# On the B-Semiampleness Conjecture

Enrica Floris and Vladimir Lazić

###### Abstract

The B-Semiampleness Conjecture of Prokhorov and Shokurov predicts that the
moduli part in a canonical bundle formula is semiample on a birational
modification. We prove that the restriction of the moduli part to any
sufficiently high divisorial valuation is semiample, assuming the conjecture
in lower dimensions.

  *     * Keywords. Canonical bundle formula, moduli divisor

2010 Mathematics Subject Classification. 14H10; 14E30; 14N30

[Français]

Titre. Sur la conjecture de B-semi-amplitude Résumé. La conjecture de B-semi-
amplitude de Prokhorov et Shokurov prédit que la partie modulaire de la
formule du fibré canonique doit être semi-ample sur une modification
birationnelle. En supposant la validité de cette conjecture en dimensions
inférieures, nous montrons que la partie modulaire est semi-ample en
restriction à toute valuation divisorielle suffisamment haute.

###### Contents

  1. 1\. Introduction
  2. 2\. Preliminaries
  3. 3\. Canonical bundle formula
  4. 4\. Semiampleness on divisorial valuations
  5. 5\. Reduction to a weaker conjecture
  6. References

##  1\. Introduction

Let $(X,\Delta)$ be a complex projective pair with log canonical singularities
and let $f\colon X\to Y$ be a morphism such that

$K_{X}+\Delta\sim_{\mathbb{Q}}f^{*}D$ (1)

for some $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor $D$ on $Y$. We say that $f$
is an _lc-trivial fibration_ ; see Section 3 below. A typical example is when
$K_{X}+\Delta$ is semiample and $f$ is the associated Iitaka fibration; a
plethora of similar situations occurs in algebraic geometry. It is a
fundamental question whether there exists a log canonical structure
$(Y,\Delta_{Y})$ such that $D\sim_{\mathbb{Q}}K_{Y}+\Delta_{Y}$: in other
words, whether the singularities of $X$ _descend_ to $Y$. This is an important
question for at least two reasons: first, an affirmative answer would show
that log canonical singularities form a “stable” category, and second, it
enables proofs by induction. The affirmative answer to the question above is
known in several important special cases: when $f$ is a fibration and
$(X,\Delta)$ has klt singularities [Amb05], when $f$ is generically finite
[FG12], and when $Y$ is a curve [Amb04, Theorem 0.1]. With notation as in (1),
it is known that

$D\sim_{\mathbb{Q}}K_{Y}+B_{Y}+M_{Y},$

where $B_{Y}$ (the _discriminant_) is closely related to the singularities of
$f$, and the divisor $M_{Y}$ (the _moduli divisor_) conjecturally carries
information on the birational variation of the fibres of $f$. A study of
formulas of this type – of _canonical bundle formulas_ – began with Kodaira’s
canonical bundle formula for elliptic surface fibrations. Much is known about
the birational behaviour of such formulas: In particular, it is known that,
after passing to a certain birational model $Y^{\prime}$ of $Y$, the divisor
$M_{Y^{\prime}}$ is nef and for any higher birational model
$Y^{\prime\prime}\to Y^{\prime}$ the induced $M_{Y^{\prime\prime}}$ on
$Y^{\prime\prime}$ is the pullback of $M_{Y^{\prime}}$ [Kaw98, Amb04, Kol07].
We call such a variety $Y^{\prime}$ an _Ambro model_ of $f$. The following is
a conjecture of Prokhorov and Shokurov [PS09, Conjecture 7.13], and it is the
most important open problem regarding canonical bundle formulas.

#### B-Semiampleness Conjecture.

Let $(X,\Delta)$ be a pair and let $f\colon(X,\Delta)\to Y$ be an lc-trivial
fibration to an $n$-dimensional variety $Y$, where the divisor $\Delta$ is
effective over the generic point of $Y$. If $Y$ is an Ambro model of $f$, then
the moduli divisor $M_{Y}$ is semiample.

A proof of this conjecture would give an affirmative answer to the question
stated at the beginning of this paper. Note that when the singularities of
$(X,\Delta)$ are only klt, it was sufficient to show a weaker version – that
the moduli part is _nef and b-good_ , as demonstrated by Ambro in [Amb05]. The
B-Semiampleness Conjecture is known when $Y$ is a curve [Amb04, Theorem 0.1],
when a general fibre of $f$ is a curve by the classical work of Kodaira and by
[PS09, Theorem 8.1], or when a general fibre of $f$ is a smooth non-rational
surface [Fuj03, Fil18]; see also [BC16] showing the semiampleness of a
“nearby” divisor. Often a crucial role in the proof is played by the existence
of a moduli space for the fibres. Nothing has been previously known in general
when $\dim Y\geq 2$. In the remainder of the paper, we say that the
B-Semiampleness Conjecture holds in dimension $n$, if it holds (in the
notation from formulation of the B-Semiampleness Conjecture) for all lc-
trivial fibrations $f\colon(X,\Delta)\to Y$ with $\dim Y=n$. The content of
the paper. The main result of this paper is that the moduli part of an lc-
trivial fibration is semiample when restricted to any divisorial valuation
over its Ambro model:

###### Theorem A

Assume the B-Semiampleness Conjecture in dimension $n-1$. Let $(X,\Delta)$ be
a pair and let $f\colon(X,\Delta)\to Y$ be an lc-trivial fibration to an
$n$-dimensional variety $Y$, where the divisor $\Delta$ is effective over the
generic point of $Y$. Assume that $Y$ is an Ambro model for $f$. Then for
every birational model $\pi\colon Y^{\prime}\to Y$ and for every prime divisor
$T$ on $Y^{\prime}$ with the normalisation $T^{\nu}$ and the induced morphism
$\nu\colon T^{\nu}\to Y^{\prime}$, the divisor $\nu^{*}\pi^{*}M_{Y}$ is
semiample on $T^{\nu}$.

If the moduli part $M_{Y}$ of the fibration is big and we consider only
components of the locus $\operatorname{\mathbf{B}}_{+}(M_{Y})$ where it is not
ample (the _augmented base locus_ of $M_{Y}$, see Definition 2.2), we can
relax the assumptions on the B-Semiampleness Conjecture:

###### Theorem B

Assume the B-Semiampleness Conjecture in dimensions at most $n-2$. Let
$(X,\Delta)$ be a pair and let $f\colon(X,\Delta)\to Y$ be an lc-trivial
fibration to an $n$-dimensional variety $Y$, where the divisor $\Delta$ is
effective over the generic point of $Y$. Assume that $Y$ is an Ambro model for
$f$ and that $M_{Y}$ is big. Then for every birational model $\pi\colon
Y^{\prime}\to Y$ and for any divisorial component $T$ of
$\operatorname{\mathbf{B}}_{+}(\pi^{*}M_{Y})$ with the normalisation $T^{\nu}$
and the induced morphism $\nu\colon T^{\nu}\to Y^{\prime}$, the divisor
$\nu^{*}\pi^{*}M_{Y}$ is semiample on $T^{\nu}$.

Immediate corollaries are:

###### Corollary C

Let $(X,\Delta)$ be a pair and let $f\colon(X,\Delta)\to Y$ be an lc-trivial
fibration to a surface $Y$, where the divisor $\Delta$ is effective over the
generic point of $Y$. Assume that $Y$ is an Ambro model for $f$. Then for
every divisor $T$ on $Y$ with the normalisation $T^{\nu}$ and the induced
morphism $\nu\colon T^{\nu}\to Y$, the divisor $\nu^{*}M_{Y}$ is semiample. If
additionally $M_{Y}$ is big, then $\nu^{*}M_{Y}\sim_{\mathbb{Q}}0$ for every
divisorial component $T$ of $\operatorname{\mathbf{B}}_{+}(M_{Y})$ with the
normalisation $T^{\nu}$ and the induced morphism $\nu\colon T^{\nu}\to Y$.

###### Corollary D

Let $(X,\Delta)$ be a pair and let $f\colon(X,\Delta)\to Y$ be an lc-trivial
fibration to a threefold $Y$, where the divisor $\Delta$ is effective over the
generic point of $Y$. Assume that $Y$ is an Ambro model for $f$ and that
$M_{Y}$ is big. Then $\nu^{*}M_{Y}$ is semiample for every divisorial
component $T$ of $\operatorname{\mathbf{B}}_{+}(M_{Y})$ with the normalisation
$T^{\nu}$ and the induced morphism $\nu\colon T^{\nu}\to Y$.

Sketch of the proof. The proof of Theorem A is very technical, and the core of
the arguments is contained in Section 4. In Proposition 4.2 we use the Minimal
Model Program to achieve, roughly, the situation where over $T$ there exists a
log canonical centre $S$ of $(X,\Delta)$ and an lc-trivial fibration
$h\colon(S,\Delta_{S})\to T$ whose moduli part is _almost_ $M_{Y}|_{T}$. This
part of the proof is somewhat similar to and inspired by the proof in [FG14].
An important difference is that we do not cut down to curves in order to use
the semistable reduction, but we study in detail those divisors which are not
contracted by the MMP. To this end, we introduce _acceptable_ lc-trivial
fibrations; roughly speaking, these fibrations might not come with a sub-
boundary $\Delta$ which is effective on a general fibre of the lc-trivial
fibration, but are obtained from such a fibration by blowing up. This suffices
to ensure that, by the results of Nakayama [Nak04], the MMP that we run
actually terminates. Finally, a careful choice of a base change in Proposition
4.4 and an analysis of the ramification loci of the finite part of the Stein
factorisation of the map $f|_{S}$ allow to finish the proof. We mention here
that this all uses many foundational results of Kawamata and Ambro. A
reduction result. As a by-product of the techniques employed in the proofs, in
Section 5 we reduce the B-Semiampleness Conjecture to a much weaker Conjecture
5.1.

###### Theorem E

Assume that the B-Semiampleness Conjecture holds for klt-trivial fibrations
$f\colon(X,\Delta)\to Y$, where $(X,\Delta)$ is a log canonical pair,
$\Delta\geq 0$, the moduli divisor $M_{Y}$ is big and $\dim Y\leq n$. Then the
B-Semiampleness Conjecture holds in dimension $n$.

The result says that it suffices to prove the B-Semiampleness Conjecture for
_klt-trivial_ fibrations $f\colon(X,\Delta)\to Y$, where $\Delta$ is effective
and the moduli divisor of $f$ is big. The method of the proof is similar to
that in Section 4, together with a quick application of a result of Ambro
[Amb05, Theorem 3.3]. This improves on the discussion and results in [Fuj15,
Section 3]. Consequently, in Theorems A and B it suffices to assume Conjecture
5.1 instead of the B-Semiampleness Conjecture.

##  2\. Preliminaries

We work over $\mathbb{C}$. We denote by $\equiv$, $\sim$ and
$\sim_{\mathbb{Q}}$ the numerical, linear and $\mathbb{Q}$-linear equivalence
of divisors respectively. For a decomposition of a Weil $\mathbb{Q}$-divisor
$D=\sum d_{i}D_{i}$ into prime components and a real number $\alpha$, we
denote

$D_{\mathrm{red}}=\sum D_{i},\quad
D^{\geq\alpha}:=\sum_{d_{i}\geq\alpha}d_{i}D_{i}\quad\text{and}\quad
D^{\leq\alpha}:=\sum_{d_{i}\leq\alpha}d_{i}D_{i},$

and similarly for $D^{>\alpha}$, $D^{<\alpha}$ and $D^{=\alpha}$. If $f\colon
X\to Y$ is a proper surjective morphism between normal varieties and $D$ is a
Weil $\mathbb{R}$-divisor on $X$, then $D_{v}$ and $D_{h}$ denote the vertical
and the horizontal part of $D$ with respect to $f$; the relevant map will be
clear from the context. For instance, the notation $D_{v}^{\geq 0}$ denotes
the non-negative part of the vertical part of $D$.

###  2.A. Base loci

We start with the following well-known result; we include the proof for the
benefit of the reader.

###### Lemma 2.1

Let $f\colon X\rightarrow Y$ be a surjective morphism between normal
projective varieties. Let $D$ be a Cartier divisor on $Y$. Then $D$ is
semiample if and only if $f^{*}D$ is semiample.

  * Proof.

Since necessity is clear, it suffices to prove sufficiency. So assume that
$f^{*}D$ is semiample, and we may assume that $f^{*}D$ is basepoint free. By
considering the Stein factorisation of $f$, it suffices to consider separately
the cases when either $f$ has connected fibres or $f$ is finite. When $f$ has
connected fibres, pick a closed point $y\in Y$ and a closed point $x\in
f^{-1}(y)$. Since $f^{*}D$ is basepoint free, there exists $D_{X}\in|f^{*}D|$
such that $x\notin\operatorname{Supp}D_{X}$. As $H^{0}(X,f^{*}D)\simeq
H^{0}(Y,D)$, there exists a divisor $D_{Y}\in|D|$ such that
$D_{X}=f^{*}D_{Y}$. Then it is clear that $y\notin\operatorname{Supp}D_{Y}$,
which shows that $D$ is basepoint free. Now assume that $f$ is finite. If
$X^{\circ}$ and $Y^{\circ}$ are the smooth loci of $X$ and $Y$, respectively,
consider the open sets $U_{Y}:=Y^{\circ}\setminus f(X\setminus
X^{\circ})\subseteq Y$ and $U_{X}:=f^{-1}(U_{Y})\subseteq X$. By normality of
$X$ and $Y$, we have

$\operatorname{codim}_{X}(X\setminus
U_{X})=\operatorname{codim}_{Y}(Y\setminus U_{Y})\geq 2,$

and the map $f|_{U_{X}}\colon U_{X}\to U_{Y}$ is flat by [Har77, Exercise
III.9.3(a)]. Then, by [Ful98, Example 1.7.4], it yields
$(f|_{U_{X}})_{*}(f|_{U_{X}})^{*}(D|_{U_{Y}})=(\deg f)(D|_{U_{Y}})$ hence

$f_{*}f^{*}D=(\deg f)D.$ (2)

Now, for any point $y\in Y$, a general element $E\in|f^{*}D|$ avoids the
finite set $f^{-1}(y)$, hence $y\notin\operatorname{Supp}f_{*}E$. Since
$f_{*}E\in|(\deg f)D|$ by (2), this shows that $(\deg f)D$ is basepoint free.
$\Box$

We recall the definition of stable and augmented base loci of
$\mathbb{Q}$-divisors.

###### Definition 2.2

Let $D$ be a $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor on a normal projective
variety $X$. If $D$ is integral, the base locus of $D$ is denoted by
$\operatorname{Bs}|D|$. The _stable base locus of $D$_ is

$\operatorname{\mathbf{B}}(D)=\bigcap_{D\sim_{\mathbb{Q}}D^{\prime}\geq
0}\operatorname{Supp}D^{\prime},$

and the _augmented base locus of $D$_ is

$\operatorname{\mathbf{B}}_{+}(D)=\bigcap_{n\in\mathbb{N}_{>0}}\textstyle\operatorname{\mathbf{B}}\big{(}D-\frac{1}{n}A\big{)},$

where $A$ is an ample divisor on $X$; this definition does not depend on the
choice of $A$. Clearly
$\operatorname{\mathbf{B}}(D)\subseteq\operatorname{\mathbf{B}}_{+}(D)$, and
both $\operatorname{\mathbf{B}}(D)$ and $\operatorname{\mathbf{B}}_{+}(D)$ are
closed subsets of $X$. The set $\operatorname{\mathbf{B}}_{+}(D)$ is empty if
and only if $D$ is ample, and is different from $X$ if and only if $D$ is big.

We need the following lemma in the proof of Theorem B.

###### Lemma 2.3

Let $X$ be a projective manifold and let $D$ be a nef and big
$\mathbb{Q}$-divisor on $X$. Let $T\subseteq\operatorname{\mathbf{B}}_{+}(D)$
be a prime divisor. Then $\mathcal{O}_{T}(D)$ is not big.

  * Proof.

Arguing by contradiction, we assume that $\mathcal{L}:=\mathcal{O}_{T}(D)$ is
big. Fix a point $x\in T\setminus\operatorname{\mathbf{B}}_{+}(\mathcal{L})$
which does not belong to any other component of
$\operatorname{\mathbf{B}}_{+}(D)$. Since
$x\in\operatorname{\mathbf{B}}_{+}(D)$, by [Nak00, Theorem 0.3] there exists a
positive dimensional subvariety
$V\subseteq\operatorname{\mathbf{B}}_{+}(D)\subseteq X$ such that $x\in V$ and
$D^{\dim V}\cdot V=0$. By the choice of $x$ we necessarily have $V\subseteq
T$, and hence

$\mathcal{L}^{\dim V}\cdot V=0.$

But then $V\subseteq\operatorname{\mathbf{B}}_{+}(\mathcal{L})$ again by
[Nak00, Theorem 0.3], hence $x\in\operatorname{\mathbf{B}}_{+}(\mathcal{L})$,
a contradiction. $\Box$

###  2.B. Pairs and resolutions

A _pair_ $(X,\Delta)$ consists of a normal variety $X$ and a Weil
$\mathbb{Q}$-divisor $\Delta$ such that $K_{X}+\Delta$ is
$\mathbb{Q}$-Cartier. A pair $(X,\Delta)$ is _log smooth_ if $X$ is smooth and
the support of $\Delta$ is a simple normal crossings divisor. In this paper,
unless explicitly stated otherwise, we do not require that $\Delta$ is an
effective divisor. A _log resolution_ of a pair $(X,\Delta)$ is a birational
morphism $f\colon Y\to X$ such that $Y$ is smooth, the exceptional locus
$\operatorname{Exc}(f)$ is a divisor and the divisor
$f_{*}^{-1}\Delta+\operatorname{Exc}(f)$ has simple normal crossings support.
In this paper we use the term _embedded resolution_ of a pair $(X,\Delta)$ in
the following strong sense: it is a log resolution of the pair which is an
isomorphism on the locus where $X$ is smooth and $\Delta$ has simple normal
crossings. The existence of embedded resolutions in this sense was proved in
[Sza94], see also [Kol13, Theorem 10.45(2)]. We will need the following lemma
in the proof of Proposition 4.4.

###### Lemma 2.4

Let $\pi\colon X^{\prime}\to X$ be a finite morphism between normal proper
varieties, and let $\widetilde{f}\colon\widetilde{Y}\to X^{\prime}$ be a
birational morphism. Then there exists a birational morphism $f\colon Y\to X$
such that, if $Y^{\prime}$ is the normalisation of the main component of the
fibre product $X^{\prime}\times_{X}Y$ and $f^{\prime}\colon
Y^{\prime}\dashrightarrow\widetilde{Y}$ is the induced birational map, then
$(f^{\prime})^{-1}$ is an isomorphism at the generic point of each
$\widetilde{f}$-exceptional prime divisor on $\widetilde{Y}$.

$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\widetilde{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\widetilde{f}}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad
f^{\prime}}$$\textstyle{X}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$

Moreover, we may assume that $f$ is a composition of blowups along proper
subvarieties.

  * Proof.

Each exceptional divisor on $\widetilde{Y}$ gives a divisorial valuation
$E_{i}^{\prime}$ over $X^{\prime}$. By the proof of [DL15, Proposition
2.14(i)] there are geometric valuations $E_{i}$ over $X$ such that the
following holds: Let $f\colon Y\to X$ be any birational model such that all
valuations $E_{i}$ are realised as prime divisors on $Y$; we can assume that
$f$ is a sequence of blowups along proper subvarieties by [KM98, Lemma 2.45].
If $Y^{\prime}$ is the normalisation of the main component of the fibre
product $X^{\prime}\times_{X}Y$, then every valuation $E_{i}^{\prime}$ is a
prime divisor on $Y^{\prime}$. Now, since the $E_{i}^{\prime}$ are prime
divisors on both $\widetilde{Y}$ and $Y^{\prime}$, the induced birational map
$f^{\prime}\colon Y^{\prime}\dashrightarrow\widetilde{Y}$ has to be an
isomorphism at the corresponding generic points, which proves the lemma.
$\Box$

###### Definition 2.5

Let $(X,\Delta)$ be a pair and let $\pi\colon Y\to X$ be a birational
morphism, where $Y$ is normal. We can write

$K_{Y}\sim_{\mathbb{Q}}\pi^{*}(K_{X}+\Delta)+\sum a(E_{i},X,\Delta)\cdot
E_{i},$

where $E_{i}\subseteq Y$ are distinct prime divisors and the numbers
$a(E_{i},X,\Delta)\in\mathbb{Q}$ are _discrepancies_. The closure of the image
of a geometric valuation $E$ on $X$ is the _centre of $E$ on $X$_, denoted by
$c_{X}(E)$. The pair $(X,\Delta)$ is _klt_ , respectively _log canonical_ , if
$a(E,X,\Delta)>-1$, respectively $a(E,X,\Delta)\geq-1$, for every geometric
valuation $E$ over $X$. The pair $(X,\Delta)$ is _dlt_ if $\Delta\geq 0$ and
there is a closed subset $Z\subseteq X$ such that $X\setminus Z$ is smooth,
$\Delta|_{X\setminus Z}$ is a simple normal crossings divisor, and if
$c_{X}(E)\subseteq Z$ for a geometric valuation $E$ over $X$, then
$a(E,X,\Delta)>-1$. A closed subset $Z\subseteq X$ is a _log canonical centre_
of a log canonical pair $(X,\Delta)$ if $Z=c_{X}(E)$ for some geometric
valuation $E$ over $X$ with $a(E,X,\Delta)=-1$. Let $f\colon(X,\Delta)\to Y$
be a proper morphism and let $Z\subseteq Y$ be a closed subset. Then
$(X,\Delta)$ is klt, respectively log canonical, over $Z$ (or over the generic
point of $Z$) if $a(E,X,\Delta)>-1$, resp. $a(E,X,\Delta)\geq-1$, for every
geometric valuation $E$ over $X$ with $f\big{(}c_{X}(E)\big{)}=Z$. We say that
a closed subvariety $S$ of $X$ is a _minimal log canonical centre of
$(X,\Delta)$ over $Z$_ if $S$ is a minimal log canonical centre of
$(X,\Delta)$ (with respect to inclusion) which dominates $Z$.

The following is [Fuj07, Proposition 3.9.2], see also the proof of [Kol13,
Theorem 4.16].

###### Proposition 2.6

Let $(X,\Delta)$ be a dlt pair. Let $\Delta^{=1}=\sum_{i\in I}D_{i}$ be the
decomposition into irreducible components. Then $S$ is a log canonical centre
of $(X,\Delta)$ with $\operatorname{codim}_{X}S=k$ if and only if $S$ is an
irreducible component of $D_{i_{1}}\cap D_{i_{2}}\cap\dots\cap D_{i_{k}}$ for
some $\\{i_{1},i_{2},\ldots,i_{k}\\}\subseteq I$. Moreover, $S$ is normal and
it is a log canonical centre of the dlt pair $\big{(}D_{i_{1}},(\Delta-
D_{i_{1}})|_{D_{i_{1}}}\big{)}$.

###### Definition 2.7

Let $(X_{1},\Delta_{1})$ and $(X_{2},\Delta_{2})$ be two pairs. A birational
map $\theta\colon X_{1}\dasharrow X_{2}$ is _crepant birational_ if
$a(E,X_{1},\Delta_{1})=a(E,X_{2},\Delta_{2})$ for every geometric valuation
$E$ over $X_{1}$ and $X_{2}$. If additionally we have a commutative diagram

$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\scriptstyle{\theta}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\scriptstyle{f_{2}}$ $\textstyle{Y}$

for some variety $Y$, then we say that $\theta$ is _crepant birational over
$Y$_.

We will need the following lemma in the proof of Proposition 4.2.

###### Lemma 2.8

Let $(X,\Delta)$ be a dlt pair such that $K_{X}+\Delta\sim_{\mathbb{Q}}F$,
where $F$ is an effective $\mathbb{Q}$-divisor having no common components
with $\Delta^{=1}$. Assume that there exists a smooth open subset $U\subseteq
X$ which intersects every log canonical centre of $(X,\Delta)$, and such that
the divisor $(\Delta+F)|_{U}$ has simple normal crossings support. Then any
$(K_{X}+\Delta)$-MMP is an isomorphism at the generic point of each log
canonical centre of $(X,\Delta)$, and moreover, it induces an inclusion-
preserving bijection from the set of log canonical centres of $(X,\Delta)$ to
the set of log canonical centres at each step of the MMP.

  * Proof.

For $i\geq 0$ let $(X_{i},\Delta_{i})$ be the pairs in the steps of this MMP
with $(X_{0},\Delta_{0}):=(X,\Delta)$, and let $F_{i}$ be the strict transform
of $F$ on $X_{i}$. _Step 1._ Let $\theta_{i}\colon X_{i}\to Z_{i}$ be the
extremal contraction at the $i$-th step of the MMP. If $\Gamma_{i}$ is a curve
on $X_{i}$ contracted by $\theta_{i}$, then

$F_{i}\cdot\Gamma_{i}=(K_{X_{i}}+\Delta_{i})\cdot\Gamma_{i}<0,$

and hence $\operatorname{Exc}(\theta_{i})\subseteq\operatorname{Supp}F_{i}$.
_Step 2._ We now show by induction on $i$ that:

    1. (a)i

$\theta_{i}$ is an isomorphism at the generic point of each log canonical
centre of $(X_{i},\Delta_{i})$, and

    2. (b)i

there exists a smooth open subset $U_{i}\subseteq X_{i}$ containing the
generic point of each log canonical centre of $(X_{i},\Delta_{i})$ such that
$(\Delta_{i}+F_{i})|_{U_{i}}$ has simple normal crossings support.

Indeed, note that (b)0 holds by assumption. Next, assume that (b)i holds for
some index $i$. Let $P$ be a log canonical centre of $(X_{i},\Delta_{i})$.
Then the generic point of $P$ belongs to $U_{i}$ and $P$ is an irreducible
component of an intersection of components of $\Delta_{i}^{=1}$ by Proposition
2.6, hence $P$ cannot be contained in $\operatorname{Supp}F_{i}$ by (b)i. By
Step 1, the map $\theta_{i}$ is an isomorphism at the generic point of $P$,
which shows (a)i. Therefore, by possibly shrinking $U_{i}$, we may assume that
$\theta_{i}|_{U_{i}}$ is an isomorphism, and we define $U_{i+1}$ as
$\theta_{i}(U_{i})$ if $\theta_{i}$ is divisorial, or as the image of $U_{i}$
by the corresponding flip if $\theta_{i}$ is a flipping contraction. This
shows (b)i+1. _Step 3._ Finally, the last statement in the lemma follows
immediately from Step 2, from the definition of dlt singularities, and the
fact that the MMP does not decrease discrepancies. $\Box$

###  2.C. Weakly exceptional divisors

We use the relative Nakayama-Zariski decomposition of pseudoeffective divisors
as in [Nak04, Chapter III]. Note that by [Les16] this is not always well-
defined; however, in all the cases we consider in this paper, the
decomposition exists and behaves as in the absolute case. Note that one can
define this decomposition on any $\mathbb{Q}$-factorial variety; below we give
additional comments when we use it on non-smooth varieties. The following
definition is crucial for applications in Sections 4 and 5. Note that part (b)
differs somewhat from that in [Nak04], see Remark 2.13.

###### Definition 2.9

Let $f\colon X\rightarrow Y$ be a projective surjective morphism of normal
varieties and let $D$ be an effective Weil $\mathbb{R}$-divisor on $X$ such
that $f(D)\neq Y$. Then $D$ is:

  1. (a)

_$f$ -exceptional_ if $\operatorname{codim}_{Y}\operatorname{Supp}f(D)\geq 2$,

  2. (b)

_of insufficient fibre type over $Y$_ if $f(D)$ has pure codimension $1$ and
for every prime divisor $\Gamma\subseteq f(D)$ there exists a divisor
$E\not\subseteq\operatorname{Supp}D$ such that $f(E)=\Gamma$.

  3. (c)

_weakly $f$-exceptional_ if there are effective divisors $D_{1}$ and $D_{2}$
such that $D=D_{1}+D_{2}$, where $D_{1}$ is $f$-exceptional and $D_{2}$ is of
insufficient fibre type over $Y$.

Let additionally $g\colon Y\rightarrow Z$ be a projective surjective morphism
of normal varieties and let $D_{h}$ and $D_{v}$ denote the horizontal and
vertical parts of $D$ with respect to $g\circ f$. If $D_{h}$ is
$f$-exceptional and $D_{v}$ is weakly $(g\circ f)$-exceptional, then we call
$D$ an _$(f,g)$ -EWE divisor_.

###### Remark 2.10

The abbreviation EWE stands for “exceptional-weakly-exceptional”.

Proposition 2.12 shows the most important property of EWE divisors.

###### Lemma 2.11

Let $f\colon X\to Y$ and $g\colon Y\rightarrow Z$ be projective surjective
morphisms with connected fibres between normal varieties, assume that $X$ is
smooth, and let $D$ be an effective $(f,g)$-EWE divisor on $X$. Then there
exists a component $\Gamma$ of $D$ such that $\mathcal{O}_{\Gamma}(D)$ is not
$(g\circ f)|_{\Gamma}$-pseudoeffective over $g\big{(}f(\Gamma)\big{)}$.

  * Proof.

Let $D_{h}$ and $D_{v}$ denote the horizontal and vertical parts of $D$ with
respect to $g\circ f$. Assume first that $D_{h}=0$. If $D_{v}$ is $(g\circ
f)$-exceptional, we are done by [Nak04, Lemma III.5.1]. Otherwise, choose a
prime divisor $\Gamma\subseteq\operatorname{Supp}D_{v}$ such that
$g\big{(}f(\Gamma)\big{)}$ is a divisor. Then $\mathcal{O}_{\Gamma}(D)$ is not
$(g\circ f)|_{\Gamma}$-pseudoeffective over $g\big{(}f(\Gamma)\big{)}$ by
[Nak04, Lemma III.5.2]. Thus we may assume that $D_{h}\neq 0$. By [Nak04,
Lemma III.5.1] there is a component $\Gamma$ of $D_{h}$ such that
$\mathcal{O}_{\Gamma}(D_{h})$ is not $(f|_{\Gamma})$-pseudoeffective over
$f(\Gamma)$, hence $\mathcal{O}_{\Gamma}(D_{h})$ is not $(g\circ
f)|_{\Gamma}$-pseudoeffective over $g\big{(}f(\Gamma)\big{)}=Z$. Since $D_{v}$
does not intersect the generic fibre of $g\circ f$, we obtain that
$\mathcal{O}_{\Gamma}(D_{h}+D_{v})$ is not $(g\circ
f)|_{\Gamma}$-pseudoeffective over $Z$, which implies the lemma. $\Box$

###### Proposition 2.12

Let $f\colon X\to Y$ and $g\colon Y\rightarrow Z$ be projective surjective
morphisms of normal varieties, assume that $X$ is smooth, and let $D$ be an
effective $(f,g)$-EWE divisor on $X$. Then $D=N_{\sigma}(D;X/Z)$.

  * Proof.

The proof goes verbatim as the proof of [Nak04, Proposition III.5.7],
replacing [Nak04, Lemma III.5.5] with Lemma 2.11. $\Box$

###### Remark 2.13

The definition of insufficient fibre type above is slightly different from
(and more precise than) the one in [Nak04, §III.5.a]. In order for [Nak04,
Corollary III.5.6 and Proposition III.5.7] to hold, one needs to work with the
definition above; this is in fact implicit from the proofs of these two
statements in op. cit. If one works with the definition as in op. cit., one
can easily construct a counterexample: let $f\colon X\to Y$ be a fibration
from a smooth surface to a smooth curve, let $F_{1}$ and $F_{2}$ be two
distinct fibres of $f$ and let $\pi\colon\widetilde{X}\to X$ be the blowup of
a point in $F_{1}$. Then $\pi_{*}^{-1}F_{1}+\pi^{*}F_{2}$ would be of
insufficient fibre type over $Y$, and [Nak04, Proposition III.5.7 and Lemma
III.4.2] would imply that $N_{\sigma}(\pi^{*}F_{2};Z/Y)=\pi^{*}F_{2}$, a
contradiction since $\pi^{*}F_{2}$ is $(f\circ\pi)$-nef.

We need in this paper the MMP with scaling as described in [Bir10, Definition
3.2], and the fact that log canonical flips exist [Bir12, HX13]. We also need
the following.

###### Lemma 2.14

Let $(X,\Delta)$ be a $\mathbb{Q}$-factorial dlt pair and let $\pi\colon X\to
U$ be a projective morphism such that $K_{X}+\Delta$ is $\pi$-pseudoeffective.
If $N_{\sigma}(K_{X}+\Delta;X/U)$ is an $\mathbb{R}$-divisor, then any Minimal
Model Program of $K_{X}+\Delta$ with scaling of an ample divisor over $U$
contracts precisely the components of $N_{\sigma}(K_{X}+\Delta;X/U)$.

  * Proof.

For $i\geq 0$, let $(X_{i},\Delta_{i})$ be the pairs in a $(K_{X}+\Delta)$-MMP
with scaling of an ample divisor over $U$, with
$(X_{0},\Delta_{0}):=(X,\Delta)$. Let $(p_{i},q_{i})\colon W_{i}\to X\times
X_{i}$ be a smooth resolution of indeterminacies of the induced birational map
$\varphi_{i}\colon X\dashrightarrow X_{i}$. Then there exists an effective
$q_{i}$-exceptional divisor $E_{i}$ on $W_{i}$ such that

$p_{i}^{*}(K_{X}+\Delta)\sim_{\mathbb{Q}}q_{i}^{*}(K_{X_{i}}+\Delta_{i})+E_{i}.$
(3)

Analogously as in the proof of [GL13, Lemma 2.16] we have

$N_{\sigma}\big{(}q_{i}^{*}(K_{X_{i}}+\Delta_{i})+E_{i};W_{i}/U\big{)}=N_{\sigma}\big{(}q_{i}^{*}(K_{X_{i}}+\Delta_{i});W_{i}/U\big{)}+E_{i},$
(4)

and the first two lines of the proof of [Nak04, Lemma III.5.15] give

$N_{\sigma}(K_{X}+\Delta;X/U)=p_{i*}N_{\sigma}\big{(}p_{i}^{*}(K_{X}+\Delta);W_{i}/U\big{)}$
(5)

and

$N_{\sigma}(K_{X_{i}}+\Delta_{i};X_{i}/U)=q_{i*}N_{\sigma}\big{(}q_{i}^{*}(K_{X_{i}}+\Delta_{i});W_{i}/U\big{)}.$
(6)

Then:

$\displaystyle N_{\sigma}(K_{X_{i}}+\Delta_{i};X_{i}/U)$
$\displaystyle=q_{i*}N_{\sigma}\big{(}q_{i}^{*}(K_{X_{i}}+\Delta_{i});W_{i}/U\big{)}$
by (6)
$\displaystyle=q_{i*}\big{(}N_{\sigma}\big{(}q_{i}^{*}(K_{X_{i}}+\Delta_{i});W_{i}/U\big{)}+E_{i}\big{)}$
$\displaystyle=\varphi_{i*}p_{i*}N_{\sigma}\big{(}q_{i}^{*}(K_{X_{i}}+\Delta_{i})+E_{i};W_{i}/U\big{)}$
by (4)
$\displaystyle=\varphi_{i*}p_{i*}N_{\sigma}\big{(}p_{i}^{*}(K_{X}+\Delta);W_{i}/U\big{)}$
by (3) $\displaystyle=\varphi_{i*}N_{\sigma}(K_{X}+\Delta;X/U).$ by (5)

Since $K_{X_{i}}+\Delta_{i}$ is in the movable cone over $U$ if and only if
$N_{\sigma}(K_{X_{i}}+\Delta_{i};X_{i}/U)=0$, we conclude by [Fuj11, Theorem
2.3]. $\Box$

##  3\. Canonical bundle formula

In this section we define the main object of this paper and state several
properties we repeatedly use.

###### Definition 3.1

Let $(X,\Delta)$ be a pair and let $\pi\colon X^{\prime}\rightarrow X$ be a
log resolution of the pair. A morphism $f\colon(X,\Delta)\rightarrow Y$ to a
normal projective variety $Y$ is a _klt-trivial_ , respectively _lc-trivial_ ,
fibration if $f$ is a surjective morphism with connected fibres, $(X,\Delta)$
has klt, respectively log canonical, singularities over the generic point of
$Y$, there exists a $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor $D$ on $Y$ such
that

$K_{X}+\Delta\sim_{\mathbb{Q}}f^{*}D,$

and if $f^{\prime}=f\circ\pi$, then

$\operatorname{rk}f^{\prime}_{*}\mathcal{O}_{X^{\prime}}\big{(}\lceil
K_{X^{\prime}}-\pi^{*}(K_{X}+\Delta)\rceil\big{)}=1,$

respectively

$\textstyle\operatorname{rk}f^{\prime}_{*}\mathcal{O}_{X^{\prime}}\big{(}\lceil
K_{X^{\prime}}-\pi^{*}(K_{X}+\Delta)+\sum_{a(E,X,\Delta)=-1}E\rceil\big{)}=1.$

###### Remark 3.2

This last condition in the previous definition is verified, for instance, if
$\Delta$ is effective on the generic fibre, which is the case in this paper.

###### Definition 3.3

Let $f\colon(X,\Delta)\to Y$ be an lc-trivial fibration, and let $P\subseteq
Y$ be a prime divisor. The _generic log canonical threshold_ of $f^{*}P$ with
respect to $(X,\Delta)$ is

$\gamma_{P}=\sup\\{t\in\mathbb{R}\mid(X,\Delta+tf^{*}P)\textrm{ is log
canonical over }P\\}.$

The _discriminant_ of $f$ is

$B_{f}=\sum_{P}(1-\gamma_{P})P.$ (7)

This is a Weil $\mathbb{Q}$-divisor on $Y$, and it is effective if $\Delta$ is
effective. Fix $\varphi\in\mathbb{C}(X)$ and the smallest positive integer $r$
such that $K_{X}+\Delta+\frac{1}{r}\operatorname{div}\varphi=f^{*}D$. Then
there exists a unique Weil $\mathbb{Q}$-divisor $M_{f}$, the _moduli part_ of
$f$, such that

$K_{X}+\Delta+\frac{1}{r}\operatorname{div}\varphi=f^{*}(K_{Y}+B_{f}+M_{f}).$
(8)

The formula (8) is the _canonical bundle formula_ associated to $f$ (and
$\varphi$).

###### Remark 3.4

It is customary in the literature to denote the discriminant and the moduli
part by $B_{Y}$ and $M_{Y}$. We adopted a different notation, since we
sometimes have to compare discriminants or moduli parts of different lc-
trivial fibrations which have the same base $Y$. However, if the fibration is
clear from the context, we still occasionally write $B_{Y}$ and $M_{Y}$.

###### Remark 3.5

If $f_{1}\colon(X_{1},\Delta_{1})\to Y$ and $f_{2}\colon(X_{2},\Delta_{2})\to
Y$ are two lc-trivial fibrations over the same base which are crepant
birational over $Y$, then $B_{f_{1}}=B_{f_{2}}$ and
$M_{f_{1}}\sim_{\mathbb{Q}}M_{f_{2}}$, see [Amb04, Lemma 2.6].

###### Remark 3.6

Let $f\colon(X,\Delta)\to Y$ be an lc-trivial (respectively klt-trivial)
fibration, where $Y$ is smooth. Then there exists a divisor $\Delta^{\circ}$
on $X$ such that the pair $(X,\Delta^{\circ})$ is log canonical and
$f\colon(X,\Delta^{\circ})\to Y$ is an lc-trivial (respectively klt-trivial)
fibration. Indeed, let $\pi\colon X^{\prime}\to X$ be a log resolution of
$(X,\Delta)$, and set $\Delta^{\prime}:=K_{X^{\prime}}-\pi^{*}(K_{X}+\Delta)$.
Let $\\{P_{i}\\}_{i\in I}$ be the finite set of prime divisors on $X^{\prime}$
such that $\operatorname{mult}_{P_{i}}\Delta^{\prime}>1$. Then each $P_{i}$ is
a vertical divisor, and let $D_{i}$ be any prime divisor on $Y$ which contains
$f\big{(}\pi(P_{i})\big{)}$. Then the divisor

$\Delta^{\circ}:=\Delta-\sum_{i\in
I}(\operatorname{mult}_{P_{i}}\Delta^{\prime})f^{*}D_{i}$

is the desired divisor.

The canonical bundle formula satisfies several desirable properties. To start
with, if $f\colon(X,\Delta)\to Y$ is a klt-trivial (respectively lc-trivial)
fibration, if $\rho\colon Y^{\prime}\to Y$ is a proper generically finite
morphism, and if we consider a base change diagram

$\textstyle{(X^{\prime},\Delta^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\scriptstyle{f^{\prime}}$$\textstyle{(X,\Delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{Y,}$

where $X^{\prime}$ is the normalisation of the main component of
$X\times_{Y}Y^{\prime}$ and
$\Delta^{\prime}:=K_{X^{\prime}}-\tau^{*}(K_{X}+\Delta)$, then
$f^{\prime}\colon(X^{\prime},\Delta^{\prime})\to Y^{\prime}$ is also a klt-
trivial (respectively lc-trivial) fibration. In the rest of the paper, we
implicitly refer to this klt-trivial (respectively lc-trivial) fibration when
writing $B_{Y^{\prime}}$ and $M_{Y^{\prime}}$ for the discriminant and moduli
part. The following is the important _base change property_ from [Amb04,
Theorem 0.2], [Kaw98, Theorem 2] and [FG14, Theorem 3.6]:

###### Theorem 3.7

Let $f\colon(X,\Delta)\to Y$ be an lc-trivial fibration. Then there exists a
proper birational morphism $Y^{\prime}\to Y$ such that for every proper
birational morphism $\pi\colon Y^{\prime\prime}\to Y^{\prime}$ we have:

  1. (i)

$K_{Y^{\prime}}+B_{Y^{\prime}}$ is a $\mathbb{Q}$-Cartier divisor and
$K_{Y^{\prime\prime}}+B_{Y^{\prime\prime}}=\pi^{*}(K_{Y^{\prime}}+B_{Y^{\prime}})$,

  2. (ii)

$M_{Y^{\prime}}$ is a nef $\mathbb{Q}$-Cartier divisor and
$M_{Y^{\prime\prime}}=\pi^{*}M_{Y^{\prime}}$.

In the context of the previous theorem, we say that _the moduli part descends_
to $Y^{\prime}$, and we call any such $Y^{\prime}$ an _Ambro model_ for $f$.
One of the reasons why base change property is important is the following
_inversion of adjunction_.

###### Theorem 3.8

Let $f\colon(X,\Delta)\rightarrow Y$ be an lc-trivial fibration, and assume
that $Y$ is an Ambro model for $f$. Then $(Y,B_{Y})$ has klt, respectively log
canonical, singularities in a neighbourhood of a point $y\in Y$ if and only if
$(X,\Delta)$ has klt, respectively log canonical, singularities in a
neighbourhood of $f^{-1}(y)$.

  * Proof.

This is [Amb04, Theorem 3.1]. This result is stated for klt-trivial
fibrations, but the proof extends verbatim to the lc-trivial case by using
Theorem 3.7. $\Box$

Theorem 3.8 says something highly non-trivial: that in certain situations, on
an Ambro model the _local_ log canonical thresholds in Definition 3.3 are
actually _global_ log canonical thresholds. The following is [Amb05, Theorem
3.3].

###### Theorem 3.9

Let $f\colon(X,\Delta)\to Y$ be a klt-trivial fibration between normal
projective varieties such that $\Delta$ is effective over the generic point of
$Y$. Then there exists a diagram

$\textstyle{(X,\Delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{(X^{\prime},\Delta^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{Y}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\scriptstyle{\tau^{\prime}}$$\textstyle{Y^{\prime}}$

such that:

  1. (i)

$f^{\prime}\colon(X^{\prime},\Delta^{\prime})\rightarrow Y^{\prime}$ is a klt-
trivial fibration,

  2. (ii)

$\tau$ is generically finite and surjective, and $\tau^{\prime}$ is
surjective,

  3. (iii)

the moduli part $M_{f^{\prime}}$ is big, and after a birational base change we
may assume that $f$ and $f^{\prime}$ are Ambro models and
$\tau^{*}M_{f}=\tau^{\prime*}M_{f^{\prime}}$.

The following notions and the lemma will be used in the proof of Theorem 5.2
and in our main technical results, Propositions 4.2 and 4.4.

###### Definition 3.10

Let $f\colon(X,\Delta)\rightarrow Y$ be an lc-trivial fibration, where
$(X,\Delta)$ is log smooth and $Y$ is smooth. Fix a prime divisor $T$ on $Y$.
An _$(f,T)$ -bad divisor_ is any reduced divisor $\Sigma_{f,T}$ on $Y$ which
contains:

  1. (a)

the locus of critical values of $f$,

  2. (b)

the closed set $f(\operatorname{Supp}\Delta_{v})\subseteq Y$, and

  3. (c)

the set $\operatorname{Supp}B_{f}\cup T$.

###### Remark 3.11

Clearly the set $\Sigma_{f,T}$ is not uniquely determined. It will be clear
from the context which precise set we consider in the proofs below.

###### Definition 3.12

Let $f\colon(X,\Delta)\to Y$ be an lc-trivial (respectively klt-trivial)
fibration. Then $f$ is _acceptable_ if there exists another lc-trivial
(respectively klt-trivial) fibration
$\bar{f}\colon(\overline{X},\overline{\Delta})\to Y$ such that
$\overline{\Delta}$ is effective on the generic fibre of $\bar{f}$, and a
birational morphism $\mu\colon X\to\overline{X}$ such that $f=\bar{f}\circ\mu$
and
$K_{X}+\Delta\sim_{\mathbb{Q}}\mu^{*}(K_{\overline{X}}+\overline{\Delta})$.
Note that then the horizontal part of $\Delta^{<0}$ with respect to $f$ is
$\mu$-exceptional. Note also that any birational base change of an acceptable
lc-trivial (respectively klt-trivial) fibration is again an acceptable lc-
trivial (respectively klt-trivial) fibration.

$\textstyle{(X,\Delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\mu}$$\textstyle{(\overline{X},\overline{\Delta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{f}}$$\textstyle{Y}$

###### Lemma 3.13

Let $f\colon(X,\Delta)\rightarrow Y$ be an acceptable lc-trivial (respectively
klt-trivial) fibration, where $(X,\Delta)$ is log smooth and $Y$ is a smooth
Ambro model for $f$. Fix a prime divisor $T$ on $Y$. Then there exists a
birational morphism $\alpha\colon Y^{\prime}\to Y$ from a smooth variety
$Y^{\prime}$ and a commutative diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{X_{\nu}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{\nu}}$$\scriptstyle{f_{\nu}}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\scriptstyle{f^{\prime}}$ $\scriptstyle{\
\beta}$$\textstyle{Y}$$\textstyle{Y^{\prime},\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$

such that $X^{\prime}$ is a smooth variety, $X_{\nu}$ is the normalisation of
the main component of $X\times_{Y}Y^{\prime}$, $\beta$ is birational and, if
$\Delta^{\prime}$ is defined by
$K_{X^{\prime}}+\Delta^{\prime}=\pi^{*}(K_{X}+\Delta)$ and
$T^{\prime}=\alpha_{*}^{-1}T$, then:

  1. (a)

$f^{\prime}\colon(X^{\prime},\Delta^{\prime})\to Y^{\prime}$ is an acceptable
lc-trivial (respectively klt-trivial) fibration,

  2. (b)

there exists an $(f^{\prime},T^{\prime})$-bad divisor
$\Sigma_{f^{\prime},T^{\prime}}\subseteq Y^{\prime}$ which has simple normal
crossings, and

  3. (c)

the divisor $\Delta^{\prime}+f^{\prime*}\Sigma_{f^{\prime},T^{\prime}}$ has
simple normal crossings support.

Moreover, we may define $\alpha$ as an embedded resolution of any $(f,T)$-bad
divisor $\Sigma_{f,T}$ in $Y$, and we may assume that $\beta$ is an
isomorphism away from
$\operatorname{Supp}(f_{\nu}^{*}\alpha^{*}\Sigma_{f,T})$.

  * Proof.

Let $\Sigma_{f,T}$ be an $(f,T)$-bad divisor on $Y$. Let $\alpha\colon
Y^{\prime}\to Y$ be an embedded resolution of the pair $(Y,\Sigma_{f,T})$ as
in §2.B; in particular, $\alpha$ is an isomorphism away from $\Sigma_{f,T}$.
Define a divisor $\Delta_{\nu}$ on $X_{\nu}$ by

$K_{X_{\nu}}+\Delta_{\nu}=\alpha_{\nu}^{*}(K_{X}+\Delta).$

Define $\beta\colon X^{\prime}\to X_{\nu}$ as an embedded resolution of the
pair $\big{(}X_{\nu},\Delta_{\nu}+f_{\nu}^{*}\alpha^{*}\Sigma_{f,T}\big{)}$,
so in particular, it is an isomorphism away from
$\operatorname{Supp}(f_{\nu}^{*}\alpha^{*}\Sigma_{f,T})$. If
$\bar{f}\colon(\overline{X},\overline{\Delta})\to Y$ is a fibration as in
Definition 3.12, then $f^{\prime}$ factors through the normalisation
$\overline{X}^{\prime}$ of the main component of the fibre product
$\overline{X}\times_{Y}Y^{\prime}$. Let
$\bar{f}^{\prime}\colon\overline{X}^{\prime}\to Y$ and
$\overline{\alpha}^{\prime}\colon\overline{X}^{\prime}\to\overline{X}$ be the
induced morphisms. If we set
$\overline{\Delta}^{\prime}:=K_{\overline{X}^{\prime}}-\overline{\alpha}^{\prime*}(K_{\overline{X}}+\overline{\Delta})$,
then $\overline{\Delta}^{\prime}$ is effective on the generic fibre of
$\bar{f}^{\prime}$. Thus, $f^{\prime}$ is acceptable, which shows (a). We
claim that $\Sigma_{f^{\prime},T^{\prime}}:=\alpha^{-1}(\Sigma_{f,T})$ is the
desired $(f^{\prime},T^{\prime})$-bad divisor; the claim clearly implies (b)
and (c) by construction. To show the claim, clearly the set
$\alpha^{-1}(\Sigma_{f,T})$ contains the non-smooth locus of $f^{\prime}$ and
the divisor $T^{\prime}$. Since $Y$ is an Ambro model, we have
$K_{Y^{\prime}}+B_{f^{\prime}}=\alpha^{*}(K_{Y}+B_{f})$, hence

$\operatorname{Supp}B_{f^{\prime}}\subseteq\alpha^{-1}(\operatorname{Supp}B_{f})\cup\operatorname{Exc}\alpha\subseteq\alpha^{-1}(\Sigma_{f,T}).$

Now, denote $\alpha^{\prime}:=\alpha_{\nu}\circ\beta\colon X^{\prime}\to X$.
Let $D^{\prime}$ be a component of the vertical part of $\Delta^{\prime}$ with
respect to $f^{\prime}$. It remains to show that

$f^{\prime}(D^{\prime})\subseteq\Sigma_{f^{\prime},T^{\prime}}.$ (9)

To this end, if $D^{\prime}$ is not $\alpha^{\prime}$-exceptional, then
$D:=\alpha^{\prime}(D^{\prime})$ is a component of $\Delta_{v}$. Thus,
$f(D)=\alpha(f^{\prime}(D^{\prime}))$ belongs to $\Sigma_{f,T}$ by definition,
which implies (9). If $D^{\prime}$ is $\alpha^{\prime}$-exceptional, then,
since $\alpha^{\prime}$ is an isomorphism away from $f^{-1}(\Sigma_{f,T})$, we
have $\alpha^{\prime}(D^{\prime})\subseteq f^{-1}(\Sigma_{f,T})$. Therefore,
$f(\alpha^{\prime}(D^{\prime}))=\alpha(f^{\prime}(D^{\prime}))$ is a subset of
$\Sigma_{f,T}$, which gives (9) and finishes the proof. $\Box$

##  4\. Semiampleness on divisorial valuations

In this section we prove the main technical results of this paper,
Propositions 4.2 and 4.4. At the end of the section, we then deduce Theorems A
and B, as well as Corollaries C and D.

###### Lemma 4.1

Let $f\colon(X,\Delta)\rightarrow Y$ be an lc-trivial fibration, where
$(X,\Delta)$ is a log smooth log canonical pair and $Y$ is a smooth Ambro
model for $f$. Fix a prime divisor $T$ on $Y$. Assume that there exists an
$(f,T)$-bad divisor $\Sigma_{f,T}\subseteq Y$ which has simple normal
crossings, and such that the divisor $\Delta+f^{*}\Sigma_{f,T}$ has simple
normal crossings support. Denote

$\Delta_{X}=\Delta+\sum_{\Gamma\subseteq\Sigma_{f,T}}\gamma_{\Gamma}f^{*}\Gamma,$

where $\gamma_{\Gamma}$ are the generic log canonical thresholds with respect
to $f$ as in Definition 3.3. Then $(X,\Delta_{X})$ is a log smooth log
canonical pair and there exists a smooth minimal log canonical centre of
$(X,\Delta_{X})$ over $T$.

  * Proof.

We first note that

$\textstyle K_{X}+\Delta_{X}\sim_{\mathbb{Q}}f^{*}(K_{Y}+\Sigma_{f,T}+M_{f}),$

and the pair $(Y,\Sigma_{f,T})$ is log canonical since $\Sigma_{f,T}$ is a
simple normal crossings divisor. Therefore, by Theorem 3.8, the pair
$(X,\Delta_{X})$ is log canonical, and it is not klt over each prime divisor
$\Gamma\subseteq\Sigma_{f,T}$. Moreover, the support of $\Delta_{X}$ has
simple normal crossings by assumption. Thus, there exists a component $D$ of
$f^{*}T$ which dominates $T$ and which has coefficient $1$ in $\Delta_{X}$. In
other words, there exists a log canonical centre of $(X,\Delta_{X})$ which
dominates $T$. In particular, there exists a minimal log canonical centre of
$(X,\Delta_{X})$ over $T$. Since all log canonical centres of $(X,\Delta_{X})$
are connected components of intersections of components of $\Delta_{X}^{=1}$,
they are all smooth. $\Box$

###### Proposition 4.2

Let $f\colon(X,\Delta)\rightarrow Y$ be an acceptable lc-trivial fibration,
where $(X,\Delta)$ is a log smooth log canonical pair and $Y$ is a smooth
Ambro model for $f$. Fix a prime divisor $T$ on $Y$. Assume that there exists
an $(f,T)$-bad divisor $\Sigma_{f,T}\subseteq Y$ which has simple normal
crossings, and such that the divisor $\Delta+f^{*}\Sigma_{f,T}$ has simple
normal crossings support. Denote

$\Delta_{X}=\Delta+\sum_{\Gamma\subseteq\Sigma_{f,T}}\gamma_{\Gamma}f^{*}\Gamma,$

where $\gamma_{\Gamma}$ are the generic log canonical thresholds with respect
to $f$ as in Definition 3.3. Denote

$\Xi_{T}:=(\Sigma_{f,T}-T)|_{T}.$

Let $S$ be a minimal log canonical centre of $(X,\Delta_{X})$ over $T$, which
exists by Lemma 4.1. Let

$f|_{S}\colon
S\overset{h}{\longrightarrow}T^{\prime}\overset{\tau}{\longrightarrow}T$

be the Stein factorisation, and let $R$ denote the ramification divisor of
$\tau$ on $T^{\prime}$. Then:

  1. (i)

if $K_{S}+\Delta_{S}=(K_{X}+\Delta_{X})|_{S}$, then
$h\colon(S,\Delta_{S})\rightarrow T^{\prime}$ is a klt-trivial fibration with
$B_{h}\geq 0$,

  2. (ii)

$\tau^{*}(M_{f}|_{T})\sim_{\mathbb{Q}}M_{h}+R^{\prime}+E$, where $M_{f}$ is
chosen so that $T\not\subseteq M_{f}$ and

$R^{\prime}=\sum\limits_{\Gamma\not\subseteq\tau^{-1}(\Xi_{T})}(\operatorname{mult}_{\Gamma}R)\cdot\Gamma\quad\text{and}\quad
E=\sum\limits_{\Gamma\not\subseteq\tau^{-1}(\Xi_{T})}(\operatorname{mult}_{\Gamma}B_{h})\cdot\Gamma.$

  * Proof.

_Step 1._ As in the proof of Lemma 4.1, we first note that

$\textstyle
K_{X}+\Delta_{X}\sim_{\mathbb{Q}}f^{*}\big{(}K_{Y}+\Sigma_{f,T}+M_{f}\big{)},$
(10)

and that $(X,\Delta_{X})$ is a log smooth log canonical pair. By restricting
the equation (10) to $S$ we obtain

$K_{S}+\Delta_{S}\sim_{\mathbb{Q}}(f|_{S})^{*}(K_{T}+\Xi_{T}+M_{f}|_{T}),$

hence $h\colon(S,\Delta_{S})\to T^{\prime}$ is an lc-trivial fibration, and
moreover, it is a klt-trivial fibration. Indeed, if there existed a log
canonical centre $\Theta$ of $(S,\Delta_{S})$ which dominated $T^{\prime}$,
then $\Theta$ would be a log canonical centre of $(X,\Delta_{X})$ by
Proposition 2.6, which contradicts the minimality of $S$. This shows the first
part of (i). _Step 2._ Let $\mathcal{G}$ be the set of all components $P$ of
$f^{*}\Sigma_{f,T}$ with $\operatorname{mult}_{P}\Delta_{X}=0$ and denote

$G:=\operatorname{Supp}\Delta_{X,h}^{<0}\cup\operatorname{Supp}\Delta_{X,v}^{<1}\cup\bigcup_{P\in\mathcal{G}}P\,;$

we consider $G$ as a reduced divisor on $X$. Let $\mu\colon X\to\overline{X}$
and $\bar{f}\colon\overline{X}\to Y$ be the maps given by Definition 3.12. We
denote by $\Delta_{X,h}$ and $\Delta_{X,v}$ the horizontal and vertical parts
of $\Delta_{X}$ with respect to $f$. We claim that the divisor $G$ is a
$(\mu,\bar{f})$-EWE divisor. Indeed, by our hypothesis, the divisor
$\operatorname{Supp}\Delta_{X,h}^{<0}=\operatorname{Supp}\Delta_{h}^{<0}$ is
$\mu$-exceptional. Now, pick a prime divisor
$P\subseteq\operatorname{Supp}\Delta_{X,v}^{<1}$. If $f(P)$ has codimension 2
in $Y$, then $P$ is $f$-exceptional. If $Q:=f(P)$ has codimension 1, then
$Q\subseteq\Sigma_{f,T}$. By the definition of $\Delta_{X}$, there exists a
prime divisor $E\subseteq f^{-1}(Q)$ dominating $Q$ such that
$\operatorname{mult}_{E}\Delta_{X}=1$. In particular,
$E\not\subseteq\operatorname{Supp}\Delta_{X,v}^{<1}$ and $f(E)=Q$. This shows
that $\operatorname{Supp}\Delta_{X,v}^{<1}$ is weakly exceptional over $Y$ and
finishes the proof of the claim. _Step 3._ Pick $\varepsilon\in\mathbb{Q}$
with $0<\varepsilon\ll 1$ such that the pair $(X,\Delta_{X}^{\geq
0}+\varepsilon G)$ is dlt and denote $F=-\Delta_{X}^{\leq 0}+\varepsilon G$.
Note that

$\textstyle K_{X}+\Delta_{X}^{\geq 0}+\varepsilon G\sim_{\mathbb{Q},Y}F\geq
0.$

By Step 2 and by Proposition 2.12 we have

$N_{\sigma}(F;X/Y)=F.$ (11)

We run the $(K_{X}+\Delta_{X}^{\geq 0}+\varepsilon G)$-MMP with scaling of an
ample divisor over $Y$. By (11) and by Lemma 2.14 this MMP terminates and
contracts all the components of $F$. Let $\rho\colon X\dashrightarrow W$ be
the resulting birational contraction and let $\psi\colon W\to Y$ be the
resulting morphism. Denote $\Delta_{W}:=\rho_{*}\Delta_{X}^{\geq 0}\geq 0$.

$\textstyle{(X,\Delta_{X}^{\geq 0}+\varepsilon
G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\
\
\rho}$$\scriptstyle{f}$$\textstyle{(W,\Delta_{W})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\scriptstyle{\psi}$ $\textstyle{Y}$

Therefore, $(W,\Delta_{W})$ is $\mathbb{Q}$-factorial dlt pair and note that
$\Delta_{W}=\rho_{*}\Delta_{X}$. Moreover, by the definition of $G$,

$\text{the divisor }\Delta_{W,v}\text{ is reduced,}$ (12)

and

$(\psi^{*}\Sigma_{f,T})_{\mathrm{red}}\leq\Delta_{W,v}.$ (13)

By (10) we have

$K_{W}+\Delta_{W}\sim_{\mathbb{Q}}\psi^{*}(K_{Y}+\Sigma_{f,T}+M_{f}),$ (14)

hence $\psi\colon(W,\Delta_{W})\rightarrow Y$ is an lc-trivial fibration and
the map $\rho\colon(X,\Delta_{X})\dasharrow(W,\Delta_{W})$ is crepant
birational. By Remark 3.5, we have

$B_{\psi}=\Sigma_{f,T}\quad\text{and}\quad M_{\psi}=M_{f}.$ (15)

_Step 4._ By Lemma 2.8 the map $\rho$ is an isomorphism at the generic point
of each log canonical centre of $(X,\Delta_{X})$. In particular, there is a
log canonical centre $S_{W}$ of $(W,\Delta_{W})$ which is the strict transform
of $S$, and it is a minimal log canonical centre of $(W,\Delta_{W})$ over $T$.
Let

$\psi|_{S_{W}}\colon
S_{W}\overset{h_{W}}{\longrightarrow}T_{W}\overset{\tau_{W}}{\longrightarrow}T$

be the Stein factorisation. We claim that $T_{W}=T^{\prime}$ and
$\tau_{W}=\tau$ (up to isomorphism). To this end, let $(p,q)\colon Z\to
S\times S_{W}$ be the resolution of indeterminacies of the birational map
$\rho|_{S}\colon S\dashrightarrow S_{W}$. Since $S$ and $S_{W}$ are normal by
Proposition 2.6, both $p$ and $q$ have connected fibres by Zariski’s main
theorem. As every curve contracted by $p$ is contracted by $h_{W}\circ q$, by
the Rigidity lemma [Deb01, Lemma 1.15] there exists a morphism $\xi\colon S\to
T_{W}$ with connected fibres such that $h_{W}\circ q=\xi\circ p$, and thus
$f|_{S}=\tau_{W}\circ\xi$. The claim follows by the uniqueness of the Stein
factorisation. _Step 5._ By restricting the equation (14) to $S_{W}$ we obtain

$K_{S_{W}}+\Delta_{S_{W}}\sim_{\mathbb{Q}}(\psi|_{S_{W}})^{*}(K_{T}+\Xi_{T}+M_{f}|_{T}),$
(16)

hence $h_{W}\colon(S_{W},\Delta_{S_{W}})\to T^{\prime}$ is a klt-trivial
fibration by a similar argument as in Step 1. Therefore, the map
$\rho|_{S}\colon(S,\Delta_{S})\dashrightarrow(S_{W},\Delta_{S_{W}})$ is
crepant birational over $T^{\prime}$, hence by Remark 3.5 we have

$B_{h_{W}}=B_{h}\quad\text{and}\quad M_{h_{W}}\sim_{\mathbb{Q}}M_{h}.$ (17)

The divisor $B_{h_{W}}$ is effective since $\Delta_{S_{W}}$ is. This finishes
the proof of (i). _Step 6._ By (13) there exists a component $D$ of
$\psi^{*}T$ which dominates $T$ and which has coefficient $1$ in $\Delta_{W}$.
Denote $\Delta_{D}:=(\Delta_{W}-D)|_{D}$, so that
$\psi|_{D}\colon(D,\Delta_{D})\to T$ is an lc-trivial fibration. Let $P$ be a
component of $(\psi|_{D})^{*}\Xi_{T}$. Since
$(\psi|_{D})^{*}\Xi_{T}=(\psi^{*}\Sigma_{f,T}-\psi^{*}T)|_{D}$, and each
component of $\psi^{*}\Sigma_{f,T}$ is a component of $\Delta_{W}^{=1}$ by
(12) and (13), this implies that $P$ is a component of
$(\Delta_{W}^{=1}-D)|_{D}=\Delta_{D}^{=1}$. In other words,

$\big{(}(\psi|_{D})^{*}\Xi_{T}\big{)}_{\mathrm{red}}\leq\Delta_{D}^{=1}.$ (18)

Now, by Proposition 2.6 there are components $S_{1},\dots,S_{k}$ of
$\Delta_{D}^{=1}$ such that $S_{W}$ is a component of $S_{1}\cap\dots\cap
S_{k}$, and note that the $S_{i}$ dominate $T$. This and (18) imply

$\big{(}(\psi|_{D})^{*}\Xi_{T}\big{)}_{\mathrm{red}}\leq\Delta_{D}^{=1}-S_{1}-\dots-
S_{k},$

hence

$\big{(}(\psi|_{S_{W}})^{*}\Xi_{T}\big{)}_{\mathrm{red}}\leq(\Delta_{D}^{=1}-S_{1}-\dots-
S_{k})|_{S_{W}}\leq\Delta_{S_{W}}^{=1}.$

Thus, for every prime divisor $P\subseteq\operatorname{Supp}\tau^{*}\Xi_{T}$,
the generic log-canonical threshold $\gamma_{P}$ of $(S_{W},\Delta_{S_{W}})$
with respect to $h_{W}^{*}P$ is zero. If we define

$E:=\sum\limits_{\Gamma\not\subseteq\tau^{-1}(\Xi_{T})}(\operatorname{mult}_{\Gamma}B_{h_{W}})\cdot\Gamma=\sum\limits_{\Gamma\not\subseteq\tau^{-1}(\Xi_{T})}(\operatorname{mult}_{\Gamma}B_{h})\cdot\Gamma,$

where the second equality follows from (17), then

$B_{h_{W}}=(\tau^{*}\Xi_{T})_{\mathrm{red}}+E.$ (19)

_Step 7._ Now we have all the ingredients to show (ii). From (16) we have

$K_{T^{\prime}}+B_{h_{W}}+M_{h_{W}}\sim_{\mathbb{Q}}\tau^{*}(K_{T}+\Xi_{T}+M_{f}|_{T}).$
(20)

Write the Hurwitz formula for $\tau$ as $K_{T^{\prime}}=\tau^{*}K_{T}+R$, and
define

$R^{\prime}:=R-\tau^{*}\Xi_{T}+(\tau^{*}\Xi_{T})_{\mathrm{red}}.$

Then (19) gives

$\tau^{*}(K_{T}+\Xi_{T})=K_{T^{\prime}}-R+\tau^{*}\Xi_{T}+B_{h_{W}}-E-(\tau^{*}\Xi_{T})_{\mathrm{red}}=K_{T^{\prime}}-R^{\prime}+B_{h_{W}}-E,$

which together with (17) and (20) yields

$\tau^{*}(M_{f}|_{T})\sim_{\mathbb{Q}}M_{h}+R^{\prime}+E.$

This finishes the proof. $\Box$

###### Remark 4.3

In the proof of Proposition 4.2, the MMP technique we use is similar to (and
inspired by) the one in the proof of [FG14, Claim on p. 1730]. The main
difference between our approach and the one of [FG14] is that we avoid
semistable reduction in codimension 1 by choosing carefully the EWE divisor in
Step 2 and by proving (12) and (13).

###### Proposition 4.4

Let $f\colon(X,\Delta)\rightarrow Y$ be an acceptable lc-trivial fibration,
where $(X,\Delta)$ is a log smooth log canonical pair and $Y$ is a smooth
Ambro model for $f$. Fix a prime divisor $T$ on $Y$. Assume that there exists
an $(f,T)$-bad divisor $\Sigma_{f,T}\subseteq Y$ which has simple normal
crossings, and such that the divisor $\Delta+f^{*}\Sigma_{f,T}$ has simple
normal crossings support. Then there exists a commutative diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{0}}$$\scriptstyle{\delta_{0,X}}$$\textstyle{Y}$$\textstyle{Y_{0},\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{0}}$

where $\delta_{0}$ and $\delta_{0,X}$ are projective birational morphisms,
such that, if $T_{0}$ and $\Delta_{0}$ are defined by
$T_{0}:=(\delta_{0})_{*}^{-1}T\subseteq Y_{0}$ and
$K_{X_{0}}+\Delta_{0}=\delta_{0,X}^{*}(K_{X}+\Delta)$, then the following
holds. There exists an $(f_{0},T_{0})$-bad divisor
$\Sigma_{f_{0},T_{0}}\subseteq Y_{0}$ which has simple normal crossings, and
such that the divisor $\Delta_{0}+f_{0}^{*}\Sigma_{f_{0},T_{0}}$ has simple
normal crossings support. Denote

$\Delta_{X_{0}}=\Delta_{0}+\sum_{\Gamma\subseteq\Sigma_{f_{0},T_{0}}}\gamma_{\Gamma}f_{0}^{*}\Gamma,$

where $\gamma_{\Gamma}$ are the generic log canonical thresholds with respect
to $f_{0}$ as in Definition 3.3. Let $S_{0}$ be a minimal log canonical centre
of $(X_{0},\Delta_{X_{0}})$ over $T_{0}$, which exists by Lemma 4.1. Let
$f_{0}|_{S_{0}}\colon
S_{0}\overset{h_{0}}{\longrightarrow}T_{0}^{\prime}\overset{\tau_{0}}{\longrightarrow}T_{0}$
be the Stein factorisation, and let
$h_{0}\colon(S_{0},\Delta_{S_{0}})\rightarrow T_{0}^{\prime}$ be the klt-
trivial fibration as in Proposition 4.2(i). Then:

  1. (i)

$T_{0}^{\prime}$ is an Ambro model for $h_{0}$,

  2. (ii)

$\tau_{0}^{*}(M_{f_{0}}|_{T_{0}})\sim_{\mathbb{Q}}M_{h_{0}}$, where
$M_{f_{0}}$ is chosen so that $T_{0}\not\subseteq M_{f_{0}}$.

  * Proof.

We use the notation from Proposition 4.2. _Step 1._ Let $R^{\prime}$ and $E$
be the divisors as in Proposition 4.2(ii), and consider the closed subset

$\Pi:=\tau\big{(}\operatorname{Supp}(R^{\prime}+E)\big{)}\subseteq T\subseteq
Y.$

Let $Y_{b}\to Y$ be the blowup of $Y$ along $\Pi$, and let $Y_{0}\to Y_{b}$ be
an embedded resolution of the strict transform of $T$ in $Y_{b}$, see §2.B.
Let

$\delta_{0}\colon Y_{0}\to Y$

be the composition. Then, in particular,

$\delta_{0}\big{(}\operatorname{Exc}(\delta_{0})\big{)}=\Pi.$ (21)

Let $X_{\nu}$ be the normalisation of the main component of the fibre product
$X\times_{Y}Y_{0}$ with the induced morphism $\delta_{\nu}\colon X_{\nu}\to
X$, and define a divisor $\Delta_{\nu}$ on $X_{\nu}$ by
$K_{X_{\nu}}+\Delta_{\nu}=\delta_{\nu}^{*}(K_{X}+\Delta)$. Let $\beta\colon
X_{0}\to X_{\nu}$ be an embedded resolution of
$\big{(}X_{\nu},\Delta_{\nu}+(f\circ\delta_{\nu})^{-1}(\Sigma_{f,T})\big{)}$;
in particular, $\beta$ is an isomorphism away from
$(f\circ\delta_{\nu})^{-1}(\Pi)$. Set
$\delta_{0,X}:=\delta_{\nu}\circ\beta\colon X_{0}\to X$, so that we obtain the
commutative diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{X_{\nu}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{\nu}}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\scriptstyle{f_{0}}$
$\scriptstyle{\beta}$$\scriptstyle{\delta_{0,X}}$$\textstyle{Y}$$\textstyle{Y_{0}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{0}}$

Define a divisor $\Delta_{0}$ on $X_{0}$ by
$K_{X_{0}}+\Delta_{0}=\beta^{*}(K_{X_{\nu}}+\Delta_{\nu})$. Then
$(X_{0},\Delta_{0})$ is a log smooth pair, and let
$f_{0}\colon(X_{0},\Delta_{0})\to Y_{0}$ be the induced lc-trivial fibration.
Let $T_{0}:=(\delta_{0})_{*}^{-1}T\subseteq Y_{0}$. Then the proof of Lemma
3.13 shows that the divisor

$\Sigma_{f_{0},T_{0}}:=\delta_{0}^{-1}(\Sigma_{f,T})$

is an $(f_{0},T_{0})$-bad divisor in $Y_{0}$ which has simple normal crossings
and such that the support of the divisor
$\Delta_{X_{0}}:=\Delta_{0}+f_{0}^{*}\Sigma_{f_{0},T_{0}}$ has simple normal
crossings. Since $Y$ is an Ambro model for $f$, we have
$M_{f_{0}}=\delta_{0}^{*}M_{f}$, hence

$M_{f_{0}}|_{T_{0}}=(\delta_{0}|_{T_{0}})^{*}(M_{f}|_{T}).$ (22)

_Step 2._ Since $\delta_{0,X}^{-1}$ is an isomorphism at the generic point of
$S$, there exists a unique log canonical centre $S_{0}$ of
$(X_{0},\Delta_{X_{0}})$ which is minimal over $T_{0}$, and such that the map
$\delta_{0,X}|_{S_{0}}\colon S_{0}\to S$ is birational. Let

$f_{0}|_{S_{0}}\colon
S_{0}\overset{h_{0}}{\longrightarrow}T_{0}^{\prime}\overset{\tau_{0}}{\longrightarrow}T_{0}$

be the Stein factorisation. Then $h_{0}\colon(S_{0},\Delta_{S_{0}})\to
T_{0}^{\prime}$ is a klt-trivial fibration as in Proposition 4.2(i), where
$K_{S_{0}}+\Delta_{S_{0}}=(K_{X_{0}}+\Delta_{X_{0}})|_{S_{0}}$. The Rigidity
lemma [Deb01, Lemma 1.15] applied to the diagram

$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{S_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{0,X}|_{S_{0}}}$$\scriptstyle{h_{0}}$$\textstyle{T^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{T_{0}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{0}}$$\textstyle{T}$$\textstyle{T_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{0}|_{T_{0}}}$

shows that there is a morphism $\delta_{0}^{\prime}\colon T_{0}^{\prime}\to
T^{\prime}$ making the diagram commutative; note that the morphism is then
necessarily birational. Thus, we obtain the commutative diagram

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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width=0.0pt,depth=0.0pt,height=0.0pt}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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width=8.45845pt,depth=0.0pt,height=0.0pt}}{\hbox{\kern 7.38632pt\vrule
width=-0.13889pt,depth=0.0pt,height=0.0pt}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\kern
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width=0.0pt,depth=4.80444pt,height=-4.80444pt}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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width=0.0pt,depth=14.16669pt,height=-4.80444pt}}{\hbox{\kern 24.0pt\vrule
width=0.0pt,depth=14.16669pt,height=-14.16669pt}}{\hbox{\kern-6.61632pt\raise-24.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{T}$}}}}}}}{\hbox{\kern
15.98367pt\raise-24.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{T_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\kern
15.98367pt\vrule
width=0.0pt,depth=24.0pt,height=-24.0pt}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
3.22571pt\raise-17.17612pt\hbox{{}\hbox{\kern 0.0pt\raise
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 6.47745pt\raise-24.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\xyatipfont\char 111\relax\xybtipfont\char
111\relax}}}}}{\hbox{\kern 6.75523pt\vrule
width=9.22844pt,depth=24.0pt,height=-24.0pt}}{\hbox{\kern 6.61633pt\vrule
width=-0.13889pt,depth=24.0pt,height=-24.0pt}}\ignorespaces}}}}\ignorespaces\end{gathered}$
(23)

and

$M_{h_{0}}$ is the moduli divisor obtained by the base change of $h$ by
$\delta_{0}^{\prime}$. (24)

_Step 3._ We claim that

$\tau_{0}^{*}(M_{f_{0}}|_{T_{0}})\sim_{\mathbb{Q}}M_{h_{0}},$ (25)

which then shows (ii). To this end, denote
$\Xi_{T_{0}}:=(\Sigma_{f_{0},T_{0}}-T_{0})|_{T_{0}}$, and let $R_{0}$ be the
ramification divisor of $\tau_{0}$. Let $\Gamma$ be a prime divisor in
$\operatorname{Supp}R_{0}\cup\operatorname{Supp}B_{h_{0}}$. Then by
Proposition 4.2(ii) it suffices to show that

$\Gamma\subseteq\tau_{0}^{-1}(\Xi_{T_{0}}).$ (26)

There are two cases. Assume first that $\tau_{0}(\Gamma)$ – viewed as a closed
subset of $Y_{0}$ – is a subset of $\operatorname{Exc}(\delta_{0})$. Then,
since $\operatorname{Exc}(\delta_{0})$ is a divisor, there exists a prime
divisor $\overline{\Gamma}\subseteq\operatorname{Exc}(\delta_{0})$ such that
$\tau_{0}(\Gamma)\subseteq\overline{\Gamma}\cap T_{0}$. Since
$\operatorname{Exc}(\delta_{0})\subseteq\operatorname{Supp}\Sigma_{f_{0},T_{0}}$
by construction, we have

$\tau_{0}(\Gamma)\subseteq\operatorname{Supp}(\Sigma_{f_{0},T_{0}}-T_{0})\cap
T_{0}=\Xi_{T_{0}},$

which implies (26). Assume now that
$\tau_{0}(\Gamma)\not\subseteq\operatorname{Exc}(\delta_{0})$. Then by (21) we
have $\delta_{0}(\tau_{0}(\Gamma))\not\subseteq\Pi$, and hence by (23),

$\delta_{0}^{\prime}(\Gamma)\not\subseteq\operatorname{Supp}(R^{\prime}+E).$
(27)

Since $\delta_{0}$ is an isomorphism at the generic point of
$\tau_{0}(\Gamma)$, in the neighbourhood of this generic point we have

$\delta_{0}^{\prime}(\operatorname{Supp}R_{0})=\operatorname{Supp}R\quad\text{and}\quad\delta_{0}^{\prime}(\operatorname{Supp}B_{h_{0}})=\operatorname{Supp}B_{h}.$

Therefore,

$\delta_{0}^{\prime}(\Gamma)\subseteq\operatorname{Supp}R\cup\operatorname{Supp}B_{h}.$
(28)

But now (27) and (28) imply that
$\delta_{0}^{\prime}(\Gamma)\subseteq\tau^{-1}(\Xi_{T})$, hence there exists a
prime divisor $\overline{\Gamma}\subseteq\operatorname{Supp}(\Sigma_{f,T}-T)$
such that $\tau(\delta_{0}^{\prime}(\Gamma))\subseteq\overline{\Gamma}\cap T$.
Since $\tau(\delta_{0}^{\prime}(\Gamma))=\delta_{0}(\tau_{0}(\Gamma))$ by
(23), and $\delta_{0}$ is an isomorphism at the generic point of
$\tau_{0}(\Gamma)$, this implies

$\tau_{0}(\Gamma)\subseteq(\delta_{0})_{*}^{-1}\overline{\Gamma}\cap
T_{0}\subseteq\operatorname{Supp}(\Sigma_{f_{0},T_{0}}-T_{0})\cap
T_{0}=\Xi_{T_{0}},$

which implies (26) and finishes the proof of (25). _Step 4._ Let
$\nu\colon\widetilde{T}_{0}\to T_{0}^{\prime}$ be an Ambro model for $h_{0}$.
Then by Lemma 2.4 there exists a birational morphism $\mu\colon T_{1}\to
T_{0}$ obtained by a sequence of blowups such that, if we denote by
$\lambda\colon\widehat{T}_{0}\dashrightarrow\widetilde{T}_{0}$ the induced
birational map from the normalisation of the main component of
$T_{0}^{\prime}\times_{T_{0}}T_{1}$ to $\widetilde{T}_{0}$, then
$\lambda^{-1}$ is an isomorphism at the generic point of each
$\nu$-exceptional prime divisor on $\widetilde{T}_{0}$. We denote by
$\delta_{1}\colon Y_{1}\to Y_{0}$ the corresponding birational map of ambient
spaces induced by $\mu$, so that $\mu=\delta_{1}|_{T_{1}}$.

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(29)

By possibly blowing up further, we may assume that $\delta_{1}$ is an embedded
resolution of the pair $(Y_{0},\Sigma_{f_{0},T_{0}})$, and in particular, that
$\delta_{1}$ is an isomorphism away from $\Sigma_{f_{0},T_{0}}$. Let
$(X_{1},\Delta_{1})$ be a log smooth pair which is a crepant pullback of
$(X_{0},\Delta_{0})$ obtained by making a base change of $f_{0}$ by
$\delta_{1}$ and taking an embedded resolution of the preimage of
$\Sigma_{f_{0},T_{0}}$ in $X_{0}\times_{Y_{0}}Y_{1}$ (similarly as in Step 1),
and let $f_{1}\colon(X_{1},\Delta_{1})\to Y_{1}$ be the induced klt-trivial
fibration.

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0.0pt\hbox{$\textstyle{Y_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\kern
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(30)

Observe that $T_{1}=(\delta_{1})_{*}^{-1}T_{0}$. Then the proof of Lemma 3.13
shows that the divisor

$\Sigma_{f_{1},T_{1}}:=\delta_{1}^{-1}(\Sigma_{f_{0},T_{0}})$

is an $(f_{1},T_{1})$-bad divisor in $Y_{1}$ which has simple normal crossings
and such that the support of the divisor
$\Delta_{X_{1}}:=\Delta_{1}+f_{1}^{*}\Sigma_{f_{1},T_{1}}$ has simple normal
crossings. Since $Y_{0}$ is also an Ambro model, we have
$M_{f_{1}}=\delta_{1}^{*}M_{f_{0}}$, hence

$M_{f_{1}}|_{T_{1}}=(\delta_{1}|_{T_{1}})^{*}(M_{f_{0}}|_{T_{0}}).$ (31)

_Step 5._ Since $\delta_{1,X}^{-1}$ is an isomorphism at the generic point of
$S_{0}$, there exists a unique log canonical centre $S_{1}$ of
$(X_{1},\Delta_{X_{1}})$ which is minimal over $T_{1}$, and such that the map
$\delta_{1,X}|_{S_{1}}\colon S_{1}\to S_{0}$ is birational. Let

$f_{1}|_{S_{1}}\colon
S_{1}\overset{h_{1}}{\longrightarrow}T_{1}^{\prime}\overset{\tau_{1}}{\longrightarrow}T_{1}$

be the Stein factorisation. Then $h_{1}\colon(S_{1},\Delta_{S_{1}})\to
T_{1}^{\prime}$ is a klt-trivial fibration as in Proposition 4.2(i), where
$K_{S_{1}}+\Delta_{S_{1}}=(K_{X_{1}}+\Delta_{X_{1}})|_{S_{1}}$. As in Step 2,
there is a birational morphism $\delta_{1}^{\prime}\colon T_{1}^{\prime}\to
T_{0}^{\prime}$ such that the diagram

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(32)

commutes, and

$M_{h_{1}}$ is the moduli divisor obtained by the base change of $h_{0}$ by
$\delta_{1}^{\prime}$. (33)

_Step 6._ We claim that

$\tau_{1}^{*}(M_{f_{1}}|_{T_{1}})\sim_{\mathbb{Q}}M_{h_{1}}.$ (34)

To this end, denote $\Xi_{T_{1}}:=(\Sigma_{f_{1},T_{1}}-T_{1})|_{T_{1}}$, and
let $R_{1}$ be the ramification divisor of $\tau_{1}$. Let $\Gamma$ be a prime
divisor in $\operatorname{Supp}R_{1}\cup\operatorname{Supp}B_{h_{1}}$. Then by
Proposition 4.2(ii) it suffices to show that

$\Gamma\subseteq\tau_{1}^{-1}(\Xi_{T_{1}}).$ (35)

There are two cases. If
$\tau_{1}(\Gamma)\subseteq\operatorname{Exc}(\delta_{1})$, then we conclude
analogously as in Step 3. Now we assume that
$\tau_{1}(\Gamma)\not\subseteq\operatorname{Exc}(\delta_{1})$. Then since
$\delta_{1}$ is an isomorphism at the generic point of $\tau_{1}(\Gamma)$, in
the neighbourhood of this generic point we have

$\delta_{1}^{\prime}(\operatorname{Supp}R_{1})=\operatorname{Supp}R_{0}\quad\text{and}\quad\delta_{1}^{\prime}(\operatorname{Supp}B_{h_{1}})=\operatorname{Supp}B_{h_{0}}.$

Therefore,

$\delta_{1}^{\prime}(\Gamma)\subseteq\operatorname{Supp}R_{0}\cup\operatorname{Supp}B_{h_{0}}.$
(36)

But now (36) and (26) imply that
$\delta_{1}^{\prime}(\Gamma)\subseteq\tau_{0}^{-1}(\Xi_{T_{0}})$, hence there
exists a prime divisor
$\overline{\Gamma}\subseteq\operatorname{Supp}(\Sigma_{f_{0},T_{0}}-T_{0})$
such that $\tau_{0}(\delta_{1}^{\prime}(\Gamma))\subseteq\overline{\Gamma}\cap
T_{0}$. Since
$\tau_{0}(\delta_{1}^{\prime}(\Gamma))=\delta_{1}(\tau_{1}(\Gamma))$ by (32),
and $\delta_{1}$ is an isomorphism at the generic point of $\tau_{1}(\Gamma)$,
this implies

$\tau_{1}(\Gamma)\subseteq(\delta_{1})_{*}^{-1}\overline{\Gamma}\cap
T_{1}\subseteq\operatorname{Supp}(\Sigma_{f_{1},T_{1}}-T_{1})\cap
T_{1}=\Xi_{T_{1}},$

which shows (35) and finishes the proof of (34). _Step 7._ Recall that
$\widehat{T}_{0}$ is the main component of
$T_{0}^{\prime}\times_{T_{0}}T_{1}$. Then from (32) there exists a birational
morphism $\xi\colon T_{1}^{\prime}\to\widehat{T}_{0}$, and denote

$\theta=\lambda\circ\xi\colon T_{1}^{\prime}\dashrightarrow\widetilde{T}_{0}.$

Then $\theta^{-1}$ is an isomorphism at the generic point of each
$\nu$-exceptional prime divisor on $\widetilde{T}_{0}$. Let us consider
$(p,q)\colon V\to\widetilde{T}_{0}\times T_{1}^{\prime}$ a resolution of
indeterminacies of $\theta$. Then by (29) and (32) we have the diagram

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