# Non-vanishing theorem for generalized log canonical pairs with a
polarization

Kenta Hashizume  Graduate School of Mathematical Sciences, The University of
Tokyo, 3-8-1 Komaba Meguro-ku Tokyo 153-8914, Japan<EMAIL_ADDRESS>
(Date: 2020/12/30)

###### Abstract.

We prove that the non-vanishing conjecture holds for generalized lc pairs with
a polarization.

###### Key words and phrases:

generalized abundance, non-vanishing theorem, generalized lc pairs

###### 2010 Mathematics Subject Classification:

14E30, 14J17, 14J40

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Generalized pairs
    2. 2.2 Property of being log abundant
  3. 3 Generalized abundance for generalized lc pairs
  4. 4 Proof of main results
    1. 4.1 Non-vanishing theorem
    2. 4.2 On the Kodaira type vanishing theorem
  5. 5 Appendix. On non-$\mathbb{R}$-Cartier analogue of non-vanishing theorem

##  1\. Introduction

We will work over an algebraically closed field of characteristic zero.

In this paper, we deal with generalized pairs, which is an extended notion of
log pairs. The notion was introduced by Birkar–Zhang [BZ] to prove the
effectivity of the Iitaka fibration in some sense. In [BZ], Birkar and Zhang
proved effectivity and boundedness of several invariants, called the effective
birationality, the ACC for generalized lc thresholds, and the global ACC.
These results played critical roles to prove the main result of [BZ]. After
that, applying them, Birkar proved the boundedness of complements [B1] and the
boundedness of Fano varieties with a fixed dimension and mild singularity
[B2], which led us to a significant development in birational geometry.
Currently, generalized pairs are also used as powerful tools to prove results
of log pairs ([M], [LT], [B4], [FS]).

Generalized pairs are also interesting objects to study their geometry. The
structure of generalized pairs naturally appears in the base variety of lc-
trivial fibrations or Iitaka fibrations, higher codimensional adjunction
formulas, and cones over lc varieties with nef anti-canonical divisors. A lot
of results in birational geometry, in particular the minimal model program and
the canonical bundle formula, is extended to the context of generalized pairs
([HM], [Fi], [HanLi], [HanLiu1], [HanLiu2], [L]). For the current status and
open problems, see [B3] by Birkar.

In this paper, we study the non-vanishing conjecture for generalized lc pairs.
In the case of log pairs, the non-vanishing conjecture for lc pairs predicts
that pseudo-effective log canonical divisor of an lc pair is effective up to
$\mathbb{R}$-linear equivalence. The conjecture is one of the most important
open problems in minimal model theory. By a similar way to the case of log
pairs, we can define generalized lc pairs, generalized klt pairs, and so on,
then we can formulate generalized lc pairs analogue of the non-vanishing
conjecture. However, unfortunately, a generalized lc pair on an elliptic curve
is a counterexample to the non-vanishing conjecture for generalized lc pairs.
Hence, a weaker version of the conjecture, called weak non-vanishing
conjecture (cf. [BH, Question 3.5]), is expected to hold. In addition to it,
in general, the class of generalized lc pairs are strictly larger than the
class of lc pairs. More precisely, there exists a normal projective variety
having a structure of generalized lc pair but not having any boundary divisor
with which the log pair is lc ([H2, Remark 4.13 (1)]). So the weak non-
vanishing conjecture for generalized lc pairs does not immediately follow from
the non-vanishing conjecture for lc pairs.

Though we have the above obstructions, the following main result of this paper
shows that the non-vanishing conjecture for generalized lc pairs holds true
when we consider generalized lc pairs with a polarization.

######  Theorem 1.1.

Let $(X,B,\boldsymbol{\rm M})$ be a projective generalized lc pair such that
$\boldsymbol{\rm M}$ is a finite $\mathbb{R}_{>0}$-linear combination of b-nef
$\mathbb{Q}$-b-Cartier $\mathbb{Q}$-b-divisors. Let $A$ be an ample
$\mathbb{R}$-divisor on $X$. Suppose that $K_{X}+B+A+\boldsymbol{\rm M}_{X}$
is pseudo-effective. Then followings hold true.

  1. (1)

There exists an effective $\mathbb{R}$-divisor $D$ on $X$ such that
$K_{X}+B+A+\boldsymbol{\rm M}_{X}\sim_{\mathbb{R}}D$.

  2. (2)

Suppose further that $\boldsymbol{\rm M}_{X}$ is $\mathbb{R}$-Cartier. Then,
for every real number $\alpha\geq 1$, there exists an effective
$\mathbb{R}$-divisor $D_{\alpha}$ on $X$ such that
$K_{X}+B+A+\alpha\boldsymbol{\rm M}_{X}\sim_{\mathbb{R}}D_{\alpha}$.

Note that the generalized pair $(X,B,\alpha\boldsymbol{\rm M})$ in Theorem 1.1
(2) may not be generalized lc. So Theorem 1.1 (2) is not a direct consequence
of Theorem 1.1 (1). In the generalized klt case, we can reduce the theorem to
the case of log pairs which follows from [BCHM]. When $\boldsymbol{\rm M}=0$,
Theorem 1.1 is the non-vanishing theorem for lc pairs with a polarization,
which was proved in [HH]. Because of gaps between generalized lc pairs and lc
pairs, Theorem 1.1 does not directly follow from [HH].

We can consider the same assertion as in Theorem 1.1 (2) without the
$\mathbb{R}$-Cartier property of $\boldsymbol{\rm M}_{X}$ because we need not
$\mathbb{R}$-Cartier property to define $\mathbb{R}$-linear equivalence. It is
not clear that the $\mathbb{R}$-Cartier property of $\boldsymbol{\rm M}_{X}$
in Theorem 1.1 (2) can be removed, but a partial affirmative answer to the
question will be given by Theorem 4.4. See Section 5 for the question of this
direction and a related topic. As a special case of Theorem 4.4 we introduce a
result for projective generalized lc pairs $(X,B,\boldsymbol{\rm M})$ with a
polarization $A$ satisfying “pseudo-effectivity of log canonical part
$K_{X}+B+A$” in some sense. In [LP1] and [LP2], Lazić and Peternell discussed
on generalized klt pairs (without polarizations) under the assumption on the
pseudo-effectivity of the log canonical part. Especially, in [LP1], they
proved connections between conjectures of minimal model theory for klt pairs
and those for generalized klt pairs. A conjecture called “Generalised
Nonvanishing Conjecture” was formulated in [LP1], and we prove a variant of
the conjecture.

######  Theorem 1.2 (see also Theorem 4.4).

Let $(X,B,\boldsymbol{\rm M})$ be a projective generalized lc pair such that
$\boldsymbol{\rm M}$ is a finite $\mathbb{R}_{>0}$-linear combination of b-nef
$\mathbb{Q}$-b-Cartier $\mathbb{Q}$-b-divisors. Let $A$ be an ample
$\mathbb{R}$-divisor on $X$. Suppose that $K_{X}+B+A$ is pseudo-effective in
the sense of Definition 4.2 (i.e., for every ample $\mathbb{R}$-divisor $H$ on
$X$, the divisor $K_{X}+B+A+H$ is $\mathbb{R}$-linearly equivalent to an
effective $\mathbb{R}$-divisor).

Then, for every real number $\alpha\geq 0$, there exists an effective
$\mathbb{R}$-divisor $D_{\alpha}$ on $X$ such that
$K_{X}+B+A+\alpha\boldsymbol{\rm M}_{X}\sim_{\mathbb{R}}D_{\alpha}$ as (not
necessarily $\mathbb{R}$-Cartier) $\mathbb{R}$-divisors.

When we consider $\mathbb{R}$-Cartier divisors, the pseudo-effectivity of
Definition 4.2 coincides with the usual pseudo-effectivity. In Theorem 1.2,
the pseudo-effectivity of $K_{X}+B+A$ holds, for example, when there is a
resolution $f\colon\tilde{X}\to X$ and an effective $f$-exceptional
$\mathbb{R}$-divisor $\tilde{E}$ on $\tilde{X}$ such that
$f^{*}(K_{X}+B+\boldsymbol{\rm M}_{X}+A)-\boldsymbol{\rm
M}_{\tilde{X}}+\tilde{E}$ is pseudo-effective in the usual sense.

Although the above theorems are concerned with only the non-vanishing theorem,
we can prove a stronger property (see Theorem 4.1 for more general setup).

######  Theorem 1.3 (see also Theorem 4.1).

Let $(X,B,\boldsymbol{\rm M})$ be a projective generalized lc pair such that
$\boldsymbol{\rm M}$ is a finite $\mathbb{R}_{>0}$-linear combination of b-nef
$\mathbb{Q}$-b-Cartier $\mathbb{Q}$-b-divisors. Let $A$ be an effective ample
$\mathbb{R}$-divisor on $X$ such that $(X,B+A,\boldsymbol{\rm M})$ is a
generalized lc pair and generalized lc centers of $(X,B+A,\boldsymbol{\rm M})$
are generalized lc centers of $(X,B,\boldsymbol{\rm M})$. Let
$(Y,\Gamma,\boldsymbol{\rm M})$ be a $\mathbb{Q}$-factorial dlt model of
$(X,B+A,\boldsymbol{\rm M})$.

Then, for any sequence of steps of a $(K_{Y}+\Gamma+\boldsymbol{\rm
M}_{Y})$-MMP

$(Y,\Gamma,\boldsymbol{\rm M})=:(Y_{0},\Gamma_{0},\boldsymbol{\rm
M})\dashrightarrow\cdots\dashrightarrow(Y_{i},\Gamma_{i},\boldsymbol{\rm
M})\dashrightarrow\cdots,$

the divisor $K_{Y_{i}}+\Gamma_{i}+\boldsymbol{\rm M}_{Y_{i}}$ is log abundant
with respect to $(Y_{i},\Gamma_{i},\boldsymbol{\rm M})$ for every $i\geq 0$.

It is important to consider the situation of Theorem 1.3 because the situation
is deeply concerned with the existence of flips for $(K_{X}+B+\boldsymbol{\rm
M}_{X})$-flipping contractions. Property of being log abundant for
$\mathbb{R}$-divisors is much stronger than effectivity up to
$\mathbb{R}$-linear equivalence (Subsection 2.2), therefore Theorem 1.3 gives
a partial answer to [BH, Question 3.5] by Birkar and Hu. In the case of log
pairs, the existence of log MMP with scaling of an ample divisor preserving
property of being log abundant implies the existence of a log minimal model
([H3, Corollary 1.2]). We believe that Theorem 1.3 and the arguments in [HH,
Section 5] (or [H3]) with the aid of [HanLi] by Han–Li will give a proof of
the existence of a minimal model in the setting of Theorem 1.3. On the other
hand, the abundance conjecture in the setting of Theorem 1.3 seems difficult
because the Kodaira type vanishing theorem ([Fu2], [Fu3]) and the finiteness
of pluri-canonical representation ([FG], [HX]) for generalized lc pairs are
still widely open.

Finally, we give a remark on Kodaira type vanishing theorem for generalized
pairs. As a generalization of [Fu4, Theorem 5.6.4], for all projective
generalized lc pairs $(X,B,\boldsymbol{\rm M})$ and $\mathbb{Q}$-Cartier Weil
divisors $D$ on $X$ such that $D-(K_{X}+B+\boldsymbol{\rm M}_{X})$ is ample,
we may expect that $H^{i}(X,\mathcal{O}_{X}(D))$ vanishes for every $i>0$.
This problem will be an important step toward the cone and contraction theorem
for generalized lc pairs. Since the non-vanishing theorem for generalized lc
pairs holds in the situation of Theorem 1.1 (2), it is natural to expect that
the Kodaira type vanishing theorem for generalized lc pairs also holds in the
situation as in Theorem 1.1 (2) or Theorem 1.2. Unfortunately, the following
result shows that the expectation cannot be realized in general.

######  Theorem 1.4 (see Theorem 4.6).

There is a projective $\mathbb{Q}$-factorial generalized klt pair
$(X,B,\boldsymbol{\rm M})$ and a Cartier divisor $D$ on $X$ satisfying the
following property: There is a rational number $t>1$ such that
$D-(K_{X}+B+t\boldsymbol{\rm M}_{X})$ is ample and
$H^{1}(X,\mathcal{O}_{X}(D))\neq\\{0\\}$.

The contents of this paper are as follows: In Section 2, we collect definition
and basic properties of generalized pairs, and we define property of being log
abundant. In Section 3, we study the generalized abundance for generalized lc
pairs. In Section 4, we prove theorems 1.1–1.4. In Section 5, which is an
appendix, we give a small remark on a non-$\mathbb{R}$-Cartier analogue of
Theorem 1.1 (2).

######  Acknowledgments.

The author was partially supported by JSPS KAKENHI Grant Number JP16J05875 and
JP19J00046. Part of the work was done while the author was visiting University
of Cambridge in October–November 2018 and January 2020. The author thanks
staffs of the university, Professor Caucher Birkar, Doctor Roberto Svaldi, and
Doctor Yanning Xu for their hospitality. He is grateful to Professor Caucher
Birkar, Professor Yoshnori Gongyo, and Doctor Yanning Xu for discussions and
giving him advice. He thanks Professors Osamu Fujino and Yoshinori Gongyo for
comments. He thanks Doctor Sho Ejiri for answering questions.

##  2\. Preliminaries

######  Notation 2.1.

Throughout this paper, $\mathbb{R}$-divisors on varieties are not assumed to
be $\mathbb{R}$-Cartier. In particular, big $\mathbb{R}$-divisors are not
necessarily $\mathbb{R}$-Cartier. In this paper, we use non-negative integers
as subscripts of varieties. For instance, varieties indexed by non-negative
integers are denoted by $\tilde{X}_{i}$ with $i\in\mathbb{Z}_{\geq 0}$. For
any variety $W$, $\mathbb{R}$-divisor $D$ on $W$ and birational maps
$W\dashrightarrow W^{\prime}$ and $W\dashrightarrow W_{i}$, if there is no
risk of confusion, the birational transform of $D$ on $W^{\prime}$ (resp.
$W_{i}$) is denoted by $D^{\prime}$ (resp. $D_{i}$).

###  2.1. Generalized pairs

In this subsection, we collect definition and properties of generalized pairs.

######  b-divisors.

We define b-divisors.

######  Definition 2.2 (Birational model, divisors over varieties).

Let $X$ be a normal variety. We call a normal variety $Y$ with a projective
birational morphism $Y\to X$ a birational model. When we want to mention the
birational morphism $Y\to X$ explicitly, we call $Y\to X$ a birational model.
A prime divisor $P$ over $X$ is a prime divisor on a birational model $Y$. For
any prime divisor $P$ over $X$, we denote the image of $P$ on $X$ by
$c_{X}(P)$. For any prime divisor $P$ over $X$, let $v_{P}$ be the
corresponding divisorial valuation. For any prime divisor $P$ on a birational
model $Y\to X$ and $P^{\prime}$ on a birational model $Y^{\prime}\to X$, the
divisorial valuations $v_{P}$ and $v_{P^{\prime}}$ coincide if and only if the
induced birational map $Y\dashrightarrow Y^{\prime}$ is an isomorphism on a
generic point of $P$ and the birational transform of $P$ on $Y^{\prime}$ is
equal to $P^{\prime}$.

######  Definition 2.3 (b-divisors).

Let $X$ be a normal variety. Then an $\mathbb{R}$-b-divisor $\boldsymbol{\rm
D}$ on $X$ is a (possibly infinitely many) $\mathbb{R}$-linear combination of
divisorial valuations $v_{P}$

$\boldsymbol{\rm D}:=\sum_{\begin{subarray}{c}P:{\rm\,prime\,divisor}\\\ {\rm
over\,}X\end{subarray}}r_{P}v_{P}\quad(r_{P}\in\mathbb{R})$

such that the set $\\{P\;|\;\text{$r_{P}\neq 0$ and $c_{X}(P)$ is a prime
divisor on $Y$}\\}$ is a finite set for all birational models $Y\to X$. When
all $r_{P}$ are rational numbers, we call $\boldsymbol{\rm D}$ a
$\mathbb{Q}$-b-divisor. For a given birational model $Y$, the trace of an
$\mathbb{R}$-b-divisor $\boldsymbol{\rm D}=\sum_{P}r_{P}v_{P}$ on $Y$, which
we denote $\boldsymbol{\rm D}_{Y}$, is defined by

$\boldsymbol{\rm
D}_{Y}:=\sum_{\begin{subarray}{c}c_{Y}(P){\rm\,is\;a\;divisor}\\\ {\rm
on\,}Y\end{subarray}}r_{P}c_{Y}(P).$

By definition, $\boldsymbol{\rm D}_{Y}$ is an $\mathbb{R}$-divisor on $Y$.

Let $\boldsymbol{\rm D}$ be an $\mathbb{R}$-b-divisor on $X$. If there exists
an $\mathbb{R}$-Cartier divisor $D$ on a birational model $Y$ such that

$\boldsymbol{\rm D}=\sum_{\begin{subarray}{c}P:{\rm\,prime\,divisor}\\\ {\rm
over\,}Y\end{subarray}}{\rm ord}_{P}(D)\cdot v_{P},$

then, we call $\boldsymbol{\rm D}$ an $\mathbb{R}$-b-Cartier
$\mathbb{R}$-b-divisor, we say that $\boldsymbol{\rm D}$ descends to $Y$ or
$\boldsymbol{\rm D}$ is the closure of $D$, and we write $\boldsymbol{\rm
D}=\overline{D}$. In this situation, we have $\boldsymbol{\rm D}_{Y}=D$. When
$\boldsymbol{\rm D}=\overline{D}$ and $D$ is $\mathbb{Q}$-Cartier, we call
$\boldsymbol{\rm D}$ a $\mathbb{Q}$-b-Cartier $\mathbb{Q}$-b-divisor. When
$\boldsymbol{\rm D}=\overline{D}$ and $X$ has a projective morphism $X\to Z$
to a variety $Z$ such that $D$ is nef over $Z$, we say that $\boldsymbol{\rm
D}$ is a b-nef$/Z$ $\mathbb{R}$-b-Cartier $\mathbb{R}$-b-divisor.

######  Remark 2.4.

Let $X$ be a normal variety and $\boldsymbol{\rm D}$ an $\mathbb{R}$-b-divisor
on $X$. Let $\phi\colon X\dashrightarrow X^{\prime}$ be a birational map to a
normal variety $X^{\prime}$. If there are proper birational morphisms $f\colon
Y\to X$ and $f^{\prime}\colon Y\to X^{\prime}$ such that $f^{\prime}=\phi\circ
f$, then we may think of $\boldsymbol{\rm D}$ as an $\mathbb{R}$-b-divisor on
$X^{\prime}$. If $\boldsymbol{\rm D}$ is an $\mathbb{R}$-b-Cartier (resp.
b-nef $\mathbb{R}$-b-Cartier) $\mathbb{R}$-b-divisor on $X$, then
$\boldsymbol{\rm D}$ which we see as an $\mathbb{R}$-b-divisor on $X^{\prime}$
is an $\mathbb{R}$-b-Cartier (resp. b-nef $\mathbb{R}$-b-Cartier)
$\mathbb{R}$-b-divisor. Similar facts hold true in the framework of
$\mathbb{Q}$-b-divisors.

######  Singularities of pairs.

A pair $(X,B)$ consists of a normal variety $X$ and effective
$\mathbb{R}$-divisor $B$ on $X$ such that $K_{X}+B$ is $\mathbb{R}$-Cartier.

Let $(X,B)$ be a pair and $P$ a prime divisor over $X$. Then $a(P,X,B)$
denotes the log discrepancy of $P$ with respect to $(X,B)$. A pair $(X,B)$ is
called a Kawamata log terminal (klt, for short) pair if $a(P,X,B)>0$ for all
prime divisors $P$ over $X$. A pair $(X,B)$ is called a log canonical (lc, for
short) pair if $a(P,X,B)\geq 0$ for all prime divisors $P$ over $X$. A pair
$(X,B)$ is called a divisorially log terminal (dlt, for short) pair if all
coefficients of $B$ belong to $[0,1]$ and there is a log resolution of
$(X,{\rm Supp}B)$ whose exceptional divisors $E$ satisfy $a(E,X,B)>0$. When
$(X,B)$ is an lc pair, an lc center of $(X,B)$ is the image on $X$ of a prime
divisor $P$ over $X$ which satisfies $a(P,X,B)=0$.

######  Generalized pairs.

We define generalized pairs and singularities of generalized pairs. In [BZ,
Definition 1.4] the boundary part of generalized pairs are assumed to have
coefficients in $[0,1]$, but, in Definition 2.5 below we only assume that the
boundary parts are effective divisors.

######  Definition 2.5 (Singularities of generalized pairs).

A generalized pair $(X,B,\boldsymbol{\rm M})/Z$ consists of a projective
morphism $X\to Z$ from a normal variety to a variety, an effective
$\mathbb{R}$-divisor $B$ on $X$, and a b-nef$/Z$ $\mathbb{R}$-b-Cartier
$\mathbb{R}$-b-divisor $\boldsymbol{\rm M}$ on $X$ such that
$K_{X}+B+\boldsymbol{\rm M}_{X}$ is $\mathbb{R}$-Cartier. When
$\boldsymbol{\rm M}=0$, the generalized pair $(X,B,\boldsymbol{\rm M})/Z$
becomes a pair. When $Z$ is a point, we simply denote $(X,B,\boldsymbol{\rm
M})$.

Let $(X,B,\boldsymbol{\rm M})/Z$ be a generalized pair and $P$ a prime divisor
over $X$. Let $f\colon\tilde{X}\to X$ be a projective birational morphism such
that $\boldsymbol{\rm M}=\overline{\boldsymbol{\rm M}_{\tilde{X}}}$ and $P$
appears as a divisor on $\tilde{X}$. Then there is an $\mathbb{R}$-divisor
$\tilde{B}$ on $\tilde{X}$ such that

$K_{\tilde{X}}+\tilde{B}+\boldsymbol{\rm
M}_{\tilde{X}}=f^{*}(K_{X}+B+\boldsymbol{\rm M}_{X}).$

Then the generalized log discrepancy $a(P,X,B+\boldsymbol{\rm M}_{X})$ of $P$
with respect to $(X,B,\boldsymbol{\rm M})/Z$ is defined by
$a(P,X,B+\boldsymbol{\rm M}_{X})=1-{\rm coeff}_{P}(\tilde{B})$. The
generalized log discrepancy does not depend on the choice of
$f\colon\tilde{X}\to X$. When $\boldsymbol{\rm M}=0$, the generalized log
discrepancies coincide with the log discrepancies of the pair $(X,B)$.

A generalized pair $(X,B,\boldsymbol{\rm M})/Z$ is called a generalized klt
(resp. generalized lc) pair if $a(P,X,B+\boldsymbol{\rm M}_{X})>0$ (resp.
$a(P,X,B+\boldsymbol{\rm M}_{X})\geq 0$) for all prime divisors $P$ over $X$.
A generalized lc center of $(X,B,\boldsymbol{\rm M})/Z$ is the image on $X$ of
a prime divisor $P$ over $X$ satisfying $a(P,X,B+\boldsymbol{\rm M}_{X})=0$.

A generalized pair $(X,B,\boldsymbol{\rm M})/Z$ is a $\mathbb{Q}$-factorial
generalized dlt pair if $X$ is $\mathbb{Q}$-factorial, $(X,B)$ is a dlt pair,
and $(X,B,(1+t)\boldsymbol{\rm M})/Z$ is generalized lc for some real number
$t>0$.

For a given $(X,B,\boldsymbol{\rm M})/Z$, we call $(Y,\Gamma,\boldsymbol{\rm
M})/Z$ in Theorem 2.6 below a $\mathbb{Q}$-factorial dlt model of
$(X,B,\boldsymbol{\rm M})/Z$.

######  Theorem 2.6 (Existence of a $\mathbb{Q}$-factorial generalized dlt
model, cf. [BZ, Lemma 4.5]).

Let $(X,B,\boldsymbol{\rm M})/Z$ be a generalized lc pair such that $Z$ is
quasi-projective. Then, there exists a projective birational morphism $f\colon
Y\to X$ and a $\mathbb{Q}$-factorial generalized dlt pair
$(Y,\Gamma,\boldsymbol{\rm M})/Z$ such that $K_{Y}+\Gamma+\boldsymbol{\rm
M}_{Y}=f^{*}(K_{X}+B+\boldsymbol{\rm M}_{X})$ and $\Gamma$ is the sum of
$f^{-1}_{*}B$ and the reduced $f$-exceptional divisor.

######  Remark 2.7.

There are two remarks on $\mathbb{Q}$-factorial generalized dlt pairs.

  1. (1)

The above definition of $\mathbb{Q}$-factorial generalized dlt pairs coincides
with the definition of generalized dlt pairs in [B1, 2.13 (2)] and [HanLi,
Definition 2.2] when $X$ is $\mathbb{Q}$-factorial. In this paper, we always
assume generalized dlt pairs to be $\mathbb{Q}$-factorial.

  2. (2)

Let $(X,B,\boldsymbol{\rm M})/Z$ be a generalized lc pair such that $Z$ is
quasi-projective. For every prime divisor $P$ over $X$ such that
$a(P,X,B+\boldsymbol{\rm M}_{X})=0$, [BZ, Lemma 4.5] shows that there is a
$\mathbb{Q}$-factorial generalized dlt model $(Y,\Gamma,\boldsymbol{\rm
M})\to(X,B,\boldsymbol{\rm M})$ on which $P$ appears as a divisor. In
particular, for any lc center $S$ of $(X,B,\boldsymbol{\rm M})/Z$, we can
construct a $\mathbb{Q}$-factorial generalized dlt model
$(Y,\Gamma,\boldsymbol{\rm M})$ such that $X^{\prime}\to X$ induces a
surjective morphism $T\to S$ from a component $T$ of $\llcorner
B^{\prime}\lrcorner$.

######  Lemma 2.8.

Let $(X,B,\boldsymbol{\rm M})/Z$ be a generalized lc pair, and let
$f\colon\tilde{X}\to X$ be a log resolution of $(X,{\rm SuppB})$ such that
${\boldsymbol{\rm M}}=\overline{\boldsymbol{\rm M}_{\tilde{X}}}$. We define
$\tilde{B}\geq 0$ and $\tilde{E}\geq 0$ on $\tilde{X}$ by

$K_{\tilde{X}}+\tilde{B}+\boldsymbol{\rm
M}_{\tilde{X}}=f^{*}(K_{X}+B+\boldsymbol{\rm M}_{X})+\tilde{E}$

such that $\tilde{B}$ and $\tilde{E}$ have no common components. Let $g\colon
Y\to\tilde{X}$ be a projective birational morphism from a smooth variety $Y$.
We define $\Gamma\geq 0$ and $F\geq 0$ on $Y$ by
$K_{Y}+\Gamma=g^{*}(K_{\tilde{X}}+\tilde{B})+F$ such that $\Gamma$ and $F$
have no common components.

Then $\Gamma$ and $g^{*}\tilde{E}+F$ have no common components. We note that
we have

$K_{Y}+\Gamma+\boldsymbol{\rm M}_{Y}=(f\circ g)^{*}(K_{X}+B+\boldsymbol{\rm
M}_{X})+g^{*}\tilde{E}+F$

by construction.

###### Proof.

By hypothesis, $\Gamma$ and $F$ have no common components. Hence, it is
sufficient to show that $\Gamma$ and $g^{*}\tilde{E}$ have no common
components. Since $\tilde{B}$ and $\tilde{E}$ have no common components, by
log smoothness of $\bigl{(}\tilde{X},{\rm Supp}(\tilde{B}+\tilde{E})\bigr{)}$
and computations of discrepancies as in [KM, Lemma 2.29] and [KM, Lemma 2.45],
for any prime divisor $P$ over $\tilde{X}$ if the center on $\tilde{X}$ is
contained in ${\rm Supp}\tilde{E}$ then $a(P,\tilde{X},\tilde{B})\geq 1$.
Since the log discrepancies of every component of $\Gamma$ with respect to
$(\tilde{X},\tilde{B})$ is less than $1$, we see that $\Gamma$ and
$g^{*}\tilde{E}$ have no common components. Therefore, $\Gamma$ and
$g^{*}\tilde{E}+F$ have no common components. ∎

######  Divisorial adjunction for generalized pairs.

We recall construction of a generalized pair by divisorial adjunction for
generalized lc pairs ([BZ, Definition 4.7]).

Let $(X,B,\boldsymbol{\rm M})/Z$ be a generalized lc pair, and let $S$ be a
component of $\llcorner B\lrcorner$ with the normalization $S^{\nu}\to S$. Let
$f\colon\tilde{X}\to X$ be a log resolution of $(X,{\rm Supp}B)$ such that
$\boldsymbol{\rm M}=\overline{\boldsymbol{\rm M}_{\tilde{X}}}$. We can write

$K_{\tilde{X}}+\tilde{B}+\boldsymbol{\rm
M}_{\tilde{X}}=f^{*}(K_{X}+B+\boldsymbol{\rm M}_{X})$

with an $\mathbb{R}$-divisor $\tilde{B}$ on $\tilde{X}$. Put
$\tilde{S}=f^{-1}_{*}S$ which is a component of $\llcorner\tilde{B}\lrcorner$.
Pick $\tilde{M}\sim_{\mathbb{R}}\boldsymbol{\rm M}_{\tilde{X}}$ so that
$\tilde{M}|_{\tilde{S}}$ is well-defined as an $\mathbb{R}$-divisor on
$\tilde{S}$. We define a b-nef$/Z$ $\mathbb{R}$-b-divisor $\boldsymbol{\rm N}$
on $S^{\nu}$ by the closure of $\tilde{M}|_{\tilde{S}}$, and we set
$B_{S^{\nu}}$ as the birational transform of
$(\tilde{B}-\tilde{S})|_{\tilde{S}}$ by the induced birational morphism
$f_{\tilde{S}}\colon\tilde{S}\to S^{\nu}$. Then it follows that
$(S^{\nu},B_{S^{\nu}},\boldsymbol{\rm N})/Z$ is a generalized pair and

$K_{\tilde{S}}+(\tilde{B}-\tilde{S})|_{\tilde{S}}+\boldsymbol{\rm
N}_{\tilde{S}}=f_{\tilde{S}}^{*}(K_{S^{\nu}}+B_{S^{\nu}}+\boldsymbol{\rm
N}_{S^{\nu}}).$

Note that $\boldsymbol{\rm N}$ is determined up to $\mathbb{R}$-linear
equivalence because $\boldsymbol{\rm N}$ depends on $\tilde{M}$.

######  Lemma 2.9.

Let $(X,B,\boldsymbol{\rm M})$ be a projective $\mathbb{Q}$-factorial
generalized dlt pair, and let $f\colon\tilde{X}\to X$ be a log resolution of
$(X,{\rm Supp}B)$ such that ${\boldsymbol{\rm M}}=\overline{\boldsymbol{\rm
M}_{\tilde{X}}}$. Let $S$ be a component of $\llcorner B\lrcorner$. We define
$\tilde{B}\geq 0$ and $\tilde{E}\geq 0$ on $\tilde{X}$ by
$K_{\tilde{X}}+\tilde{S}+\tilde{B}+\boldsymbol{\rm
M}_{\tilde{X}}=f^{*}(K_{X}+B+\boldsymbol{\rm M}_{X})+\tilde{E}$ such that
$\tilde{S}=f_{*}^{-1}S$ and $\tilde{B}$, $\tilde{E}$, and $\tilde{S}$ have no
common components one another. We define $\mathbb{R}$-divisors
$B_{\tilde{S}}$, $B_{S}$, $M_{\tilde{S}}$, and $M_{S}$ by divisorial
adjunction for generalized lc pairs. In other words, we define
$B_{\tilde{S}}=\tilde{B}|_{\tilde{S}}$, $B_{S}=f|_{\tilde{S}*}B_{\tilde{S}}$,
$M_{\tilde{S}}=\tilde{M}|_{\tilde{S}}$ for some
$\tilde{M}\sim_{\mathbb{R}}\boldsymbol{\rm M}_{\tilde{X}}$, and
$M_{S}=f|_{\tilde{S}*}M_{\tilde{S}}$.

Then $(S,B_{S},\overline{M_{\tilde{S}}})$ and
$(\tilde{S},\tilde{B},\overline{M_{\tilde{S}}})$ are generalized lc pairs.
Furthermore, we have

$a(P,S,B_{S}+M_{S})=a(P,\tilde{S},B_{\tilde{S}}+M_{\tilde{S}})$

for all prime divisors $P$ on $S$.

###### Proof.

By definition of $B_{\tilde{S}}$ and $M_{\tilde{S}}$, we have
$K_{\tilde{S}}+B_{\tilde{S}}+M_{\tilde{S}}=f|_{\tilde{S}}^{*}(K_{X}+B+\boldsymbol{\rm
M}_{X})|_{S}+\tilde{E}|_{\tilde{S}}.$ The first assertion follows from [BZ,
Remark 4.8]. For the second assertion, the proof of [HH, Lemma 2.4] works with
no changes. More precisely, by the same argument as in [HH, Proof of Lemma
2.4], we see that $B_{\tilde{S}}$ and $\tilde{E}|_{\tilde{S}}$ have no common
component and $\tilde{E}|_{\tilde{S}}$ is exceptional over $S$. Then the
second assertion follows from the definition of log discrepancies for
generalized pairs. ∎

######  MMP for generalized pairs.

Let $(X,B,\boldsymbol{\rm M})/Z$ be a $\mathbb{Q}$-factorial generalized lc
pair such that $Z$ is quasi-projective and $(X,0)$ is a klt pair. As in [BZ,
Section 4], we can run a $(K_{X}+B+\boldsymbol{\rm M}_{X})$-MMP over $Z$ with
scaling of an ample divisor. We often denote the sequence of the MMP by

$(X,B,\boldsymbol{\rm M})=:(X_{0},B_{0},\boldsymbol{\rm
M})\dashrightarrow\cdots\dashrightarrow(X_{i},B_{i},\boldsymbol{\rm
M})\dashrightarrow\cdots.$

Here, for each $i$ we think of $\boldsymbol{\rm M}$ as an
$\mathbb{R}$-b-divisor on $X_{i}$.

For fundamental results of MMP for generalized pairs, see [HanLi]. In this
paper, we will freely use results in [HanLi].

###  2.2. Property of being log abundant

In this subsection, we introduce the notion of being log abundant for
generalized pairs.

######  Definition 2.10 (Invariant Iitaka dimension).

Let $X$ be a normal projective variety, and let $D$ be an $\mathbb{R}$-Cartier
divisor on $X$. We define the invariant Iitaka dimension of $D$, denoted by
$\kappa_{\iota}(X,D)$, as follows ([C, Definition 2.2.1], see also [Fu4,
Definition 2.5.5]): If there is an $\mathbb{R}$-divisor $E\geq 0$ such that
$D\sim_{\mathbb{R}}E$, set $\kappa_{\iota}(X,D)=\kappa(X,E)$. Here, the right
hand side is the usual Iitaka dimension of $E$. Otherwise, we set
$\kappa_{\iota}(X,D)=-\infty$. We can check that $\kappa_{\iota}(X,D)$ is
well-defined, i.e., when there is $E\geq 0$ such that $D\sim_{\mathbb{R}}E$,
the invariant Iitaka dimension $\kappa_{\iota}(X,D)$ does not depend on the
choice of $E$. By definition, we have $\kappa_{\iota}(X,D)\geq 0$ if and only
if $D$ is $\mathbb{R}$-linearly equivalent to an effective
$\mathbb{R}$-divisor.

######  Definition 2.11 (Numerical dimension).

Let $X$ be a normal projective variety, and let $D$ be an $\mathbb{R}$-Cartier
divisor on $X$. We define the numerical dimension of $D$, denoted by
$\kappa_{\sigma}(X,D)$, as follows ([N, V, 2.5 Definition]): For any Cartier
divisor $A$ on $X$, we set

$\sigma(D;A)={\rm max}\\!\Set{\\!k\in\mathbb{Z}_{\geq
0}}{\underset{m\to\infty}{\rm lim\,sup}\frac{{\rm
dim}H^{0}(X,\mathcal{O}_{X}(\llcorner mD\lrcorner+A))}{m^{k}}>0\\!}$

if ${\rm dim}H^{0}(X,\mathcal{O}_{X}(\llcorner mD\lrcorner+A))>0$ for
infinitely many $m\in\mathbb{Z}_{>0}$, and otherwise we set
$\sigma(D;A):=-\infty$. Then, we define

$\kappa_{\sigma}(X,D):={\rm max}\\!\set{\sigma(D;A)}{\text{$A$ is a Cartier
divisor on $X$}\\!}.$

See [HH, Remark 2.8] for basic properties of the invariant Iitaka dimension
and the numerical dimension.

######  Definition 2.12 (Abundant divisors and log abundant divisors).

Let $X$ be a normal projective variety, and let $D$ be an $\mathbb{R}$-Cartier
divisor on $X$. We say that $D$ is abundant if the equality
$\kappa_{\iota}(X,D)=\kappa_{\sigma}(X,D)$ holds.

Let $X$ and $D$ be as above, and let $(X,B,\boldsymbol{\rm M})$ be a
generalized lc pair. We say that $D$ is log abundant with respect to
$(X,B,\boldsymbol{\rm M})$ if $D$ is abundant and for any lc center $S$ of
$(X,B,\boldsymbol{\rm M})$ with the normalization $S^{\nu}\to S$, the pullback
$D|_{S^{\nu}}$ is abundant.

We can define the relative version of the invariant Iitaka dimension, the
numerical dimension, and the property of being log abundant ([H3, Section 5]).
But we will not use them in this paper.

##  3\. Generalized abundance for generalized lc pairs

######  Lemma 3.1.

Let $(X,B,\boldsymbol{\rm M})$ be a projective generalized lc pair. Let
$\phi\colon X\dashrightarrow X^{\prime}$ be a birational contraction to a
normal projective variety $X^{\prime}$, and let $f\colon\tilde{X}\to X$ be a
log resolution of $(X,{\rm Supp}B)$ such that $\boldsymbol{\rm
M}=\overline{\boldsymbol{\rm M}_{\tilde{X}}}$ and the induced birational map
$\tilde{X}\to X^{\prime}$ is a morphism. Let
$(\tilde{X},\tilde{B},\boldsymbol{\rm M})$ be a generalized lc pair. Suppose
that

  * •

$K_{X^{\prime}}+\phi_{*}B+\boldsymbol{\rm M}_{X^{\prime}}$ is
$\mathbb{R}$-Cartier,

  * •

$a(P,X,B+\boldsymbol{\rm M}_{X})\leq a(P,X^{\prime},\phi_{*}B+\boldsymbol{\rm
M}_{X^{\prime}})$ for all prime divisors $P$ on $\tilde{X}$, and

  * •

the relation $K_{\tilde{X}}+\tilde{B}+\boldsymbol{\rm
M}_{\tilde{X}}=f^{*}(K_{X}+B+\boldsymbol{\rm M}_{X})+\tilde{E}$ holds for an
effective $f$-exceptional $\mathbb{R}$-divisor $\tilde{E}$.

Then, by running a $(K_{\tilde{X}}+\tilde{B}+\boldsymbol{\rm
M}_{\tilde{X}})$-MMP over $X^{\prime}$ with scaling of an ample divisor we get
a projective birational morphism $f^{\prime}\colon\tilde{X}^{\prime}\to
X^{\prime}$ such that

$K_{\tilde{X}^{\prime}}+\tilde{B}^{\prime}+\boldsymbol{\rm
M}_{\tilde{X}^{\prime}}=f^{\prime*}(K_{X^{\prime}}+\phi_{*}B+\boldsymbol{\rm
M}_{X^{\prime}}),$

where $\tilde{B}^{\prime}$ is the birational transforms of $\tilde{B}$ on
$\tilde{X}^{\prime}$.

###### Proof.

We denote $\tilde{X}\to X^{\prime}$ by $g$. By the hypothesis of relation on
discrepancies of the generalized pairs, there is a $g$-exceptional
$\mathbb{R}$-divisor $\tilde{F}\geq 0$ on $\tilde{X}$ such that

$K_{\tilde{X}}+\tilde{B}+\boldsymbol{\rm
M}_{\tilde{X}}=f^{*}(K_{X}+B+\boldsymbol{\rm
M}_{X})+\tilde{E}=g^{*}(K_{X^{\prime}}+\phi_{*}B+\boldsymbol{\rm
M}_{X^{\prime}})+\tilde{E}+\tilde{F}.$

By construction, $\tilde{E}+\tilde{F}$ is effective and $g$-exceptional. By
[HanLi, Proposition 3.8] and running a
$(K_{\tilde{X}}+\tilde{B}+\boldsymbol{\rm M}_{\tilde{X}})$-MMP over
$X^{\prime}$ with scaling of an ample divisor, we get a birational contraction
$\tilde{X}\dashrightarrow\tilde{X}^{\prime}$ over $X^{\prime}$ which contracts
$\tilde{E}+\tilde{F}$. Then it is clear that the morphism
$\tilde{X}^{\prime}\to X^{\prime}$ is the desired one. ∎

######  Proposition 3.2.

Let $(X,B,\boldsymbol{\rm M})/Z$ be a generalized lc pair such that $Z$ is
quasi-projective. Let $f\colon\tilde{X}\to X$ be a log resolution of $(X,{\rm
Supp}B)$ such that ${\boldsymbol{\rm M}}=\overline{\boldsymbol{\rm
M}_{\tilde{X}}}$, and let $\tilde{B}$ be an $\mathbb{R}$-divisor on
$\tilde{X}$ such that $(\tilde{X},\tilde{B})$ is a log smooth lc pair and
$f_{*}\tilde{B}=B$. Suppose that

  * •

$K_{\tilde{X}}+\tilde{B}+\boldsymbol{\rm M}_{\tilde{X}}$ is pseudo-effective
over $Z$,

  * •

$K_{X}+B+\boldsymbol{\rm M}_{X}\sim_{\mathbb{R},Z}0$, and

  * •

there is a klt pair $(X,\Delta)$ such that $\Delta$ is big over $Z$ and
$K_{X}+\Delta\sim_{\mathbb{R},Z}0$.

Then there is a birational contraction
$\tilde{X}\dashrightarrow\tilde{X}^{\prime}$ over $Z$ such that if
$\tilde{B}^{\prime}$ is the birational transforms of $\tilde{B}$ on
$\tilde{X}^{\prime}$, then

  * •

$K_{\tilde{X}^{\prime}}+\tilde{B}^{\prime}+\boldsymbol{\rm
M}_{\tilde{X}^{\prime}}$ is semi-ample over $Z$, and

  * •

$a(P,\tilde{X},\tilde{B}+\boldsymbol{\rm M}_{\tilde{X}})\leq
a(P,\tilde{X}^{\prime},\tilde{B}^{\prime}+\boldsymbol{\rm
M}_{\tilde{X}^{\prime}})$ for all prime divisors $P$ over $\tilde{X}$.

###### Proof.

We prove it with several steps.

######  Step 1.

By hypothesis, we can write

$K_{\tilde{X}}+\tilde{B}+\boldsymbol{\rm
M}_{\tilde{X}}=f^{*}(K_{X}+B+\boldsymbol{\rm
M}_{X})+\tilde{E}-\tilde{F}\sim_{\mathbb{R},Z}\tilde{E}-\tilde{F},$

where $\tilde{E}\geq 0$ and $\tilde{F}\geq 0$ are $f$-exceptional
$\mathbb{R}$-divisors having no common components. Let $\tilde{G}$ be the sum
of all $f$-exceptional prime divisors which are neither components of
$\tilde{B}$ nor $\tilde{F}$. By definition, the divisor
$\tilde{B}+\tilde{G}+\tilde{F}$ contains all $f$-exceptional prime divisors.
Since $(X,B,\boldsymbol{\rm M})$ is generalized lc, we have ${\rm
coeff}_{P}(\tilde{B}+\tilde{F}-\tilde{E})\leq 1$ for all prime divisors $P$ on
$\tilde{X}$. If $P$ is a component of $\tilde{F}$, then

${\rm coeff}_{P}(\tilde{B}+\tilde{F})={\rm
coeff}_{P}(\tilde{B}+\tilde{F}-\tilde{E})\leq 1$

because $\tilde{E}$ and $\tilde{F}$ have no common components. If $P$ is not a
component of $\tilde{F}$, then

${\rm coeff}_{P}(\tilde{B}+\tilde{F})={\rm coeff}_{P}(\tilde{B})\leq 1$

by definition of $\tilde{B}$. Thus, all prime divisors $P$ on $\tilde{X}$
satisfy ${\rm coeff}_{P}(\tilde{B}+\tilde{F})\leq 1$. Since
$\bigl{(}\tilde{X},{\rm Supp}\tilde{B}\cup{\rm Ex}(f)\bigr{)}$ is log smooth,
there is $t_{0}\in(0,1)$ such that
$\bigl{(}\tilde{X},\tilde{B}+t_{0}(\tilde{G}+\tilde{F})\bigr{)}$ is a log
smooth lc pair. Then
$\bigl{(}\tilde{X},\tilde{B}+t_{0}(\tilde{G}+\tilde{F}),{\boldsymbol{\rm M}})$
is a generalized lc pair with the nef part ${\boldsymbol{\rm
M}}=\overline{\boldsymbol{\rm M}_{\tilde{X}}}$, and we have

$K_{\tilde{X}}+\tilde{B}+t_{0}(\tilde{G}+\tilde{F})+\boldsymbol{\rm
M}_{\tilde{X}}\sim_{\mathbb{R},Z}+\tilde{E}+t_{0}\tilde{G}-(1-t_{0})\tilde{F}.$

Because $\tilde{E}$ and $\tilde{F}$ have no common components and the same
property holds for $\tilde{G}$ and $\tilde{F}$ by definition of $\tilde{G}$,
the divisors $\tilde{E}+t_{0}\tilde{G}$ and $\tilde{F}$ have no common
components.

######  Step 2.

We run a $(K_{\tilde{X}}+\tilde{B}+\boldsymbol{\rm M}_{\tilde{X}})$-MMP over
$X$ with scaling of an ample divisor. Since $\tilde{E}\geq 0$ is
$f$-exceptional, by [HanLi, Proposition 3.8] and regarding
$(\tilde{X},\tilde{B},{\boldsymbol{\rm M}})$ as a generalized lc pair over
$X$, after finitely many steps we get a morphism $f_{0}\colon\tilde{X}_{0}\to
X$ such that $\tilde{E}$ is contracted by the map
$\tilde{X}\dashrightarrow\tilde{X}_{0}$ of the MMP. Let $\tilde{B}_{0}$,
$\tilde{F}_{0}$ and $\tilde{G}_{0}$ be the birational transforms of
$\tilde{B}$, $\tilde{F}$ and $\tilde{G}$ on $\tilde{X}_{0}$, respectively.
Then the relation $K_{\tilde{X}_{0}}+\tilde{B}_{0}+\boldsymbol{\rm
M}_{\tilde{X}_{0}}\sim_{\mathbb{R},Z}-\tilde{F}_{0}$ holds and the inequality

($*$) $a\bigl{(}P,\tilde{X},\tilde{B}+\boldsymbol{\rm
M}_{\tilde{X}}\bigr{)}\leq
a\bigl{(}P,\tilde{X}_{0},\tilde{B}_{0}+\boldsymbol{\rm
M}_{\tilde{X}_{0}}\bigr{)}$

holds for all prime divisors $P$ over $\tilde{X}$. By the first condition of
Proposition 3.2, we see that $K_{\tilde{X}_{0}}+\tilde{B}_{0}+\tilde{M}_{0}$
is pseudo-effective over $Z$, so $\tilde{F}_{0}$ is vertical over $Z$. In
particular, there is $\tilde{H}_{0}\geq 0$ on $\tilde{X}_{0}$ such that
$K_{\tilde{X}_{0}}+\tilde{B}_{0}+\boldsymbol{\rm
M}_{\tilde{X}_{0}}\sim_{\mathbb{R},Z}\tilde{H}_{0}$. For any $t\in(0.t_{0}]$,
we have

($\spadesuit$)
$K_{\tilde{X}_{0}}+\tilde{B}_{0}+t(\tilde{G}_{0}+\tilde{F}_{0})+\boldsymbol{\rm
M}_{\tilde{X}_{0}}\sim_{\mathbb{R},Z}t\tilde{G}_{0}-(1-t)\tilde{F}_{0}\sim_{\mathbb{R},Z}t(\tilde{G}_{0}+\tilde{F}_{0})+\tilde{H}_{0}.$

Replacing $t_{0}>0$ in Step 1 with a small one, we may assume that the
birational map $\tilde{X}\dashrightarrow\tilde{X}_{0}$ is a sequence of steps
of a $(K_{\tilde{X}}+\tilde{B}+t_{0}(\tilde{G}+\tilde{F})+\boldsymbol{\rm
M}_{\tilde{X}})$-MMP. Then the generalized pair
$\bigl{(}\tilde{X}_{0},\tilde{B}_{0}+t_{0}(\tilde{G}_{0}+\tilde{F}_{0}),\boldsymbol{\rm
M}\bigr{)}$ is a $\mathbb{Q}$-factorial generalized dlt pair.

######  Step 3.

Pick a strictly decreasing sequence of positive real numbers
$\\{t_{i}\\}_{i\geq 1}$ such that $0<t_{i}<t_{0}$ for any $i$ and ${\rm
lim}_{i\to\infty}t_{i}=0$. In this step, we prove that for each $i\geq 1$
there exists a sequence of steps of a
$\bigl{(}K_{\tilde{X}_{0}}+\tilde{B}_{0}+t_{i}(\tilde{G}_{0}+\tilde{F}_{0})+\boldsymbol{\rm
M}_{\tilde{X}_{0}}\bigr{)}$-MMP over $Z$

$\bigl{(}\tilde{X}_{0},\tilde{B}_{0}+t_{i}(\tilde{G}_{0}+\tilde{F}_{0}),\boldsymbol{\rm
M}\bigr{)}\dashrightarrow\bigl{(}\tilde{X}_{i},\tilde{B}_{i}+t_{i}(\tilde{G}_{i}+\tilde{F}_{i}),\boldsymbol{\rm
M}\bigr{)}$

such that $\tilde{G}_{0}$ is contracted by the map
$\tilde{X}_{0}\dashrightarrow\tilde{X}_{i}$ (i.e., $\tilde{G}_{i}=0$) and
$K_{\tilde{X}_{i}}+\tilde{B}_{i}+t_{i}\tilde{F}_{i}+\boldsymbol{\rm
M}_{\tilde{X}_{i}}$ is semi-ample over $Z$. From now to the end of this step,
we fix $i$.

We run a
$\bigl{(}K_{\tilde{X}_{0}}+\tilde{B}_{0}+t_{i}(\tilde{G}_{0}+\tilde{F}_{0})+\boldsymbol{\rm
M}_{\tilde{X}_{0}}\bigr{)}$-MMP over $X$ with scaling of an ample divisor. By
($\spadesuit$ ‣ 2) in Step 2, this MMP is an MMP for
$t_{i}\tilde{G}_{0}-(1-t_{i})\tilde{F}_{0}$ over $X$. Regarding
$\bigl{(}\tilde{X}_{0},\tilde{B}_{0}+t_{i}(\tilde{G}_{0}+\tilde{F}_{0}),\boldsymbol{\rm
M}\bigr{)}$ as a $\mathbb{Q}$-factorial generalized dlt pair over $X$ and
applying [HanLi, Proposition 3.8], after finitely many steps we get a morphism
$f^{\prime}_{0}\colon\tilde{X}^{\prime}_{0}\to X$ such that $\tilde{G}_{0}$ is
contracted by the birational map
$\tilde{X}_{0}\dashrightarrow\tilde{X}^{\prime}_{0}$ of the MMP. In this way,
we obtain a sequence of steps of a
$\bigl{(}K_{\tilde{X}_{0}}+\tilde{B}_{0}+t_{i}(\tilde{G}_{0}+\tilde{F}_{0})+\boldsymbol{\rm
M}_{\tilde{X}_{0}}\bigr{)}$-MMP over $Z$

$\bigl{(}\tilde{X}_{0},\tilde{B}_{0}+t_{i}(\tilde{G}_{0}+\tilde{F}_{0}),\boldsymbol{\rm
M}\bigr{)}\dashrightarrow\bigl{(}\tilde{X}^{\prime}_{0},\tilde{B}^{\prime}_{0}+t_{i}\tilde{F}^{\prime}_{0},\boldsymbol{\rm
M}\bigr{)}.$

Here, the divisors $\tilde{B}^{\prime}_{0}$ and $\tilde{F}^{\prime}_{0}$ are
the birational transforms of $\tilde{B}_{0}$ and $\tilde{F}_{0}$,
respectively.

Recall that $\tilde{B}+\tilde{G}+\tilde{F}$ contains all $f$-exceptional prime
divisors. Therefore, $\tilde{B}^{\prime}_{0}+t_{i}\tilde{F}^{\prime}_{0}$
contains all $f^{\prime}_{0}$-exceptional prime divisors in its support. Let
$\tilde{\Delta}^{\prime}_{0}$ be an $\mathbb{R}$-divisor on
$\tilde{X}^{\prime}_{0}$ defined by an equation
$K_{\tilde{X}^{\prime}_{0}}+\tilde{\Delta}^{\prime}_{0}=f^{\prime*}_{0}(K_{X}+\Delta)$,
where $\Delta$ is as in the third condition of Proposition 3.2. Since $\Delta$
is big over $Z$ and $\tilde{B}^{\prime}_{0}+t_{i}\tilde{F}^{\prime}_{0}$
contains all $f^{\prime}_{0}$-exceptional prime divisors in its support, there
is $u>0$ such that

$\tilde{\Psi}^{\prime}_{0}:=\frac{1}{1+u}(\tilde{B}^{\prime}_{0}+t_{i}\tilde{F}^{\prime}_{0})+\frac{u}{1+u}\tilde{\Delta}^{\prime}_{0}$

is effective and big over $Z$. Since $(X,\Delta)$ is klt and
$\bigl{(}\tilde{X}^{\prime}_{0},\tilde{B}^{\prime}_{0}+t_{i}\tilde{F}^{\prime}_{0},\boldsymbol{\rm
M}\bigr{)}$ is generalized lc, the generalized pair
$\bigl{(}\tilde{X}^{\prime}_{0},\tilde{\Psi}^{\prime}_{0},\frac{1}{1+u}\boldsymbol{\rm
M}\bigr{)}$ is generalized klt. Since $\tilde{\Psi}^{\prime}_{0}$ is big over
$Z$ and
$\bigl{(}\tilde{X}^{\prime}_{0},\tilde{\Psi}^{\prime}_{0},\frac{1}{1+u}\boldsymbol{\rm
M}\bigr{)}$ is generalized klt, there exists an $\mathbb{R}$-divisor
$\tilde{\Gamma}^{\prime}_{0}\geq 0$ on $\tilde{X}^{\prime}_{0}$ which is big
over $Z$ such that
$\tilde{\Gamma}^{\prime}_{0}\sim_{\mathbb{R},Z}\tilde{\Psi}^{\prime}_{0}+\tfrac{1}{1+u}\boldsymbol{\rm
M}_{\tilde{X}^{\prime}_{0}}$ and the pair
$\bigl{(}\tilde{X}^{\prime}_{0},\tilde{\Gamma}^{\prime}_{0})$ is klt. By
[BCHM], there is a sequence of steps of a
$(K_{\tilde{X}^{\prime}_{0}}+\tilde{\Gamma}^{\prime}_{0})$-MMP over $Z$
terminating with a good minimal model. By construction, we have

$\begin{split}K_{\tilde{X}^{\prime}_{0}}+\tilde{\Gamma}^{\prime}_{0}\sim_{\mathbb{R},Z}&K_{\tilde{X}^{\prime}_{0}}+\tilde{\Psi}^{\prime}_{0}+\frac{1}{1+u}\boldsymbol{\rm
M}_{\tilde{X}^{\prime}_{0}}\\\
=&K_{\tilde{X}^{\prime}_{0}}+\frac{1}{1+u}(\tilde{B}^{\prime}_{0}+t_{i}\tilde{F}^{\prime}_{0})+\frac{u}{1+u}\tilde{\Delta}^{\prime}_{0}+\frac{1}{1+u}\boldsymbol{\rm
M}_{\tilde{X}^{\prime}_{0}}\\\
=&\frac{1}{1+u}\bigl{(}K_{\tilde{X}^{\prime}_{0}}+\tilde{B}^{\prime}_{0}+t_{i}\tilde{F}^{\prime}_{0}+\boldsymbol{\rm
M}_{\tilde{X}^{\prime}_{0}}\bigr{)}+\frac{u}{1+u}\bigl{(}K_{\tilde{X}^{\prime}_{0}}+\tilde{\Delta}^{\prime}_{0}\bigr{)}\\\
\sim_{\mathbb{R},Z}&\frac{1}{1+u}\bigl{(}K_{\tilde{X}^{\prime}_{0}}+\tilde{B}^{\prime}_{0}+t_{i}\tilde{F}^{\prime}_{0}+\boldsymbol{\rm
M}_{\tilde{X}^{\prime}_{0}}\bigr{)}.\end{split}$

The final relation follows from
$K_{\tilde{X}^{\prime}_{0}}+\tilde{\Delta}^{\prime}_{0}=f^{\prime*}_{0}(K_{X}+\Delta)\sim_{\mathbb{R},Z}0$.
From this discussion, there is a sequence of steps of a
$\bigl{(}K_{\tilde{X}^{\prime}_{0}}+\tilde{B}^{\prime}_{0}+t_{i}\tilde{F}^{\prime}_{0}+\boldsymbol{\rm
M}_{\tilde{X}^{\prime}_{0}}\bigr{)}$-MMP over $Z$

$\bigl{(}\tilde{X}^{\prime}_{0},\tilde{B}^{\prime}_{0}+t_{i}\tilde{F}^{\prime}_{0},\boldsymbol{\rm
M}\bigr{)}\dashrightarrow\bigl{(}\tilde{X}_{i},\tilde{B}_{i}+t_{i}\tilde{F}_{i},\boldsymbol{\rm
M}\bigr{)}$

such that $K_{\tilde{X}_{i}}+\tilde{B}_{i}+t_{i}\tilde{F}_{i}+\boldsymbol{\rm
M}_{\tilde{X}_{i}}$ is semi-ample over $Z$. Then the composition

$\bigl{(}\tilde{X}_{0},\tilde{B}_{0}+t_{i}(\tilde{G}_{0}+\tilde{F}_{0}),\boldsymbol{\rm
M}\bigr{)}\dashrightarrow\bigl{(}\tilde{X}^{\prime}_{0},\tilde{B}^{\prime}_{0}+t_{i}\tilde{F}^{\prime}_{0},\boldsymbol{\rm
M}\bigr{)}\dashrightarrow\bigl{(}\tilde{X}_{i},\tilde{B}_{i}+t_{i}\tilde{F}_{i},\boldsymbol{\rm
M}\bigr{)}$

is the desired
$\bigl{(}K_{\tilde{X}_{0}}+\tilde{B}_{0}+t_{i}(\tilde{G}_{0}+\tilde{F}_{0})+\boldsymbol{\rm
M}_{\tilde{X}_{0}}\bigr{)}$-MMP over $Z$. We finish this step.

######  Step 4.

With this step we complete the proof.

For any $i\geq 1$, we denote a sequence of steps of a
$\bigl{(}K_{\tilde{X}_{0}}+\tilde{B}_{0}+t_{i}(\tilde{G}_{0}+\tilde{F}_{0})+\boldsymbol{\rm
M}_{\tilde{X}_{0}}\bigr{)}$-MMP over $Z$ constructed in Step 3 by

$\bigl{(}\tilde{X}_{0},\tilde{B}_{0}+t_{i}(\tilde{G}_{0}+\tilde{F}_{0}),\boldsymbol{\rm
M}\bigr{)}\dashrightarrow\bigl{(}\tilde{X}_{i},\tilde{B}_{i}+t_{i}\tilde{F}_{i},\boldsymbol{\rm
M}).$

We recall that $\tilde{G}_{0}$ is contracted by the map
$\tilde{X}_{0}\dashrightarrow\tilde{X}_{i}$ and
$K_{\tilde{X}_{i}}+\tilde{B}_{i}+t_{i}\tilde{F}_{i}+\boldsymbol{\rm
M}_{\tilde{X}_{i}}$ is semi-ample over $Z$. By ($\spadesuit$ ‣ 2) in Step 2,
we see that prime divisors contracted by the map
$\tilde{X}_{0}\dashrightarrow\tilde{X}_{i}$ are components of
$\tilde{G}_{0}+\tilde{F}_{0}+\tilde{H}_{0}$, which are independent of $t_{i}$.
Therefore, by replacing $\\{t_{i}\\}_{i\geq 1}$ with a subsequence, we may
assume that all maps $\tilde{X}_{0}\dashrightarrow\tilde{X}_{i}$ contract the
same divisors, so all $X_{i}$ are isomorphic in codimension one.

We prove that the birational contraction
$\tilde{X}\dashrightarrow\tilde{X}_{1}$ satisfies the two conditions of
Proposition 3.2. We first check that
$K_{\tilde{X}_{1}}+\tilde{B}_{1}+\boldsymbol{\rm M}_{\tilde{X}_{1}}$ is semi-
ample over $Z$. Because $\tilde{G}$ is contracted by the map
$\tilde{X}\dashrightarrow\tilde{X}_{1}$, with ($\spadesuit$ ‣ 2) in Step 2, we
have

$\begin{split}K_{\tilde{X}_{1}}+\tilde{B}_{1}+t_{1}\tilde{F}_{1}+\boldsymbol{\rm
M}_{\tilde{X}_{1}}\sim_{\mathbb{R},Z}-(1-t_{1})\tilde{F}_{1}.\end{split}$

From this relation and the fact that
$K_{\tilde{X}_{1}}+\tilde{B}_{1}+t_{1}\tilde{F}_{1}+\boldsymbol{\rm
M}_{\tilde{X}_{1}}$ is semi-ample over $Z$, we see that

$\begin{split}K_{\tilde{X}_{1}}+\tilde{B}_{1}+\boldsymbol{\rm
M}_{\tilde{X}_{1}}\sim_{\mathbb{R},Z}-\tilde{F}_{1}\sim_{\mathbb{R},Z}\frac{1}{1-t_{1}}\bigl{(}K_{\tilde{X}_{1}}+\tilde{B}_{1}+t_{1}\tilde{F}_{1}+\boldsymbol{\rm
M}_{\tilde{X}_{1}}\bigr{)}\end{split}$

is semi-ample over $Z$. Therefore, the first condition of Proposition 3.2
holds true.

Next, we prove $a(P,\tilde{X},\tilde{B}+\boldsymbol{\rm M}_{\tilde{X}})\leq
a(P,\tilde{X}_{1},\tilde{B}_{1}+\boldsymbol{\rm M}_{\tilde{X}_{1}})$ for all
prime divisors $P$ over $\tilde{X}$. Since
$K_{\tilde{X}_{1}}+\tilde{B}_{1}+\boldsymbol{\rm M}_{\tilde{X}_{1}}$ is semi-
ample over $Z$, we see that

$\begin{split}K_{\tilde{X}_{1}}+\tilde{B}_{1}+t_{i}\tilde{F}_{1}+\boldsymbol{\rm
M}_{\tilde{X}_{1}}\sim_{\mathbb{R},Z}-(1-t_{i})\tilde{F}_{1}\sim_{\mathbb{R},Z}(1-t_{i})\bigl{(}K_{\tilde{X}_{1}}+\tilde{B}_{1}+\boldsymbol{\rm
M}_{\tilde{X}_{1}}\bigr{)}\end{split}$

is semi-ample over $Z$ for any $i$. By construction, the induced birational
map $\tilde{X}_{1}\dashrightarrow\tilde{X}_{i}$ are isomorphic in codimension
one and $\tilde{B}_{i}$, $\tilde{F}_{i}$, and $\boldsymbol{\rm
M}_{\tilde{X}_{i}}$ are the birational transforms of $\tilde{B}_{1}$,
$\tilde{F}_{1}$, and $\boldsymbol{\rm M}_{\tilde{X}_{1}}$ on $\tilde{X}_{i}$,
respectively. Since
$K_{\tilde{X}_{i}}+\tilde{B}_{i}+t_{i}\tilde{F}_{i}+\boldsymbol{\rm
M}_{\tilde{X}_{i}}$ is semi-ample over $Z$, by taking a common resolution of
$\tilde{X}_{1}\dashrightarrow\tilde{X}_{i}$ and the negativity lemma, we
obtain

$a(P,\tilde{X}_{i},\tilde{B}_{i}+t_{i}\tilde{F}_{i}+\boldsymbol{\rm
M}_{\tilde{X}_{i}})=a(P,\tilde{X}_{1},\tilde{B}_{1}+t_{i}\tilde{F}_{1}+\boldsymbol{\rm
M}_{\tilde{X}_{1}})$

for any $i>1$. Combining this with construction of the map
$\tilde{X}_{0}\dashrightarrow\tilde{X}_{i}$ in Step 3, we obtain
$a(P,\tilde{X}_{0},\tilde{B}_{0}+t_{i}(\tilde{G}_{0}+\tilde{F}_{0})+\boldsymbol{\rm
M}_{\tilde{X}_{0}})\leq
a(P,\tilde{X}_{1},\tilde{B}_{1}+t_{i}\tilde{F}_{1}+\boldsymbol{\rm
M}_{\tilde{X}_{1}})$ for any $i$. By taking the limit $i\to\infty$, we have

$a(P,\tilde{X}_{0},\tilde{B}_{0}+\boldsymbol{\rm M}_{\tilde{X}_{0}})\leq
a(P,\tilde{X}_{1},\tilde{B}_{1}+\boldsymbol{\rm M}_{\tilde{X}_{1}}).$

We combine this inequality and ($*$ ‣ 2) in Step 2, then we obtain

$a\bigl{(}P,\tilde{X},\tilde{B}+\boldsymbol{\rm M}_{\tilde{X}}\bigr{)}\leq
a(P,\tilde{X}_{0},\tilde{B}_{0}+\boldsymbol{\rm M}_{\tilde{X}_{0}})\leq
a(P,\tilde{X}_{1},\tilde{B}_{1}+\boldsymbol{\rm M}_{\tilde{X}_{1}}).$

In this way, we see that $\tilde{X}\dashrightarrow\tilde{X}_{1}$ is the
desired birational contraction.

We complete the proof. ∎

######  Remark 3.3.

When $B$ and ${\boldsymbol{\rm M}}$ have rational coefficients, Proposition
3.2 follows from [B4, Lemma 4.10] and arguments in [B4, Proof of Lemma 4.14].
These arguments in [B4, Proofs of Lemma 4.10 and Lemma 4.14] work in our
situation. We note that the arguments are different from our proof.

######  Proposition 3.4.

Let $\pi\colon X\to Z$ be a morphism of normal projective varieties, and let
$(X,B,\boldsymbol{\rm M})$ be a projective generalized klt pair. Suppose that

  * •

$\boldsymbol{\rm M}$ is a finite $\mathbb{R}_{>0}$-linear combination of b-nef
$\mathbb{Q}$-b-Cartier $\mathbb{Q}$-b-divisors, and

  * •

there exists an open subset $U\subset Z$ such that $(K_{X}+B+\boldsymbol{\rm
M}_{X})|_{\pi^{-1}(U)}\sim_{\mathbb{R},U}0$.

Let $A_{Z}$ be an ample $\mathbb{R}$-divisor on $Z$.

Then, $K_{X}+B+\boldsymbol{\rm M}_{X}+\pi^{*}A_{Z}$ is abundant. Furthermore,
when $K_{X}+B+\boldsymbol{\rm M}_{X}+\pi^{*}A_{Z}$ is pseudo-effective, the
divisor birationally has the Nakayama–Zariski decomposition with semi-ample
positive part.

###### Proof.

The argument is very similar to [H1, Proof of Lemma 3.1].

######  Step 1.

By taking a Stein factorization of $\pi$, we may assume that $\pi$ is a
contraction. Let $\overline{f}\colon\overline{X}\to X$ be a log resolution of
$(X,{\rm Supp}B)$ such that $\boldsymbol{\rm M}=\overline{\boldsymbol{\rm
M}_{\overline{X}}}$. By construction, we can find $\mathbb{R}$-divisors
$\overline{B}$ and $\overline{E}$ on $\overline{X}$ such that
$(\overline{X},\overline{B})$ is a log smooth klt pair, $\overline{E}$ is
$\overline{f}$-exceptional, and

$K_{\overline{X}}+\overline{B}+\boldsymbol{\rm
M}_{\overline{X}}=\overline{f}^{*}(K_{X}+B+\boldsymbol{\rm
M}_{X})+\overline{E}.$

Note that $\overline{B}$ and $\overline{E}$ may have common components. We
apply the weak semistable reduction ([AK, Proof of Theorem 2.1], [AK,
Proposition 4.4] and [AK, Remark 4.5]) to the morphism
$(\overline{X},\overline{B})\to Z$. Then we have the following diagram

$\textstyle{\overline{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi\circ\overline{f}}$$\textstyle{\tilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{f}}$$\scriptstyle{\tilde{\pi}}$$\textstyle{Z}$$\textstyle{\tilde{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$

with normal projective varieties $\tilde{X}$ and a smooth projective variety
$\tilde{Z}$ such that

  * •

$\tilde{f}\colon\tilde{X}\to\overline{X}$ and $g\colon\tilde{Z}\to Z$ are
birational morphisms,

  * •

$\tilde{\pi}\colon\tilde{X}\to\tilde{Z}$ is a contraction and all fibers of
$\tilde{\pi}$ have the same dimensions, and

  * •

$(\tilde{X},0)$ is $\mathbb{Q}$-factorial klt and $\bigl{(}\tilde{X},{\rm
Supp}\tilde{f}_{*}^{-1}\overline{B}\cup{\rm Ex}(\tilde{f})\bigr{)}$ is an lc
pair, here we regard ${\rm Supp}\tilde{f}_{*}^{-1}\overline{B}\cup{\rm
Ex}(\tilde{f})$ as a reduced divisor on $\tilde{X}$.

By the third condition, there is an $\mathbb{R}$-divisor $\tilde{B}$ on
$\tilde{X}$ such that $(\tilde{X},\tilde{B})$ is a $\mathbb{Q}$-factorial klt
pair and
$K_{\tilde{X}}+\tilde{B}-\tilde{f}^{*}(K_{\overline{X}}+\overline{B})$ is
effective and $\tilde{f}$-exceptional. Set
$f=\overline{f}\circ\tilde{f}\colon\tilde{X}\to X$. Because $\boldsymbol{\rm
M}=\overline{\boldsymbol{\rm M}_{\overline{X}}}$ and $\boldsymbol{\rm M}$ is
b-nef, $\boldsymbol{\rm M}_{\tilde{X}}$ is nef, so
$\bigl{(}\tilde{X},\tilde{B},\boldsymbol{\rm M}\bigr{)}$ is a generalized klt
pair. Furthermore, by construction of $\overline{B}$ and $\tilde{B}$, we may
write

$K_{\tilde{X}}+\tilde{B}+\boldsymbol{\rm
M}_{\tilde{X}}=f^{*}(K_{X}+B+\boldsymbol{\rm M}_{X})+\tilde{E}$

for some $f$-exceptional $\mathbb{R}$-divisor $\tilde{E}\geq 0$.

In this way, we obtain a diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\tilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\tilde{\pi}}$$\textstyle{Z}$$\textstyle{\tilde{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$

and a projective $\mathbb{Q}$-factorial generalized klt pair
$\bigl{(}\tilde{X},\tilde{B},\boldsymbol{\rm M}\bigr{)}$ with $\boldsymbol{\rm
M}=\overline{\boldsymbol{\rm M}_{\tilde{X}}}$ such that

  1. (1)

$f\colon\tilde{X}\to\overline{X}$ and $g\colon\tilde{Z}\to Z$ are birational
morphisms and $\tilde{Z}$ is smooth,

  2. (2)

$\tilde{\pi}\colon\tilde{X}\to\tilde{Z}$ is a contraction and all fibers of
$\tilde{\pi}$ have the same dimensions, and

  3. (3)

$K_{\tilde{X}}+\tilde{B}+\boldsymbol{\rm
M}_{\tilde{X}}=f^{*}(K_{X}+B+\boldsymbol{\rm M}_{X})+\tilde{E}$ for some
$f$-exceptional $\mathbb{R}$-divisor $\tilde{E}\geq 0$.

By property (3), it follows that $K_{X}+B+\boldsymbol{\rm M}_{X}+\pi^{*}A_{Z}$
is abundant and birationally has the Nakayama–Zariski decomposition with semi-
ample positive part if and only if the same assertion holds for
$K_{\tilde{X}}+\tilde{B}+\boldsymbol{\rm
M}_{\tilde{X}}+\tilde{\pi}^{*}g^{*}A_{Z}$.

######  Step 2.

By (3) in Step 1 and the hypothesis that $(K_{X}+B+\boldsymbol{\rm
M}_{X})|_{\pi^{-1}(U)}\sim_{\mathbb{R},U}0$ for some open $U\subset Z$, there
exist an $\mathbb{R}$-divisor $\tilde{F}\geq 0$ on $\tilde{X}$ such that
$K_{\tilde{X}}+\tilde{B}+\boldsymbol{\rm
M}_{\tilde{X}}\sim_{\mathbb{R},\tilde{Z}}\tilde{F}$. We can write
$\tilde{F}=\tilde{F}_{h}+\tilde{F}_{v}$ such that all components of
$\tilde{F}_{h}$ dominates $\tilde{Z}$ and all components $\tilde{F}_{v}$ are
vertical over $\tilde{Z}$. We recall condition (2) in Step 1, which says that
all fibers of $\tilde{\pi}$ have the same dimensions. Therefore, the image of
any component of $\tilde{F}_{v}$ on $\tilde{Z}$ is a divisor. Since
$\tilde{Z}$ is smooth, we can consider

$\nu_{P}:={\rm sup}\set{\nu\in\mathbb{R}_{\geq
0}\mid\text{$\tilde{F}_{v}-\nu\tilde{\pi}^{*}P\geq 0$}}$

for any prime divisor $P$ on $\tilde{Z}$. Then it is easy to see that
$\nu_{P}>0$ with only finitely many prime divisors $P$. By (2) in Step 1, we
see that $\tilde{F}_{v}-\sum_{P}\nu_{P}\tilde{\pi}^{*}P$ is an effective
$\mathbb{R}$-divisor which is very exceptional over $\tilde{Z}$. By replacing
$\tilde{F}_{v}$ with $\tilde{F}_{v}-\sum_{P}\nu_{P}\tilde{\pi}^{*}P$, we may
assume that $\tilde{F}_{v}$ is very exceptional over $\tilde{Z}$.

We run a $(K_{\tilde{X}}+\tilde{B}+\boldsymbol{\rm M}_{\tilde{X}})$-MMP over
$\tilde{Z}$ with scaling of an ample divisor

$(\tilde{X},\tilde{B},\boldsymbol{\rm
M})=(\tilde{X}_{0},\tilde{B}_{0},\boldsymbol{\rm
M})\dashrightarrow\cdots\dashrightarrow(\tilde{X}_{i},\tilde{B}_{i},\boldsymbol{\rm
M})\dashrightarrow\cdots.$

We prove that after finitely many steps we reach a model
$(\tilde{X}_{m},\tilde{B}_{m},\boldsymbol{\rm M})$ such that
$K_{\tilde{X}_{m}}+\tilde{B}_{m}+\boldsymbol{\rm
M}_{\tilde{X}_{m}}\sim_{\mathbb{R},\tilde{Z}}0$. We recall the hypothesis that
$(K_{X}+B+\boldsymbol{\rm M}_{X})|_{\pi^{-1}(U)}\sim_{\mathbb{R},U}0$ for some
open $U\subset Z$. By shrinking $U$, we may assume $g\colon\tilde{Z}\to Z$ is
an isomorphism over $U$, so $U$ can be thought of an open subset of
$\tilde{Z}$. For each $i$, let $\tilde{V}_{i}$ be the inverse image of $U$ to
$\tilde{X}_{i}$. Then, we see that

$(\tilde{V}_{0},\tilde{B}_{0}|_{\tilde{V}_{0}},\boldsymbol{\rm
M}|_{\tilde{V}_{0}})\dashrightarrow\cdots\dashrightarrow(\tilde{V}_{i},\tilde{B}_{i}|_{\tilde{V}_{i}},\boldsymbol{\rm
M}|_{\tilde{V}_{i}})\dashrightarrow\cdots$

is a sequence of steps of a
$(K_{\tilde{V}_{0}}+\tilde{B}_{0}|_{\tilde{V}_{0}}+\boldsymbol{\rm
M}_{\tilde{X}_{0}}|_{\tilde{V}_{0}})$-MMP over $U$, and the generalized klt
pair $(\tilde{V}_{0},\tilde{B}_{0}|_{\tilde{V}_{0}},\boldsymbol{\rm
M}|_{\tilde{V}_{0}})$ has a relatively trivial minimal model over $U$ because
of (3) in Step 1 and the relation $(K_{X}+B+\boldsymbol{\rm
M}_{X})|_{\pi^{-1}(U)}\sim_{\mathbb{R},U}0$. By [HanLi, Theorem 4.1], for any
$i\gg 0$ we obtain
$K_{\tilde{V}_{i}}+\tilde{B}_{i}|_{\tilde{V}_{i}}+\boldsymbol{\rm
M}_{\tilde{X}_{i}}|_{\tilde{V}_{i}}\sim_{\mathbb{R},U}0$. Because we have
$K_{\tilde{X}}+\tilde{B}+\boldsymbol{\rm
M}_{\tilde{X}}\sim_{\mathbb{R},\tilde{Z}}\tilde{F}_{h}+\tilde{F}_{v}$ and each
component of $\tilde{F}_{h}$ dominates over $\tilde{Z}$, we see that
$\tilde{F}_{h}$ is contracted by the map
$\tilde{X}\dashrightarrow\tilde{X}_{i}$ for any $i\gg 0$. Pick $m>0$ such that
$\tilde{F}_{h}$ is contracted by $\tilde{X}\dashrightarrow\tilde{X}_{m}$.
Since the $(K_{\tilde{X}}+\tilde{B}+\boldsymbol{\rm M}_{\tilde{X}})$-MMP over
$\tilde{Z}$ occurs only in ${\rm Supp}(\tilde{F}_{h}+\tilde{F}_{v})$, the
birational transform of $\tilde{F}_{v}$ on $\tilde{X}_{m}$ is very exceptional
over $\tilde{Z}$. Applying [HanLi, Proposition 3.8] to
$\bigl{(}\tilde{X}_{m},\tilde{B}_{m},\boldsymbol{\rm M}\bigr{)}/\tilde{Z}$ and
replacing $m$, we see that $\tilde{F}_{v}$ is also contracted by
$\tilde{X}\dashrightarrow\tilde{X}_{m}$.

From this discussion, there exists $m>0$ such that
$K_{\tilde{X}_{m}}+\tilde{B}_{m}+\boldsymbol{\rm
M}_{\tilde{X}_{m}}\sim_{\mathbb{R},\tilde{Z}}0$.

######  Step 3.

We denote the natural murphism $\tilde{X}\to\tilde{Z}$ by $\tilde{\pi}_{m}$.
Now we have the following diagram

---
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\tilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\tilde{\pi}}$$\textstyle{\tilde{X}_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\pi}_{m}}$$\textstyle{Z}$$\textstyle{\tilde{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$

and a projective generalized klt pair
$\bigl{(}\tilde{X}_{m},\tilde{B}_{m},\boldsymbol{\rm M}\bigr{)}$ such that
$K_{\tilde{X}_{m}}+\tilde{B}_{m}+\boldsymbol{\rm
M}_{\tilde{X}_{m}}\sim_{\mathbb{R},\tilde{Z}}0$. Since
$\tilde{X}\dashrightarrow\tilde{X}_{m}$ is also a sequence of steps of a
$(K_{\tilde{X}}+\tilde{B}+\tilde{\pi}^{*}g^{*}A_{Z}+\boldsymbol{\rm
M}_{\tilde{X}})$-MMP, the divisor $K_{\tilde{X}}+\tilde{B}+\boldsymbol{\rm
M}_{\tilde{X}}+\tilde{\pi}^{*}g^{*}A_{Z}$ is abundant and birationally has the
Nakayama–Zariski decomposition with semi-ample positive part if and only if
the same assertion holds for
$K_{\tilde{X}_{m}}+\tilde{B}_{m}+\tilde{\pi}_{m}^{*}g^{*}A_{Z}+\boldsymbol{\rm
M}_{\tilde{X}_{m}}$.

We recall the first condition of Proposition 3.4 saying that $\boldsymbol{\rm
M}$ is a finite $\mathbb{R}_{>0}$-linear combination of b-nef
$\mathbb{Q}$-b-Cartier $\mathbb{Q}$-b-divisors. By an argument using convex
geometry, we can find finitely many positive real numbers $r_{1},\cdots,r_{l}$
and projective generalized klt pairs
$\bigl{(}\tilde{X}_{m},\tilde{B}^{(1)}_{m},\boldsymbol{\rm
M}^{(1)}\bigr{)},\cdots,\bigl{(}\tilde{X}_{m},\tilde{B}^{(l)}_{m},\boldsymbol{{\rm
M}}^{(l)}\bigr{)}$ such that

  * •

$\tilde{B}^{(j)}_{m}$ is a $\mathbb{Q}$-divisors and $\boldsymbol{\rm
M}^{(j)}$ is a $\mathbb{Q}$-b-divisors for every $1\leq j\leq l$,

  * •

$\sum_{j=1}^{l}r_{j}=1$,
$\sum_{j=1}^{l}r_{j}\tilde{B}^{(j)}_{m}=\tilde{B}_{m}$ and
$\sum_{j=1}^{l}r_{j}\boldsymbol{{\rm M}}^{(j)}=\boldsymbol{\rm M}$, and

  * •

$K_{\tilde{X}_{m}}+\tilde{B}^{(j)}_{m}+\boldsymbol{\rm
M}^{(j)}\sim_{\mathbb{Q},\tilde{Z}}0$ for any $1\leq j\leq l$.

By applying the canonical bundle formula for generalized pairs [Fi, Theorem
1.4] to each $\bigl{(}\tilde{X}_{m},\tilde{B}^{(j)}_{m},\boldsymbol{{\rm
M}}^{(j)}\bigr{)}\to\tilde{Z}$ and using the above conditions, we can find a
generalized klt pair $(\tilde{Z},B_{\tilde{Z}},\boldsymbol{\rm N})$ such that
$K_{\tilde{X}_{m}}+\tilde{B}_{m}+\boldsymbol{\rm
M}_{\tilde{X}_{m}}\sim_{\mathbb{R}}\tilde{\pi}_{m}^{*}(K_{\tilde{Z}}+B_{\tilde{Z}}+\boldsymbol{\rm
N}_{\tilde{Z}})$. Since $g^{*}A_{Z}$ is nef and big, we can find a big
$\mathbb{R}$-divisor $\Delta_{\tilde{Z}}$ on $\tilde{Z}$ such that
$K_{\tilde{Z}}+B_{\tilde{Z}}+\boldsymbol{\rm
N}_{\tilde{Z}}+g^{*}A_{Z}\sim_{\mathbb{R}}K_{\tilde{Z}}+\Delta_{\tilde{Z}}$
and $(\tilde{Z},\Delta_{\tilde{Z}})$ is klt. By [BCHM],
$(\tilde{Z},\Delta_{\tilde{Z}})$ has a good minimal model or a Mori fiber
space. In particular, $K_{\tilde{Z}}+\Delta_{\tilde{Z}}$ is abundant, and the
divisor birationally has the Nakayama–Zariski decomposition with semi-ample
positive part when it is pseudo-effective. Because

$K_{\tilde{X}_{m}}+\tilde{B}_{m}+\tilde{\pi}_{m}^{*}g^{*}A_{Z}+\boldsymbol{\rm
M}_{\tilde{X}_{m}}\sim_{\mathbb{R}}\tilde{\pi}_{m}^{*}(K_{\tilde{Z}}+\Delta_{\tilde{Z}}),$

we see that
$K_{\tilde{X}_{m}}+\tilde{B}_{m}+\tilde{\pi}_{m}^{*}g^{*}A_{Z}+\boldsymbol{\rm
M}_{\tilde{X}_{m}}$ is abundant. When
$K_{\tilde{X}_{m}}+\tilde{B}_{m}+\tilde{\pi}_{m}^{*}g^{*}A_{Z}+\boldsymbol{\rm
M}_{\tilde{X}_{m}}$ is pseudo-effective, applying [N, Corollary 5.17], we see
that
$K_{\tilde{X}_{m}}+\tilde{B}_{m}+\tilde{\pi}_{m}^{*}g^{*}A_{Z}+\boldsymbol{\rm
M}_{\tilde{X}_{m}}$ birationally has the Nakayama–Zariski decomposition with
semi-ample positive part.

By the first paragraph of Step 3 and the final sentence of Step 1,
$K_{X}+B+\boldsymbol{\rm M}_{X}+\pi^{*}A_{Z}$ is abundant, and the divisor
birationally has the Nakayama–Zariski decomposition with semi-ample positive
part when it is pseudo-effective. We finish the proof. ∎

######  Theorem 3.5.

Let $\pi\colon X\to Z$ be a morphism of normal projective varieties, and let
$(X,B,\boldsymbol{\rm M})$ be a generalized lc pair. Suppose that

  1. (I)

$\boldsymbol{\rm M}$ is a finite $\mathbb{R}_{>0}$-linear combination of b-nef
$\mathbb{Q}$-b-Cartier $\mathbb{Q}$-b-divisors, and

  2. (II)

there is an effective $\mathbb{R}$-Cartier divisor $C$ on $X$ such that

    1. (II-a)

the generalized pair $(X,B+tC,\boldsymbol{\rm M})$ is generalized lc for some
$t>0$, and

    2. (II-b)

$K_{X}+B+C+\boldsymbol{\rm M}_{X}\sim_{\mathbb{R},Z}0$.

Let $A_{Z}$ be an ample $\mathbb{R}$-divisor on $Z$, and pick $0\leq
A\sim_{\mathbb{R}}\pi^{*}A_{Z}$ such that $(X,B+A,\boldsymbol{\rm M})$ is a
generalized lc pair.

Then, $K_{X}+B+A+\boldsymbol{\rm M}_{X}$ is abundant.

Before proving the theorem, we prove the following lemma.

######  Lemma 3.6.

Assume Theorem 3.5 for all projective generalized lc pairs of dimension at
most $n-1$. Let $\pi\colon X\to Z$ be a morphism, $(X,B,\boldsymbol{\rm M})$ a
projective generalized lc pair, and let $C$, $A_{Z}$ and $A$ be
$\mathbb{R}$-Cartier divisors as in Theorem 3.5 such that ${\rm dim}X\leq
n-1$. Let $f\colon\tilde{X}\to X$ be a log resolution of $(X,{\rm Supp}B)$
such that $\boldsymbol{\rm M}=\overline{\boldsymbol{\rm M}_{\tilde{X}}}$. Let
$\tilde{B}\geq 0$ be an $\mathbb{R}$-divisor on $\tilde{X}$ such that the pair
$(\tilde{X},\tilde{B}+f^{*}A)$ is an lc pair and the effective part of the
divisor $K_{\tilde{X}}+\tilde{B}+\boldsymbol{\rm
M}_{\tilde{X}}-f^{*}(K_{X}+B+\boldsymbol{\rm M}_{X})$ is $f$-exceptional.

Then, $K_{\tilde{X}}+\tilde{B}+f^{*}A+\boldsymbol{\rm M}_{\tilde{X}}$ is
abundant.

###### Proof.

We set $\tilde{A}=f^{*}A$. We can write

$K_{\tilde{X}}+\tilde{B}+\boldsymbol{\rm
M}_{\tilde{X}}-f^{*}(K_{X}+B+\boldsymbol{\rm
M}_{X})+\tilde{E}_{+}-\tilde{E}_{-}$

where $\tilde{E}_{+}\geq 0$ and $\tilde{E}_{-}\geq 0$ are $f$-exceptional
$\mathbb{R}$-divisors having no common components. We run a
$(K_{\tilde{X}}+\tilde{B}+\tilde{A}+\boldsymbol{\rm M}_{\tilde{X}})$-MMP over
$X$ with scaling of an ample divisor. By applying [HanLi, Proposition 3.8] to
$(\tilde{X},\tilde{B}+\tilde{A},\boldsymbol{\rm M})/X$, after finitely many
steps we get a projective morphism $f^{\prime}\colon\tilde{X}^{\prime}\to X$
such that

$K_{\tilde{X}^{\prime}}+\tilde{B}^{\prime}+\tilde{A}^{\prime}+\boldsymbol{\rm
M}_{\tilde{X}^{\prime}}=f^{\prime*}(K_{X}+B+\boldsymbol{\rm
M}_{X}+A)-\tilde{E}^{\prime}_{-},$

where $\tilde{B}^{\prime}$, $\tilde{A}^{\prime}$, and $\tilde{E}^{\prime}_{-}$
are the birational transforms of $\tilde{B}$, $\tilde{A}$, and $\tilde{E}_{-}$
on $\tilde{X}^{\prime}$, respectively. Then $\tilde{A}^{\prime}=f^{\prime*}A$,
and we have

$K_{\tilde{X}^{\prime}}+\tilde{B}^{\prime}+(uf^{\prime*}C+\tilde{E}^{\prime}_{-})+\boldsymbol{\rm
M}_{\tilde{X}^{\prime}}=f^{\prime*}(K_{X}+B+uC+\boldsymbol{\rm M}_{X})$

for all real numbers $u\geq 0$. By (II-a) in Theorem 3.5, we see that the
generalized pair
$\bigl{(}\tilde{X^{\prime}},\tilde{B}^{\prime}+tf^{\prime*}C+\tilde{E}^{\prime}_{-},\boldsymbol{\rm
M}\bigr{)}$ is generalized lc for some $t>0$. Putting $t^{\prime}={\rm
min}\\{1,t\\}$, we see that
$\bigl{(}\tilde{X^{\prime}},\tilde{B}^{\prime}+t^{\prime}(f^{\prime*}C+\tilde{E}^{\prime}_{-}),\boldsymbol{\rm
M}\bigr{)}$ is a generalized lc pair. Furthermore, by (II-b) in Theorem 3.5,
we obtain the relation

$K_{\tilde{X}^{\prime}}+\tilde{B}^{\prime}+(f^{\prime*}C+\tilde{E}^{\prime}_{-})+\boldsymbol{\rm
M}_{\tilde{X}^{\prime}}=f^{\prime*}(K_{X}+B+C+\boldsymbol{\rm
M}_{X})\sim_{\mathbb{R},Z}0.$

By construction, it is obvious that
$\bigl{(}\tilde{X}^{\prime},\tilde{B}^{\prime}+\tilde{A}^{\prime},\boldsymbol{\rm
M}\bigr{)}$ is a generalized lc pair. From these arguments, we can apply
Theorem 3.5 to $\pi\circ f^{\prime}\colon\tilde{X}\to Z$,
$\bigl{(}\tilde{X^{\prime}},\tilde{B}^{\prime},\boldsymbol{\rm M}\bigr{)}$,
$f^{\prime*}C+\tilde{E}^{\prime}_{-}$, $A_{Z}$ and $\tilde{A}^{\prime}$. We
see that
$K_{\tilde{X}^{\prime}}+\tilde{B}^{\prime}+\tilde{A}^{\prime}+\boldsymbol{\rm
M}_{\tilde{X}^{\prime}}$ is abundant. Since $\tilde{X}^{\prime}$ is
constructed by running a $(K_{\tilde{X}}+\tilde{B}+\tilde{A}+\boldsymbol{\rm
M}_{\tilde{X}})$-MMP, the divisor
$K_{\tilde{X}}+\tilde{B}+\tilde{A}+\boldsymbol{\rm M}_{\tilde{X}}$ is also
abundant. ∎

From now on, we prove Theorem 3.5.

###### Proof of Theorem 3.5.

The argument is very similar to [HH, Proof of Theorem 5.4]. We prove Theorem
3.5 by induction on the dimension of $X$. Assume Theorem 3.5 for all
projective generalized lc pairs of dimension $\leq n-1$. Let $\pi\colon X\to
Z$ be a projective morphism, $(X,B,\boldsymbol{\rm M})$ a projective
generalized lc pair, and let $C$, $A_{Z}$ and $A$ be $\mathbb{R}$-Cartier
divisors as in Theorem 3.5 such that ${\rm dim}X=n$. We may assume that
$K_{X}+B+A+\boldsymbol{\rm M}_{X}$ is pseudo-effective. By replacing $A$ with
a general one, we may assume that $B$ and $A$ have no common components and
all generalized lc centers of $(X,B+A,\boldsymbol{\rm M})$ are those of
$(X,B,\boldsymbol{\rm M})$. By replacing $(X,B,\boldsymbol{\rm M})$ with a
generalized $\mathbb{Q}$-factorial dlt model and replacing $C$ and $A$ with
the pullbacks, we may assume that $(X,B,\boldsymbol{\rm M})$ is a generalized
$\mathbb{Q}$-factorial dlt pair. Note that the conditions (I), (II-a), and
(II-b) still hold after replacing $(X,B,\boldsymbol{\rm M})$, $C$, and $A$.

From now on, we divide the proof into several steps. Each step corresponds to
that of [HH, Proof of Theorem 5.4].

######  Step 1.

In this step, we prove Theorem 3.5 in the case where
$K_{X}+B-\epsilon\llcorner B\lrcorner+A+\boldsymbol{\rm M}_{X}$ is pseudo-
effective for some real number $\epsilon>0$. Note that this case includes the
generalized klt case.

Since we have $K_{X}+B+C+\boldsymbol{\rm M}_{X}\sim_{\mathbb{R},Z}0$, by
restricting $K_{X}+B-\epsilon\llcorner B\lrcorner+A+\boldsymbol{\rm M}_{X}$ to
a general fiber of the Stein factorization of $\pi$, we see that $\llcorner
B\lrcorner$ and $C$ are vertical over $Z$. Then there is an open subset
$U\subset Z$ such that $(K_{X}+B-\epsilon\llcorner B\lrcorner+\boldsymbol{\rm
M}_{X})|_{\pi^{-1}(U)}\sim_{\mathbb{R},U}0$. By Proposition 3.4,
$K_{X}+B-\epsilon\llcorner B\lrcorner+A+\boldsymbol{\rm M}_{X}$ is abundant,
so $K_{X}+B-\epsilon\llcorner B\lrcorner+A+\boldsymbol{\rm
M}_{X}\sim_{\mathbb{R}}G$ for an $\mathbb{R}$-divisor $G\geq 0$. Similarly, we
see that $K_{X}+B-\tfrac{\epsilon}{2}\llcorner B\lrcorner+A+\boldsymbol{\rm
M}_{X}$ is abundant. Now we have

$\begin{split}K_{X}+B-\tfrac{\epsilon}{2}\llcorner
B\lrcorner+A+\boldsymbol{\rm
M}_{X}\sim_{\mathbb{R}}G+\tfrac{\epsilon}{2}\llcorner B\lrcorner\quad{\rm
and}\quad K_{X}+B+A+\boldsymbol{\rm M}_{X}\sim_{\mathbb{R}}G+\llcorner
B\lrcorner.\end{split}$

By [HH, Remark 2.15], we have

$\begin{split}\kappa_{\sigma}(X,K_{X}+B+A+\boldsymbol{\rm
M}_{X})&=\kappa_{\sigma}(X,K_{X}+B-\tfrac{\epsilon}{2}\llcorner
B\lrcorner+A+\boldsymbol{\rm M}_{X})\\\
&=\kappa_{\iota}(X,K_{X}+B-\tfrac{\epsilon}{2}\llcorner
B\lrcorner+A+\boldsymbol{\rm M}_{X})\\\
&=\kappa_{\iota}(X,K_{X}+B+A+\boldsymbol{\rm M}_{X}).\end{split}$

In this way, we see that $K_{X}+B+A+\boldsymbol{\rm M}_{X}$ is abundant.

######  Step 2.

By Step 1, we may assume that the divisor $K_{X}+B-\epsilon\llcorner
B\lrcorner+A+\boldsymbol{\rm M}_{X}$ is not pseudo-effective for any real
number $\epsilon>0$. Then there exists a component $S$ of $\llcorner
B\lrcorner$ such that $K_{X}+B-\epsilon S+A+\boldsymbol{\rm M}_{X}$ is not
pseudo-effective for any real number $\epsilon>0$. In this step, we will apply
[HanLiu1, Proof of Lemma 4.4] (see also [B4, Lemma 2.9]).

For any $\epsilon^{\prime}\in(0,1]$, by running a
$(K_{X}+B-\epsilon^{\prime}S+A+\boldsymbol{\rm M}_{X})$-MMP with scaling of an
ample divisor, we obtain a birational contraction $X\dashrightarrow
X^{\prime}$ and a projective generalized lc pair
$(X^{\prime},B^{\prime}-\epsilon^{\prime}S^{\prime}+A^{\prime},\boldsymbol{\rm
M})$ admitting a structure of Mori fiber space $X^{\prime}\to Z^{\prime}$,
that is, ${\rm dim}Z^{\prime}<{\rm dim}X^{\prime}$,
$-(K_{X^{\prime}}+B^{\prime}-\epsilon^{\prime}S^{\prime}+A^{\prime}+\boldsymbol{\rm
M}_{X^{\prime}})$ is ample over $Z^{\prime}$, and the relative Picard number
is $1$. With the ACC for generalized lc thresholds ([BZ, Theorem 1.5]), we can
find $\epsilon_{0}\in(0,1]$ such that if
$\epsilon^{\prime}\in(0,\epsilon_{0})$, then
$(X^{\prime},B^{\prime}+A^{\prime},\boldsymbol{\rm M})$ is generalized lc.

By (I) of Theorem 3.5, we may write $\boldsymbol{\rm
M}=\sum_{i=1}^{l}r_{l}\boldsymbol{\rm M}^{(i)}$ where $r_{i}$ are positive
real numbers and $\boldsymbol{\rm M}^{(i)}$ are b-nef $\mathbb{Q}$-b-Cartier
$\mathbb{Q}$-b-divisors. Since $X^{\prime}$ is $\mathbb{Q}$-factorial,
$\boldsymbol{\rm M}^{(i)}_{X^{\prime}}$ are $\mathbb{Q}$-Cartier divisors on
$X^{\prime}$. By construction, we may find a positive real number
$t_{\epsilon^{\prime}}<\epsilon^{\prime}$ such that
$K_{X^{\prime}}+B^{\prime}-t_{\epsilon^{\prime}}S^{\prime}+A^{\prime}+\boldsymbol{\rm
M}_{X^{\prime}}$ is numerically trivial over $Z^{\prime}$. We pick a general
fiber $F^{\prime}$ of $X^{\prime}\to Z^{\prime}$. Then
$K_{F^{\prime}}+(B^{\prime}-t_{\epsilon^{\prime}}S^{\prime}+A^{\prime}+\boldsymbol{\rm
M}_{X^{\prime}})|_{F^{\prime}}\equiv 0$ and $\boldsymbol{\rm
M}|_{F^{\prime}}=\sum_{i=1}^{l}r_{l}\boldsymbol{\rm M}^{(i)}|_{F^{\prime}}$.
Here, $\boldsymbol{\rm M}^{(i)}|_{F^{\prime}}$ are b-nef
$\mathbb{Q}$-b-Cartier $\mathbb{Q}$-b-divisors on $F^{\prime}$ and the trace
of each $\boldsymbol{\rm M}^{(i)}|_{F^{\prime}}$ on $F^{\prime}$ is a
$\mathbb{Q}$-Cartier divisor. By the negativity lemma, for each $i$, if
$\boldsymbol{\rm M}^{(i)}|_{F^{\prime}}$ is the closure of a numerically
trivial $\mathbb{Q}$-Cartier divisor then the trace of $\boldsymbol{\rm
M}^{(i)}|_{F^{\prime}}$ on $F^{\prime}$ is numerically trivial and
$\boldsymbol{\rm M}^{(i)}|_{F^{\prime}}$ descends to $F^{\prime}$. After
discarding $\boldsymbol{\rm M}^{(i)}|_{F^{\prime}}$ which are the closures of
numerically trivial divisors (such $\boldsymbol{\rm M}^{(i)}|_{F^{\prime}}$
depend on $\epsilon^{\prime}$), we can apply the global ACC [BZ, Theorem 1.6]
to the set
$\bigl{\\{}\bigl{(}F^{\prime},(B^{\prime}-t_{\epsilon^{\prime}}S^{\prime}+A^{\prime})|_{F^{\prime}},\boldsymbol{\rm
M}_{X^{\prime}}|_{F^{\prime}}\bigr{)}\,\big{|}\,\epsilon^{\prime}\in(0,1]\bigr{\\}}$.
We can find $\epsilon^{\prime}\in(0,\epsilon_{0})$ such that $X\dashrightarrow
X^{\prime}$ and $X^{\prime}\to Z^{\prime}$ as above satisfy that
$(X^{\prime},B^{\prime}+A^{\prime},\boldsymbol{\rm M})$ is generalized lc and
$K_{X^{\prime}}+B^{\prime}+A^{\prime}+\boldsymbol{\rm
M}_{X^{\prime}}\sim_{\mathbb{R},Z^{\prime}}0$.

By the above discussions, by choosing $\epsilon^{\prime}\in(0,1]$ sufficiently
small, we get a birational contraction $X\dashrightarrow X^{\prime}$ and a
contraction $X^{\prime}\to Z^{\prime}$ such that $(X^{\prime},0)$ is
$\mathbb{Q}$-factorial klt, $(X^{\prime},B^{\prime}+A^{\prime},\boldsymbol{\rm
M})$ is generalized lc, and
$K_{X^{\prime}}+B^{\prime}+A^{\prime}+\boldsymbol{\rm
M}_{X^{\prime}}\sim_{\mathbb{R},Z^{\prime}}0$.

######  Step 3.

The goal of this step is to construct a diagram

$\textstyle{\tilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\tilde{X}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$$\textstyle{\tilde{Z}}$

having good properties, where $f$ is a log resolution of $(X,{\rm Supp}B)$
such that $\boldsymbol{\rm M}=\overline{\boldsymbol{\rm M}_{\tilde{X}}}$, the
map $\tilde{X}\dashrightarrow\tilde{X}^{\prime}$ is a sequence of steps of an
MMP over $Z^{\prime}$ of a divisor
$K_{\tilde{X}}+\tilde{B}+\tilde{A}+\boldsymbol{\rm M}_{\tilde{X}}$ such that
the property of being abundant for $K_{X}+B+A+\boldsymbol{\rm M}_{X}$ follows
from that for $K_{\tilde{X}}+\tilde{B}+\tilde{A}+\boldsymbol{\rm
M}_{\tilde{X}}$, and $\tilde{X}^{\prime}\to\tilde{Z}$ is a contraction over
$Z^{\prime}$ induced by the birational transform of
$K_{\tilde{X}}+\tilde{B}+\tilde{A}+\boldsymbol{\rm M}_{\tilde{X}}$ on
$\tilde{X}^{\prime}$.

Let $\overline{f}\colon\overline{X}\to X$ and
$\overline{f}^{\prime}\colon\overline{X}\to X^{\prime}$ be a common resolution
of $X\dashrightarrow X^{\prime}$ such that $\overline{f}$ and
$\overline{f}^{\prime}$ are log resolutions of $(X,{\rm Supp}B)$ and
$(X^{\prime},{\rm Supp}B^{\prime})$ respectively and $\boldsymbol{\rm
M}=\overline{\boldsymbol{\rm M}_{\overline{X}}}$. Since $A$ is general, we may
replace $A$ by a general member of its $\mathbb{R}$-linear system without
changing the coefficients. The ACC for generalized lc thresholds guarantees
that the property of $(X^{\prime},B^{\prime}+A^{\prime},\boldsymbol{\rm M})$
being generalized lc is preserved after we replace $A$. Thus, we can replace
$A$ with a general one keeping that
$(X^{\prime},B^{\prime}+A^{\prime},\boldsymbol{\rm M})$ is generalized lc and
$K_{X^{\prime}}+B^{\prime}+A^{\prime}+\boldsymbol{\rm
M}_{X^{\prime}}\sim_{\mathbb{R},Z^{\prime}}0$. By [HH, Lemma 2.2] and
replacing $A$, we may assume that $\overline{f}$ (resp.
$\overline{f}^{\prime}$) is a log resolution of $(X,{\rm Supp}(B+A))$ (resp.
$(X^{\prime},{\rm Supp}(B^{\prime}+A^{\prime}))$) and that ${\rm
Supp}\overline{f}^{*}A$ and ${\rm Supp}\overline{f}_{*}^{-1}B\cup{\rm
Ex}(\overline{f})\cup{\rm Ex}(\overline{f}^{\prime})$ have no common
divisorial components. Then $\overline{f}^{*}A=\overline{f}_{*}^{-1}A$.
Putting $\overline{A}=\overline{f}^{*}A$, we can write

(1) $K_{\overline{X}}+\overline{B}+\overline{A}+\boldsymbol{\rm
M}_{\overline{X}}=\overline{f}^{*}(K_{X}+B+A+\boldsymbol{\rm
M}_{X})+\overline{E}$

with $\overline{B}\geq 0$ and $\overline{E}\geq 0$ which have no common
components. Then $\overline{B}+\overline{A}$ and $\overline{E}$ have no common
components and $(\overline{X},\overline{B}+\overline{A},\boldsymbol{\rm M})$
is a $\mathbb{Q}$-factorial generalized dlt pair with the nef part
$\boldsymbol{\rm M}=\overline{\boldsymbol{\rm M}_{\overline{X}}}$.

We check that the generalized lc pair
$(X^{\prime},B^{\prime}+A^{\prime},\boldsymbol{\rm M})/Z$, the morphisms
$\overline{f}^{\prime}\colon\overline{X}\to X^{\prime}$ and $X^{\prime}\to
Z^{\prime}$, and the generalized lc pair
$(\overline{X},\overline{B}+\overline{A},\boldsymbol{\rm M})$ satisfy all the
conditions of Proposition 3.2. Note that we may regard
$(X^{\prime},B^{\prime}+A^{\prime},\boldsymbol{\rm M})$ as a generalized lc
pair over $Z$ because $\boldsymbol{\rm M}$ is b-nef. By construction,
$\overline{f}^{\prime}$ is a log resolution of $(X^{\prime},{\rm
Supp}(B^{\prime}+A^{\prime}))$ and $X^{\prime}\to Z^{\prime}$ is a
contraction. Since $K_{X}+B+A+\boldsymbol{\rm M}_{X}$ is pseudo-effective,
$K_{\overline{X}}+\overline{B}+\overline{A}+\boldsymbol{\rm M}_{\overline{X}}$
is also pseudo-effective, which is the first condition of Proposition 3.2. The
relation $K_{X^{\prime}}+B^{\prime}+A^{\prime}+\boldsymbol{\rm
M}_{X^{\prime}}\sim_{\mathbb{R},Z^{\prime}}0$ is nothing but the second
condition of Proposition 3.2. By construction of $X\dashrightarrow X^{\prime}$
and $X^{\prime}\to Z$, the divisor $S^{\prime}$ is ample over $Z^{\prime}$ and
$(X^{\prime},0)$ is a klt pair. Since $X^{\prime}$ is $\mathbb{Q}$-factorial,
there is $\Delta^{\prime}$ on $X^{\prime}$ such that
$(X^{\prime},\Delta^{\prime})$ is klt and
$K_{X^{\prime}}+\Delta^{\prime}\sim_{\mathbb{R},Z^{\prime}}K_{X^{\prime}}+B^{\prime}+A^{\prime}+\boldsymbol{\rm
M}_{X^{\prime}}\sim_{\mathbb{R},Z^{\prime}}0$. Thus, we can apply Proposition
3.2 to the morphisms $(\overline{X},\overline{B}+\overline{A},\boldsymbol{\rm
M})\to X^{\prime}$ and $X^{\prime}\to Z^{\prime}$, amd there is a birational
contraction $\overline{X}\dashrightarrow\overline{X}^{\prime}$ over
$Z^{\prime}$ such that if we set $\overline{B}^{\prime}$ and
$\overline{A}^{\prime}$ as the birational transforms of $\overline{B}$ and
$\overline{A}$ on $\overline{X}^{\prime}$ respectively, then we have

  1. (i)

$K_{\overline{X}^{\prime}}+\overline{B}^{\prime}+\overline{A}^{\prime}+\boldsymbol{\rm
M}_{\overline{X}^{\prime}}$ is semi-ample over $Z^{\prime}$, and

  2. (ii)

$a(P,\overline{X},\overline{B}+\overline{A}+\boldsymbol{\rm
M}_{\overline{X}})\leq
a(P,\overline{X}^{\prime},\overline{B}^{\prime}+\overline{A}^{\prime}+\boldsymbol{\rm
M}_{\overline{X}^{\prime}})$ for any prime divisor $P$ over $\overline{X}$.

Now we have the following diagram.

$\textstyle{\overline{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{f}}$$\textstyle{\overline{X}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z}$$\textstyle{Z^{\prime}}$

We take a log resolution $\tilde{f}\colon\tilde{X}\to\overline{X}$ of
$(\overline{X},{\rm Supp}\overline{B})$ such that $\boldsymbol{\rm
M}=\overline{\boldsymbol{\rm M}_{\tilde{X}}}$ and the induced birational map
$\tilde{X}\dashrightarrow\overline{X}^{\prime}$ is a morphism. We set
$\tilde{A}=\tilde{f}^{*}\overline{A}$. We can write

(2)
$K_{\tilde{X}}+\tilde{B}+\tilde{A}=\tilde{f}^{*}(K_{\overline{X}}+\overline{B}+\overline{A})+\tilde{E}$

with $\tilde{B}\geq 0$ and $\tilde{E}\geq 0$ having no common components.
Replacing $A$, we may assume $\tilde{A}=\tilde{f}_{*}^{-1}\overline{A}$ and
that $(\tilde{X},\tilde{B}+\tilde{A})$ is a log smooth lc pair. We may apply
Lemma 3.1 to $\bigl{(}\overline{X},\overline{B}+\overline{A},\boldsymbol{\rm
M}\bigr{)}$, $\overline{X}\dashrightarrow\overline{X}^{\prime}$,
$\tilde{f}\colon\tilde{X}\to\overline{X}$, and
$\bigl{(}\tilde{X},\tilde{B}+\tilde{A},\boldsymbol{\rm M}\bigr{)}$. Indeed,
the first condition of Lemma 3.1 follows from the above (i), the second
condition of Lemma 3.1 is nothing but (ii) stated above, and the third
condition of Lemma 3.1 follows from the relation (2). By applying Lemma 3.1,
we obtain a birational contraction
$\tilde{X}\dashrightarrow\tilde{X}^{\prime}$ over $\overline{X}^{\prime}$
which is a sequence of steps of a
$(K_{\tilde{X}}+\tilde{B}+\tilde{A}+\boldsymbol{\rm M}_{\tilde{X}})$-MMP and a
projective $\mathbb{Q}$-factorial generalized dlt pair
$\bigl{(}\tilde{X}^{\prime},\tilde{B}^{\prime}+\tilde{A}^{\prime},\boldsymbol{\rm
M}\bigr{)}$ such that
$K_{\tilde{X}^{\prime}}+\tilde{B}^{\prime}+\tilde{A}^{\prime}+\boldsymbol{\rm
M}_{\tilde{X}^{\prime}}$ is equal to the pullback of
$K_{\overline{X}^{\prime}}+\overline{B}^{\prime}+\overline{A}^{\prime}+\boldsymbol{\rm
M}_{\overline{X}^{\prime}}$. By the property (i), the divisor
$K_{\tilde{X}^{\prime}}+\tilde{B}^{\prime}+\tilde{A}^{\prime}+\boldsymbol{\rm
M}_{\tilde{X}^{\prime}}$ is semi-ample over $Z^{\prime}$. Let
$\tilde{X}^{\prime}\to\tilde{Z}$ be a contraction over $Z^{\prime}$ induced by
$K_{\tilde{X}^{\prime}}+\tilde{B}^{\prime}+\tilde{A}^{\prime}+\boldsymbol{\rm
M}_{\tilde{X}^{\prime}}$. It follows that the induced morphism $\tilde{Z}\to
Z^{\prime}$ is birational because the birational transform of
$K_{\tilde{X}^{\prime}}+\tilde{B}^{\prime}+\tilde{A}^{\prime}+\boldsymbol{\rm
M}_{\tilde{X}^{\prime}}$ on $X^{\prime}$ is
$K_{X^{\prime}}+B^{\prime}+A^{\prime}+\boldsymbol{\rm
M}_{X^{\prime}}\sim_{\mathbb{R},Z}0$ which shows that the restriction of
$K_{\tilde{X}^{\prime}}+\tilde{B}^{\prime}+\tilde{A}^{\prime}+\boldsymbol{\rm
M}_{\tilde{X}^{\prime}}$ to a general fiber of $\tilde{X}^{\prime}\to
Z^{\prime}$ is numerically trivial.

######  Step 4.

We put $f=\overline{f}\circ\tilde{f}\colon\tilde{X}\to X$. We have constructed
the following diagram

---
$\textstyle{\tilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\tilde{X}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\tilde{Z}}$$\textstyle{Z}$

and projective $\mathbb{Q}$-factorial generalized dlt pairs
$\bigl{(}\tilde{X},\tilde{B}+\tilde{A},\boldsymbol{\rm M}\bigr{)}$ and
$\bigl{(}\tilde{X}^{\prime},\tilde{B}^{\prime}+\tilde{A}^{\prime},\boldsymbol{\rm
M}\bigr{)}$ such that $(\tilde{X},\tilde{B}+\tilde{A})$ is a log smooth lc
pair and the relation
$K_{\tilde{X}^{\prime}}+\tilde{B}^{\prime}+\tilde{A}^{\prime}+\boldsymbol{\rm
M}_{\tilde{X}^{\prime}}\sim_{\mathbb{R},\tilde{Z}}0$ holds. We have also
proved that the property of being abundant for $K_{X}+B+A+\boldsymbol{\rm
M}_{X}$ is reduced to that for
$K_{\tilde{X}^{\prime}}+\tilde{B}^{\prime}+\tilde{A}^{\prime}+\boldsymbol{\rm
M}_{\tilde{X}^{\prime}}$. In this step, using divisorial adjunction for
generalized pairs, we will construct new generalized lc pairs whose dimensions
are ${\rm dim}X-1$.

By the relations (1) and (2) in Step 3 and Lemma 2.8, we can write

(3) $K_{\tilde{X}}+\tilde{B}+\tilde{A}+\boldsymbol{\rm
M}_{\tilde{X}}=f^{*}(K_{X}+B+A+\boldsymbol{\rm
M}_{X})+\tilde{f}^{*}\overline{E}+\tilde{E}$

and $\tilde{B}+\tilde{A}$ and $\tilde{f}^{*}\overline{E}+\tilde{E}$ have no
common components. By (3) and that
$\tilde{X}\dashrightarrow\tilde{X}^{\prime}$ is a sequence of steps of a
$(K_{\tilde{X}}+\tilde{B}+\tilde{A}+\boldsymbol{\rm M}_{\tilde{X}})$-MMP, to
prove Theorem 3.5, it is sufficient to prove that
$K_{\tilde{X}^{\prime}}+\tilde{B}^{\prime}+\tilde{A}^{\prime}+\boldsymbol{\rm
M}_{\tilde{X}}$ is abundant. We recall that $S$ is a component of $\llcorner
B\lrcorner$ and the birational transform $S^{\prime}$ on $X^{\prime}$ is ample
over $Z^{\prime}$. Let $\tilde{S}$ be the birational transform of $S$ on
$\tilde{X}$. Then $\tilde{S}$ is not contracted by the map
$\tilde{X}\dashrightarrow\tilde{X}^{\prime}$ because
$K_{\tilde{X}}+\tilde{B}-\epsilon\tilde{S}+\tilde{A}+\boldsymbol{\rm
M}_{\tilde{X}}$ is not pseudo-effective for any $\epsilon>0$ (this follows
from the second sentence in Step 2). Furthermore, the bitarional transform
$\tilde{S}^{\prime}$ of $S$ on $\tilde{X^{\prime}}$ dominates $\tilde{Z}$
because $S^{\prime}$ is ample over $Z^{\prime}$ and $\tilde{Z}\to Z^{\prime}$
is birational. Then the natural morphism $\tilde{S}^{\prime}\to\tilde{Z}$ is
surjective and
$K_{\tilde{X}^{\prime}}+\tilde{B}^{\prime}+\tilde{A}^{\prime}+\boldsymbol{\rm
M}_{\tilde{X}^{\prime}}\sim_{\mathbb{R},\tilde{Z}}0$. Thus, it is sufficient
to prove that
$(K_{\tilde{X}^{\prime}}+\tilde{B}^{\prime}+\tilde{A}^{\prime}+\boldsymbol{\rm
M}_{\tilde{X}^{\prime}})|_{\tilde{S}^{\prime}}$ is abundant ([HH, Remark 2..8
(2)]).

By taking a suitable resolution of the normalization of the graph of
$\tilde{X}\dashrightarrow\tilde{X}^{\prime}$, we can find a common resolution
$Y\to\tilde{X}$ and $Y\to\tilde{X}^{\prime}$ and a subvariety $T\subset Y$
which is birational to $\tilde{S}$ and $\tilde{S}^{\prime}$ such that the
induced morphisms $T\to\tilde{S}$ and $T\to\tilde{S}^{\prime}$ form a common
resolution of $\tilde{S}\dashrightarrow\tilde{S}^{\prime}$. Since $T$ has
codimension one in $Y$, replacing $Y$ by a higher resolution if necessary, we
may assume $\boldsymbol{\rm M}=\overline{\boldsymbol{\rm M}_{Y}}$. Pick
$M_{T}\sim_{\mathbb{R}}\boldsymbol{\rm M}_{Y}|_{T}$ which is an
$\mathbb{R}_{>0}$-linear combination of nef Cartier divisors on $T$, and
denote the b-divisor $\overline{M_{T}}$ by $\boldsymbol{\rm N}$. With this
$\boldsymbol{\rm N}$, by applying divisorial adjunction to
$(X,B,\boldsymbol{\rm M})$, $\bigl{(}\tilde{X},\tilde{B},\boldsymbol{\rm
M}\bigr{)}$, and
$\bigl{(}\tilde{X}^{\prime},\tilde{B}^{\prime},\boldsymbol{\rm M}\bigr{)}$, we
may construct generalized pairs

$\begin{split}(S,B_{S},\boldsymbol{\rm
N}),\quad(\tilde{S},B_{\tilde{S}},\boldsymbol{\rm N}),\quad{\rm
and}\quad(\tilde{S}^{\prime},B_{\tilde{S}^{\prime}},\boldsymbol{\rm
N}).\end{split}$

Here, we regard $\boldsymbol{\rm N}=\overline{M_{T}}$ as b-divisors on $S$,
$\tilde{S}$, and $\tilde{S}^{\prime}$. We take a common resolution
$\tau\colon\tilde{T}\to\tilde{S}$ and
$\tau^{\prime}\colon\tilde{T}\to\tilde{S}^{\prime}$ of the birational map
$\tilde{S}\dashrightarrow\tilde{S}^{\prime}$ such that $\tau$ and
$\tau^{\prime}$ are log resolutions of $(\tilde{S},{\rm Supp}B_{\tilde{S}})$
and $(\tilde{S}^{\prime},{\rm Supp}B_{\tilde{S}^{\prime}})$ respectively,
$\tau^{\prime-1}_{*}B_{\tilde{S}^{\prime}}\cup{\rm Ex}(\tau)\cup{\rm
Ex}(\tau^{\prime})$ is a simple normal crossing divisor, and the induced map
$\tilde{T}\dashrightarrow T$ is a morphism. We can regarded $\boldsymbol{\rm
N}$ as a b-divisor on $\tilde{T}$. We set

$A_{S}=A|_{S},\quad A_{\tilde{S}}=\tilde{A}|_{\tilde{S}},\quad{\rm and}\quad
A_{\tilde{S}^{\prime}}=\tilde{A}^{\prime}|_{\tilde{S}^{\prime}}.$

By construction of divisorial adjunction for generalized pairs [BZ, Definition
4.7], the generalized pair $(S,B_{S}+A_{S},\boldsymbol{\rm N})$ is equal to
the generalized pair constructed with the divisorial adjunction for
$(X,B+A,\boldsymbol{\rm M})$. Therefore, $(S,B_{S}+A_{S},\boldsymbol{\rm N})$
is generalized lc. By the same reasons,
$(\tilde{S},B_{\tilde{S}}+A_{\tilde{S}},\boldsymbol{\rm N})$ and
$(\tilde{S}^{\prime},B_{\tilde{S}^{\prime}}+A_{\tilde{S}^{\prime}},\boldsymbol{\rm
N})$ are generalized lc. Since

$K_{\tilde{S}^{\prime}}+B_{\tilde{S}^{\prime}}+A_{\tilde{S}^{\prime}}+\boldsymbol{\rm
N}_{\tilde{S}^{\prime}}\sim_{\mathbb{R}}(K_{\tilde{X}^{\prime}}+\tilde{B}^{\prime}+\tilde{A}^{\prime}+\boldsymbol{\rm
M}_{\tilde{X}^{\prime}})|_{\tilde{S}^{\prime}},$

to prove Theorem 3.5, it is sufficient to prove that
$K_{\tilde{S}^{\prime}}+B_{\tilde{S}^{\prime}}+A_{\tilde{S}^{\prime}}+\boldsymbol{\rm
N}_{\tilde{S}^{\prime}}$ is abundant. Since
$A_{\tilde{S}}=f^{*}A|_{\tilde{S}}$, by [HH, Lemma 2.1] and [HH, Lemma 2.2]
and replacing $A$ with a general one, we may assume $A_{\tilde{S}}\geq 0$,
$A_{\tilde{S}^{\prime}}\geq 0$,
$\tau^{*}A_{\tilde{S}}\leq\tau^{\prime-1}_{*}A_{\tilde{S}^{\prime}}$, and
$\tau^{\prime}$ is a log resolution of $\bigl{(}\tilde{S}^{\prime},{\rm
Supp}(B_{\tilde{S}^{\prime}}+A_{\tilde{S}^{\prime}})\bigr{)}$.

######  Step 5.

Let $\pi_{S}\colon S\to Z$ be the restriction of $\pi\colon X\to Z$ to $S$,
and let $f_{\tilde{S}}\colon\tilde{S}\to S$ be the restriction of
$f\colon\tilde{X}\to X$ to $\tilde{S}$. We have the following diagram

$\textstyle{\tilde{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\scriptstyle{\tau^{\prime}}$$\textstyle{\tilde{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\tilde{S}}}$$\textstyle{\tilde{S}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{S}}$$\textstyle{\tilde{Z}}$$\textstyle{Z}$

and generalized lc pairs $(S,B_{S}+A_{S},\boldsymbol{\rm N})$,
$(\tilde{S},B_{\tilde{S}}+A_{\tilde{S}},\boldsymbol{\rm N})$, and
$(\tilde{S}^{\prime},B_{\tilde{S}^{\prime}}+A_{\tilde{S}^{\prime}},\boldsymbol{\rm
N})$. By Step 4, the property of being abundant for
$K_{\tilde{X}^{\prime}}+\tilde{B}^{\prime}+\tilde{A}^{\prime}+\boldsymbol{\rm
M}_{\tilde{X}^{\prime}}$ was reduced to that for
$K_{\tilde{S}^{\prime}}+B_{\tilde{S}^{\prime}}+A_{\tilde{S}^{\prime}}+\boldsymbol{\rm
N}_{\tilde{S}^{\prime}}$. In this step, we will study the morphism
$(S,B_{S}+A_{S},\boldsymbol{\rm N})\to Z$, and we will construct a generalized
lc pair on $\tilde{T}$ by using
$(\tilde{S}^{\prime},B_{\tilde{S}^{\prime}}+A_{\tilde{S}^{\prime}},\boldsymbol{\rm
N})$.

We check that $\pi_{S}\colon S\to Z$, $(S,B_{S},\boldsymbol{\rm N})$,
$C|_{S}$, and $A_{S}\sim_{\mathbb{R}}\pi_{S}^{*}A_{Z}$ satisfy conditions of
Theorem 3.5. By construction, $(S,B_{S},\boldsymbol{\rm N})$ is a generalized
lc pair such that $\boldsymbol{\rm N}=\overline{M_{T}}$ is a finite
$\mathbb{R}_{>0}$-linear combination of b-nef $\mathbb{Q}$-b-Cartier
$\mathbb{Q}$-b-divisors. Thus, condition (I) of Theorem 3.5 is satisfied.
Since $(X,B+tC,\boldsymbol{\rm M})$ is a generalized lc pair for some $t>0$
and $S$ is a component of $\llcorner B\lrcorner$, the divisor $C|_{S}$ is a
well-defined effective $\mathbb{R}$-Cartier divisor on $S$. By construction of
divisorial adjunction for generalized pairs [BZ, Definition 4.7], the
generalized pair $(S,B_{S}+tC|_{S},\boldsymbol{\rm N})$ is equal to the
generalized pair constructed with the divisorial adjunction for
$(X,B+tC,\boldsymbol{\rm M})$. Since $(X,B+tC,\boldsymbol{\rm M})$ is
generalized lc, we see that $(S,B_{S}+tC|_{S},\boldsymbol{\rm N})$ is
generalized lc. Thus, condition (II-a) of Theorem 3.5 is satisfied. By
construction, we have

$K_{S}+B_{S}+C|_{S}+\boldsymbol{\rm
N}_{S}\sim_{\mathbb{R}}(K_{X}+B+C+\boldsymbol{\rm
M}_{X})|_{S}\sim_{\mathbb{R},Z}0.$

Thus, condition (II-b) of Theorem 3.5 is satisfied. Finally,
$(S,B_{S}+A_{S},\boldsymbol{\rm N})$ is generalized lc. From these
discussions, we see that $\pi_{S}\colon S\to Z$, $(S,B_{S},\boldsymbol{\rm
N})$, $C|_{S}$, and $A_{S}\sim_{\mathbb{R}}\pi_{S}^{*}A_{Z}$ satisfy
conditions of Theorem 3.5. By induction hypothesis of Theorem 3.5, the divisor
$K_{S}+B_{S}+A_{S}+\boldsymbol{\rm N}_{S}$ is abundant.

We put $A_{\tilde{T}}=\tau^{*}A_{\tilde{S}}$, which is equal to the pullback
of $A_{S}=A|_{S}$ to $\tilde{T}$. Thanks to the relation
$\tau^{*}A_{\tilde{S}}\leq\tau^{\prime-1}_{*}A_{\tilde{S}^{\prime}}$, we can
write

(4) $K_{\tilde{T}}+\Psi_{\tilde{T}}+A_{\tilde{T}}+\boldsymbol{\rm
N}_{\tilde{T}}=\tau^{\prime*}(K_{\tilde{S}^{\prime}}+B_{\tilde{S}^{\prime}}+A_{\tilde{S}^{\prime}}+\boldsymbol{\rm
N}_{\tilde{S}^{\prime}})+E_{\tilde{T}}$

where $\Psi_{\tilde{T}}+A_{\tilde{T}}$ and $E_{\tilde{T}}$ have no common
components. Note that $(\tilde{T},\Psi_{\tilde{T}}+A_{\tilde{T}})$ is a log
smooth lc pair because $\tau^{\prime}\colon\tilde{T}\to\tilde{S}^{\prime}$ is
a log resolution of $\bigl{(}\tilde{S}^{\prime},{\rm
Supp}(B_{\tilde{S}^{\prime}}+A_{\tilde{S}^{\prime}})\bigr{)}$. By (4) above,
it is sufficient to prove that
$K_{\tilde{T}}+\Psi_{\tilde{T}}+A_{\tilde{T}}+\boldsymbol{\rm N}_{\tilde{T}}$
is abundant.

######  Step 6.

In this step, we prove that
$K_{\tilde{T}}+\Psi_{\tilde{T}}+A_{\tilde{T}}+\boldsymbol{\rm N}_{\tilde{T}}$
is abundant. We may write

$K_{\tilde{T}}+\Psi_{\tilde{T}}+A_{\tilde{T}}+\boldsymbol{\rm
N}_{\tilde{T}}=(f_{\tilde{S}}\circ\tau)^{*}(K_{S}+B_{S}+A_{S}+\boldsymbol{\rm
N}_{S})+\Xi_{+}-\Xi_{-},$

where $\Xi_{+}\geq 0$ and $\Xi_{-}\geq 0$ have no common components. Since
$A_{\tilde{T}}=(f_{\tilde{S}}\circ\tau)^{*}A_{S}$, to apply Lemma 3.6, we show
that $\Xi_{+}$ is exceptional over $S$.

Suppose by contradiction that there is a component $Q$ of $\Xi_{+}$ which is
not exceptional over $S$. Then

$a(Q,\tilde{T},\Psi_{\tilde{T}}+A_{\tilde{T}}+\boldsymbol{\rm
N}_{\tilde{T}})<a(Q,S,B_{S}+A_{S}+\boldsymbol{\rm N}_{S})\leq 1.$

By (3) in Step 4 and Lemma 2.9, we have

$a(Q,S,B_{S}+A_{S}+\boldsymbol{\rm
N}_{S})=a(Q,\tilde{S},B_{\tilde{S}}+A_{\tilde{S}}+\boldsymbol{\rm
N}_{\tilde{S}})$

Since the birational map $\tilde{X}\dashrightarrow\tilde{X}^{\prime}$ is a
sequence of steps of a $(K_{\tilde{X}}+\tilde{B}+\tilde{A}+\boldsymbol{\rm
M}_{\tilde{X}})$-MMP, by taking a suitable common resolution of
$\tilde{X}\dashrightarrow\tilde{X}^{\prime}$ and the negativity lemma (that is
the generalized pair analogue of [Fu1, Lemma 4.2.10]), we have

$a(Q,\tilde{S},B_{\tilde{S}}+A_{\tilde{S}}+\boldsymbol{\rm N}_{\tilde{S}})\leq
a(Q,\tilde{S}^{\prime},B_{\tilde{S}^{\prime}}+A_{\tilde{S}^{\prime}}+\boldsymbol{\rm
N}_{\tilde{S}^{\prime}}).$

Finally, since $\Psi_{\tilde{T}}+A_{\tilde{T}}$ and $E_{\tilde{T}}$ have no
common components (see (4) in Step 5), we can write

$a(Q,\tilde{T},\Psi_{\tilde{T}}+A_{\tilde{T}}+\boldsymbol{\rm
N}_{\tilde{T}})={\rm
min}\\{a(Q,\tilde{S}^{\prime},B_{\tilde{S}^{\prime}}+A_{\tilde{S}^{\prime}}+\boldsymbol{\rm
N}_{\tilde{S}^{\prime}}),1\\}.$

By combining these relations, we obtain

$\begin{split}a(Q,S,B_{S}+A_{S}+\boldsymbol{\rm N}_{S})&={\rm
min}\\{a(Q,S,B_{S}+A_{S}+\boldsymbol{\rm N}_{S}),1\\}\\\ &={\rm
min}\\{a(Q,\tilde{S},B_{\tilde{S}}+A_{\tilde{S}}+\boldsymbol{\rm
N}_{\tilde{S}}),1\\}\\\ &\leq{\rm
min}\\{a(Q,\tilde{S}^{\prime},B_{\tilde{S}^{\prime}}+A_{\tilde{S}^{\prime}}+\boldsymbol{\rm
N}_{\tilde{S}^{\prime}}),1\\}\\\
&=a(Q,\tilde{T},\Psi_{\tilde{T}}+A_{\tilde{T}}+\boldsymbol{\rm
N}_{\tilde{T}})\\\ &<a(Q,S,B_{S}+A_{S}+\boldsymbol{\rm N}_{S}),\end{split}$

which is a contradiction. In this way, we see that $\Xi_{+}$ is exceptional
over $S$. By applying Lemma 3.6 to $(S,B_{S},\boldsymbol{\rm N})\to Z$,
$A_{S}\sim_{\mathbb{R}}\pi^{*}_{S}A_{Z}$, $\tilde{T}\to S$, and
$\Psi_{\tilde{T}}$, we see that the divisor
$K_{\tilde{T}}+\Psi_{\tilde{T}}+A_{\tilde{T}}+\boldsymbol{\rm N}_{\tilde{T}}$
is abundant. We finish this step.

By (4) in Step 5, the divisor
$K_{\tilde{S}^{\prime}}+B_{\tilde{S}^{\prime}}+A_{\tilde{S}^{\prime}}+\boldsymbol{\rm
N}_{\tilde{S}^{\prime}}$ is abundant. By the arguments in Step 4, the divisor
$K_{\tilde{X}^{\prime}}+\tilde{B}^{\prime}+\tilde{A}^{\prime}+\boldsymbol{\rm
M}_{\tilde{X}^{\prime}}$ is abundant. By construction of
$\tilde{X}\dashrightarrow\tilde{X}^{\prime}$ in Step 3, the divisor
$K_{\tilde{X}}+\tilde{B}+\tilde{A}+\boldsymbol{\rm M}_{\tilde{X}}$ is
abundant. Finally, using (3) in Step 4, we see that the divisor
$K_{X}+B+A+\boldsymbol{\rm M}_{X}$ is abundant. We are done. ∎

##  4\. Proof of main results

In this section, we prove the main result of this paper.

###  4.1. Non-vanishing theorem

In this subsection, we firstly prove Theorem 1.3 , then we prove Theorem 1.1,
and finally we prove Theorem 4.4 which is a non-$\mathbb{R}$-Cartier analogue
of Theorem 1.1. Theorem 1.2 is a special case of Theorem 4.4.

######  Theorem 4.1.

Let $\pi\colon X\to Z$ be a morphism of normal projective varieties, and let
$(X,B,\boldsymbol{\rm M})$ be a $\mathbb{Q}$-factorial generalized dlt pair.
Suppose that

  1. (I)

$\boldsymbol{\rm M}$ is a finite $\mathbb{R}_{>0}$-linear combination of b-nef
$\mathbb{Q}$-b-Cartier $\mathbb{Q}$-b-divisors, and

  2. (II)

there is an effective $\mathbb{R}$-divisor $C$ on $X$ such that

    1. (II-a)

the generalized pair $(X,B+tC,\boldsymbol{\rm M})$ is generalized lc for some
$t>0$, and

    2. (II-b)

$K_{X}+B+C+\boldsymbol{\rm M}_{X}\sim_{\mathbb{R},Z}0$.

Let $A_{Z}$ be an ample $\mathbb{R}$-divisor on $Z$. Pick
$A\sim_{\mathbb{R}}\pi^{*}A_{Z}$ so that $A\geq 0$ and $(X,B+A,\boldsymbol{\rm
M})$ is a generalized lc pair whose generalized lc centers are those of
$(X,B,\boldsymbol{\rm M})$.

We put $\Delta=B+A$. Then, for any sequence of steps of a
$(K_{X}+\Delta+\boldsymbol{\rm M}_{X})$-MMP

$(X,\Delta,\boldsymbol{\rm M})=:(X_{0},\Delta_{0},\boldsymbol{\rm
M})\dashrightarrow\cdots\dashrightarrow(X_{i},\Delta_{i},\boldsymbol{\rm
M})\dashrightarrow\cdots,$

the divisor $K_{X_{i}}+\Delta_{i}+\boldsymbol{\rm M}_{X_{i}}$ is log abundant
with respect to $(X_{i},\Delta_{i},\boldsymbol{\rm M})$ for every $i$.
Furthermore, if $(X,B,\boldsymbol{\rm M})$ is generalized klt and
$K_{X}+B+A+\boldsymbol{\rm M}_{X}$ is pseudo-effective, then
$K_{X}+B+A+\boldsymbol{\rm M}_{X}$ birationally has the Nakayama–Zariski
decomposition with semi-ample positive part.

###### Proof.

When $(X,B,\boldsymbol{\rm M})$ is generalized klt and
$K_{X}+B+A+\boldsymbol{\rm M}_{X}$ is pseudo-effective, the divisor $C$ of
(II) is vertical over $Z$. Thus, we can find an open subset $U\subset Z$ such
that $(K_{X}+B+\boldsymbol{\rm M}_{X})|_{\pi^{-1}(U)}\sim_{\mathbb{R},U}0$.
Then the second assertions directly follows from Proposition 3.4. So it is
enough to show the property of being log abundant.

The strategy is very similar to [HH, Proof of Theorem 5.5]. Let
$(X,B,\boldsymbol{\rm M})\to Z$ and $A$ be as in Theorem 4.1. Put
$\Delta=B+A$, and let

$(X,\Delta,\boldsymbol{\rm M})=:(X_{0},\Delta_{0},\boldsymbol{\rm
M})\dashrightarrow\cdots\dashrightarrow(X_{i},\Delta_{i},\boldsymbol{\rm
M})\dashrightarrow\cdots$

be a sequence of steps of a $(K_{X}+\Delta+\boldsymbol{\rm M}_{X})$-MMP. Fix
an index $i$. By Theorem 3.5, the divisor $K_{X}+B+A+\boldsymbol{\rm M}_{X}$
is abundant, so $K_{X_{i}}+\Delta_{i}+\boldsymbol{\rm M}_{X_{i}}$ is also
abundant. Therefore, it is sufficient to prove that
$(K_{X_{i}}+\Delta_{i}+\boldsymbol{\rm M}_{X_{i}})|_{S_{i}}$ is abundant for
every generalized lc center $S_{i}$ of $(X_{i},\Delta_{i},\boldsymbol{\rm
M})$. We fix a generalized lc center $S_{i}$ of
$(X_{i},\Delta_{i},\boldsymbol{\rm M})$. Then there is a generalized lc center
$S$ of $(X,\Delta,\boldsymbol{\rm M})$ such that the map $X\dashrightarrow
X_{i}$ induces a birational map $S\dashrightarrow S_{i}$.

Let $f\colon(X^{\prime},\Delta^{\prime},\boldsymbol{\rm
M})\to(X,B+A,\boldsymbol{\rm M})$ be a $\mathbb{Q}$-factorial generalized dlt
model such that there is a component $S^{\prime}$ of
$\llcorner\Delta^{\prime}\lrcorner$ from which $f$ induces a surjective
morphism $S^{\prime}\to S$. Such model always exists (Remark 2.7 (2)). Taking
a lift of the $(K_{X}+\Delta+\boldsymbol{\rm M}_{X})$-MMP as in [HanLi, 3.5],
we get a sequence of steps of a
$(K_{X^{\prime}}+\Delta^{\prime}+\boldsymbol{\rm M}_{X^{\prime}})$-MMP

$(X^{\prime},\Delta^{\prime},\boldsymbol{\rm
M})=:(X^{\prime}_{0},\Delta^{\prime}_{0},\boldsymbol{\rm
M})\dashrightarrow\cdots\dashrightarrow(X^{\prime}_{k_{i}},\Delta^{\prime}_{k_{i}},\boldsymbol{\rm
M})$

and $\mathbb{Q}$-factorial generalized dlt models
$f_{i}\colon(X^{\prime}_{k_{i}},\Delta^{\prime}_{k_{i}},\boldsymbol{\rm
M})\to(X_{i},\Delta_{i},\boldsymbol{\rm M})$ such that the birational
transform $S^{\prime}_{k_{i}}$ of $S^{\prime}$ on $X^{\prime}_{k_{i}}$ is a
component of $\llcorner\Delta^{\prime}_{k_{i}}\lrcorner$ and $f_{i}\colon
X^{\prime}_{k_{i}}\to X_{i}$ induces a surjective morphism
$S^{\prime}_{k_{i}}\to S_{i}$. By construction, we have

$K_{X^{\prime}_{k_{i}}}+\Delta^{\prime}_{k_{i}}+\boldsymbol{\rm
M}_{X^{\prime}_{k_{i}}}=f_{i}^{*}(K_{X_{i}}+\Delta_{i}+\boldsymbol{\rm
M}_{X_{i}}),$

thus $(K_{X_{i}}+\Delta_{i}+\boldsymbol{\rm M}_{X_{i}})|_{S_{i}}$ is abundant
if and only if
$(K_{X^{\prime}_{k_{i}}}+\Delta^{\prime}_{k_{i}}+\boldsymbol{\rm
M}_{X^{\prime}_{k_{i}}})|_{S^{\prime}_{k_{i}}}$ is abundant. Furthermore, it
is easy to check that the conditions (I), (II-a), and (II-b) in Theorem 4.1
hold after replacing $(X,B+A,\boldsymbol{\rm M})$ by
$(X^{\prime},\Delta^{\prime},\boldsymbol{\rm M})$. From these arguments, by
replacing the $(K_{X}+\Delta+\boldsymbol{\rm M}_{X})$-MMP and $S_{i}$, we may
assume that $S_{i}$ is a component of $\llcorner\Delta_{i}\lrcorner$. By the
assumption of Theorem 4.1 that $(X,B,\boldsymbol{\rm M})$ is a
$\mathbb{Q}$-factorial generalized dlt pair and $(X,B+A,\boldsymbol{\rm M})$
is a generalized lc pair whose generalized lc centers are those of
$(X,B,\boldsymbol{\rm M})$, it follows that $S$ is a component of $\llcorner
B\lrcorner$. Let $B_{i}$ and $A_{i}$ be the birational transforms of $B$ and
$A$ on $X_{i}$, respectively. Then $S_{i}$ is a component of $\llcorner
B_{i}\lrcorner$.

Taking a suitable resolution of the normalization of the graph of
$X\dashrightarrow X_{i}$, we can find a common resolution $Y\to X$ and $Y\to
X_{i}$ of $X\dashrightarrow X_{i}$ and a subvariety $T\subset Y$ which is
birational to $S$ and $S_{i}$ such that the induced morphisms $T\to S$ and
$T\to S_{i}$ form a common resolution of $S\dashrightarrow S_{i}$. Since $T$
has codimension one in $Y$, replacing $Y$ by a higher resolution if necessary,
we may assume $\boldsymbol{\rm M}=\overline{\boldsymbol{\rm M}_{Y}}$. We pick
$M_{T}\sim_{\mathbb{R}}\boldsymbol{\rm M}_{Y}|_{T}$ which is an
$\mathbb{R}_{>0}$-linear combination of nef Cartier divisors on $T$, and we
denote the b-divisor $\overline{M_{T}}$ by $\boldsymbol{\rm N}$. With this
$\boldsymbol{\rm N}$, by applying divisorial adjunction to
$(X,B,\boldsymbol{\rm M})$ and $(X_{i},B_{i},\boldsymbol{\rm M})$, we
construct generalized pairs

$(S,B_{S},\boldsymbol{\rm N})\quad{\rm
and}\quad(S_{i},B_{S_{i}},\boldsymbol{\rm N}).$

Here, we regard $\boldsymbol{\rm N}$ as b-divisors on $S$ and $S_{i}$. We take
a common resolution $\tau\colon\tilde{T}\to S$ and
$\tau^{\prime}\colon\tilde{T}\to S_{i}$ of the birational map
$S\dashrightarrow S_{i}$ such that $\tau$ and $\tau^{\prime}$ are log
resolutions of $(S,{\rm Supp}B_{S})$ and $(S_{i},{\rm Supp}B_{S_{i}})$
respectively, $\tau^{\prime-1}_{*}B_{S_{i}}\cup{\rm Ex}(\tau)\cup{\rm
Ex}(\tau^{\prime})$ is a simple normal crossing divisor, and the induced
birational map $\tilde{T}\dashrightarrow T$ is a morphism. We can regard
$\boldsymbol{\rm N}$ as a b-divisor on $\tilde{T}$. We set

$A_{S}=A|_{S}\quad{\rm and}\quad A_{S_{i}}=A_{i}|_{S_{i}}.$

By construction of divisorial adjunction for generalized pairs [BZ, Definition
4.7], the generalized pair $(S,B_{S}+A_{S},\boldsymbol{\rm N})$ is equal to
the generalized pair constructed with the divisorial adjunction for
$(X,B+A,\boldsymbol{\rm M})$. Therefore, $(S,B_{S}+A_{S},\boldsymbol{\rm N})$
is generalized lc. By the same reason,
$(S_{i},B_{S_{i}}+A_{S_{i}},\boldsymbol{\rm N})$ is also generalized lc. By
[HH, Lemma 2.1] and [HH, Lemma 2.2] and replacing $A$ with a general one, we
may assume $A_{S}\geq 0$, $A_{S_{i}}\geq 0$,
$\tau^{*}A_{S}\leq\tau^{\prime-1}_{*}A_{S_{i}}$, and
$\tau^{\prime}\colon\tilde{T}\to S_{i}$ is a log resolution of
$\bigl{(}S_{i},{\rm Supp}(B_{S_{i}}+A_{S_{i}})\bigr{)}$. Since

$K_{S_{i}}+B_{S_{i}}+A_{S_{i}}+\boldsymbol{\rm
N}_{S_{i}}\sim_{\mathbb{R}}(K_{X_{i}}+\Delta_{i}+\boldsymbol{\rm
M}_{X_{i}})|_{S_{i}},$

it is sufficient to prove that $K_{S_{i}}+B_{S_{i}}+A_{S_{i}}+\boldsymbol{\rm
N}_{S_{i}}$ is abundant.

Since $\tau^{*}A_{S}\leq\tau^{\prime-1}_{*}A_{S_{i}}$, we can write

($\clubsuit$)
$\begin{split}K_{\tilde{T}}+\Psi_{\tilde{T}}+\tau^{*}A_{S}+\boldsymbol{\rm
N}_{\tilde{T}}=\tau^{\prime*}(K_{S_{i}}+B_{S_{i}}+A_{S_{i}}+\boldsymbol{\rm
N}_{S_{i}})+E_{\tilde{T}}\end{split}$

with $\Psi_{\tilde{T}}\geq 0$ and $E_{\tilde{T}}\geq 0$ such that
$\Psi_{\tilde{T}}+\tau^{*}A_{S}$ and $E_{\tilde{T}}$ have no common
components. Since $\tau^{\prime}\colon\tilde{T}\to S_{i}$ is a log resolution
of $\bigl{(}S_{i},{\rm Supp}(B_{S_{i}}+A_{S_{i}})\bigr{)}$, the pair
$(\tilde{T},\Psi_{\tilde{T}}+\tau^{*}A_{S})$ is a log smooth lc pair. Now we
can write

$K_{\tilde{T}}+\Psi_{\tilde{T}}+\tau^{*}A_{S}+\boldsymbol{\rm
N}_{\tilde{T}}=\tau^{*}(K_{S}+B_{S}+A_{S}+\boldsymbol{\rm
N}_{S})+\Xi_{+}-\Xi_{-}$

such that $\Xi_{+}\geq 0$ and $\Xi_{-}\geq 0$ have no common components. Pick
a component $Q$ of $\Xi_{+}$. Since $X\dashrightarrow X_{i}$ is a sequence of
steps of a $(K_{X}+B+A+\boldsymbol{\rm M}_{X})$-MMP, by taking a suitable
common resolution of $X\dashrightarrow X_{i}$ and the negativity lemma (or
applying the generalized pair analogue of [Fu1, Lemma 4.2.10]), we have

$a(Q,S,B_{S}+A_{S}+\boldsymbol{\rm N}_{S})\leq
a(Q,S_{i},B_{S_{i}}+A_{S_{i}}+\boldsymbol{\rm N}_{S_{i}}).$

By construction of $\Psi_{\tilde{T}}$ and $E_{\tilde{T}}$, we can write

$a(Q,\tilde{T},\Psi_{\tilde{T}}+\tau^{*}A_{S}+\boldsymbol{\rm
N}_{\tilde{T}})={\rm min}\\{a(Q,S_{i},B_{S_{i}}+A_{S_{i}}+\boldsymbol{\rm
N}_{S_{i}}),1\\}.$

Thus, we obtain

$\begin{split}a(Q,S,B_{S}+A_{S}+\boldsymbol{\rm
N}_{S})&>a(Q,\tilde{T},\Psi_{\tilde{T}}+\tau^{*}A_{S}+\boldsymbol{\rm
N}_{\tilde{T}})\\\ &={\rm min}\\{a(Q,S_{i},B_{S_{i}}+A_{S_{i}}+\boldsymbol{\rm
N}_{S_{i}}),1\\}\\\ &\geq{\rm min}\\{a(Q,S,B_{S}+A_{S}+\boldsymbol{\rm
N}_{S}),1\\}.\end{split}$

Thus, it follows that $a(Q,S,B_{S}+A_{S}+\boldsymbol{\rm N}_{S})>1$ which
shows that $Q$ is $\tau$-exceptional. Therefore, $\Xi_{+}$ is
$\tau$-exceptional.

Now we can apply Lemma 3.6 to $(S,B_{S},\boldsymbol{\rm N})\to Z$, $C|_{S}$,
$A_{S}$, $\tau\colon\tilde{T}\to S$, and $\Psi_{\tilde{T}}$. Then we see that
$K_{\tilde{T}}+\Psi_{\tilde{T}}+\tau^{*}A_{S}+\boldsymbol{\rm N}_{\tilde{T}}$
is abundant. By ($\clubsuit$ ‣ 4.1), the divisor
$K_{S_{i}}+B_{S_{i}}+A_{S_{i}}+\boldsymbol{\rm N}_{S_{i}}$ is abundant,
therefore $(K_{X_{i}}+\Delta_{i}+\boldsymbol{\rm M}_{X_{i}})|_{S_{i}}$ is
abundant. Since $i$ is arbitrary index and $S_{i}$ is arbitrary generalized lc
center of $(X_{i},\Delta_{i},\boldsymbol{\rm M})$, we see that the divisor
$K_{X_{i}}+\Delta_{i}+\boldsymbol{\rm M}_{X_{i}}$ is log abundant with respect
to $(X_{i},\Delta_{i},\boldsymbol{\rm M})$ for every $i$. So we are done. ∎

###### Proof of Theorem 1.3.

The theorem is the case of Theorem 4.1 where $\pi$ is the morphism of a
$\mathbb{Q}$-factorial dlt model and $C=0$. ∎

###### Proof of Theorem 1.1.

The first statement immediately follows from Theorem 3.5. Thus, it is enough
to prove the second statement. Suppose that $\boldsymbol{\rm M}_{X}$ is
$\mathbb{R}$-Cartier. Fix a real number $\alpha\geq 1$. We pick a real number
$\epsilon\in(0,1)$ such that $\epsilon\boldsymbol{\rm M}_{X}+A$ is ample, and
we pick general $A_{\epsilon}\sim_{\mathbb{R}}\epsilon\boldsymbol{\rm
M}_{X}+A$. Then $K_{X}+B+A_{\epsilon}+(1-\epsilon)\boldsymbol{\rm M}_{X}$ is
pseudo-effective. Let $f\colon\tilde{X}\to X$ be a log resolution of $(X,{\rm
Supp}B)$ such that $\boldsymbol{\rm M}=\overline{\boldsymbol{\rm
M}_{\tilde{X}}}$. We may write

$K_{\tilde{X}}+\tilde{B}+(1-\epsilon)\boldsymbol{\rm
M}_{\tilde{X}}=f^{*}(K_{X}+B+(1-\epsilon)\boldsymbol{\rm M}_{X})+\tilde{E}$

with effective $\mathbb{R}$-divisors $\tilde{B}$ and $\tilde{E}$ on
$\tilde{X}$ having no common components. Furthermore, since $\boldsymbol{\rm
M}_{X}$ is $\mathbb{R}$-Cartier, we may write

$\boldsymbol{\rm M}_{\tilde{X}}+\tilde{F}=f^{*}\boldsymbol{\rm M}_{X}$

for some effective $f$-exceptional $\mathbb{R}$-divisor $\tilde{F}$ on
$\tilde{X}$. Since $(X,B,\boldsymbol{\rm M})$ is generalized lc, we can find a
real number $\delta>0$ such that $(\tilde{X},\tilde{B}+\delta\tilde{F})$ is a
log smooth lc pair. Put $\alpha^{\prime}=\alpha-\epsilon$. Note that
$\alpha^{\prime}\geq 1-\epsilon$. Since
$\alpha^{\prime}=(\alpha-1)+(1-\epsilon)$, we have

$K_{\tilde{X}}+\tilde{B}+f^{*}A_{\epsilon}+\alpha^{\prime}\boldsymbol{\rm
M}_{\tilde{X}}=f^{*}(K_{X}+B+A_{\epsilon}+\alpha^{\prime}\boldsymbol{\rm
M}_{X})+\tilde{E}-(\alpha-1)\tilde{F}.$

Since $K_{X}+B+A_{\epsilon}+(1-\epsilon)\boldsymbol{\rm M}_{X}$ is pseudo-
effective, we see that
$K_{\tilde{X}}+\tilde{B}+f^{*}A_{\epsilon}+(1-\epsilon)\boldsymbol{\rm
M}_{\tilde{X}}$ is pseudo-effective. Since $\alpha^{\prime}\geq 1-\epsilon$,
we see that the divisor
$K_{\tilde{X}}+\tilde{B}+f^{*}A_{\epsilon}+\alpha^{\prime}\boldsymbol{\rm
M}_{\tilde{X}}$ is pseudo-effective.

We run a $(K_{\tilde{X}}+\tilde{B}+\alpha^{\prime}\boldsymbol{\rm
M}_{\tilde{X}})$-MMP over $X$ with scaling of an ample divisor. By the
negativity lemma, after finitely many steps we obtain a projective birational
morphism $f^{\prime}\colon\tilde{X}^{\prime}\to X$ and a projective
$\mathbb{Q}$-factorial generalized dlt pair
$(\tilde{X}^{\prime},\tilde{B}^{\prime},\alpha^{\prime}\boldsymbol{\rm M})$
such that the effective part of $\tilde{E}-(\alpha-1)\tilde{F}$ is contracted
by the birational map $\tilde{X}\dashrightarrow\tilde{X}^{\prime}$. By
replacing $\delta$ with a smaller one, we may assume that
$\tilde{X}\dashrightarrow\tilde{X}^{\prime}$ is a sequence of steps of a
$(K_{\tilde{X}}+\tilde{B}+\delta\tilde{F}+\alpha^{\prime}\boldsymbol{\rm
M}_{\tilde{X}})$-MMP. Let $\tilde{E}^{\prime}$ and $\tilde{F}^{\prime}$ be the
birational transforms of $\tilde{E}$ and $\tilde{F}$ on $\tilde{X}^{\prime}$,
respectively. By construction, the generalized pair
$(\tilde{X}^{\prime},\tilde{B}^{\prime}+\delta\tilde{F}^{\prime},\alpha^{\prime}\boldsymbol{\rm
M})$ is generalized lc, $(\alpha-1)\tilde{F}^{\prime}-\tilde{E}^{\prime}$ is
effective, and

$K_{\tilde{X}^{\prime}}+\tilde{B}^{\prime}+\alpha^{\prime}\boldsymbol{\rm
M}_{\tilde{X}^{\prime}}+\bigl{(}(\alpha-1)\tilde{F}^{\prime}-\tilde{E}^{\prime}\bigr{)}=f^{\prime*}(K_{X}+B+\alpha^{\prime}\boldsymbol{\rm
M}_{X}).$

Now the $\mathbb{Q}$-factorial generalized dlt pair
$(\tilde{X}^{\prime},\tilde{B}^{\prime},\alpha^{\prime}\boldsymbol{\rm M})$
and the morphism $f^{\prime}\colon\tilde{X}^{\prime}\to X$ satisfy the
conditions of Theorem 3.5. Indeed, condition (I) of Theorem 3.5 is obvious,
the effective $\mathbb{R}$-divisor
$(\alpha-1)\tilde{F}^{\prime}-\tilde{E}^{\prime}$ satisfies (II-b) of Theorem
3.5 because of the above relation, and
$(\alpha-1)\tilde{F}^{\prime}-\tilde{E}^{\prime}$ satisfies (II-a) of Theorem
3.5 because
$(\tilde{X}^{\prime},\tilde{B}^{\prime}+\delta\tilde{F}^{\prime},\alpha^{\prime}\boldsymbol{\rm
M})$ is a generalized lc pair and ${\rm
Supp}\bigl{(}(\alpha-1)\tilde{F}^{\prime}-\tilde{E}^{\prime}\bigr{)}\subset{\rm
Supp}\tilde{F}^{\prime}$ which show the existence of a real number $t>0$ such
that the generalized pair
$\bigl{(}\tilde{X}^{\prime},\tilde{B}^{\prime}+t((\alpha-1)\tilde{F}^{\prime}-\tilde{E}^{\prime}),\alpha^{\prime}\boldsymbol{\rm
M}\bigr{)}$ is generalized lc. Applying Theorem 3.5 to
$(\tilde{X}^{\prime},\tilde{B}^{\prime},\alpha^{\prime}\boldsymbol{\rm M})\to
X$ and $A_{\epsilon}$, we see that the divisor
$K_{\tilde{X}^{\prime}}+\tilde{B}^{\prime}+\alpha^{\prime}\boldsymbol{\rm
M}_{\tilde{X}^{\prime}}+f^{\prime*}A_{\epsilon}$ is in particular abundant.
Since
$K_{\tilde{X}}+\tilde{B}+f^{*}A_{\epsilon}+\alpha^{\prime}\boldsymbol{\rm
M}_{\tilde{X}}$ is pseudo-effective,
$K_{\tilde{X}^{\prime}}+\tilde{B}^{\prime}+f^{\prime*}A_{\epsilon}+\alpha^{\prime}\boldsymbol{\rm
M}_{\tilde{X}^{\prime}}$ is also pseudo-effective. Thus, we can find an
effective $\mathbb{R}$-divisor $\tilde{D}^{\prime}_{\alpha}$ on
$\tilde{X}^{\prime}$ such that

$\tilde{D}^{\prime}_{\alpha}\sim_{\mathbb{R}}K_{\tilde{X}^{\prime}}+\tilde{B}^{\prime}+f^{\prime*}A_{\epsilon}+\alpha^{\prime}\boldsymbol{\rm
M}_{\tilde{X}^{\prime}}.$

By putting $D_{\alpha}=f^{\prime}_{*}\tilde{D}^{\prime}_{\alpha}$ and using
$\alpha^{\prime}=\alpha-\epsilon$ and
$A_{\epsilon}\sim_{\mathbb{R}}\epsilon\boldsymbol{\rm M}_{X}+A$, we obtain

$D_{\alpha}\sim_{\mathbb{R}}K_{X}+B+A_{\epsilon}+\alpha^{\prime}\boldsymbol{\rm
M}_{X}\sim_{\mathbb{R}}K_{X}+B+A+\alpha\boldsymbol{\rm M}_{X}.$

Thus, the second statement of Theorem 1.1 holds. We are done. ∎

From now on, we discuss a non-$\mathbb{R}$-Cartier analogue of Theorem 1.1.

######  Definition 4.2 (Pseudo-effective $\mathbb{R}$-divisor).

Let $X$ be a normal projective variety, and let $D$ be an $\mathbb{R}$-divisor
on $X$, which is not necessarily $\mathbb{R}$-Cartier. We say that $D$ is
pseudo-effective if $D+A$ is $\mathbb{R}$-linearly equivalent to an effective
$\mathbb{R}$-divisor for all ample $\mathbb{R}$-divisors $A$ on $X$.

######  Lemma 4.3.

Let $X$ be a normal projective variety, and let $D$ be an $\mathbb{R}$-divisor
on $X$. Let $f\colon Y\to X$ be a small birational morphism from a normal
projective variety $Y$ such that $f^{-1}_{*}D$ is $\mathbb{R}$-Cartier. Then
$D$ is pseudo-effective if and only if $f^{-1}_{*}D$ is pseudo-effective in
the usual sense.

###### Proof.

Since $f^{-1}_{*}D$ is $\mathbb{R}$-Cartier, $f^{-1}_{*}D$ is pseudo-effective
if and only if $f^{-1}_{*}D+f^{*}A$ is $\mathbb{R}$-linearly equivalent to an
effective $\mathbb{R}$-divisor for all ample $\mathbb{R}$-divisors $A$ on $X$.
Since $f$ is small, for all ample $A$, the existence of $E_{Y}\geq 0$
satisfying $E_{Y}\sim_{\mathbb{R}}f^{-1}_{*}D+f^{*}A$ is equivalent to the
existence of $E\geq 0$ satisfying $E\sim_{\mathbb{R}}D+A$. By Definition 4.2,
we see that $D$ is pseudo-effective if and only if $f^{-1}_{*}D$ is pseudo-
effective. ∎

######  Theorem 4.4.

Let $(X,B,\boldsymbol{\rm M})$ be a projective generalized lc pair such that
$\boldsymbol{\rm M}$ is a finite $\mathbb{R}_{>0}$-linear combination of b-nef
$\mathbb{Q}$-b-Cartier $\mathbb{Q}$-b-divisors. Let $A$ be an ample
$\mathbb{R}$-divisor on $X$. Suppose that there is $t\in[0,1)$ such that
$K_{X}+B+A+t\boldsymbol{\rm M}_{X}$ is pseudo-effective in the sense of
Definition 4.2.

Then, for every real number $\alpha\geq t$, there exists an effective
$\mathbb{R}$-divisor $D_{\alpha}$ on $X$ such that
$K_{X}+B+A+\alpha\boldsymbol{\rm M}_{X}\sim_{\mathbb{R}}D_{\alpha}$.

###### Proof.

Fix a real number $\alpha\geq t$. By [H2, Proposition 4.12], with notations as
in [H2], the pair of $X$ and $B$ is pseudo-lc defined in [H2, Definition 4.2].
By [H2, Theorem 1.2], there exists a small birational morphism $h\colon Y\to
X$ from a normal projective variety $Y$ such that if we put
$B_{Y}=h^{-1}_{*}B$, then $K_{Y}+B_{Y}$ is an $h$-ample $\mathbb{R}$-Cartier
divisor on $Y$. We can find a real number $\epsilon>0$ such that
$\tfrac{t}{1-\epsilon}<1$ and $\epsilon(K_{Y}+B_{Y})+h^{*}A_{Y}$ is an ample
$\mathbb{R}$-divisor on $Y$. Putting
$A_{Y}=\tfrac{1}{1-\epsilon}(\epsilon(K_{Y}+B_{Y})+h^{*}A_{Y})$, we have

$K_{Y}+B_{Y}+h^{*}A+t\boldsymbol{\rm
M}_{Y}=(1-\epsilon)\left(K_{Y}+B_{Y}+A_{Y}+\frac{t}{1-\epsilon}\boldsymbol{\rm
M}_{Y}\right).$

Then $\boldsymbol{\rm M}_{Y}$ is $\mathbb{R}$-Cartier and pseudo-effective.
Since $K_{X}+B+A+t\boldsymbol{\rm M}_{X}$ is pseudo-effective by hypothesis of
Theorem 4.4, by Lemma 4.3, we see that
$K_{Y}+B_{Y}+A_{Y}+\tfrac{t}{1-\epsilon}\boldsymbol{\rm M}_{Y}$ is pseudo-
effective with $\frac{t}{1-\epsilon}<1$. Therefore, for any $\alpha\geq t$,
the relation

$K_{Y}+B_{Y}+h^{*}A+\alpha\boldsymbol{\rm
M}_{Y}=(1-\epsilon)\left(K_{Y}+B_{Y}+A_{Y}+\frac{\alpha}{1-\epsilon}\boldsymbol{\rm
M}_{Y}\right)$

holds and $K_{Y}+B_{Y}+A_{Y}+\frac{\alpha}{1-\epsilon}\boldsymbol{\rm M}_{Y}$
is pseudo-effective. Since $h\colon Y\to X$ is small, the generalized pair
$(Y,B_{Y},\boldsymbol{\rm M})$ is generalized lc. If there is an effective
$\mathbb{R}$-divisor $D_{Y}$ on $Y$ such that
$D_{Y}\sim_{\mathbb{R}}K_{Y}+B_{Y}+A_{Y}+\frac{\alpha}{1-\epsilon}\boldsymbol{\rm
M}_{Y}$, then $(1-\epsilon)h_{*}D_{Y}$ is effective and it satisfies
$(1-\epsilon)h_{*}D_{Y}\sim_{\mathbb{R}}K_{X}+B+A+\alpha\boldsymbol{\rm
M}_{X}$. Thus, it is sufficient to show the existence of an effective
$\mathbb{R}$-divisor $D_{Y}$ on $Y$ such that
$D_{Y}\sim_{\mathbb{R}}K_{Y}+B_{Y}+A_{Y}+\frac{\alpha}{1-\epsilon}\boldsymbol{\rm
M}_{Y}$. In this way, by replacing $X$ (resp. $B$, $A$, $t$, $\alpha$) with
$Y$ (resp. $B_{Y}$, $A_{Y}$, $\frac{t}{1-\epsilon}$,
$\frac{\alpha}{1-\epsilon}$), we may assume that $\boldsymbol{\rm M}_{X}$ is
$\mathbb{R}$-Cartier. Note that this condition shows that the pair $(X,B)$ is
lc.

If $\alpha=0$, then Theorem 4.4 follows from [HH, Theorem 1.5]. Therefore, we
may assume $\alpha>0$. Put $t^{\prime}={\rm min}\\{1,\alpha\\}$. Then, the
generalized pair $(X,B,t^{\prime}\boldsymbol{\rm M})$ is generalized lc. By
hypothesis, $K_{X}+B+A+t^{\prime}\boldsymbol{\rm M}_{X}$ is pseudo-effective.
By Theorem 1.1 (2), we can find an effective $\mathbb{R}$-divisor $D_{\alpha}$
on $X$ such that $K_{X}+B+A+\alpha\boldsymbol{\rm
M}_{X}\sim_{\mathbb{R}}D_{\alpha}$. ∎

###### Proof of Theorem 1.2.

It is the case of $t=0$ of Theorem 4.4. ∎

The following result is out of interests of this paper, but we write down for
possibility of future use.

######  Theorem 4.5.

Let $(X,B,\boldsymbol{\rm M})$ be a projective generalized lc pair, and let
$A$ be an ample $\mathbb{R}$-divisor on $X$. Then the followings hold.

  1. (1)

If $K_{X}+B+A$ is pseudo-effective in the sense of Definition 4.2, then there
exists an effective $\mathbb{R}$-divisor $D$ on $X$ such that
$K_{X}+B+A\sim_{\mathbb{R}}D$.

  2. (2)

Let $G$ be a $\mathbb{Q}$-divisor on $X$ such that
$G\sim_{\mathbb{R}}K_{X}+B+A$. Then the graded ring

$\mathcal{R}(X,G)=\underset{m\in\mathbb{Z}_{\geq
0}}{\bigoplus}H^{0}(X,\mathcal{O}_{X}(\llcorner mG\lrcorner))$

is a finitely generated.

###### Proof.

The first assertion is the case of $\alpha=0$ of Theorem 4.4. We will prove
the second assertion. By [H2, Proposition 4.12], notations as in [H2], the
pair $\langle X,B\rangle$ of $X$ and $B$ is pseudo-lc in the sense of [H2,
Definition 4.2]. By [H2, Theorem 1.2], there exists a small birational
morphism $h\colon Y\to X$ from a normal projective variety $Y$ such that if we
put $B_{Y}=h^{-1}_{*}B$, then $K_{Y}+B_{Y}$ is an $h$-ample
$\mathbb{R}$-Cartier divisor on $Y$. Then $t(K_{Y}+B_{Y})+h^{*}A_{Y}$ is an
ample $\mathbb{R}$-divisor on $Y$ for some $t\in(0,1)$. Taking a general
member $A_{Y}\sim_{\mathbb{R}}\tfrac{1}{1-t}(t(K_{Y}+B_{Y})+h^{*}A_{Y})$, we
have

$K_{Y}+B_{Y}+h^{*}A\sim_{\mathbb{R}}(1-t)(K_{Y}+B_{Y}+A_{Y}).$

Since $h$ is small, for each $m$ we have

$H^{0}(X,\mathcal{O}_{X}(\llcorner mG\lrcorner))\simeq
H^{0}(Y,\mathcal{O}_{Y}(\llcorner mh^{-1}_{*}G\lrcorner)).$

Therefore, the graded ring $\mathcal{R}(X,G)$ is finitely generated if and
only if

$\mathcal{R}(Y,h^{-1}_{*}G)=\underset{m\in\mathbb{Z}_{\geq
0}}{\bigoplus}H^{0}(Y,\mathcal{O}_{Y}(\llcorner mh^{-1}_{*}G\lrcorner))$

is finitely generated. Furthermore, we have
$h^{-1}_{*}G\sim_{\mathbb{R}}(1-t)(K_{Y}+B_{Y}+A_{Y})$. Applying [HH, Theorem
1.5] and [HH, Theorem 1.7], we see that $\mathcal{R}(Y,h^{-1}_{*}G)$ is
finitely generated, thus $\mathcal{R}(X,G)$ is finitely generated. We finish
the proof. ∎

###  4.2. On the Kodaira type vanishing theorem

We give an example which shows that the Kodaira type vanishing theorem does
not necessarily hold in the situation of Theorem 1.1 (2) or Theorem 1.2. The
following theorem is a stronger statement than Theorem 1.4.

######  Theorem 4.6.

Let $d$ be a positive integer. Then, there is a $\mathbb{Q}$-factorial
generalized klt pair $(X,B,\boldsymbol{\rm M})$, an ample $\mathbb{Q}$-divisor
$A$ on $X$, and a Cartier divisor $D$ on $X$ such that

  * •

${\rm dim}X=3$ and $K_{X}+B+A$ is pseudo-effective,

  * •

$D\sim_{\mathbb{Q}}K_{X}+B+A+t\boldsymbol{\rm M}_{X}$ for some $t>1$, and

  * •

${\rm dim}H^{1}(X,\mathcal{O}_{X}(D))\geq d$.

###### Proof.

We prove the theorem in several steps.

######  Step 1.

We construct some varieties and divisors.

We put $V=\mathbb{P}^{1}$, and fix a very ample Cartier divisor $H_{V}$ such
that $\mathcal{O}_{V}(H_{V})=\mathcal{O}_{V}(1)$. Let $W$ be an elliptic
curve, and fix a very ample Cartier divisor $H_{W}$ on $W$. We set
$p_{V}\colon V\times W\to V$ and $p_{W}\colon V\times W\to W$ as projections.
The divisor $p_{V}^{*}H_{V}+p_{W}^{*}H_{W}$ is a very ample Cartier divisor on
$V\times W$. We construct a $\mathbb{P}^{1}$-bundle

$Y:=\mathbb{P}_{V\times W}(\mathcal{O}_{V\times W}\oplus\mathcal{O}_{V\times
W}(-p_{V}^{*}H_{V}-p_{W}^{*}H_{W}))\overset{f}{\longrightarrow}V\times W.$

Let $T$ be the unique section corresponding to $\mathcal{O}_{Y}(1)$. It is
obvious that $(Y,T)$ is a log smooth lc pair. We define a Cartier divisor $L$
on $Y$ by $L=T+f^{*}(p_{V}^{*}H_{V}+p_{W}^{*}H_{W})$. By construction, we have

$K_{V\times W}=p_{V}^{*}K_{V}+p_{W}^{*}K_{W}\sim-2p_{V}^{*}H_{V}.$

From this, we see that $L$ is basepoint free and the relation

$K_{Y}+T+L\sim-2f^{*}p_{V}^{*}H_{V}$

holds. Let $Y\to Z$ be the contraction induced by $L$. Since
$L+f^{*}p_{W}^{*}H_{W}$ is basepoint free, we get a contraction $\pi\colon
Y\to X$ induced by $L+f^{*}p_{W}^{*}H_{W}$. By construction, these morphisms
are isomorphisms outside $T$. By calculations of intersection numbers of
curves contracted by $\pi$, we see that all curves contracted by $\pi$ are
also contracted by $Y\to Z$. Thus, the induced birational map
$X\dashrightarrow Z$ is a morphism. We denote $Y\to Z$ by $\phi$ and $X\to Z$
by $\psi$. Then $L=\phi^{*}\phi_{*}L=\pi^{*}\psi^{*}\phi_{*}L$. Therefore,
$\pi_{*}L$ is a nef and big $\mathbb{Q}$-Cartier divisor on $X$ and
$L=\pi^{*}\pi_{*}L$. Since
$L+f^{*}p_{W}^{*}H_{W}=\pi^{*}\pi_{*}(L+f^{*}p_{W}^{*}H_{W})$, the divisor
$\pi_{*}f^{*}p_{W}^{*}H_{W}$ is $\mathbb{Q}$-Cartier and
$f^{*}p_{W}^{*}H_{W}=\pi^{*}\pi_{*}f^{*}p_{W}^{*}H_{W}$. Since $H_{W}$ is very
ample, $\pi_{*}f^{*}p_{W}^{*}H_{W}$ is semi-ample and
$\pi_{*}f^{*}p_{W}^{*}H_{W}$ induces a contraction $g\colon X\to W$ such that
$g\circ\pi=p_{W}\circ f$. We put $H_{X}=\pi_{*}f^{*}p_{V}H_{V}$. Since
$L=\pi^{*}\pi_{*}L$ and
$f^{*}p_{W}^{*}H_{W}=\pi^{*}\pi_{*}f^{*}p_{W}^{*}H_{W}$, we see that $H_{X}$
is $\mathbb{Q}$-Cartier and $\pi^{*}H_{X}=T+f^{*}p_{V}^{*}H_{V}$.

By these discussion, we obtain the following diagram

---
$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\pi}$$\textstyle{V\times
W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{V}}$$\scriptstyle{p_{W}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{V}$$\textstyle{W}$

and divisors $H_{V}$, $H_{W}$, $T$,
$L=T+f^{*}(p_{V}^{*}H_{V}+p_{W}^{*}H_{W})$, and
$H_{X}=\pi_{*}f^{*}p_{V}^{*}H_{V}$ such that

  1. (1)

$(Y,T)$ is a log smooth lc pair,

  2. (2)

$H_{V}$ and $H_{W}$ are very ample and $K_{V\times W}\sim-2p_{V}^{*}H_{V}$.

  3. (3)

$K_{Y}+T+L+2f^{*}p_{V}^{*}H_{V}\sim 0$,

  4. (4)

$\pi_{*}L$ is a nef and big $\mathbb{Q}$-Cartier divisor on $X$ such that
$\pi^{*}\pi_{*}L=L$, and

  5. (5)

$H_{X}$ is $\mathbb{Q}$-Cartier and $\pi^{*}H_{X}=T+f^{*}p_{V}^{*}H_{V}$.

######  Step 2.

In this step, we study properties of $X$ constructed in Step 1. More
precisely, we show that $X$ is $\mathbb{Q}$-factorial and
$\pi^{*}K_{X}=K_{Y}-T$.

Pick any Weil divisor $D$ on $X$. Then $\pi^{-1}_{*}D$ is linearly equivalent
to the sum of a multiple of $T$ and the pullback of a Cartier divisor $G$ on
$V\times W$. Since $V=\mathbb{P}^{1}$, we can find an integer $\alpha$ and a
Cartier divisor $G_{W}$ on $W$ such that $G\sim\alpha
p_{V}^{*}H_{V}+p_{W}^{*}G_{W}$. Thus, we have

$D=\pi_{*}\pi^{-1}_{*}D\sim\pi_{*}f^{*}G\sim\alpha\pi_{*}f^{*}p_{V}^{*}H_{V}+\pi_{*}f^{*}p_{W}^{*}G_{W}=\alpha
H_{X}+g^{*}G_{W}.$

Since $H_{X}$ is $\mathbb{Q}$-Cartier, $D$ is also $\mathbb{Q}$-Cartier.
Therefore, $X$ is $\mathbb{Q}$-factorial.

By (3) in Step 1, we have $K_{X}+\pi_{*}L+2H_{X}\sim 0$. The negativity lemma
shows

$K_{Y}+T+L+2f^{*}p_{V}^{*}H_{V}=\pi^{*}(K_{X}+\pi_{*}L+2H_{X}).$

By using (4) and (5) in Step 1, we obtain $\pi^{*}K_{X}=K_{Y}-T$.

######  Step 3.

In this step, we prove ${\rm dim}H^{2}(Y,\mathcal{O}_{Y}(-mK_{Y}+(m-1)T))=0$
for all positive integers $m$.

We fix a positive integer $m$. Put $h=p_{W}\circ f\colon Y\to W$. Applying
Relay spectral sequence to $\mathcal{O}_{Y}(-mK_{Y}+(m-1)T)$ and $h\colon Y\to
W$, we see that it is sufficient to prove the following facts.

  * •

$H^{2}\bigl{(}W,h_{*}\mathcal{O}_{Y}(-mK_{Y}+(m-1)T)\bigr{)}=0$,

  * •

$H^{1}\bigl{(}W,R^{1}h_{*}\mathcal{O}_{Y}(-mK_{Y}+(m-1)T)\bigr{)}=0,$ and

  * •

$H^{0}\bigl{(}W,R^{2}h_{*}\mathcal{O}_{Y}(-mK_{Y}+(m-1)T)\bigr{)}=0$.

The first fact is clear because ${\rm dim}W=1$. We also see that

$\begin{split}-mK_{Y}+(m-1)T=&K_{Y}-(m+1)K_{Y}+(m-1)T\\\
\sim&K_{Y}+(m+1)(T+L+2f^{*}p_{V}^{*}H_{V})+(m-1)T\quad(\text{(\ref{proof--thm
--van-example-(3)}) in Step \ref{step2-example}})\\\
=&K_{Y}+2mT+(m+1)L+(2m+2)f^{*}p_{V}^{*}H_{V}\\\
=&K_{Y}+(3m+1)L+2f^{*}p^{*}_{V}H_{V}-2mf^{*}p_{W}^{*}H_{W},\end{split}$

where the final relation follows from
$L=T+f^{*}(p_{V}^{*}H_{V}+p_{W}^{*}H_{W})$. Since $H_{V}$ is very ample and
$L$ is nef and big, we can apply relative Kawamata–Viehweg vanishing theorem
to the sheaf $\mathcal{O}_{Y}(-mK_{Y}+(m-1)T)$ and the morphism $Y\to W$.
Thus, we see that

$R^{1}h_{*}\mathcal{O}_{Y}(-mK_{Y}+(m-1)T)=R^{2}h_{*}\mathcal{O}_{Y}(-mK_{Y}+(m-1)T)=0.$

From this, the second and the third facts stated above follow. In this way, we
obtain the equality ${\rm dim}H^{2}(Y,\mathcal{O}_{Y}(-mK_{Y}+(m-1)T))=0,$
which is what we wanted to prove in this step.

######  Step 4.

In this step, we prove

${\rm dim}H^{1}(X,\mathcal{O}_{X}(-lK_{X}))\geq{\rm
dim}H^{0}(W,\mathcal{O}_{W}(2lH_{W}))$

for every positive integer $l$ such that $lK_{X}$ is Cartier.

Since $X$ is $\mathbb{Q}$-factorial (see Step 2) and $T$ is the unique
$\pi$-exceptional divisor, $-T$ is ample over $X$. Since
$\pi^{*}K_{X}=K_{Y}-T$ (Step 2), it follows that $Y$ is Fano over $X$. Thus,
we have $R^{i}\pi_{*}\mathcal{O}_{Y}=0$ for every $i>0$. By Leray spectral
sequence, it follows that

${\rm dim}H^{1}(X,\mathcal{O}_{X}(-lK_{X}))={\rm
dim}H^{1}(Y,\mathcal{O}_{Y}(-l\pi^{*}K_{X}))={\rm
dim}H^{1}(Y,\mathcal{O}_{Y}(-lK_{Y}+lT)).$

We consider the exact sequence

$0\longrightarrow\mathcal{O}_{Y}(-lK_{Y}+(l-1)T)\longrightarrow\mathcal{O}_{Y}(-lK_{Y}+lT)\longrightarrow\mathcal{O}_{T}((-lK_{Y}+lT)|_{T})\longrightarrow
0.$

Since we have $H^{2}(Y,\mathcal{O}_{Y}(-lK_{Y}+(l-1)T))=0$ by Step 3, taking
the cohomology long exact sequence, we obtain the exact sequence

($*$) $\begin{split}H^{1}(Y,\mathcal{O}_{Y}(-lK_{Y}+lT))&\longrightarrow
H^{1}(T,\mathcal{O}_{T}((-lK_{Y}+lT)|_{T}))\longrightarrow 0.\end{split}$

Since $(T+f^{*}(p_{V}^{*}H_{V}+p_{W}^{*}H_{W}))|_{T}\sim 0$, we have

$(-lK_{Y}+lT)|_{T}\sim-lK_{T}-2lf^{*}(p_{V}^{*}H_{V}+p_{W}^{*}H_{W})|_{T}.$

By identifying $T$ with $V\times W$ and using $K_{V\times
W}\sim-2p_{V}^{*}H_{V}$, we obtain

${\rm dim}H^{1}\bigl{(}T,\mathcal{O}_{T}((-lK_{Y}+lT)|_{T})\bigr{)}={\rm
dim}H^{1}\bigl{(}V\times W,\mathcal{O}_{V\times
W}(-2lp_{W}^{*}H_{W})\bigr{)}.$

Since $V=\mathbb{P}^{1}$, it is easy to check that
$R^{i}p_{W*}\mathcal{O}_{V\times W}=0$ for every $i>0$. By applying Leray
spectral sequence to $p_{W}\colon V\times W\to W$, we have

${\rm dim}H^{1}\bigl{(}V\times W,\mathcal{O}_{V\times
W}(-2lp_{W}^{*}H_{W})\bigr{)}={\rm dim}H^{1}(W,\mathcal{O}_{W}(-2lH_{W})).$

Since $W$ is an elliptic curve, by Serre duality, we have

${\rm dim}H^{1}(W,\mathcal{O}_{W}(-2lH_{W}))={\rm
dim}H^{0}(W,\mathcal{O}_{W}(2lH_{W})).$

Combining these relations, we obtain

$\begin{split}{\rm dim}H^{1}(X,\mathcal{O}_{X}(-lK_{X}))=&{\rm
dim}H^{1}(Y,\mathcal{O}_{Y}(-lK_{Y}+lT))\\\ \geq&{\rm
dim}H^{1}\bigl{(}T,\mathcal{O}_{T}((-lK_{Y}+lT)|_{T})\bigr{)}\qquad\quad(\text{by
(\ref{proof--thm--van-example-(***)})})\\\ =&{\rm dim}H^{1}\bigl{(}V\times
W,\mathcal{O}_{V\times W}(-2lp_{W}^{*}H_{W})\bigr{)}\\\ =&{\rm
dim}H^{1}(W,\mathcal{O}_{W}(-2lH_{W}))\\\ =&{\rm
dim}H^{0}(W,\mathcal{O}_{W}(2lH_{W})).\end{split}$