have
$\lambda_{+}=0$, and so we may freely replace $(X,B)$ by
$(X^{\prime},B^{\prime})$ and therefore may assume in addition that
$K_{X}\sim_{\mathbb{Q}}0/Y$.

Now, if $B=0$, we may conclude by [Kaw85, Theorem 1.1], see also Remark 5.5.
Thus, we may assume that $B\neq 0$, and we wish to reduce to [Kaw85]. Now,
consider a general ample divisor $H_{0}\subset Y_{0}$, and let $H$ denote the
divisor $H_{0}\times Z\subset Y$. Furthermore, we let $D$ denote the pull-back
of $H$ to $X$. For some $m\gg 0$ sufficiently large, and for general enough
$H$, we may find cyclic a Galois cover $\tilde{Y}\rightarrow Y$ and
$\tilde{X}\rightarrow W$ of degree $=m$, which are ramified along $H$ and $D$,
respectively.

Since $K_{X}\sim_{\mathbb{Q}}0/Y$ and $H_{0}$ is sufficiently ample,
$\tilde{X}\rightarrow\tilde{Y}$ is the relative Iitaka fibration of
$\tilde{X}$ over $Z$. Furthermore, by [Kol22, Proposition 2.13],
$\tilde{X}\rightarrow Z$ is a locally stable family. By construction, the
variation of the relatively ample model of $\tilde{X}$ is 0, since both
$(Y,B_{Y}+M_{Y})$ and $H$ have no variation. Then, by [Kaw85, Theorem 1.1], it
follows that $\tilde{X}\rightarrow Z$ is isotrivial, and hence (up to a base
change) a product.

Write $\tilde{X}=\tilde{X}_{0}\times Z$, and $\tilde{Y}=\tilde{Y}_{0}\times
Z$. Observe that by construction if $g$ generates the cyclic Galois action on
$\tilde{X}$, then $g$ is of the form $g^{\prime}\times 1_{Z}$ where
$g^{\prime}\in{\mathrm{A}ut}(\tilde{X}_{0})$, in fact, $g^{\prime}$ is
contained in
${\mathrm{A}ut}(\tilde{Y}_{0})\subset{\mathrm{A}ut}(\tilde{X}_{0})$. It
follows that we have an isomorpism $\tilde{X}/\langle
g\rangle\cong\tilde{X}_{0}/\langle g^{\prime}\rangle\times Z$, and so $X$
splits as a product. We then have $(X,B)\rightarrow Z$ is a product, since we
reduced to the case when $B$ is the pull-back of a divisor supported on
$B_{Y}$.

Case 2: In this case we assume that $B$ is big over $Y$.

Suppose for sake of contradiction that ${\mathrm{v}ar}(X/Z,B)=1$.

Let $(X^{\prime},(1+\epsilon)B^{\prime})$ denote the log canonical model of
$(X,(1+\epsilon)B)$ over $T$ for $0<\epsilon\ll 1$. Note that
$(X^{\prime},(1+\epsilon)B^{\prime})$ is a stable family over some open subset
$U\subset Z$. Let $(X^{\prime\prime},(1+\epsilon)B^{\prime\prime})$ be a
compactification of this stable family over $U$ to a stable family over $Z$
(perhaps after replacing $Z$ by a finite cover). By [KP17, Corollary 8.3] and
our assumption that ${\mathrm{v}ar}(X/Z,B)=1$ we see that
$K_{X^{\prime\prime}/Z}+(1+\epsilon)B^{\prime\prime}$ is big.

For general $y\in Y_{0}$, let $Z_{y}$ denote the fibre of $Y_{0}\times
Z\rightarrow Y_{0}$, and let
$(X_{y}^{\prime\prime},(1+\epsilon)B^{\prime\prime}_{y})$ denote the
restricted family over $Z_{y}$. We note that by [Kol22, Proposition 2.13]
$(X^{\prime\prime}_{y},(1+\epsilon)B^{\prime\prime}_{y})$ is locally stable
over $Z_{y}$. By Proposition 2.13 we have that
$(X^{\prime\prime}_{y},(1+\epsilon)B^{\prime\prime}_{y})\rightarrow Z_{y}$ is
isotrivial over $U$, hence isotrivial. This implies that
$K_{X^{\prime\prime}_{y}}+(1+\epsilon)B^{\prime\prime}_{y}$ is not big.
Finally, observe that
$(K_{X^{\prime\prime}/Z}+(1+\epsilon)B^{\prime\prime})|_{X^{\prime\prime}_{y}}\sim
K_{X^{\prime\prime}_{y}/Z_{y}}+(1+\epsilon)B^{\prime\prime}_{y}$, which
contradicts the bigness of
$K_{X^{\prime\prime}/Z}+(1+\epsilon)B^{\prime\prime}$, giving our required
contradiction.

The map from $Z$ to the moduli space parametrising the fibres of
$(X^{\prime\prime},(1+\epsilon)B^{\prime\prime})\rightarrow Z$ is therefore
constant, and so $(X^{\prime\prime},(1+\epsilon)B^{\prime\prime})$ splits as a
product after a generically finite base change, hence by Lemma 5.3 $(X,B)$
itself splits as a product.

Case 3: In this case we assume that $K_{X}$ is not pseudo-effective over $Y$
and we reduce to Case 2.

We run a $K_{X}$-MMP with scaling of any ample divisor over $Y$. Since $K_{X}$
is not pseudo-effective, this MMP terminates with a Mori fiber space
$X^{\prime}\rightarrow W$ over $Y$. In particular, we have
$\operatorname{dim}W<\operatorname{dim}X$, and hence $X^{\prime}\rightarrow W$
is not birational. Since $K_{X}+B\sim_{\mathbb{Q}}0/Y$, the final outcome
$(X^{\prime},B^{\prime})\rightarrow Z$ is still a locally stable family of
good minimal models, and it is crepant birational to $(X,B)$. Thus, up to
replacing $(X,B)$ with $(X^{\prime},B^{\prime})$, we may assume that
$X\rightarrow W$ is a morphism. If $\operatorname{dim}W=\operatorname{dim}Y$,
then $W\rightarrow Y$ is birational and $B$ is big over $Y$ and so we may
reduce to Case 2. Therefore, we may assume that
$\operatorname{dim}Y<\operatorname{dim}W<\operatorname{dim}X$.

Let $(W,B_{W},\mathbf{M}_{W})$ denote the generalized pair induced by the
canonical bundle formula for $(X,B)\rightarrow W$. By [Amb05], we may choose
$0\leq\Gamma\sim_{\mathbb{Q}}\mathbf{M}_{W}$ such that $(W,B_{W}+\Gamma)$ is
klt. It follows that $(W,B_{W}+\Gamma)\rightarrow Z$ is generically a locally
stable family of good minimal models, and so, up to base change of $Z$, we may
assume that this family admits a compactification
$(W^{\prime},B_{W^{\prime}}+\Gamma^{\prime})\rightarrow Z$ that is everywhere
locally stable and that agrees with $(W,B_{W}+\Gamma)\rightarrow Z$ over a
non-empty open subset of $Z$. Let $\tilde{W}$ denote the normalization of the
graph of $W\rightarrow W^{\prime}$, and let $p\colon\tilde{W}\rightarrow W$
and $q\colon\tilde{W}\rightarrow W^{\prime}$ denote the corresponding
morphisms. Following the proof of [Kol22, Theorem 11.40], it follows that

(6) $p^{*}(K_{W}+B_{W}+\Gamma)\geq
q^{*}(K_{W^{\prime}}+B_{W^{\prime}}+\Gamma^{\prime}).$

Since $(W,B_{W}+\Gamma)$ is crepant to $(X,B)$, it follows that
$\lambda_{-}=\lambda_{+}=0$ for $(W,B_{W}+\Gamma)\rightarrow Z$. Then, by (6)
and the fact that $(W^{\prime},B_{W^{\prime}}+\Gamma^{\prime})\rightarrow Z$
is a locally stable family of good minimal models, it follows that
$\lambda_{-}=\lambda_{+}=0$ for
$(W^{\prime},B_{W^{\prime}}+\Gamma^{\prime})\rightarrow Z$. By our induction
hypothesis, up to a base change,
$(W^{\prime},B_{W^{\prime}}+\Gamma^{\prime})\rightarrow Z$ splits as a
product.

Since $(Y,B_{Y}+\mathbf{M}_{Y})$ and
$(W^{\prime},B_{W^{\prime}}+\Gamma^{\prime})$ are products, and since for a
general $z$ we have a morphism $g_{z}\colon W^{\prime}_{z}\rightarrow Y_{z}$
with
$g_{z}^{*}(K_{Y,z}+B_{Y,z}+\mathbf{M}_{Y,z})\sim_{\mathbb{Q}}K_{W^{\prime},z}+B_{W^{\prime},z}+\Gamma^{\prime}_{z}$
it follows that there exists a morphism $g\colon W^{\prime}\rightarrow Y$ such
that
$g^{*}(K_{Y}+B_{Y}+\mathbf{M}_{Y})\sim_{\mathbb{Q}}K_{W^{\prime}}+B_{W^{\prime}}+\Gamma^{\prime}$.
By construction we have a morphism $h\colon W\rightarrow Y$ such that
$K_{W}+B_{W}+\Gamma\sim_{\mathbb{Q}}h^{*}(K_{Y}+B_{Y}+\mathbf{M}_{Y})$ and so
$(W,B_{W}+\Gamma)$, and $(W^{\prime},B_{W^{\prime}}+\Gamma^{\prime})$ are
crepant birational to each other. By Lemma 5.3 it follows that
$(W,B_{W}+\Gamma)$ is a product.

We may then argue as in Case (2), applied to the morphism $(X,B)\rightarrow W$
to deduce that $(X,B)$ is itself a product. ∎

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