# Adjunction and Inversion of Adjunction

###### Abstract.

We establish adjunction and inversion of adjunction for log canonical centers
of arbitrary codimension in full generality.

###### Key words and phrases:

adjunction, inversion of adjunction, log canonical centers, generalized pairs

###### 2010 Mathematics Subject Classification:

Primary 14E30; Secondary 14N30

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 On basic slc-trivial fibrations
  4. 4 Fundamental theorem for basic $\mathbb{Q}$-slc-trivial fibrations
  5. 5 Fundamental theorem for basic $\mathbb{R}$-slc-trivial fibrations
  6. 6 Proof of Theorem 1.2
  7. 7 Adjunction for codimension two log canonical centers

##  1\. Introduction

Throughout this paper, we will work over $\mathbb{C}$, the complex number
filed. We establish the following adjunction and inversion of adjunction for
log canonical centers of arbitrary codimension.

######  Theorem 1.1.

Let $X$ be a normal variety and let $\Delta$ be an effective
$\mathbb{R}$-divisor on $X$ such that $K_{X}+\Delta$ is $\mathbb{R}$-Cartier.
Let $W$ be a log canonical center of $(X,\Delta)$ and let $\nu\colon Z\to W$
be the normalization of $W$. Then we have the adjunction formula

$\nu^{*}(K_{X}+\Delta)=K_{Z}+B_{Z}+M_{Z}$

with the following properties:

  * (A)

$(X,\Delta)$ is log canonical in a neighborhood of $W$ if and only if
$(Z,B_{Z}+M_{Z})$ is an NQC generalized log canonical pair, and

  * (B)

$(X,\Delta)$ is log canonical in a neighborhood of $W$ and $W$ is a minimal
log canonical center of $(X,\Delta)$ if and only if $(Z,B_{Z}+M_{Z})$ is an
NQC generalized kawamata log terminal pair.

For the definition of NQC generalized log canonical pairs and NQC generalized
kawamata log terminal pairs, see [13, Section 2].

For the formulation of adjunction and inversion of adjunction for log
canonical centers of arbitrary codimension in full generality, the notion of
b-divisors, which was first introduced by Shokurov, is very useful. In fact,
the $\mathbb{R}$-divisors $B_{Z}$ and $M_{Z}$ in Theorem 1.1 are the traces of
certain $\mathbb{R}$-b-divisors $\mathbf{B}$ and $\mathbf{M}$ on $Z$,
respectively. The precise version of Theorem 1.1 is:

######  Theorem 1.2 (Adjunction and Inversion of Adjunction).

Let $X$ be a normal variety and let $\Delta$ be an effective
$\mathbb{R}$-divisor on $X$ such that $K_{X}+\Delta$ is $\mathbb{R}$-Cartier.
Let $W$ be a log canonical center of $(X,\Delta)$ and let $\nu\colon Z\to W$
be the normalization of $W$. Then there exist a b-potentially nef
$\mathbb{R}$-b-divisor $\mathbf{M}$ and an $\mathbb{R}$-b-divisor $\mathbf{B}$
on $Z$ such that $\mathbf{B}_{Z}$ is effective with

$\nu^{*}(K_{X}+\Delta)=\mathbf{K}_{Z}+\mathbf{M}_{Z}+\mathbf{B}_{Z}.$

More precisely, there exists a projective birational morphism $p\colon
Z^{\prime}\to Z$ from a smooth quasi-projective variety $Z^{\prime}$ such that

  * (i)

$\mathbf{M}=\overline{\mathbf{M}_{Z^{\prime}}}$ and $\mathbf{M}_{Z^{\prime}}$
is a potentially nef $\mathbb{R}$-divisor on $Z^{\prime}$,

  * (ii)

$\mathbf{K}+\mathbf{B}=\overline{\mathbf{K}_{Z^{\prime}}+\mathbf{B}_{Z^{\prime}}}$,

  * (iii)

$\operatorname{Supp}\mathbf{B}_{Z^{\prime}}$ is a simple normal crossing
divisor on $Z^{\prime}$,

  * (iv)

$\nu\circ
p\left(\mathbf{B}^{>1}_{Z^{\prime}}\right)=W\cap\operatorname{Nlc}(X,\Delta)$
holds set theoretically, where $\operatorname{Nlc}(X,\Delta)$ denotes the non-
lc locus of $(X,\Delta)$, and

  * (v)

$\nu\circ p\left(\mathbf{B}^{\geq
1}_{Z^{\prime}}\right)=W\cap\left(\operatorname{Nlc}(X,\Delta)\cup\bigcup_{W\not\subset
W^{\dagger}}W^{\dagger}\right)$, where $W^{\dagger}$ runs over log canonical
centers of $(X,\Delta)$ which do not contain $W$, holds set theoretically.

Hence, $(Z,\mathbf{B}_{Z}+\mathbf{M}_{Z})$ is generalized log canonical, that
is, $\mathbf{B}^{>1}_{Z^{\prime}}=0$, if and only if $(X,\Delta)$ is log
canonical in a neighborhood of $W$. Moreover,
$(Z,\mathbf{B}_{Z}+\mathbf{M}_{Z})$ is generalized kawamata log terminal, that
is, $\mathbf{B}^{\geq 1}_{Z^{\prime}}=0$, if and only if $(X,\Delta)$ is log
canonical in a neighborhood of $W$ and $W$ is a minimal log canonical center
of $(X,\Delta)$. We note that $\mathbf{M}_{Z^{\prime}}$ is semi-ample when
$\dim W=1$. We also note that if $K_{X}+\Delta$ is $\mathbb{Q}$-Cartier then
$\mathbf{B}$ and $\mathbf{M}$ become $\mathbb{Q}$-b-divisors by construction.

In this paper, the $\mathbb{R}$-b-divisors $\mathbf{B}$ and $\mathbf{M}$ in
Theorem 1.2 are defined by using the notion of basic $\mathbb{R}$-slc-trivial
fibrations. Here, we explain an alternative definition of $\mathbf{B}$ and
$\mathbf{M}$ for the reader’s convenience. For the details of Definition 1.3,
see [10, Section 5] and [11, Definition 2.1].

######  Definition 1.3 (see [10, Section 5], [11, Definition 2.1], and Remark
6.1).

Let $(X,\Delta)$, $W$, and $\nu\colon Z\to W$ be as in Theorem 1.2. For any
higher birational model $\rho\colon\tilde{Z}\to Z$, we consider all prime
divisors $T$ over $X$ such that $a(T,X,\Delta)=-1$ and the center of $T$ on
$X$ is $W$. We take a log resolution $f\colon Y\to X$ of $(X,\Delta)$ so that
$T$ is a prime divisor on $Y$ and the induced map $f_{T}\colon
T\dashrightarrow\tilde{Z}$ is a morphism. We put
$\Delta_{T}=(\Delta_{Y}-T)|_{T}$, where $\Delta_{Y}$ is defined by
$K_{Y}+\Delta_{Y}=f^{*}(K_{X}+\Delta)$. For any prime divisor $P$ on
$\tilde{Z}$, we define a real number $\alpha_{P,T}$ by

$\alpha_{P,T}=\sup\\{\lambda\in\mathbb{R}\,|\,(T,\Delta_{T}+\lambda
f^{*}_{T}P)\ \text{is sub log canonical over the generic point of}\ P\\}.$

Then the trace $\mathbf{B}_{\tilde{Z}}$ of $\mathbf{B}$ on $\tilde{Z}$ is
defined by

$\mathbf{B}_{\tilde{Z}}=\sum_{P}(1-\underset{T}{\inf}\alpha_{P,T})P$

where $P$ runs over prime divisors on $\tilde{Z}$ and $T$ runs over prime
divisors over $X$ such that $a(T,X,\Delta)=-1$ and the center of $T$ on $X$ is
$W$. When $W$ is a prime divisor on $X$, $T$ is the strict transform of $W$ on
$Y$. In this case, we can easily check that
$\mathbf{B}_{\tilde{Z}}=(f_{T})_{*}\Delta_{T}$ holds. We consider the
$\mathbb{R}$-line bundle $\mathcal{L}$ on $X$ associated to $K_{X}+\Delta$. We
fix an $\mathbb{R}$-Cartier $\mathbb{R}$-divisor $D_{\tilde{Z}}$ on
$\tilde{Z}$ whose associated $\mathbb{R}$-line bundle is
$\rho^{*}\nu^{*}(\mathcal{L}|_{W})$. Then the trace $\mathbf{M}_{\tilde{Z}}$
of $\mathbf{M}$ on $\tilde{Z}$ is defined by

$\mathbf{M}_{\tilde{Z}}=D_{\tilde{Z}}-K_{\tilde{Z}}-\mathbf{B}_{\tilde{Z}}.$

We simply write

$\rho^{*}\nu^{*}(K_{X}+\Delta)=K_{\tilde{Z}}+\mathbf{B}_{\tilde{Z}}+\mathbf{M}_{\tilde{Z}}$

if there is no danger of confusion (see also Remark 6.1).

As we saw in Definition 1.3, the $\mathbb{R}$-b-divisor $\mathbf{B}$ on $Z$
depends only on the singularities of $(X,\Delta)$ near $W$. Conversely,
Theorem 1.2 (ii)–(v) implies that $\mathbf{B}$ remembers properties of the
singularities of $(X,\Delta)$ near $W$. If we put $B_{Z}=\mathbf{B}_{Z}$ and
$M_{Z}=\mathbf{M}_{Z}$, then Theorem 1.1 directly follows from Theorem 1.2.
Our new formulation of adjunction and inversion of adjunction includes some
classical results as special cases. The following corollary is the case of
${\rm dim}W={\rm dim}X-1$ which recovers the classical adjunction and
inversion of adjunction.

######  Corollary 1.4 (Classical Adjunction and Inversion of Adjunction).

In Theorem 1.1, we further assume that $\dim W=\dim X-1$, that is, $W$ is a
prime divisor on $X$. Then $M_{Z}$ and $B_{Z}$ become zero and Shokurov’s
different, respectively. Then (A) recovers Kawakita’s inversion of adjunction
on log canonicity. By (B), we have that $(X,\Delta)$ is purely log terminal in
a neighborhood of $W$ if and only if $(Z,B_{Z})$ is kawamata log terminal.

We know that we have already had many related results. We only make some
remarks on [12] and [3].

######  Remark 1.5 (Hacon’s inversion of adjunction).

In [12, Theorem 1], Hacon treated inversion of adjunction on log canonicity
for log canonical centers of arbitrary codimension under the extra assumption
that $\Delta$ is a boundary $\mathbb{Q}$-divisor. We note that the b-divisor
$\mathbf{B}(V;X,\Delta)$ in [12] coincides with $\mathbf{B}$ in Theorem 1.2 by
[11, Theorem 1.2]. In [10, Theorem 5.4], we proved a generalization of [12,
Theorem 1]. We note that $\mathbf{B}$ in [10, Theorem 5.4] coincides with
$\mathbf{B}$ in Theorem 1.2. Hence Theorem 1.2 can be seen as a complete
generalization of [10, Theorem 5.4] and [12, Theorem 1].

######  Remark 1.6 (Generalized adjunction and inversion of adjunction by
Filipazzi).

In [3], Filipazzi established some related results for generalized pairs (see,
for example, [3, Theorem 1.6]). Although they are more general than Theorems
1.1 and 1.2 in some sense, they do not include Theorem 1.1.

The main ingredients of Theorem 1.2 are the existence theorem of log canonical
modifications established in [10] and the theory of basic slc-trivial
fibrations in [6] and [7]. Hence this paper can be seen as a continuation of
[7] and [10]. Moreover, the theory of partial resolutions of singularities of
pairs in [2] is indispensable. We do not use Kawakita’s inversion of
adjunction (see [15, Theorem]) nor the Kawamata–Viehweg vanishing theorem. If
$K_{X}+\Delta$ is $\mathbb{Q}$-Cartier, then Theorem 1.2 easily follows from
[6], [7], and [10]. Unfortunately, however, the framework of basic slc-trivial
fibrations discussed in [6] is not sufficient for our purposes in this paper.
Hence we establish the following result.

######  Theorem 1.7 (Corollary 5.2).

Let $f\colon(X,B)\to Y$ be a basic $\mathbb{R}$-slc-trivial fibration and let
$\mathbf{B}$ and $\mathbf{M}$ be the discriminant and moduli
$\mathbb{R}$-b-divisors associated to $f\colon(X,B)\to Y$, respectively. Then
we have the following properties:

  * (i)

$\mathbf{K}+\mathbf{B}$ is $\mathbb{R}$-b-Cartier, where $\mathbf{K}$ is the
canonical b-divisor of $Y$, and

  * (ii)

$\mathbf{M}$ is b-potentially nef, that is, there exists a proper birational
morphism $\sigma\colon Y^{\prime}\to Y$ from a normal variety $Y^{\prime}$
such that $\mathbf{M}_{Y^{\prime}}$ is a potentially nef $\mathbb{R}$-divisor
on $Y^{\prime}$ and that $\mathbf{M}=\overline{\mathbf{M}_{Y^{\prime}}}$
holds.

If $f\colon(X,B)\to Y$ is a basic $\mathbb{Q}$-slc-trivial fibration, then
Theorem 1.7 is nothing but [6, Theorem 1.2], which is the main result of [6].
More precisely, we establish:

######  Theorem 1.8 (see Theorem 5.1).

Let $f\colon(X,B)\to Y$ be a projective surjective morphism from a simple
normal crossing pair $(X,B)$ to a smooth quasi-projective variety $Y$ such
that every stratum of $X$ is dominant onto $Y$ and
$f_{*}\mathcal{O}_{X}\simeq\mathcal{O}_{Y}$ with

  * •

$B=B^{\leq 1}$ holds over the generic point of $Y$,

  * •

there exists an $\mathbb{R}$-Cartier $\mathbb{R}$-divisor $D$ on $Y$ such that
$K_{Y}+B\sim_{\mathbb{R}}f^{*}D$ holds, and

  * •

$\operatorname{rank}f_{*}\mathcal{O}_{X}(\lceil-(B^{<1})\rceil)=1$.

We assume that there exists a simple normal crossing divisor $\Sigma$ on $Y$
such that $\operatorname{Supp}D\subset\Sigma$ and that every stratum of
$(X,\operatorname{Supp}B)$ is smooth over $Y\setminus\Sigma$. Let $\mathbf{B}$
and $\mathbf{M}$ be the discriminant and moduli $\mathbb{R}$-b-divisors
associated to $f\colon(X,B)\to Y$, respectively. Then

  * (i)

$\mathbf{K}+\mathbf{B}=\overline{\mathbf{K}_{Y}+\mathbf{B}_{Y}}$ holds, where
$\mathbf{K}$ is the canonical b-divisor of $Y$, and

  * (ii)

$\mathbf{M}_{Y}$ is a potentially nef $\mathbb{R}$-divisor on $Y$ with
$\mathbf{M}=\overline{\mathbf{M}_{Y}}$.

Note that Theorem 1.8 completely generalizes [14, Lemma 2.8]. By Theorem 1.8,
we can use the framework of basic slc-trivial fibrations in [6] for
$\mathbb{R}$-divisors. We also note that the main part of this paper is
devoted to the proof of Theorem 1.8. In the proof of Theorem 1.2, we naturally
construct a basic $\mathbb{R}$-slc-trivial fibration $f\colon(V,\Delta_{V})\to
Z$ by taking a suitable resolution of singularities of the pair $(X,\Delta)$.
The $\mathbb{R}$-b-divisors $\mathbf{B}$ and $\mathbf{M}$ on $Z$ in Theorem
1.2 are the discriminant and moduli $\mathbb{R}$-b-divisors associated to
$f\colon(V,\Delta_{V})\to Z$, respectively.

######  Conjecture 1.9.

In Theorem 1.8, $\mathbf{M}_{Y}$ is semi-ample.

If Conjecture 1.9 holds true, then $\mathbf{M}$ in Theorem 1.2 is b-semi-
ample, that is, $\mathbf{M}_{Z^{\prime}}$ is semi-ample. Note that Conjecture
1.9 follows from [6, Conjecture 1.4]. When $\dim Y=1$, we can easily check
that $\mathbf{M}_{Y}$ is semi-ample by [9, Corollary 1.4]. Unfortunately,
however, it is still widely open. In this paper, we prove Conjecture 1.9 for
basic slc-trivial fibrations of relative dimension one under some extra
assumption (see Theorem 7.2). Then we establish:

######  Theorem 1.10 (see Corollary 7.3).

If $W$ is a codimension two log canonical center of $(X,\Delta)$ in Theorem
1.2, then $\mathbf{M}$ is b-semi-ample.

Theorem 1.10 generalizes Kawamata’s result (see [16, Theorem 1]). For the
details, see Corollary 7.3.

We briefly look at the organization of this paper. In Section 2, we recall
some basic definitions and results. In Section 3, we introduce the notion of
basic $\mathbb{R}$-slc-trivial fibrations and recall the main result of [6].
In Section 4, we slightly generalize the main result of [6]. This
generalization (see Theorem 4.1) seems to be indispensable in order to treat
basic $\mathbb{R}$-slc-trivial fibrations. In Section 5, we establish a
fundamental theorem for basic $\mathbb{R}$-slc-trivial fibrations (see
Theorems 1.8 and 5.1). In Section 6, we prove the main result, that is,
adjunction and inversion of adjunction for log canonical centers of arbitrary
codimension, in full generality. More precisely, we first establish Theorem
1.2. Then we see that Theorem 1.1 and Corollary 1.4 easily follow from Theorem
1.2. In Section 7, we treat adjunction and inversion of adjunction for log
canonical centers of codimension two.

######  Acknowledgments.

The first author was partially supported by JSPS KAKENHI Grant Numbers
JP16H03925, JP16H06337, JP19H01787, JP20H00111, JP21H00974. The second author
was partially supported by JSPS KAKENHI Grant Numbers JP16J05875, JP19J00046.
The authors thank Christopher Hacon very much for answering their question.

##  2\. Preliminaries

In this paper, we will freely use the standard notation as in [4], [5], [6],
and [7]. A scheme means a separated scheme of finite type over $\mathbb{C}$. A
variety means an integral scheme, that is, an irreducible and reduced
separated scheme of finite type over $\mathbb{C}$. We note that $\mathbb{Q}$
and $\mathbb{R}$ denote the sets of rational numbers and real numbers,
respectively. We also note that $\mathbb{Q}_{>0}$ and $\mathbb{R}_{>0}$ are
the sets of positive rational numbers and positive real numbers, respectively.
Similarly, $\mathbb{Q}_{\geq 0}$ denotes the set of nonnegative rational
numbers.

Here, we collect some basic definitions for the reader’s convenience. Let us
start with the definition of potentially nef divisors.

######  Definition 2.1 (Potentially nef divisors, see [6, Definition 2.5]).

Let $X$ be a normal variety and let $D$ be a divisor on $X$. If there exist a
completion $X^{\dagger}$ of $X$, that is, $X^{\dagger}$ is a complete normal
variety and contains $X$ as a dense Zariski open subset, and a nef divisor
$D^{\dagger}$ on $X^{\dagger}$ such that $D=D^{\dagger}|_{X}$, then $D$ is
called a potentially nef divisor on $X$. A finite $\mathbb{Q}_{>0}$-linear
(resp. $\mathbb{R}_{>0}$-linear) combination of potentially nef divisors is
called a potentially nef $\mathbb{Q}$-divisor (resp. $\mathbb{R}$-divisor).

We give two important remarks on potentially nef $\mathbb{R}$-divisors.

######  Remark 2.2.

Let $D$ be a nef $\mathbb{R}$-divisor on a smooth projective variety $X$. Then
$D$ is not necessarily a potentially nef $\mathbb{R}$-divisor. This means that
$D$ is not always a finite $\mathbb{R}_{>0}$-linear combination of nef Cartier
divisors on $X$.

######  Remark 2.3.

Let $X$ be a normal variety and let $D$ be a potentially nef
$\mathbb{R}$-divisor on $X$. Then $D\cdot C\geq 0$ for every projective curve
$C$ on $X$. In particular, $D$ is $\pi$-nef for every proper morphism
$\pi\colon X\to S$ to a scheme $S$.

It is convenient to use b-divisors to explain several results. Here we do not
repeat the definition of b-divisors. For the details, see [6, Section 2].

######  Definition 2.4 (Canonical b-divisors).

Let $X$ be a normal variety and let $\omega$ be a top rational differential
form of $X$. Then $(\omega)$ defines a b-divisor $\mathbf{K}$. We call
$\mathbf{K}$ the canonical b-divisor of $X$.

######  Definition 2.5 ($\mathbb{R}$-Cartier closures).

The $\mathbb{R}$-Cartier closure of an $\mathbb{R}$-Cartier
$\mathbb{R}$-divisor $D$ on a normal variety $X$ is the $\mathbb{R}$-b-divisor
$\overline{D}$ with trace

$\overline{D}_{Y}=f^{*}D,$

where $f\colon Y\to X$ is a proper birational morphism from a normal variety
$Y$.

We use the following definition in order to state our results (see Theorem
1.2).

######  Definition 2.6 ([6, Definition 2.12]).

Let $X$ be a normal variety. An $\mathbb{R}$-b-divisor $\mathbf{D}$ of $X$ is
b-potentially nef (resp. b-semi-ample) if there exists a proper birational
morphism $X^{\prime}\to X$ from a normal variety $X^{\prime}$ such that
$\mathbf{D}=\overline{\mathbf{D}_{X^{\prime}}}$, that is, $\mathbf{D}$ is the
$\mathbb{R}$-Cartier closure of $\mathbf{D}_{X^{\prime}}$, and that
$\mathbf{D}_{X^{\prime}}$ is potentially nef (resp. semi-ample). An
$\mathbb{R}$-b-divisor $\mathbf{D}$ of $X$ is $\mathbb{R}$-b-Cartier if there
is a proper birational morphism $X^{\prime}\to X$ from a normal variety
$X^{\prime}$ such that $\mathbf{D}=\overline{\mathbf{D}_{X^{\prime}}}$.
Obviously, $\mathbf{D}$ is said to be $\mathbb{Q}$-b-Cartier when
$\mathbf{D}_{X^{\prime}}$ is $\mathbb{Q}$-Cartier and
$\mathbf{D}=\overline{\mathbf{D}_{X^{\prime}}}$.

For the reader’s convenience, let us recall the definition of singularities of
pairs. The following definition is standard and is well known.

######  Definition 2.7 (Singularities of pairs).

Let $X$ be a variety and let $E$ be a prime divisor on $Y$ for some birational
morphism $f\colon Y\to X$ from a normal variety $Y$. Then $E$ is called a
divisor over $X$. A normal pair $(X,\Delta)$ consists of a normal variety $X$
and an $\mathbb{R}$-divisor $\Delta$ on $X$ such that $K_{X}+\Delta$ is
$\mathbb{R}$-Cartier. Let $(X,\Delta)$ be a normal pair and let $f\colon Y\to
X$ be a projective birational morphism from a normal variety $Y$. Then we can
write

$K_{Y}=f^{*}(K_{X}+\Delta)+\sum_{E}a(E,X,\Delta)E$

with

$f_{*}\left(\underset{E}{\sum}a(E,X,\Delta)E\right)=-\Delta,$

where $E$ runs over prime divisors on $Y$. We call $a(E,X,\Delta)$ the
discrepancy of $E$ with respect to $(X,\Delta)$. Note that we can define the
discrepancy $a(E,X,\Delta)$ for any prime divisor $E$ over $X$ by taking a
suitable resolution of singularities of $X$. If $a(E,X,\Delta)\geq-1$ (resp.
$>-1$) for every prime divisor $E$ over $X$, then $(X,\Delta)$ is called sub
log canonical (resp. sub kawamata log terminal). We further assume that
$\Delta$ is effective. Then $(X,\Delta)$ is called log canonical and kawamata
log terminal if it is sub log canonical and sub kawamata log terminal,
respectively. When $\Delta$ is effective and $a(E,X,\Delta)>-1$ holds for
every exceptional divisor $E$ over $X$, we say that $(X,\Delta)$ is purely log
terminal.

Let $(X,\Delta)$ be a log canonical pair. If there exists a projective
birational morphism $f\colon Y\to X$ from a smooth variety $Y$ such that both
$\operatorname{Exc}(f)$, the exceptional locus of $f$, and
$\operatorname{Exc}(f)\cup\operatorname{Supp}f^{-1}_{*}\Delta$ are simple
normal crossing divisors on $Y$ and that $a(E,X,\Delta)>-1$ holds for every
$f$-exceptional divisor $E$ on $Y$, then $(X,\Delta)$ is called divisorial log
terminal (dlt, for short). It is well known that if $(X,\Delta)$ is purely log
terminal then it is divisorial log terminal.

In this paper, the notion of non-lc loci and log canonical centers is
indispensable.

######  Definition 2.8 (Non-lc loci and log canonical centers).

Let $(X,\Delta)$ be a normal pair. If there exist a projective birational
morphism $f\colon Y\to X$ from a normal variety $Y$ and a prime divisor $E$ on
$Y$ such that $(X,\Delta)$ is sub log canonical in a neighborhood of the
generic point of $f(E)$ and that $a(E,X,\Delta)=-1$, then $f(E)$ is called a
log canonical center of $(X,\Delta)$.

From now on, we further assume that $\Delta$ is effective. The non-lc locus of
$(X,\Delta)$, denoted by $\operatorname{Nlc}(X,\Delta)$, is the smallest
closed subset $Z$ of $X$ such that the complement $(X\setminus
Z,\Delta|_{X\setminus Z})$ is log canonical. We can define a natural scheme
structure on $\operatorname{Nlc}(X,\Delta)$ by the non-lc ideal sheaf
$\mathcal{J}_{\operatorname{NLC}}(X,\Delta)$ of $(X,\Delta)$. For the
definition of $\mathcal{J}_{\operatorname{NLC}}(X,\Delta)$, see [4, Section
7].

We omit the precise definition of NQC generalized log canonical pairs and NQC
generalized kawamata log terminal pairs here since we need it only in Theorem
1.1 and the statement of Theorem 1.2 is sharper than that of Theorem 1.1. For
the basic definitions and properties of generalized polarized pairs, we
recommend the reader to see [13, Section 2]. Note that the notion of
generalized pairs plays a crucial role in the recent study of higher-
dimensional algebraic varieties.

######  Definition 2.9.

Let $X$ be an equidimensional reduced scheme. Note that $X$ is not necessarily
regular in codimension one. Let $D$ be an $\mathbb{R}$-divisor (resp. a
$\mathbb{Q}$-divisor), that is, $D$ is a finite formal sum
$\sum_{i}d_{i}D_{i}$, where $D_{i}$ is an irreducible reduced closed subscheme
of $X$ of pure codimension one and $d_{i}\in\mathbb{R}$ (resp.
$d_{i}\in\mathbb{Q}$) for every $i$ such that $D_{i}\neq D_{j}$ for $i\neq j$.
We put

$D^{<1}=\sum_{d_{i}<1}d_{i}D_{i},\quad D^{=1}=\sum_{d_{i}=1}D_{i},\quad
D^{>1}=\sum_{d_{i}>1}d_{i}D_{i},\quad\text{and}\quad\lceil
D\rceil=\sum_{i}\lceil d_{i}\rceil D_{i},$

where $\lceil d_{i}\rceil$ is the integer defined by $d_{i}\leq\lceil
d_{i}\rceil<d_{i}+1$. We note that $\lfloor D\rfloor=-\lceil-D\rceil$ and
$\\{D\\}=D-\lfloor D\rfloor$. Similarly, we put

$D^{\geq 1}=\sum_{d_{i}\geq 1}d_{i}D_{i}.$

Let $D$ be an $\mathbb{R}$-divisor (resp. a $\mathbb{Q}$-divisor) as above. We
call $D$ a subboundary $\mathbb{R}$-divisor (resp. $\mathbb{Q}$-divisor) if
$D=D^{\leq 1}$ holds. When $D$ is effective and $D=D^{\leq 1}$ holds, we call
$D$ a boundary $\mathbb{R}$-divisor (resp. $\mathbb{Q}$-divisor).

We further assume that $f\colon X\to Y$ is a surjective morphism onto a
variety $Y$ such that every irreducible component of $X$ is dominant onto $Y$.
Then we put

$D^{v}=\sum_{f(D_{i})\subsetneq Y}d_{i}D_{i}\quad\text{and}\quad
D^{h}=\sum_{f(D_{i})=Y}d_{i}D_{i}.$

We call $D^{v}$ (resp. $D^{h}$) the vertical part (resp. horizontal part) of
$D$ with respect to $f\colon X\to Y$.

##  3\. On basic slc-trivial fibrations

Roughly speaking, a basic slc-trivial fibration is a canonical bundle formula
for simple normal crossing pairs. It was first introduced in [6] based on [8].
Let us start with the definition of simple normal crossing pairs.

######  Definition 3.1 (Simple normal crossing pairs).

A pair $(X,B)$ consists of an equidimensional reduced scheme $X$ and an
$\mathbb{R}$-divisor $B$ on $X$. We say that the pair $(X,B)$ is simple normal
crossing at a point $x\in X$ if $X$ has a Zariski open neighborhood $U$ of $x$
that can be embedded in a smooth variety $M$, where $M$ has a regular system
of parameters $(x_{1},\ldots,x_{p},y_{1},\ldots,y_{r})$ at $x=0$ in which $U$
is defined by a monomial equation

$x_{1}\cdots x_{p}=0$

and

$B|_{U}=\sum_{i=1}^{r}\alpha_{i}(y_{i}=0)|_{U},\quad\alpha_{i}\in\mathbb{R}.$

We say that $(X,B)$ is a simple normal crossing pair if it is simple normal
crossing at every point of $X$.

Let $(X,B)$ be a simple normal crossing pair and let $\nu\colon X^{\nu}\to X$
be the normalization. We define $B^{\nu}$ by
$K_{X^{\nu}}+B^{\nu}=\nu^{*}(K_{X}+B)$, that is, $B^{\nu}$ is the sum of the
inverse images of $B$ and the singular locus of $X$. Then a stratum of $(X,B)$
is an irreducible component of $X$ or the $\nu$-image of some log canonical
center of $(X^{\nu},B^{\nu})$.

Let $(X,B)$ be a simple normal crossing pair and let $X=\bigcup_{i\in I}X_{i}$
be the irreducible decomposition of $X$. Then a stratum of $X$ means an
irreducible component of $X_{i_{1}}\cap\cdots\cap X_{i_{k}}$ for some
$\\{i_{1},\ldots,i_{k}\\}\subset I$. It is easy to see that $W$ is a stratum
of $X$ if and only if $W$ is a stratum of $(X,0)$.

We introduce the notion of basic slc-trivial fibrations. In [6], we only treat
basic $\mathbb{Q}$-slc-trivial fibrations.

######  Definition 3.2 (Basic slc-trivial fibrations, see [6, Definition
4.1]).

A pre-basic $\mathbb{Q}$-slc-trivial (resp. $\mathbb{R}$-slc-trivial)
fibration $f\colon(X,B)\to Y$ consists of a projective surjective morphism
$f\colon X\to Y$ and a simple normal crossing pair $(X,B)$ satisfying the
following properties:

  * (1)

$Y$ is a normal variety,

  * (2)

every stratum of $X$ is dominant onto $Y$ and
$f_{*}\mathcal{O}_{X}\simeq\mathcal{O}_{Y}$,

  * (3)

$B$ is a $\mathbb{Q}$-divisor (resp. an $\mathbb{R}$-divisor) such that
$B=B^{\leq 1}$ holds over the generic point of $Y$, and

  * (4)

there exists a $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor (resp. an
$\mathbb{R}$-Cartier $\mathbb{R}$-divisor) $D$ on $Y$ such that
$K_{X}+B\sim_{\mathbb{Q}}f^{*}D$ (resp. $K_{X}+B\sim_{\mathbb{R}}f^{*}D$),
that is, $K_{X}+B$ is $\mathbb{Q}$-linearly (resp. $\mathbb{R}$-linearly)
equivalent to $f^{*}D$.

If a pre-basic $\mathbb{Q}$-slc-trivial (resp. $\mathbb{R}$-slc-trivial)
fibration $f\colon(X,B)\to Y$ also satisfies

  * (5)

$\operatorname{rank}f_{*}\mathcal{O}_{X}(\lceil-(B^{<1})\rceil)=1$,

then it is called a basic $\mathbb{Q}$-slc-trivial (resp. $\mathbb{R}$-slc-
trivial) fibration.

If there is no danger of confusion, we sometimes use (pre-)basic slc-trivial
fibrations to denote (pre-)basic $\mathbb{Q}$-slc-trivial fibrations or
(pre-)basic $\mathbb{R}$-slc-trivial fibrations.

######  Remark 3.3 (see Remark 4.5).

The condition $f_{*}\mathcal{O}_{X}\simeq\mathcal{O}_{Y}$ in (2) in Definition
3.2 does not play an important role. Moreover, we have to treat the case where
$\mathcal{O}_{Y}\subsetneq f_{*}\mathcal{O}_{X}$ in this paper. The reader can
find that we do not need the condition
$f_{*}\mathcal{O}_{X}\simeq\mathcal{O}_{Y}$ in many places in [6]. Hence it
may be better to remove the condition
$f_{*}\mathcal{O}_{X}\simeq\mathcal{O}_{Y}$ from the definition of pre-basic
slc-trivial fibrations (see [6, Definition 4.1] and Definition 3.2). However,
we keep it here not to cause unnecessary confusion.

Note that the condition $f_{*}\mathcal{O}_{X}\simeq\mathcal{O}_{Y}$ always
holds for basic slc-trivial fibrations even when we remove it from the
definition of pre-basic slc-trivial fibrations. We will see it more precisely.
It is sufficient to see that if every stratum of $X$ is dominant onto $Y$ with
$\operatorname{rank}f_{*}\mathcal{O}_{X}(\lceil-(B^{<1})\rceil)=1$ then the
natural map $\mathcal{O}_{Y}\to f_{*}\mathcal{O}_{X}$ must be an isomorphism.
We note that there are natural inclusions

$\mathcal{O}_{Y}\hookrightarrow f_{*}\mathcal{O}_{X}\hookrightarrow
f_{*}\mathcal{O}_{X}(\lceil-(B^{<1})\rceil)$

since $\lceil-(B^{<1})\rceil$ is effective. Hence
$\mathcal{O}_{Y}\hookrightarrow f_{*}\mathcal{O}_{X}$ is an isomorphism over
some nonempty Zariski open subset of $Y$ and
$\operatorname{rank}f_{*}\mathcal{O}_{X}=1$ holds. We consider the Stein
factorization

$f\colon X\longrightarrow
Z:=\operatorname{Spec}_{Y}f_{*}\mathcal{O}_{X}\overset{\alpha}{\longrightarrow}Y$

of $f\colon X\to Y$. Since every irreducible component of $X$ is dominant onto
$Y$, $Z$ is a variety. Moreover, $\alpha\colon Z\to Y$ is birational since
$\operatorname{rank}f_{*}\mathcal{O}_{X}=1$. By Zariski’s main theorem,
$\alpha\colon Z\to Y$ is an isomorphism. Hence the natural map
$\mathcal{O}_{Y}\to f_{*}\mathcal{O}_{X}$ is an isomorphism.

In order to define discriminant $\mathbb{R}$-b-divisors and moduli
$\mathbb{R}$-b-divisors for basic slc-trivial fibrations, we need the notion
of induced $($pre-$)$basic slc-trivial fibrations.

######  Definition 3.4 (Induced (pre-)basic slc-trivial fibrations, [6, 4.3]).

Let $f\colon(X,B)\to Y$ be a (pre-)basic slc-trivial fibration and let
$\sigma\colon Y^{\prime}\to Y$ be a generically finite surjective morphism
from a normal variety $Y^{\prime}$. Then we have an induced (pre-)basic slc-
trivial fibration $f^{\prime}\colon(X^{\prime},B_{X^{\prime}})\to Y^{\prime}$,
where $B_{X^{\prime}}$ is defined by
$\mu^{*}(K_{X}+B)=K_{X^{\prime}}+B_{X^{\prime}}$, with the following
commutative diagram:

$\textstyle{(X^{\prime},B_{X^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{f^{\prime}}$$\textstyle{(X,B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{Y,}$

where $X^{\prime}$ coincides with $X\times_{Y}Y^{\prime}$ over a nonempty
Zariski open subset of $Y^{\prime}$. More precisely,
$(X^{\prime},B_{X^{\prime}})$ is a simple normal crossing pair with a morphism
$X^{\prime}\to X\times_{Y}Y^{\prime}$ that is an isomorphism over a nonempty
Zariski open subset of $Y^{\prime}$ such that $X^{\prime}$ is projective over
$Y^{\prime}$ and that every stratum of $X^{\prime}$ is dominant onto
$Y^{\prime}$.

Now we are ready to define discriminant $\mathbb{R}$-b-divisors and moduli
$\mathbb{R}$-b-divisors for basic slc-trivial fibrations.

######  Definition 3.5 (Discriminant and moduli $\mathbb{R}$-b-divisors, [6,
4.5]).

Let $f\colon(X,B)\to Y$ be a (pre-)basic slc-trivial fibration as in
Definition 3.2. Let $P$ be a prime divisor on $Y$. By shrinking $Y$ around the
generic point of $P$, we assume that $P$ is Cartier. We set

$b_{P}=\max\left\\{t\in\mathbb{R}\,\left|\,\begin{array}[]{l}{\text{$(X^{\nu},B^{\nu}+t\nu^{*}f^{*}P)$
is sub log canonical}}\\\ {\text{over the generic point of
$P$}}\end{array}\right.\right\\},$

where $\nu\colon X^{\nu}\to X$ is the normalization and
$K_{X^{\nu}}+B^{\nu}=\nu^{*}(K_{X}+B)$, that is, $B^{\nu}$ is the sum of the
inverse images of $B$ and the singular locus of $X$, and set

$B_{Y}=\sum_{P}(1-b_{P})P,$

where $P$ runs over prime divisors on $Y$. Then it is easy to see that $B_{Y}$
is a well-defined $\mathbb{R}$-divisor on $Y$ and is called the discriminant
$\mathbb{R}$-divisor of $f\colon(X,B)\to Y$. We set

$M_{Y}=D-K_{Y}-B_{Y}$

and call $M_{Y}$ the moduli $\mathbb{R}$-divisor of $f\colon(X,B)\to Y$. By
definition, we have

$K_{X}+B\sim_{\mathbb{R}}f^{*}(K_{Y}+B_{Y}+M_{Y}).$

Let $\sigma\colon Y^{\prime}\to Y$ be a proper birational morphism from a
normal variety $Y^{\prime}$ and let
$f^{\prime}\colon(X^{\prime},B_{X^{\prime}})\to Y^{\prime}$ be an induced
(pre-)basic slc-trivial fibration by $\sigma\colon Y^{\prime}\to Y$. We can
define $B_{Y^{\prime}}$, $K_{Y^{\prime}}$ and $M_{Y^{\prime}}$ such that
$\sigma^{*}D=K_{Y^{\prime}}+B_{Y^{\prime}}+M_{Y^{\prime}}$,
$\sigma_{*}B_{Y^{\prime}}=B_{Y}$, $\sigma_{*}K_{Y^{\prime}}=K_{Y}$ and
$\sigma_{*}M_{Y^{\prime}}=M_{Y}$. We note that $B_{Y^{\prime}}$ is independent
of the choice of $(X^{\prime},B_{X^{\prime}})$, that is, $B_{Y^{\prime}}$ is
well defined. Hence there exist a unique $\mathbb{R}$-b-divisor $\mathbf{B}$
such that $\mathbf{B}_{Y^{\prime}}=B_{Y^{\prime}}$ for every $\sigma\colon
Y^{\prime}\to Y$ and a unique $\mathbb{R}$-b-divisor $\mathbf{M}$ such that
$\mathbf{M}_{Y^{\prime}}=M_{Y^{\prime}}$ for every $\sigma\colon Y^{\prime}\to
Y$. Note that $\mathbf{B}$ is called the discriminant $\mathbb{R}$-b-divisor
and that $\mathbf{M}$ is called the moduli $\mathbb{R}$-b-divisor associated
to $f\colon(X,B)\to Y$. We sometimes simply say that $\mathbf{M}$ is the
moduli part of $f\colon(X,B)\to Y$.

Let $g\colon V\to Y$ be a proper surjective morphism from an equidimensional
normal scheme $V$ onto a normal variety $Y$ such that every irreducible
component of $V$ is dominant onto $Y$. Let $G$ be an $\mathbb{R}$-divisor on
$V$ such that $K_{V}+G$ is $\mathbb{R}$-Cartier. Assume that $(V,G)$ is sub
log canonical over the generic point of $Y$. Let $\sigma\colon Y^{\prime}\to
Y$ be a generically finite surjective morphism from a normal variety
$Y^{\prime}$. Then we have the following commutative diagram:

$\textstyle{(V^{\prime},G^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{g^{\prime}}$$\textstyle{(V,G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{Y,}$

where $V^{\prime}$ is the normalization of the main components of
$V\times_{Y}Y^{\prime}$ and $G^{\prime}$ is defined by
$K_{V^{\prime}}+G^{\prime}=\mu^{*}(K_{V}+G)$. Then we can define the
discriminant $\mathbb{R}$-divisor $B_{Y}$ on $Y$ and the discriminant
$\mathbb{R}$-b-divisor $\mathbf{B}$ as in Definition 3.5. Let $f\colon(X,B)\to
Y$ be a (pre-)basic slc-trivial fibration and let $\nu\colon X^{\nu}\to X$ be
the normalization with $K_{X^{\nu}}+B^{\nu}=\nu^{*}(K_{X}+B)$. Then the
discriminant $\mathbb{R}$-b-divisor $\mathbf{B}$ associated to
$f\colon(X,B)\to Y$ defined in Definition 3.5 obviously coincides with that of
$f\circ\nu\colon(X^{\nu},B^{\nu})\to Y$ by definition.

Let us see the main result of [6].

######  Theorem 3.6 ([6, Theorem 1.2]).

Let $f\colon(X,B)\to Y$ be a basic $\mathbb{Q}$-slc-trivial fibration and let
$\mathbf{B}$ and $\mathbf{M}$ be the discriminant and moduli
$\mathbb{Q}$-b-divisors associated to $f\colon(X,B)\to Y$, respectively. Then
we have the following properties:

  * (i)

$\mathbf{K}+\mathbf{B}$ is $\mathbb{Q}$-b-Cartier, where $\mathbf{K}$ is the
canonical b-divisor of $Y$, and

  * (ii)

$\mathbf{M}$ is b-potentially nef, that is, there exists a proper birational
morphism $\sigma\colon Y^{\prime}\to Y$ from a normal variety $Y^{\prime}$
such that $\mathbf{M}_{Y^{\prime}}$ is a potentially nef $\mathbb{Q}$-divisor
on $Y^{\prime}$ and that $\mathbf{M}=\overline{\mathbf{M}_{Y^{\prime}}}$.

The following result was established in [9].

######  Theorem 3.7 ([9, Corollary 1.4]).

In Theorem 3.6, if $Y$ is a curve, then $\mathbf{M}_{Y}$ is semi-ample.

We close this section with important remarks on [6].

######  Remark 3.8.

In (d) in [6, Section 6], we assume that
$\operatorname{Supp}M_{Y}\subset\operatorname{Supp}\Sigma_{Y}$. However, this
conditions is unnecessary. This is because if $P$ is not an irreducible
component of $\operatorname{Supp}\Sigma_{Y}$ then we can always take a prime
divisor $Q$ on $V$ such that $\operatorname{mult}_{Q}(-B_{V}+h^{*}B_{Y})=0$,
$h(Q)=P$, and $\operatorname{mult}_{Q}h^{*}P=1$ (see [6, Proposition 6.3
(iv)]).

######  Remark 3.9.

In [6, 6.1], we assume that
$\operatorname{Supp}\left(B-f^{*}(B_{Y}+M_{Y})\right)$ is a simple normal
crossing divisor on $X$. However, we do not need this assumption. All we need
in [6, 6.1] is the fact that the support of $\\{\Delta\\}$ is a simple normal
crossing divisor on $X$. We note that

$\operatorname{Supp}\\{\Delta\\}\subset\operatorname{Supp}\left(B-f^{*}(B_{Y}+M_{Y})\right)$

always holds since $\Delta=K_{X/Y}+B-f^{*}(B_{Y}+M_{Y})$.

##  4\. Fundamental theorem for basic $\mathbb{Q}$-slc-trivial fibrations

In this section, we will slightly generalize the main theorem of [6] (see
Theorem 3.6). The following theorem is the main result of this section.

######  Theorem 4.1 (see [6, Theorem 1.2]).

Let $f\colon(X,B)\to Y$ be a basic $\mathbb{Q}$-slc-trivial fibration such
that $Y$ is a smooth quasi-projective variety. We write
$K_{X}+B\sim_{\mathbb{Q}}f^{*}D$. Assume that there exists a simple normal
crossing divisor $\Sigma$ on $Y$ such that $\operatorname{Supp}D\subset\Sigma$
and that every stratum of $(X,\operatorname{Supp}B)$ is smooth over
$Y\setminus\Sigma$. Then

  * (i)

$\mathbf{K}+\mathbf{B}=\overline{\mathbf{K}_{Y}+\mathbf{B}_{Y}}$ holds, and

  * (ii)

$\mathbf{M}_{Y}$ is a potentially nef $\mathbb{Q}$-divisor on $Y$ with
$\mathbf{M}=\overline{\mathbf{M}_{Y}}$.

In Section 5, Theorem 4.1 will be generalized for basic $\mathbb{R}$-slc-
trivial fibrations (see Theorems 1.8 and 5.1). We note that Theorem 4.1 is
indispensable for the proof of Theorem 5.1 in Section 5. For the proof of
Theorem 4.1, we prepare a lemma on simultaneous partial resolutions of
singularities of pairs. Let us recall the main result of [2].

######  Theorem 4.2 ([2, Theorem 1.4]).

Let $X$ be a reduced scheme, and let $D$ be a $\mathbb{Q}$-divisor on $X$. Let
$U$ be the largest open subset of $X$ such that $(U,D|_{U})$ is a simple
normal crossing pair. Then there is a morphism $f\colon\tilde{X}\to X$, which
is a composition of blow-ups, such that

  * •

the exceptional locus $\operatorname{Exc}(f)$ is of pure codimension one,

  * •

putting $\tilde{D}=f^{-1}_{*}D+\operatorname{Exc}(f)$ then
$(\tilde{X},\tilde{D})$ is a simple normal crossing pair, and

  * •

$f$ is an isomorphism over $U$.

######  Remark 4.3 (Functoriality, see [2, Remark 1.5 (3)]).

By [2, Remark 1.5 (3)], for every reduced scheme $X$ and a
$\mathbb{Q}$-divisor $D$ on $X$ we may take $f_{X}\colon\tilde{X}\to X$ of
Theorem 4.2 satisfying the following funtoriality. Suppose that we are given
an étale or a smooth morphism $\phi\colon X\to Y$ of reduced schemes and
$\mathbb{Q}$-divisors $D_{X}$ and $D_{Y}$ on $X$ and $Y$ respectively such
that

  * •

$\phi^{*}D_{Y}=D_{X}$, and

  * •

the number of irreducible components of $X$ (resp. $\operatorname{Supp}D_{X}$)
at a point $x\in X$ coincides with that of $Y$ (resp.
$\operatorname{Supp}D_{Y}$) at $\phi(x)\in Y$ for every $x\in X$.

Then, the morphisms $f_{X}\colon\tilde{X}\to X$ and $f_{Y}\colon\tilde{Y}\to
Y$ as in Theorem 4.2 form the diagram of the fiber product

$\textstyle{\tilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{X}}$$\scriptstyle{\tilde{\phi}}$$\textstyle{\tilde{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{Y}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{\square}$

that is, $\tilde{X}=X\times_{Y}\tilde{Y}$.

The following lemma is a key lemma for the proof of Theorem 4.1.

######  Lemma 4.4.

Let $(X,B)$ be a simple normal crossing pair such that $B$ is a
$\mathbb{Q}$-divisor. Let $f\colon X\to Y$ be a surjective morphism onto a
smooth variety $Y$ such that every stratum of $(X,\operatorname{Supp}B)$ is
smooth over $Y$. We put $\Delta=K_{X}+B$ and assume that $b\Delta\sim 0$ for
some positive integer $b$. We consider a $b$-fold cyclic cover

$\pi\colon\widetilde{X}=\operatorname{Spec}_{X}\bigoplus_{i=0}^{b-1}\mathcal{O}_{X}(\lfloor
i\Delta\rfloor)\longrightarrow X$

associated to $b\Delta\sim 0$. We put
$K_{\widetilde{X}}+B_{\widetilde{X}}=\pi^{*}(K_{X}+B)$. Let $\widetilde{U}$ be
the largest Zariski open subset of $\widetilde{X}$ such that
$(\widetilde{U},B_{\widetilde{X}}|_{\widetilde{U}})$ is a simple normal
crossing pair. Then there exists a morphism $d\colon V\to\widetilde{X}$ given
by a composition of blow-ups such that

  * (i)

$d$ is an isomorphism over $\widetilde{U}$,

  * (ii)

$(V,B_{V})$ is a simple normal crossing pair, where
$K_{V}+B_{V}=d^{*}(K_{\widetilde{X}}+B_{\widetilde{X}})$, and

  * (iii)

every stratum of $(V,\operatorname{Supp}B_{V})$ is smooth over $Y$.

###### Proof.

Let us quickly recall the $b$-fold cyclic cover $\pi\colon\widetilde{X}\to X$.
We fix a rational function $\phi$ on $X$ such that
$b\Delta=\operatorname{div}(\phi).$ As usual, we can define an
$\mathcal{O}_{X}$-algebra structure of
$\bigoplus_{i=0}^{b-1}\mathcal{O}_{X}(\lfloor i\Delta\rfloor)$ by
$b\Delta={\operatorname{div}}(\phi)$. We note that

$\mathcal{O}_{X}(\lfloor i\Delta\rfloor)\times\mathcal{O}_{X}(\lfloor
j\Delta\rfloor)\to\mathcal{O}_{X}(\lfloor(i+j)\Delta\rfloor)$

is well defined for $0\leq i,j\leq b-1$ by $\lfloor i\Delta\rfloor+\lfloor
j\Delta\rfloor\leq\lfloor(i+j)\Delta\rfloor$ and that

$\mathcal{O}_{X}(\lfloor(i+j)\Delta\rfloor)\simeq\mathcal{O}_{X}(\lfloor(i+j-b)\Delta\rfloor)$

for $i+j\geq b$ defined by the multiplication with $\phi^{-1}$. We put

$\pi\colon\widetilde{X}=\operatorname{Spec}_{X}\bigoplus_{i=0}^{b-1}\mathcal{O}_{X}(\lfloor
i\Delta\rfloor)$

and call it a $b$-fold cyclic cover associated to $b\Delta\sim 0$. By
construction, $\pi\colon\widetilde{X}\to X$ is étale outside
$\operatorname{Supp}\\{\Delta\\}$. We note that $\widetilde{X}$ is normal over
a neighborhood of the generic point of every irreducible component of
$\operatorname{Supp}\\{\Delta\\}$. We also note that
$(\widetilde{X},B_{\widetilde{X}})$ is simple normal crossing in codimension
one. Throughout this proof, we will freely use the following commutative
diagram.

$\textstyle{(X,B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{(\widetilde{X},B_{\widetilde{X}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{f}}$$\scriptstyle{\pi}$$\textstyle{(V,B_{V})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{d}$$\textstyle{Y}$

######  Step 1.

Let $U$ and $Z$ be affine open neighborhood of $x\in X$ and $y=f(x)\in Y$,
respectively. Without loss of generality, we may assume that $U$ is a simple
normal crossing divisor on a smooth affine variety $W$ since $(X,B)$ is a
simple normal crossing pair. By shrinking $W$, $U$, and $Z$ suitably, we get
the following commutative diagram

$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f|_{U}}$$\scriptstyle{\iota}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{Z}$

where $\iota$ is the natural closed embedding $U\hookrightarrow W$. From now
on, we will repeatedly shrink $W$, $U$, and $Z$ suitably without mentioning it
explicitly. Since every stratum of $X$ is smooth over $Y$, we may assume that
$p$ is a smooth morphism between smooth affine varieties.

######  Step 2.

Since $p\colon W\to Z$ is a smooth morphism, there exists a commutative
diagram

$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{g}$$\textstyle{Z\times\mathbb{C}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\textstyle{Z}$

where $g$ is étale and $p_{1}$ is the first projection (see, for example, [1,
Chapter VII, Definition (1.1) and Theorem (1.8)]). By choosing a coordinate
system $(z_{1},\ldots,z_{n})$ of $\mathbb{C}^{n}$ suitably and shrinking $U$
and $W$ if necessary, we may further assume that $U$ is defined by a monomial

$x_{1}\cdots x_{p}=0$

on $W$, where $x_{i}=g^{*}z_{i}$ for $1\leq i\leq p$, and

$B|_{U}=\sum_{i=1}^{r}\alpha_{i}(y_{i}=0)|_{U}\quad\text{with}\quad\alpha_{i}\in\mathbb{Q}$

holds, where $y_{i}=g^{*}z_{p+i}$ for $1\leq i\leq r$. Here, we used the
hypothesis that every stratum of $(X,\operatorname{Supp}B)$ is smooth over
$Y$.

######  Step 3.

We put $L=(z_{1}\cdots z_{p}=0)$ in $\mathbb{C}^{n}$. Then we have the
following commutative diagram.

$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f|_{U}}$$\scriptstyle{g|_{U}}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\times
L}$$\scriptstyle{p_{1}}$$\textstyle{Z}$

Note that $g|_{U}$ is étale because it is the base change of $g$ by
$L\hookrightarrow\mathbb{C}^{n}$. We put

$D=\sum_{i=1}^{r}\alpha_{i}(z_{p+i}=0)$

on $\mathbb{C}^{n}$. Let $p_{2}\colon Z\times\mathbb{C}^{n}\to\mathbb{C}^{n}$
be the second projection. Then $B|_{U}=g^{*}p^{*}_{2}D|_{U}$ holds.

######  Step 4.

Without loss of generality, we may assume that $K_{Z}\sim 0$ by shrinking $Z$
suitably. Then $K_{U}\sim 0$ holds. Hence, by using the second projection
$p_{2}\colon Z\times\mathbb{C}^{n}\to\mathbb{C}^{n}$, we have

$0\sim b\Delta|_{U}=b(K_{U}+B|_{U})\sim
bg|^{*}_{U}\bigl{(}p_{2}^{*}D|_{Z\times L}\bigr{)}.$

Since $g|_{U}$ is étale, we see that all the coefficients of
$bp_{2}^{*}D|_{Z\times L}$ are integers. Since $p_{2}$ is the second
projection and $D+L$ is a simple normal crossing divisor on $\mathbb{C}^{n}$,
all the coefficients of $bD$ are integers. Therefore, we have $bD\sim 0$. We
fix a rational function $\sigma$ on $\mathbb{C}^{n}$ such that
$bD=\operatorname{div}(\sigma)$. We consider the $b$-fold cyclic cover
$\alpha\colon M\to\mathbb{C}^{n}$ associated to
$bD=\operatorname{div}(\sigma)$. We put $N=\alpha^{-1}L$. We define $B_{N}$ by
$K_{N}+B_{N}=(\alpha|_{N})^{*}(K_{L}+D|_{L})$ and put $B_{Z\times
N}=p^{*}_{2}B_{N}$, where $p_{2}\colon Z\times N\to N$ is the second
projection. Then we get the following commutative diagram:

|
---|---
$\textstyle{U^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$$\textstyle{Z\times
N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
id}_{Z}\times(\alpha|_{N})}$$\scriptstyle{p_{2}}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha|_{N}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f|_{U}}$$\scriptstyle{g|_{U}}$$\textstyle{Z\times
L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{C}^{n}}$$\textstyle{Z}$

where $g^{\prime}\colon U^{\prime}\to Z\times N$ is the base change of
$g|_{U}\colon U\to Z\times L$ by ${\rm id}_{Z}\times(\alpha|_{N})$. We put
$B_{U^{\prime}}=g^{\prime*}B_{Z\times N}$. Then
$K_{U^{\prime}}+B_{U^{\prime}}$ is equal to the pullback of $K_{U}+B|_{U}$ to
$U^{\prime}$.

######  Step 5.

Since $\alpha\colon M\to\mathbb{C}^{n}$ is the $b$-fold cyclic cover
associated to $bD=\operatorname{div}(\sigma)$, we see that

$M=\operatorname{Spec}_{\mathbb{C}^{n}}\bigoplus_{i=0}^{b-1}\mathcal{O}_{\mathbb{C}^{n}}(\lfloor
iD\rfloor).$

Since $p_{2}\circ g\colon W\to Z\times\mathbb{C}^{n}\to\mathbb{C}^{n}$ is the
composition of an étale morphism and the second projection, we have
$g^{*}p_{2}^{*}\lfloor iD\rfloor|_{U}=\lfloor
ig^{*}p_{2}^{*}D\rfloor|_{U}=\lfloor iB\rfloor|_{U}=\lfloor iB|_{U}\rfloor$,
where the last equality follows from that $(U,B|_{U})$ is a simple normal
crossing pair. Let $\sigma_{U}$ be a rational function on $U$ which is the
pullback of $\sigma$. Then $bB|_{U}=\operatorname{div}(\sigma_{U})$ because we
have $bD=\operatorname{div}(\sigma)$. By the construction of $U^{\prime}\to
U$, we see that

$U^{\prime}=\operatorname{Spec}_{U}\bigoplus_{i=0}^{b-1}\mathcal{O}_{U}(\lfloor
iB|_{U}\rfloor)$

and $U^{\prime}\to U$ is the $b$-fold cyclic cover associated to
$bB|_{U}=\operatorname{div}(\sigma_{U})$.

We recall that $\Delta=K_{X}+B$ and $\widetilde{X}\to X$ is the $b$-fold
cyclic cover associated to $b\Delta=\operatorname{div}(\phi)$. We put
$\phi_{U}$ as the restriction of $\phi$ to $U$. Then, the morphism
$\pi^{-1}(U)\to U$ is the $b$-fold cyclic cover associated to
$b\Delta|_{U}=\operatorname{div}(\phi_{U})$. Now $\Delta|_{U}-B|_{U}$ is a
Cartier divisor on $U$ and
$b(\Delta|_{U}-B|_{U})=\operatorname{div}(\phi_{U}\cdot\sigma_{U}^{-1})$. With
this relation, we construct a $b$-fold cyclic cover $\tau\colon\overline{U}\to
U$. Then $\tau$ is étale, $\tau^{*}(\Delta|_{U}-B|_{U})$ is Cartier and
$\tau^{*}(\Delta|_{U}-B|_{U})\sim 0$. So there exists a rational function
$\xi$ on $\overline{U}$ such that
$\xi^{b}=\tau^{*}(\phi_{U}\cdot\sigma^{-1}_{U})$, equivalently,
$\tau^{*}(\Delta|_{U}-B|_{U})=\operatorname{div}(\xi)$. From this, the
$b$-fold cyclic cover $U^{\dagger}_{1}\to\overline{U}$ associated to
$b\tau^{*}\Delta|_{U}=\operatorname{div}(\tau^{*}\phi_{U})$ is isomorphic to
the $b$-fold cyclic cover $U^{\dagger}_{2}\to\overline{U}$ associated to
$b\tau^{*}B|_{U}=\operatorname{div}(\tau^{*}\sigma_{U})=\operatorname{div}(\tau^{*}\phi_{U}\cdot\xi^{-b})$.
Since $\tau\colon\overline{U}\to U$ is étale, the construction of
$U^{\dagger}_{2}$ shows that $U_{2}^{\dagger}\to\overline{U}$ is the base
change of $U^{\prime}\to U$ by $\overline{U}\to U$. Similarly, we see that
$U^{\dagger}_{1}\to\overline{U}$ is the base change of $\pi^{-1}(U)\to U$ by
$\overline{U}\to U$.

$\textstyle{U^{\dagger}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{2}}$$\textstyle{U^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U^{\dagger}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{1}}$$\textstyle{\pi^{-1}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Box}$$\textstyle{\overline{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Box}$

We put $a_{1}\colon U^{\dagger}_{1}\to\pi^{-1}(U)$ and $a_{2}\colon
U^{\dagger}_{2}\to U^{\prime}$. By construction, $a_{1}$ and $a_{2}$ are
étale. We see that the composition $U^{\dagger}_{1}\to\pi^{-1}(U)\to U$ is
isomorphic to the composition $U^{\dagger}_{2}\to U^{\prime}\to U$ by
construction. By this isomorphism, we obtain that
$a_{1}^{*}(B_{\widetilde{X}}|_{\pi^{-1}(U)})$ is isomorphic to
$a_{2}^{*}B_{U^{\prime}}$.

In this way, there exist étale morphisms $a\colon U^{\dagger}\to\pi^{-1}(U)$
and $a^{\prime}\colon U^{\dagger}\to U^{\prime}$ over $Z$ such that
$U^{\dagger}_{1}\simeq U^{\dagger}\simeq U^{\dagger}_{2}$ with the following
commutative diagram:

$\textstyle{U^{\dagger}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{a^{\prime}}$$\textstyle{\pi^{-1}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{U^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U}$

such that $a^{*}(B_{\widetilde{X}}|_{\pi^{-1}(U)})=a^{\prime*}B_{U^{\prime}}$.

######  Step 6.

We apply [2, Theorem 1.4] (see Theorem 4.2) to the pair
$(\widetilde{X},B_{\widetilde{X}})$. Then we obtain a morphism $d\colon
V\to\widetilde{X}$ given by a composite of blow-ups satisfying (i) and (ii).
Hence, all we have to do is to check that $d\colon V\to\widetilde{X}$
satisfies (iii).

######  Step 7.

Recall that $B_{Z\times N}=p^{*}_{2}B_{N}$, where $p_{2}\colon Z\times N\to
N$, and $B_{U^{\prime}}=g^{\prime*}B_{Z\times N}$. Recall also the relation
$a^{*}(B_{\widetilde{X}}|_{\pi^{-1}(U)})=a^{\prime*}B_{U^{\prime}}$. We apply
[2, Theorem 1.4] (see Theorem 4.2) to $N$ and $B_{N}$, and we obtain a
morphism $\beta\colon N^{\prime}\to N$ given by a composition of blow-ups. We
apply [2, Theorem 1.4] again to $Z\times N$ and $B_{Z\times N}$. Then we get a
morphism ${\rm id}_{Z}\times\beta\colon Z\times N^{\prime}\to Z\times N$ by
the functoriality of [2, Theorem 1.4] (see Remark 4.3). We put
$\widehat{V}=d^{-1}(\pi^{-1}(U))\subset V$, and we apply [2, Theorem 1.4] to
the pair of $U^{\dagger}$ and $a^{*}(B_{\widetilde{X}}|_{\pi^{-1}(U)})$, and
the pair of $U^{\prime}$ and $B_{U^{\prime}}$. Then we obtain morphisms
$V^{\dagger}\to U^{\dagger}$ and $V^{\prime}\to U^{\prime}$.

We check that we may apply the functoriality (see Remark 4.3) to the morphisms

$g^{\prime}\colon U^{\prime}\to Z\times N$, $a^{\prime}\colon U^{{\dagger}}\to
U^{\prime}$, and $a\colon U^{{\dagger}}\to\pi^{-1}(U)$

(see the diagram in the next paragraph) and divisors

$B_{Z\times N}$ on $Z\times N$, $B_{U^{\prime}}$ on $U^{\prime}$, and
$B_{\widetilde{X}}|_{\pi^{-1}(U)}$ on $\pi^{-1}(U)$ and their pullbacks.

We only check the second condition of Remark 4.3 for schemes because the case
of divisors can be proved by the same way. By construction, $g^{\prime}$ is
the base change of $g|_{U}\colon U\to Z\times L$ by the morphism $Z\times N\to
Z\times L$. Because $Z\times L$ is a simple normal crossing divisor on
$Z\times\mathbb{C}^{n}$ and $g|_{U}$ is étale, by arguing locally, we see that
$g|_{U}$ satisfies the second condition of Remark 4.3. Then so does
$g^{\prime}$ since $g^{\prime}$ is constructed by the base change of $g|_{U}$.
Similarly, $a^{\prime}$ (resp. $a$) is constructed with the base change of
$\tau\colon\overline{U}\to U$ by $U^{\prime}\to U$ (resp. $\pi^{-1}(U)\to U$),
and $U$ is a simple normal crossing divisor on $W$. Thus, the same argument as
above implies that $a^{\prime}$ and $a$ satisfy the second condition of Remark
4.3. Thus, we may apply the functoriality (see Remark 4.3) to the above
morphisms and divisors.

Applying the functoriality (see Remark 4.3), we have the following diagram:

$\textstyle{\widehat{V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d|_{\widehat{V}}}$$\textstyle{V^{\dagger}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\times
N^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{-1}(U)}$$\textstyle{U^{\dagger}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{a^{\prime}}$$\scriptstyle{\square}$$\textstyle{U^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$$\scriptstyle{\square}$$\textstyle{Z\times
N,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\square}$

where each square is the fiber product. By construction, all the upper
horizontal morphisms are étale. Let $B_{V^{\dagger}}$ (resp.
$B_{\widehat{V}}$) be the sum of the birational transform of
$a^{\prime*}B_{U^{\prime}}$ (resp. $B_{\widetilde{X}}|_{\pi^{-1}(U)}$) and the
exceptional locus of $V^{\dagger}\to U^{\dagger}$ (resp.
$\widehat{V}\to\pi^{-1}(U)$). Then, every stratum of
$(V^{\dagger},\operatorname{Supp}B_{V^{\dagger}})$ is smooth over $Z$. Since
each irreducible component of $V^{\dagger}$ is smooth over $Z$ and
$V^{\dagger}\to\widehat{V}$ is étale, we see that each irreducible component
of $\widehat{V}$ is smooth over $Z$. By a similar argument, we see that every
stratum of $(\widehat{V},\operatorname{Supp}B_{\widehat{V}})$ is smooth over
$Z$. This implies that $d\colon V\to\widetilde{X}$ satisfies (iii).

We finish the proof of Lemma 4.4. ∎

Before we start the proof of Theorem 4.1, we make an important remark on [6].

######  Remark 4.5 (see Remark 3.3).

In Theorem 4.1, we can write

$K_{X}+B+\frac{1}{b}\operatorname{div}(\varphi)=f^{*}D$

for some positive integer $b$ and a rational function
$\varphi\in\Gamma(X,\mathcal{K}^{*}_{X})$, where $\mathcal{K}_{X}$ is the
sheaf of total quotient rings of $\mathcal{O}_{X}$ and $\mathcal{K}^{*}_{X}$
denotes the sheaf of invertible elements in $\mathcal{K}_{X}$, such that
$b(K_{X}+B-f^{*}D)\sim 0$. In general, $b$ is larger than $b(F,B_{F})$ in [6,
Section 6]. We take a $b$-fold cyclic cover $\pi\colon\widetilde{X}\to X$
associated to $b\Delta\sim 0$, where $\Delta=K_{X}+B-f^{*}D$, as in [6,
Section 6]. Then the general fiber of $h\colon V\to Y$ is not necessarily
connected in [6, Section 6]. Moreover, $V$ is not necessarily connected. This
means that [6, Proposition 6.3 (ii)] does not hold true since the natural map
$\mathcal{O}_{Y}\to h_{*}\mathcal{O}_{V}$ is not always an isomorphism.
Fortunately, the condition $h_{*}\mathcal{O}_{V}\simeq\mathcal{O}_{Y}$ is not
necessary for the proof of the other properties of [6, Proposition 6.3]. We
note that the condition $h_{*}\mathcal{O}_{V}\simeq\mathcal{O}_{Y}$ is
unnecessary in [6, Lemma 7.3 and Theorem 8.1]. Hence it may be better to
remove the condition $f_{*}\mathcal{O}_{X}\simeq\mathcal{O}_{Y}$ from (2) in
Definition 3.2.

Let $b$ be the smallest positive integer such that $b(K_{X}+B-f^{*}D)\sim 0$.
Then we can write

$K_{X}+B+\frac{1}{b}\operatorname{div}(\varphi)=f^{*}D.$

As usual, we consider the $b$-fold cyclic cover $\pi\colon\widetilde{X}\to X$
associated to $b\Delta=\operatorname{div}(\varphi^{-1})$, where
$\Delta=K_{X}+B-f^{*}D$. Let $b^{\sharp}$ be any positive integer with
$b^{\sharp}\geq 2$. We put $\varphi^{\sharp}=\varphi^{b^{\sharp}}$. Then we
get

$K_{X}+B+\frac{1}{bb^{\sharp}}\operatorname{div}(\varphi^{\sharp})=f^{*}D.$

Let $\pi^{\sharp}\colon X^{\sharp}\to X$ be the $bb^{\sharp}$-fold cyclic
cover associated to
$bb^{\sharp}\Delta=\operatorname{div}\left((\varphi^{\sharp})^{-1}\right)$. We
take the $H$-invariant part of $\pi^{\sharp}\colon X^{\sharp}\to X$, where $H$
is the subgroup of the Galois group
$\operatorname{Gal}(X^{\sharp}/X)\simeq\mathbb{Z}/bb^{\sharp}\mathbb{Z}$ of
$\pi^{\sharp}\colon X^{\sharp}\to X$ corresponding to
$b\mathbb{Z}/bb^{\sharp}\mathbb{Z}$. Then we can recover
$\pi\colon\widetilde{X}\to X$. Note that $\pi^{\sharp}\colon X^{\sharp}\to X$
is decomposed into $b^{\sharp}$ components and that each component is
isomorphic to $\pi\colon\widetilde{X}\to X$.

Let us prove Theorem 4.1.

###### Proof of Theorem 4.1.

Here, we only explain how to modify the proof of [6, Theorem 1.2] by using
Lemma 4.4.

By taking a completion as in [6, Lemma 4.12], we may further assume that $Y$
is projective. By Lemma 4.4, we can construct a commutative diagram (6.4) in
[6, Section 6] satisfying (a)–(g) such that $\Sigma_{Y}=\Sigma$ holds without
taking birational modifications of $Y$. Here, we do not require the condition
$\operatorname{Supp}M_{Y}\subset\operatorname{Supp}\Sigma_{Y}$ in (d) in [6,
Section 6] (see Remark 3.8). We also do not require the condition that the
general fiber of $h\colon V\to Y$ is connected (see Remark 4.5). The covering
arguments and [6, Proposition 6.3] work without any modifications. We note
that $Y$ is a smooth projective variety. In what follows, we apply the proof
of [6, Theorem 8.1]. Let $\gamma\colon Y^{\prime}\to Y$ be a projective
birational morphism from a normal variety $Y^{\prime}$. By replacing
$Y^{\prime}$ with a higher model if necessary, we may assume that $Y^{\prime}$
is smooth and that $\gamma^{-1}\Sigma_{Y}$ is a simple normal crossing divisor
on $Y^{\prime}$. With [6, Lemma 7.3], we construct $\tau\colon\overline{Y}\to
Y$ a unipotent reduction of the local monodromies around $\Sigma_{Y}$. Then
the induced fibration over $\overline{Y}$ satisfies [6, Proposition 6.3 (iv),
(v)]. As in the proof of [6, Theorem 8.1], we get a diagram:

$\textstyle{\overline{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\overline{Y}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma^{\prime}}$$\scriptstyle{\tau^{\prime}}$$\textstyle{Y}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$

such that $\tau^{\prime}$ is finite and the induced fibration over
$\overline{Y}^{\prime}$ satisfies [6, Proposition 6.3 (iv), (v)]. By [6,
Theorem 3.1], we see that $\mathbf{M}_{\overline{Y}}$ is a nef Cartier divisor
and
$\gamma^{\prime*}\mathbf{M}_{\overline{Y}}=\mathbf{M}_{\overline{Y}^{\prime}}$.
Moreover, we have $\tau^{*}\mathbf{M}_{Y}=\mathbf{M}_{\overline{Y}}$ and
$\tau^{\prime*}\mathbf{M}_{Y^{\prime}}=\mathbf{M}_{\overline{Y}^{\prime}}$
because $\tau$ and $\tau^{\prime}$ are both finite (see [6, Lemma 4.10]).
Thus, we have that $\mathbf{M}_{Y}$ is a nef $\mathbb{Q}$-divisor and
$\gamma^{*}\mathbf{M}_{Y}=\mathbf{M}_{Y^{\prime}}$. This is Theorem 4.1 (ii).
Theorem 4.1 (i) immediately follows from Theorem 4.1 (ii). So we are done. ∎

##  5\. Fundamental theorem for basic $\mathbb{R}$-slc-trivial fibrations

In this section, we will establish the following fundamental theorem for basic
$\mathbb{R}$-slc-trivial fibrations.

######  Theorem 5.1 (see Theorem 1.8).

Let $f\colon(X,B)\to Y$ be a basic $\mathbb{R}$-slc-trivial fibration such
that $Y$ is a smooth quasi-projective variety. We write
$K_{X}+B\sim_{\mathbb{R}}f^{*}D$. Assume that there exists a simple normal
crossing divisor $\Sigma$ on $Y$ such that $\operatorname{Supp}D\subset\Sigma$
and that every stratum of $(X,\operatorname{Supp}B)$ is smooth over
$Y\setminus\Sigma$. Then

  * (i)

$\mathbf{K}+\mathbf{B}=\overline{\mathbf{K}_{Y}+\mathbf{B}_{Y}}$ holds, and

  * (ii)

$\mathbf{M}_{Y}$ is a potentially nef $\mathbb{R}$-divisor on $Y$ with
$\mathbf{M}=\overline{\mathbf{M}_{Y}}$.

By Theorem 5.1, which is obviously a generalization of Theorem 4.1, we can use
the theory of basic slc-trivial fibrations in [6] and [7] for
$\mathbb{R}$-divisors. The following formulation may be useful. Hence we state
it explicitly here for the reader’s convenience. We note that if
$f\colon(X,B)\to Y$ is a basic $\mathbb{Q}$-slc-trivial fibration then
Corollary 5.2 is nothing but [6, Theorem 1.2].

######  Corollary 5.2 ([6, Theorem 1.2]).

Let $f\colon(X,B)\to Y$ be a basic $\mathbb{R}$-slc-trivial fibration and let
$\mathbf{B}$ and $\mathbf{M}$ be the discriminant and moduli
$\mathbb{R}$-b-divisors associated to $f\colon(X,B)\to Y$, respectively. Then
we have the following properties:

  * (i)

$\mathbf{K}+\mathbf{B}$ is $\mathbb{R}$-b-Cartier, where $\mathbf{K}$ is the
canonical b-divisor of $Y$, and

  * (ii)

$\mathbf{M}$ is b-potentially nef, that is, there exists a proper birational
morphism $\sigma\colon Y^{\prime}\to Y$ from a normal variety $Y^{\prime}$
such that $\mathbf{M}_{Y^{\prime}}$ is a potentially nef $\mathbb{R}$-divisor
on $Y^{\prime}$ and that $\mathbf{M}=\overline{\mathbf{M}_{Y^{\prime}}}$
holds.

######  Remark 5.3 (see [9, Corollary 1.4]).

In Theorem 5.1 and Corollary 5.2, we can easily see that $\mathbf{M}_{Y}$ is
semi-ample when $Y$ is a curve by Theorem 3.7 and Lemma 5.4 below.

Let us start with an easy lemma.

######  Lemma 5.4.

Let $f\colon(X,B)\to Y$ be a basic $\mathbb{R}$-slc-trivial fibration with
$K_{X}+B\sim_{\mathbb{R}}f^{*}D$. Then there exist a $\mathbb{Q}$-divisor
$B_{i}$ on $X$, a $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor $D_{i}$ on $Y$,
and a positive real number $r_{i}$ for $1\leq i\leq k$ such that

  * (1)

$\sum_{i=1}^{k}r_{i}=1$ with $\sum_{i=1}^{k}r_{i}B_{i}=B$ and
$\sum_{i=1}^{k}r_{i}D_{i}=D$,

  * (2)

$\operatorname{Supp}B=\operatorname{Supp}B_{i}$, $\lfloor
B^{>1}\rfloor=\lfloor B^{>1}_{i}\rfloor$, and
$\lceil-(B^{<1})\rceil=\lceil-(B^{<1}_{i})\rceil$ hold for every $i$,

  * (3)

if $\operatorname{coeff}_{S}(B)\in\mathbb{Q}$ for a prime divisor $S$ on $X$,
then $\operatorname{coeff}_{S}(B)=\operatorname{coeff}_{S}(B_{i})$ holds for
every $i$,

  * (4)

$\operatorname{Supp}D=\operatorname{Supp}D_{i}$ holds for every $i$,

  * (5)

if $\operatorname{coeff}_{T}(D)\in\mathbb{Q}$ for a prime divisor $T$ on $Y$,
then $\operatorname{coeff}_{T}(D)=\operatorname{coeff}_{T}(D_{i})$ holds for
every $i$, and

  * (6)

$K_{X}+B_{i}\sim_{\mathbb{Q}}f^{*}D_{i}$ holds for every $i$.

In particular, $f\colon(X,B_{i})\to Y$ is a basic $\mathbb{Q}$-slc-trivial
fibration with $K_{X}+B_{i}\sim_{\mathbb{Q}}f^{*}D_{i}$ for every $i$.
Moreover, if $t_{1},\ldots,t_{k}$ are real numbers such that $0\leq t_{i}\leq
1$ for every $i$ with $\sum_{i=1}^{k}t_{i}=1$, then
$f\colon\left(X,\sum_{i=1}^{k}t_{i}B_{i}\right)\to Y$ is a basic
$\mathbb{R}$-slc-trivial fibration with
$K_{X}+\sum_{i=1}^{k}t_{i}B_{i}\sim_{\mathbb{R}}f^{*}\left(\sum_{i=1}^{k}t_{i}D_{i}\right)$.

###### Proof.

The proof of [6, Lemma 11.1] works with some suitable minor modifications.
Therefore, we can take $B_{i}$, $D_{i}$, and $r_{i}$ for $1\leq i\leq k$
satisfying (1)–(6). By (2), $B_{i}=B^{\leq 1}_{i}$ holds over the generic
point of $Y$ for every $i$. By (2) again,
$\operatorname{rank}f_{*}\mathcal{O}_{X}(\lceil-(B^{<1}_{i})\rceil)=\operatorname{rank}f_{*}\mathcal{O}_{X}(\lceil-(B^{<1})\rceil)=1$.
Hence $f\colon(X,B_{i})\to Y$ is a basic $\mathbb{Q}$-slc-trivial fibration
with $K_{X}+B_{i}\sim_{\mathbb{Q}}f^{*}D_{i}$ for every $i$. We put
$\widetilde{B}=\sum_{i=1}^{k}t_{i}B_{i}$. Then
$\widetilde{B}=\widetilde{B}^{\leq 1}$ holds over the generic point of $Y$ by
(2). By (2) again, we see that
$\lceil-(\widetilde{B}^{<1})\rceil=\lceil-(B^{<1})\rceil$ holds. Therefore,
$f\colon(X,\widetilde{B})\to Y$ is a basic $\mathbb{R}$-slc-trivial fibration.
∎

We also need the following lemma.

######  Lemma 5.5.

Let $f\colon(X,B)\to Y$ be a basic $\mathbb{R}$-slc-trivial fibration. Let
$\mathbf{B}$ denote the discriminant $\mathbb{R}$-b-divisor associated to
$f\colon(X,B)\to Y$. Suppose that there are $\mathbb{Q}$-divisors
$B_{1},\dots,B_{k}$ on $X$ and real numbers $r_{1},\dots,r_{k}$ such that
$\sum_{i=1}^{k}r_{i}=1$ and $\sum_{i=1}^{k}r_{i}B_{i}=B$. We put

$\mathcal{P}=\left\\{\left.\sum_{i=1}^{k}t_{i}B_{i}\,\right|\,\text{$0\leq
t_{i}\leq 1$ for every $i$ with $\sum_{i=1}^{k}t_{i}=1$}\right\\}.$

Assume that $f\colon(X,\Delta)\to Y$ has the structure of a basic
$\mathbb{R}$-slc-trivial fibration for every $\Delta\in\mathcal{P}$. For
$\Delta\in\mathcal{P}$, $\mathbf{B}^{\Delta}$ denotes the discriminant
$\mathbb{R}$-b-divisor of the basic $\mathbb{R}$-slc-trivial fibration
$f\colon(X,\Delta)\to Y$. Then, we can find
$\Delta_{1},\dots,\Delta_{l}\in\mathcal{P}$ which are $\mathbb{Q}_{\geq
0}$-linear combinations of $B_{1},\dots,B_{k}$ and positive real numbers
$s_{1},\dots,s_{l}$ such that

  * •

$\sum_{j=1}^{l}s_{j}=1$ and $\sum_{j=1}^{l}s_{j}\Delta_{j}=B$, and

  * •

$\mathbf{B}_{Y}=\sum_{j=1}^{l}s_{j}\mathbf{B}^{\Delta_{j}}_{Y}$.

Here, $\mathbf{B}_{Y}$ (resp. $\mathbf{B}^{\Delta_{j}}_{Y}$) is the trace of
the discriminant $\mathbb{R}$-b-divisor $\mathbf{B}$ (resp.
$\mathbf{B}^{\Delta_{j}}$) on $Y$.

###### Proof.

Since $\mathbf{B}$ is an $\mathbb{R}$-b-divisor, it is sufficient to prove the
lemma for a resolution of $Y^{\prime}\to Y$ and the induced basic slc-trivial
fibrations $f^{\prime}\colon(X^{\prime},B_{X^{\prime}})\to Y$ and
$f^{\prime}\colon(X^{\prime},(B_{i})_{X^{\prime}})\to Y^{\prime}$. Moreover,
by Definition 3.5 and taking the normalization of $X$, we may assume that $X$
is a disjoint union of smooth varieties. Therefore, by replacing $X$, $Y$,
$B$, and $B_{i}$, we may assume that $Y$ is smooth and there are simple normal
crossing divisors $\Sigma_{X}$ on $X$ and $\Sigma_{Y}$ on $Y$ such that

  * •

$\operatorname{Supp}B\subset\Sigma_{X}$ and
$\operatorname{Supp}B_{i}\subset\Sigma_{X}$ for every $i$,

  * •

$\Sigma_{X}^{v}\subset f^{-1}\Sigma_{Y}\subset\Sigma_{X}$, where
$\Sigma_{X}^{v}$ is the vertical part of $\Sigma_{X}$,

  * •

$f$ is smooth over $Y\setminus\Sigma_{Y}$, and

  * •

$\Sigma_{X}$ is relatively simple normal crossing over $Y\setminus\Sigma_{Y}$.

Then it is clear that $\operatorname{Supp}\mathbf{B}_{Y}\subset\Sigma_{Y}$ and
$\operatorname{Supp}\mathbf{B}^{\Delta}_{Y}\subset\Sigma_{Y}$ for all
$\Delta\in\mathcal{P}$. We consider a rational convex polytope

$\mathcal{C}=\left\\{\boldsymbol{v}=(v_{1},\ldots,v_{k})\in[0,1]^{k}\,\left|\,\text{$\sum_{j}v_{j}=1$}\right.\right\\}\subset[0,1]^{k}.$

Then we may identify $\mathcal{C}$ with $\mathcal{P}$ by putting
$\Delta_{\boldsymbol{v}}=\sum_{i}v_{i}B_{i}\in\mathcal{P}$ for
$\boldsymbol{v}=(v_{1},\ldots,v_{k})\in\mathcal{C}$. We define
$\boldsymbol{v}_{0}\in\mathcal{C}$ to be the point such that
$\Delta_{\boldsymbol{v}_{0}}=B$.

Fix a prime divisor $Q$ on $Y$ which is a component of $\Sigma_{Y}$. We shrink
$Y$ near the generic point of $Q$ so that all components of $f^{*}Q$ dominate
$Q$. We can write $f^{*}Q=\sum_{i}m_{P_{i}}P_{i}$, where $P_{i}$ are
components of $\Sigma_{X}$ such that $f(P_{i})=Q$, and
$m_{P_{i}}=\operatorname{coeff}_{P_{i}}(f^{*}Q)$. We fix a component
$P_{(B,Q)}$ of $f^{*}Q$ such that

$\frac{1-\operatorname{coeff}_{P_{(B,Q)}}(B)}{m_{P_{(B,Q)}}}=\underset{P_{i}}{\min}\left\\{\frac{1-\operatorname{coeff}_{P_{i}}(B)}{m_{P_{i}}}\right\\}.$

Note that $\frac{1-\operatorname{coeff}_{P_{(B,Q)}}(B)}{m_{P_{(B,Q)}}}$ is the
log canonical threshold of $(X,B)$ with respect to $f^{*}Q$ over the generic
point of $Q$ because $(X,B+\mu f^{*}Q)$ is sub log canonical over the generic
point of $Q$ if and only if $\operatorname{coeff}_{P_{i}}(B)+\mu m_{P_{i}}\leq
1$ for all $P_{i}$. For every component $P_{i}$ of $f^{*}Q$, we can define a
function

$H^{(P_{i})}(\boldsymbol{v}):=\frac{1-\operatorname{coeff}_{P_{(B,Q)}}(\Delta_{\boldsymbol{v}})}{m_{P_{(B,Q)}}}-\frac{1-\operatorname{coeff}_{P_{i}}(\Delta_{\boldsymbol{v}})}{m_{P_{i}}}$

and the half space

$H^{(P_{i})}_{\leq
0}:=\\{\boldsymbol{v}\in\mathcal{C}\;|\;H^{(P_{i})}(\boldsymbol{v})\leq 0\\}.$

It is easy to check that $H^{(P_{i})}$ are rational affine functions and the
half spaces $H^{(P_{i})}_{\leq 0}$ contain $\boldsymbol{v}_{0}$ since
$\boldsymbol{v}_{0}$ is the point such that $\Delta_{\boldsymbol{v}_{0}}=B$.
Therefore, the set

$\mathcal{C}_{Q}:=\mathcal{C}\cap\left(\bigcap_{P_{i}}H^{(P_{i})}_{\leq
0}\right)$

is a rational polytope in $\mathcal{C}$ containing $\boldsymbol{v}_{0}$, where
$P_{i}$ runs over components of $f^{*}Q$. We put

$\begin{split}t(\Delta_{\boldsymbol{v}},Q):=&1-\operatorname{coeff}_{Q}\left(\mathbf{B}^{\Delta_{\boldsymbol{v}}}_{Y}\right)\\\
=&\sup\\{\mu\in\mathbb{R}\,|\,\text{$(X,\Delta_{\boldsymbol{v}}+\mu f^{*}Q)$
is sub log canonical over the generic point of $Q$}\\}.\end{split}$

Then, by the definitions of $H^{(P_{i})}_{\leq 0}$, every
$\boldsymbol{v}\in\mathcal{C}_{Q}$ satisfies

(5.1)
$t(\Delta_{\boldsymbol{v}},Q)=\underset{P_{i}}{\min}\left\\{\frac{1-\operatorname{coeff}_{P_{i}}(\Delta_{\boldsymbol{v}})}{m_{P_{i}}}\right\\}=\frac{1-\operatorname{coeff}_{P_{(B,Q)}}(\Delta_{\boldsymbol{v}})}{m_{P_{(B,Q)}}}.$

Here, to prove the first equality we used the fact that
$(X,\Delta_{\boldsymbol{v}}+\mu f^{*}Q)$ is sub log canonical over the generic
point of $Q$ if and only if
$\operatorname{coeff}_{P_{i}}(\Delta_{\boldsymbol{v}})+\mu m_{P_{i}}\leq 1$
for all $P_{i}$.

Finally, we define

$\mathcal{C^{\prime}}:=\bigcap_{Q}\mathcal{C}_{Q},$

where $Q$ runs over all irreducible components of $\Sigma_{Y}$. It is easy to
see that $\mathcal{C^{\prime}}$ is a rational polytope in $\mathcal{C}$ and
$\mathcal{C^{\prime}}$ contains $\boldsymbol{v}_{0}$. Thus, we can find
rational points $\boldsymbol{v}_{1},\ldots,\boldsymbol{v}_{l}$ and positive
real numbers $s_{1},\ldots,s_{l}$ such that $\sum_{j=1}^{l}s_{j}=1$ and
$\sum_{j=1}^{l}s_{j}\boldsymbol{v}_{j}=\boldsymbol{v}_{0}$. We put
$\Delta_{j}=\Delta_{\boldsymbol{v}_{j}}$ for each $1\leq j\leq l$. Then
$B=\Delta_{\boldsymbol{v}_{0}}=\sum_{j=1}^{l}s_{j}\Delta_{j}$. For every
component $Q$ of $\Sigma_{Y}$, the equation (5.1) implies that

$\begin{array}[]{llll}t(B,Q)&=&\dfrac{1-\operatorname{coeff}_{P_{(B,Q)}}(B)}{m_{P_{(B,Q)}}}&\qquad\qquad(\text{see
\eqref{x-eq5.1}})\\\ \\\
&=&\dfrac{1-\operatorname{coeff}_{P_{(B,Q)}}\left(\sum_{j=1}^{l}s_{j}\Delta_{j}\right)}{m_{P_{(B,Q)}}}&\qquad\qquad(B=\sum_{j=1}^{l}s_{j}\Delta_{j})\\\
\\\
&=&\sum_{j=1}^{l}s_{j}\cdot\dfrac{1-\operatorname{coeff}_{P_{(B,Q)}}(\Delta_{j})}{m_{P_{(B,Q)}}}&\qquad\qquad(\sum_{j=1}^{l}s_{j}=1)\\\
\\\ &=&\sum_{j=1}^{l}s_{j}\cdot t(\Delta_{j},Q)&\qquad\qquad(\text{see
\eqref{x-eq5.1}}).\end{array}$

Since
$t(\Delta_{j},Q)=1-\operatorname{coeff}_{Q}\left(\mathbf{B}^{\Delta_{j}}_{Y}\right)$
for every $1\leq j\leq l$ and every irreducible component $Q$ of $\Sigma_{Y}$,
we see that $\mathbf{B}_{Y}=\sum_{j=1}^{l}s_{j}\mathbf{B}^{\Delta_{j}}_{Y}$. ∎

We are ready to prove Theorem 5.1.

###### Proof of Theorem 5.1.

Fix an arbitrary projective birational morphism $\sigma\colon Y^{\prime}\to Y$
from a normal quasi-projective variety $Y^{\prime}$, and let

$\textstyle{(X^{\prime},B_{X^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{f^{\prime}}$$\textstyle{(X,B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{Y}$

be the induced basic $\mathbb{R}$-slc-trivial fibration (see Definition 3.4).
It is sufficient to show that
$\sigma^{*}(K_{Y}+\mathbf{B}_{Y})=K_{Y^{\prime}}+\mathbf{B}_{Y^{\prime}}$ and
$\mathbf{M}_{Y}$ is a potentially nef $\mathbb{R}$-divisor on $Y$ with
$\sigma^{*}\mathbf{M}_{Y}=\mathbf{M}_{Y^{\prime}}$.

We pick $\mathbb{Q}$-divisors $B_{1},\dots,B_{k}$ on $X$,
$\mathbb{Q}$-divisors $D_{1},\dots,D_{k}$ on $Y$ and positive real numbers
$r_{1},\dots,r_{k}$ as in Lemma 5.4. Then, the following properties hold.

  * •

$\sum_{i=1}^{k}r_{i}=1$ with $\sum_{i=1}^{k}r_{i}B_{i}=B$ and
$\sum_{i=1}^{k}r_{i}D_{i}=D$,

  * •

$\operatorname{Supp}B=\operatorname{Supp}B_{i}$ and
$\operatorname{Supp}D=\operatorname{Supp}D_{i}$ hold for every $i$, and

  * •

$K_{X}+B_{i}\sim_{\mathbb{Q}}f^{*}D_{i}$ holds for every $i$.

We put $D^{\prime}_{i}=\sigma^{*}D_{i}$ and we define $B^{\prime}_{i}$ by
$K_{X^{\prime}}+B^{\prime}_{i}=\mu^{*}(K_{X}+B_{i})$ for any $1\leq i\leq k$.
Then $f^{\prime}\colon(X^{\prime},B^{\prime}_{i})\to Y^{\prime}$ are basic
$\mathbb{Q}$-slc-trivial fibrations with
$K_{X^{\prime}}+B^{\prime}_{i}\sim_{\mathbb{Q}}f^{\prime*}D^{\prime}_{i}$. As
in Lemma 5.5, we put

$\mathcal{P}^{\prime}=\left\\{\left.\sum_{i=1}^{k}t_{i}B^{\prime}_{i}\,\right|\,\text{$0\leq
t_{i}\leq 1$ for every $i$ with $\sum_{i=1}^{k}t_{i}=1$}\right\\}.$

We may assume that $f^{\prime}\colon(X^{\prime},\Delta)\to Y^{\prime}$ is a
basic $\mathbb{R}$-slc-trivial fibration for every
$\Delta\in\mathcal{P}^{\prime}$. We define $\mathcal{P}^{\prime}_{\mathbb{Q}}$
by

$\mathcal{P}^{\prime}_{\mathbb{Q}}:=\left\\{\left.\sum_{i=1}^{k}t_{i}B^{\prime}_{i}\,\right|\,\text{$t_{i}\in\mathbb{Q}$
and $0\leq t_{i}\leq 1$ for every $i$ with $\sum_{i=1}^{k}t_{i}=1$}\right\\}.$

Note that $B_{X^{\prime}}\in\mathcal{P}^{\prime}$.

Pick any
$\Delta=\sum_{i=1}^{k}t_{i}B^{\prime}_{i}\in\mathcal{P}^{\prime}_{\mathbb{Q}}$.
Since $\mu_{*}B^{\prime}_{i}=B_{i}$, we have
$\mu_{*}\Delta=\sum_{i=1}^{k}t_{i}B_{i}$ such that $t_{i}\in\mathbb{Q}$.
Therefore, the morphism $f\colon(X,\mu_{*}\Delta)\to Y$ is a basic
$\mathbb{Q}$-slc-trivial fibration such that
$K_{X}+\mu_{*}\Delta\sim_{\mathbb{Q}}f^{*}\bigl{(}\sum_{i=1}^{k}t_{i}D_{i}\bigr{)}$.
Let $\mathbf{B}^{\Delta}$ and $\mathbf{M}^{\Delta}$ be the discriminant
$\mathbb{Q}$-b-divisor and the moduli $\mathbb{Q}$-b-divisor of the basic
$\mathbb{Q}$-slc-trivial fibration $f\colon(X,\mu_{*}\Delta)\to Y$,
respectively. Because we have
$\operatorname{Supp}\bigl{(}\sum_{i=1}^{k}t_{i}D_{i}\bigr{)}\subset\operatorname{Supp}D$
and $\operatorname{Supp}\mu_{*}\Delta\subset\operatorname{Supp}B$, we may
apply Theorem 4.1. Therefore, for every
$\Delta\in\mathcal{P}^{\prime}_{\mathbb{Q}}$ it follows that
$\sigma^{*}(K_{Y}+\mathbf{B}^{\Delta}_{Y})=K_{Y^{\prime}}+\mathbf{B}^{\Delta}_{Y^{\prime}}$
and $\mathbf{M}^{\Delta}_{Y}$ is a potentially nef $\mathbb{Q}$-divisor on $Y$
with $\sigma^{*}\mathbf{M}^{\Delta}_{Y}=\mathbf{M}^{\Delta}_{Y^{\prime}}$. It
also follows from the construction that
$f^{\prime}\colon(X^{\prime},\Delta)\to Y^{\prime}$ is the basic
$\mathbb{Q}$-slc-trivial fibration induced from $f\colon(X,\mu_{*}\Delta)\to
Y$ such that
$K_{X^{\prime}}+\Delta\sim_{\mathbb{Q}}f^{\prime*}\bigl{(}\sum_{i=1}^{k}t_{i}D^{\prime}_{i}\bigr{)}$.
It is because $K_{X^{\prime}}+\Delta=\mu^{*}(K_{X}+\mu_{*}\Delta)$ by
construction.

We apply Lemma 5.5 to $f^{\prime}\colon(X^{\prime},B_{X^{\prime}})\to
Y^{\prime}$ and $\mathcal{P}^{\prime}$. Then, we can find
$\Delta_{1},\dots,\Delta_{l}\in\mathcal{P}^{\prime}_{\mathbb{Q}}$ and positive
real numbers $s_{1},\dots,s_{l}$ such that

  * •

$\sum_{j=1}^{l}s_{j}=1$ and $\sum_{j=1}^{l}s_{j}\Delta_{j}=B_{X^{\prime}}$,
and

  * •

$\mathbf{B}_{Y^{\prime}}=\sum_{j=1}^{l}s_{j}\mathbf{B}^{\Delta_{j}}_{Y^{\prime}}$.

Since $\mathbf{B}$ and $\mathbf{B}^{\Delta_{j}}$ are $\mathbb{R}$-b-divisors,
we have $\mathbf{B}_{Y}=\sum_{j=1}^{l}s_{j}\mathbf{B}^{\Delta_{j}}_{Y}$. Then

$\begin{split}\sigma^{*}(K_{Y}+\mathbf{B}_{Y})&=\sigma^{*}\left(K_{Y}+\sum_{j=1}^{l}s_{j}\mathbf{B}^{\Delta_{j}}_{Y}\right)=\sum_{j=1}^{l}s_{j}\sigma^{*}(K_{Y}+\mathbf{B}^{\Delta_{j}}_{Y})\\\
&=\sum_{j=1}^{l}s_{j}(K_{Y^{\prime}}+\mathbf{B}^{\Delta_{j}}_{Y^{\prime}})=K_{Y^{\prime}}+\sum_{j=1}^{l}s_{j}\mathbf{B}^{\Delta_{j}}_{Y^{\prime}}\\\
&=K_{Y^{\prime}}+\mathbf{B}_{Y^{\prime}}.\end{split}$

Therefore, we have
$\sigma^{*}(K_{Y}+\mathbf{B}_{Y})=K_{Y^{\prime}}+\mathbf{B}_{Y^{\prime}}$.
Recalling that $\sigma\colon Y^{\prime}\to Y$ is an arbitrary projective
birational morphism, we see that (i) of Theorem 5.1 holds, i.e.,

$\mathbf{K}+\mathbf{B}=\overline{\mathbf{K}_{Y}+\mathbf{B}_{Y}}.$

As in the third paragraph, for each $j$ we define $D^{\prime}_{\Delta_{j}}$ to
be the $\mathbb{Q}$-divisor on $Y^{\prime}$ associated to the basic
$\mathbb{Q}$-slc-trivial fibration $f^{\prime}\colon(X^{\prime},\Delta_{j})\to
Y^{\prime}$. Note that
$K_{X^{\prime}}+\Delta_{j}\sim_{\mathbb{Q}}f^{\prime*}D^{\prime}_{\Delta_{j}}$
for all $j$. Since $\sum_{j=1}^{l}s_{j}=1$ and
$\sum_{j=1}^{l}s_{j}\Delta_{j}=B_{X^{\prime}}$, we have
$\sigma^{*}D=\sum_{j=1}^{l}s_{j}D^{\prime}_{\Delta_{j}}$. By the relation
$\mathbf{B}_{Y^{\prime}}=\sum_{j=1}^{l}s_{j}\mathbf{B}^{\Delta_{j}}_{Y^{\prime}}$
and the definition of the moduli $\mathbb{R}$-b-divisors (see Definition 3.5),
we have

$\mathbf{M}_{Y^{\prime}}=\sum_{j=1}^{l}s_{j}\mathbf{M}^{\Delta_{j}}_{Y^{\prime}}\qquad\text{and}\qquad\mathbf{M}_{Y}=\sum_{j=1}^{l}s_{j}\mathbf{M}^{\Delta_{j}}_{Y}$

Then $\mathbf{M}_{Y}$ is a potentially nef $\mathbb{R}$-divisor on $Y$ and

$\begin{split}\sigma^{*}\mathbf{M}_{Y}&=\sigma^{*}\left(\sum_{j=1}^{l}s_{j}\mathbf{M}^{\Delta_{j}}_{Y}\right)=\sum_{j=1}^{l}s_{j}\mathbf{M}^{\Delta_{j}}_{Y^{\prime}}=\mathbf{M}_{Y^{\prime}}.\end{split}$

Here, we used
$\sigma^{*}\mathbf{M}^{\Delta_{j}}_{Y}=\mathbf{M}^{\Delta_{j}}_{Y^{\prime}}$
for every $j$, which follows from the third paragraph. We complete the proof.
∎

The following result is essentially obtained in the proof of Theorem 5.1. We
explicitly state it here for future use.

######  Theorem 5.6.

Let $f\colon(X,B)\to Y$ be a basic $\mathbb{R}$-slc-trivial fibration with
$K_{X}+B\sim_{\mathbb{R}}f^{*}D$. Then there are $\mathbb{Q}$-divisors
$B_{1},\dots,B_{l}$ on $X$, $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisors
$D_{1},\dots,D_{l}$ on $Y$ and positive real numbers $r_{1},\dots,r_{l}$
satisfying the following properties.

  * •

$\sum_{j=1}^{l}r_{j}=1$ with $\sum_{j=1}^{l}r_{j}B_{j}=B$ and
$\sum_{j=1}^{l}r_{j}D_{j}=D$,

  * •

$\operatorname{Supp}B=\operatorname{Supp}B_{j}$, $\lfloor
B^{>1}\rfloor=\lfloor B^{>1}_{j}\rfloor$, and
$\lceil-(B^{<1})\rceil=\lceil-(B^{<1}_{j})\rceil$ hold for every $j$,

  * •

if $\operatorname{coeff}_{S}(B)\in\mathbb{Q}$ for a prime divisor $S$ on $X$,
then $\operatorname{coeff}_{S}(B)=\operatorname{coeff}_{S}(B_{j})$ holds for
every $j$,

  * •

$\operatorname{Supp}D=\operatorname{Supp}D_{j}$ holds for every $j$,

  * •

if $\operatorname{coeff}_{T}(D)\in\mathbb{Q}$ for a prime divisor $T$ on $Y$,
then $\operatorname{coeff}_{T}(D)=\operatorname{coeff}_{T}(D_{j})$ holds for
every $j$,

  * •

$K_{X}+B_{j}\sim_{\mathbb{Q}}f^{*}D_{j}$ holds for every $j$,

  * •

$\mathbf{B}=\sum_{j=1}^{l}r_{j}\mathbf{B}_{j}$ as b-divisors, where
$\mathbf{B}$ (resp. $\mathbf{B}_{j}$) is the discriminant
$\mathbb{R}$-b-divisor (resp. the discriminant $\mathbb{Q}$-b-divisor) of
$f\colon(X,B)\to Y$ (resp. $f\colon(X,B_{j})\to Y$), and

  * •

$\mathbf{M}=\sum_{j=1}^{l}r_{i}\mathbf{M}_{j}$ as b-divisors, where
$\mathbf{M}$ (resp. $\mathbf{M}_{j}$) is the moduli $\mathbb{R}$-b-divisor
(the moduli $\mathbb{Q}$-b-divisor) associated to $f\colon(X,B)\to Y$ (resp.
$f\colon(X,B_{j})\to Y$).

###### Sketch of Proof.

It can be proved by Theorem 5.1, Lemma 5.4 and Lemma 5.5. We only outline the
proof.

We note that the properties of Theorem 5.6 except the last two properties
correspond to (1)–(6) of Lemma 5.4 respectively. By Lemma 5.4, we can find
$\mathbb{Q}$-divisors $\widetilde{B}_{1},\dots,\widetilde{B}_{k}$ on $X$,
$\mathbb{Q}$-Cartier $\mathbb{Q}$-divisors
$\widetilde{D}_{1},\dots,\widetilde{D}_{k}$ on $Y$ and positive real numbers
$\widetilde{r}_{1},\dots,\widetilde{r}_{k}$ satisfying (1)–(6) of Lemma 5.4.
Then $\widetilde{B}_{i}$, $\widetilde{D}_{i}$, and $\widetilde{r}_{i}$ satisfy
all the properties of Theorem 5.6 except the last two properties. More
specifically, $\widetilde{B}_{i}$, $\widetilde{D}_{i}$, and
$\widetilde{r}_{i}$ satisfy

  * •

$\sum_{i=1}^{k}\widetilde{r}_{i}=1$ with
$\sum_{i=1}^{k}\widetilde{r}_{i}\widetilde{B}_{i}=B$ and
$\sum_{i=1}^{k}\widetilde{r}_{i}\widetilde{D}_{i}=D$ (see (1) of Lemma 5.4),

  * •

$\operatorname{Supp}B=\operatorname{Supp}\widetilde{B}_{i}$ and
$\operatorname{Supp}D=\operatorname{Supp}\widetilde{D}_{i}$ hold for every
$i$,

  * •

$K_{X}+\widetilde{B}_{i}\sim_{\mathbb{Q}}f^{*}\widetilde{D}_{i}$ holds for
every $i$ (see (6) of Lemma 5.4),

and (2)–(5) in Lemma 5.4. We take a smooth higher model $\sigma\colon
Y^{\prime}\to Y$ so that the induced basic $\mathbb{R}$-slc-trivial fibration
$f^{\prime}\colon(X^{\prime},B^{\prime})\to Y^{\prime}$ satisfies the property
that there exists a simple normal crossing divisor $\Sigma^{\prime}$ on
$Y^{\prime}$ such that $\operatorname{Supp}\sigma^{*}D\subset\Sigma^{\prime}$
and that every stratum of $(X^{\prime},\operatorname{Supp}B^{\prime})$ is
smooth over $Y^{\prime}\setminus\Sigma^{\prime}$. The morphism $X^{\prime}\to
X$ is denoted by $\mu$. For each $1\leq i\leq k$, let
$\widetilde{B}^{\prime}_{i}$ be a $\mathbb{Q}$-divisor on $X^{\prime}$ defined
by
$K_{X^{\prime}}+\widetilde{B}^{\prime}_{i}=\mu^{*}(K_{X}+\widetilde{B}_{i})$.
Note that
$K_{X^{\prime}}+\widetilde{B}^{\prime}_{i}\sim_{\mathbb{Q}}f^{\prime*}\sigma^{*}\widetilde{D}_{i}$.
We may assume that
$\operatorname{Supp}\sigma^{*}\widetilde{D}_{i}\subset\Sigma^{\prime}$ and
that every stratum of
$(X^{\prime},\operatorname{Supp}\widetilde{B}^{\prime}_{i})$ is smooth over
$Y^{\prime}\setminus\Sigma^{\prime}$ for every $i$ by taking $\sigma\colon
Y^{\prime}\to Y$ suitably. We define

$\mathcal{P}=\left\\{\left.\sum_{i=1}^{k}t_{i}\widetilde{B}^{\prime}_{i}\,\right|\,\text{$0\leq
t_{i}\leq 1$ for every $i$ with $\sum_{i=1}^{k}t_{i}=1$}\right\\}.$

By Lemma 5.5, we can find $B^{\prime}_{1},\dots,B^{\prime}_{l}\in\mathcal{P}$
which are $\mathbb{Q}_{\geq 0}$-linear combinations of
$\widetilde{B}^{\prime}_{1},\dots,\widetilde{B}^{\prime}_{l}$ and positive
real numbers $r_{1},\dots,r_{l}$ such that

  * •

$\sum_{j=1}^{l}r_{j}=1$ and $\sum_{j=1}^{l}r_{j}B^{\prime}_{j}=B^{\prime}$,
and

  * •

$\mathbf{B}_{Y^{\prime}}=\sum_{j=1}^{l}r_{j}\mathbf{B}_{jY^{\prime}}$.

Here, $\mathbf{B}_{j}$ is the discriminant $\mathbb{Q}$-b-divisor associated
to $f^{\prime}\colon(X^{\prime},B^{\prime}_{j})\to Y^{\prime}$. By Theorem
5.1, we have
$\mathbf{K}+\mathbf{B}=\overline{\mathbf{K}_{Y^{\prime}}+\mathbf{B}_{Y^{\prime}}}$
and
$\mathbf{K}+\mathbf{B}_{j}=\overline{\mathbf{K}_{Y^{\prime}}+\mathbf{B}_{jY^{\prime}}}$.
We put $B_{j}=\mu_{*}B^{\prime}_{j}$ for each $1\leq j\leq l$. Then we can
find $\mathbb{Q}$-divisors $D_{1},\dots,D_{l}$ on $Y$ such that
$K_{X}+B_{j}\sim_{\mathbb{Q}}f^{*}D_{j}$ and $\sum_{j=1}^{l}r_{j}D_{j}=D$. By
construction, we can easily see that $B_{1}\ldots,B_{l}$,
$D_{1},\ldots,D_{l}$, and $r_{1},\ldots,r_{l}$ constructed above satisfy the
desired properties. ∎

##  6\. Proof of Theorem 1.2

In this section, we will prove Theorem 1.2, which is the main result of this
paper. Then we will treat Theorem 1.1 and Corollary 1.4. We note that we will
freely use the framework of quasi-log schemes in the proof of Theorem 1.2. For
the details of quasi-log schemes, see [5, Chapter 6]. Let us start with the
proof of Theorem 1.2.

###### Proof of Theorem 1.2.

From Step 1 to Step 3, we will define a natural quasi-log scheme structure on
$Z$. This part is essentially contained in [5, Chapter 6] and [7].

######  Step 1.

In this step, we will give a natural quasi-log scheme structure on
$W^{\prime}:=W\cup\operatorname{Nlc}(X,\Delta)$. This step is essentially the
adjunction for quasi-log schemes (see [5, Theorem 6.3.5 (i)]).

We put $W^{\prime}:=W\cup\operatorname{Nlc}(X,\Delta)$ as above. We will
sketch how to define a natural quasi-log scheme structure on $W^{\prime}$. Let
$f\colon Y\to X$ be a projective birational morphism from a smooth quasi-
projective variety $Y$ such that $K_{Y}+\Delta_{Y}=f^{*}(K_{X}+\Delta)$ and
that $\operatorname{Supp}\Delta_{Y}$ is a simple normal crossing divisor on
$Y$. By taking some more blow-ups, we may assume that the union of all log
canonical centers of $(Y,\Delta_{Y})$ mapped to $W^{\prime}$ by $f$, which is
denoted by $V^{\prime}$, is a union of some irreducible components of
$\Delta^{=1}_{Y}$. As usual, we put $A=\lceil-(\Delta^{<1}_{Y})\rceil$ and
$N=\lfloor\Delta^{>1}_{Y}\rfloor$ and consider the following short exact
sequence:

$0\to\mathcal{O}_{Y}(A-N-V^{\prime})\to\mathcal{O}_{Y}(A-N)\to\mathcal{O}_{V^{\prime}}(A-N)\to
0.$

By taking $R^{i}f_{*}$, we obtain:

$\begin{split}0&\longrightarrow
f_{*}\mathcal{O}_{Y}(A-N-V^{\prime})\longrightarrow
f_{*}\mathcal{O}_{Y}(A-N)\longrightarrow f_{*}\mathcal{O}_{V^{\prime}}(A-N)\\\
&\overset{\delta}{\longrightarrow}R^{1}f_{*}\mathcal{O}_{Y}(A-N-V^{\prime})\longrightarrow\cdots\end{split}$

The connecting homomorphism $\delta$ is zero since no associated prime of
$R^{1}f_{*}\mathcal{O}_{Y}(A-N-V^{\prime})$ is contained in
$W^{\prime}=f(V^{\prime})$ (see [4, theorem 6.3 (i)] and [5, Theorem 5.6.2
(i)]). Hence we have:

$0\to f_{*}\mathcal{O}_{Y}(A-N-V^{\prime})\to f_{*}\mathcal{O}_{Y}(A-N)\to
f_{*}\mathcal{O}_{V^{\prime}}(A-N)\to 0.$

Note that
$\mathcal{J}_{\operatorname{NLC}}(X,\Delta)=f_{*}\mathcal{O}_{Y}(A-N)$ by
definition. We put
$\mathcal{I}_{W^{\prime}}=f_{*}\mathcal{O}_{Y}(A-N-V^{\prime})$ and
$\mathcal{I}_{W^{\prime}_{-\infty}}=f_{*}\mathcal{O}_{V^{\prime}}(A-N)$. We
define $\Delta_{V^{\prime}}$ by
$(K_{Y}+\Delta_{Y})|_{V^{\prime}}=K_{V^{\prime}}+\Delta_{V^{\prime}}$. Then

$\left(W^{\prime},(K_{X}+\Delta)|_{W^{\prime}},f\colon(V^{\prime},\Delta_{V^{\prime}})\to
W^{\prime}\right)$

is a quasi-log scheme. By construction,
$\operatorname{Nqlc}(W^{\prime},(K_{X}+\Delta)|_{W^{\prime}})=\operatorname{Nlc}(X,\Delta)$
holds. By construction again, a subset $C\subset X$ is a qlc stratum of
$[W^{\prime},(K_{X}+\Delta)|_{W^{\prime}}]$ if and only if $C$ is a log
canonical center of $(X,\Delta)$ included in $W$. We note that the above
construction is independent of the choice of $f\colon Y\to X$ by [5,
Proposition 6.3.1].

######  Step 2.

In this step, we will give a natural quasi-log scheme structure on
$[W,(K_{X}+\Delta)|_{W}]$. This step is essentially [7, Lemma 4.19].

In Step 1, we may further assume that the union of all strata of
$(V^{\prime},\Delta_{V^{\prime}})$ mapped to
$W\cap\operatorname{Nlc}(X,\Delta)$ is also a union of some irreducible
components of $V^{\prime}$. Let $\widehat{V}$ be the union of the irreducible
components of $V^{\prime}$ mapped to $W$ by $f$. We put $\Delta_{\widehat{V}}$
by $(K_{Y}+\Delta_{Y})|_{\widehat{V}}=K_{\widehat{V}}+\Delta_{\widehat{V}}$.
Then, by the proof of [7, Lemmas 4.18 and 4.19],

$\left(W,(K_{X}+\Delta)|_{W},f\colon(\widehat{V},\Delta_{\widehat{V}})\to
W\right)$

is a quasi-log scheme. By [7, Lemma 4.19], we obtain that
$\mathcal{I}_{W_{-\infty}}=\mathcal{I}_{W^{\prime}_{-\infty}}$ holds and that
a subset $C\subset X$ is a qlc stratum of
$[W^{\prime},(K_{X}+\Delta)|_{W^{\prime}}]$ if and only if $C$ is a qlc
stratum of $[W,(K_{X}+\Delta)|_{W}]$. Hence
$W\cap\operatorname{Nlc}(X,\Delta)=W_{-\infty}$ and

$W\cap\left(\operatorname{Nlc}(X,\Delta)\cup\bigcup_{W\not\subset
W^{\dagger}}W^{\dagger}\right)=\operatorname{Nqklt}(W,(K_{X}+\Delta)|_{W})$

hold set theoretically, where $W^{\dagger}$ runs over log canonical centers of
$(X,\Delta)$ which do not contain $W$.

######  Step 3.

In this step, we will give a natural quasi-log scheme structure on $Z$. This
step is nothing but [7, Theorem 1.9].

In Step 2, we may further assume that the union of all strata of
$(\widehat{V},\Delta_{\widehat{V}})$ mapped to
$\operatorname{Nqklt}(W,(K_{X}+\Delta)|_{W})$ is a union of some irreducible
components of $\widehat{V}$. Let $V$ be the union of the irreducible
components of $\widehat{V}$ which are dominant onto $W$. Then, by the proof of
[7, Theorem 1.9], $f\colon V\to W$ factors through $Z$ and

$\left(Z,\nu^{*}(K_{X}+\Delta),f\colon(V,\Delta_{V})\to Z\right)$

becomes a quasi-log scheme, where $\Delta_{V}$ is defined by
$(K_{Y}+\Delta_{Y})|_{V}=K_{V}+\Delta_{V}$. By construction, we have
$\nu_{*}\mathcal{I}_{\operatorname{Nqklt}(Z,\nu^{*}(K_{X}+\Delta))}=\mathcal{I}_{\operatorname{Nqklt}(W,(K_{X}+\Delta)|_{W})}$.
Hence

$\operatorname{Nqklt}(Z,\nu^{*}(K_{X}+\Delta))=\nu^{-1}\operatorname{Nqklt}(W,(K_{X}+\Delta)|_{W})$

holds.

######  Step 4.

Then $f\colon(V,\Delta_{V})\to Z$ is a basic $\mathbb{R}$-slc-trivial
fibration. Hence we can apply Corollary 5.2 and Remark 5.3 to
$f\colon(V,\Delta_{V})\to Z$. We note that $f\colon(V,\Delta_{V})\to Z$ is a
basic $\mathbb{Q}$-slc-trivial fibration when $K_{X}+\Delta$ is
$\mathbb{Q}$-Cartier. In that case, Theorem 3.6 with Theorem 3.7 is
sufficient.

######  Step 5.

By [7, Theorem 7.1] and Steps 1, 2, and 3 in its proof, we can construct a
projective birational morphism $p\colon Z^{\prime}\to Z$ from a smooth quasi-
projective variety $Z^{\prime}$ satisfying (i), (ii), (iii), and (v). We note
that we can directly apply Step 3 in the proof of [7, Theorem 7.1] to basic
$\mathbb{R}$-slc-trivial fibrations by Corollary 5.2. We also note that
$\mathbf{B}$ is a well-defined $\mathbb{R}$-b-divisor on $Z$ and is
independent of $f\colon Y\to X$ (see [10, Lemma 5.1] and [11, Theorem 1.2]).

######  Step 6 (see [10, Theorem 5.4]).

In this final step, we will prove (iv). This step is essentially [10, Theorem
5.4]. We explain it here for the reader’s convenience.

Without loss of generality, we may assume that $X$ is affine by taking a
finite affine open cover of $X$. Let $g_{dlt}\colon X_{dlt}\to X$ be a good
dlt blow-up of $(X,\Delta)$ such that
$K_{X_{dlt}}+\Delta_{X_{dlt}}=g^{*}_{dlt}(K_{X}+\Delta)$ (see [10, Lemma
3.5]). We may assume that there is an irreducible component $S$ of
$\Delta^{=1}_{X_{dlt}}$ with $g_{dlt}(S)=W$. We put

$D=\Delta^{\geq 1}_{X_{dlt}}-\operatorname{Supp}\Delta^{\geq
1}_{X_{dlt}}=\Delta^{>1}_{X_{dlt}}-\operatorname{Supp}\Delta^{>1}_{X_{dlt}}.$

Then $-D$ is semi-ample over $X$ and
$\operatorname{Supp}D=\operatorname{Nlc}(X_{dlt},\Delta_{X_{dlt}})$ holds set
theoretically (see [10, Lemma 3.5]). By taking the contraction morphism
$\varphi\colon X_{dlt}\to X_{lc}$ associated to $-D$ over $X$, we get a log
canonical modification $g_{lc}\colon X_{lc}\to X$ with
$K_{X_{lc}}+\Delta_{X_{lc}}=g^{*}_{lc}(K_{X}+\Delta)$ (see [10, Theorem 1.3]).

$\textstyle{X_{dlt}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{dlt}}$$\scriptstyle{\varphi}$$\textstyle{X_{lc}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{lc}}$$\textstyle{X}$

We put $D^{\prime}=\varphi_{*}D$. Then $-D^{\prime}$ is ample over $X$, and

$g^{-1}_{lc}\operatorname{Nlc}(X,\Delta)=\operatorname{Nlc}(X_{lc},\Delta_{X_{lc}})=\operatorname{Supp}D^{\prime}$

holds set theoretically. We note that

$\operatorname{Nlc}(X_{dlt},\Delta_{X_{dlt}})=\varphi^{-1}\operatorname{Nlc}(X_{lc},\Delta_{X_{lc}})=g^{-1}_{dlt}\operatorname{Nlc}(X,\Delta)$

holds set theoretically. Let $\tilde{W}$ be the strict transform of $W$ on
$X_{lc}$. Let $\tilde{\nu}\colon\tilde{Z}\to\tilde{W}$ be the normalization.
Then we can easily see that

$\operatorname{Supp}\mathbf{B}^{>1}_{\tilde{Z}}=\widetilde{\nu}^{*}D^{\prime}=\tilde{\nu}^{-1}\left(\operatorname{Nlc}(X_{lc},\Delta_{X_{lc}})\cap\tilde{W}\right)=(g_{lc}\circ\tilde{\nu})^{-1}\left(\operatorname{Nlc}(X,\Delta)\cap
W\right)$

holds set theoretically. We note that $\mathbf{B}^{>1}=0$ over
$X\setminus\operatorname{Nlc}(X,\Delta)$ by construction. Hence we obtain
$\nu\circ p(\mathbf{B}^{>1}_{Z^{\prime}})=W\cap\operatorname{Nlc}(X,\Delta)$
set theoretically.

We finish the proof of Theorem 1.2. ∎

Finally, we prove Theorem 1.1 and Corollary 1.4.

###### Proof of Theorem 1.1.

Here, we use the same notation as in Theorem 1.2. We put
$B_{Z}=\mathbf{B}_{Z}$ and $M_{Z}=\mathbf{M}_{Z}$ in Theorem 1.2. We note that
$\mathbf{M}_{Z^{\prime}}$ is a finite $\mathbb{R}_{>0}$-linear combination of
potentially nef Cartier divisors on $Z^{\prime}$ with
$p_{*}\mathbf{M}_{Z^{\prime}}=M_{Z}$. Hence the desired statement follows from
Theorem 1.2. ∎

###### Proof of Corollary 1.4.

By the definition of $\mathbf{B}$ in Theorem 1.2 (see the proof of Theorem 1.2
and Definition 1.3), we can easily check that $B_{Z}$ is nothing but
Shokurov’s different (see [4, Section 14]) and
$\nu^{*}(K_{X}+\Delta)=K_{Z}+B_{Z}$ holds, where $\nu\colon Z\to W$ is the
normalization of $W$. In particular, we have $M_{Z}=0$. By (A) in Theorem 1.1,
we obtain that $(X,\Delta)$ is log canonical in a neighborhood of $W$ if and
only if $(Z,B_{Z})$ is log canonical in the usual sense. It recovers
Kawakita’s inversion of adjunction (see [15, Theorem]). By (B), we see that
$(Z,B_{Z})$ is kawamata log terminal if and only if $(X,\Delta)$ is log
canonical in a neighborhood of $W$ and $W$ is a minimal log canonical center
of $(X,\Delta)$ (see [4, Theorem 9.1] and [5, Theorem 6.3.11]). Note that
$(X,\Delta)$ is purely log terminal in a neighborhood of $W$ if and only if
$(X,\Delta)$ is log canonical in a neighborhood of $W$ and $W$ is a minimal
log canonical center of $(X,\Delta)$. ∎

We close this section with the following remark which summarizes the
construction of the $\mathbb{R}$-b-divisors $\mathbf{B}$ and $\mathbf{M}$ on
$Z$.

######  Remark 6.1.

Let $X$ be a normal variety and let $\Delta$ be an effective
$\mathbb{R}$-divisor on $X$ such that $K_{X}+\Delta$ is $\mathbb{R}$-Cartier.
Let $W$ be a log canonical center of $(X,\Delta)$ and let $\nu\colon Z\to W$
be the normalization of $W$.

We take a log resolution $f\colon Y\to X$ of $(X,\Delta)$ which is a
sufficiently high birational model. We define $\Delta_{Y}$ by
$K_{Y}+\Delta_{Y}=f^{*}(K_{X}+\Delta)$, and let $V$ be the union of the
irreducible components of $\Delta_{Y}^{=1}$ which map onto $W$. Let
$\Delta_{V}$ be an $\mathbb{R}$-divisor on $V$ defined by
$K_{V}+\Delta_{V}=(K_{Y}+\Delta_{Y})|_{V}$, then we get the morphism
$f\colon(V,\Delta_{V})\to Z$ which has the structure of a basic
$\mathbb{R}$-slc-trivial fibration. Then $\mathbf{B}$ and $\mathbf{M}$ are
defined to be the discriminant $\mathbb{R}$-b-divisor and the moduli
$\mathbb{R}$-b-divisor as in Definition 3.5. By construction, we can easily
check that the construction in the proof of Theorem 1.2 and the one in
Definition 1.3 define the same $\mathbb{R}$-b-divisor $\mathbf{B}$ on $Z$ (see
[10, Lemma 5.1] and [11, Theorem 1.2]). Precisely speaking, when $\dim
W\leq\dim X-2$, we consider the $\mathbb{R}$-line bundle $\mathcal{L}$ on $X$
associated to $K_{X}+\Delta$. We fix an $\mathbb{R}$-Cartier
$\mathbb{R}$-divisor $D$ on $Z$ whose associated $\mathbb{R}$-line bundle is
the pullback of $\mathcal{L}$. Then we put
$\mathbf{M}=\overline{D}-\mathbf{K}-\mathbf{B}$, where $\overline{D}$ is the
$\mathbb{R}$-Cartier closure of $D$ and $\mathbf{K}$ is the canonical
b-divisor of $Z$.

##  7\. Adjunction for codimension two log canonical centers

In this final section, we first discuss basic slc-trivial fibrations under
some extra assumption and then prove adjunction for codimension two log
canonical centers.

######  Theorem 7.1.

Let $f\colon(X,B)\to Y$ be a basic $\mathbb{R}$-slc-trivial fibration. Assume
that there exists a stratum $S$ of $(X,B)$ such that the induced morphism
$S\to Y$ is generically finite and surjective. Then there exists a proper
birational morphism $p\colon Y^{\prime}\to Y$ from a smooth quasi-projective
variety $Y^{\prime}$ such that $\mathbf{M}=\overline{\mathbf{M}_{Y^{\prime}}}$
with $\mathbf{M}_{Y^{\prime}}\sim_{\mathbb{R}}0$. In particular, $\mathbf{M}$
is b-semi-ample.

###### Proof.

By Theorem 5.6, we may assume that $f\colon(X,B)\to Y$ is a basic
$\mathbb{Q}$-slc-trivial fibration. Let $\nu\colon X^{\nu}\to X$ be the
normalization. We define a $\mathbb{Q}$-divisor $B^{\nu}$ on $X^{\nu}$ by
$K_{X^{\nu}}+B^{\nu}=\nu^{*}(K_{X}+B)$. Note that after the reduction we may
find a log canonical center $S$ of $(X^{\nu},B^{\nu})$ such that the induced
morphism $S\to Y$ is generically finite and surjective. By [6, Lemma 4.12], we
may further assume that $Y$ is a complete variety. By replacing $Y$ with a
smooth higher birational model and $f\colon(X,B)\to Y$ with the induced basic
$\mathbb{Q}$-slc-trivial fibration, we may assume that $Y$ is a smooth
projective variety, $\mathbf{M}=\overline{{\mathbf{M}}_{Y}}$, and
${\mathbf{M}}_{Y}$ is nef. The induced morphism $S\to Y$ is denoted by
$f_{S}$. We define a $\mathbb{Q}$-divisor $B_{S}$ on $S$ by
$K_{S}+B_{S}=(K_{X^{\nu}}+B^{\nu})|_{S}$.

From now on, we will show that $-{\mathbf{M}}_{Y}$ is $\mathbb{Q}$-linearly
equivalent to an effective $\mathbb{Q}$-divisor. We consider the divisor
$\nu^{*}f^{*}{\mathbf{M}}_{Y}\sim_{\mathbb{Q}}K_{X^{\nu}}+B^{\nu}-\nu^{*}f^{*}(K_{Y}+\mathbf{B}_{Y})$.
By restricting it to $S$, we get the relation
$f_{S}^{*}{\mathbf{M}}_{Y}\sim_{\mathbb{Q}}K_{S}+B_{S}-f_{S}^{*}(K_{Y}+\mathbf{B}_{Y}).$
Let $g\colon S\to T$ be the Stein factorization of $f_{S}$. The finite
morphism $T\to Y$ is denoted by $f_{T}$. We put $B_{T}=g_{*}B_{S}$. Then the
relation $K_{S}+B_{S}=g^{*}(K_{T}+B_{T})$ holds because $K_{S}+B_{S}$ is
$\mathbb{Q}$-linearly trivial over $Y$. We also have the relation

$f_{T}^{*}{\mathbf{M}}_{Y}\sim_{\mathbb{Q}}K_{T}+B_{T}-f_{T}^{*}(K_{Y}+\mathbf{B}_{Y}).$

To show that $-{\mathbf{M}}_{Y}$ is $\mathbb{Q}$-linearly equivalent to an
effective $\mathbb{Q}$-divisor, it is sufficient to prove that
$-\bigl{(}K_{T}+B_{T}-f_{T}^{*}(K_{Y}+\mathbf{B}_{Y})\bigr{)}$ is
$\mathbb{Q}$-linearly equivalent to an effective $\mathbb{Q}$-divisor.

By the definition of the discriminant $\mathbb{Q}$-b-divisor (see Definition
3.5), for every prime divisor $P$ on $Y$, we have
$\operatorname{coeff}_{P}(\mathbf{B}_{Y})=1-b_{P}$ where $b_{P}$ is the log
canonical threshold of $(X^{\nu},B^{\nu})$ with respect to $\nu^{*}f^{*}P$
over the generic point of $P$. Since $f_{T}$ is finite, we may write
$f_{T}^{*}P=\sum_{Q_{i}}m_{i}Q_{i}$, where $Q_{i}$ runs over prime divisors on
$T$ with $f_{T}(Q_{i})=P$ and $m_{i}$ is the multiplicity of $Q_{i}$ with
respect to $f_{T}$. By the ramification formula, over a neighborhood of the
generic point of $P$ we may write

$\begin{split}f_{T}^{*}(K_{Y}+\mathbf{B}_{Y})=&f_{T}^{*}(K_{Y}+(1-b_{P})P)\\\
=&K_{T}-\sum_{Q_{i}}(m_{i}-1)Q_{i}+(1-b_{P})\sum_{Q_{i}}m_{i}Q_{i}\\\
=&K_{T}+\sum_{Q_{i}}(1-m_{i}b_{P})Q_{i}.\end{split}$

We define
$E:=\sum_{Q_{i}}(\operatorname{coeff}_{Q_{i}}(B_{T})-(1-m_{i}b_{P}))Q_{i}$.
Then, over a neighborhood of the generic point of $P$, we have

$f_{T}^{*}{\mathbf{M}}_{Y}\sim_{\mathbb{Q}}K_{T}+B_{T}-f_{T}^{*}(K_{Y}+\mathbf{B}_{Y})=\sum_{Q_{i}}(\operatorname{coeff}_{Q_{i}}(B_{T})-(1-m_{i}b_{P}))Q_{i}=E.$

On the other hand, by the definition of $b_{P}$ (see Definition 3.5) and the
fact that $S$ is a log canonical center of $(X^{\nu},B^{\nu})$, the pair
$(S,B_{S}+b_{P}f_{S}^{*}P)$ is sub log canonical over the generic point of
$P$. Since $g\colon S\to T$ is birational and
$K_{S}+B_{S}=g^{*}(K_{T}+B_{T})$, the pair $(T,B_{T}+b_{P}f_{T}^{*}P)$ is sub
log canonical over the generic point of $P$. This shows
$\operatorname{coeff}_{Q_{i}}(B_{T})+m_{i}b_{P}\leq 1$ for all $Q_{i}$ such
that $f_{T}(Q_{i})=P$. Thus, $-E$ is effective. Hence $-{\mathbf{M}}_{Y}$ is
$\mathbb{Q}$-linearly equivalent to an effective $\mathbb{Q}$-divisor.

Finally, since ${\mathbf{M}}_{Y}$ is nef, we see that
${\mathbf{M}}_{Y}\sim_{\mathbb{Q}}0$. ∎

We prove the b-semi-ampleness of $\mathbf{M}$ for basic slc-trivial fibrations
of relative dimension one under some extra assumption.

######  Theorem 7.2.

Let $f\colon(X,B)\to Y$ be a basic $\mathbb{R}$-slc-trivial fibration with
$\dim X=\dim Y+1$ such that the horizontal part $B^{h}$ of $B$ is effective.
Then the moduli $\mathbb{R}$-b-divisor $\mathbf{M}$ is b-semi-ample.

###### Proof.

By Theorem 5.6, we may assume that $f\colon(X,B)\to Y$ is a basic
$\mathbb{Q}$-slc-trivial fibration. By [6, Lemma 4.12], we may further assume
that $Y$ is a complete variety. When $X$ is reducible, by the definition of
basic slc-trivial fibrations (see Definition 3.2), there is a stratum $S$ of
$X$ such that the morphism $S\to Y$ is generically finite and surjective since
$\dim X=\dim Y+1$. Thus, we can apply Theorem 7.1. By Theorem 7.1, the moduli
$\mathbb{Q}$-b-divisor $\mathbf{M}$ is b-semi-ample when $X$ is reducible. So
we may assume that $X$ is irreducible. Let $F$ be a general fiber of $f$. Then
$B|_{F}\geq 0$ by the assumption $B^{h}\geq 0$. If $(F,B|_{F})$ is not
kawamata log terminal, then there is a log canonical center $S^{\prime}$ of
$(X,B)$, that is, $S^{\prime}$ is a stratum of $(X,B)$, such that the morphism
$S^{\prime}\to Y$ is generically finite and surjective. As in the reducible
case, by applying Theorem 7.1, we see that the moduli $\mathbb{Q}$-b-divisor
$\mathbf{M}$ is b-semi-ample. If $(F,B|_{F})$ is kawamata log terminal, then
the morphism $f\colon(X,B)\to Y$ satisfies [17, Assumption 7.11]. Therefore,
by [17, Theorem 8.1], the moduli $\mathbb{Q}$-b-divisor $\mathbf{M}$ is
b-semi-ample. In this way, in any case, the moduli $\mathbb{Q}$-b-divisor
$\mathbf{M}$ is b-semi-ample. ∎

By combining Theorem 7.2 with the proof of Theorem 1.2, we obtain the
following result, which generalizes Kawamata’s theorem (see [16, Theorem 1]).

######  Corollary 7.3 (Adjunction and Inversion of Adjunction in codimension
two).

Under the same notation as in Theorem 1.2, we further assume that $\dim W=\dim
X-2$. Then $\mathbf{M}$ is b-semi-ample. Equivalently, $M_{Z^{\prime}}$ is
semi-ample. In particular, there exists an effective $\mathbb{R}$-divisor
$\Delta_{Z}$ on $Z$ such that

  * •

$\nu^{*}(K_{X}+\Delta)\sim_{\mathbb{R}}K_{Z}+\Delta_{Z}$,

  * •

$(Z,\Delta_{Z})$ is log canonical if and only if $(X,\Delta)$ is log canonical
near $W$, and

  * •

$(Z,\Delta_{Z})$ is kawamata log terminal if and only if $(X,\Delta)$ is log
canonical near $W$ and $W$ is a minimal log canonical center of $(X,\Delta)$.

When $K_{X}+\Delta$ is $\mathbb{Q}$-Cartier, we further make $\Delta_{Z}$ an
effective $\mathbb{Q}$-divisor on $Z$ such that
$\nu^{*}(K_{X}+\Delta)\sim_{\mathbb{Q}}K_{Z}+\Delta_{Z}$ in the above
statement.

###### Proof.

We use the same notation as in Theorem 1.2. Note that $W$ is a codimension two
log canonical center of $(X,\Delta)$ by assumption. Let $f\colon Y\to X$ be a
projective birational morphism from a smooth quasi-projective variety $Y$ such
that $K_{Y}+\Delta_{Y}=f^{*}(K_{X}+\Delta)$ and that
$\operatorname{Supp}\Delta_{Y}$ is a simple normal crossing divisor on $Y$.
Without loss of generality, we may assume that $f^{-1}(W)$ is a simple normal
crossing divisor on $Y$ such that $f^{-1}(W)=\sum_{i}E_{i}$ is the irreducible
decomposition. We put

$E=\sum_{a(E_{i},X,\Delta)=-1}E_{i}.$

We define $\Delta_{E}$ by $K_{E}+\Delta_{E}=(K_{Y}+\Delta_{Y})|_{E}$. In this
situation, we can check that $\Delta_{E}$ is effective over the generic point
of $W$. Indeed, if $X$ is a surface then we can check this fact by using the
minimal resolution. In the general case, by shrinking $X$ and cutting $X$ by
general hyperplanes, we can reduce the problem to the case where $X$ is a
surface.

Let $Z$ be the normalization of $W$. By the same arguments as in Steps 1, 2,
and 3 in the proof of Theorem 1.2, we can construct a basic $\mathbb{R}$-slc-
trivial fibration $f\colon(V,\Delta_{V})\to Z$. Then $\dim V=\dim Z+1$ because
$\dim V=\dim X-1$ and $W$ is a codimension two log canonical center of
$(X,\Delta)$. Furthermore, by the discussion in the first paragraph, we see
that the horizontal part $\Delta^{h}_{V}$ of $\Delta_{V}$ with respect to
$f\colon V\to Z$ is effective. By the same arguments as in Steps 4, 5, and 6
in the proof of Theorem 1.2, we get a projective birational morphism $p\colon
Z^{\prime}\to Z$ from a smooth quasi-projective variety $Z^{\prime}$
satisfying (i)–(v) of Theorem 1.2. Moreover, by Theorem 7.2, $\mathbf{M}$ is
b-semi-ample, that is, $\mathbf{M}_{Z^{\prime}}$ is semi-ample.

Let $N\sim_{\mathbb{R}}\mathbf{M}_{Z^{\prime}}$ be a general effective
$\mathbb{R}$-divisor such that $N$ and $\mathbf{B}_{Z^{\prime}}$ have no
common components, $\operatorname{Supp}(N+\mathbf{B}_{Z^{\prime}})$ is a
simple normal crossing divisor on $Z^{\prime}$, and all the coefficients of
$N$ are less than one. We put $\Delta_{Z}=p_{*}N+\mathbf{B}_{Z}$. Then, it is
easy to see that $\Delta_{Z}$ satisfies the desired three conditions of
Corollary 7.3. By the above construction, we can make $\Delta_{Z}$ an
effective $\mathbb{Q}$-divisor such that
$K_{Z}+\Delta_{Z}\sim_{\mathbb{Q}}\nu^{*}(K_{X}+\Delta)$ when $K_{X}+\Delta$
is $\mathbb{Q}$-Cartier. So we are done. ∎

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