# Finiteness of log abundant log canonical pairs in log minimal model program
with scaling

Kenta Hashizume  Graduate School of Mathematical Sciences, The University of
Tokyo, 3-8-1 Komaba Meguro-ku Tokyo 153-8914, Japan<EMAIL_ADDRESS>
(Date: 2020/12/21, version 0.16)

###### Abstract.

We study relations between the property of being log abundant for lc pairs and
the termination of log MMP with scaling. We prove that any log MMP with
scaling of an ample divisor starting with a projective dlt pair contains only
finitely many log abundant dlt pairs.

###### Key words and phrases:

property of being log abundant, termination of log MMP with scaling

###### 2010 Mathematics Subject Classification:

14E30

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Divisors, morphisms and singularities of pairs
    2. 2.2 Log MMP and models
    3. 2.3 Invariant Iitaka dimension and numerical dimension
    4. 2.4 On log MMP and existence of good minimal models
  3. 3 Log MMP containing log abundant log canonical pairs
    1. 3.1 Proof of main result
    2. 3.2 Corollaries
  4. 4 On log MMP for semi log canonical pairs
  5. 5 Varieties over an algebraically closed field of characteristic zero

##  1\. Introduction

Throughout this paper we will work over the complex number field $\mathbb{C}$
(see Section 5 for generalization to varieties over an algebraically closed
field of characteristic zero).

In the minimal model theory, the properties of being abundant and being log
abundant are closely related to the existence of good minimal models. For the
definitions of these two properties and log abundant lc pairs, see Definition
2.15 and Definition 2.16, respectively. The two properties are the same
notions when we deal with klt pairs. It is well-known that the log MMP
preserves property of being abundant. Hence, for any lc pair $(X,\Delta)$ it
is well-known that the divisor $K_{X}+\Delta$, which we call the log canonical
divisor, is abundant if the minimal model theory holds for the given lc pair.
When the given lc pair is klt, results by Birkar–Cascini–Hacon–McKernan [4]
and Lai [33] (see also [29] and [19]) show that the converse also holds. On
the other hand, we cannot know directly from construction that log MMP
preserves property of being log abundant, but it is known that log canonical
divisors of all lc pairs are log abundant if and only if the minimal model
theory holds for all lc pairs. The situations of the main results of [3] and
[22] (cf. [26]) are the cases where the log MMP preserves the property after
finitely many steps, whereas there is a situation in which log MMP breaks
property of being log big, a special kind of property of being log abundant
([6, Example 5.2], [2]). Because induction on dimension of varieties works
well under the assumption of property of being log abundant, the abundance
conjecture for log abundant lc pairs was thoroughly studied ([36], [17], [8],
[12], [16], [23]) and proved by Hacon–Xu [23] (see also [16] by Fujino–Gongyo
for projective case). In contrast, we know relations between the log MMP and
the property of being log abundant only in special cases ([3], [22], [26],
[28]).

In this paper, we study how the property of being log abundant affects
termination of log MMP. The following theorem is the main result of this
paper.

######  Theorem 1.1 (=Theorem 3.6).

Let $\pi\colon X\to Z$ be a projective morphism of normal quasi-projective
varieties, and let $(X,\Delta)$ be a $\mathbb{Q}$-factorial dlt pair such that
$K_{X}+\Delta$ is pseudo-effective over $Z$. Let $A$ be an ample
$\mathbb{R}$-divisor on $X$ such that $(X,\Delta+A)$ is lc and
$K_{X}+\Delta+A$ is nef over $Z$. Then there is an integer $l\geq 0$
satisfying the following: Let

$(X_{0}:=X,\Delta_{0}:=\Delta)\dashrightarrow(X_{1},\Delta_{1})\dashrightarrow\cdots\dashrightarrow(X_{i},\Delta_{i})\dashrightarrow\cdots$

be a sequence of steps of a $(K_{X}+\Delta)$-MMP over $Z$ with scaling of $A$.
If $(X_{i},\Delta_{i})$ is log abundant over $Z$ for some $i\geq l$, then the
$(K_{X}+\Delta)$-MMP over $Z$ terminates with a good minimal model.

######  Corollary 1.2 (=Corollary 3.8).

Any log MMP with scaling of an ample divisor starting with a projective dlt
pair contains only finitely many log abundant dlt pairs.

As mentioned above, the case of klt pairs of Theorem 1.1 is known by [4] and
[33]. Theorem 1.1 can be applied to the situations of [3, Theorem 1.1], [22,
Theorem 1.1], [26], [28, Theorem 1.1], and [27, Theorem 1.3] (see Remark 3.7).
Thus, Theorem 1.1 is the unification of those results as well as a relation
between the log MMP and the property of being log abundant. Furthermore,
Theorem 1.1 shows the existence of a good minimal model in the following new
case.

######  Corollary 1.3 (=Corollary 3.10).

Let $(X,\Delta)$ be a projective lc pair. Suppose that

  * •

$(X,\Delta)$ is log abundant and $K_{X}+\Delta$ is pseudo-effective, and

  * •

the stable base locus of $K_{X}+\Delta$ does not contain the image of any
prime divisor $P$ over $X$ satisfying $a(P,X,\Delta)<0$.

Then $(X,\Delta)$ has a good minimal model.

Corollary 1.3 has a further generalization to semi log canonical pairs
(Theorem 4.1). For the generalization, the result by Ambro–Kollár [2] plays a
crucial role. In general, we cannot run a log MMP for semi log canonical
pairs. Nevertheless, the main result of [2] and Corollary 1.3 enables us to
run a log MMP terminating with a good minimal model in a similar situation to
Corollary 1.3. For details, see Section 4.

We explain the ideas of the proofs of Theorem 1.1 and Corollary 1.2. We will
first prove Corollary 1.2 (in fact, we will prove Theorem 3.5, an lc analogue
of Corollary 1.2), then we will prove Theorem 1.1. To outline the proof of
Corollary 1.2, fix a projective dlt pair $(X,\Delta)$, and suppose by
contradiction that there is an infinite sequence of steps of a
$(K_{X}+\Delta)$-MMP with scaling of an ample divisor which contains
infinitely many log abundant dlt pairs. In particular, the log MMP does not
terminate, so $K_{X}+\Delta$ is pseudo-effective by [4]. We will prove the
existence of a log minimal model of $(X,\Delta)$ to get a contradiction with
[3, Theorem 4.1]. By the special termination ([10]), replacing $(X,\Delta)$,
we may assume that $(X,\Delta)$ is log abundant and the restrictions of
$K_{X}+\Delta$ to all lc centers of $(X,\Delta)$ are nef. Despite of the two
properties, as shown in the following example, it is still difficult to prove
the existence of a log minimal model of $(X,\Delta)$.

######  Example 1.4 ([6, Example 5.2]).

Fix a smooth projective variety $S$ with $\kappa(S,K_{S})\geq 0$. With
notations as in [6, Example 5.2], we can pick $\epsilon\in\mathbb{R}_{>0}$ and
an ample Cartier divisor $H$ so that $K_{S}+\epsilon H$ is ample. Then the
projective dlt pair $(X,B)$ in [6, Example 5.2] is log abundant and the
restrictions of $K_{X}+B$ to all lc centers of $(X,B)$ are nef. Moreover, if
$(X,B)$ has a log minimal model, then so does $(S,0)$. From this, the
existence of a log minimal model of projective dlt pairs with the above two
properties is more difficult than that of smooth projective varieties with
non-negative Kodaira dimensions.

Hence, the two properties stated above are insufficient for our proof. One
more key property is that the non-nef locus of $K_{X}+\Delta$ is disjoint from
any lc center of $(X,\Delta)$. In fact, we need the three properties to apply
the following theorem.

######  Theorem 1.5 (=Theorem 3.4).

Let $(X,\Delta)$ be a projective dlt pair. Suppose that

  * •

$K_{X}+\Delta$ is pseudo-effective and abundant,

  * •

for any lc center $S$ of $(X,\Delta)$, the restriction $(K_{X}+\Delta)|_{S}$
is nef, and

  * •

$\sigma_{P}(K_{X}+\Delta)=0$ for all prime divisors $P$ over $X$ such that
$a(P,X,\Delta)<0$ and $c_{X}(P)$ intersects an lc center of $(X,\Delta)$,
where $\sigma_{P}(\,\cdot\,)$ is the asymptotic vanishing order of $P$ defined
in Definition 2.1.

Then $(X,\Delta)$ has a log minimal model.

The third condition of Theorem 1.5 has not been mentioned in the papers
related to Theorem 1.1 ([3], [22], [26], [28], [27]). The condition enables us
to unify the hypothesis of the results in [3], [22], [26], [28], and [27,
Section 3] into one situation. We will prove Theorem 1.5 with a special kind
of log MMP used in [3], [22], [26], [28], and [27]. The third condition of
Theorem 1.5 plays a crucial role for the termination of the special kind of
log MMP. Once Theorem 1.5 is proved, Corollary 1.2 follows from Theorem 1.5
and the special termination ([10]) discussed above. Finally, we will prove
Theorem 1.1 using Corollary 1.2 and the following idea: For a given
$\mathbb{Q}$-factorial dlt pair $(X,\Delta)\to Z$, with the finiteness of
models ([4, Theorem E]) we will find an integer $l$ such that for every
$(K_{X}+\Delta)$-MMP over $Z$ with scaling of an ample divisor, if the
$(K_{X}+\Delta)$-MMP contains a log abundant dlt pair after the $l$-th step
then the $(K_{X}+\Delta)$-MMP terminates with a nef and log abundant dlt pair
or the $(K_{X}+\Delta)$-MMP contains infinitely many log abundant dlt pairs.
Then the relative case of Corollary 1.2 (see Theorem 3.5) shows that the
latter case cannot happen. Moreover, the abundance theorem for nef and log
abundant lc pairs ([12], [23]) shows that the resulting dlt pair of the
$(K_{X}+\Delta)$-MMP is a good minimal model. Therefore, the integer $l$
satisfies the property of Theorem 1.1.

We close the introduction with an application of Theorem 1.5 (see Corollary
3.13 for an application for higher-dimensional lc pairs).

######  Corollary 1.6 (cf. Corollary 3.12).

Let $(X,\Delta)$ be a projective lc pair of dimension $6$ such that
$\llcorner\Delta\lrcorner=0$ and the invariant Iitaka dimension
$\kappa_{\iota}(X,K_{X}+\Delta)$ is greater than or equal to $3$. Then
$(X,\Delta)$ has a log minimal model. In particular, all projective lc pairs
$(X,0)$ satisfying ${\rm dim}X=6$ and $\kappa(X,K_{X})\geq 3$ have a log
minimal model.

The contents of this paper are as follows: In Section 2, we collect
definitions, notations and basic results. In Section 3, we prove the main
results and corollaries. In Section 4, we discuss a generalization of
Corollary 1.3 to semi log canonical pairs. In Section 5, we give a remark on
generalizations of results of this paper to varieties over an algebraically
closed field of characteristic zero.

######  Acknowledgments.

The author was partially supported by JSPS KAKENHI Grant Number JP19J00046.
Part of the work was done while the author was visiting University of
Cambridge in January 2020. The author thanks Professor Caucher Birkar for his
hospitality. He thanks Professors Caucher Birkar and Yoshinori Gongyo for
discussions. He thanks Professor János Kollár for answering questions. He
thanks the referee for many suggestions which improve the paper considerably.

##  2\. Preliminaries

In this section, we collect definitions and some results.

###  2.1. Divisors, morphisms and singularities of pairs

We collect notations and definitions on divisors, morphisms and singularities
of pairs. We also show some lemmas.

######  Divisors and morphisms.

Let $X$ be a normal projective variety. We use the standard definitions of nef
$\mathbb{R}$-divisor, ample $\mathbb{R}$-divisor, semi-ample
$\mathbb{R}$-divisor, and pseudo-effective $\mathbb{R}$-divisor. All big
$\mathbb{R}$-divisors appearing in this paper are $\mathbb{R}$-Cartier. For
any $\mathbb{R}$-divisor $D=\sum_{i}a_{i}D_{i}$ on $X$ where $D_{i}$ are
distinct prime divisors, an $\mathbb{R}$-divisor $\llcorner D\lrcorner$ is
defined by $\llcorner D\lrcorner=\sum_{i}\llcorner a_{i}\lrcorner D_{i}$,
where $\llcorner a_{i}\lrcorner$ is the largest integer not exceeding $a_{i}$.
For any morphism $f\colon X\to Y$ and any $\mathbb{R}$-Cartier divisor
$D^{\prime}$ on $Y$, we sometimes denote $f^{*}D^{\prime}$ by
$D^{\prime}|_{X}$. For any prime divisor $P$ over $X$, the image of $P$ on $X$
is denoted by $c_{X}(P)$.

A projective morphism $f\colon X\to Y$ of varieties is called a contraction if
it is surjective and has connected fibers. For a variety $X$ and an
$\mathbb{R}$-divisor $D$ on it, a log resolution of $(X,D)$ denotes a
projective birational morphism $f\colon Y\to X$ from a smooth variety such
that the exceptional locus ${\rm Ex}(f)$ is pure codimension one and ${\rm
Ex}(f)\cup{\rm Supp}f_{*}^{-1}D$ is a simple normal crossing divisor.

######  Definition 2.1 (Asymptotic vanishing order).

Let $X$ be a normal projective variety and $D$ a pseudo-effective
$\mathbb{R}$-Cartier divisor on $X$. Let $P$ be a prime divisor over $X$. We
define asymptotic vanishing order of $P$ with respect to $D$, which we denote
$\sigma_{P}(D)$, as follows: We take a projective birational morphism $f\colon
Y\to X$ so that $P$ appears as a prime divisor on $Y$. When $D$ is big, we
define $\sigma_{P}(D)$ by

$\sigma_{P}(D)={\rm inf}\\!\set{{\rm
coeff}_{P}(D^{\prime})}{f^{*}D\sim_{\mathbb{R}}D^{\prime}\geq 0}.$

When $D$ is not necessarily big, $\sigma_{P}(D)$ is defined by

$\sigma_{P}(D)=\underset{\epsilon\to+0}{\rm lim}\sigma_{P}(D+\epsilon A)$

for an ample $\mathbb{R}$-divisor $A$ on $X$. It is easy to see that
$\sigma_{P}(D)$ is independent of $f\colon Y\to X$. We have $\sigma_{P}(D)\geq
0$, and $\sigma_{P}(D)$ is also independent of $A$ ([35, III, 1.5 (2) Lemma]).
We can easily check $\sigma_{P}(D)={\rm sup}\\!\set{\sigma_{P}(D+H)}{H\text{
is ample}}$.

######  Definition 2.2 (Negative and positive part of Nakayama–Zariski
decomposition).

For $X$ and $D$ as in Definition 2.1, the negative part of Nakayama–Zariski
decomposition of $D$, denoted by $N_{\sigma}(D)$, is defined by

$N_{\sigma}(D)=\sum_{\begin{subarray}{c}P:{\rm\,prime\,divisor}\\\ {\rm
on\,}X\end{subarray}}\sigma_{P}(D)P.$

We call the divisor $D-N_{\sigma}(D)$ the positive part of Nakayama–Zariski
decomposition of $D$. Note that $N_{\sigma}(D)$ and $D-N_{\sigma}(D)$ are not
necessarily $\mathbb{R}$-Cartier in this paper.

When $X$ is smooth, the definition of $N_{\sigma}(D)$ coincides with [35, III,
1.12 Definition]. When $X$ is not necessarily smooth, $N_{\sigma}(D)$ in the
definition can be written as $g_{*}N_{\sigma}(g^{*}D)$ where $g\colon W\to X$
is a resolution and $N_{\sigma}(g^{*}D)$ is defined in [35, III, 1.12
Definition].

We write down some properties of asymptotic vanishing order and negative part
of Nakayama–Zariski decomposition.

######  Remark 2.3.

Let $X$ be a normal projective variety, $D$ a pseudo-effective
$\mathbb{R}$-Cartier divisor on $X$, and $P$ a prime divisor over $X$.

  1. (1)

(Lower convexity, see [35, III, 1.6 Definition]). For any pseudo-effective
$\mathbb{R}$-Cartier divisor $D^{\prime}$ on $X$, we have

$\sigma_{P}(D+D^{\prime})\leq\sigma_{P}(D)+\sigma_{P}(D^{\prime}).$

  2. (2)

([35, III, 1.7 (2) Lemma]). For any pseudo-effective $\mathbb{R}$-Cartier
divisor $G$ on $X$, we have

$\underset{\epsilon\to+0}{\rm lim}\sigma_{P}(D+\epsilon G)=\sigma_{P}(D).$

  3. (3)

([35, III, 5.14 Lemma]). Let $f\colon Y\to X$ be a projective birational
morphism from a normal variety $Y$ and $E$ an effective $f$-exceptional
$\mathbb{R}$-divisor on $Y$. If $f^{*}D+E$ is $\mathbb{R}$-Cartier (in this
situation $E$ is also $\mathbb{R}$-Cartier), then

$\sigma_{P}(f^{*}D+E)=\sigma_{P}(f^{*}D)+{\rm ord}_{P}(E).$

  4. (4)

(cf. [35, III, 5.16 Theorem]). Let $f\colon Y\to X$ be a projective birational
morphism from a normal variety and $D^{\prime\prime}$ an $\mathbb{R}$-Cartier
divisor on $X$ such that $D^{\prime\prime}\leq N_{\sigma}(D)$. Then

$N_{\sigma}(f^{*}D)=f^{*}D^{\prime\prime}+F$

for an effective $\mathbb{R}$-divisor $F$ such that
$D^{\prime\prime}+f_{*}F=N_{\sigma}(D)$.

We outline proof of (4). Fix an ample $\mathbb{R}$-divisor $A$ on $X$. By
definition, for any $t<1$ there is $\epsilon_{t}>0$ such that
$N_{\sigma}(f^{*}D+\epsilon f^{*}A)\geq tN_{\sigma}(f^{*}D)$ for all
$\epsilon\in(0,\epsilon_{t})$. For these $\epsilon$, all $G\in|f^{*}D+\epsilon
f^{*}A|_{\mathbb{R}}$ satisfy $G\geq tN_{\sigma}(f^{*}D)$. Then $f_{*}G\geq
tf_{*}N_{\sigma}(f^{*}D)=tN_{\sigma}(D)\geq tD^{\prime\prime}$. Because
$G=f^{*}f_{*}G$, we have $G\geq tf^{*}D^{\prime\prime}$ for all
$G\in|f^{*}D+\epsilon f^{*}A|_{\mathbb{R}}$. By considering the limit
$\epsilon\to 0$, we have $N_{\sigma}(f^{*}D)\geq tf^{*}D^{\prime\prime}$. Then
we obtain $N_{\sigma}(f^{*}D)\geq f^{*}D^{\prime\prime}$ by taking the limit
$t\to 1$, and therefore $F:=N_{\sigma}(f^{*}D)-f^{*}D^{\prime\prime}$ is the
desired divisor.

######  Lemma 2.4.

Let $X$ be a normal projective variety. Let $D$ and $D^{\prime}$ be pseudo-
effective $\mathbb{R}$-Cartier divisors on $X$. Then, there is a real number
$t_{0}>0$ such that the support of $N_{\sigma}(D+tD^{\prime})$ does not depend
on $t$ for any $t\in(0,t_{0}]$.

###### Proof.

By property of lower convexity as in Remark 2.3 (1), components of
$N_{\sigma}(D+tD^{\prime})$ are components of
$N_{\sigma}(D)+N_{\sigma}(D^{\prime})$ for any $t\in(0,1]$. Let
$D_{1},\cdots,D_{k}$ be all components of
$N_{\sigma}(D)+N_{\sigma}(D^{\prime})$, and we consider the set
$J_{i}:=\\{s\in(0,1]|\sigma_{D_{i}}(D+sD^{\prime})=0\\}$ for all $1\leq i\leq
k$. Fix $i$. When $J_{i}$ is not empty and ${\rm inf}J_{i}=0$, we pick an
element $s_{i}$ of $J_{i}$. Then $\sigma_{D_{i}}(D+sD^{\prime})=0$ for all
$s\in(0,s_{i}]$ by Remark 2.3 (1). We put $s_{i}=1$ when $J_{i}$ is empty, and
we put $s_{i}=\frac{1}{2}\cdot{\rm inf}J_{i}$ when $J_{i}$ is not empty and
${\rm inf}J_{i}>0$, then $\sigma_{D_{i}}(D+sD^{\prime})>0$ for all
$s\in(0,s_{i}]$. In this way, we can find $s_{i}\in(0,1]$ such that
$\sigma_{D_{i}}(D+sD^{\prime})=0$ for all $s\in(0,s_{i}]$ or
$\sigma_{D_{i}}(D+sD^{\prime})>0$ for all $s\in(0,s_{i}]$. We set $t_{0}={\rm
min}\\{s_{i}\\}_{1\leq i\leq k}$. By definition of $D_{1},\cdots,D_{k}$, we
have
$N_{\sigma}(D+tD^{\prime})=\sum_{i=1}^{k}\sigma_{D_{i}}(D+tD^{\prime})D_{i}$,
thus construction of $t_{0}$ shows that the support of
$N_{\sigma}(D+tD^{\prime})$ does not depend on $t\in(0,t_{0}]$. ∎

######  Singularities of pairs.

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

Let $(X,\Delta)$ be a pair, and let $P$ be a prime divisor over $X$. We denote
by $a(P,X,\Delta)$ the discrepancy of $P$ with respect to $(X,\Delta)$. We
will freely use the standard definitions of Kawamata log terminal (klt, for
short) pair, log canonical (lc, for short) pair and divisorial log terminal
(dlt, for short) pair as in [32] or [4]. In [32], pairs and classes of
singularity are defined in the framework of $\mathbb{Q}$-divisors, but we can
similarly define those classes of singularity for pairs of a normal variety
and an effective $\mathbb{R}$-divisor. When $(X,\Delta)$ is an lc pair, an lc
center of $(X,\Delta)$ is $c_{X}(P)$ for a prime divisor $P$ over $X$ such
that $a(P,X,\Delta)=-1$.

######  Lemma 2.5.

Let $(X,\Delta)$ be a dlt pair and $S$ an lc center of $(X,\Delta)$. Fix a log
resolution $f\colon Y\to X$ of $(X,\Delta)$ which is isomorphic over an open
subset of $X$ intersecting $S$. Let $T\subset Y$ be a subvariety such that the
restriction of $f$ to $T$ induces a birational morphism $f|_{T}\colon T\to S$,
and let $M$ be an $f$-exceptional divisor on $Y$ such that all components $P$
of $M$ satisfy $a(P,X,\Delta)>0$. Then all components of $M|_{T}$ are
exceptional over $S$.

###### Proof.

We may write $K_{Y}+\Gamma=f^{*}(K_{X}+\Delta)+E$ such that $\Gamma\geq 0$ and
$E\geq 0$ have no common components and all components of $M$ are components
of $E$. Since $(X,\Delta)$ is dlt, there are components $D_{1},\cdots,D_{k}$
of $\llcorner\Delta\lrcorner$ such that $S$ is an irreducible component of
$D_{1}\cap\cdots\cap D_{k}$ (see [9, Proposition 3.9.2] or [30, Theorem
4.16]). Let $D^{\prime}_{1},\cdots,D^{\prime}_{k}$ be the birational
transforms of $D_{1},\cdots,D_{k}$ on $Y$, respectively. By definitions of
$\Gamma$ and $T$, the divisors $D^{\prime}_{1},\cdots,D^{\prime}_{k}$ are
components of $\llcorner\Gamma\lrcorner$ and $T$ is an irreducible component
of $D^{\prime}_{1}\cap\cdots\cap D^{\prime}_{k}$. We put
$G=\Gamma-\sum_{i=1}^{k}D^{\prime}_{i}$. We apply [28, Proof of Lemma 2.4] to
check that $E|_{T}$ is exceptional over $S$. Suppose that $G|_{T}$ and
$E|_{T}$ have a common component $Q$. Then $Q\subset G^{\prime}\cap
E^{\prime}\cap D^{\prime}_{1}\cap\cdots\cap D^{\prime}_{k}$ for some
components $G^{\prime}$ of $G$ and $E^{\prime}$ of $E$, thus the left hand
side has codimension $k+1$ in $Y$ but the right hand side has codimension
$k+2$ in $Y$ which is impossible. Therefore, $G|_{T}$ and $E|_{T}$ have no
common components. We put $f_{T}=f|_{T}\colon T\to S$ and define $\Delta_{S}$
on $S$ by adjunction $K_{S}+\Delta_{S}=(K_{X}+\Delta)|_{S}$. Then
$\Delta_{S}\geq 0$ and $\Delta_{S}=f_{T*}G|_{T}-f_{T*}E|_{T}$. Since
$f_{T}\colon T\to S$ is birational, $f_{T*}G|_{T}$ and $f_{T*}E|_{T}$ have no
common components. From these facts, it follows that $f_{T*}E|_{T}=0$.
Therefore $E|_{T}$ is exceptional over $S$, so is $M|_{T}$. ∎

######  Lemma 2.6.

Let $(X,\Delta)$ and $(X^{\prime},\Delta^{\prime})$ be projective dlt pairs,
let $S$ and $S^{\prime}$ be lc centers of $(X,\Delta)$ and
$(X^{\prime},\Delta^{\prime})$ respectively, and let $f\colon X\dashrightarrow
X^{\prime}$ be a birational map such that $f$ is an isomorphism on an open set
intersecting $S$ and the restriction of $f$ to $S$ induces a birational map
$f|_{S}\colon S\dashrightarrow S^{\prime}$. Suppose that $K_{X}+\Delta$ is
pseudo-effective. Suppose in addition that

  * •

$a(D^{\prime},X^{\prime},\Delta^{\prime})\leq a(D^{\prime},X,\Delta)$ for all
prime divisors $D^{\prime}$ on $X^{\prime}$, and

  * •

$\sigma_{P}(K_{X}+\Delta)=0$ for all prime divisors $P$ over $X$ such that
$a(P,X,\Delta)<0$ and $c_{X}(P)$ intersects $S$, where $\sigma_{P}(\,\cdot\,)$
is the asymptotic vanishing order of $P$ defined in Definition 2.1.

Let $(S,\Delta_{S})$ and $(S^{\prime},\Delta_{S^{\prime}})$ be projective dlt
pairs where $\Delta_{S}$ and $\Delta_{S^{\prime}}$ are constructed with
adjunctions $K_{S}+\Delta_{S}=(K_{X}+\Delta)|_{S}$ and
$K_{S^{\prime}}+\Delta_{S^{\prime}}=(K_{X^{\prime}}+\Delta^{\prime})|_{S^{\prime}}$
respectively. Then, the inequality $a(Q,S^{\prime},\Delta_{S^{\prime}})\leq
a(Q,S,\Delta_{S})$ holds for all prime divisors $Q$ on $S^{\prime}$.

###### Proof.

By taking an appropriate resolution of the graph of $f\colon X\dashrightarrow
X^{\prime}$, we may find a common log resolution $g\colon Y\to X$ and
$g^{\prime}\colon Y\to{X^{\prime}}$ of the map
$(X,\Delta)\dashrightarrow(X^{\prime},\Delta^{\prime})$ and a subvariety
$T\subset Y$ birational to $S$ and $S^{\prime}$ such that the induced
morphisms $g|_{T}\colon T\to S$ and $g^{\prime}|_{T}\colon T\to S^{\prime}$
form a common resolution of $f|_{S}\colon S\dashrightarrow S^{\prime}$. We may
write

($*$) $g^{*}(K_{X}+\Delta)=g^{\prime*}(K_{X^{\prime}}+\Delta^{\prime})+M-N$

such that $M\geq 0$ and $N\geq 0$ have no common components. By the first
condition of the lemma, $M$ is $g^{\prime}$-exceptional. Since the morphism
$g^{\prime}\colon Y\to X^{\prime}$ is an isomorphism over an open subset which
contains general points of $S^{\prime}$, we see that $T\not\subset{\rm
Supp}M$. We also see that $T\not\subset{\rm Supp}N$. Indeed, we can find a
prime divisor $P^{\prime}$ over $X$ such that $c_{Y}(P^{\prime})=T$ and
$a(P^{\prime},X,\Delta)=-1$. If $T\subset{\rm Supp}N$, then the fact
$T\not\subset{\rm Supp}M$ shows that
$a(P^{\prime},X^{\prime},\Delta^{\prime})=a(P^{\prime},X,\Delta)+{\rm
ord}_{P^{\prime}}(M)-{\rm ord}_{P^{\prime}}(N)<-1$ which contradicts that
$(X^{\prime},\Delta^{\prime})$ is dlt. Thus, we see that $T\not\subset{\rm
Supp}N$. Restricting ($*$ ‣ 2.1) to $T$, we obtain

($**$)
$g|_{T}^{*}(K_{S}+\Delta_{S})=g^{\prime}|_{T}^{*}(K_{S^{\prime}}+\Delta_{S^{\prime}})+M|_{T}-N|_{T}$

where $M|_{T}$ and $N|_{T}$ are effective.

We decompose $M=M_{0}+M_{1}$ for $M_{0}\geq 0$ and $M_{1}\geq 0$ such that any
component of $M_{0}$ does not intersect $T$ and all components of $M_{1}$
intersect $T$. By ($**$ ‣ 2.1), for any prime divisor $Q$ on $S^{\prime}$ we
obtain

($*\\!*\\!*$) $a(Q,S^{\prime},\Delta_{S^{\prime}})-a(Q,S,\Delta_{S})={\rm
ord}_{Q}(M|_{T}-N|_{T})\leq{\rm ord}_{Q}(M_{1}|_{T}).$

Pick any component $E$ of $M_{1}$. Then $\sigma_{E}(N)={\rm coeff}_{E}(N)=0$
since $M$ and $N$ have no common components. Since $K_{X}+\Delta$ is pseudo-
effective by hypothesis of Lemma 2.6 and since
$K_{X^{\prime}}+\Delta^{\prime}=g^{\prime}_{*}(g^{*}(K_{X}+\Delta)+N)$ by ($*$
‣ 2.1), the divisor $K_{X^{\prime}}+\Delta^{\prime}$ is pseudo-effective. By
Remark 2.3 (1), (3) and relation ($*$ ‣ 2.1), we have

$\begin{split}\sigma_{E}(K_{X}+\Delta)&=\sigma_{E}(g^{*}(K_{X}+\Delta))=\sigma_{E}(g^{*}(K_{X}+\Delta))+\sigma_{E}(N)\\\
&\geq\sigma_{E}(g^{*}(K_{X}+\Delta)+N)=\sigma_{E}(g^{\prime*}(K_{X^{\prime}}+\Delta^{\prime})+M)\\\
&=\sigma_{E}(g^{\prime*}(K_{X^{\prime}}+\Delta^{\prime}))+{\rm
coeff}_{E}(M)>0.\end{split}$

By definition of $M_{1}$, the image $g(E)$ intersects $S$. If
$a(E,X,\Delta)<0$, then the inequality $\sigma_{E}(K_{X}+\Delta)>0$
contradicts the second hypothesis of Lemma 2.6. Therefore, we see that
$a(E,X,\Delta)\geq 0$. Using ($*$ ‣ 2.1) again, we obtain

$a(E,X^{\prime},\Delta^{\prime})>a(E,X,\Delta)\geq 0.$

Thus, we see that all components $E$ of $M_{1}$ satisfy
$a(E,X^{\prime},\Delta^{\prime})>0$. By Lemma 2.5, all components of
$M_{1}|_{T}$ are exceptional over $S^{\prime}$. Recalling that $Q$ is a prime
divisor on $S^{\prime}$, we see that ${\rm ord}_{Q}(M_{1}|_{T})=0$. By
($*\\!*\\!*$ ‣ 2.1), we have $a(Q,S^{\prime},\Delta_{S^{\prime}})\leq
a(Q,S,\Delta_{S})$. ∎

###  2.2. Log MMP and models

In this paper, we will freely use log MMP with scaling in [3, Definition 2.4]
and [14, 4.4.11]. Let $X\to Z$ be a projective morphism from a normal quasi-
projective variety to a quasi-projective variety, let $(X,\Delta)$ be an lc
pair, and let $A\geq 0$ be an $\mathbb{R}$-divisor on $X$ such that
$(X,\Delta+A)$ is lc and $K_{X}+\Delta+A$ is nef over $Z$. Let

$(X_{0}:=X,\Delta_{0}:=\Delta)\dashrightarrow(X_{1},\Delta_{1})\dashrightarrow\cdots\dashrightarrow(X_{i},\Delta_{i})\dashrightarrow\cdots$

be a sequence of steps of a $(K_{X}+\Delta)$-MMP over $Z$ with scaling of $A$.
By construction, the following properties immediately follow.

  * •

$K_{X_{i}}+\Delta_{i}+\lambda_{i-1}A_{i}$ is nef over $Z$ for every $i>0$. In
particular, $\lambda_{i}\leq\lambda_{i-1}$.

  * •

Pick any $t>{\rm lim}_{i\to\infty}\lambda_{i}$, and take the minimum $l$ such
that $K_{X_{l}}+\Delta_{l}+tA_{l}$ is nef over $Z$. Then the birational map
$(X,\Delta+tA)\dashrightarrow(X_{l},\Delta_{l}+tA_{l})$ is finitely many steps
of a $(K_{X}+\Delta+tA)$-MMP over $Z$ to a log minimal model
$(X_{l},\Delta_{l}+tA_{l})$.

######  Definition 2.7 (Models, [25, Definition 2.2]).

Let $\pi\colon X\to Z$ be a projective morphism from a normal variety to a
variety, and let $(X,\Delta)$ be an lc pair. Let $\pi^{\prime}\colon
X^{\prime}\to Z$ be a projective morphism from a normal variety to $Z$ and
$\phi\colon X\dashrightarrow X^{\prime}$ a birational map over $Z$. Let $E$ be
the reduced $\phi^{-1}$-exceptional divisor on $X^{\prime}$, that is, $E=\sum
E_{j}$ where $E_{j}$ are $\phi^{-1}$-exceptional prime divisors on
$X^{\prime}$. When $K_{X^{\prime}}+\phi_{*}\Delta+E$ is $\mathbb{R}$-Cartier,
the pair $(X^{\prime},\Delta^{\prime}=\phi_{*}\Delta+E)$ is called a log
birational model of $(X,\Delta)$ over $Z$. A log birational model
$(X^{\prime},\Delta^{\prime})$ of $(X,\Delta)$ over $Z$ is a weak log
canonical model (weak lc model, for short) if

  * •

$K_{X^{\prime}}+\Delta^{\prime}$ is nef over $Z$, and

  * •

for any prime divisor $D$ on $X$ which is exceptional over $X^{\prime}$, we
have

$a(D,X,\Delta)\leq a(D,X^{\prime},\Delta^{\prime}).$

A weak lc model $(X^{\prime},\Delta^{\prime})$ of $(X,\Delta)$ over $Z$ is a
log minimal model if

  * •

$X^{\prime}$ is $\mathbb{Q}$-factorial, and

  * •

the above inequality on discrepancies is strict.

A log minimal model $(X^{\prime},\Delta^{\prime})$ of $(X,\Delta)$ over $Z$ is
called a good minimal model if $K_{X^{\prime}}+\Delta^{\prime}$ is semi-ample
over $Z$.

In particular, we do not assume that log minimal models are dlt.

######  Remark 2.8 ([28, Remark 2.5], see also [3, Remark 2.7]).

Let $\pi\colon X\to Z$ be a projective morphism from a normal variety to a
variety and $(X,\Delta)$ an lc pair. Let $(X^{\prime},\Delta^{\prime})$ be a
log minimal model of $(X,\Delta)$ over $Z$, and let
$(X^{\prime\prime},\Delta^{\prime\prime})$ be a $\mathbb{Q}$-factorial lc pair
with a small birational map $X^{\prime}\dashrightarrow X^{\prime\prime}$ such
that $K_{X^{\prime\prime}}+\Delta^{\prime\prime}$ is nef over $Z$ and the
birational transform of $\Delta^{\prime}$ on $X^{\prime\prime}$ is equal to
$\Delta^{\prime\prime}$. Then $(X^{\prime\prime},\Delta^{\prime\prime})$ is
also a log minimal model of $(X,\Delta)$ over $Z$. Furthermore, if
$(X^{\prime},\Delta^{\prime})$ is a good minimal model of $(X,\Delta)$ over
$Z$, then $(X^{\prime\prime},\Delta^{\prime\prime})$ is also a good minimal
model of $(X,\Delta)$ over $Z$.

######  Remark 2.9.

Let $(X,\Delta)$ be a projective lc pair, and let
$(X^{\prime},\Delta^{\prime})$ be a weak lc model of $(X,\Delta)$. By taking a
common resolution of $X\dashrightarrow X^{\prime}$ and the negativity lemma,
we can easily check the following facts.

  1. (1)

By Remark 2.3 (3), we have the equality

$\sigma_{P}(K_{X}+\Delta)=a(P,X^{\prime},\Delta^{\prime})-a(P,X,\Delta)$

for all prime divisors $P$ over $X$.

  2. (2)

Suppose that $(X^{\prime},\Delta^{\prime})$ is a log minimal model and
$(X,\Delta)\dashrightarrow(X^{\prime},\Delta^{\prime})$ be a sequence of steps
of a $(K_{X}+\Delta)$-MMP. Let $D$ be a prime divisor on $X$. Then $D$ is
contracted by the birational map $X\dashrightarrow X^{\prime}$ if and only if
$D$ is a component of $N_{\sigma}(K_{X}+\Delta)$.

We call $(Y,\Gamma)$ in Theorem 2.10 below a $\mathbb{Q}$-factorial dlt model
of $(X,\Delta)$.

######  Theorem 2.10 (Dlt blow-up, [11, Theorem 10.4], [31, Theorem 3.1]).

Let $X$ be a normal quasi-projective variety, and let $(X,\Delta)$ be an lc
pair. Then, there is a projective birational morphism $f\colon Y\to X$ from a
normal quasi-projective variety $Y$ such that defining $\Gamma$ to be the sum
of $f_{*}^{-1}\Delta$ and the reduced $f$-exceptional divisor, then
$(Y,\Gamma)$ is a $\mathbb{Q}$-factorial dlt pair and
$K_{Y}+\Gamma=f^{*}(K_{X}+\Delta)$.

######  Lemma 2.11 ($\mathbb{Q}$-factorial dlt closure).

Let $\pi\colon X\to Z$ be a projective morphism from a normal quasi-projective
variety to a quasi-projective variety, let $(X,\Delta)$ be a
$\mathbb{Q}$-factorial dlt pair, and let $Z\hookrightarrow Z^{c}$ be an open
immersion to a projective variety. Then there is a projective morphism
$\pi^{c}\colon X^{c}\to Z^{c}$ and a projective $\mathbb{Q}$-factorial dlt
pair $(X^{c},\Delta^{c})$ such that $X$ is an open subset of $X^{c}$,
$\pi^{c}|_{X}=\pi$, $(X^{c}|_{X},\Delta^{c}|_{X})=(X,\Delta)$, and all lc
centers of $(X^{c},\Delta^{c})$ intersects $X$.

###### Proof.

We take an open immersion $X\hookrightarrow\overline{X}$ to a projective
variety with a projective morphism $\overline{X}\to Z^{c}$. Since $(X,\Delta)$
is dlt, we can take a log resolution $f\colon X^{\prime}\to X$ of $(X,\Delta)$
such that $a(E,X,\Delta)>-1$ for all $f$-exceptional prime divisors $E$. Let
$\Delta^{\prime}$ be the sum of $f^{-1}_{*}\Delta$ and the reduced
$f$-exceptional divisor. By [11, Theorem 4.17] and taking blow-ups if
necessary, we may take an open immersion
$X^{\prime}\hookrightarrow\overline{X}^{\prime}$ to a smooth projective
variety and a birational morphism $\overline{X}^{\prime}\to\overline{X}$ such
that putting $\overline{\Delta}^{\prime}$ as the closure of $\Delta^{\prime}$
in $\overline{X}^{\prime}$ then
$(\overline{X}^{\prime},\overline{\Delta}^{\prime})$ is log smooth and all lc
centers of $(\overline{X}^{\prime},\overline{\Delta}^{\prime})$ intersects
$X^{\prime}$.

We run a $(K_{\overline{X}^{\prime}}+\overline{\Delta}^{\prime})$-MMP over
$\overline{X}$ with scaling of an ample divisor. After finitely many steps, we
get a projective $\mathbb{Q}$-factorial dlt pair $(X^{c},\Delta^{c})$ whose
restriction over $X$, which we denote $(\tilde{X}^{c},\tilde{\Delta}^{c})$, is
a $\mathbb{Q}$-factorial dlt model of $(X,\Delta)$ (cf. [3, Theorem 3.5]).
Since $a(E,X,\Delta)>-1$ for all $f$-exceptional prime divisors $E$, the
birational morphism $\tilde{X}^{c}\to X$ is small. Since $X$ is
$\mathbb{Q}$-factorial, the morphism is an isomorphism. We see that
$\Delta^{c}|_{X}=\Delta$ by construction. Since all lc centers of
$(\overline{X}^{\prime},\overline{\Delta}^{\prime})$ intersects $X^{\prime}$,
all lc centers of $(X^{c},\Delta^{c})$ intersects $X$. Thus,
$(X^{c},\Delta^{c})$ is the desired $\mathbb{Q}$-factorial dlt pair. ∎

###  2.3. Invariant Iitaka dimension and numerical dimension

We define invariant Iitaka dimension ([7]) and numerical dimension ([35]) for
$\mathbb{R}$-Cartier divisors on normal projective varieties, then we collect
some properties.

######  Definition 2.12 (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 ([7, Definition 2.2.1], see also [14,
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
$\kappa_{\iota}(X,D)=-\infty$. 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.

Let $X\to Z$ be a projective morphism from a normal variety to a variety and
$D$ an $\mathbb{R}$-Cartier divisor on $X$. Then the relative invariant Iitaka
dimension of $D$, denoted by $\kappa_{\iota}(X/Z,D)$, is similarly defined: If
there is an $\mathbb{R}$-divisor $E\geq 0$ such that $D\sim_{\mathbb{R},Z}E$
then we set $\kappa_{\iota}(X/Z,D)=\kappa_{\iota}(F,D|_{F})$, where $F$ is a
sufficiently general fiber of the Stein factorization of $X\to Z$, and
otherwise we set $\kappa_{\iota}(X/Z,D)=-\infty$. By definition, the
inequality $\kappa_{\iota}(X/Z,D)\geq 0$ holds if and only if
$D\sim_{\mathbb{R},Z}E$ for some $E\geq 0$.

######  Definition 2.13 (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 ([35, 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 $X$}\\!}.$

Let $X\to Z$ be a projective morphism from a normal variety to a variety, and
let $D$ be an $\mathbb{R}$-Cartier divisor on $X$. Then, the relative
numerical dimension of $D$ over $Z$ is defined by $\kappa_{\sigma}(F,D|_{F})$,
where $F$ is a sufficiently general fiber of the Stein factorization of $X\to
Z$. Then $\kappa_{\sigma}(F,D|_{F})$ does not depend on the choice of $F$, so
the relative numerical dimension is well-defined. In this paper, we denote
$\kappa_{\sigma}(F,D|_{F})$ by $\kappa_{\sigma}(X/Z,D)$.

######  Remark 2.14 ([35, V, 2.7 Proposition], [27, Remark 3.2], [28, Remark
2.8]).

We write down basic properties of the invariant Iitaka dimension and the
numerical dimension.

  1. (1)

Let $D_{1}$ and $D_{2}$ be $\mathbb{R}$-Cartier divisors on a normal
projective variety $X$.

     * •

Suppose that $D_{1}\sim_{\mathbb{R}}D_{2}$ holds. Then the equalities
$\kappa_{\iota}(X,D_{1})=\kappa_{\iota}(X,D_{2})$ and
$\kappa_{\sigma}(X,D_{1})=\kappa_{\sigma}(X,D_{2})$ hold.

     * •

Suppose that $D_{1}\sim_{\mathbb{R}}N_{1}$ and $D_{2}\sim_{\mathbb{R}}N_{2}$
for some $\mathbb{R}$-divisors $N_{1}\geq 0$ and $N_{2}\geq 0$ such that ${\rm
Supp}N_{1}={\rm Supp}N_{2}$. Then we have
$\kappa_{\iota}(X,D_{1})=\kappa_{\iota}(X,D_{2})$ and
$\kappa_{\sigma}(X,D_{1})=\kappa_{\sigma}(X,D_{2})$.

  2. (2)

Let $f\colon Y\to X$ be a surjective morphism of normal projective varieties,
and let $D$ be an $\mathbb{R}$-Cartier divisor on $X$.

     * •

The equalities $\kappa_{\iota}(X,D)=\kappa_{\iota}(Y,f^{*}D)$ and
$\kappa_{\sigma}(X,D)=\kappa_{\sigma}(Y,f^{*}D)$ hold.

     * •

Suppose that $f$ is birational. Let $D^{\prime}$ be an $\mathbb{R}$-Cartier
divisor on $Y$ such that $D^{\prime}-f^{*}D$ is effective and $f$-exceptional.
Then we have $\kappa_{\iota}(X,D)=\kappa_{\iota}(Y,D^{\prime})$ and
$\kappa_{\sigma}(X,D)=\kappa_{\sigma}(Y,D^{\prime})$.

######  Definition 2.15 (Relatively abundant divisor and relatively log
abundant divisor).

Let $\pi\colon X\to Z$ be a projective morphism from a normal variety to a
variety, and let $D$ be an $\mathbb{R}$-Cartier divisor on $X$. We say that
$D$ is $\pi$-abundant (or abundant over $Z$) if the equality
$\kappa_{\iota}(X/Z,D)=\kappa_{\sigma}(X/Z,D)$ holds. When $Z$ is a point, we
simply say that $D$ is abundant.

Let $\pi\colon X\to Z$ and $D$ be as above, and let $(X,\Delta)$ be an lc
pair. We say that $D$ is $\pi$-log abundant (or log abundant over $Z$) with
respect to $(X,\Delta)$ if $D$ is abundant over $Z$ and for any lc center $S$
of $(X,\Delta)$ with the normalization $S^{\nu}\to S$, the pullback
$D|_{S^{\nu}}$ is abundant over $Z$.

######  Definition 2.16 (Log abundant lc pairs).

Let $\pi\colon X\to Z$ be a projective morphism from a normal variety to a
variety and $(X,\Delta)$ an lc (resp. dlt) pair. When $K_{X}+\Delta$ is log
abundant over $Z$ with respect to $(X,\Delta)$, we call $(X,\Delta)$ a log
abundant lc (resp. log abundant dlt) pair over $Z$ or we say that $(X,\Delta)$
is log abundant over $Z$. When $Z$ is a point, we simply say $(X,\Delta)$ is
log abundant.

######  Lemma 2.17 ([28, Lemma 2.11]).

Let $(X,\Delta)$ be a projective lc pair with an $\mathbb{R}$-divisor
$\Delta$. Suppose that $K_{X}+\Delta$ is abundant and there is an effective
$\mathbb{R}$-divisor $D\sim_{\mathbb{R}}K_{X}+\Delta$. Let $X\dashrightarrow
V$ be the Iitaka fibration associated to $D$. Pick a log resolution $f\colon
Y\to X$ of $(X,\Delta)$ such that the induced map $Y\dashrightarrow V$ is a
morphism, and let $(Y,\Gamma)$ be a projective lc pair such that we can write
$K_{Y}+\Gamma=f^{*}(K_{X}+\Delta)+E$ for an effective $f$-exceptional divisor
$E$. Then, we have $\kappa_{\sigma}(Y/V,K_{Y}+\Gamma)=0$.

######  Lemma 2.18.

Let $\pi\colon X\to Z$ and $\pi^{\prime}\colon X^{\prime}\to Z$ be projective
morphisms of normal quasi-projective varieties. Let $(X,\Delta)$ and
$(X^{\prime},\Delta^{\prime})$ be dlt pairs, and let $S$ and $S^{\prime}$ be
lc centers of $(X,\Delta)$ and $(X^{\prime},\Delta^{\prime})$ respectively.
Suppose that there is a small birational map $f\colon X\dashrightarrow
X^{\prime}$ over $Z$ such that $f_{*}\Delta=\Delta^{\prime}$, $f$ is an
isomorphism on an open set intersecting $S$ and the restriction of $f$ to $S$
induces a birational map $f|_{S}\colon S\dashrightarrow S^{\prime}$. If there
is an $\mathbb{R}$-divisor $A$ on $X$ such that $K_{X}+\Delta+aA$ and
$K_{X^{\prime}}+\Delta^{\prime}+a^{\prime}f_{*}A$ are nef over $Z$ for some
$a>a^{\prime}>0$, then the following inequalities hold true:

$\begin{split}\kappa_{\iota}(S/Z,(K_{X}+\Delta)|_{S})&\geq\kappa_{\iota}(S^{\prime}/Z,(K_{X^{\prime}}+\Delta^{\prime})|_{S^{\prime}})\quad{\rm
and}\\\
\kappa_{\sigma}(S/Z,(K_{X}+\Delta)|_{S})&\geq\kappa_{\sigma}(S^{\prime}/Z,(K_{X^{\prime}}+\Delta^{\prime})|_{S^{\prime}}).\end{split}$

###### Proof.

By taking an appropriate resolution of the graph of $f\colon X\dashrightarrow
X^{\prime}$, we may find a common log resolution $g\colon Y\to X$ and
$g^{\prime}\colon Y\to{X^{\prime}}$ of the map
$(X,\Delta)\dashrightarrow(X^{\prime},\Delta^{\prime})$ and a subvariety
$T\subset Y$ birational to $S$ and $S^{\prime}$ such that the induced
morphisms $g|_{T}\colon T\to S$ and $g^{\prime}|_{T}\colon T\to S^{\prime}$
form a common resolution of $f|_{S}\colon S\dashrightarrow S^{\prime}$. We
note that $A$ and $f_{*}A$ are $\mathbb{R}$-Cartier by hypothesis. Since $f$
is small, by the negativity lemma we may write

$\begin{split}&E+g^{*}(K_{X}+\Delta+aA)=g^{\prime*}(K_{X^{\prime}}+\Delta^{\prime}+af_{*}A)\quad{\rm
and}\\\
&g^{*}(K_{X}+\Delta+a^{\prime}A)=g^{\prime*}(K_{X^{\prime}}+\Delta^{\prime}+a^{\prime}f_{*}A)+E^{\prime}\end{split}$

for a $g$-exceptional $\mathbb{R}$-divisor $E\geq 0$ and a
$g^{\prime}$-exceptional $\mathbb{R}$-divisor $E^{\prime}\geq 0$. Thus there
is a $g^{\prime}$-exceptional $\mathbb{R}$-divisor $F\geq 0$ such that
$g^{*}(K_{X}+\Delta)=g^{\prime*}(K_{X^{\prime}}+\Delta^{\prime})+F$. Since
$g\colon Y\to X$ is an isomorphism over an open subset which contains general
points of $S$, we have $F|_{T}\geq 0$. By restricting to $T$, we obtain

$g|_{T}^{*}((K_{X}+\Delta)|_{S})=g^{\prime}|_{T}^{*}((K_{X^{\prime}}+\Delta^{\prime})|_{S^{\prime}})+F|_{T}.$

From this relation, the inequalities of Lemma 2.18 follow. ∎

######  Lemma 2.19.

Let $\pi\colon X\to Z$ be a morphism of normal projective varieties and
$(X,\Delta)$ an lc pair such that $K_{X}+\Delta$ is abundant over $Z$. Let
$A_{Z}$ be an ample Cartier divisor on $Z$, and pick $\alpha>2\cdot{\rm
dim}X$. Then $K_{X}+\Delta+\alpha\pi^{*}A_{Z}$ is (globally) abundant.

###### Proof.

We may assume that $\pi$ is a contraction and $(X,\Delta)$ is
$\mathbb{Q}$-factorial dlt by taking the Stein factorization of $\pi$ and a
dlt blow-up, respectively. We may assume that $K_{X}+\Delta$ is pseudo-
effective over $Z$ because otherwise we can check that
$K_{X}+\Delta+\alpha\pi^{*}A_{Z}$ is not pseudo-effective by the definition of
numerical dimension.

First we show that $K_{X}+\Delta+\alpha\pi^{*}A_{Z}$ is pseudo-effective. Pick
any ample $\mathbb{R}$-divisor $H$ on $X$. By [4], there is a good minimal
model $(\tilde{X},\tilde{\Delta}+\tilde{H})$ of $(X,\Delta+H)$ over $Z$. Let
$\tilde{\pi}\colon\tilde{X}\to Z$ be the induced morphism. With the length of
extremal rays, we can check that
$K_{\tilde{X}}+\tilde{\Delta}+\tilde{H}+\alpha\tilde{\pi}^{*}A_{Z}$ is
(globally) nef. This shows that $K_{X}+\Delta+H+\alpha\pi^{*}A_{Z}$ is pseudo-
effective for any ample $H$. Thus $K_{X}+\Delta+\alpha\pi^{*}A_{Z}$ is pseudo-
effective.

We fix $\alpha^{\prime}$ such that $2\cdot{\rm dim}X<\alpha^{\prime}<\alpha$.
By the previous argument, $K_{X}+\Delta+\alpha^{\prime}\pi^{*}A_{Z}$ is
pseudo-effective. Pick $E\geq 0$ such that $K_{X}+\Delta\sim_{\mathbb{R},Z}E$.
We take the relative Iitaka fibration $X\dashrightarrow V$ over $Z$ induced by
$E$. By Remark 2.14 and replacing $(X,\Delta)$, we may assume that
$X\dashrightarrow V$ is a morphism. By Lemma 2.17, we have
$\kappa_{\sigma}(X/V,K_{X}+\Delta)=0$. Furthermore, the equality
$\kappa_{\sigma}(X/Z,K_{X}+\Delta)={\rm dim}V-{\rm dim}Z$ holds by
construction. By [26, Lemma 3.1] and [26, Remark 3.2] which are applications
of weak semistable reduction ([1]), we obtain a projective lc pair
$(X^{\prime},\Delta^{\prime})$ and the following diagram of normal projective
varieties

---
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{p^{\prime}}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{\prime}}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$$\textstyle{Z}$

where three horizontal morphisms are birational and other morphisms are
contractions such that
$K_{X^{\prime}}+\Delta^{\prime}\sim_{\mathbb{R},V^{\prime}}0$ and
$p^{*}(K_{X}+\Delta)+F_{p}=p^{\prime*}(K_{X^{\prime}}+\Delta^{\prime})+F_{p^{\prime}}$
for a $p$-exceptional $\mathbb{R}$-divisor $F_{p}\geq 0$ and a
$p^{\prime}$-exceptional $\mathbb{R}$-divisor $F_{p^{\prime}}\geq 0$. Then we
can find an $\mathbb{R}$-Cartier divisor $D^{\prime}$ on $V^{\prime}$ such
that
$\varphi^{\prime*}D^{\prime}\sim_{\mathbb{R}}K_{X^{\prime}}+\Delta^{\prime}$.
Then $\kappa_{\sigma}(V^{\prime}/Z,D^{\prime})={\rm dim}V^{\prime}-{\rm
dim}Z$, so $D^{\prime}$ is big over $Z$. By construction and Remark 2.14,
$D^{\prime}+\alpha^{\prime}g^{\prime*}A_{Z}$ is pseudo-effective, and
$K_{X}+\Delta+\alpha\pi^{*}A_{Z}$ is abundant if and only if
$D^{\prime}+\alpha g^{\prime*}A_{Z}$ is abundant. Now

$\begin{split}D^{\prime}+\alpha
g^{\prime*}A_{Z}=(1-\delta)(D^{\prime}+\alpha^{\prime}g^{\prime*}A_{Z})+\delta
D^{\prime}+(\alpha-\alpha^{\prime})g^{\prime*}A_{Z}+\delta\alpha^{\prime}g^{\prime*}A_{Z}\end{split}$

for any $\delta>0$. Since $D^{\prime}$ is big over $Z$ and $A_{Z}$ is ample,
taking a sufficiently small $\delta>0$ we see that $D^{\prime}+\alpha
g^{\prime*}A_{Z}$ is big. Therefore $K_{X}+\Delta$ is abundant. ∎

######  Lemma 2.20.

Let $\pi\colon X\to Z$ be a projective morphism of normal quasi-projective
varieties, and let $(X,\Delta)$ be an lc pair. If $K_{X}+\Delta$ is nef and
log abundant over $Z$ with respect to $(X,\Delta)$, then it is semi-ample over
$Z$.

###### Proof.

When $\Delta$ is a $\mathbb{Q}$-divisor, the lemma follows by induction using
[12, Theorem 1.1] and [23, Theorem 1.4]. When $\Delta$ is not necessarily a
$\mathbb{Q}$-divisor, we may reduce to the case of $\mathbb{Q}$-divisors by
the argument of [27, Proof of Lemma 3.4] with minor changes. ∎

###  2.4. On log MMP and existence of good minimal models

In this subsection, we collect some results on log MMP and existence of good
minimal models for lc pairs.

Firstly, we introduce a remark and a lemma on log MMP with scaling.

######  Remark 2.21 ([10], [3, Remark 2.10]).

In this remark, we recall a part of argument of special termination which is
important for the proof of the main result of this paper.

Let $\pi\colon X\to Z$ be a projective morphism of normal quasi-projective
varieties and $(X,\Delta)$ a dlt pair. Fix an lc center $S$ of $(X,\Delta)$,
and define a dlt pair $(S,\Delta_{S})$ where $\Delta_{S}$ is constructed with
adjunction $K_{S}+\Delta_{S}=(K_{X}+\Delta)|_{S}$. Let $A$ be an effective
$\mathbb{R}$-divisor on $X$ such that $(X,\Delta+A)$ is lc and
$K_{X}+\Delta+A$ is nef over $Z$. For a sequence of steps of a
$(K_{X}+\Delta)$-MMP over $Z$ with scaling of $A$

$(X_{0}:=X,\Delta_{0}:=\Delta)\dashrightarrow(X_{1},\Delta_{1})\dashrightarrow\cdots\dashrightarrow(X_{i},\Delta_{i})\dashrightarrow\cdots,$

suppose that each birational map $X\dashrightarrow X_{i}$ induces a small
birational map $S\dashrightarrow S_{i}$ to an lc center $S_{i}$ of
$(X_{i},\Delta_{i})$ such that the birational transform of $\Delta_{S}$ on
$S_{i}$ is equal to the $\mathbb{R}$-divisor $\Delta_{S_{i}}$ defined by
adjunction $K_{S_{i}}+\Delta_{S_{i}}=(K_{X_{i}}+\Delta_{i})|_{S_{i}}$. Let
$(T,\Psi)\to(S,\Delta_{S})$ be a dlt blow-up, and we put $A_{T}=A|_{T}$. Let
$X\to V$ be the $(K_{X}+\Delta)$-negative extremal contraction of the first
step of the $(K_{X}+\Delta)$-MMP over $Z$. It is obvious that
$(S_{1},\Delta_{S_{1}})$ is a weak lc model of $(T,\Psi)$ over $V$ with
relatively ample log canonical divisor. Then $(T,\Psi)$ has a good minimal
model ([26, Remark 2.10]). Using [3, Theorem 4.1], we get a sequence of steps
of a $(K_{T}+\Psi)$-MMP over $V$ that terminates with a good minimal model
$(T_{1},\Psi_{1})$ over $V$. By construction, this is a $(K_{T}+\Psi)$-MMP
over $Z$ with scaling of $A_{T}$ and $(T_{1},\Psi_{1})$ is a
$\mathbb{Q}$-factorial dlt model of $(S_{1},\Delta_{S_{1}})$ as in Theorem
2.10. Repeating this discussion, over the sequence of birational maps
$S\dashrightarrow\cdots\dashrightarrow S_{i}\dashrightarrow\cdots$, we can
construct a sequence of steps of a $(K_{T}+\Psi)$-MMP over $Z$ with scaling of
$A_{T}$

$(T_{0}:=T,\Psi_{0}:=\Psi)\dashrightarrow(T_{1},\Psi_{1})\dashrightarrow\cdots\dashrightarrow(T_{i},\Psi_{i})\dashrightarrow\cdots$

such that each $(T_{i},\Psi_{i})$ is a $\mathbb{Q}$-factorial dlt model of
$(S_{i},\Delta_{S_{i}})$. We note that the birational map
$(T_{i},\Psi_{i})\dashrightarrow(T_{i+1},\Psi_{i+1})$ is not necessarily one
step of the $(K_{T_{i}}+\Psi_{i})$-MMP and it can be an isomorphism for some
$i$.

We define

$\lambda_{i}={\rm inf}\set{\mu\in\mathbb{R}_{\geq
0}\\!}{\\!K_{X_{i}}+\Delta_{i}+\mu A_{i}\text{\rm\, is nef over }Z},$

and suppose that ${\rm lim}_{i\to\infty}\lambda_{i}=0$. Here, $A_{i}$ is the
birational transform of $A$ on $X_{i}$. By the negativity lemma, for any $i$,
the birational map $X_{i}\dashrightarrow X_{i+1}$ is an isomorphism in a
neighborhood of $S_{i}$ if and only if the induced birational map
$(S_{i},\Delta_{S_{i}})\dashrightarrow(S_{i+1},\Delta_{S_{i+1}})$ is an
isomorphism. The isomorphism holds for all $i\gg 0$ if the $(K_{T}+\Psi)$-MMP
over $Z$ terminates with a log minimal model. Let $A_{T_{i}}$ be the
birational transform of $A_{T}$ on $T_{i}$. We define

$\lambda^{\prime}_{i}={\rm inf}\set{\mu\in\mathbb{R}_{\geq
0}\\!}{\\!K_{T_{i}}+\Psi_{i}+\mu A_{T_{i}}\text{\rm\, is nef over }Z}.$

Then $\lambda^{\prime}_{i}\leq\lambda_{i}$, so ${\rm
lim}_{i\to\infty}\lambda^{\prime}_{i}=0$, and $K_{T}+\Psi$ is pseudo-effective
over $Z$. In this way, when ${\rm lim}_{i\to\infty}\lambda_{i}=0$, the
termination of the $(K_{T}+\Psi)$-MMP over $Z$ implies that the birational map
$X_{i}\dashrightarrow X_{i+1}$ is an isomorphism in a neighborhood of $S_{i}$
for all $i\gg 0$. By [3, Theorem 4.1 (iii)] and since $(T,\Psi)$ is a
$\mathbb{Q}$-factorial dlt model of $(S,\Delta_{S})$, the existence of a log
minimal model of $(S,\Delta_{S})$ over $Z$ shows the termination of the
$(K_{T}+\Psi)$-MMP over $Z$.

######  Lemma 2.22.

Let $\pi\colon X\to Z$ be a projective morphism of normal quasi-projective
varieties, let $(X,\Delta)$ be an lc pair, and let $S$ be a subvariety of $X$.
Let

$(X_{0}:=X,\Delta_{0}:=\Delta)\dashrightarrow(X_{1},\Delta_{1})\dashrightarrow\cdots\dashrightarrow(X_{i},\Delta_{i})\dashrightarrow\cdots$

be a sequence of steps of a $(K_{X}+\Delta)$-MMP over $Z$ with scaling of some
$\mathbb{R}$-divisor $A\geq 0$. We define $\lambda_{i}={\rm
inf}\set{\mu\in\mathbb{R}_{\geq 0}\\!}{\\!K_{X_{i}}+\Delta_{i}+\mu
A_{i}\text{\rm\, is nef over }Z}$ where $A_{i}$ is the birational transform of
$A$ on $X_{i}$. Suppose that each step of the $(K_{X}+\Delta)$-MMP is an
isomorphism on a neighborhood of $S$ and ${\rm
lim}_{i\to\infty}\lambda_{i}=0$. Then, for any $\pi$-ample
$\mathbb{R}$-divisor $H$ on $X$ and closed point $x\in S$, there is $E\geq 0$
such that $E\sim_{\mathbb{R},Z}K_{X}+\Delta+H$ and ${\rm Supp}E\not\ni x$.

###### Proof.

Fix a $\pi$-ample $\mathbb{R}$-divisor $H$ on $X$ and a closed point $x\in S$.
Fix $i\gg 0$ such that $\frac{1}{2}H-\lambda_{i}A$ is $\pi$-ample and
$X\dashrightarrow X_{i}$ is a sequence of steps of a
$(K_{X}+\Delta+\lambda_{i}A+\epsilon H)$-MMP over $Z$ for some
$\epsilon\in(0,\frac{1}{2})$. Since $H-\epsilon H-\lambda_{i}A$ is ample over
$Z$ and

$K_{X}+\Delta+H=K_{X}+\Delta+\lambda_{i}A+\epsilon H+(H-\epsilon
H-\lambda_{i}A),$

it is sufficient to find $E_{0}\geq 0$ such that
$E_{0}\sim_{\mathbb{R},Z}K_{X}+\Delta+\lambda_{i}A+\epsilon H$ and ${\rm
Supp}E_{0}\not\ni x$. Note that $X\dashrightarrow X_{i}$ is a birational
contraction and the birational transform $H_{i}$ of $H$ on $X_{i}$ is
$\mathbb{R}$-Cartier ([28, Remark 6.1]).

Let $U$ be the largest open subset of $X$ on which $X\dashrightarrow X_{i}$ is
an isomorphism, and let $U_{i}$ be its image on $X_{i}$. By hypothesis the
inclusion $S\subset U$ holds, and the codimension of $X_{i}\setminus U_{i}$ in
$X_{i}$ is at least $2$ and $H_{i}|_{U_{i}}$ is ample over $Z$. Let $S_{i}$
(resp. $x_{i}$) be the image of $S$ (resp. $x$) on $X_{i}$. Note that
$x_{i}\in S_{i}\subset U_{i}$. Pick an ample $\mathbb{R}$-divisor $B_{i}$ on
$X_{i}$ such that the restriction $(\epsilon H_{i}-B_{i})|_{U_{i}}$ is ample
over $Z$. Because $K_{X_{i}}+\Delta_{i}+\lambda_{i}A_{i}+B_{i}$ is ample over
$Z$, we can find $E_{U_{i}}\geq 0$ on $U_{i}$ such that

$E_{U_{i}}\sim_{\mathbb{R},Z}(K_{X_{i}}+\Delta_{i}+\lambda_{i}A_{i}+B_{i})|_{U_{i}}+(\epsilon
H_{i}-B_{i})|_{U_{i}}$

and ${\rm Supp}E_{U_{i}}\not\ni x_{i}$. Let $E_{i}$ be the closure of
$E_{U_{i}}$ in $X_{i}$. Then $E_{i}\geq 0$ and ${\rm Supp}E_{i}\not\ni x_{i}$.
Since the codimension of $X_{i}\setminus U_{i}$ in $X_{i}$ is at least $2$, we
have $E_{i}\sim_{\mathbb{R},Z}K_{X_{i}}+\Delta_{i}+\lambda_{i}A_{i}+\epsilon
H_{i}.$ By combining this and that $H_{i}$ is $\mathbb{R}$-Cartier, we see
that $E_{i}$ is $\mathbb{R}$-Cartier.

We take the normalization $Y$ of the graph of $X\dashrightarrow X_{i}$, and we
denote the natural morphisms $Y\to X$ and $Y\to X_{i}$ by $f$ and $f_{i}$,
respectively. Since $X\dashrightarrow X_{i}$ is a sequence of steps of a
$(K_{X}+\Delta+\lambda_{i}A+\epsilon H)$-MMP, we have

$f^{*}(K_{X}+\Delta+\lambda_{i}A+\epsilon
H)=f_{i}^{*}(K_{X_{i}}+\Delta_{i}+\lambda_{i}A_{i}+\epsilon H_{i})+F$

for some $F\geq 0$, and ${\rm Supp}F$ does not intersect $f^{-1}(U)$. We set
$E_{0}=f_{*}(f_{i}^{*}E_{i}+F)$. By definition of $x_{i}$ and $U$, it follows
that $E_{0}\sim_{\mathbb{R},Z}K_{X}+\Delta+\lambda_{i}A+\epsilon H$ and ${\rm
Supp}E_{0}\not\ni x$. Thus $E_{0}$ is the desired divisor discussed in the
first paragraph, so Lemma 2.22 holds. ∎

Secondly, we introduce good minimal model analogue of [5, Theorem 1.1]. It is
known to the experts, and the proof of [5, Theorem 1.1] works with no changes
to prove the case of the existence of a good minimal model. So we skip the
proof.

######  Theorem 2.23 (cf. [5, Theorem 1.1]).

Let $(X,\Delta)$ be a projective lc pair such that $K_{X}+\Delta$ is pseudo-
effective. Then $(X,\Delta)$ has a log minimal model (resp. a good minimal
model) if and only if there exists a resolution $f\colon Y\to X$ such that
$f^{*}(K_{X}+\Delta)$ has the Nakayama–Zariski decomposition with nef (resp.
semi-ample) positive part.

Thirdly, we prepare results on equivalence about existence log minimal models
and good minimal models for given two lc pairs. For the proof of Lemma 2.24,
see [26].

######  Lemma 2.24 ([26, Lemma 2.15]).

Let $\pi\colon X\to Z$ be a projective morphism of normal quasi-projective
varieties, and let $(X,\Delta)$ be an lc pair. Let $(Y,\Gamma)$ be an lc pair
with a projective birational morphism $f\colon Y\to X$ such that
$K_{Y}+\Gamma=f^{*}(K_{X}+\Delta)+E$ for an effective $f$-exceptional divisor
$E$. Then $(X,\Delta)$ has a weak lc model (resp. a log minimal model, a good
minimal model) over $Z$ if and only if $(Y,\Gamma)$ has a weak lc model (resp.
a log minimal model, a good minimal model) over $Z$.

######  Lemma 2.25 (cf. [20, Lemma 5.3]).

Let $(X,\Delta)$ be a projective lc pair, and let $(Y,\Gamma)$ be an lc pair
with a projective birational morphism $f\colon Y\to X$. Suppose that
$K_{X}+\Delta$ is pseudo-effective and all prime divisors $D$ on $Y$ satisfy

$0\leq a(D,Y,\Gamma)-a(D,X,\Delta)\leq\sigma_{D}(K_{X}+\Delta).$

Then $K_{Y}+\Gamma$ is pseudo-effective. Furthermore, $(X,\Delta)$ has a log
minimal model (resp. a good minimal model) if and only if $(Y,\Gamma)$ has a
log minimal model (resp. a good minimal model).

###### Proof.

Let $\Delta_{Y}$ be an $\mathbb{R}$-divisor defined by
$K_{Y}+\Delta_{Y}=f^{*}(K_{X}+\Delta)$. Then the hypothesis on discrepancies
is equivalent to $0\leq\Delta_{Y}-\Gamma\leq N_{\sigma}(K_{Y}+\Delta_{Y})$.
Thus we have $\Delta_{Y}\geq 0$. We may replace $(X,\Delta)$ by
$(Y,\Delta_{Y})$, therefore we may assume $X=Y$.

Pick any resolution $g\colon W\to X$. By Remark 2.3 (4), we have

$N_{\sigma}(g^{*}(K_{X}+\Delta))\geq
g^{*}(\Delta-\Gamma)=g^{*}((K_{X}+\Delta)-(K_{X}+\Gamma)).$

Therefore, $g^{*}(K_{X}+\Gamma)\geq
g^{*}(K_{X}+\Delta)-N_{\sigma}(g^{*}(K_{X}+\Delta))$ and the right hand side
is the limit of effective divisors by definition of $N_{\sigma}(\,\cdot\,)$.
Then $g^{*}(K_{X}+\Gamma)$ is pseudo-effective, hence $K_{X}+\Gamma$ is also
pseudo-effective. By [35, III, 1.8 Lemma], we have

$\begin{split}N_{\sigma}(g^{*}(K_{X}+\Gamma))&=N_{\sigma}\bigl{(}g^{*}(K_{X}+\Delta)-g^{*}((K_{X}+\Delta)-(K_{X}+\Gamma))\bigr{)}\\\
&=N_{\sigma}(g^{*}(K_{X}+\Delta))-g^{*}(K_{X}+\Delta)+g^{*}(K_{X}+\Gamma).\end{split}$

Thus, positive parts of Nakayama–Zariski decompositions of
$g^{*}(K_{X}+\Delta)$ and $g^{*}(K_{Y}+\Gamma)$ are the same divisors. Because
$g\colon W\to X$ is any resolution, $K_{X}+\Delta$ birationally has
Nakayama–Zariski decomposition with nef (resp. semi-ample) positive part if
and only if $K_{X}+\Gamma$ birationally has Nakayama–Zariski decomposition
with nef (resp. semi-ample) positive part. Then Lemma 2.25 follows from
Theorem 2.23. ∎

######  Lemma 2.26.

Let $\pi\colon X\to Z$ and $\pi^{\prime}\colon X^{\prime}\to Z$ be projective
morphisms of normal quasi-projective varieties, and let $X\dashrightarrow
X^{\prime}$ be a birational map over $Z$. Let $(X,\Delta)$ and
$(X^{\prime},\Delta^{\prime})$ be lc pairs. Suppose that

  * •

$a(P,X,\Delta)\leq a(P,X^{\prime},\Delta^{\prime})$ for all prime divisors $P$
on $X$, and

  * •

$a(P^{\prime},X^{\prime},\Delta^{\prime})\leq a(P^{\prime},X,\Delta)$ for all
prime divisors $P^{\prime}$ on $X^{\prime}$.

Then, $K_{X}+\Delta$ is abundant over $Z$ if and only if
$K_{X^{\prime}}+\Delta^{\prime}$ is abundant over $Z$, and furthermore,
$(X,\Delta)$ has a log minimal model (resp. a good minimal model) over $Z$ if
and only if $(X^{\prime},\Delta^{\prime})$ has a log minimal model (resp. a
good minimal model) over $Z$.

###### Proof.

Let $f\colon Y\to X$ and $f^{\prime}\colon Y\to X^{\prime}$ be a common log
resolution of the map $(X,\Delta)\dashrightarrow(X^{\prime},\Delta^{\prime})$.
We define an $\mathbb{R}$-divisor $\Gamma$ on $Y$ by

$\Gamma=-\sum_{E}{\rm
min}\\{0,a(E,X,\Delta),a(E,X^{\prime},\Delta^{\prime})\\}E,$

where $E$ runs over all prime divisors on $Y$. By construction of $Y$ and
since $(X,\Delta)$ and $(X^{\prime},\Delta^{\prime})$ are lc, $(Y,\Gamma)$ is
log smooth and lc. We put $F=K_{Y}+\Gamma-f^{*}(K_{X}+\Delta)$ and
$F^{\prime}=K_{Y}+\Gamma-f^{\prime*}(K_{X^{\prime}}+\Delta^{\prime}).$ Then,
for any prime divisor $D$ on $Y$ we have

${\rm coeff}_{D}(F)=a(D,X,\Delta)-{\rm
min}\\{0,a(D,X,\Delta),a(D,X^{\prime},\Delta^{\prime})\\}\geq 0,$

and ${\rm coeff}_{D}(F)=0$ when $D$ is a prime divisor on $X$ because we have
$a(D,X,\Delta)\leq 0$ and $a(D,X,\Delta)\leq a(D,X^{\prime},\Delta^{\prime})$
by the first condition of Lemma 2.26. Thus $F$ is effective and
$f$-exceptional. The same argument shows that $F^{\prime}$ is effective and
$f^{\prime}$-exceptional. Applying Remark 2.14 (2) or Lemma 2.24 to the
morphisms $(Y,\Gamma)\to(X,\Delta)$ and
$(Y,\Gamma)\to(X^{\prime},\Delta^{\prime})$, we see that Lemma 2.26 holds. ∎

Finally, we prove a key lemma in the proof of Theorem 1.1.

######  Lemma 2.27.

Let $\pi\colon X\to Z$ be a projective morphism of normal quasi-projective
varieties and $(X,\Delta)$ a $\mathbb{Q}$-factorial dlt pair such that
$K_{X}+\Delta$ is pseudo-effective over $Z$. Let $A$ be an ample
$\mathbb{R}$-divisor on $X$. Then there exists a real number $\epsilon>0$ such
that for any two real numbers $t,t^{\prime}\in(0,\epsilon]$, any sequence of
steps of a $(K_{X}+\Delta+tA)$-MMP
$(X,\Delta+tA)\dashrightarrow(Y,\Delta_{Y}+tA_{Y})$ over $Z$ to a good minimal
model $(Y,\Delta_{Y}+tA_{Y})$ and any sequence of steps of a
$(K_{X}+\Delta+t^{\prime}A)$-MMP
$(X,\Delta+t^{\prime}A)\dashrightarrow(Y^{\prime},\Delta_{Y^{\prime}}+t^{\prime}A_{Y^{\prime}})$
over $Z$ to a good minimal model
$(Y^{\prime},\Delta_{Y^{\prime}}+t^{\prime}A_{Y^{\prime}})$, the following
properties hold.

  * •

The birational map $Y\dashrightarrow Y^{\prime}$ is small,

  * •

$Y\dashrightarrow Y^{\prime}$ is an isomorphism on an open subset $U\subset Y$
such that $U$ intersects all lc centers of $(Y,\Delta_{Y})$ and the image
$U^{\prime}\subset Y^{\prime}$ intersects all lc centers of
$(Y^{\prime},\Delta_{Y^{\prime}})$, in particular, $Y\dashrightarrow
Y^{\prime}$ induces one-to-one correspondence between lc centers of
$(Y,\Delta_{Y})$ and lc centers of $(Y^{\prime},\Delta_{Y^{\prime}})$, and

  * •

for any lc center $S_{Y}$ of $(Y,\Delta_{Y})$ with the induced birational map
$S_{Y}\dashrightarrow S_{Y^{\prime}}$ to the corresponding lc center
$S_{Y^{\prime}}$ of $(Y^{\prime},\Delta_{Y^{\prime}})$, the following
equalities

$\begin{split}\kappa_{\iota}(S_{Y}/Z,(K_{Y}+\Delta_{Y})|_{S_{Y}})&=\kappa_{\iota}(S_{Y^{\prime}}/Z,(K_{Y^{\prime}}+\Delta_{Y^{\prime}})|_{S_{Y^{\prime}}})\quad{\rm
and}\\\
\kappa_{\sigma}(S_{Y}/Z,(K_{Y}+\Delta_{Y})|_{S_{Y}})&=\kappa_{\sigma}(S_{Y^{\prime}}/Z,(K_{Y^{\prime}}+\Delta_{Y^{\prime}})|_{S_{Y^{\prime}}})\end{split}$

hold true. In particular, $(Y,\Delta_{Y})$ is log abundant over $Z$ if and
only if $(Y^{\prime},\Delta_{Y^{\prime}})$ is log abundant over $Z$.

###### Proof.

By rescaling $A$, we may assume that $K_{X}+\Delta+A$ is nef over $Z$.
Furthermore, by replacing $A$ with a general one, we may also assume $A\geq 0$
and that $(X,\Delta+A)$ is a dlt pair whose lc centers of $(X,\Delta+A)$ are
lc centers of $(X,\Delta)$. Fix a sequence of steps of a $(K_{X}+\Delta)$-MMP
over $Z$ with scaling of $A$

$(X_{0}:=X,\Delta_{0}:=\Delta)\dashrightarrow(X_{1},\Delta_{1})\dashrightarrow\cdots\dashrightarrow(X_{i},\Delta_{i})\dashrightarrow\cdots.$

We set $\lambda_{i}={\rm inf}\set{\mu\in\mathbb{R}_{\geq
0}\\!}{\\!K_{X_{i}}+\Delta_{i}+\mu A_{i}\text{\rm\, is nef over }Z},$ where
$A_{i}$ is the birational transform of $A$ on $X_{i}$. Then ${\rm
lim}_{i\to\infty}\lambda_{i}=0$ by [4] (see also [3, Theorem 4.1 (ii)]).

######  Step 1.

In this step, we find an index $m$ such that the birational map
$X_{m}\dashrightarrow X_{i}$ satisfies the three properties of Lemma 2.27 for
any $i\geq m$. When the $(K_{X}+\Delta)$-MMP over $Z$ terminates with a log
minimal model $X_{m^{\prime}}$, there is nothing to prove because we may take
$m^{\prime}$ as $m$. When the $(K_{X}+\Delta)$-MMP over $Z$ does not
terminate, there is $i^{\prime}$ such that for any $i\geq i^{\prime}$, the
birational map $X_{i}\dashrightarrow X_{i+1}$ is small and for any lc center
$S_{i}$ of $(X_{i},\Delta_{i})$, the restriction of $X_{i}\dashrightarrow
X_{i+1}$ to $S_{i}$ induces a birational map $S_{i}\dashrightarrow S_{i+1}$ to
an lc center of $(X_{i+1},\Delta_{i+1})$. For any $i\geq i^{\prime}$ and any
lc center $S_{i}$ of $(X_{i},\Delta_{i})$ with the birational map
$S_{i}\dashrightarrow S_{i+1}$, there is a common resolution $Y\to X_{i}$ and
$Y\to X_{i+1}$ of $X_{i}\dashrightarrow X_{i+1}$ and a subvariety $T\subset Y$
birational to $S_{i}$ and $S_{i+1}$ such that the induced morphisms $T\to
S_{i}$ and $T\to S_{i+1}$ form a common resolution of $S_{i}\dashrightarrow
S_{i+1}$. Comparing coefficients of $(K_{X_{i}}+\Delta_{i})|_{T}$ and
$(K_{X_{i+1}}+\Delta_{i+1})|_{T}$, we have
$(K_{X_{i}}+\Delta_{i})|_{T}=(K_{X_{i+1}}+\Delta_{i+1})|_{T}+E$ for some
$E\geq 0$ on $T$. Then, for the sequence
$S_{i^{\prime}}\dashrightarrow\cdots\dashrightarrow
S_{i}\dashrightarrow\cdots$ of birational maps of lc centers, the two
sequences
$\\{\kappa_{\iota}(S_{i}/Z,(K_{X_{i}}+\Delta_{i})|_{S_{i}})\\}_{i\geq
i^{\prime}}$ and
$\\{\kappa_{\sigma}(S_{i}/Z,(K_{X_{i}}+\Delta_{i})|_{S_{i}})\\}_{i\geq
i^{\prime}}$ consisting of elements of $\mathbb{Z}_{\geq 0}\cup\\{-\infty\\}$
are non-increasing. From this, for the sequence
$S_{i^{\prime}}\dashrightarrow\cdots$ of birational maps of lc centers,
$\kappa_{\iota}(S_{i}/Z,(K_{X_{i}}+\Delta_{i})|_{S_{i}})$ and
$\kappa_{\sigma}(S_{i}/Z,(K_{X_{i}}+\Delta_{i})|_{S_{i}})$ do not depend on
$i$ for all $i\gg i^{\prime}$.

From these discussion, we can find an index $m$ such that for any $i\geq m$,
the birational map $X_{m}\dashrightarrow X_{i}$ satisfies the three properties
of Lemma 2.27, in other words, we have

  * •

the birational map $X_{m}\dashrightarrow X_{i}$ is small,

  * •

$X_{m}\dashrightarrow X_{i}$ is an isomorphism on an open set $U_{m}\subset
X_{m}$ such that $U_{m}$ intersects all lc centers of $(X_{m},\Delta_{m})$ and
the image $U_{i}\subset X_{i}$ intersects all lc centers of
$(X_{i},\Delta_{i})$, and

  * •

for any lc center $S_{m}$ of $(X_{m},\Delta_{m})$ with the induced birational
map $S_{m}\dashrightarrow S_{i}$ to the corresponding lc center $S_{i}$ of
$(X_{i},\Delta_{i})$, the following equalities hold.

$\begin{split}\kappa_{\iota}(S_{m}/Z,(K_{X_{m}}+\Delta_{m})|_{S_{m}})&=\kappa_{\iota}(S_{i}/Z,(K_{X_{i}}+\Delta_{i})|_{S_{i}})\quad{\rm
and}\\\
\kappa_{\sigma}(S_{m}/Z,(K_{X_{m}}+\Delta_{m})|_{S_{m}})&=\kappa_{\sigma}(S_{i}/Z,(K_{X_{i}}+\Delta_{i})|_{S_{i}}).\end{split}$

######  Step 2.

We will complete the proof with this step. We may assume $\lambda_{0}>0$
because otherwise there is nothing to prove. We may also assume $m>0$. We set
$\epsilon=\frac{\lambda_{m-1}}{2}$. We prove that this $\epsilon$ satisfies
the conditions of Lemma 2.27.

To complete the proof, it is sufficient to prove that for any
$t\in(0,\epsilon]$ and any sequence of steps of a $(K_{X}+\Delta+tA)$-MMP

$(X,\Delta+tA)\dashrightarrow(Y,\Delta_{Y}+tA_{Y})$

over $Z$ to a good minimal model $(Y,\Delta_{Y}+tA_{Y})$, the birational map
$Y\dashrightarrow X_{m}$ satisfies the three conditions of Lemma 2.27. Fix
$t\in(0,\epsilon]$ and $(X,\Delta+tA)\dashrightarrow(Y,\Delta_{Y}+tA_{Y})$ the
log MMP over $Z$ as above. Let $j$ be the minimum index such that
$K_{X_{j}}+\Delta_{j}+tA_{j}$ is nef over $Z$. Such $j$ exists since ${\rm
lim}_{i\to\infty}\lambda_{i}=0$, and we have $j\geq m$ since
$\lambda_{m-1}>\epsilon\geq t$. By construction, the birational map
$(X,\Delta+tA)\dashrightarrow(X_{j},\Delta_{j}+tA_{j})$ is a sequence of steps
of a $(K_{X}+\Delta+tA)$-MMP over $Z$ to a good minimal model
$(X_{j},\Delta_{j}+tA_{j})$.

Both $(X,\Delta+tA)\dashrightarrow(X_{j},\Delta_{j}+tA_{j})$ and
$(X,\Delta+tA)\dashrightarrow(Y,\Delta_{Y}+tA_{Y})$ are sequences of steps of
the $(K_{X}+\Delta+tA)$-MMP over $Z$ to good minimal models, so the induced
birational map $Y\dashrightarrow X_{j}$ is small and it is an isomorphism on
an open set $V\subset Y$ such that $V$ intersects all lc centers of
$(Y,\Delta_{Y}+tA_{Y})$ and the image $V_{j}\subset X_{j}$ intersects all lc
centers of $(X_{j},\Delta_{j}+tA_{j})$. Since $A$ is general, lc centers of
$(X_{j},\Delta_{j}+tA_{j})$ (resp. lc centers of $(Y,\Delta_{Y}+tA_{Y})$) are
the same as lc centers of $(X_{j},\Delta_{j})$ (resp. lc centers of
$(Y,\Delta_{Y})$). By this fact, applying the first and the second conditions
in Step 1 to $X_{m}\dashrightarrow X_{j}$ and $X_{m}\dashrightarrow X_{i}$ for
any $i\geq m$, we see that the birational map $Y\dashrightarrow X_{i}$ is
small and we can find an open set $V^{(i)}\subset Y$ such that
$Y\dashrightarrow X_{i}$ is an isomorphism on $V^{(i)}$, all lc centers of
$(Y,\Delta_{Y})$ intersects $V^{(i)}$ and its image $V_{i}\subset X_{i}$
intersects all lc centers of $(X_{i},\Delta_{i})$. In this way, the first and
the second condition of Lemma 2.27 hold for $Y\dashrightarrow X_{i}$ for all
$i\geq m$. In particular, those conditions hold for $Y\dashrightarrow X_{m}$.

Pick any lc center $S_{Y}$ of $(Y,\Delta_{Y})$. Then $Y\dashrightarrow X_{m}$
is small and there is an lc center $S_{m}$ on $(X_{m},\Delta_{m})$ such that
the restriction of $Y\dashrightarrow X_{m}$ induces a birational map
$S_{Y}\dashrightarrow S_{m}$. It also follows that the divisor
$K_{X_{m}}+\Delta_{m}+\lambda_{m-1}A_{m}$ is nef over $Z$. Thus, we can apply
Lemma 2.18 to $X_{m}\dashrightarrow Y$ over $Z$, $S_{m}\dashrightarrow S_{Y}$,
$K_{X_{m}}+\Delta_{m}+\lambda_{m-1}A_{m}$ and $K_{Y}+\Delta_{Y}+tA_{Y}$. Since
$\lambda_{m-1}>\epsilon\geq t>0$, we have

$\begin{split}\kappa_{\iota}(S_{m}/Z,(K_{X_{m}}+\Delta_{m})|_{S_{m}})&\geq\kappa_{\iota}(S_{Y}/Z,(K_{Y}+\Delta_{Y})|_{S_{Y}})\quad{\rm
and}\\\
\kappa_{\sigma}(S_{m}/Z,(K_{X_{m}}+\Delta_{m})|_{S_{m}})&\geq\kappa_{\sigma}(S_{Y}/Z,(K_{Y}+\Delta_{Y})|_{S_{Y}}).\end{split}$

Let $l$ be the minimum index such that $K_{X_{l}}+\Delta_{l}+\frac{t}{2}A_{l}$
is nef over $Z$. Then $l\geq m$ since $\lambda_{m-1}>\frac{t}{2}$, and the
birational map
$(X,\Delta+\frac{t}{2}A)\dashrightarrow(X_{l},\Delta_{l}+\frac{t}{2}A_{l})$ is
a sequence of steps of a $(K_{X}+\Delta+\frac{t}{2}A)$-MMP over $Z$ to a good
minimal model. By the discussion of the previous paragraph, $Y\dashrightarrow
X_{l}$ is small and there is an lc center $S_{l}$ on $(X_{l},\Delta_{l})$ such
that the restriction of $Y\dashrightarrow X_{l}$ induces a birational map
$S_{Y}\dashrightarrow S_{l}$. Applying Lemma 2.18 to $Y\dashrightarrow X_{l}$
over $Z$, $S_{Y}\dashrightarrow S_{l}$, $K_{Y}+\Delta_{Y}+tA_{Y}$ and
$K_{X_{l}}+\Delta_{l}+\frac{t}{2}A_{l}$, we obtain

$\begin{split}\kappa_{\iota}(S_{Y}/Z,(K_{Y}+\Delta_{Y})|_{S_{Y}})&\geq\kappa_{\iota}(S_{l}/Z,(K_{X_{l}}+\Delta_{l})|_{S_{l}})\quad{\rm
and}\\\
\kappa_{\sigma}(S_{Y}/Z,(K_{Y}+\Delta_{Y})|_{S_{Y}})&\geq\kappa_{\sigma}(S_{l}/Z,(K_{X_{l}}+\Delta_{l})|_{S_{l}}).\end{split}$

By construction, the birational map $X_{m}\dashrightarrow X_{l}$ induces a
birational map $S_{m}\dashrightarrow S_{l}$. By the above inequalities and the
third condition of Step 1 for $S_{m}\dashrightarrow S_{l}$, we obtain

$\begin{split}\kappa_{\iota}(S_{m}/Z,(K_{X_{m}}+\Delta_{m})|_{S_{m}})\geq&\kappa_{\iota}(S_{Y}/Z,(K_{Y}+\Delta_{Y})|_{S_{Y}})\\\
\geq&\kappa_{\iota}(S_{l}/Z,(K_{X_{l}}+\Delta_{l})|_{S_{l}})\\\
=&\kappa_{\iota}(S_{m}/Z,(K_{X_{m}}+\Delta_{m})|_{S_{m}})\end{split}$

for any lc center $S_{Y}$, and the same relation holds between
$\kappa_{\sigma}(S_{Y}/Z,(K_{Y}+\Delta_{Y})|_{S_{Y}})$ and
$\kappa_{\sigma}(S_{m}/Z,(K_{X_{m}}+\Delta_{m})|_{S_{m}})$. In this way, the
third condition of Lemma 2.27 holds for any lc center $S_{Y}$ with the
birational map $S_{Y}\dashrightarrow S_{m}$ induced by $Y\dashrightarrow
X_{m}$.

By the conclusion of Step 2, we see that for any $t\in(0,\epsilon]$ and any
sequence of steps of a $(K_{X}+\Delta+tA)$-MMP
$(X,\Delta+tA)\dashrightarrow(Y,\Delta_{Y}+tA_{Y})$ over $Z$ to a good minimal
model $(Y,\Delta_{Y}+tA_{Y})$, the three properties of Lemma 2.27 holds for
$Y\dashrightarrow X_{m}$. Therefore, it follows that $\epsilon>0$ satisfies
the condition of Lemma 2.27. ∎

##  3\. Log MMP containing log abundant log canonical pairs

In this section, we study log MMP with scaling containing log abundant lc
pairs.

###  3.1. Proof of main result

The goal of this subsection is to prove Theorem 3.5 and Theorem 3.6. We start
with recalling the special case of [26, Proposition 3.3], which was proved by
applying weak semistable reduction ([1]).

######  Lemma 3.1 (cf. [26, Proposition 3.3]).

Let $(X,\Delta)$ be a projective lc pair. Suppose that there is a contraction
$X\to V$ to a normal projective variety $V$ such that

  * •

$\kappa_{\sigma}(X/V,K_{X}+\Delta)\\!=0$,

  * •

$\kappa_{\sigma}(X,K_{X}+\Delta)={\rm dim}\,V$, and

  * •

all lc centers of $(X,\Delta)$ dominate $V$.

Then, $(X,\Delta)$ has a good minimal model.

We prove three results with Lemma 3.1 and arguments in [28, Proof of Theorem
1.1].

######  Proposition 3.2.

Let $(X,\Delta)$ be a projective lc pair such that $K_{X}+\Delta$ is pseudo-
effective and abundant. Then there is a dlt blow-up
$\widetilde{f}\colon(\widetilde{X},\widetilde{\Delta})\to(X,\Delta)$ and
effective $\mathbb{R}$-divisors $\widetilde{G}$ and $\widetilde{H}$ on
$\widetilde{X}$ satisfying the following properties.

  1. (I)

$K_{\widetilde{X}}+\widetilde{\Delta}\sim_{\mathbb{R}}\widetilde{G}+\widetilde{H}$,

  2. (II)

${\rm Supp}\widetilde{G}\subset{\rm
Supp}\llcorner\widetilde{\Delta}\lrcorner$, and

  3. (III)

there exists a real number $t_{0}>0$ such that for any $t\in(0,t_{0}]$,
followings hold:

    1. (III-a)

The pair $(\widetilde{X},\widetilde{\Delta}+t\widetilde{H})$ is
$\mathbb{Q}$-factorial dlt and the support of
$N_{\sigma}(K_{\widetilde{X}}+\widetilde{\Delta}+t\widetilde{H})$ does not
depend on $t$, and

    2. (III-b)

the pair $(\widetilde{X},\widetilde{\Delta}-t\widetilde{G})$ has a good
minimal model.

###### Proof.

We apply arguments in [28, Steps 2–3 in the proof of Theorem 1.1]. Since
$K_{X}+\Delta$ is pseudo-effective and abundant, we can find an effective
$\mathbb{R}$-divisor $D$ on $X$ such that $K_{X}+\Delta\sim_{\mathbb{R}}D$. We
take the Iitaka fibration $X\dashrightarrow V$ associated to $D$. Then we have
${\rm dim}V=\kappa_{\sigma}(X,K_{X}+\Delta)$. We take a log resolution
$\bar{f}\colon\bar{X}\to X$ of $(X,\Delta)$ such that the induced map
$\bar{X}\dashrightarrow V$ is a morphism. Let $(\bar{X},\bar{\Delta})$ be a
log smooth model of $(X,\Delta)$ as in [25, Definition 2.9], which coincides
with “log smooth model of type (2)” defined in [3, Definition 2.3]. Then
$(\bar{X},\bar{\Delta})$ is a log smooth lc pair such that

  1. (i)

$K_{\bar{X}}+\bar{\Delta}=\bar{f}^{*}(K_{X}+\Delta)+\bar{E}$ for an
$\bar{f}$-exceptional $\mathbb{R}$-divisor $\bar{E}\geq 0$ whose support
contains all $\bar{f}$-exceptional prime divisors $\bar{P}$ satisfying
$a(\bar{P},X,\Delta)>-1$, and

  2. (ii)

$\kappa_{\sigma}(\bar{X},K_{\bar{X}}+\bar{\Delta})={\rm dim}V$ and
$\kappa_{\sigma}(\bar{X}/V,K_{\bar{X}}+\bar{\Delta})=0$ (Lemma 2.17).

By construction of $X\dashrightarrow V$ and $(\bar{X},\bar{\Delta})$, the
divisor $K_{\bar{X}}+\bar{\Delta}$ is $\mathbb{R}$-linearly equivalent to the
sum of an effective $\mathbb{R}$-divisor and the pullback of an ample divisor
on $V$. Therefore, we can find an effective $\mathbb{R}$-divisor
$\bar{D}\sim_{\mathbb{R}}K_{\bar{X}}+\bar{\Delta}$ such that ${\rm
Supp}\bar{D}$ contains all lc centers of $(\bar{X},\bar{\Delta})$ which are
vertical over $V$. By applying [25, Lemma 2.10] to the morphism
$(\bar{X},\bar{\Delta})\to V$, we may assume that

  1. (iii)

$\bar{\Delta}=\bar{\Delta}^{\prime}+\bar{\Delta}^{\prime\prime}$ such that
$\bar{\Delta}^{\prime}$ is effective, $\bar{\Delta}^{\prime\prime}=0$ or
$\bar{\Delta}^{\prime\prime}$ is a reduced divisor which is vertical over $V$,
and all lc centers of $(\bar{X},\bar{\Delta}-\bar{\Delta}^{\prime\prime})$
dominate $V$.

By taking a log resolution of $(\bar{X},\bar{\Delta}+\bar{D})$ and replacing
$(\bar{X},\bar{\Delta})$ with a log smooth model in [25, Definition 2.9] (and
replacing $\bar{D}$ accordingly), we may assume that
$(\bar{X},\bar{\Delta}+\bar{D})$ is log smooth. Note that the condition (iii)
is preserved after this replacement. We have ${\rm Supp}\bar{D}\supset{\rm
Supp}\bar{\Delta}^{\prime\prime}$ since ${\rm Supp}\bar{D}$ contains all lc
centers of $(\bar{X},\bar{\Delta})$ which are vertical over $V$. By
decomposing $\bar{D}$ appropriately, we obtain effective $\mathbb{R}$-divisors
$\bar{G}$ and $\bar{H}$ such that

  1. (iv)

$K_{\bar{X}}+\bar{\Delta}\sim_{\mathbb{R}}\bar{G}+\bar{H}$,

  2. (v)

${\rm Supp}\bar{\Delta}^{\prime\prime}\subset{\rm Supp}\bar{G}\subset{\rm
Supp}\llcorner\bar{\Delta}\lrcorner$, and

  3. (vi)

no component of $\bar{H}$ is a component of $\llcorner\bar{\Delta}\lrcorner$
and $(\bar{X},{\rm Supp}(\bar{\Delta}+\bar{H}))$ is log smooth.

We fix a real number $t_{0}\in(0,1)$ such that $\bar{\Delta}-t_{0}\bar{G}\geq
0$. For any $t\in(0,t_{0}]$, we consider the pair
$(\bar{X},\bar{\Delta}-t\bar{G})$. By (v), there is a real number
$t^{\prime}>0$ such that $t^{\prime}\bar{\Delta}^{\prime\prime}\leq t\bar{G}$.
Then all lc centers of $(\bar{X},\bar{\Delta}-t\bar{G})$ are lc centers of
$(\bar{X},\bar{\Delta}-t^{\prime}\bar{\Delta}^{\prime\prime})$. Since
$(\bar{X},\bar{\Delta})$ is lc, all lc centers of
$(\bar{X},\bar{\Delta}-t^{\prime}\bar{\Delta}^{\prime\prime})$ are lc centers
of $(\bar{X},\bar{\Delta}-\bar{\Delta}^{\prime\prime})$. By these facts and
(iii), we see that all lc centers of $(\bar{X},\bar{\Delta}-t\bar{G})$
dominate $V$. Since we have

$K_{\bar{X}}+\bar{\Delta}\sim_{\mathbb{R}}\bar{G}+\bar{H}\quad\text{and}\quad
K_{\bar{X}}+\bar{\Delta}-t\bar{G}\sim_{\mathbb{R}}(1-t)\bar{G}+\bar{H},$

by Remark 2.14 (1) we have
$\kappa_{\sigma}(\bar{X},K_{\bar{X}}+\bar{\Delta}-t\bar{G})=\kappa_{\sigma}(\bar{X},K_{\bar{X}}+\bar{\Delta}).$
By (ii), we obtain

$\kappa_{\sigma}(\bar{X},K_{\bar{X}}+\bar{\Delta}-t\bar{G})={\rm dim}V.$

Similarly, we obtain
$\kappa_{\sigma}(\bar{X}/V,K_{\bar{X}}+\bar{\Delta}-t\bar{G})=0$. Thus, the
morphism $(\bar{X},\bar{\Delta}-t\bar{G})\to V$ satisfies all conditions of
Lemma 3.1. By Lemma 3.1, the pair $(\bar{X},\bar{\Delta}-t\bar{G})$ has a good
minimal model for any $t\in(0,t_{0}]$.

We run a $(K_{\bar{X}}+\bar{\Delta})$-MMP over $X$. By definition of
$\bar{\Delta}$ (and (i)), we get a dlt blow-up
$\widetilde{f}\colon(\widetilde{X},\widetilde{\Delta})\to(X,\Delta)$. Let
$\widetilde{G}$ (resp. $\widetilde{H}$) be the birational transform of
$\bar{G}$ (resp. $\bar{H}$) on $\widetilde{X}$. We show that the dlt blow-up
$\widetilde{f}\colon(\widetilde{X},\widetilde{\Delta})\to(X,\Delta)$ and the
$\mathbb{R}$-divisors $\widetilde{G}$ and $\widetilde{H}$ satisfy all
conditions of Proposition 3.2. Condition (iv) shows relation
$K_{\widetilde{X}}+\widetilde{\Delta}\sim_{\mathbb{R}}\widetilde{G}+\widetilde{H}$,
so (I) of Proposition 3.2 holds. Condition (v) shows (II) of Proposition 3.2.
By (vi), replacing $t_{0}$ if necessary, we may assume that the pair
$(\bar{X},\bar{\Delta}+t_{0}\bar{H})$ is dlt. Since
$\bar{X}\dashrightarrow\widetilde{X}$ is a sequence of steps of a
$(K_{\bar{X}}+\bar{\Delta})$-MMP, replacing $t_{0}$ again, we may assume the
following.

  * •

$\bar{X}\dashrightarrow\widetilde{X}$ is a sequence of steps of a
$(K_{\bar{X}}+\bar{\Delta}+t\bar{H})$-MMP and a sequence of steps of a
$(K_{\bar{X}}+\bar{\Delta}-t\bar{G})$-MMP for any $t\in(0,t_{0}]$.

By this statement, $(\widetilde{X},\widetilde{\Delta}+t_{0}\widetilde{H})$ is
$\mathbb{Q}$-factorial dlt which shows the first half of (III-a) of
Proposition 3.2, and $(\widetilde{X},\widetilde{\Delta}-t\widetilde{G})$ has a
good minimal model for all $t\in(0,t_{0}]$ because
$(\bar{X},\bar{\Delta}-t\bar{G})$ has a good minimal model. In particular,
(III-b) of Proposition 3.2 holds. Applying Lemma 2.4 to
$K_{\widetilde{X}}+\widetilde{\Delta}$ and $\widetilde{H}$ and replacing
$t_{0}$, we may assume that $t_{0}$ satisfies the latter half of (III-a) of
Proposition 3.2. Thus $t_{0}$ satisfies (III) of Proposition 3.2.

Thus, $\widetilde{f}\colon(\widetilde{X},\widetilde{\Delta})\to(X,\Delta)$,
$\widetilde{G}$ and $\widetilde{H}$ satisfy all conditions of Proposition 3.2.
∎

######  Proposition 3.3.

Let $(X,\Delta)$ be a projective $\mathbb{Q}$-factorial dlt pair such that
there are two effective $\mathbb{R}$-divisors $G$ and $H$ on $X$ satisfying
(I), (II), (III), (III-a) and (III-b) in Proposition 3.2. Then there is a real
number $\lambda_{0}>0$ and a sequence of birational maps

$X\dashrightarrow X_{1}\dashrightarrow
X_{2}\dashrightarrow\cdots\dashrightarrow X_{i}\dashrightarrow\cdots$

such that putting $\Delta_{i}$ and $H_{i}$ as the birational transforms of
$\Delta$ and $H$ on $X_{i}$ respectively, the following properties hold.

  1. (1)

The pair $(X,\Delta+\lambda_{0}H)$ is $\mathbb{Q}$-factorial dlt and all lc
centers of $(X,\Delta+\lambda_{0}H)$ are lc centers of $(X,\Delta)$,

  2. (2)

the birational map $X\dashrightarrow X_{1}$ is a sequence of steps of a
$(K_{X}+\Delta+\lambda_{0}H)$-MMP to a good minimal model
$(X_{1},\Delta_{1}+\lambda_{0}H_{1})$,

  3. (3)

the sequence of birational maps $X_{1}\dashrightarrow\cdots\dashrightarrow
X_{i}\dashrightarrow\cdots$ is a sequence of steps of a
$(K_{X_{1}}+\Delta_{1})$-MMP with scaling of $\lambda_{0}H_{1}$ such that if
we set

$\lambda_{i}:={\rm inf}\set{\mu\in\mathbb{R}_{\geq
0}\\!}{\\!K_{X_{i}}+\Delta_{i}+\mu H_{i}\text{\rm\;is nef}}$

for each $i\geq 1$, then we have ${\rm lim}_{i\to\infty}\lambda_{i}=0$,

  4. (4)

for all $i\geq 1$ and all positive real numbers
$u\in[\lambda_{i},\lambda_{i-1}]$, the pair $(X_{i},\Delta_{i}+uH_{i})$ is a
good minimal model of $(X,\Delta+uH)$ and $(X_{1},\Delta_{1}+uH_{1})$, and

  5. (5)

the $(K_{X_{1}}+\Delta_{1})$-MMP with scaling of $\lambda_{0}H_{1}$ which is
proved in (3) occurs only in ${\rm Supp}\llcorner\Delta_{1}\lrcorner$.

###### Proof.

Throughout this proof, unless otherwise stated, (I), (II), (III), (III-a) and
(III-b) mean the properties (I), (II), (III), (III-a) and (III-b) in
Proposition 3.2, respectively. We follow [28, Step 4 in the proof of Theorem
1.1].

By (I), for any $s\in\mathbb{R}_{\geq 0}$ we have

($\spadesuit$)
$K_{X}+\Delta+sH\sim_{\mathbb{R}}(1+s)\Bigl{(}K_{X}+\Delta-\frac{s}{1+s}G\Bigr{)}.$

Fix $\lambda_{0}\in(0,t_{0})$, where $t_{0}$ is as in (III). By (III-a), we
can easily check that $(X,\Delta+\lambda_{0}H)$ is a $\mathbb{Q}$-factorial
dlt pair and lc centers of $(X,\Delta+\lambda_{0}H)$ are lc centers of
$(X,\Delta)$, hence (1) of Proposition 3.3 holds.

By (III-b) and [3, Theorem 4.1], we can run a
$(K_{X}+\Delta-\frac{\lambda_{0}}{1+\lambda_{0}}G)$-MMP that terminates with a
good minimal model
$(X_{1},\Delta_{1}-\frac{\lambda_{0}}{1+\lambda_{0}}G_{1})$, where
$\Delta_{1}$ (resp. $G_{1}$) is the birational transform of $\Delta$ (resp.
$G$) on $X_{1}$. Note that $\frac{\lambda_{0}}{1+\lambda_{0}}<t_{0}$,
therefore we may use (III-b). By ($\spadesuit$ ‣ 3.1), the birational map

$(X,\Delta+\lambda_{0}H)\dashrightarrow(X_{1},\Delta_{1}+\lambda_{0}H_{1})$

is a sequence of steps of a $(K_{X}+\Delta+\lambda_{0}H)$-MMP and
$(X_{1},\Delta_{1}+\lambda_{0}H_{1})$ is a good minimal model of
$(X,\Delta+\lambda_{0}H)$, where $H_{1}$ is the birational transform of $H$ on
$X_{1}$. It is obvious that the birational map $X\dashrightarrow X_{1}$
satisfies (2) of Proposition 3.3.

By the same argument as above, by (III-b), [3, Theorem 4.1] and ($\spadesuit$
‣ 3.1), for all $t\in(0,\lambda_{0})$ we get a sequence of steps of a
$(K_{X}+\Delta+tH)$-MMP

($\clubsuit$)
$(X,\Delta+tH)\dashrightarrow(\tilde{X}_{t},\tilde{\Delta}_{t}+t\tilde{H}_{t})$

that terminates with a good minimal model
$(\tilde{X}_{t},\tilde{\Delta}_{t}+t\tilde{H}_{t})$. The latter half of
(III-a) and Remark 2.9 (2) show that $X\dashrightarrow X_{1}$ and
$X\dashrightarrow\tilde{X}_{t}$ contract the same divisors for all
$t\in(0,\lambda_{0})$, therefore the induced birational map
$X_{1}\dashrightarrow\tilde{X}_{t}$ is small. This implies that
$(\tilde{X}_{t},\tilde{\Delta}_{t}+t\tilde{H}_{t})$ is a good minimal model of
$(X_{1},\Delta_{1}+tH_{1})$ for all $t\in(0,\lambda_{0})$. Therefore, we see
that $(X_{1},\Delta_{1}+tH_{1})$ has a good minimal model for all
$t\in(0,\lambda_{0})$ and $K_{X_{1}}+\Delta_{1}$ is the limit of movable
divisors. With [26, Lemma 2.14], we can construct a sequence of steps of a
$(K_{X_{1}}+\Delta_{1})$-MMP with scaling of $\lambda_{0}H_{1}$

$(X_{1},\Delta_{1})\dashrightarrow(X_{2},\Delta_{2})\dashrightarrow\cdots\dashrightarrow(X_{i},\Delta_{i})\dashrightarrow\cdots$

such that if we define $\lambda_{i}$ by $\lambda_{i}={\rm
inf}\\!\set{\\!\mu\in\mathbb{R}_{\geq 0}}{\text{$K_{X_{i}}+\Delta_{i}+\mu
H_{i}$ is nef}\\!},$ where $H_{i}$ is the birational transform of $H_{1}$ on
$X_{i}$, then the $(K_{X_{1}}+\Delta_{1})$-MMP terminates after finitely many
steps or we have ${\rm lim}_{i\to\infty}\lambda_{i}=0$ even if it does not
terminate. Since $K_{X_{1}}+\Delta_{1}$ is pseudo-effective, in both cases we
have ${\rm lim}_{i\to\infty}\lambda_{i}=0$. Therefore, the
$(K_{X_{1}}+\Delta_{1})$-MMP satisfies (3) of Proposition 3.3.

We pick any $i\geq 1$ and any positive real number
$u\in[\lambda_{i},\lambda_{i-1}]$. By properties of log MMP with scaling, it
is easy to see that $(X_{i},\Delta_{i}+uH_{i})$ is lc and
$K_{X_{i}}+\Delta_{i}+uH_{i}$ is nef. Since $K_{X_{1}}+\Delta_{1}$ is the
limit of movable divisors, the $(K_{X_{1}}+\Delta_{1})$-MMP contains only
flips. Since the birational map $X_{1}\dashrightarrow\tilde{X}_{u}$ is small,
where $\tilde{X}_{u}$ is as in ($\clubsuit$ ‣ 3.1), the induced birational map
$X_{i}\dashrightarrow\tilde{X}_{u}$ is small. By ($\clubsuit$ ‣ 3.1) the pair
$(\tilde{X}_{u},\tilde{\Delta}_{u}+u\tilde{H}_{u})$ is a good minimal model of
$(X,\Delta+uH)$, from which and Remark 2.8 we see that
$(X_{i},\Delta_{i}+uH_{i})$ is a good minimal model of $(X,\Delta+uH)$. Then
$K_{X_{i}}+\Delta_{i}+uH_{i}$ is semi-ample. Since $X_{1}\dashrightarrow
X_{i}$ is small, the pair $(X_{i},\Delta_{i}+uH_{i})$ is a good minimal model
of $(X_{1},\Delta_{1}+uH_{1})$. In this way, we see that (4) of Proposition
3.3 holds.

By properties of log MMP with scaling, any curve which is contracted in the
$i$-th step of the $(K_{X_{1}}+\Delta_{1})$-MMP with scaling of
$\lambda_{0}H_{1}$ has a negative (resp. positive) intersection number with
$K_{X_{i}}+\Delta_{i}$ (resp. $H_{i}$). On the other hand, the relation
$G_{i}\sim_{\mathbb{R}}-H_{i}+K_{X_{i}}+\Delta_{i}$ holds by (I), where
$G_{i}$ is the birational transform of $G$ on $X_{i}$. Thus, curves contracted
in the $i$-th step of the $(K_{X_{1}}+\Delta_{1})$-MMP with scaling of
$\lambda_{0}H_{1}$ have negative intersection numbers with $G_{i}$. Since
$G_{i}\geq 0$, the $(K_{X_{1}}+\Delta_{1})$-MMP occurs only in ${\rm
Supp}G_{1}$. Now the inclusion ${\rm Supp}G_{1}\subset{\rm
Supp}\llcorner\Delta_{1}\lrcorner$ holds by (II), hence (5) of Proposition 3.3
holds. ∎

######  Theorem 3.4.

Let $(X,\Delta)$ be a projective dlt pair. Suppose that

  * •

$K_{X}+\Delta$ is pseudo-effective and abundant,

  * •

for any lc center $S$ of $(X,\Delta)$, the restriction $(K_{X}+\Delta)|_{S}$
is nef, and

  * •

$\sigma_{P}(K_{X}+\Delta)=0$ for all prime divisors $P$ over $X$ such that
$a(P,X,\Delta)<0$ and $c_{X}(P)$ intersects an lc center of $(X,\Delta)$,
where $\sigma_{P}(\,\cdot\,)$ is the asymptotic vanishing order of $P$ defined
in Definition 2.1.

Then $(X,\Delta)$ has a log minimal model.

###### Proof.

We prove the theorem in several steps.

######  Step 1.

First, we reduce Theorem 3.4 to special termination of a log MMP with scaling.
Idea is very similar to [28, Steps 2–6 in the proof of Theorem 1.1].

Using Proposition 3.2, we construct a dlt blow-up
$\widetilde{f}\colon(\widetilde{X},\widetilde{\Delta})\to(X,\Delta)$ and
effective $\mathbb{R}$-divisors $\widetilde{G}$ and $\widetilde{H}$ on
$\widetilde{X}$ satisfying conditions of Proposition 3.2. Then we can replace
$(X,\Delta)$ with $(\widetilde{X},\widetilde{\Delta})$. Indeed, after
replacing $(X,\Delta)$ by $(\widetilde{X},\widetilde{\Delta})$, the second
condition of Theorem 3.4 is clearly preserved and the first (resp. third)
condition of Theorem 3.4 is preserved by Remark 2.14 (2) (resp. definition of
$\sigma_{P}(\,\cdot\,)$). From this, replacing $(X,\Delta)$ by
$(\widetilde{X},\widetilde{\Delta})$, we may assume that there are effective
$\mathbb{R}$-divisors $G$ and $H$ on $X$ satisfying (I), (II), (III), (III-a)
and (III-b) in Proposition 3.2. With Proposition 3.3, there is a positive real
number $\lambda_{0}$ and a sequence of birational maps

$X\dashrightarrow X_{1}\dashrightarrow
X_{2}\dashrightarrow\cdots\dashrightarrow X_{i}\dashrightarrow\cdots$

such that

  1. (1)

the pair $(X,\Delta+\lambda_{0}H)$ is $\mathbb{Q}$-factorial dlt and all lc
centers of $(X,\Delta+\lambda_{0}H)$ are lc centers of $(X,\Delta)$,

  2. (2)

the birational map $X\dashrightarrow X_{1}$ is a sequence of steps of a
$(K_{X}+\Delta+\lambda_{0}H)$-MMP to a good minimal model
$(X_{1},\Delta_{1}+\lambda_{0}H_{1})$,

  3. (3)

the sequence of birational maps $X_{1}\dashrightarrow\cdots\dashrightarrow
X_{i}\dashrightarrow\cdots$ is a sequence of steps of a
$(K_{X_{1}}+\Delta_{1})$-MMP with scaling of $\lambda_{0}H_{1}$ such that if
we set

$\lambda_{i}:={\rm inf}\set{\mu\in\mathbb{R}_{\geq
0}\\!}{\\!K_{X_{i}}+\Delta_{i}+\mu H_{i}\text{\rm\;is nef}}$

for each $i\geq 1$, then we have ${\rm lim}_{i\to\infty}\lambda_{i}=0$,

  4. (4)

for all $i\geq 1$ and all positive real numbers
$u\in[\lambda_{i},\lambda_{i-1}]$, the pair $(X_{i},\Delta_{i}+uH_{i})$ is a
good minimal model of $(X,\Delta+uH)$ and $(X_{1},\Delta_{1}+uH_{1})$, and

  5. (5)

the $(K_{X_{1}}+\Delta_{1})$-MMP with scaling of $\lambda_{0}H_{1}$ in (3)
occurs only in ${\rm Supp}\llcorner\Delta_{1}\lrcorner$.

Suppose that the $(K_{X_{1}}+\Delta_{1})$-MMP with scaling of
$\lambda_{0}H_{1}$ terminates. Then $\lambda_{l-1}>\lambda_{l}=0$ for some
$l>0$. For all real numbers $u\in(0,\lambda_{l-1}]$, it follows by (4) that
$K_{X_{l}}+\Delta_{l}+uH_{l}$ is nef and $a(P^{\prime},X,\Delta+uH)\leq
a(P^{\prime},X_{l},\Delta_{l}+uH_{l})$ for all prime divisors $P^{\prime}$ on
$X$ which are exceptional over $X_{l}$. By taking the limit $u\to 0$, it
follows that $K_{X_{l}}+\Delta_{l}$ is nef and $a(P^{\prime},X,\Delta)\leq
a(P^{\prime},X_{l},\Delta_{l})$. Therefore $(X_{l},\Delta_{l})$ is a weak lc
model of $(X,\Delta)$. From this, the pair $(X,\Delta)$ has a log minimal
model ([3, Corollary 3.7] or [26, Remark 2.10 (i)]). In this way, to prove the
existence of a log minimal model of $(X,\Delta)$, it is sufficient to prove
the termination of the $(K_{X_{1}}+\Delta_{1})$-MMP. By (5), it is sufficient
to prove that the non-isomorphic locus of the $(K_{X_{1}}+\Delta_{1})$-MMP is
disjoint from the birational transform of $\llcorner\Delta_{1}\lrcorner$ after
finitely many steps. In the rest of the proof, we will carry out the arguments
of the special termination as in [10].

There exists $m>0$ such that for any lc center $S_{m}$ of $(X_{m},\Delta_{m})$
and any $i\geq m$, the birational map $X_{m}\dashrightarrow X_{i}$ induces a
birational map $S_{m}\dashrightarrow S_{i}$ to an lc center $S_{i}$ of
$(X_{i},\Delta_{i})$. For any $i\geq m$ and any lc center $S_{i}$ of
$(X_{i},\Delta_{i})$, we define $\mathbb{R}$-divisors $\Delta_{S_{i}}$ and
$H_{S_{i}}$ on $S_{i}$ by
$K_{S_{i}}+\Delta_{S_{i}}=(K_{X_{i}}+\Delta_{i})|_{S_{i}}$ and
$H_{S_{i}}=H_{i}|_{S_{i}}$, respectively. Similarly, on every lc center $S$ of
$(X,\Delta)$, we define $\mathbb{R}$-divisors $\Delta_{S}$ and $H_{S}$ by
$K_{S}+\Delta_{S}=(K_{X}+\Delta)|_{S}$ and $H_{S}=H|_{S}$, respectively. Then
$H_{S}$ and $H_{S_{i}}$ are effective $\mathbb{R}$-Cartier divisors, and
$(S_{i},\Delta_{S_{i}})$ is dlt for all $i\geq m$ and all $S_{i}$.

From now on, we prove that for any integer $d\geq 0$, there is $m_{d}\geq m$
such that after $m_{d}$ steps non-isomorphic locus of the
$(K_{X_{1}}+\Delta_{1})$-MMP is disjoint from all $d$-dimensional lc centers
of $(X_{m_{d}},\Delta_{m_{d}})$. Indeed, the case of $d={\rm dim}X-1$ of this
statement and (5) imply the termination of the $(K_{X_{1}}+\Delta_{1})$-MMP.
Therefore, it is sufficient to prove the statement, and we prove it by
induction on $d$. In the case of $d=0$, we can take $m$ as $m_{0}$. We assume
the existence of $m_{d-1}$ in the statement. By replacing $m$ by $m_{d-1}$, we
may assume $m_{d-1}=m$. By arguments as in [10], replacing $m$ again, we may
assume that for any $d$-dimensional lc center $S_{m}$ of $(X_{m},\Delta_{m})$
the induced birational map $S_{m}\dashrightarrow S_{i}$ is small and the
birational transform of $\Delta_{S_{m}}$ on $S_{i}$ is equal to
$\Delta_{S_{i}}$. Then $H_{S_{i}}$ is equal to the birational transform of
$H_{S_{m}}$ on $S_{i}$. Since ${\rm lim}_{i\to\infty}\lambda_{i}=0$, as in the
second paragraph of Remark 2.21, to prove the existence of $m_{d}$ it is
sufficient to prove that $(S_{m},\Delta_{S_{m}})$ has a log minimal model.

######  Step 2.

In the rest of the proof, all $i$ are assumed to be $i\geq m$ unless otherwise
stated.

We pick any $d$-dimensional lc center $S_{m}$ of $(X_{m},\Delta_{m})$. By Step
1, the birational map $S_{m}\dashrightarrow S_{i}$ is small and
$\Delta_{S_{i}}$ is equal to the birational transform of $\Delta_{S_{m}}$ on
$S_{i}$ for all $i$. From now on, we prove that $(S_{m},\Delta_{S_{m}})$ has a
log minimal model. Idea is very similar to [28, Steps 7–9 in the proof of
Theorem 1.1]. By construction of the birational map $X\dashrightarrow X_{i}$,
we can find an lc center $S$ of $(X,\Delta)$ such that $X\dashrightarrow
X_{i}$ induces a birational map $S\dashrightarrow S_{i}$ ((1), (2) and (3) in
Step 1). Using the lc center $S$, in the rest of this step we establish the
following variety, divisor and inequalities.

  1. (a)

A projective birational morphism $\psi\colon T\to S_{m}$ from a projective
$\mathbb{Q}$-factorial variety $T$ such that all prime divisors $\bar{D}$ on
$S$ satisfying $a(\bar{D},S_{m},\Delta_{S_{m}})<a(\bar{D},S,\Delta_{S})$ are
prime divisors on $T$,

  2. (b)

an effective $\mathbb{R}$-divisor $\Psi$ on $T$ defined by
$\Psi=-\sum_{\begin{subarray}{c}D\end{subarray}}a(D,S,\Delta_{S})D$, where $D$
runs over all prime divisors on $T$,

  3. (c)

the inequality $a(Q,S,\Delta_{S}+\lambda_{i}H_{S})\leq
a(Q,S_{i},\Delta_{S_{i}}+\lambda_{i}H_{S_{i}})$ for all $i$ and prime divisors
$Q$ over $S$, and

  4. (d)

the inequality $a(Q^{\prime},S_{m},\Delta_{S_{m}})\leq a(Q^{\prime},T,\Psi)$
for all prime divisors $Q^{\prime}$ over $S_{m}$, in particular, the pair
$(T,\Psi)$ is lc.

By taking a suitable resolution of the graph of $X\dashrightarrow X_{i}$, we
can construct a common resolution $\overline{X}\to X$ and
$\overline{X}\to{X_{i}}$ and a subvariety $\overline{S}\subset\overline{X}$
birational to $S$ and $S_{i}$ such that the induced birational morphisms
$\overline{S}\to S$ and $\overline{S}\to S_{i}$ form a common resolution of
the birational map $S\dashrightarrow S_{i}$. By (4) in Step 1, comparing
coefficients of divisors $(K_{X}+\Delta+\lambda_{i}H)|_{\overline{S}}$ and
$(K_{X_{i}}+\Delta_{i}+\lambda_{i}H_{i})|_{\overline{S}}$, the inequality

$a(Q,S,\Delta_{S}+\lambda_{i}H_{S})\leq
a(Q,S_{i},\Delta_{S_{i}}+\lambda_{i}H_{S_{i}})$

follows for all prime divisors $Q$ over $S$. We have proved (c).

Let $\widetilde{D}$ be a prime divisor on $S_{m}$. Then $\widetilde{D}$ is a
prime divisor on $S_{i}$ for any $i$ because the birational maps
$S_{m}\dashrightarrow S_{i}$ are small. For all $i$, we can easily check that
the birational map $(X,\Delta)\dashrightarrow(X_{i},\Delta_{i})$ and dlt pairs
$(S,\Delta_{S})$ and $(S_{i},\Delta_{S_{i}})$ satisfy the two conditions of
Lemma 2.6. Indeed, the birational map $X\dashrightarrow X_{i}$ is a birational
contraction and $\Delta_{i}$ is the birational transform of $\Delta$ on
$X_{i}$, so the first condition of Lemma 2.6 holds. Since $S$ is an lc center
of $(X,\Delta)$, the second condition of Lemma 2.6 follows from the third
condition of Theorem 3.4. In this way, we can apply Lemma 2.6 to
$(X,\Delta)\dashrightarrow(X_{i},\Delta_{i})$, $(S,\Delta_{S})$ and
$(S_{i},\Delta_{S_{i}})$. Thus, we have
$a(\widetilde{D},S_{i},\Delta_{S_{i}})\leq a(\widetilde{D},S,\Delta_{S})$ for
all $i$. Furthermore, the inequality
$a(\widetilde{D},S_{i},\Delta_{S_{i}}+\lambda_{i}H_{S_{i}})\leq
a(\widetilde{D},S_{i},\Delta_{S_{i}})$ also holds by a basic property of
discrepancies. Combining these inequalities with (c), we have

$a(\widetilde{D},S,\Delta_{S}+\lambda_{i}H_{S})\leq
a(\widetilde{D},S_{i},\Delta_{S_{i}}+\lambda_{i}H_{S_{i}})\leq
a(\widetilde{D},S,\Delta_{S}).$

for all $i$. Since $\Delta_{S_{i}}+\lambda_{i}H_{S_{i}}$ is equal to the
birational transform of $\Delta_{S_{m}}+\lambda_{i}H_{S_{m}}$ on $S_{i}$, we
see that
$a(\widetilde{D},S_{i},\Delta_{S_{i}}+\lambda_{i}H_{S_{i}})=a(\widetilde{D},S_{m},\Delta_{S_{m}}+\lambda_{i}H_{S_{m}})$
for all $i$. Since ${\rm lim}_{i\to\infty}\lambda_{i}=0$ by (3) in Step 1, by
taking the limit $i\to\infty$, the equality

($\star$)
$a(\widetilde{D},S_{m},\Delta_{S_{m}})=a(\widetilde{D},S,\Delta_{S})$

holds for all prime divisors $\widetilde{D}$ on $S_{m}$.

We set

$\mathcal{C}=\Set{\bar{D}}{\begin{array}[]{l}\\!\\!\text{$\bar{D}$ is a prime
divisor on $S$ such that
$a(\bar{D},S_{m},\Delta_{S_{m}})<a(\bar{D},S,\Delta_{S})$}\end{array}\\!\\!}.$

By ($\star$ ‣ 2), all elements of $\mathcal{C}$ are exceptional over $S_{m}$.
By a basic property of discrepancies we have
$a(\bar{D},S_{m},\Delta_{S_{m}}+\lambda_{m}H_{S_{m}})\leq
a(\bar{D},S_{m},\Delta_{S_{m}})$. Combining this with (c), we obtain

$a(\bar{D},S,\Delta_{S}+\lambda_{m}H_{S})\leq
a(\bar{D},S_{m},\Delta_{S_{m}})<a(\bar{D},S,\Delta_{S})\leq 0$

for all $\bar{D}\in\mathcal{C}$. Since every element of $\mathcal{C}$ is a
prime divisor on $S$, we see that all elements of $\mathcal{C}$ are components
of $H_{S}$. Thus $\mathcal{C}$ is a finite set, and furthermore, any
$\bar{D}\in\mathcal{C}$ satisfies
$-1<a(\bar{D},S,\Delta_{S}+\lambda_{m}H_{S})$. From them, the relation
$-1<a(\bar{D},S_{m},\Delta_{S_{m}})<0$ holds for all $\bar{D}\in\mathcal{C}$.
Since $(S_{m},\Delta_{S_{m}})$ is dlt, by [28, Lemma 2.16], we get a
projective birational morphism $\psi\colon T\to S_{m}$ such that $T$ is
$\mathbb{Q}$-factorial and $\psi^{-1}$ exactly extracts elements of
$\mathcal{C}$. We have constructed the desired variety as in (a).

Let $D$ be a prime divisor on $T$. When $D$ is $\psi$-exceptional, by
definitions of $\mathcal{C}$ and $\psi$ we obtain
$a(D,S_{m},\Delta_{S_{m}})<a(D,S,\Delta_{S})\leq 0$. When $D$ is not
$\psi$-exceptional, by ($\star$ ‣ 2) we obtain
$a(D,S,\Delta_{S})=a(D,S_{m},\Delta_{S_{m}})\leq 0$. In any case, the relation

($\star$$\star$) $a(D,S_{m},\Delta_{S_{m}})\leq a(D,S,\Delta_{S})\leq 0.$

holds for all prime divisors $D$ on $T$. Note that only finitely many prime
divisors $D$ on $T$ satisfy $a(D,S,\Delta_{S})<0$ because
$a(D,S,\Delta_{S})=0$ when $D$ is a prime divisor on $S$ and not a component
of $\Delta_{S}$. Therefore, we may define an $\mathbb{R}$-divisor $\Psi\geq 0$
on $T$ by

$\Psi=-\sum_{\begin{subarray}{c}D\end{subarray}}a(D,S,\Delta_{S})D,$

where $D$ runs over all prime divisors on $T$. This is the divisor stated in
(b).

Finally we prove (d). Since $T$ is $\mathbb{Q}$-factorial, $K_{T}+\Psi$ is
$\mathbb{R}$-Cartier. By ($\star$$\star$ ‣ 2), we obtain
$K_{T}+\Psi\leq\psi^{*}(K_{S_{m}}+\Delta_{S_{m}})$. From this, we have

$-1\leq a(Q^{\prime},S_{m},\Delta_{S_{m}})\leq a(Q^{\prime},T,\Psi).$

for any prime divisor $Q^{\prime}$ over $S_{m}$. This shows (d).

######  Step 3.

In this step, we prove that $(T,\Psi)$ has a log minimal model.

By the second condition of Theorem 3.4, $(S,\Delta_{S})$ is a log minimal
model of $(S,\Delta_{S})$ itself. To apply Lemma 2.26 to the birational map
$(S,\Delta_{S})\dashrightarrow(T,\Psi)$, we will prove the following claim.

######  Claim.

Let $\tilde{Q}$ be a prime divisor over $S$. Then the following two statements
hold:

  * •

If $\tilde{Q}$ is a prime divisor on $S$, then $a(\tilde{Q},S,\Delta_{S})\leq
a(\tilde{Q},T,\Psi)$, and

  * •

if $\tilde{Q}$ is a prime divisor on $T$, then $a(\tilde{Q},T,\Psi)\leq
a(\tilde{Q},S,\Delta_{S})$.

###### Proof of Claim.

Let $\tilde{Q}$ be a prime divisor over $S$. When $\tilde{Q}$ is a prime
divisor on $T$ or a prime divisor on $S$ which appears as a prime divisor on
$T$, by (b) in Step 2 we have

$a(\tilde{Q},T,\Psi)=-{\rm
coeff}_{\tilde{Q}}(\Psi)=a(\tilde{Q},S,\Delta_{S}).$

Therefore the second assertion of the claim holds, and, to prove the first
assertion of the claim we may assume that $\tilde{Q}$ is a prime divisor on
$S$ but not a prime divisor on $T$. In this case, (a) in Step 2 shows
$a(\tilde{Q},S,\Delta_{S})\leq a(\tilde{Q},S_{m},\Delta_{m})$, then (d) in
Step 2 shows

$a(\tilde{Q},S,\Delta_{S})\leq a(\tilde{Q},S_{m},\Delta_{m})\leq
a(\tilde{Q},T,\Psi).$

Therefore the first assertion of the claim also holds. In this way, the claim
holds. ∎

By the second condition of Theorem 3.4 and Lemma 2.26, we see that $(T,\Psi)$
has a log minimal model.

######  Step 4.

With this step we complete the proof of the existence of a log minimal model
of $(X,\Delta)$. In other words, we complete the argument of special
termination in Step 1 by showing that $(S_{m},\Delta_{S_{m}})$ has a log
minimal model.

For each $i$, the pair $(S_{i},\Delta_{S_{i}}+\lambda_{i}H_{S_{i}})$ is a weak
lc model of $(S_{m},\Delta_{S_{m}}+\lambda_{i}H_{S_{m}})$ because the
birational map $S_{m}\dashrightarrow S_{i}$ is small, the divisor
$\Delta_{S_{i}}+\lambda_{i}H_{S_{i}}$ is equal to the birational transform of
$\Delta_{S_{m}}+\lambda_{i}H_{S_{m}}$ on $S_{i}$, and the divisor
$K_{S_{i}}+\Delta_{S_{i}}+\lambda_{i}H_{S_{i}}=(K_{X_{i}}+\Delta_{i}+\lambda_{i}H_{i})|_{S_{i}}$
is nef. Pick any prime divisor $D$ on $T$. By Remark 2.9 (1) and (c) in Step
2, we obtain

$\begin{split}\sigma_{D}(K_{S_{m}}+\Delta_{S_{m}}+\lambda_{i}H_{S_{m}})&=a(D,S_{i},\Delta_{S_{i}}+\lambda_{i}H_{S_{i}})-a(D,S_{m},\Delta_{S_{m}}+\lambda_{i}H_{S_{m}})\\\
&\geq
a(D,S,\Delta_{S}+\lambda_{i}H_{S})-a(D,S_{m},\Delta_{S_{m}}+\lambda_{i}H_{S_{m}}).\end{split}$

By (b) in Step 2, it follows that $a(D,S,\Delta_{S})=a(D,T,\Psi)$. By Remark
2.3 (2), applying ${\rm lim}_{i\to\infty}\lambda_{i}=0$ and taking the limit
$i\to\infty$, we obtain

$\begin{split}\sigma_{D}(K_{S_{m}}+\Delta_{S_{m}})&\geq\underset{i\to\infty}{\rm
lim}\bigl{(}a(D,S,\Delta_{S}+\lambda_{i}H_{S})-a(D,S_{m},\Delta_{S_{m}}+\lambda_{i}H_{S_{m}})\bigr{)}\\\
&=a(D,T,\Psi)-a(D,S_{m},\Delta_{S_{m}})\end{split}$

for all prime divisors $D$ on $T$. Furthermore, we have
$a(D,T,\Psi)-a(D,S_{m},\Delta_{S_{m}})\geq 0$ by (d) in Step 2. These facts
imply the relation between difference of two discrepancies and asymptotic
vanishing order in Lemma 2.25. Because $(T,\Psi)$ has a log minimal model, we
can apply Lemma 2.25 to $(S_{m},\Delta_{S_{m}})$, $(T,\Psi)$ and $T\to S_{m}$.
In this way, we see that $(S_{m},\Delta_{S_{m}})$ has a log minimal model.

By arguments in the last paragraph of Step 1 (see also Remark 2.21), we
complete the argument of special termination, therefore $(X,\Delta)$ has a log
minimal model.

We complete the proof. ∎

We are ready to prove main results (Theorem 3.5 and Theorem 3.6).

######  Theorem 3.5.

Let $\pi\colon X\to Z$ be a projective morphism of normal quasi-projective
varieties and $(X,\Delta)$ an lc pair. Let $A\geq 0$ be an
$\mathbb{R}$-divisor on $X$ such that $(X,\Delta+A)$ is lc and
$K_{X}+\Delta+A$ is nef over $Z$. Then, no infinite sequence of a
$(K_{X}+\Delta)$-MMP over $Z$ with scaling of $A$

$(X_{0}:=X,\Delta_{0}:=\Delta)\dashrightarrow(X_{1},\Delta_{1})\dashrightarrow\cdots\dashrightarrow(X_{i},\Delta_{i})\dashrightarrow\cdots$

satisfies the following properties.

  * •

If we define $\lambda_{i}={\rm inf}\set{\mu\in\mathbb{R}_{\geq
0}\\!}{\\!K_{X_{i}}+\Delta_{i}+\mu A_{i}\text{\rm\, is nef over }Z}$, where
$A_{i}$ is the birational transform of $A$ on $X_{i}$, then ${\rm
lim}_{i\to\infty}\lambda_{i}=0$, and

  * •

there are infinitely many $i$ such that $(X_{i},\Delta_{i})$ is log abundant
over $Z$.

###### Proof.

We prove the theorem by induction on dimension of $X$. Let $(X,\Delta)$ and
$A$ be as in Theorem 3.5, and assume Theorem 3.5 for all lc pairs of dimension
less than ${\rm dim}X$. Suppose by contradiction that there is an infinite
sequence of steps of a $(K_{X}+\Delta)$-MMP over $Z$ with scaling of $A$

$(X_{0}:=X,\Delta_{0}:=\Delta)\dashrightarrow(X_{1},\Delta_{1})\dashrightarrow\cdots\dashrightarrow(X_{i},\Delta_{i})\dashrightarrow\cdots$

satisfying the two properties of Theorem 3.5. Since the two properties of
Theorem 3.5 are preserved after replacing each $(X_{i},\Delta_{i})$ with a
$\mathbb{Q}$-factorial dlt model, by lifting this log MMP ([3, Remark 2.9]),
we may assume that $(X_{i},\Delta_{i})$ are $\mathbb{Q}$-factorial dlt. By
using the negativity lemma, we see that $K_{X}+\Delta+\lambda_{i}A$ is pseudo-
effective over $Z$ for all $i$. Therefore, the divisor $K_{X}+\Delta$ is
pseudo-effective over $Z$.

In the rest of the proof, we will prove that $(X,\Delta)$ has a log minimal
model over $Z$, then we will get a contradiction.

######  Step 1.

In this step, we prove that non-isomorphic locus of the $(K_{X}+\Delta)$-MMP
does not intersect the birational transform of $\llcorner\Delta\lrcorner$
after finitely many steps.

For any $i$ and any lc center $S_{i}$ of $(X_{i},\Delta_{i})$, we define a dlt
pair $(S_{i},\Delta_{S_{i}})$ by adjunction
$K_{S_{i}}+\Delta_{S_{i}}=(K_{X_{i}}+\Delta_{i})|_{S_{i}}$. We prove by
induction that for any integer $0\leq d<{\rm dimX}$, there is $m_{d}\geq 0$
such that the non-isomorphic locus of the $(K_{X}+\Delta)$-MMP is disjoint
from all $d$-dimensional lc centers of $(X_{m_{d}},\Delta_{m_{d}})$ after
$m_{d}$ steps. We assume the existence of $m_{d-1}$, and we put $m=m_{d-1}$.
By arguments as in [10], replacing $m$ we may assume that for any $i\geq m$
and any $d$-dimensional lc center $S_{m}$ on $(X_{m},\Delta_{m})$, the
birational map $X_{i}\dashrightarrow X_{i+1}$ induces a small birational map
$S_{i}\dashrightarrow S_{i+1}$ such that the birational transform of
$\Delta_{S_{i}}$ on $S_{i+1}$ is equal to $\Delta_{S_{i+1}}$. Let
$(T_{m},\Psi_{m})\to(S_{m},\Delta_{S_{m}})$ be a dlt blow-up, and put
$A_{T_{m}}=A_{m}|_{T_{m}}$. By ${\rm lim}_{i\to\infty}\lambda_{i}=0$ and
arguments in Remark 2.21, the sequence of birational maps
$S_{m}\dashrightarrow\cdots\dashrightarrow S_{i}\dashrightarrow\cdots$ induces
a $(K_{{T}_{m}}+\Psi_{m})$-MMP over $Z$ with scaling of $A_{{T}_{m}}$

$({T}_{m},\Psi_{m})\dashrightarrow\cdots\dashrightarrow({T}_{i},\Psi_{i})\dashrightarrow\cdots$

such that $({T}_{i},\Psi_{i})$ is a $\mathbb{Q}$-factorial dlt model of
$(S_{i},\Delta_{S_{i}})$ for each $i\geq m$ and termination of the
$(K_{{T}_{m}}+\Psi_{m})$-MMP over $Z$ implies the existence of $m_{d}$ stated
above. Since there exist infinitely many log abundant dlt pairs
$(X_{i},\Delta_{i})$ over $Z$, we see that $(T_{i},\Psi_{i})$ is log abundant
over $Z$ for infinitely many $i$. By construction, the
$(K_{{T}_{m}}+\Psi_{m})$-MMP over $Z$ with scaling of $A_{{T}_{m}}$ satisfies
two conditions about log MMP in Theorem 3.5. By the induction hypothesis of
Theorem 3.5, we see that the $(K_{{T}_{m}}+\Psi_{m})$-MMP terminates.
Therefore, we may find $m_{d}$ as in the above statement, and the non-
isomorphic locus of the $(K_{X}+\Delta)$-MMP does not intersect the birational
transform of $\llcorner\Delta\lrcorner$ after finitely many steps.

In this way, by replacing $(X,\Delta)$ with $(X_{i},\Delta_{i})$ for some
$i\gg 0$, we may assume that the non-isomorphic locus of the
$(K_{X}+\Delta)$-MMP does not intersect ${\rm Supp}\llcorner\Delta\lrcorner$.
By the second condition about log MMP in Theorem 3.5, we may also assume that
$(X,\Delta)$ is log abundant over $Z$.

######  Step 2.

In this step, using Theorem 3.4 we prove that $(X,\Delta)$ has a log minimal
model over $Z$ in the case when $Z$ is a point.

The first condition of Theorem 3.4 is clear. By the hypothesis of the
$(K_{X}+\Delta)$-MMP and Lemma 2.22, for any ample $\mathbb{R}$-divisor
$A^{\prime}$ on $X$, it follows that the stable base locus of
$K_{X}+\Delta+A^{\prime}$ is disjoint from ${\rm
Supp}\llcorner\Delta\lrcorner$. Hence, the restriction of $K_{X}+\Delta$ to
any component of ${\rm Supp}\llcorner\Delta\lrcorner$ is nef and
$\sigma_{P}(K_{X}+\Delta)=0$ for all prime divisors $P$ over $X$ such that
$c_{X}(P)$ intersects ${\rm Supp}\llcorner\Delta\lrcorner$. By these facts,
the second and the third conditions of Theorem 3.4 hold. By Theorem 3.4,
$(X,\Delta)$ has a log minimal model.

######  Step 3.

We prove the existence of a log minimal model of $(X,\Delta)$ over $Z$ in
general case. Taking the Stein factorization of $\pi$, we may assume $\pi$ is
a contraction.

We take an open immersion $Z\hookrightarrow\bar{Z}$ to a normal projective
variety. By Lemma 2.11, we can get a $\mathbb{Q}$-factorial dlt closure
$(\bar{X},\bar{\Delta})$ of $(X,\Delta)$, that is, a projective
$\mathbb{Q}$-factorial dlt pair $(\bar{X},\bar{\Delta})$ with a projective
morphism $\bar{\pi}\colon\bar{X}\to\bar{Z}$ such that $X$ is an open subset of
$\bar{X}$, $(\bar{X}|_{X},\bar{\Delta}|_{X})=(X,\Delta)$ and all lc centers of
$(\bar{X},\bar{\Delta})$ intersects $X$. By construction, we have
${\bar{\pi}}^{-1}(Z)=X$. Since all lc centers of $(\bar{X},\bar{\Delta})$
intersects $X$ and property of being relatively abundant for
$\mathbb{R}$-Cartier divisors is determined on sufficiently general fibers of
$\bar{X}\to\bar{Z}$ ([28, Lemma 2.10]), the pair $(\bar{X},\bar{\Delta})$ is
log abundant over $\bar{Z}$. If $(\bar{X},\bar{\Delta})$ has a log minimal
model over $\bar{Z}$ then its restriction over $Z$ is a log minimal model of
$(X,\Delta)$ over $Z$. In this way, to prove the existence of a log minimal
model of $(X,\Delta)$ over $Z$, it is sufficient to prove that
$(\bar{X},\bar{\Delta})$ has a log minimal model over $\bar{Z}$.

Pick $H_{\bar{Z}}$ an ample Cartier divisor on $\bar{Z}$. Let $\bar{H}$ be a
general member of the $\mathbb{R}$-linear system $|(3\cdot{\rm
dim}X)\bar{\pi}^{*}H_{\bar{Z}}|_{\mathbb{R}}$. We may assume that
$(\bar{X},\bar{\Delta}+\bar{H})$ is dlt and its lc centers are lc centers of
$(\bar{X},\bar{\Delta})$. We run a $(K_{\bar{X}}+\bar{\Delta}+\bar{H})$-MMP
with scaling of an ample divisor

$(\bar{X}_{0}:=\bar{X},\bar{\Delta}_{0}+\bar{H}_{0}:=\bar{\Delta}+\bar{H})\dashrightarrow\cdots\dashrightarrow(\bar{X}_{i},\bar{\Delta}_{i}+\bar{H}_{i})\dashrightarrow\cdots.$

The $(K_{\bar{X}}+\bar{\Delta}+\bar{H})$-MMP satisfies the first condition
about log MMP in Theorem 3.5. We prove that
$(\bar{X}_{i},\bar{\Delta}_{i}+\bar{H}_{i})$ is log abundant over $\bar{Z}$
for all $i$. By definition of $\bar{H}$ and the length of extremal rays, the
$(K_{\bar{X}}+\bar{\Delta}+\bar{H})$-MMP is a $(K_{\bar{X}}+\bar{\Delta})$-MMP
over $\bar{Z}$. Because the non-isomorphic locus of the $(K_{X}+\Delta)$-MMP
over $Z$ is disjoint from ${\rm Supp}\llcorner\Delta\lrcorner$ and ${\rm
lim}_{i\to\infty}\lambda_{i}=0$, we may apply Lemma 2.22. For any ample
$\mathbb{R}$-divisor $H^{\prime}$ on $X$ and any closed point $x\in{\rm
Supp}\llcorner\Delta\lrcorner$, we can find $E\geq 0$ such that
$E\sim_{\mathbb{R},Z}K_{X}+\Delta+H^{\prime}$ and ${\rm Supp}E\not\ni x$.
Since $X=\bar{\pi}^{-1}(Z)$, for any curve which is contracted by $\bar{\pi}$
and intersects $\bar{\pi}^{-1}(Z)\cap{\rm
Supp}\llcorner\bar{\Delta}\lrcorner$, its intersection number with
$K_{\bar{X}}+\bar{\Delta}$ is non-negative. By this and induction on $i$, the
non-isomorphic locus of the $(K_{\bar{X}}+\bar{\Delta})$-MMP over $\bar{Z}$ is
disjoint from $\bar{\pi}^{-1}(Z)\cap{\rm Supp}\llcorner\bar{\Delta}\lrcorner$.
Then, for any lc center $\bar{S}$ of $(\bar{X},\bar{\Delta}+\bar{H})$, the
$(K_{\bar{X}}+\bar{\Delta})$-MMP over $\bar{Z}$ does not modify sufficiently
general fibers of the morphism $\bar{S}\to\bar{\pi}(\bar{S})$. For any lc
center $\bar{S}_{i}$ of $(\bar{X}_{i},\bar{\Delta}_{i})$, we define a dlt pair
$(\bar{S}_{i},\Delta_{\bar{S}_{i}})$ by adjunction
$K_{\bar{S}_{i}}+\Delta_{\bar{S}_{i}}=(K_{\bar{X}_{i}}+\bar{\Delta}_{i})|_{\bar{S}_{i}}$.
Since $(\bar{X},\bar{\Delta})$ is log abundant over $\bar{Z}$, applying [28,
Lemma 2.10] we see that $K_{\bar{S}_{i}}+\Delta_{\bar{S}_{i}}$ is abundant
over $\bar{Z}$. By definition of $\bar{H}$ and Lemma 2.19, the divisor
$(K_{\bar{X}_{i}}+\bar{\Delta}_{i}+\bar{H}_{i})|_{\bar{S}_{i}}$ is abundant.
Because all lc centers of $(\bar{X}_{i},\bar{\Delta}_{i}+\bar{H}_{i})$ are lc
centers of $(\bar{X}_{i},\bar{\Delta}_{i})$, it follows that
$(\bar{X}_{i},\bar{\Delta}_{i}+\bar{H}_{i})$ is log abundant over $\bar{Z}$
for all $i$.

The $(K_{\bar{X}}+\bar{\Delta}+\bar{H})$-MMP with scaling satisfies the two
conditions of Theorem 3.5 about log MMP. By the conclusion of Step 2 in this
proof, $(\bar{X},\bar{\Delta}+\bar{H})$ has a log minimal model. By definition
of $\bar{H}$, the pair $(X,\Delta)$ has a log minimal model over $Z$.

Since ${\rm lim}_{i\to\infty}\lambda_{i}=0$ and $(X,\Delta)$ has a log minimal
model over $Z$, the $(K_{X}+\Delta)$-MMP with scaling of $A$ must terminate
([3, Theorem 4.1 (iii)]). This contradicts construction of the
$(K_{X}+\Delta)$-MMP. In this way, Theorem 3.5 holds true. ∎

######  Theorem 3.6.

Let $\pi\colon X\to Z$ be a projective morphism of normal quasi-projective
varieties, and let $(X,\Delta)$ be a $\mathbb{Q}$-factorial dlt pair such that
$K_{X}+\Delta$ is pseudo-effective over $Z$. Let $A$ be an ample
$\mathbb{R}$-divisor on $X$ such that $(X,\Delta+A)$ is lc and
$K_{X}+\Delta+A$ is nef over $Z$. Then there is an integer $l\geq 0$
satisfying the following: Let

$(X_{0}:=X,\Delta_{0}:=\Delta)\dashrightarrow(X_{1},\Delta_{1})\dashrightarrow\cdots\dashrightarrow(X_{i},\Delta_{i})\dashrightarrow\cdots$

be a sequence of steps of a $(K_{X}+\Delta)$-MMP over $Z$ with scaling of $A$.
If $(X_{i},\Delta_{i})$ is log abundant over $Z$ for some $i\geq l$, then the
$(K_{X}+\Delta)$-MMP over $Z$ terminates with a good minimal model.

###### Proof.

By Lemma 2.27, there exists $\epsilon>0$ such that for any two real numbers
$t,t^{\prime}\in(0,\epsilon]$, any sequence of steps of a
$(K_{X}+\Delta+tA)$-MMP $(X,\Delta+tA)\dashrightarrow(Y,\Delta_{Y}+tA_{Y})$
over $Z$ to a good minimal model $(Y,\Delta_{Y}+tA_{Y})$ and any sequence of
steps of a $(K_{X}+\Delta+t^{\prime}A)$-MMP
$(X,\Delta+t^{\prime}A)\dashrightarrow(Y^{\prime},\Delta_{Y^{\prime}}+t^{\prime}A_{Y^{\prime}})$
over $Z$ to a good minimal model
$(Y^{\prime},\Delta_{Y^{\prime}}+t^{\prime}A_{Y^{\prime}})$, the three
properties of Lemma 2.27 hold true. By applying the finiteness of models as in
[4, Theorem E] to a suitable convex rational polytope of ${\rm
WDiv}_{\mathbb{R}}(X)$ which contains the set
$\set{\Delta+tA}{\tfrac{\epsilon}{2}\leq t\leq 1}$, we can find finitely many
birational maps $X\dashrightarrow Y_{k}$ such that if
$t\in[\frac{\epsilon}{2},1]$ then birational contraction from $X$ to any weak
lc model of $(X,\Delta+tA)$ over $Z$ is isomorphic to one of $X\dashrightarrow
Y_{k}$.

Let $l$ be the number of those varieties $Y_{k}$. We prove that this $l$ is
the desired integer. Consider any sequence of steps of a $(K_{X}+\Delta)$-MMP
over $Z$ with scaling of $A$

$(X_{0}:=X,\Delta_{0}:=\Delta)\dashrightarrow(X_{1},\Delta_{1})\dashrightarrow\cdots\dashrightarrow(X_{i},\Delta_{i})\dashrightarrow\cdots,$

and suppose that $(X_{i_{0}},\Delta_{i_{0}})$ is log abundant over $Z$ for
some $i_{0}\geq l$. We define

$\lambda_{i}={\rm inf}\set{\mu\in\mathbb{R}_{\geq
0}\\!}{\\!K_{X_{i}}+\Delta_{i}+\mu A_{i}\text{\rm\, is nef over }Z},$

where $A_{i}$ is the birational transform of $A$ on $X_{i}$. Since
$(X_{i},\Delta_{i}+\lambda_{i}A_{i})$ is a weak lc model of
$(X,\Delta+\lambda_{i}A)$ over $Z$ for all $0\leq i<l$, by the choice of $l$,
we see that $\lambda_{i_{0}}<\frac{\epsilon}{2}$. If $\lambda_{i_{0}}=0$, then
$(X_{i_{0}},\Delta_{i_{0}})$ is a log minimal model of $(X,\Delta)$ over $Z$
and it is log abundant over $Z$, so $(X_{i_{0}},\Delta_{i_{0}})$ is a good
minimal model by Lemma 2.20. We assume $\lambda_{i_{0}}>0$. Pick indices
$l^{\prime}$ and $l^{\prime\prime}$ such that
$\lambda_{l^{\prime}-1}>\lambda_{l^{\prime}}=\cdots=\lambda_{i_{0}}=\cdots=\lambda_{l^{\prime\prime}-1}>\lambda_{l^{\prime\prime}}$
(put $l^{\prime}=0$ when $\lambda_{0}=\lambda_{i_{0}}$). Then we can find
$t^{\prime},t^{\prime\prime}\in(0,\epsilon]$ such that
$t^{\prime\prime}<\lambda_{i_{0}}<t^{\prime}$, the birational map
$(X,\Delta+t^{\prime}A)\dashrightarrow(X_{l^{\prime}},\Delta_{l^{\prime}}+t^{\prime}A_{l^{\prime}})$
is a sequence of steps of a $(K_{X}+\Delta+t^{\prime}A)$-MMP over $Z$ to a
good minimal model, and
$(X,\Delta+t^{\prime\prime}A)\dashrightarrow(X_{l^{\prime\prime}},\Delta_{l^{\prime\prime}}+t^{\prime\prime}A_{l^{\prime\prime}})$
is a sequence of steps of a $(K_{X}+\Delta+t^{\prime\prime}A)$-MMP over $Z$ to
a good minimal model. Then $X_{l^{\prime}}\dashrightarrow
X_{l^{\prime\prime}}$ satisfies the three properties of Lemma 2.27. By
construction of log MMP, we see that the birational map
$X_{l^{\prime}}\dashrightarrow X_{i_{0}}$ is small, and this is an isomorphism
on an open set $U^{\prime}\subset X_{l^{\prime}}$ such that $U^{\prime}$
intersects all lc centers of $(X_{l^{\prime}},\Delta_{l^{\prime}})$ and the
image $U\subset X_{i_{0}}$ intersects all lc centers of
$(X_{i_{0}},\Delta_{i_{0}})$. The same property holds for
$X_{i_{0}}\dashrightarrow X_{l^{\prime\prime}}$. Furthermore, by the same
arguments as in Step 2 of the proof of Lemma 2.27, by applying Lemma 2.18 we
obtain the following condition.

  * •

For any lc center $S_{i_{0}}$ of $(X_{i_{0}},\Delta_{i_{0}})$ with the
birational map $S_{l^{\prime}}\dashrightarrow S_{i_{0}}\dashrightarrow
S_{l^{\prime\prime}}$ induced by $X_{l^{\prime}}\dashrightarrow
X_{i_{0}}\dashrightarrow X_{l^{\prime\prime}}$, where $S_{l^{\prime}}$ and
$S_{l^{\prime\prime}}$ are corresponding lc centers, we have

$\begin{split}\kappa_{\iota}(S_{l^{\prime}}/Z,(K_{X_{l^{\prime}}}+\Delta_{l^{\prime}})|_{S_{l^{\prime}}})&\geq\kappa_{\iota}(S_{i_{0}}/Z,(K_{X_{i_{0}}}+\Delta_{i_{0}})|_{S_{i_{0}}})\\\
&\geq\kappa_{\iota}(S_{l^{\prime\prime}}/Z,(K_{X_{l^{\prime\prime}}}+\Delta_{l^{\prime\prime}})|_{S_{l^{\prime\prime}}}),\quad{\rm
and}\\\
\kappa_{\sigma}(S_{l^{\prime}}/Z,(K_{X_{l^{\prime}}}+\Delta_{l^{\prime}})|_{S_{l^{\prime}}})&\geq\kappa_{\sigma}(S_{i_{0}}/Z,(K_{X_{i_{0}}}+\Delta_{i_{0}})|_{S_{i_{0}}})\\\
&\geq\kappa_{\sigma}(S_{l^{\prime\prime}}/Z,(K_{X_{l^{\prime\prime}}}+\Delta_{l^{\prime\prime}})|_{S_{l^{\prime\prime}}}).\end{split}$

Now we apply the third condition of Lemma 2.27 to
$X_{l^{\prime}}\dashrightarrow X_{l^{\prime\prime}}$ and
$S_{l^{\prime}}\dashrightarrow S_{l^{\prime\prime}}$. Using the above
relations, for any lc center $S_{i_{0}}$ with the birational map
$S_{l^{\prime}}\dashrightarrow S_{i_{0}}$, we have

$\begin{split}\kappa_{\iota}(S_{l^{\prime}}/Z,(K_{X_{l^{\prime}}}+\Delta_{l^{\prime}})|_{S_{l^{\prime}}})&=\kappa_{\iota}(S_{i_{0}}/Z,(K_{X_{i_{0}}}+\Delta_{i_{0}})|_{S_{i_{0}}})\quad{\rm
and}\\\
\kappa_{\sigma}(S_{l^{\prime}}/Z,(K_{X_{l^{\prime}}}+\Delta_{l^{\prime}})|_{S_{l^{\prime}}})&=\kappa_{\sigma}(S_{i_{0}}/Z,(K_{X_{i_{0}}}+\Delta_{i_{0}})|_{S_{i_{0}}}).\end{split}$

By recalling that $(X_{i_{0}},\Delta_{i_{0}})$ is log abundant over $Z$, we
see that $(X_{l^{\prime}},\Delta_{l^{\prime}})$ is log abundant over $Z$. For
any $j\geq l^{\prime\prime}$ such that $\lambda_{j-1}>\lambda_{j}$, by taking
$u\in(\lambda_{j},\lambda_{j-1})$ we get a sequence of steps of a
$(K_{X}+\Delta+uA)$-MMP
$(X,\Delta+uA)\dashrightarrow(X_{j},\Delta_{j}+uA_{j})$ over $Z$ to a good
minimal model. Since $t^{\prime},u\in(0,\epsilon]$ and
$(X_{l^{\prime}},\Delta_{l^{\prime}})$ is log abundant over $Z$, by applying
Lemma 2.27 to
$(X,\Delta+t^{\prime}A)\dashrightarrow(X_{l^{\prime}}+\Delta_{l^{\prime}}+t^{\prime}A_{l^{\prime}})$
and $(X,\Delta+uA)\dashrightarrow(X_{j},\Delta_{j}+uA_{j})$, we see that
$(X_{j},\Delta_{j})$ is log abundant over $Z$. Therefore, $(X_{j},\Delta_{j})$
is log abundant over $Z$ for any $j\geq l^{\prime\prime}$ such that
$\lambda_{j-1}>\lambda_{j}$.

By Theorem 3.5, the $(K_{X}+\Delta)$-MMP over $Z$ with scaling of $A$ must
terminate with a log minimal model, and the final dlt pair
$(X_{m},\Delta_{m})$ is log abundant over $Z$ because
$\lambda_{m-1}>\lambda_{m}$ and $m\geq l^{\prime\prime}$ which follows from
$\lambda_{l^{\prime\prime}-1}=\lambda_{i_{0}}>0$. Therefore, by Lemma 2.20,
the pair $(X_{m},\Delta_{m})$ is a good minimal model of $(X,\Delta)$ over
$Z$. So we are done. ∎

######  Remark 3.7.

We check that the main results of [3], [22] and [26], [28, Theorem 1.1] and
[27, Theorem 1.3] follow from Theorem 3.6. Let $\pi\colon X\to Z$ be a
projective morphism of normal quasi-projective varieties and $(X,\Delta)$ an
lc pair. Without loss of generality, we may assume that $(X,\Delta)$ is
$\mathbb{Q}$-factorial dlt and $K_{X}+\Delta$ is pseudo-effective over $Z$.

If $\pi$ and $(X,\Delta)$ are as in [3, Theorem 1.1] (see also [22, Theorem
1.6]), then the relative numerical dimension of the restriction of
$K_{X}+\Delta$ to any stratum is either $0$ or $-\infty$, therefore
$K_{X}+\Delta$ is log abundant over $Z$ ([18]). Because any
$(K_{X}+\Delta)$-MMP over $Z$ preserves the condition of [3, Theorem 1.1], any
$(K_{X}+\Delta)$-MMP over $Z$ preserves property of being log abundant over
$Z$.

If $\pi$ and $(X,\Delta)$ are as in [22, Theorem 1.1] (or [3, Theorem 1.4]),
with notations as in [22, Theorem 1.1], by running a $(K_{X}+\Delta)$-MMP over
$Z$ with scaling of an ample divisor
$(X,\Delta)\dashrightarrow\cdots\dashrightarrow(X_{i},\Delta_{i})\dashrightarrow\cdots$
and replacing $(X,\Delta)$ by $(X_{i},\Delta_{i})$ for some $i\gg 0$, we may
assume that $(K_{X}+\Delta)|_{f^{-1}(U)}$ is semi-ample over $U$. Then
$(X,\Delta)$ is log abundant over $Z$ ([28, Lemma 2.10]). Since any
$(K_{X}+\Delta)$-MMP over $Z$ does not modify $f^{-1}(U)$, it preserves
property of being log abundant over $Z$.

The above arguments are valid for the case of the main results of [26] because
they are $\mathbb{R}$-boundary divisor analogs of [3, Theorem 1.1] and [22,
Theorem 1.1].

If $\pi$ and $(X,\Delta)$ are as in [28, Theorem 1.1], for any finite steps
$(X,\Delta)\dashrightarrow(X_{i},\Delta_{i})$ of a $(K_{X}+\Delta)$-MMP over
$Z$ and any lc center $S_{i}$ of $(X_{i},\Delta_{i})$, there is an lc center
$S$ of $(X,\Delta)$ such that $X\dashrightarrow X_{i}$ induces a birational
map $S\dashrightarrow S_{i}$. Let $(S,\Delta_{S})$ and
$(S_{i},\Delta_{S_{i}})$ be dlt pairs defined by adjunctions. Then
$a(Q,S,\Delta_{S})\leq a(Q,S_{i},\Delta_{S_{i}})$ for all prime divisors $Q$
over $S$ ([10, Lemma 4.2.10]). By [28, Lemma 2.17] and the hypothesis of [28,
Theorem 1.1] on lc centers, we see that $K_{S_{i}}+\Delta_{S_{i}}$ is abundant
over $Z$, therefore $K_{X_{i}}+\Delta_{i}$ is log abundant over $Z$. We note
that we need [26, Theorem 1.1] to prove [28, Lemma 2.17], and [26, Theorem
1.1] have been already recovered in the above discussion. Thus, any
$(K_{X}+\Delta)$-MMP over $Z$ preserves property of being log abundant over
$Z$.

If $\pi$ and $(X,\Delta)$ are as [27, Theorem 1.3], then the relative
numerical dimension of the restriction of $K_{X}+\Delta$ to any stratum is
either $0$ or $-\infty$. Therefore, this situation is a special case (i.e.,
the case of $n=3$) of [28, Theorem 1.1].

In any case, by Theorem 3.6, any $(K_{X}+\Delta)$-MMP over $Z$ with scaling of
an ample divisor terminates with a log minimal model $(X_{m},\Delta_{m})$
which is log abundant over $Z$. By Lemma 2.20, we see that
$(X_{m},\Delta_{m})$ is a good minimal model of $(X,\Delta)$ over $Z$.

###  3.2. Corollaries

In this subsection, we prove corollaries.

######  Corollary 3.8.

Any log MMP with scaling of an ample divisor starting with a projective dlt
pair contains only finitely many log abundant dlt pairs.

###### Proof.

Fix a projective dlt pair $(X,\Delta)$ and an ample $\mathbb{R}$-divisor $A$
on $X$. By using the perturbation of coefficients of $\Delta$ and [9, Theorem
3.5.1], we can find an $\mathbb{R}$-divisor $B$ on $X$ such that $(X,B)$ is
klt. We may assume that $K_{X}+\Delta$ is pseudo-effective because otherwise
any $(K_{X}+\Delta)$-MMP with scaling of $A$ terminates with a Mori fiber
space. Consider any sequence of steps of a $(K_{X}+\Delta)$-MMP with scaling
of $A$

$(X_{0}:=X,\Delta_{0}:=\Delta)\dashrightarrow(X_{1},\Delta_{1})\dashrightarrow\cdots\dashrightarrow(X_{i},\Delta_{i})\dashrightarrow\cdots.$

We set $\lambda_{i}={\rm inf}\set{\mu\in\mathbb{R}_{\geq
0}\\!}{\\!K_{X_{i}}+\Delta_{i}+\mu A_{i}\text{\rm\, is nef}}$ and
$\lambda_{\infty}:={\rm lim}_{i\to\infty}\lambda_{i}$, where $A_{i}$ is the
birational transform of $A$ on $X_{i}$. Suppose that the $(K_{X}+\Delta)$-MMP
is an infinite sequence. For any $i\neq j$, the birational map
$X_{i}\dashrightarrow X_{j}$ is not an isomorphism. Since each lc pair
$(X_{i},\Delta_{i}+\lambda_{i}A_{i})$ is a weak lc model of
$(X,\Delta+\lambda_{i}A)$, if $\lambda_{\infty}>0$ then we get a contradiction
by finiteness of models ([4, Theorem E]), so $\lambda_{\infty}=0$. By Theorem
3.5, the $(K_{X}+\Delta)$-MMP contains only finitely many log abundant dlt
pairs. ∎

######  Corollary 3.9.

For any projective lc pair $(X,\Delta)$ and any ample $\mathbb{R}$-divisor $A$
on $X$, there is a sequence of steps of a $(K_{X}+\Delta)$-MMP with scaling of
$A$ which contains only finitely many log abundant lc pairs.

###### Proof.

Let $(X,\Delta)$ and $A$ be as in the corollary. When $K_{X}+\Delta$ is not
pseudo-effective, there is a finite sequence of steps of a
$(K_{X}+\Delta)$-MMP with scaling of $A$ terminating with a Mori fiber space
([28, Theorem 1.7]). When $K_{X}+\Delta$ is pseudo-effective, $(X,\Delta+tA)$
has a good minimal model for all $t>0$ ([28, Theorem 1.5]). By applying
arguments as in [26, Proof of Lemma 2.14] (see also [28, Proof of Theorem
1.7]) with the length of extremal rays ([11, Section 18]) and [28, Proposition
6.1], we can construct a sequence of steps of a $(K_{X}+\Delta)$-MMP with
scaling of $A$

$(X_{0}:=X,\Delta_{0}:=\Delta)\dashrightarrow(X_{1},\Delta_{1})\dashrightarrow\cdots\dashrightarrow(X_{i},\Delta_{i})\dashrightarrow\cdots$

such that if we define $\lambda_{i}={\rm inf}\set{\mu\in\mathbb{R}_{\geq
0}\\!}{\\!K_{X_{i}}+\Delta_{i}+\mu A_{i}\text{\rm\, is nef}}$ and
$\lambda_{\infty}:={\rm lim}_{i\to\infty}\lambda_{i}$, where $A_{i}$ is the
birational transform of $A$ on $X_{i}$, then the $(K_{X}+\Delta)$-MMP
terminates or $\lambda_{\infty}=0$. Therefore, the $(K_{X}+\Delta)$-MMP
satisfies the first condition about log MMP in Theorem 3.5 when it is an
infinite sequence. By Theorem 3.5, only finitely many log abundant lc pairs
appear in the log MMP. ∎

######  Corollary 3.10.

Let $(X,\Delta)$ be a projective lc pair. Suppose that

  * •

$(X,\Delta)$ is log abundant and $K_{X}+\Delta$ is pseudo-effective, and

  * •

the stable base locus of $K_{X}+\Delta$ does not contain the image of any
prime divisor $P$ over $X$ satisfying $a(P,X,\Delta)<0$.

Then $(X,\Delta)$ has a good minimal model.

###### Proof.

By taking a dlt blow-up, we may assume that $(X,\Delta)$ is
$\mathbb{Q}$-factorial dlt. Let

$(X_{0}:=X,\Delta_{0}:=\Delta)\dashrightarrow(X_{1},\Delta_{1})\dashrightarrow\cdots\dashrightarrow(X_{i},\Delta_{i})\dashrightarrow\cdots$

be a sequence of steps of a $(K_{X}+\Delta)$-MMP with scaling of an ample
divisor. For any $i\geq 0$ and any lc center $S_{i}$ of $(X_{i},\Delta_{i})$,
there is an lc center $S$ of $(X,\Delta)$ such that the map $X\dashrightarrow
X_{i}$ induces a birational map $S\dashrightarrow S_{i}$. We define projective
dlt pairs $(S,\Delta_{S})$ and $(S_{i},\Delta_{S_{i}})$ by adjunctions
$K_{S}+\Delta_{S}=(K_{X}+\Delta)|_{S}$ and
$K_{S_{i}}+\Delta_{S_{i}}=(K_{X_{i}}+\Delta_{i})|_{S_{i}}$, respectively. By
[10, Lemma 4.2.10], we have $a(P,S,\Delta_{S})\leq a(P,S_{i},\Delta_{S_{i}})$
for all prime divisors $P$ over $X$. By Lemma 2.6 and the second condition of
Corollary 3.10, we have $a(D,S_{i},\Delta_{S_{i}})\leq a(D,S,\Delta_{S})$ for
all prime divisors $D$ on $S_{i}$. By Lemma 2.26, the divisor
$K_{S_{i}}+\Delta_{S_{i}}$ is abundant. Therefore, $(X_{i},\Delta_{i})$ is log
abundant for all $i$. By Theorem 3.6, the $(K_{X}+\Delta)$-MMP terminates with
a log minimal model $(X_{m},\Delta_{m})$, and $K_{X_{m}}+\Delta_{m}$ is log
abundant. By Lemma 2.20, we see that $(X,\Delta)$ has a good minimal model. ∎

We close this subsection with applications of Theorem 3.4.

######  Lemma 3.11.

Let $(X,\Delta)$ be a projective lc pair and $\pi\colon X\to Z$ a projective
morphism from a normal variety to a projective variety such that ${\rm