# Cone and contraction theorem for projective morphisms between complex
analytic spaces

Osamu Fujino  Department of Mathematics, Graduate School of Science, Kyoto
University, Kyoto 606-8502, Japan<EMAIL_ADDRESS>
(Date: 2023/8/12, version 0.13)

###### Abstract.

We discuss the cone and contraction theorem in a suitable complex analytic
setting. More precisely, we establish the cone and contraction theorem of
normal pairs for projective morphisms between complex analytic spaces. This
result is a starting point of the minimal model program for complex analytic
log canonical pairs. In this paper, we are mainly interested in normal pairs
whose singularities are worse than kawamata log terminal singularities.

###### Key words and phrases:

minimal model program, log canonical pairs, complex analytic spaces, cone and
contraction theorem, basepoint-free theorem, extremal rational curves,
vanishing theorems, strict support condition

###### 2010 Mathematics Subject Classification:

Primary 14E30; Secondary 32C15

###### Contents

  1. 1 Introduction
    1. 1.1 Main theorem
  2. 2 Preliminaries
  3. 3 Basic properties of relatively ample and relatively nef $\mathbb{R}$-line bundles
  4. 4 Non-lc ideal sheaves
  5. 5 Quick review of vanishing theorems
    1. 5.1 Vanishing theorems of Reid–Fukuda type
  6. 6 Vanishing theorems for normal pairs
  7. 7 On log canonical centers
  8. 8 Non-vanishing theorem
  9. 9 Basepoint-free theorem
  10. 10 Rationality theorem
  11. 11 Kleiman–Mori cones
    1. 11.1 Nakayama’s finiteness
  12. 12 Cone theorem
    1. 12.1 Proof of Theorem 1.4 and Corollary 1.5
  13. 13 Lengths of extremal rational curves
  14. 14 On Shokurov’s polytopes
  15. 15 Basepoint-free theorem for $\mathbb{R}$-divisors
  16. 16 Proof of Main theorem

##  1\. Introduction

In his epoch-making paper [Mo], Shigefumi Mori established the cone theorem
for smooth projective varieties defined over any algebraically closed field
$k$ of arbitrary characteristic by his ingenious method of bend and break.
Then he established the contraction theorem for smooth projective threefolds
when the characteristic of the base field $k$ is zero. After that, in
characteristic zero, the cone and contraction theorem was generalized for so-
called log-terminal pairs in any dimension by using Hironaka’s resolution of
singularities and the Kawamata–Viehweg vanishing theorem. For the details, see
[KMM], [KM] and references therein. Now we know that, in characteristic zero,
the cone and contraction theorem holds for more general settings (see [Fu2],
[Fu3, Chapter 6], and references therein). In this paper, we will discuss the
cone and contraction theorem of normal pairs for projective morphisms between
complex analytic spaces. For kawamata log terminal pairs, it was known and has
played an important role in [Na1], [Na2], and [Fu8]. In [Fu8], we have already
discussed the minimal model program for kawamata log terminal pairs in a
complex analytic setting (see also [Fu13]). Roughly speaking, we showed that
[BCHM] and [HM] can work for projective morphisms between complex analytic
spaces. We note that the Kawamata–Viehweg vanishing theorem can be formulated
and proved for projective morphisms of complex analytic spaces and is
sufficient for the study of kawamata log terminal pairs. We also note that
$L^{2}$-methods can work for kawamata log terminal pairs. For an alternative
approach to the minimal model program of kawamata log terminal pairs for
projective morphisms between complex analytic spaces, see [DHP], which uses
the idea of [CL]. The reader can find a new approach to the relative minimal
model program in [LM], which can work in larger categories of spaces. In
[Fu10], we established some vanishing theorems and related results necessary
for the study of complex analytic log canonical pairs and quasi-log structures
on complex analytic spaces. Note that [Fu10] depends on Morihiko Saito’s
theory of mixed Hodge modules (see [Sa1], [Sa2], [Sa3], [FFS], and [Sa4]) and
Takegoshi’s generalization of Kollár’s torsion-free and vanishing theorem (see
[Ta]). For an approach without using the theory of mixed Hodge modules, see
[FF]. In this paper, we will discuss the cone and contraction theorem of
normal pairs for projective morphisms between complex analytic spaces as an
application of [Fu10]. This paper can be seen as a complex analytic
generalization of [Fu2] and as a generalization of Nakayama’s paper [Na1]. We
note that Nakayama only treated kawamata log terminal pairs and
$\mathbb{Q}$-divisors in [Na1]. Finally, this paper is independent of [Fu8]
and does not use any results obtained in [Fu8].

######  1.1Standard setting.

One of the main difficulties to discuss the minimal model theory for complex
analytic spaces is how to formulate it.

Let $\pi\colon X\to Y$ be a projective morphism between complex analytic
spaces such that $X$ is a normal complex variety and let $W$ be a compact
subset of $Y$. In this paper, we formulate and prove almost everything over
some open neighborhood of $W$. Let $\Delta$ be an $\mathbb{R}$-divisor on $X$
such that $K_{X}+\Delta$ is $\mathbb{R}$-Cartier. The number of the
irreducible components of $\operatorname{Supp}\Delta$ is only locally finite.
In general, the support of $\Delta$ may have infinitely many irreducible
components. By shrinking $Y$ around $W$ suitably, that is, by replacing $Y$
with a suitable relatively compact open neighborhood of $W$, we can always
assume that $\operatorname{Supp}\Delta$ has only finitely many irreducible
components. Moreover, we can assume that a given $\mathbb{R}$-Cartier divisor
on $X$ is a finite $\mathbb{R}$-linear combination of Cartier divisors.
Therefore, by considering some relatively compact open neighborhood of $W$, we
can avoid subtle problems caused by the difference between the Zariski
topology and the Euclidean topology. In [Fu8], we almost always assume that
$W$ is a Stein compact subset of $Y$ such that $\Gamma(W,\mathcal{O}_{Y})$ is
noetherian. In this paper, however, we usually assume that $W$ is only a
compact subset of $Y$. When we consider the Kleiman–Mori cone
$\operatorname{\overline{NE}}(X/Y;W)$ of $\pi\colon X\to Y$ and $W$, we
further assume that the dimension of $N^{1}(X/Y;W)$ is finite. For the details
of $\operatorname{\overline{NE}}(X/Y;W)$ and $N^{1}(X/Y;W)$, see Section 11.
Note that if $W\cap V$ has only finitely many connected components for any
analytic subset $V$ which is defined over an open neighborhood of $W$ then the
dimension of $N^{1}(X/Y;W)$ is finite by Nakayama’s finiteness (see Theorem
11.10). Therefore, if $W$ is a compact semianalytic subset of $Y$, then the
dimension of $N^{1}(X/Y;W)$ is always finite. Thus, we can find many compact
subsets $W$ with $\dim_{\mathbb{R}}N^{1}(X/Y;W)<\infty$.

###  1.1. Main theorem

In this paper, we call $(X,\Delta)$ a normal pair if it consists of a normal
complex variety $X$ and an effective $\mathbb{R}$-divisor $\Delta$ on $X$ such
that $K_{X}+\Delta$ is $\mathbb{R}$-Cartier. The main purpose of this paper is
to establish the following cone and contraction theorem of normal pairs for
projective morphisms between complex analytic spaces.

######  Theorem 1.2 (Cone and contraction theorem, see Theorems 12.1, 12.2,
13.2, and 14.4).

Let $\pi\colon X\to Y$ be a projective morphism of complex analytic spaces
such that $X$ is a normal complex variety and let $W$ be a compact subset of
$Y$. Assume that the dimension of $N^{1}(X/Y;W)$ is finite. Let $\Delta$ be an
effective $\mathbb{R}$-divisor on $X$ such that $K_{X}+\Delta$ is
$\mathbb{R}$-Cartier. Then we have

$\operatorname{\overline{NE}}(X/Y;W)=\operatorname{\overline{NE}}(X/Y;W)_{(K_{X}+\Delta)\geq
0}+\operatorname{\overline{NE}}(X/Y;W)_{\operatorname{Nlc}(X,\Delta)}+\sum
R_{j}$

with the following properties.

  * (1)

$\operatorname{Nlc}(X,\Delta)$ is the non-lc locus of $(X,\Delta)$ and
$\operatorname{\overline{NE}}(X/Y;W)_{\operatorname{Nlc}(X,\Delta)}$ is the
subcone of $\operatorname{\overline{NE}}(X/Y;W)$ which is the closure of the
convex cone spanned by the projective integral curves $C$ on
$\operatorname{Nlc}(X,\Delta)$ such that $\pi(C)$ is a point of $W$.

  * (2)

$R_{j}$ is a $(K_{X}+\Delta)$-negative extremal ray of
$\operatorname{\overline{NE}}(X/Y;W)$ which satisfies

$R_{j}\cap\operatorname{\overline{NE}}(X/Y;W)_{\operatorname{Nlc}(X,\Delta)}=\\{0\\}$

for every $j$.

  * (3)

Let $\mathcal{A}$ be a $\pi$-ample $\mathbb{R}$-line bundle on $X$. Then there
are only finitely many $R_{j}$’s included in
$\operatorname{\overline{NE}}(X/Y;W)_{(K_{X}+\Delta+\mathcal{A})<0}$. In
particular, the $R_{j}$’s are discrete in the half-space
$\operatorname{\overline{NE}}(X/Y;W)_{(K_{X}+\Delta)<0}$.

  * (4)

Let $F$ be any face of $\operatorname{\overline{NE}}(X/Y;W)$ such that

$F\cap\left(\operatorname{\overline{NE}}(X/Y;W)_{(K_{X}+\Delta)\geq
0}+\operatorname{\overline{NE}}(X/Y;W)_{\operatorname{Nlc}(X,\Delta)}\right)=\\{0\\}.$

Then, after shrinking $Y$ around $W$ suitably, there exists a contraction
morphism $\varphi_{F}\colon X\to Z$ over $Y$ satisfying the following
properties.

    * (i)

Let $C$ be a projective integral curve on $X$ such that $\pi(C)$ is a point of
$W$. Then $\varphi_{F}(C)$ is a point if and only if the numerical equivalence
class $[C]$ of $C$ is in $F$.

    * (ii)

The natural map $\mathcal{O}_{Z}\to(\varphi_{F})_{*}\mathcal{O}_{X}$ is an
isomorphism.

    * (iii)

Let $\mathcal{L}$ be a line bundle on $X$ such that $\mathcal{L}\cdot C=0$ for
every curve $C$ with $[C]\in F$. Then, after shrinking $Y$ around $W$ suitably
again, there exists a line bundle $\mathcal{L}_{Z}$ on $Z$ such that
$\mathcal{L}\simeq\varphi^{*}_{F}\mathcal{L}_{Z}$ holds.

  * (5)

Every $(K_{X}+\Delta)$-negative extremal ray $R$ with

$R\cap\operatorname{\overline{NE}}(X/Y;W)_{\operatorname{Nlc}(X,\Delta)}=\\{0\\}$

is spanned by a (possibly singular) rational curve $C$ with

$0<-(K_{X}+\Delta)\cdot C\leq 2\dim X.$

From now on, we further assume that $(X,\Delta)$ is log canonical,
equivalently, $\operatorname{Nlc}(X,\Delta)=\emptyset$. Then we have the
following properties.

  * (6)

Let $H$ be an effective $\mathbb{R}$-Cartier $\mathbb{R}$-divisor on $X$ such
that $K_{X}+\Delta+H$ is $\pi$-nef over $W$ and $(X,\Delta+H)$ is log
canonical. Then, either $K_{X}+\Delta$ is also $\pi$-nef over $W$ or there
exists a $(K_{X}+\Delta)$-negative extremal ray $R$ of
$\operatorname{\overline{NE}}(X/Y;W)$ such that

$(K_{X}+\Delta+\lambda H)\cdot R=0,$

where

$\lambda:=\inf\\{t\geq 0\,|\,K_{X}+\Delta+tH\ {\text{is $\pi$-nef over
$W$}}\\}.$

Of course, $K_{X}+\Delta+\lambda H$ is $\pi$-nef over $W$.

Similarly, we have:

  * (7)

Let $\mathcal{H}$ be an $\mathbb{R}$-line bundle on $X$ which is $\pi$-ample
over $W$ such that $K_{X}+\Delta+\mathcal{H}$ is $\pi$-nef over $W$. Then,
either $K_{X}+\Delta$ is also $\pi$-nef over $W$ or there exists a
$(K_{X}+\Delta)$-negative extremal ray $R$ of
$\operatorname{\overline{NE}}(X/Y;W)$ such that

$(K_{X}+\Delta+\lambda\mathcal{H})\cdot R=0,$

where

$\lambda:=\inf\\{t\geq 0\,|\,K_{X}+\Delta+t\mathcal{H}\ {\text{is $\pi$-nef
over $W$}}\\}.$

Note that $K_{X}+\Delta+\lambda\mathcal{H}$ is $\pi$-nef over $W$.

######  Remark 1.3.

In Theorem 1.2, the proof of (5) needs Mori’s bend and break method and (6) is
an application of (5). On the other hand, (7) is an easy consequence of (3).
Note that $\pi$-very ample line bundles do not always have global sections.
Hence (7) is not a special case of (6). We need (7) in order to discuss the
minimal model program of log canonical pairs with ample scaling for projective
morphisms between complex analytic spaces.

For the minimal model program, the following theorem, which is a supplement to
Theorem 1.2, may be useful (see [Fu8]).

######  Theorem 1.4.

Let $(X,\Delta)$ be a log canonical pair. Let $\pi\colon X\to Y$ be a
projective morphism of complex analytic spaces and let $W$ be a compact subset
of $Y$ such that the dimension of $N^{1}(X/Y;W)$ is finite. Suppose that
$\pi\colon X\to Y$ is decomposed as

$\textstyle{\pi\colon
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y^{\flat}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{Y}$

such that $Y^{\flat}$ is projective over $Y$. Let $\mathcal{A}_{Y^{\flat}}$ be
a $g$-ample line bundle on $Y^{\flat}$. Let $R$ be a $\left(K_{X}+\Delta+(\dim
X+1)f^{*}\mathcal{A}_{Y^{\flat}}\right)$-negative extremal ray of
$\operatorname{\overline{NE}}(X/Y;W)$. Then $R$ is a $(K_{X}+\Delta)$-negative
extremal ray of
$\operatorname{\overline{NE}}\left(X/{Y^{\flat}};g^{-1}(W)\right)$, that is,
$R\cdot f^{*}\mathcal{A}_{Y^{\flat}}=0$.

We prove Theorem 1.4 as an application of the vanishing theorem for projective
quasi-log schemes. We do not need Theorem 1.2 (5) for the proof of Theorem
1.4. We have the following result as an easy consequence of Theorem 1.4.

######  Corollary 1.5 (see [Fu5, Corollary 1.2]).

Let $(X,\Delta)$ be a log canonical pair. Let $\pi\colon X\to Y$ be a
projective morphism of complex analytic spaces and let $\mathcal{A}$ be any
$\pi$-ample line bundle on $X$. Then $K_{X}+\Delta+(\dim X+1)\mathcal{A}$ is
always nef over $Y$.

We make an important remark on Theorem 1.2. By Remark 1.6, we see that the
cone and contraction theorem of normal pairs holds for projective morphisms
between compact analytic spaces.

######  Remark 1.6.

Let $\pi\colon X\to Y$ be a projective morphism of complex analytic spaces and
let $W$ be a compact subset of $Y$ as in Theorem 1.2. Then the dimension of
$N^{1}(X/Y;W)$ is not always finite (see Example 11.9). In [Na2, Chapter II.
5.19. Lemma] (see Theorem 11.10), Noboru Nakayama proved that if

  * •

$W$ is a compact subset of $Y$ such that $W\cap V$ has only finitely many
connected components for any analytic subset $V$ which is defined over an open
neighborhood of $W$,

then the dimension of $N^{1}(X/Y;W)$ is finite. We note that the above
assumption is satisfied in the following cases:

  * (i)

$W$ is a point of $Y$.

  * (ii)

$W$ is a compact semianalytic subset of $Y$.

  * (iii)

$W=Y$ when $Y$ is compact.

Case (i) is obvious. In Case (ii), $W\cap V$ is a compact semianalytic subset
of $Y$. Thus we see that $W\cap V$ has only finitely many connected components
(see, for example, [BM1, Corollary 2.7 (2)]). In Case (iii), $W\cap V=V$ is a
compact analytic subset of $Y$. Hence it has only finitely many connected
components.

By Remark 1.6, we see that there are many compact subsets $W$ of $Y$ such that
the dimension of $N^{1}(X/Y;W)$ is finite.

We note that we can formulate and prove the basepoint-free theorem for
projective morphisms of complex analytic spaces as follows. In Theorem 1.7,
$L$ is only assumed to be $\pi$-nef over $W$, that is, $L|_{\pi^{-1}(w)}$ is
nef in the usual sense for every $w\in W$. Equivalently, $L\cdot C\geq 0$ for
every projective integral curve $C$ on $X$ such that $\pi(C)$ is a point of
$W$. However, Theorem 1.7 claims that it is $\pi$-semiample over some open
neighborhood of $W$.

######  Theorem 1.7 (Basepoint-free theorem: Theorem 9.1).

Let $\pi\colon X\to Y$ be a projective morphism of complex analytic spaces
such that $X$ is a normal complex variety and let $W$ be a compact subset of
$Y$. Let $\Delta$ be an effective $\mathbb{R}$-divisor on $X$ such that
$K_{X}+\Delta$ is $\mathbb{R}$-Cartier. Let $L$ be a Cartier divisor on $X$
which is $\pi$-nef over $W$. We assume that

  * (i)

$aL-(K_{X}+\Delta)$ is $\pi$-ample over $W$ for some positive real number $a$,
and

  * (ii)

$\mathcal{O}_{\operatorname{Nlc}(X,\Delta)}(mL)$ is
$\pi|_{\operatorname{Nlc}(X,\Delta)}$-generated over some open neighborhood of
$W$ for every $m\gg 0$.

Then there exists a relatively compact open neighborhood $U$ of $W$ such that
$\mathcal{O}_{X}(mL)$ is $\pi$-generated over $U$ for every $m\gg 0$.

In Theorem 1.7, $W$ is only assumed to be a compact subset of $Y$. We do not
need the assumption that $\dim_{\mathbb{R}}N^{1}(X/Y;W)<\infty$ holds. When
$(X,\Delta)$ is log canonical, we will also prove the basepoint-free theorem
for $\mathbb{R}$-Cartier divisors (see Theorem 15.1). In Theorem 15.1, we have
to assume that the dimension of $N^{1}(X/Y;W)$ is finite since we need the
cone theorem for the proof of Theorem 15.1.

In the proof of Theorems 1.2, 1.7, and so on, the following basic properties
of log canonical centers play an important role.

######  Theorem 1.8 (Basic properties of log canonical centers: Theorem 7.1).

Let $(X,\Delta)$ be a log canonical pair. Then the following properties hold.

  * (1)

The number of log canonical centers of $(X,\Delta)$ is locally finite.

  * (2)

The intersection of two log canonical centers is a union of some log canonical
centers.

  * (3)

Let $x\in X$ be any point such that $(X,\Delta)$ is log canonical but is not
kawamata log terminal at $x$. Then there exists a unique minimal (with respect
to the inclusion) log canonical center $C_{x}$ passing through $x$. Moreover,
$C_{x}$ is normal at $x$.

Theorem 1.8 is new for complex analytic log canonical pairs although it is
well known when $(X,\Delta)$ is algebraic. It will be useful for the study of
complex analytic log canonical singularities (see also [Fu9]).

Theorem 1.2 is a starting point of the minimal model program of log canonical
pairs for projective morphisms between complex analytic spaces. We can
formulate the minimal model theory of log canonical pairs for projective
morphisms between complex analytic spaces by using Theorem 1.2 as in the
algebraic case. On the other hand, one of the main goals of the minimal model
theory for projective morphisms between complex analytic spaces is the
following conjecture.

######  Conjecture 1.9 (Finite generation).

Let $\pi\colon X\to Y$ be a projective morphism of complex analytic spaces and
let $\Delta$ be an effective $\mathbb{Q}$-divisor on $X$ such that
$(X,\Delta)$ is log canonical. Then

$R(X/Y,K_{X}+\Delta):=\bigoplus_{m\in\mathbb{N}}\pi_{*}\mathcal{O}_{X}(\lfloor
m(K_{X}+\Delta)\rfloor)$

is a locally finitely generated graded $\mathcal{O}_{Y}$-algebra.

We note that in [Fu8] Conjecture 1.9 was already solved completely when
$(X,\Delta)$ is kawamata log terminal. We also note that Conjecture 1.9 is
still widely open even when $\pi\colon X\to Y$ is algebraic (see [FG]).

The author first prepared a short manuscript which only explains how to modify
arguments in [Fu2]. Unfortunately, however, it seemed to be hard to read.
Hence he made great efforts to make this paper as self-contained as possible
except for the results established in [Fu10] (see also [Fu12]). He sometimes
repeats arguments in [Fu2] and [Fu3]. Thus, some parts of this paper are very
similar to those of [Fu2] and [Fu3].

######  Remark 1.10 (Quasi-log structures).

By [Fu10, Theorems 1.1 and 1.2] (see Theorems 5.5 and 5.7), we can formulate
and discuss quasi-log structures on complex analytic spaces (see [Fu3, Chapter
6]). Hence we can establish the cone and contraction theorem for highly
singular complex analytic spaces. However, in this paper, we will only discuss
the cone and contraction theorem of normal pairs (see Theorem 1.2). This is
because Theorem 1.2 is sufficient for many geometric applications and it is
not so easy psychologically to treat reducible complex analytic spaces. We
will describe the theory of quasi-log complex analytic spaces in [Fu11].

We look at the organization of this paper. In Section 2, we collect some
necessary definitions and results for the reader’s convenience. Since we have
to work in the complex analytic setting, some of them become much more subtle
than the usual ones in the algebraic setting. In Section 3, we collect some
basic properties of relatively nef and relatively ample $\mathbb{R}$-line
bundles for the sake of completeness. They are indispensable in subsequent
sections. In Section 4, we define non-lc ideal sheaves in the complex analytic
setting and prove some elementary lemmas. In Section 5, we quickly recall the
main result of [Fu10] without proof. Note that the proof of the main result in
[Fu10] depends on Saito’s theory of mixed Hodge modules and Takegoshi’s
generalization of Kollár’s torsion-free and vanishing theorem. However, we can
make the main result of [Fu10] free from the theory of mixed Hodge modules by
[FF]. In Section 6, we prepare some necessary vanishing theorems as
applications of the vanishing result explained in Section 5. In Section 7, we
establish the basic properties of log canonical centers. They are new and very
important in the theory of minimal models in the complex analytic setting. In
Section 8, we prove the non-vanishing theorem in the complex analytic setting
with the aid of the theory of quasi-log schemes. Note that Lemma 8.2 is new
and will be useful for the study of quasi-log structures. In Section 9, we
establish the basepoint-free theorem for normal pairs in the complex analytic
setting by using the non-vanishing theorem proved in Section 8. It is well
known and is not difficult to prove for kawamata log terminal pairs. In
Sections 10, we prove the rationality theorem for normal pairs in the complex
analytic setting. The proof is essentially the same as the one for algebraic
varieties explained in [Fu2]. In Section 11, we define Kleiman–Mori cones for
projective morphisms of complex analytic spaces. In Subsection 11.1, we
briefly explain Nakayama’s finiteness without proof for the reader’s
convenience. Note that in this paper we do not need it except in the proof of
Corollary 1.5. Then, in Section 12, we prove the cone theorem for normal pairs
in the complex analytic setting. The results in Section 12 are easy
consequences of the basepoint-free theorem in Section 9 and the rationality
theorem in Section 10. In Subsection 12.1, we prove Theorem 1.4 as an easy
application of the vanishing theorem for projective quasi-log schemes. In
Section 13, we discuss lengths of extremal rational curves. The result in
Section 13 seems to be indispensable for the minimal model program with
scaling. Here, we use the framework of quasi-log schemes. In Section 14, we
discuss Shokurov’s polytopes and some applications. The results in this
section are well known and have already played an important role in the usual
algebraic setting. In Section 15, we prove the basepoint-free theorem of log
canonical pairs for $\mathbb{R}$-Cartier divisors. It can be seen as an
application of the cone theorem. In Section 16, which is the final section, we
prove the main result of this paper, that is, the cone and contraction theorem
of normal pairs for projective morphisms between complex analytic spaces:
Theorem 1.2.

######  Acknowledgments.

The author was partially supported by JSPS KAKENHI Grant Numbers JP19H01787,
JP20H00111, JP21H00974, JP21H04994. He thanks Professor Noboru Nakayama very
much for useful suggestions and answering his questions. He also would like to
thank Professors Taro Fujisawa, Shigefumi Mori, and Morihiko Saito very much.

In this paper, every complex analytic space is assumed to be Hausdorff and
second-countable. An irreducible and reduced complex analytic space is called
a complex variety. We will freely use the standard notation in [Fu2], [Fu3],
[Fu8], and so on. We will also freely use the basic results on complex
analytic geometry in [BS] and [Fi]. For the minimal model program for
projective morphisms between complex analytic spaces, see [Na1], [Na2], and
[Fu8]. For the traditional framework of the minimal model program, see [KMM]
and [KM]. We note that $\mathbb{Z}$, $\mathbb{Q}$, and $\mathbb{R}$ denote the
set of integers, rational numbers, and real numbers, respectively. We also
note that $\mathbb{N}$ (resp. $\mathbb{Z}_{>0}$) is the set of non-negative
integers (resp. positive integers).

##  2\. Preliminaries

In this section, we collect basic definitions and results necessary for this
paper. For the details, see [Fu2], [Fu3], [Fu8], and so on. Since we are
working in the complex analytic setting, some of them become subtle.

Let us start with the definition of singularities of pairs, which is
indispensable in the theory of minimal models.

######  2.1Singularities of pairs, log canonical centers, and non-lc loci.

We consider a normal complex variety $X$. Let $X_{\mathrm{sm}}$ denote the
smooth locus of $X$. Then the canonical sheaf $\omega_{X}$ of $X$ is the
unique reflexive sheaf whose restriction to $X_{\mathrm{sm}}$ is isomorphic to
the sheaf $\Omega^{n}_{X_{\mathrm{sm}}}$, where $n=\dim X$. Let $\Delta$ be an
$\mathbb{R}$-divisor on $X$, that is, $\Delta$ is a locally finite
$\mathbb{R}$-linear combination of prime divisors on $X$. We say that
$K_{X}+\Delta$ is $\mathbb{R}$-Cartier at $x\in X$ if there exist an open
neighborhood $U_{x}$ of $x$ and a Weil divisor $K_{U_{x}}$ on $U_{x}$ with
$\mathcal{O}_{U_{x}}(K_{U_{x}})\simeq\omega_{X}|_{U_{x}}$ such that
$K_{U_{x}}+\Delta|_{U_{x}}$ is $\mathbb{R}$-Cartier, that is,
$K_{U_{x}}+\Delta|_{U_{x}}$ is a finite $\mathbb{R}$-linear combination of
Cartier divisors on $U_{x}$. For any subset $L$ of $X$, we say that
$K_{X}+\Delta$ is $\mathbb{R}$-Cartier at $L$ if it is $\mathbb{R}$-Cartier at
any point $x$ of $L$. We simply say that $K_{X}+\Delta$ is
$\mathbb{R}$-Cartier when $K_{X}+\Delta$ is $\mathbb{R}$-Cartier at any point
$x\in X$. Unfortunately, however, we can not always define $K_{X}$ globally
with $\mathcal{O}_{X}(K_{X})\simeq\omega_{X}$. In general, it only exists
locally on $X$. We usually use the symbol $K_{X}$ as a formal divisor class
with an isomorphism $\mathcal{O}_{X}(K_{X})\simeq\omega_{X}$ and call it the
canonical divisor of $X$ if there is no danger of confusion.

Let $f\colon Y\to X$ be a proper bimeromorphic morphism from a normal complex
variety $Y$. Suppose that $K_{X}+\Delta$ is $\mathbb{R}$-Cartier in the above
sense. We take a small Stein open subset $U$ of $X$ where $K_{U}+\Delta|_{U}$
is a well-defined $\mathbb{R}$-Cartier $\mathbb{R}$-divisor on $U$. In this
situation, we can define $K_{f^{-1}(U)}$ and $K_{U}$ such that
$f_{*}K_{f^{-1}(U)}=K_{U}$. Then we can write

$K_{f^{-1}(U)}=f^{*}(K_{U}+\Delta|_{U})+E_{U}$

as usual. Note that $E_{U}$ is a well-defined $\mathbb{R}$-divisor on
$f^{-1}(U)$ such that $f_{*}E_{U}=\Delta|_{U}$. Then we have the following
formula

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

as in the algebraic case. We note that $\sum_{E}a(E,X,\Delta)E$ is a globally
well-defined $\mathbb{R}$-divisor on $Y$ such that
$\left(\sum_{E}a(E,X,\Delta)E\right)|_{f^{-1}(U)}=E_{U}$ although $K_{X}$ and
$K_{Y}$ are well defined only locally.

If $\Delta$ is a boundary $\mathbb{R}$-divisor, that is, all the coefficients
of $\Delta$ are in $[0,1]\cap\mathbb{R}$, and $a(E,X,\Delta)\geq-1$ holds for
any $f\colon Y\to X$ and every $f$-exceptional divisor $E$, then $(X,\Delta)$
is called a log canonical pair. If $(X,\Delta)$ is log canonical and
$a(E,X,\Delta)>-1$ for any $f\colon Y\to X$ and every $f$-exceptional divisor
$E$, then $(X,\Delta)$ is called a purely log terminal pair. If $(X,\Delta)$
is purely log terminal and $\lfloor\Delta\rfloor=0$, that is, the coefficients
of $\Delta$ are in $[0,1)\cap\mathbb{R}$, then $(X,\Delta)$ is called a
kawamata log terminal pair. When $\Delta=0$ and $a(E,X,0)\geq 0$ (resp. $>0$)
for any $f\colon Y\to X$ and every $f$-exceptional divisor $E$, we simply say
that $X$ has only canonical singularities (resp. terminal singularities). In
this paper, we will only use log canonical pairs and kawamata log terminal
pairs.

More generally, let $X$ be a normal complex variety and let $\Delta$ be an
effective $\mathbb{R}$-divisor on $X$. We say that $(X,\Delta)$ is log
canonical (resp. kawamata log terminal) at $x\in X$ if there exists an open
neighborhood $U_{x}$ of $x$ such that $(U_{x},\Delta|_{U_{x}})$ is a log
canonical pair (resp. kawamata log terminal pair). Let $L$ be any subset of
$X$. We say that $(X,\Delta)$ is log canonical (resp. kawamata log terminal)
at $L$ if $(X,\Delta)$ is log canonical (resp. kawamata log terminal) at any
point $x$ of $L$. We note that $(X,\Delta)$ is log canonical (resp. kawamata
log terminal) in the above sense if and only if $(X,\Delta)$ is log canonical
(resp. kawamata log terminal) at any point $x$ of $X$.

Let $X$ be a normal complex variety and let $\Delta$ be an effective
$\mathbb{R}$-divisor on $X$ such that $K_{X}+\Delta$ is $\mathbb{R}$-Cartier.
The image of $E$ with $a(E,X,\Delta)=-1$ for some $f\colon Y\to X$ such that
$(X,\Delta)$ is log canonical around general points of $f(E)$ is called a log
canonical center of $(X,\Delta)$. The non-lc locus of $(X,\Delta)$, denoted by
$\operatorname{Nlc}(X,\Delta)$, is the smallest closed subset $Z$ of $X$ such
that the complement $(X\setminus Z,\Delta|_{X\setminus Z})$ is log canonical.
We can define a natural complex analytic space structure on
$\operatorname{Nlc}(X,\Delta)$ by the non-lc ideal sheaf
$\mathcal{J}_{\operatorname{NLC}}(X,\Delta)$ of $(X,\Delta)$. For the
definition of $\mathcal{J}_{\operatorname{NLC}}(X,\Delta)$, see Section 4
below.

The above definition is compatible with the usual definition for algebraic
varieties.

######  Remark 2.2.

Let $(X,\Delta)$ be a pair consisting of a normal algebraic variety $X$ and an
effective $\mathbb{R}$-divisor on $X$ such that $K_{X}+\Delta$ is
$\mathbb{R}$-Cartier. Then $(X,\Delta)$ is kawamata log terminal (resp. log
canonical) in the usual sense (see [Fu2], [Fu3], and so on) if and only if
$(X^{\mathrm{an}},\Delta^{\mathrm{an}})$ is kawamata log terminal (resp. log
canonical) in the above sense, where $X^{\mathrm{an}}$ is the complex analytic
space naturally associated to $X$ and let $\Delta^{\mathrm{an}}$ be the
$\mathbb{R}$-divisor on $X^{\mathrm{an}}$ associated to $\Delta$.

The following lemma is well known for algebraic varieties.

######  Lemma 2.3.

Let $X$ be a normal complex variety and let $\Delta$ be an effective
$\mathbb{R}$-divisor on $X$ such that $K_{X}+\Delta$ is $\mathbb{R}$-Cartier.
Let $P$ be a point of $X$ and let $D_{i}$ be an effective Cartier divisor on
$X$ with $P\in\operatorname{Supp}D_{i}$ for every $i$. If
$\left(X,\Delta+\sum_{i=1}^{k}D_{i}\right)$ is log canonical at $P$, then
$k\leq\dim X$ holds.

We omit the proof of Lemma 2.3 here since the usual proof for algebraic
varieties can work without any changes (see, for example, [Fu2, Lemma 13.2]).
We will use Lemma 2.3 in order to create a new log canonical center.

In this paper, we sometimes implicitly use Serre’s GAGA.

######  2.4Serre’s GAGA.

Let $\pi\colon X\to Y$ be a projective morphism of complex analytic spaces and
let $F$ be a fiber of $\pi\colon X\to Y$. Then $F$ is projective. Hence we can
apply various results of projective schemes to $F$ with the aid of Serre’s
GAGA (see [Se]).

In the theory of minimal models, we need the notion of $\mathbb{R}$-line
bundles and $\mathbb{Q}$-line bundles.

######  2.5Line bundles, $\mathbb{R}$-line bundles, and $\mathbb{Q}$-line
bundles.

Let $X$ be a complex analytic space and let $\operatorname{Pic}(X)$ denote the
group of line bundles on $X$, that is, the Picard group of $X$. An element of
$\operatorname{Pic}(X)\otimes_{\mathbb{Z}}\mathbb{R}$ (resp.
$\operatorname{Pic}(X)\otimes_{\mathbb{Z}}\mathbb{Q}$) is called an
$\mathbb{R}$-line bundle (resp. a $\mathbb{Q}$-line bundle) on $X$. In this
paper, we usually write the group law of
$\operatorname{Pic}(X)\otimes_{\mathbb{Z}}\mathbb{R}$ additively for
simplicity of notation. Hence we sometimes use $m\mathcal{L}$ to denote
$\mathcal{L}^{\otimes m}$ for $\mathcal{L}\in\operatorname{Pic}(X)$ and
$m\in\mathbb{Z}$.

We also need the notion of $\mathbb{R}$-divisors and $\mathbb{Q}$-divisors.

######  2.6Divisors, $\mathbb{R}$-divisors, and $\mathbb{Q}$-divisors.

Let $X$ be a reduced equidimensional complex analytic space. A prime divisor
on $X$ is an irreducible and reduced closed analytic subspace of codimension
one. An $\mathbb{R}$-divisor $D$ on $X$ is a formal sum

$D=\sum_{i}a_{i}D_{i},$

where $D_{i}$ is a prime divisor on $X$ with $D_{i}\neq D_{j}$ for $i\neq j$,
$a_{i}\in\mathbb{R}$ for every $i$, and the support

$\operatorname{Supp}D:=\bigcup_{a_{i}\neq 0}D_{i}$

is a closed analytic subset of $X$. In other words, the formal sum
$\sum_{i}a_{i}D_{i}$ is locally finite. If $a_{i}\in\mathbb{Z}$ (resp.
$a_{i}\in\mathbb{Q}$) for every $i$, then $D$ is called a divisor (resp.
$\mathbb{Q}$-divisor) on $X$. Note that a divisor is sometimes called an
integral Weil divisor in order to emphasize the condition that
$a_{i}\in\mathbb{Z}$ for every $i$. If $0\leq a_{i}\leq 1$ (resp. $a_{i}\leq
1$) holds for every $i$, then an $\mathbb{R}$-divisor $D$ is called a boundary
(resp. subboundary) $\mathbb{R}$-divisor.

Let $D=\sum_{i}a_{i}D_{i}$ be an $\mathbb{R}$-divisor on $X$ such that $D_{i}$
is a prime divisor for every $i$ with $D_{i}\neq D_{j}$ for $i\neq j$. The
round-down $\lfloor D\rfloor$ of $D$ is defined to be the divisor

$\lfloor D\rfloor=\sum_{i}\lfloor a_{i}\rfloor D_{i},$

where $\lfloor x\rfloor$ is the integer defined by $x-1<\lfloor x\rfloor\leq
x$ for every real number $x$. The round-up and the fractional part of $D$ are
defined to be

$\lceil D\rceil:=-\lfloor-D\rfloor,\quad\text{and}\quad\\{D\\}:=D-\lfloor
D\rfloor,$

respectively. We put

$D^{=1}:=\sum_{a_{i}=1}D_{i},\quad
D^{<1}:=\sum_{a_{i}<1}a_{i}D_{i},\quad\text{and}\quad
D^{>1}:=\sum_{a_{i}>1}a_{i}D_{i}.$

Let $D$ be an $\mathbb{R}$-divisor on $X$ and let $x$ be a point of $X$. If
$D$ is written as a finite $\mathbb{R}$-linear (resp. $\mathbb{Q}$-linear)
combination of Cartier divisors on some open neighborhood of $x$, then $D$ is
said to be $\mathbb{R}$-Cartier at $x$ (resp. $\mathbb{Q}$-Cartier at $x$). If
$D$ is $\mathbb{R}$-Cartier (resp. $\mathbb{Q}$-Cartier) at $x$ for every
$x\in X$, then $D$ is said to be $\mathbb{R}$-Cartier (resp.
$\mathbb{Q}$-Cartier). More generally, for any subset $L$ of $X$, if $D$ is
$\mathbb{R}$-Cartier (resp. $\mathbb{Q}$-Cartier) at $x$ for every $x\in L$,
then $D$ is said to be $\mathbb{R}$-Cartier (resp. $\mathbb{Q}$-Cartier) at
$L$. Note that a $\mathbb{Q}$-Cartier $\mathbb{R}$-divisor $D$ is
automatically a $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor by definition. If
$D$ is a finite $\mathbb{R}$-linear (resp. $\mathbb{Q}$-linear) combination of
Cartier divisors on $X$, then we sometimes say that $D$ is a globally
$\mathbb{R}$-Cartier $\mathbb{R}$-divisor (resp. globally $\mathbb{Q}$-Cartier
$\mathbb{Q}$-divisor).

Two $\mathbb{R}$-divisors $D_{1}$ and $D_{2}$ are said to be linearly
equivalent if $D_{1}-D_{2}$ is a principal Cartier divisor. The linear
equivalence is denoted by $D_{1}\sim D_{2}$. Two $\mathbb{R}$-divisors $D_{1}$
and $D_{2}$ are said to be $\mathbb{R}$-linearly equivalent (resp.
$\mathbb{Q}$-linearly equivalent) if $D_{1}-D_{2}$ is a finite
$\mathbb{R}$-linear (resp. $\mathbb{Q}$-linear) combination of principal
Cartier divisors. When $D_{1}$ is $\mathbb{R}$-linearly (resp.
$\mathbb{Q}$-linearly) equivalent to $D_{2}$, we write
$D_{1}\sim_{\mathbb{R}}D_{2}$ (resp. $D_{1}\sim_{\mathbb{Q}}D_{2}$).

######  Example 2.7.

Let $X$ be a non-compact Riemann surface and let $\\{P_{k}\\}_{k=1}^{\infty}$
be a set of mutually distinct discrete points of $X$. We put
$D:=\sum_{k=1}^{\infty}\frac{1}{k}P_{k}$. Then $D$ is obviously a
$\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor on $X$. However, $D$ is not a finite
$\mathbb{Q}$-linear combination of Cartier divisors on $X$.

We note that in this paper we can almost always assume that
$\operatorname{Supp}D$ has only finitely many irreducible components.

######  2.8Hybrids of $\mathbb{R}$-line bundles and $\mathbb{R}$-Cartier
divisors.

In this paper, we usually treat hybrids of $\mathbb{R}$-line bundles and
$\mathbb{R}$-Cartier divisors.

Let $\pi\colon X\to Y$ be a projective morphism between complex analytic
spaces and let $W$ be a compact subset of $Y$. Let $A$ and $B$ be
$\mathbb{R}$-Cartier divisors on $X$ and let $\mathcal{L}$ and $\mathcal{M}$
be $\mathbb{R}$-line bundles on $X$.

We sometimes say that

$\mathcal{L}+A\sim_{\mathbb{R}}\mathcal{M}+B$

holds over some open neighborhood $U$ of $W$. This means:

  * (i)

We implicitly assume that $A|_{\pi^{-1}(U)}$ and $B|_{\pi^{-1}(U)}$ are finite
$\mathbb{R}$-linear combinations of Cartier divisors on $\pi^{-1}(U)$. Thus we
can obtain $\mathbb{R}$-line bundles $\mathcal{A}$ and $\mathcal{B}$ naturally
associated to $A|_{\pi^{-1}(U)}$ and $B|_{\pi^{-1}(U)}$, respectively.

  * (ii)

In $\operatorname{Pic}(\pi^{-1}(U))\otimes_{\mathbb{Z}}\mathbb{R}$, the
following equality

$\mathcal{L}|_{\pi^{-1}(U)}+\mathcal{A}=\mathcal{M}|_{\pi^{-1}(U)}+\mathcal{B}$

holds.

If $X$ is a normal complex variety and $U$ is a relatively compact open subset
of $Y$, then $A|_{\pi^{-1}(U)}$ and $B|_{\pi^{-1}(U)}$ are automatically
finite $\mathbb{R}$-linear combinations of Cartier divisors on $\pi^{-1}(U)$.
Therefore, (i) is harmless for applications.

Similarly, we say that $\mathcal{L}+A$ is $\pi$-ample over some open
neighborhood $U$ of $W$ if $A|_{\pi^{-1}(U)}$ is a finite $\mathbb{R}$-linear
combination of Cartier divisors on $\pi^{-1}(U)$, $\mathcal{A}$ is the
$\mathbb{R}$-line bundle naturally associated to $A|_{\pi^{-1}(U)}$, and
$\mathcal{L}|_{\pi^{-1}(U)}+\mathcal{A}$ is $\pi$-ample over $U$, that is,
$\mathcal{L}|_{\pi^{-1}(U)}+\mathcal{A}$ is a finite positive
$\mathbb{R}$-linear combination of $\pi$-ample line bundles on $\pi^{-1}(U)$.

######  2.9.

Let $\pi\colon X\to Y$ be a projective morphism of complex analytic spaces
such that $X$ is a normal complex variety and let $\Delta$ be an
$\mathbb{R}$-divisor on $X$ such that $K_{X}+\Delta$ is $\mathbb{R}$-Cartier.
Let $y$ be an arbitrary point of $Y$ and let $U_{y}$ be any relatively compact
Stein open neighborhood of $y\in Y$. In this case, we can always find a Weil
divisor $K_{\pi^{-1}(U_{y})}$ on $\pi^{-1}(U_{y})$ such that
$\mathcal{O}_{\pi^{-1}(U_{y})}(K_{\pi^{-1}(U_{y})})\simeq\omega_{\pi^{-1}(U_{y})}$
holds since $\pi$ is projective and $U_{y}$ is Stein. Since $U_{y}$ is
relatively compact, $\operatorname{Supp}\Delta|_{\pi^{-1}(U_{y})}$ has only
finitely many irreducible components. Thus, we can easily see that
$K_{\pi^{-1}(U_{y})}+\Delta|_{\pi^{-1}(U_{y})}$ is a globally
$\mathbb{R}$-Cartier $\mathbb{R}$-divisor on $\pi^{-1}(U_{y})$. Moreover, for
any $\mathbb{R}$-line bundle $\mathcal{L}$ on $X$, we can take a globally
$\mathbb{R}$-Cartier $\mathbb{R}$-divisor $L$ on $\pi^{-1}(U_{y})$ such that
$\mathcal{L}|_{\pi^{-1}(U_{y})}$ is the $\mathbb{R}$-line bundle naturally
associated to $L$.

In the theory of minimal models, we often use the following formulation. We
will repeatedly use it in subsequent sections.

######  2.10.

Let $X$ be a normal complex variety. A real vector space spanned by the prime
divisors on $X$ is denoted by $\operatorname{WDiv}_{\mathbb{R}}(X)$, which has
a canonical basis given by the prime divisors. Let $D$ be an element of
$\operatorname{WDiv}_{\mathbb{R}}(X)$. Then the sup norm of $D$ with respect
to this basis is denoted by $|\\!|D|\\!|$. Note that an $\mathbb{R}$-divisor
$D$ on $X$ is an element of $\operatorname{WDiv}_{\mathbb{R}}(X)$ if and only
if $\operatorname{Supp}D$ has only finitely many irreducible components.

Let $V$ be a finite-dimensional affine subspace of
$\operatorname{WDiv}_{\mathbb{R}}(X)$, which is defined over the rationals. We
put

$\mathcal{R}(V;x):=\\{\Delta\in V\mid{\text{$K_{X}+\Delta$ is
$\mathbb{R}$-Cartier at $x$}}\\}.$

It is obvious that $\mathcal{R}(V;x)$ is an affine subspace of $V$. We take an
arbitrary element $\Delta$ of $\mathcal{R}(V;x)$. Then $K_{X}+\Delta$ is
$\mathbb{R}$-Cartier at $x$ by definition. Therefore, there exist a small open
neighborhood $U_{x}$ of $x$ such that

$K_{U_{x}}+\Delta|_{U_{x}}=\sum_{i=1}^{k}a_{i}D_{i},$

where $D_{i}$ is a Cartier divisor on $U_{x}$ and $a_{i}$ is a real number for
every $i$. By this description, we can easily see that there exists an affine
subspace $\mathcal{T}$ of $V$ defined over the rationals such that
$\Delta\in\mathcal{T}\subset\mathcal{R}(V;x)$. Hence $\mathcal{R}(V;x)$ itself
is an affine subspace of $V$ defined over the rationals. Let $L$ be a compact
subset of $X$. We put

$\mathcal{R}(V;L):=\\{\Delta\in V\mid{\text{$K_{X}+\Delta$ is
$\mathbb{R}$-Cartier at $L$}}\\}.$

Then the following equality

$\mathcal{R}(V;L)=\bigcap_{x\in L}\mathcal{R}(V;x)$

obviously holds. Therefore, $\mathcal{R}(V;L)$ is an affine subspace of $V$
defined over the rationals. After shrinking $X$ around $L$ suitably, we may
assume that $K_{X}+\Delta$ is $\mathbb{R}$-Cartier for every
$\Delta\in\mathcal{R}(V;L)$ since $V$ is finite-dimensional and $L$ is
compact. Let $\Theta$ be the union of the support of any element of
$\mathcal{R}(V;L)$. By [BM2, Theorem 13.2], after shrinking $X$ around $L$
suitably, we can construct a projective bimeromorphic morphism $f\colon Y\to
X$ from a smooth complex analytic space $Y$ such that $\operatorname{Exc}(f)$
and $\operatorname{Exc}(f)\cup\operatorname{Supp}f^{-1}_{*}\Theta$ are simple
normal crossing divisors on $Y$, where $\operatorname{Exc}(f)$ denotes the
exceptional locus of $f\colon Y\to X$. Thus, for any
$\Delta\in\mathcal{R}(V;L)$, we can write

$K_{Y}+\Delta_{Y}:=f^{*}(K_{X}+\Delta)$

such that $\operatorname{Supp}\Delta_{Y}$ is a simple normal crossing divisor
on $Y$. In this situation, $(X,\Delta)$ is log canonical at $L$ if and only if
$\Delta$ is effective at $L$ and the coefficients of $\Delta_{Y}$ are less
than or equal to one over $L$. Hence, we can easily check that

$\mathcal{L}(V;L):=\\{\Delta\in V\,|\,\text{$K_{X}+\Delta$ is log canonical at
$L$}\\}$

is a rational polytope contained in $\mathcal{R}(V;L)$. We can also check that
there exists an open neighborhood $U$ of $L$ such that $(U,\Delta|_{U})$ is
log canonical for every $\Delta\in\mathcal{L}(V;L)$.

######  2.11.

Let $X$ be a complex analytic space. An analytic subset (resp. A locally
closed analytic subset) of $X$ is the support of a closed analytic subspace
(resp. a locally closed analytic subspace) of $X$. A Zariski open subset of
$X$ means the complement of an analytic subset. We note the following easy
example.

######  Example 2.12.

We consider $\Delta:=\\{z\in\mathbb{C}\mid|z|<1\\}$ and
$\Delta^{*}:=\Delta\setminus\\{0\\}$. Then $\Delta^{*}$ is a Zariski open
subset of $\Delta$. We put

$U:=\Delta^{*}\setminus\left\\{\frac{1}{n}\,\middle|\,{\text{$n\in\mathbb{Z}$
with $n\geq 2$}}\right\\}.$

Then $U$ is a Zariski open subset of $\Delta^{*}$ since

$\left\\{\frac{1}{n}\,\middle|\,{\text{$n\in\mathbb{Z}$ with $n\geq
2$}}\right\\}$

is a closed analytic subset of $\Delta^{*}$. However, $U$ is not a Zariski
open subset of $\Delta$. This is because

$\\{0\\}\cup\left\\{\frac{1}{n}\,\middle|\,{\text{$n\in\mathbb{Z}$ with $n\geq
2$}}\right\\}$

is not a closed analytic subset of $\Delta$.

######  2.13.

A subset $\mathcal{S}$ of a complex analytic space $X$ is said to be
analytically meagre if

$\mathcal{S}\subset\bigcup_{n\in\mathbb{N}}Y_{n},$

where each $Y_{n}$ is a locally closed analytic subset of $X$ of codimension
$\geq 1$.

Let $X$ be a complex analytic space. We say that a property $P$ holds for an
analytically sufficiently general point $x\in X$ when $P$ holds for every
point $x$ contained in $X\setminus\mathcal{S}$ for some analytically meagre
subset $\mathcal{S}$ of $X$.

Let $\pi\colon X\to Y$ be a morphism of analytic spaces. Similarly, we say
that a property $P$ holds for an analytically sufficiently general fiber of
$\pi\colon X\to Y$ when $P$ holds for $\pi^{-1}(y)$ for every $y\in
Y\setminus\mathcal{S}$, where $\mathcal{S}$ is some analytically meagre subset
of $Y$.

In this paper, we will freely use the following facts, which can be found in
[BS, Chapter III].

######  2.14.

Let $\pi\colon X\to Y$ be a projective surjective morphism of complex analytic
spaces and let $\mathcal{L}$ be a line bundle on $X$. If
$R^{p}\pi_{*}\mathcal{L}=0$ holds, then $H^{p}(F,\mathcal{L}|_{F})=0$ for an
analytically sufficiently general fiber $F$ of $\pi\colon X\to Y$. If
$H^{0}(F,\mathcal{L}|_{F})\neq 0$ for an analytically sufficiently general
fiber $F$ of $\pi\colon X\to Y$, then $\pi_{*}\mathcal{L}\neq 0$ holds.

We will use the following convention throughout this paper.

######  2.15.

The expression ‘… for every $m\gg 0$’ means that ‘there exists a positive real
number $m_{0}$ such that … for every $m\geq m_{0}$.’

##  3\. Basic properties of relatively ample and relatively nef
$\mathbb{R}$-line bundles

In this section, we will collect some basic properties of relatively nef and
relatively ample $\mathbb{R}$-line bundles for the reader’s convenience. We
will frequently use them in subsequent sections.

Let us recall the definition of projective morphisms of complex analytic
spaces for the sake of completeness.

######  Definition 3.1 (Projective morphisms of complex analytic spaces).

Let $\pi\colon X\to Y$ be a proper morphism of complex analytic spaces and let
$\mathcal{L}$ be a line bundle on $X$. Then $\mathcal{L}$ is said to be
$\pi$-very ample or relatively very ample over $Y$ if $\mathcal{L}$ is
$\pi$-free, that is,

$\pi^{*}\pi_{*}\mathcal{L}\to\mathcal{L}$

is surjective, and the induced morphism

$X\to\mathbb{P}_{Y}(\pi_{*}\mathcal{L})$

over $Y$ is a closed embedding. A line bundle $\mathcal{L}$ on $X$ is called
$\pi$-ample or ample over $Y$ if for any point $y\in Y$ there are an open
neighborhood $U$ of $y$ and a positive integer $m$ such that
$\mathcal{L}^{\otimes m}|_{\pi^{-1}(U)}$ is relatively very ample over $U$.
Let $D$ be a Cartier divisor on $X$. Then we say that $D$ is $\pi$-very ample,
$\pi$-free, and $\pi$-ample if the line bundle $\mathcal{O}_{X}(D)$ is so,
respectively. We note that $\pi\colon X\to Y$ is said to be projective when
there exists a $\pi$-ample line bundle on $X$.

For the basic properties of $\pi$-ample line bundles, see [BS, Chapter IV] and
[Na2, Chapter II. §1.c. Ample line bundles]. Since we are mainly interested in
$\mathbb{R}$-line bundles in this paper, the following easy lemma is
indispensable.

######  Lemma 3.2.

Let $\pi\colon X\to Y$ be a projective morphism between complex analytic
spaces and let $W$ be a compact subset of $Y$. Let $\mathcal{L}$ be an
$\mathbb{R}$-line bundle on $X$. Then the following two conditions are
equivalent.

  * (i)

$\mathcal{L}$ is $\pi$-ample over $W$, that is, $\mathcal{L}|_{\pi^{-1}(w)}$
is ample in the usual sense for every $w\in W$.

  * (ii)

$\mathcal{L}$ is $\pi$-ample over some open neighborhood $U$ of $W$, that is,
$\mathcal{L}|_{\pi^{-1}(U)}$ is a finite positive $\mathbb{R}$-linear
combination of $\pi|_{\pi^{-1}(U)}$-ample line bundles.

###### Sketch of Proof of Lemma 3.2.

It is obvious that (i) follows from (ii). Hence it is sufficient to prove that
(ii) follows from (i). It is an easy exercise to modify the proof of [FM2,
Lemmas 6.1 and 6.2] suitably with the aid of [Na1, Proposition 1.4]. ∎

Throughout this paper, we will freely use Lemma 3.2 without mentioning it
explicitly. The following lemma is more or less well known to the experts. We
describe it here for the sake of completeness.

######  Lemma 3.3.

Let $\pi\colon X\to Y$ be a projective surjective morphism of complex analytic
spaces such that $X$ and $Y$ are both irreducible. Let $\mathcal{L}$ be a line
bundle on $X$. Assume that $\mathcal{L}|_{\pi^{-1}(y)}$ is ample for some
$y\in Y$. Then there exists a Zariski open neighborhood $U$ of $y$ in $Y$ and
a positive integer $m$ such that $\mathcal{L}^{\otimes m}|_{\pi^{-1}(U)}$ is
$\pi$-very ample over $U$. In particular, $\mathcal{L}|_{\pi^{-1}(U)}$ is
$\pi$-ample over $U$.

###### Proof.

It is well known that there exists a small open neighborhood $U_{1}$ of $y$ in
$Y$ such that $\mathcal{L}|_{\pi^{-1}(U_{1})}$ is $\pi$-ample over $U_{1}$
(see [Na1, Proposition 1.4]). Therefore, we can take some positive integer $m$
such that $\mathcal{L}^{\otimes m}$ is $\pi$-very ample over some small open
neighborhood $U_{2}$ of $y$ in $Y$. We consider
$\pi^{*}\pi_{*}\mathcal{L}^{\otimes m}\to\mathcal{L}^{\otimes m}$. It is
obviously surjective over $U_{2}$. Therefore,

$\pi\left(\operatorname{Supp}\operatorname{Coker}(\pi^{*}\pi_{*}\mathcal{L}^{\otimes
m}\to\mathcal{L}^{\otimes m})\right)\cap U_{2}=\emptyset.$

Then we put

$U_{3}:=Y\setminus\pi\left(\operatorname{Supp}\operatorname{Coker}(\pi^{*}\pi_{*}\mathcal{L}^{\otimes
m}\to\mathcal{L}^{\otimes m})\right).$

Hence $U_{3}$ is a non-empty Zariski open subset of $Y$ such that $y\in U_{3}$
and that $\pi^{*}\pi_{*}\mathcal{L}^{\otimes m}\to\mathcal{L}^{\otimes m}$ is
surjective over $U_{3}$. We put

$\mathcal{I}:=\operatorname{Im}\left(\pi^{*}\pi_{*}\mathcal{L}^{\otimes
m}\to\mathcal{L}^{\otimes
m}\right)\otimes\mathcal{L}^{\otimes(-m)}\subset\mathcal{O}_{X}.$

Then $\mathcal{I}$ is a coherent ideal sheaf on $X$. We take the blow-up
$p\colon Z\to X$ of $X$ along the ideal sheaf $\mathcal{I}$, that is, $p\colon
Z:=\operatorname{Projan}_{X}\bigoplus_{d=0}^{\infty}\mathcal{I}^{d}\to X$. By
construction,

$\mathcal{M}:=\operatorname{Im}\left(p^{*}\pi^{*}\pi_{*}\mathcal{L}^{\otimes
m}\to p^{*}\mathcal{L}^{\otimes m}\right)$

becomes a line bundle on $Z$. This gives a closed embedding

$Z\simeq\mathbb{P}_{Z}(\mathcal{M})\hookrightarrow\mathbb{P}_{Y}(\pi_{*}\mathcal{L}^{\otimes
m})\times_{Y}Z.$

Thus we obtain a morphism $\alpha\colon
Z\to\mathbb{P}_{Y}(\pi_{*}\mathcal{L}^{\otimes m})$ over $Y$. By construction
again, $p$ is an isomorphism over $U_{3}$ and $\alpha$ is a closed embedding
over $U_{2}$. We can take a non-empty Zariski open subset $V$ of $\alpha(Z)$
such that $\alpha$ is flat over $V$. Without loss of generality, we may assume
that $V$ contains $q^{-1}(U_{2})$, where $q\colon\alpha(Z)\to Y$, and that
$\alpha$ is an isomorphism over $V$.

$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\scriptstyle{\alpha}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\circ
p^{-1}}$$\scriptstyle{\pi}$$\textstyle{\alpha(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{Y}$

We put

$U:=U_{3}\cap\left(Y\setminus q\left(\alpha(Z)\setminus V\right)\right).$

Then $U$ is a non-empty Zariski open subset of $Y$ such that $y\in U$ and
$\alpha\circ p^{-1}\colon
X\dashrightarrow\mathbb{P}_{Y}(\pi_{*}\mathcal{L}^{\otimes m})$ is a closed
embedding over $U$. Therefore, $\mathcal{L}^{\otimes m}$ is $\pi$-very ample
over $U$. ∎

As an application of Lemma 3.3, we have:

######  Lemma 3.4.

Let $\pi\colon X\to Y$ be a projective surjective morphism of complex analytic
spaces. Let $\mathcal{L}$ be an $\mathbb{R}$-line bundle on $X$. Assume that
$\mathcal{L}|_{\pi^{-1}(y)}$ is ample for some $y\in Y$. Then there exists a
Zariski open neighborhood $U$ of $y$ in $Y$ such that
$\mathcal{L}|_{\pi^{-1}(U)}$ is $\pi$-ample over $U$.

In the theory of minimal models, we have to treat $\mathbb{R}$-line bundles.
Therefore, Lemma 3.4 is indispensable. Since we can not directly apply
geometric arguments to $\mathbb{R}$-line bundles, Lemma 3.4 is not so obvious.

###### Proof of Lemma 3.4.

We can write $\mathcal{L}=\sum_{i\in I}a_{i}\mathcal{L}_{i}$ in
$\operatorname{Pic}(X)\otimes_{\mathbb{Z}}\mathbb{R}$ such that $a_{i}$ is a
positive real number, $\mathcal{L}_{i}\in\operatorname{Pic}(X)$, and
$\mathcal{L}_{i}|_{\pi^{-1}(y)}$ is ample for every $i\in I$. Let
$X=\bigcup_{j\in J}X_{j}$ be the irreducible decomposition. We put

$J_{1}:=\\{j\in J\,|\,y\in\pi(X_{j})\\}\quad\text{and}\quad J_{2}:=\\{j\in
J\,|\,y\not\in\pi(X_{j})\\}.$

We take an irreducible component $X_{j}$ of $X$ with $j\in J_{1}$. By applying
Lemma 3.3 to $\mathcal{L}_{i}|_{X_{j}}$, we can find a Zariski closed subset
$\Sigma_{j}$ of $\pi(X_{j})$ such that $y\in\pi(X_{j})\setminus\Sigma_{j}$,
$\mathcal{L}_{i}|_{X_{j}}$ is ample over $\pi(X_{j})\setminus\Sigma_{j}$ for
every $i\in I$. This implies that $\mathcal{L}|_{X_{j}}$ is ample over
$\pi(X_{j})\setminus\Sigma_{j}$. We put

$\Sigma:=\left(\bigcup_{i\in J_{1}}\Sigma_{j}\right)\cup\pi\left(\bigcup_{j\in
J_{2}}X_{j}\right).$

Then $\Sigma$ is a Zariski closed subset of $Y$ such that $y\in
Y\setminus\Sigma$ and that $\mathcal{L}$ is $\pi$-ample over
$Y\setminus\Sigma$. Therefore, $U:=Y\setminus\Sigma$ is a desired Zariski open
neighborhood of $y$ in $Y$. ∎

By Lemma 3.4, we can easily obtain:

######  Lemma 3.5.

Let $\pi\colon X\to Y$ be a projective surjective morphism of complex analytic
spaces. Let $\mathcal{L}$ be an $\mathbb{R}$-line bundle on $X$. Assume that
$\mathcal{L}|_{\pi^{-1}(y_{0})}$ is nef for some $y_{0}\in Y$. Then there
exists an analytically meagre subset $\mathcal{S}$ such that
$\mathcal{L}|_{\pi^{-1}(y)}$ is nef for every $y\in Y\setminus\mathcal{S}$.

Although Lemma 3.5 is easy, it will play a very important role in our
framework of the minimal model program of complex analytic spaces. We note
that we can not make $\mathcal{S}$ a Zariski closed subset of $Y$ in Lemma
3.5.

###### Proof of Lemma 3.5.

We take a $\pi$-ample line bundle $\mathcal{H}$ on $X$. Then
$(m\mathcal{L}+\mathcal{H})|_{\pi^{-1}(y_{0})}$ is ample for every positive
integer $m$. Therefore, by Lemma 3.4, for each $m\in\mathbb{Z}_{>0}$, we can
take a Zariski open neighborhood $U_{m}$ of $y_{0}$ in $Y$ such that
$m\mathcal{L}+\mathcal{H}$ is $\pi$-ample over $U_{m}$. We put
$\mathcal{S}:=\bigcup_{m\in\mathbb{Z}_{>0}}(Y\setminus U_{m})$. Then
$(m\mathcal{L}+\mathcal{H})|_{\pi^{-1}(y)}$ is ample for every
$m\in\mathbb{Z}_{>0}$ and every $y\in Y\setminus\mathcal{S}$. This means that
$\mathcal{L}|_{\pi^{-1}(y)}$ is nef for every $y\in Y\setminus\mathcal{S}$. ∎

The following obvious corollary of Lemma 3.5 is also useful for geometric
applications. We note that we sometimes have to treat a countably infinite set
of line bundles.

######  Corollary 3.6.

Let $\pi\colon X\to Y$ be a projective surjective morphism of complex analytic
spaces. Let $\mathcal{L}_{i}$ be an $\mathbb{R}$-line bundle on $X$ for
$i\in\mathbb{N}$. Assume that $\mathcal{L}_{i}|_{\pi^{-1}(y_{0})}$ is nef for
some $y_{0}\in Y$ and for every $i\in\mathbb{N}$. Then there exists an
analytically meagre subset $\mathcal{S}$ such that
$\mathcal{L}_{i}|_{\pi^{-1}(y)}$ is nef for every $y\in Y\setminus\mathcal{S}$
and every $i\in\mathbb{N}$. Therefore, if $\mathcal{H}$ is any $\pi$-ample
$\mathbb{R}$-line bundle on $X$, then
$(\mathcal{H}+\mathcal{L}_{i})|_{\pi^{-1}(y)}$ is ample for every $y\in
Y\setminus\mathcal{S}$ and every $i\in\mathbb{N}$.

###### Proof.

By Lemma 3.5, for each $i\in\mathbb{N}$, we can find an analytically meagre
subset $\mathcal{S}_{i}$ of $Y$ such that $\mathcal{L}_{i}|_{\pi^{-1}(y)}$ is
nef for every $y\in Y\setminus\mathcal{S}_{i}$. We put
$\mathcal{S}:=\bigcup_{i\in\mathbb{N}}\mathcal{S}_{i}$. Then it is easy to see
that $\mathcal{S}$ is a desired analytically meagre subset of $Y$. ∎

By the proof of Lemma 3.5 and Corollary 3.6, we have:

######  Remark 3.7.

In Lemma 3.5 and Corollary 3.6, we can make $Y\setminus\mathcal{S}$ a
countable intersection of non-empty Zariski open subsets of $Y$.

##  4\. Non-lc ideal sheaves

Let us recall the notion of non-lc ideal sheaves. It is well defined even in
the complex analytic setting.

######  Definition 4.1 (Non-lc ideal sheaves, see [Fu2, Definition 7.1]).

Let $X$ be a normal complex variety and let $\Delta$ be an effective
$\mathbb{R}$-divisor on $X$ such that $K_{X}+\Delta$ is $\mathbb{R}$-Cartier.
Let $f\colon Z\to X$ be a projective bimeromorphic morphism from a smooth
complex variety $Z$ with $K_{Z}+\Delta_{Z}:=f^{*}(K_{X}+\Delta)$ such that
$\operatorname{Supp}\Delta_{Z}$ is a simple normal crossing divisor on $Z$.
Then we put

$\mathcal{J}_{\operatorname{NLC}}(X,\Delta):=f_{*}\mathcal{O}_{Z}(\lceil-(\Delta^{<1}_{Z})\rceil-\lfloor\Delta^{>1}_{Z}\rfloor)=f_{*}\mathcal{O}_{Z}(-\lfloor\Delta_{Z}\rfloor+\Delta^{=1}_{Z})$

and call it the non-lc ideal sheaf associated to $(X,\Delta)$. We put

$\mathcal{J}(X,\Delta):=f_{*}\mathcal{O}_{Z}(-\lfloor\Delta_{Z}\rfloor).$

Then $\mathcal{J}(X,\Delta)$ is the well-known multiplier ideal sheaf
associated to $(X,\Delta)$. By definition, the following inclusion

$\mathcal{J}(X,\Delta)\subset\mathcal{J}_{\operatorname{NLC}}(X,\Delta)$

always holds. By definition again, we can easily see that the support of
$\mathcal{O}_{X}/\mathcal{J}_{\operatorname{NLC}}(X,\Delta)$ is the non-lc
locus $\operatorname{Nlc}(X,\Delta)$ of $(X,\Delta)$.

By the standard argument (see, for example, [Fu2, Lemma 7.2]), there are no
difficulties to check the following lemma.

######  Lemma 4.2.

In Definition 4.1, $\mathcal{J}_{\operatorname{NLC}}(X,\Delta)$ and
$\mathcal{J}(X,\Delta)$ are independent of the choice of the resolution
$f\colon Z\to X$. Hence $\mathcal{J}_{\operatorname{NLC}}(X,\Delta)$ and
$\mathcal{J}(X,\Delta)$ are well-defined coherent ideal sheaves on $X$.

###### Sketch of Proof of Lemma 4.2.

Since we do not use $\mathcal{J}(X,\Delta)$ in this paper and the proof for
$\mathcal{J}(X,\Delta)$ is simpler than for
$\mathcal{J}_{\operatorname{NLC}}(Z,\Delta)$, we only treat
$\mathcal{J}_{\operatorname{NLC}}(X,\Delta)$ here. Let $f_{1}\colon Z_{1}\to
X$ and $f_{2}\colon Z_{2}\to X$ be two resolutions with
$K_{Z_{1}}+\Delta_{Z_{1}}=f^{*}_{1}(K_{X}+\Delta)$ and
$K_{Z_{2}}+\Delta_{Z_{2}}=f^{*}_{2}(K_{X}+\Delta)$ as in Definition 4.1. We
take an arbitrary point $x\in X$. It is sufficient to prove that

${f_{1}}_{*}\mathcal{O}_{Z_{1}}(-\lfloor\Delta_{Z_{1}}\rfloor+\Delta^{=1}_{Z_{1}})={f_{2}}_{*}\mathcal{O}_{Z_{2}}(-\lfloor\Delta_{Z_{2}}\rfloor+\Delta^{=1}_{Z_{2}})$

holds on some open neighborhood of $x$. Therefore, by shrinking $X$ around $x$
and taking an elimination of indeterminacy of $Z_{2}\dashrightarrow Z_{1}$, we
may further assume that $f_{2}$ decomposes as

$\textstyle{f_{2}\colon
Z_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\textstyle{X.}$

Then, by [Fu3, Proposition 6.3.1], we can directly check that
${f_{1}}_{*}\mathcal{O}_{Z_{1}}(-\lfloor\Delta_{Z_{1}}\rfloor+\Delta^{=1}_{Z_{1}})={f_{2}}_{*}\mathcal{O}_{Z_{2}}(-\lfloor\Delta_{Z_{2}}\rfloor+\Delta^{=1}_{Z_{2}})$
holds. We finish the proof. ∎

In this paper, we need the following Bertini-type theorem for
$\mathcal{J}_{\operatorname{NLC}}(X,\Delta)$.

######  Lemma 4.3 ([Fu2, Proposition 7.5]).

Let $X$ be a normal complex variety and let $\Delta$ be an effective
$\mathbb{R}$-divisor on $X$ such that $K_{X}+\Delta$ is $\mathbb{R}$-Cartier.
Let $\Lambda$ ($\simeq\mathbb{P}^{N}$) be a finite-dimensional linear system
on $X$. Let $X^{\dagger}$ be any relatively compact open subset of $X$. Then
there exists an analytically meagre subset $\mathcal{S}$ of $\Lambda$ such
that

$\mathcal{J}_{\operatorname{NLC}}(X^{\dagger},\Delta+tD)=\mathcal{J}_{\operatorname{NLC}}(X^{\dagger},\Delta)$

holds outside the base locus $\operatorname{Bs}\Lambda$ of $\Lambda$ for every
element $D$ of $\Lambda\setminus\mathcal{S}$ and every $0\leq t\leq 1$.

###### Proof.

Without loss of generality, we can freely replace $X$ with a relatively
compact open neighborhood of $\overline{X^{\dagger}}$. Therefore, by the
desingularization theorem (see [BM2, Theorem 13.2]), we can take a projective
bimeromorphic morphism $f\colon Z\to X$ from a smooth complex variety $Z$ with
$K_{Z}+\Delta_{Z}=f^{*}(K_{X}+\Delta)$ such that
$\operatorname{Supp}\Delta_{Z}$ is a simple normal crossing divisor on $Z$. By
replacing $X$ with $X\setminus\operatorname{Bs}\Lambda$, we may further assume
that $\operatorname{Bs}\Lambda=\emptyset$. By Bertini’s theorem, there exists
an analytically meagre subset $\mathcal{S}$ of $\Lambda$ such that $f^{*}D$ is
smooth, $f^{*}D=f^{-1}_{*}D$, $f^{*}D$ and $\operatorname{Supp}\Delta_{Z}$
have no common irreducible components, and the support of
$f^{*}D+\operatorname{Supp}\Delta_{Z}$ is a simple normal crossing divisor on
$Z$ for every element $D$ of $\Lambda\setminus\mathcal{S}$. Then
$K_{Z}+\Delta_{Z}+f^{*}tD=f^{*}(K_{X}+\Delta+tD)$ holds over $X^{\dagger}$
with $f^{*}tD=tf^{-1}_{*}D$. Thus,

$\lceil-(\Delta^{<1}_{Z})\rceil-\lfloor\Delta^{>1}_{Z}\rfloor=\lceil-(\Delta_{Z}+f^{*}tD)^{<1}\rceil-\lfloor(\Delta_{Z}+f^{*}tD)^{>1}\rfloor$

holds over $X^{\dagger}$ for every $0\leq t\leq 1$ and every element $D$ of
$\Lambda\setminus\mathcal{S}$. Thus, we obtain

$\mathcal{J}_{\operatorname{NLC}}(X^{\dagger},\Delta+tD)=\mathcal{J}_{\operatorname{NLC}}(X^{\dagger},\Delta)$

by definition. This is what we wanted. ∎

We need the following lemma in order to reduce the problems for
$\mathbb{R}$-divisors to simpler problems for $\mathbb{Q}$-divisors.

######  Lemma 4.4.

Let $X$ be a normal complex variety and let $L$ be a compact subset of $X$.
Let $\Delta$ be an effective $\mathbb{R}$-divisor on $X$ such that
$K_{X}+\Delta$ is $\mathbb{R}$-Cartier at $L$. Then, after shrinking $X$
around $L$ suitably, there exist effective $\mathbb{Q}$-divisors
$\Delta_{1},\ldots,\Delta_{k}$ on $X$ and positive real numbers
$r_{1},\ldots,r_{k}$ with $\sum_{i=1}^{k}r_{i}=1$ such that $K_{X}+\Delta_{i}$
is $\mathbb{Q}$-Cartier for every $i$, $\Delta=\sum_{i=1}^{k}r_{i}\Delta_{i}$,
and
$\mathcal{J}_{\operatorname{NLC}}(X,\Delta_{i})=\mathcal{J}_{\operatorname{NLC}}(X,\Delta)$
holds for every $i$. In particular, if $(X,\Delta)$ is log canonical, then
$(X,\Delta_{i})$ is log canonical for every $i$.

###### Proof.

By shrinking $X$ around $L$ suitably, we may assume that
$\operatorname{Supp}\Delta$ has only finitely many irreducible components. Let
$\operatorname{Supp}\Delta:=\sum_{j=1}^{l}D_{j}$ be the irreducible
decomposition. We consider the $\mathbb{R}$-vector space
$V:=\bigoplus_{j=1}^{l}\mathbb{R}D_{j}$. We put

$\mathcal{R}(V;L):=\\{D\in V\,|\,{\text{$K_{X}+D$ is $\mathbb{R}$-Cartier at
$L$}}\\}.$

Then $\mathcal{R}(V;L)$ is an affine subspace of $V$ defined over the
rationals (see 2.10). By shrinking $X$ around $L$ suitably again, we may
assume that $K_{X}+D$ is $\mathbb{R}$-Cartier for every
$D\in\mathcal{R}(V;L)$. By [BM2, Theorem 13.2], we may further assume that
there exists a projective bimeromorphic morphism $f\colon Z\to X$ from a
smooth complex analytic space $Z$ such that $\operatorname{Exc}(f)$ and
$\operatorname{Exc}(f)\cup\sum_{j=1}^{l}\operatorname{Supp}f^{-1}_{*}D_{j}$
are simple normal crossing divisors on $Z$. We put

$\mathcal{S}_{\Delta}(V;L):=\left\\{D\in\mathcal{R}(V;L)\middle|\begin{array}[]{l}{\text{$a(E,X,D)=a(E,X,\Delta)$
holds for every}}\\\ {\text{divisor $E$ on $Z$ with
$a(E,X,\Delta)\in\mathbb{Q}$}}\end{array}\right\\}.$

Then $\mathcal{S}_{\Delta}(V;L)$ is an affine subspace of $V$ defined over the
rationals with $\Delta\in\mathcal{S}_{\Delta}(V;L)$. Since
$\mathcal{S}_{\Delta}(V;L)$ is defined over the rationals, we can take
effective $\mathbb{Q}$-divisors $\Delta_{1},\ldots,\Delta_{k}$ from
$\mathcal{S}_{\Delta}(V;L)$ such that they are close to $\Delta$ in
$\mathcal{S}_{\Delta}(V;L)$ and positive real numbers $r_{1},\ldots,r_{k}$
with all the desired properties. ∎

Although we do not use multiplier ideal sheaves in this paper, we note:

######  Remark 4.5.

In Lemma 4.4, we see that $\mathcal{J}(X,\Delta_{i})=\mathcal{J}(X,\Delta)$
holds for every $i$ by construction. In particular, $(X,\Delta_{i})$ is
kawamata log terminal for every $i$ if $(X,\Delta)$ is kawamata log terminal.

##  5\. Quick review of vanishing theorems

In this section, let us quickly recall the vanishing theorems established in
[Fu10] (see also [FF] and [Fu12]). Let us start with the definition of
analytic simple normal crossing pairs.

######  Definition 5.1 (Analytic simple normal crossing pairs).

Let $X$ be a simple normal crossing divisor on a smooth complex analytic space
$M$ and let $B$ be an $\mathbb{R}$-divisor on $M$ such that
$\operatorname{Supp}(B+X)$ is a simple normal crossing divisor on $M$ and that
$B$ and $X$ have no common irreducible components. Then we put $D:=B|_{X}$ and
consider the pair $(X,D)$. We call $(X,D)$ an analytic globally embedded
simple normal crossing pair and $M$ the ambient space of $(X,D)$.

If the pair $(X,D)$ is locally isomorphic to an analytic globally embedded
simple normal crossing pair at any point of $X$ and the irreducible components
of $X$ and $D$ are all smooth, then $(X,D)$ is called an analytic simple
normal crossing pair.

When $(X,D)$ is an analytic simple normal crossing pair, $X$ has an invertible
dualizing sheaf $\omega_{X}$ since it is Gorenstein. We use the symbol $K_{X}$
as a formal divisor class with an isomorphism
$\mathcal{O}_{X}(K_{X})\simeq\omega_{X}$ if there is no danger of confusion.
Note that we can not always define $K_{X}$ globally with
$\mathcal{O}_{X}(K_{X})\simeq\omega_{X}$. In general, it only exists locally
on $X$.

######  Remark 5.2.

Let $X$ be a smooth complex analytic space and let $D$ be an
$\mathbb{R}$-divisor on $X$ such that $\operatorname{Supp}D$ is a simple
normal crossing divisor on $X$. Then, by considering $M:=X\times\mathbb{C}$,
we can see $(X,D)$ as an analytic globally embedded simple normal crossing
pair.

The notion of strata, which is a generalization of that of log canonical
centers, plays a crucial role.

######  Definition 5.3 (Strata).

Let $(X,D)$ be an analytic simple normal crossing pair such that $D$ is
effective. Let $\nu\colon X^{\nu}\to X$ be the normalization. We put

$K_{X^{\nu}}+\Theta=\nu^{*}(K_{X}+D).$

This means that $\Theta$ is the union of $\nu^{-1}_{*}D$ and the inverse image
of the singular locus of $X$. If $S$ is an irreducible component of $X$ or the
$\nu$-image of some log canonical center of $(X^{\nu},\Theta)$, then $S$ is
called a stratum of $(X,D)$.

We recall Siu’s theorem on coherent analytic sheaves, which is a special case
of [Si, Theorem 4].

######  Theorem 5.4.

Let $\mathcal{F}$ be a coherent sheaf on a complex analytic space $X$. Then
there exists a locally finite family $\\{Y_{i}\\}_{i\in I}$ of complex
analytic subvarieties of $X$ such that

$\operatorname{Ass}_{\mathcal{O}_{X,x}}(\mathcal{F}_{x})=\\{\mathfrak{p}_{x,1},\ldots,\mathfrak{p}_{x,r(x)}\\}$

holds for every point $x\in X$, where
$\mathfrak{p}_{x,1},\ldots,\mathfrak{p}_{x,r(x)}$ are the prime ideals of
$\mathcal{O}_{X,x}$ associated to the irreducible components of the germs
$Y_{i,x}$ of $Y_{i}$ at $x$ with $x\in Y_{i}$. We note that each $Y_{i}$ is
called an associated subvariety of $\mathcal{F}$.

Now we are ready to state the main result of [Fu10].

######  Theorem 5.5 ([Fu10, Theorem 1.1]).

Let $(X,\Delta)$ be an analytic simple normal crossing pair such that $\Delta$
is a boundary $\mathbb{R}$-divisor on $X$. Let $f\colon X\to Y$ be a
projective morphism to a complex analytic space $Y$ and let $\mathcal{L}$ be a
line bundle on $X$. Let $q$ be an arbitrary non-negative integer. Then we have
the following properties.

  * (i)

$($Strict support condition$)$. If $\mathcal{L}-(\omega_{X}+\Delta)$ is
$f$-semiample, then every associated subvariety of $R^{q}f_{*}\mathcal{L}$ is
the $f$-image of some stratum of $(X,\Delta)$.

  * (ii)

$($Vanishing theorem$)$. If
$\mathcal{L}-(\omega_{X}+\Delta)\sim_{\mathbb{R}}f^{*}\mathcal{H}$ holds for
some $\pi$-ample $\mathbb{R}$-line bundle $\mathcal{H}$ on $Y$, where
$\pi\colon Y\to Z$ is a projective morphism to a complex analytic space $Z$,
then we have

$R^{p}\pi_{*}R^{q}f_{*}\mathcal{L}=0$

for every $p>0$.

We make a supplementary remark on Theorem 5.5.

######  Remark 5.6.

In Theorem 5.5 (and Theorem 5.7 below), we always assume that $\Delta$ is
globally $\mathbb{R}$-Cartier, that is, $\Delta$ is a finite
$\mathbb{R}$-linear combination of Cartier divisors. We note that if the
support of $\Delta$ has only finitely many irreducible components then it is
globally $\mathbb{R}$-Cartier. Since we are mainly interested in the standard
setting explained in 1.1, this assumption is harmless to geometric
applications. Under this assumption, we can obtain an $\mathbb{R}$-line bundle
$\mathcal{N}$ on $X$ naturally associated to
$\mathcal{L}-(\omega_{X}+\Delta)$. The assumption in (i) means that
$\mathcal{N}$ is a finite positive $\mathbb{R}$-linear combination of
$\pi$-semiample line bundles on $X$. The assumption in (ii) says that
$\mathcal{N}=f^{*}\mathcal{H}$ holds in
$\operatorname{Pic}(X)\otimes_{\mathbb{Z}}\mathbb{R}$.

We do not prove Theorem 5.5 here. For the details of the proof of Theorem 5.5,
see [Fu10], which depends on Saito’s theory of mixed Hodge modules (see [Sa1],
[Sa2], [Sa3], [FFS], and [Sa4]) and Takegoshi’s analytic generalization of
Kollár’s torsion-free and vanishing theorem (see [Ta]). The reader can find an
alternative approach to Theorem 5.5 without using Saito’s theory of mixed
Hodge modules in [FF]. We note that Theorem 5.5 is one of the main ingredients
of this paper. Or, we can see this paper as an application of Theorem 5.5.

###  5.1. Vanishing theorems of Reid–Fukuda type

Although we do not need vanishing theorems of Reid–Fukuda type in this paper,
we will shortly discuss them here for future usage.

######  Theorem 5.7 (Vanishing theorem of Reid–Fukuda type, see [Fu10, Theorem
1.2]).

Let $(X,\Delta)$ be an analytic simple normal crossing pair such that $\Delta$
is a boundary $\mathbb{R}$-divisor on $X$. Let $f\colon X\to Y$ and $\pi\colon
Y\to Z$ be projective morphisms between complex analytic spaces and let
$\mathcal{L}$ be a line bundle on $X$. If
$\mathcal{L}-(\omega_{X}+\Delta)\sim_{\mathbb{R}}f^{*}\mathcal{H}$ holds such
that $\mathcal{H}$ is an $\mathbb{R}$-line bundle, which is nef and log big
over $Z$ with respect to $f\colon(X,\Delta)\to Y$, on $Y$, then

$R^{p}\pi_{*}R^{q}f_{*}\mathcal{L}=0$

holds for every $p>0$ and every $q$.

Theorem 5.7 is obviously a generalization of Theorem 5.5 (ii). The reader can
find the detailed proof of Theorem 5.7 in [Fu10], which is harder than that of
Theorem 5.5 (ii). As an easy application of Theorem 5.7, we can establish the
vanishing theorem of Reid–Fukuda type for log canonical pairs in the complex
analytic setting.

######  Theorem 5.8 (Vanishing theorem of Reid–Fukuda type for log canonical
pairs).

Let $(X,\Delta)$ be a log canonical pair and let $\pi\colon X\to Y$ be a
projective morphism of complex analytic spaces. Let $L$ be a
$\mathbb{Q}$-Cartier integral Weil divisor on $X$. Assume that
$L-(K_{X}+\Delta)$ is nef and big over $Y$ and that $(L-(K_{X}+\Delta))|_{C}$
is big over $\pi(C)$ for every log canonical center $C$ of $(X,\Delta)$. Then

$R^{q}\pi_{*}\mathcal{O}_{X}(L)=0$

holds for every $q>0$.

###### Proof.

The proof of [Fu3, Theorem 5.7.6] works by Theorem 5.7. ∎

We leave the details of Theorems 5.7 and 5.8 for the interested readers.

##  6\. Vanishing theorems for normal pairs

In this section, we will prepare some vanishing theorems, which are suitable
for geometric applications. They will play a crucial role in subsequent
sections. We note that the results in this section follow from Theorem 5.5.

######  Theorem 6.1 (see [Fu2, Theorem 8.1]).

Let $\pi\colon X\to Y$ be a projective morphism of complex analytic spaces
such that $X$ is a normal complex variety and let $W$ be a compact subset of
$Y$. Let $\Delta$ be an effective $\mathbb{R}$-divisor on $X$ such that
$K_{X}+\Delta$ is $\mathbb{R}$-Cartier and let $\mathcal{L}$ be a line bundle
on $X$. We assume that $\mathcal{L}-(K_{X}+\Delta)$ is $\pi$-ample over $W$,
that is, $\left(\mathcal{L}-(K_{X}+\Delta)\right)|_{\pi^{-1}(w)}$ is ample for
every $w\in W$. Let $\\{C_{i}\\}_{i\in I}$ be any set of log canonical centers
of the pair $(X,\Delta)$. We put $V:=\bigcup_{i\in I}C_{i}$ with the reduced
structure. We further assume that $V$ is disjoint from the non-lc locus
$\operatorname{Nlc}(X,\Delta)$ of $(X,\Delta)$. Then there exists some open
neighborhood $U$ of $W$ such that

$R^{i}\pi_{*}(\mathcal{J}\otimes\mathcal{L})=0$

holds on $U$ for every $i>0$, where
$\mathcal{J}:=\mathcal{I}_{V}\cdot\mathcal{J}_{\operatorname{NLC}}(X,\Delta)\subset\mathcal{O}_{X}$
and $\mathcal{I}_{V}$ is the defining ideal sheaf of $V$ on $X$. Therefore,
the restriction map

$\pi_{*}\mathcal{L}\to\pi_{*}(\mathcal{L}|_{V})\oplus\pi_{*}(\mathcal{L}|_{\operatorname{Nlc}(X,\Delta)})$

is surjective on $U$ and

$R^{i}\pi_{*}(\mathcal{L}|_{V})=0$

holds on $U$ for every $i>0$. In particular, the restriction maps

$\pi_{*}\mathcal{L}\to\pi_{*}(\mathcal{L}|_{V})$

and

$\pi_{*}\mathcal{L}\to\pi_{*}(\mathcal{L}|_{\operatorname{Nlc}(X,\Delta)})$

are surjective on $U$.

The result and argument in Step 1 in the proof of Theorem 6.1 is the most
important part of this paper. We will use them repeatedly in subsequent
sections.

###### Proof of Theorem 6.1.

In Steps 1 and 2, we will use the strict support condition (see Theorem 5.5
(i)) and the vanishing theorem (see Theorem 5.5 (ii)), respectively. The
assumption that $\mathcal{L}-(K_{X}+\Delta)$ is $\pi$-ample over $W$ will be
used only in Step 2.

We take an arbitrary point $w\in W$. Then it is sufficient to prove the
desired vanishing theorem over some open neighborhood of $w$ by the
compactness of $W$. Therefore, we may replace $Y$ with a relatively compact
Stein open neighborhood of $w$ and may assume that
$\mathcal{L}-(K_{X}+\Delta)$ is $\pi$-ample over $Y$.

######  Step 1.

We can take a resolution of singularities $f\colon Z\to X$ of $X$ such that
$f$ is projective and that
$\operatorname{Supp}f^{-1}_{*}\Delta\cup\operatorname{Exc}(f)$ is a simple
normal crossing divisor on $Z$. We may further assme that $f^{-1}(V)$ is a
simple normal crossing divisor on $Z$. Then we can write

$K_{Z}+\Delta_{Z}=f^{*}(K_{X}+\Delta).$

Let $T$ be the union of the irreducible components of $\Delta^{=1}_{Z}$ that
are mapped into $V$ by $f$. We consider the following short exact sequence

$0\to\mathcal{O}_{Z}(A-N-T)\to\mathcal{O}_{Z}(A-N)\to\mathcal{O}_{T}(A-N)\to
0,$

where $A:=\lceil-(\Delta^{<1}_{Z})\rceil$ and
$N:=\lfloor\Delta^{>1}_{Z}\rfloor$. By definition, $A$ is an effective
$f$-exceptional divisor on $Z$. We obtain the following long exact sequence

$\begin{split}0&\to f_{*}\mathcal{O}_{Z}(A-N-T)\to
f_{*}\mathcal{O}_{Z}(A-N)\to f_{*}\mathcal{O}_{T}(A-N)\\\
&\overset{\delta}{\to}R^{1}f_{*}\mathcal{O}_{Z}(A-N-T)\to\cdots.\end{split}$

Since

$A-N-T-(K_{Z}+\\{\Delta_{Z}\\}+\Delta^{=1}_{Z}-T)=-(K_{Z}+\Delta_{Z})\sim_{\mathbb{R}}-f^{*}(K_{X}+\Delta),$

every associated subvariety of $R^{1}f_{*}\mathcal{O}_{Z}(A-N-T)$ is the
$f$-image of some stratum of $(Z,\\{\Delta_{Z}\\}+\Delta^{=1}_{Z}-T)$ by the
strict support condition in Theorem 5.5 (i). Since $f^{-1}(V)$ is a simple
normal crossing divisor, there are no strata of
$(Z,\\{\Delta_{Z}\\}+\Delta^{=1}_{Z}-T)$ that are mapped into $V$ by $f$. On
the other hand, $V=f(T)$ holds by construction. Thus, the connecting
homomorphism $\delta$ is a zero map. Hence we have a short exact sequence

(6.1) $0\to f_{*}\mathcal{O}_{Z}(A-N-T)\to f_{*}\mathcal{O}_{Z}(A-N)\to
f_{*}\mathcal{O}_{T}(A-N)\to 0.$

We put $\mathcal{J}:=f_{*}\mathcal{O}_{Z}(A-N-T)\subset\mathcal{O}_{X}$. Since
$V$ is disjoint from $\operatorname{Nlc}(X,\Delta)$ by assumption, the ideal
sheaf $\mathcal{J}$ coincides with $\mathcal{I}_{V}$ and
$\mathcal{J}_{\operatorname{NLC}}(X,\Delta)$ in a neighborhood of $V$ and
$\operatorname{Nlc}(X,\Delta)$, respectively. Therefore, we have
$\mathcal{J}=\mathcal{I}_{V}\cdot\mathcal{J}_{\operatorname{NLC}}(X,\Delta)$.
We note that if $V$ is empty then $\mathcal{I}_{V}=\mathcal{O}_{X}$ and
$\mathcal{J}=\mathcal{J}_{\operatorname{NLC}}(X,\Delta)$. We put
$X^{*}:=X\setminus\operatorname{Nlc}(X,\Delta)$ and $Z^{*}:=f^{-1}(X^{*})$. By
restricting (6.1) to $X^{*}$, we obtain

$0\to f_{*}\mathcal{O}_{Z^{*}}(A-T)\to f_{*}\mathcal{O}_{Z^{*}}(A)\to
f_{*}\mathcal{O}_{T}(A)\to 0.$

Since $f_{*}\mathcal{O}_{Z^{*}}(A)\simeq\mathcal{O}_{X^{*}}$, we have
$f_{*}\mathcal{O}_{T}(A)\simeq\mathcal{O}_{V}$. This implies that
$\mathcal{O}_{V}\simeq f_{*}\mathcal{O}_{T}$ holds.

######  Step 2.

Since

$f^{*}\mathcal{L}+A-N-T-(K_{Z}+\\{\Delta_{Z}\\}+\Delta^{=1}_{Z}-T)\sim_{\mathbb{R}}f^{*}(\mathcal{L}-(K_{X}+\Delta)),$

we have

$R^{i}\pi_{*}(\mathcal{J}\otimes\mathcal{L})\simeq
R^{i}\pi_{*}(f_{*}\mathcal{O}_{Z}(A-N-T)\otimes\mathcal{L})=0$

for every $i>0$ by the vanishing theorem in Theorem 5.5 (ii). If we put
$V=\emptyset$, then we have
$\mathcal{J}=\mathcal{J}_{\operatorname{NLC}}(X,\Delta)$. Therefore,

$R^{i}\pi_{*}\left(\mathcal{J}_{\operatorname{NLC}}(X,\Delta)\otimes\mathcal{L}\right)=0$

holds for every $i>0$ as a special case. By considering the short exact
sequence

$0\to\mathcal{J}\to\mathcal{J}_{\operatorname{NLC}}(X,\Delta)\to\mathcal{O}_{V}\to
0,$

we obtain

$\cdots\to
R^{i}\pi_{*}(\mathcal{J}_{\operatorname{NLC}}(X,\Delta)\otimes\mathcal{L})\to
R^{i}\pi_{*}(\mathcal{L}|_{V})\to
R^{i+1}\pi_{*}\mathcal{(}\mathcal{J}\otimes\mathcal{L})\to\cdots.$

Since we have already checked

$R^{i}\pi_{*}(\mathcal{J}_{\operatorname{NLC}}(X,\Delta)\otimes\mathcal{L})=R^{i}\pi_{*}(\mathcal{J}\otimes\mathcal{L})=0$

for every $i>0$, we have $R^{i}\pi_{*}(\mathcal{L}|_{V})=0$ for all $i>0$.
Finally, we consider the following short exact sequence

$0\to\mathcal{J}\to\mathcal{O}_{X}\to\mathcal{O}_{V}\oplus\mathcal{O}_{\operatorname{Nlc}(X,\Delta)}\to
0.$

By taking $\otimes\mathcal{L}$ and $R^{i}\pi_{*}$, we obtain that

$0\to\pi_{*}(\mathcal{J}\otimes\mathcal{L})\to\pi_{*}\mathcal{L}\to\pi_{*}(\mathcal{L}|_{V})\oplus\pi_{*}(\mathcal{L}|_{\operatorname{Nlc}(X,\Delta)})\to
0$

is exact.

We finish the proof. ∎

The following remark is obvious by the proof of Theorem 6.1.

######  Remark 6.2.

If $\left(\mathcal{L}-(K_{X}+\Delta)\right)|_{\pi^{-1}(y)}$ is ample for every
$y\in Y$, then the proof of Theorem 6.1 shows that Theorem 6.1 holds over $Y$.
This means that we can take $U=Y$ in Theorem 6.1.

We prepare one more vanishing theorem.

######  Theorem 6.3 (see [Fu2, Theorem 11.1]).

Let $\pi\colon X\to Y$ be a projective morphism of complex analytic spaces
such that $X$ is a normal complex variety and let $W$ be a compact subset of
$Y$. Let $\Delta$ be an effective $\mathbb{R}$-divisor on $X$ such that
$K_{X}+\Delta$ is $\mathbb{R}$-Cartier. Let $\\{C_{i}\\}_{i\in I}$ be any set
of log canonical centers of the pair $(X,\Delta)$. We put $V:=\bigcup_{i\in
I}C_{i}$ with the reduced structure. We assume that $V$ is disjoint from the
non-lc locus $\operatorname{Nlc}(X,\Delta)$ of $(X,\Delta)$. Let $\mathcal{M}$
be a line bundle on $V$ such that $\mathcal{M}-(K_{X}+\Delta)|_{V}$ is
$\pi$-ample over $W$. Then there exists some open neighborhood $U$ of $W$ such
that $R^{i}\pi_{*}\mathcal{M}=0$ holds on $U$ for every $i>0$.

###### Proof.

As in Theorem 6.1, it is sufficient to prove the desired vanishing theorem for
some open neighborhood of any point $w\in W$. We will use the same notation as
in the proof of Theorem 6.1. We note that

$A-N-(K_{Z}+\\{\Delta_{Z}\\}+\Delta^{=1}_{Z})\sim_{\mathbb{R}}-f^{*}(K_{X}+\Delta)$

holds. We put $f_{T}:=f|_{T}\colon T\to V$. Then

$f^{*}_{T}\mathcal{M}+A|_{T}-(K_{T}+(\\{\Delta_{Z}\\}+\Delta^{=1}_{Z}-T)|_{T})\sim_{\mathbb{R}}f^{*}_{T}(\mathcal{M}-(K_{X}+\Delta)|_{V})$

holds. Note that $(T,(\\{\Delta_{Z}\\}+\Delta^{=1}_{Z}-T)|_{T})$ is an
analytic globally embedded simple normal crossing pair. Thus, by the vanishing
theorem in Theorem 5.5 (ii),

$R^{i}\pi_{*}\mathcal{M}\simeq
R^{i}\pi_{*}\left(\mathcal{M}\otimes(f_{T})_{*}\mathcal{O}_{T}(A|_{T})\right)=0$

for every $i>0$. Here, we used the following isomorphism
$(f|_{T})_{*}\mathcal{O}_{T}(A|_{T})\simeq\mathcal{O}_{V}$ obtained in Step 1
in the proof of Theorem 6.1. ∎

We make two remarks on Theorem 6.3.

######  Remark 6.4.

If $\left(\mathcal{M}-(K_{X}+\Delta)|_{V}\right)|_{\pi^{-1}_{V}(y)}$ is ample
for every $y\in\pi(V)$, where $\pi_{V}:=\pi|_{V}$, then Theorem 6.3 holds true
over $Y$, that is, we can take $U=Y$ in Theorem 6.3. We can check it by the
proof of Theorems 6.1 and 6.3.

######  Remark 6.5.

In [Fu2, Theorem 11.1], $V$ is assumed to be a minimal log canonical center of
$(X,\Delta)$ which is disjoint from $\operatorname{Nlc}(X,\Delta)$. Moreover,
the proof of [Fu2, Theorem 11.1] depends on [BCHM]. For the details, see [Fu2,
Remark 11.2].

##  7\. On log canonical centers

The main purpose of this section is to prove the following very fundamental
theorem on log canonical centers, which is an easy application of Theorem 6.1
and its proof. It will play an important role in this paper.

######  Theorem 7.1 (Basic properties of log canonical centers).

Let $(X,\Delta)$ be a log canonical pair. Then the following properties hold.

  * (1)

The number of log canonical centers of $(X,\Delta)$ is locally finite.

  * (2)

The intersection of two log canonical centers is a union of some log canonical
centers.

  * (3)

Let $x\in X$ be any point such that $(X,\Delta)$ is log canonical but is not
kawamata log terminal at $x$. Then there exists a unique minimal (with respect
to the inclusion) log canonical center $C_{x}$ passing through $x$. Moreover,
$C_{x}$ is normal at $x$.

###### Proof.

We note that (1) is almost obvious by definition. We take an arbitrary point
$x\in X$ and shrink $X$ around $x$ suitably. Then we may assume that there
exists a projective bimeromorphic morphism $f\colon Y\to X$ from a smooth
complex analytic space $Y$ such that $K_{Y}+\Delta_{Y}:=f^{*}(K_{X}+\Delta)$,
$\operatorname{Supp}\Delta_{Y}$ is a simple normal crossing divisor on $Y$,
and $\operatorname{Supp}\Delta_{Y}$ has only finitely many irreducible
components (see [BM2, Theorem 13.2]). Let $\Delta^{=1}_{Y}:=\sum_{i\in
I}\Delta_{i}$ be the irreducible decomposition. Then $C$ is a log canonical
center of $(X,\Delta)$ if and only if $C=f(S)$, where $S$ is an irreducible
component of $\Delta_{i_{1}}\cap\cdots\cap\Delta_{i_{k}}$ for some
$\\{i_{1},\ldots,i_{k}\\}\subset I$. Therefore, there exists only finitely
many log canonical centers on some open neighborhood of $x$. Thus we obtain
(1).

From now on, we will use the same notation as in the proof of Theorem 6.1 with
$Y=X$. Let $C_{1}$ and $C_{2}$ be two log canonical centers of $(X,\Delta)$.
We fix a closed point $P\in C_{1}\cap C_{2}$. We replace $X$ with a relatively
compact Stein open neighborhood of $P\in X$ and apply the argument in the
proof of Theorem 6.1. For the proof of (2), it is enough to find a log
canonical center $C$ such that $P\in C\subset C_{1}\cap C_{2}$. We put
$V:=C_{1}\cup C_{2}$. By Step 1 in the proof of Theorem 6.1, we obtain
$f_{*}\mathcal{O}_{T}\simeq\mathcal{O}_{V}$. This means that $f\colon T\to V$
has connected fibers. We note that $T$ is a simple normal crossing divisor on
$Z$. Thus, there exist irreducible components $T_{1}$ and $T_{2}$ of $T$ such
that $T_{1}\cap T_{2}\cap f^{-1}(P)\neq\emptyset$ and that $f(T_{i})\subset
C_{i}$ for $i=1,2$. Therefore, we can find a log canonical center $C$ with
$P\in C\subset C_{1}\cap C_{2}$. We finish the proof of (2). Finally, we will
prove (3). The existence and the uniqueness of the minimal log canonical
center follow from (2). We take the unique minimal log canonical center
$C=C_{x}$ passing through $x$. We put $V:=C$. We may replace $X$ with a
relatively compact Stein open neighborhood of $x\in X$. Then, by Step 1 in the
proof of Theorem 6.1, we have $f_{*}\mathcal{O}_{T}\simeq\mathcal{O}_{V}$. By
shrinking $V$ around $x$, we can assume that every stratum of $T$ dominates
$V$. Let $\nu\colon V^{\nu}\to V$ be the normalization of $V$. By applying
Hironaka’s flattening theorem (see [Hi]) to the graph of $T\dashrightarrow
V^{\nu}$ and then using the desingularization theorem (see [BM2, Theorems 13.3
and 12.4]), we can obtain the following commutative diagram:

$\textstyle{T^{\dagger}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{p}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{V^{\nu}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\textstyle{V,}$

where $p\colon T^{\dagger}\to T$ is a projective bimeromorphic morphism such
that $T^{\dagger}$ is simple normal crossing with
$p_{*}\mathcal{O}_{T^{\dagger}}\simeq\mathcal{O}_{T}$ (see [Fu10, Lemma
2.15]). Hence

$\mathcal{O}_{V}\hookrightarrow\nu_{*}\mathcal{O}_{V^{\nu}}\hookrightarrow\nu_{*}q_{*}\mathcal{O}_{T^{\dagger}}\simeq
f_{*}p_{*}\mathcal{O}_{T^{\dagger}}\simeq
f_{*}\mathcal{O}_{T}\simeq\mathcal{O}_{V}.$

This implies that $\mathcal{O}_{V}\simeq\nu_{*}\mathcal{O}_{V^{\nu}}$ holds,
that is, $V$ is normal. Thus we obtain (3). ∎

By the above proof of Theorem 7.1, we see that Theorem 7.1 (2) and (3) are
consequences of the strict support condition in Theorem 5.5 (i).

##  8\. Non-vanishing theorem

In this section, we will explain the non-vanishing theorem for projective
morphisms between complex analytic spaces.

######  Theorem 8.1 (see [Fu2, Theorem 12.1]).

Let $\pi\colon X\to Y$ be a projective morphism of complex analytic spaces
such that $X$ is a normal complex variety and let $W$ be a compact subset of
$Y$. Let $\Delta$ be an effective $\mathbb{R}$-divisor on $X$ such that
$K_{X}+\Delta$ is $\mathbb{R}$-Cartier. Let $L$ be a Cartier divisor on $X$
which is $\pi$-nef over $W$, that is, $L|_{\pi^{-1}(w)}$ is nef for every
$w\in W$. We assume that

  * (i)

$aL-(K_{X}+\Delta)$ is $\pi$-ample over $W$ for some positive real number $a$,
and

  * (ii)

$\mathcal{O}_{\operatorname{Nlc}(X,\Delta)}(mL)$ is
$\pi|_{\operatorname{Nlc}(X,\Delta)}$-generated over some open neighborhood of
$W$ for every $m\gg 0$.

Then for every $m\gg 0$ there exists a relatively compact open neighborhood
$U_{m}$ of $W$ over which the relative base locus
$\operatorname{Bs}_{\pi}|mL|$ contains no log canonical centers of
$(X,\Delta)$ and is disjoint from $\operatorname{Nlc}(X,\Delta)$. We note that
the open subset $U_{m}$ depends on $m$.

We first prepare the following useful lemma, which is new, for the proof of
Theorem 8.1. For the details of the theory of quasi-log schemes, see [Fu3,
Chapter 6], [Fu6], and [Fu7].

######  Lemma 8.2.

Let $\pi\colon X\to Y$ be a projective morphism between complex analytic
spaces such that $X$ is a normal complex variety and let
$f\colon(Z,\Delta_{Z})\to X$ be a projective morphism from an analytic
globally embedded simple normal crossing pair $(Z,\Delta_{Z})$ such that
$\Delta_{Z}$ is a subboundary $\mathbb{R}$-divisor on $Z$ and is a finite
$\mathbb{R}$-linear combination of Cartier divisors, the natural map
$\mathcal{O}_{X}\to f_{*}\mathcal{O}_{Z}(\lceil-(\Delta^{<1}_{Z})\rceil)$ is
an isomorphism, and $K_{Z}+\Delta_{Z}\sim_{\mathbb{R}}f^{*}\omega$ holds for
some $\mathbb{R}$-line bundle $\omega$ on $X$. Let $y$ be an analytically
sufficiently general point of $\pi(X)$. Then

$\left(X_{y},\omega|_{X_{y}},f_{y}\colon(Z_{y},\Delta_{Z_{y}})\to
X_{y}\right)$

is a projective quasi-log canonical pair, where $X_{y}:=\pi^{-1}(y)$,
$Z_{y}:=(\pi\circ f)^{-1}(y)$, $f_{y}:=f|_{X_{y}}$, and
$\Delta_{Z_{y}}:=\Delta_{Z}|_{Z_{y}}$.

This lemma is also a consequence of the strict support condition in Theorem
5.5 (i).

###### Proof of Lemma 8.2.

By replacing $Y$ with $\pi(X)$, we may assume that $\pi$ is surjective and $Y$
is a complex variety. By replacing $Y$ with a Zariski open subset of $Y$, we
may further assume that $Y$ is smooth. By replacing $Y$ with a suitable
Zariski open subset, we may assume that $\pi\circ f$ is flat. Then, by
replacing $Y$ with a suitable Zariski open subset again, we may assume that
every stratum of $(Z,\operatorname{Supp}\Delta_{Z})$ is smooth over $Y$. We
take an arbitrary point $y\in Y$. Then $(Z_{y},\Delta_{Z_{y}})$ is an analytic
simple normal crossing pair. From now on, we will prove that

$\left(X_{y},\omega|_{X_{y}},f_{y}\colon(Z_{y},\Delta_{Z_{y}})\to
X_{y}\right)$

is a projective quasi-log canonical pair. Without loss of generality, we may
assume that $Y$ is a polydisc $\Delta^{m}$ with $y=0\in\Delta^{m}$. Let
$(z_{1},\ldots,z_{m})$ be the local coordinate system of $\Delta^{m}$. Then
$\left((\pi\circ f)^{*}z_{i}=0\right)$ does not contain any strata of
$(Z,\operatorname{Supp}\Delta_{Z})$. Therefore,

$\pi^{*}z_{i}\times\colon
R^{p}f_{*}\mathcal{O}_{Z}(\lceil-(\Delta^{<1}_{Z})\rceil)\to
R^{p}f_{*}\mathcal{O}_{Z}(\lceil-(\Delta^{<1}_{Z})\rceil)$

is injective for every $i$ and every $p$ since
$\lceil-(\Delta^{<1}_{Z})\rceil-(K_{Z}+\\{\Delta_{Z}\\}+\Delta^{=1}_{Z})\sim_{\mathbb{R}}-f^{*}\omega$.
We put $X_{1}:=(\pi^{*}z_{1}=0)$ and $Z_{1}:=\left((\pi\circ
f)^{*}z_{1}=0\right)$. Since

$\pi^{*}z_{1}\times\colon
R^{1}f_{*}\mathcal{O}_{Z}(\lceil-(\Delta^{<1}_{Z})\rceil)\to
R^{1}f_{*}\mathcal{O}_{Z}(\lceil-(\Delta^{<1}_{Z})\rceil)$

is injective, we obtain the following short exact sequence:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{f_{*}\mathcal{O}_{Z}(\lceil-(\Delta^{<1}_{Z})\rceil)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\times\pi^{*}z_{1}}$$\textstyle{f_{*}\mathcal{O}_{Z}(\lceil-(\Delta^{<1}_{Z})\rceil)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{f_{*}\mathcal{O}_{Z_{1}}(\lceil-(\Delta^{<1}_{Z_{1}})\rceil)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where $\Delta_{Z_{1}}=\Delta_{Z}|_{Z_{1}}$. This implies that the natural map
$\mathcal{O}_{X_{1}}\to
f_{*}\mathcal{O}_{Z_{1}}(\lceil-(\Delta^{<1}_{Z_{1}})\rceil)$ is an
isomorphism. By repeating this argument, we finally obtain that
$\mathcal{O}_{X_{y}}\simeq(f_{y})_{*}\mathcal{O}_{Z_{y}}(\lceil-(\Delta^{<1}_{Z_{y}})\rceil)$
holds. By [Fu4, Theorem 4.9], this means that

$\left(X_{y},\omega|_{X_{y}},f_{y}\colon(Z_{y},\Delta_{Z_{y}})\to
X_{y}\right)$

is a projective quasi-log canonical pair. ∎

Let us prove Theorem 8.1.

###### Proof of Theorem 8.1.

We divide the proof into several small steps.

######  Step 1.

By shrinking $Y$ suitably, we may assume that there exists a positive integer
$m_{1}$ such that $\mathcal{O}_{\operatorname{Nlc}(X,\Delta)}(mL)$ is
$\pi|_{\operatorname{Nlc}(X,\Delta)}$-generated for every $m\geq m_{1}$ by
(ii). We may further assume that $aL-(K_{X}+\Delta)$ is $\pi$-ample over $Y$.

######  Step 2.

In this step, we will prove the following claim.

######  Claim.

There exists a positive integer $m_{2}$ such that
$\pi_{*}\mathcal{O}_{V}(mL)\neq 0$ holds for every $m\geq m_{2}$, where $V$ is
any minimal log canonical center of $(X,\Delta)$ such that $\pi(V)\cap
W\neq\emptyset$ and that $V\cap\operatorname{Nlc}(X,\Delta)=\emptyset$ over
some open neighborhood of $W$.

###### Proof of Claim.

We note that the number of the minimal log canonical centers $V$ of
$(X,\Delta)$ with $\pi(V)\cap W\neq\emptyset$ is finite. We take a minimal log
canonical center $V$ such that $\pi(V)\cap W\neq\emptyset$ and that
$V\cap\operatorname{Nlc}(X,\Delta)=\emptyset$ over some open neighborhood of
$W$. Let $y$ be an arbitrary point of $\pi(V)\cap W$. It is sufficient to
prove $\pi_{*}\mathcal{O}_{V}(mL)\neq 0$ on a small open neighborhood of $y$.
Therefore, we can replace $Y$ with a small relatively compact Stein open
neighborhood of $y$. Thus, by Step 1 in the proof of Theorem 6.1, we can
construct a projective surjective morphism $f\colon(T,\Delta_{T})\to V$ from
an analytic globally embedded simple normal crossing pair $(T,\Delta_{T})$
such that $\Delta_{T}$ is a subboundary $\mathbb{R}$-divisor on $T$, the
natural map $\mathcal{O}_{V}\to
f_{*}\mathcal{O}_{T}(\lceil-(\Delta^{<1}_{T})\rceil)$ is an isomorphism, and
$K_{T}+\Delta_{T}\sim_{\mathbb{R}}f^{*}(K_{X}+\Delta)|_{V}$ holds. Thus, by
Lemma 8.2, an analytically sufficiently general fiber $F$ of $\pi\colon
V\to\pi(V)$ is a projective quasi-log canonical pair. By Lemma 3.5, we may
assume that $L|_{F}$ is nef. Therefore, by the basepoint-free theorem for
quasi-log canonical pairs (see [Fu3, Theorem 6.5.1]), there exists a positive
integer $m_{2}$ such that $|mL|_{F}|$ is basepoint-free for every $m\geq
m_{2}$. This implies that $\pi_{*}\mathcal{O}_{V}(mL)\neq 0$ for every $m\geq
m_{2}$. This is what we wanted. ∎

######  Step 3.

We put $m_{0}:=\max\\{a,m_{1},m_{2}\\}$. Let $m$ be any positive integer with
$m\geq m_{0}$. Since $aL-(K_{X}+\Delta)$ is $\pi$-ample over $W$ and $L$ is
$\pi$-nef over $W$, $mL-(K_{X}+\Delta)$ is $\pi$-ample over $W$. By Theorem
6.1, we can find an open neighborhood $U_{m}$ of $W$ such that the vanishing
theorem holds for $mL$ over $U_{m}$. Without loss of generality, we may assume
that every minimal log canonical center $V$ of $(X,\Delta)$ with $\pi(V)\cap
U_{m}\neq\emptyset$ always satisfies $\pi(V)\cap W\neq\emptyset$ by shrinking
$U_{m}$ suitably.

######  Step 4.

In this final step, we will prove that over $U_{m}$ the relative base locus
$\operatorname{Bs}_{\pi}|mL|$ contains no log canonical centers of
$(X,\Delta)$ and is disjoint from $\operatorname{Nlc}(X,\Delta)$ for every
$m\geq m_{0}$.

By the vanishing theorem (see Theorem 6.1), we have
$R^{1}\pi_{*}\left(\mathcal{J}_{\operatorname{NLC}}(X,\Delta)\otimes\mathcal{O}_{X}(mL)\right)=0$
on $U_{m}$. Thus the restriction map

$\pi_{*}\mathcal{O}_{X}(mL)\to\pi_{*}\mathcal{O}_{\operatorname{Nlc}(X,\Delta)}(mL)$

is surjective on $U_{m}$. This implies that the relative base locus
$\operatorname{Bs}_{\pi}|mL|$ is disjoint from $\operatorname{Nlc}(X,\Delta)$
over $U_{m}$. Let $V$ be a minimal log canonical center of $(X,\Delta)$ with
$\pi(V)\cap U_{m}\neq\emptyset$. If
$V\cap\operatorname{Nlc}(X,\Delta)\neq\emptyset$ over $U_{m}$, then
$V\not\subset\operatorname{Bs}_{\pi}|mL|$ since
$\operatorname{Nlc}(X,\Delta)\cap\operatorname{Bs}_{\pi}|mL|=\emptyset$ over
$U_{m}$. Hence we may assume that
$V\cap\operatorname{Nlc}(X,\Delta)=\emptyset$ over $U_{m}$. In this case,
$\pi_{*}\mathcal{O}_{V}(mL)\neq 0$ by Claim in Step 2. On the other hand, by
the vanishing theorem (see Theorem 6.1), the restriction map

$\pi_{*}\mathcal{O}_{X}(mL)\to\pi_{*}\mathcal{O}_{V}(mL)$

is surjective on $U_{m}$. This implies that
$V\not\subset\operatorname{Bs}_{\pi}|mL|$. Hence $\operatorname{Bs}_{\pi}|mL|$
contains no log canonical centers of $(X,\Delta)$ over $U_{m}$.

We finish the proof. ∎

We make an important remark on Theorem 8.1.

######  Remark 8.3.

In Step 3 in the proof of Theorem 8.1, the condition that $mL-(K_{X}+\Delta)$
is $\pi$-ample over $w\in W$ only implies that $mL-(K_{X}+\Delta)$ is
$\pi$-ample over some open neighborhood $\mathcal{U}^{m}_{w}$ of $w$ in $Y$.
We note that $\mathcal{U}^{m}_{w}$ depends on $m$. Therefore, $U_{m}$ in
Theorem 8.1 also depends on $m$.

For kawamata log terminal pairs, the non-vanishing theorem is formulated as
follows.

######  Theorem 8.4 (Non-vanishing theorem for kawamata log terminal pairs).

Let $\pi\colon X\to Y$ be a projective morphism of complex analytic spaces
such that $X$ is a normal complex variety and let $W$ be a compact subset of
$Y$. Let $\Delta$ be an effective $\mathbb{R}$-divisor on $X$ such that
$(X,\Delta)$ is kawamata log terminal. Let $L$ be a Cartier divisor on $X$
which is $\pi$-nef over $W$. We assume that $aL-(K_{X}+\Delta)$ is $\pi$-ample
over $W$ for some positive real number $a$. Then
$\pi_{*}\mathcal{O}_{X}(mL)\neq 0$ holds for every $m\gg 0$

###### Proof.

By shrinking $Y$ around $W$ suitably, we may assume that $aL-(K_{X}+\Delta)$
is $\pi$-ample over $Y$. Let $F$ be an analytically sufficiently general fiber
of $\pi\colon X\to\pi(X)$. Then $(F,\Delta|_{F})$ is kawamata log terminal. By
Lemma 3.5, we may assume that $L|_{F}$ is nef. Hence $|mL|_{F}|$ is basepoint-
free for every $m\gg 0$ by the usual Kawamata–Shokurov basepoint-free theorem
for projective kawamata log terminal pairs. Thus, we obtain that
$\pi_{*}\mathcal{O}_{X}(mL)\neq 0$ for every $m\gg 0$. This is what we wanted.
∎

We will use Theorems 8.1 and 8.4 in the proof of the basepoint-freeness in
Section 9.

##  9\. Basepoint-free theorem

In this section, we will explain the basepoint-free theorem in the complex
analytic setting.

######  Theorem 9.1 (see [Fu2, Theorem 13.1]).

Let $\pi\colon X\to Y$ be a projective morphism of complex analytic spaces
such that $X$ is a normal complex variety and let $W$ be a compact subset of
$Y$. Let $\Delta$ be an effective $\mathbb{R}$-divisor on $X$ such that
$K_{X}+\Delta$ is $\mathbb{R}$-Cartier. Let $L$ be a Cartier divisor on $X$
which is $\pi$-nef over $W$. We assume that

  * (i)

$aL-(K_{X}+\Delta)$ is $\pi$-ample over $W$ for some positive real number $a$,
and

  * (ii)

$\mathcal{O}_{\operatorname{Nlc}(X,\Delta)}(mL)$ is
$\pi|_{\operatorname{Nlc}(X,\Delta)}$-generated over some open neighborhood of
$W$ for every $m\gg 0$.

Then there exists a relatively compact open neighborhood $U$ of $W$ such that
$\mathcal{O}_{X}(mL)$ is $\pi$-generated over $U$ for every $m\gg 0$.

Theorem 9.1 is a consequence of the vanishing theorem (see Theorem 6.1) and
the non-vanishing theorem (see Theorems 8.1 and 8.4).

###### Proof of Theorem 9.1.

We take an arbitrary point $y\in W$. It is sufficient to prove that
$\mathcal{O}_{X}(mL)$ is $\pi$-generated for every $m\gg 0$ over some
relatively compact Stein open neighborhood of $y$. Hence, we will sometimes
freely replace $Y$ with a suitable relatively compact Stein open neighborhood
of $y$ without mentioning it explicitly throughout this proof. So, from now
on, we assume that $Y$ is Stein and that $\pi$ is surjective.

######  Step 1.

Let $p$ be a prime number. In this step, we will prove that there exists a
positive integer $k$ such that $\operatorname{Bs}_{\pi}|p^{k}L|=\emptyset$
holds over some open neighborhood of $y$.

By putting $W:=\\{y\\}$ and using the non-vanishing theorem (see Theorems 8.1
and 8.4), we obtain $|p^{k_{1}}L|\neq\emptyset$ for some positive integer
$k_{1}$. If $\operatorname{Bs}_{\pi}|p^{k_{1}}L|=\emptyset$, then there is
nothing to prove. Hence we may assume that
$\operatorname{Bs}_{\pi}|p^{k_{1}}L|\neq\emptyset$. By Theorem 8.1, we may
assume that $\operatorname{Bs}_{\pi}|p^{k_{1}}L|$ contains no log canonical
centers of $(X,\Delta)$ and is disjoint from $\operatorname{Nlc}(X,\Delta)$
after shrinking $Y$ suitably. By shrinking $Y$ around $W=\\{y\\}$, we may
assume that $\pi(V)\cap W\neq\emptyset$, where $V$ is any irreducible
component of $\operatorname{Bs}_{\pi}|p^{k_{1}}L|$. Without loss of
generality, we may further assume that $aL-(K_{X}+\Delta)$ is $\pi$-ample over
$Y$. We take general members $D_{1},\ldots,D_{n+1}$ of $|p^{k_{1}}L|$ with
$n=\dim X$. We put $D:=\sum_{i=1}^{n+1}D_{i}$. We may assume that
$(X,\Delta+D)$ is log canonical outside
$\operatorname{Bs}_{\pi}|p^{k_{1}}L|\cup\operatorname{Nlc}(X,\Delta)$. Let
$x\in X$ be any point of $\operatorname{Bs}_{\pi}|p^{k_{1}}L|$. Then, by Lemma
2.3, $(X,\Delta+D)$ is not log canonical at $x$. We put

$c:=\sup\left\\{t\in\mathbb{R}\mid\text{$(X,\Delta+tD)$ is log canonical at
$\pi^{-1}(y)\cap\left(X\setminus\operatorname{Nlc}(X,\Delta)\right)$}\right\\}.$

Then we can check that $0<c<1$. By shrinking $Y$ around $W=\\{y\\}$ suitably
again, we may assume that $(X,\Delta+cD)$ is log canonical outside
$\operatorname{Nlc}(X,\Delta)$. By Lemma 4.3 and its proof, we see that
$\mathcal{J}_{\operatorname{NLC}}(X,\Delta+cD)=\mathcal{J}_{\operatorname{NLC}}(X,\Delta)$
holds. By construction,

$\left(c(n+1)p^{k_{1}}+a\right)L-(K_{X}+\Delta+cD)\sim_{\mathbb{R}}aL-(K_{X}+\Delta)$

is $\pi$-ample over $Y$. By construction again, there exists a log canonical
center $V$ of $(X,\Delta+cD)$ which is contained in
$\operatorname{Bs}_{\pi}|p^{k_{1}}L|$ such that $\pi(V)\cap W\neq\emptyset$.
By the non-vanishing theorem (see Theorem 8.1), we can find $k_{2}>k_{1}$ such
that
$\operatorname{Bs}_{\pi}|p^{k_{2}}L|\subsetneq\operatorname{Bs}_{\pi}|p^{k_{1}}L|$.
Here, we replaced $Y$ with a smaller open neighborhood of $y$. By repeating
this process finitely many times, we can find $k$ such that
$\operatorname{Bs}_{\pi}|p^{k}L|=\emptyset$ over some open neighborhood of
$y$.

######  Step 2.

We take another prime number $p^{\prime}$. Then there exists $k^{\prime}$ such
that $\operatorname{Bs}_{\pi}|p^{\prime k^{\prime}}L|=\emptyset$ over some
open neighborhood of $y$ by Step 1. This implies that there exist a positive
integer $m_{0}$ and some open neighborhood $U_{y}$ of $y$ such that for every
$m\geq m_{0}$ the relative base locus $\operatorname{Bs}_{\pi}|mL|$ is empty
over $U_{y}$.

Since $W$ is compact, we obtain a desired open neighborhood $U$ of $W$. We
finish the proof. ∎

######  Remark 9.2.

Although the non-vanishing theorem (see Theorems 8.1 and 8.4) and the
basepoint-free theorem (see Theorem 9.1) were formulated for Cartier divisors
$L$, it is obvious that they hold true even for line bundles $\mathcal{L}$. We
will sometimes use the basepoint-free theorem for line bundles in subsequent
sections.

##  10\. Rationality theorem

In this section, we will explain the rationality theorem in the complex
analytic setting. Although the proof of [Fu2, Theorem 15.1], which is the
rationality theorem in the algebraic setting, works with some suitable
modifications, we will explain the details for the reader’s convenience. This
is because the proof of the rationality theorem is complicated.

######  Theorem 10.1 (Rationality theorem, see [Fu2, Theorem 15.1]).

Let $\pi\colon X\to Y$ be a projective morphism of complex analytic spaces
such that $X$ is a normal complex variety and let $W$ be a compact subset of
$Y$. Let $\Delta$ be an effective $\mathbb{Q}$-divisor on $X$ such that
$K_{X}+\Delta$ is $\mathbb{Q}$-Cartier. Let $H$ be a $\pi$-ample Cartier
divisor on $X$. Assume that $K_{X}+\Delta$ is not $\pi$-nef over $W$ and that
$r$ is a positive number such that

  * (i)

$H+r(K_{X}+\Delta)$ is $\pi$-nef over $W$ but is not $\pi$-ample over $W$, and

  * (ii)

$\left(H+r(K_{X}+\Delta)\right)|_{\operatorname{Nlc}(X,\Delta)}$ is
$\pi|_{\operatorname{Nlc}(X,\Delta)}$-ample over $W$.

Then $r$ is a rational number, and in reduced form, it has denominator at most
$a(d+1)$, where $d:=\max_{w\in W}\dim\pi^{-1}(w)$ and $a$ is a positive
integer such that $a(K_{X}+\Delta)$ is a Cartier divisor in a neighborhood of
$\pi^{-1}(W)$.

In the proof of Theorem 10.1, we will use the following elementary lemmas. We
do not prove Lemma 10.2 here. For the proof, see, for example, [Fu2].

######  Lemma 10.2 ([KM, Lemma 3.19]).

Let $P(x,y)$ be a non-trivial polynomial of degree $\leq d$ and assume that
$P$ vanishes for all sufficiently large integral solutions of $0<ay-
rx<\varepsilon$ for some fixed positive integer $a$ and positive $\varepsilon$
for some $r\in\mathbb{R}$. Then $r$ is rational, and in reduced form, $r$ has
denominator $\leq a(d+1)/\varepsilon$.

######  Lemma 10.3.

Let $F$ be a projective variety and let $D_{1}$ and $D_{2}$ be Cartier
divisors on $X$. Let us consider the Hilbert polynomial

$P(u_{1},u_{2}):=\chi(F,\mathcal{O}_{F}(u_{1}D_{1}+u_{2}D_{2})).$

If $D_{1}$ is ample, then $P(u_{1},u_{2})$ is a non-trivial polynomial of
total degree $\leq\dim F$. It is because
$P(u_{1},0)=\dim_{\mathbb{C}}H^{0}(F,\mathcal{O}_{F}(u_{1}D_{1}))\not\equiv 0$
if $u_{1}$ is sufficiently large.

Let us start the proof of Theorem 10.1.

###### Proof of Theorem 10.1.

Throughout this proof, we can freely shrink $Y$ around $W$ suitably. Hence we
sometimes will replace $Y$ with a small open neighborhood of $W$ without
mentioning it explicitly.

Let $m$ be a positive integer such that $H^{\prime}:=mH$ is $\pi$-very ample
after shrinking $Y$ around $W$ suitably. If
$H^{\prime}+r^{\prime}(K_{X}+\Delta)$ is $\pi$-nef over $W$ but is not
$\pi$-ample over $W$, and
$\left(H^{\prime}+r^{\prime}(K_{X}+\Delta)\right)|_{\operatorname{Nlc}(X,\Delta)}$
is $\pi|_{\operatorname{Nlc}(X,\Delta)}$-ample over $W$, then

$H+r(K_{X}+\Delta)=\frac{1}{m}(H^{\prime}+r^{\prime}(K_{X}+\Delta))$

holds. This implies that $r=(1/m)r^{\prime}$ holds. Therefore, it is obvious
that $r$ is rational if and only if $r^{\prime}$ is rational. We further
assume that $r^{\prime}$ has denominator $v$. Then $r$ has denominator
dividing $mv$. Since $m$ can be an arbitrary sufficiently large positive
integer, this means that $r$ has denominator dividing $v$. Hence, by replacing
$H$ with $mH$, we may assume that $H$ is $\pi$-very ample.

For each $(p,q)\in\mathbb{Z}^{2}$, we put $M(p,q):=pH+qa(K_{X}+\Delta)$ and
define

$L(p,q):=\operatorname{Supp}(\operatorname{Coker}(\pi^{*}\pi_{*}\mathcal{O}_{X}(M(p,q))\to\mathcal{O}_{X}(M(p,q)))).$

By definition, $L(p,q)=X$ holds if and only if
$\pi_{*}\mathcal{O}_{X}(M(p,q))=0$.

######  Claim 1.

Let $\varepsilon$ be a positive number. For $(p,q)$ sufficiently large and
$0<aq-rp<\varepsilon$, $L(p,q)$ is the same subset of $X$ after shrinking $Y$
around $W$ suitably. We call this subset $L_{0}$. Let $I\subset\mathbb{Z}^{2}$
be the set of $(p,q)$ for which $0<aq-rp<1$ and $L(p,q)=L_{0}$. Then we note
that $I$ contains all sufficiently large $(p,q)$ with $0<aq-rp<1$.

###### Proof of Claim 1.

We fix $(p_{0},q_{0})\in\mathbb{Z}^{2}$ such that $p_{0}>0$ and
$0<aq_{0}-rp_{0}<1$. Since $H$ is $\pi$-very ample, there exists a positive
integer $m_{0}$ such that $\mathcal{O}_{X}(mH+ja(K_{X}+\Delta))$ is
$\pi$-generated for every $m>m_{0}$ and every $0\leq j\leq q_{0}-1$ after
shrinking $Y$ around $W$ suitably. Let $M$ be the round-up of

$\left.\left(m_{0}+\frac{1}{r}\right)\right/\left(\frac{a}{r}-\frac{p_{0}}{q_{0}}\right).$

If $(p^{\prime},q^{\prime})\in\mathbb{Z}^{2}$ such that
$0<aq^{\prime}-rp^{\prime}<1$ and $q^{\prime}\geq M+q_{0}-1$, then we can
write

$p^{\prime}H+q^{\prime}a(K_{X}+\Delta)=k(p_{0}H+q_{0}a(K_{X}+\Delta))+(lH+ja(K_{X}+\Delta))$

for some $k\geq 0$, $0\leq j\leq q_{0}-1$ with $l>m_{0}$. It is because we can
uniquely write $q^{\prime}=kq_{0}+j$ with $0\leq j\leq q_{0}-1$. Thus, we have
$kq_{0}\geq M$. So, we obtain

$l=p^{\prime}-kp_{0}>\frac{a}{r}q^{\prime}-\frac{1}{r}-(kq_{0})\frac{p_{0}}{q_{0}}\geq\left(\frac{a}{r}-\frac{p_{0}}{q_{0}}\right)M-\frac{1}{r}\geq
m_{0}.$

Therefore, $L(p^{\prime},q^{\prime})\subset L(p_{0},q_{0})$. We note that we
can use the noetherian induction over a relatively compact open neighborhood
of $W$ (see [Fi, 0.40. Corollary]). Therefore, after shrinking $Y$ around $W$
suitably again, we obtain the desired closed subset $L_{0}\subset X$. We can
check that the subset $I\subset\mathbb{Z}^{2}$ contains all sufficiently large
$(p,q)$ with $0<aq-rp<1$ without any difficulties. ∎

######  Claim 2.

We have $L_{0}\cap\operatorname{Nlc}(X,\Delta)=\emptyset$.

###### Proof of Claim 2.

We take $(\alpha,\beta)\in\mathbb{Q}^{2}$ such that $\alpha>0$, $\beta>0$, and
$\beta a/\alpha>r$ is sufficiently close to $r$. Then $(\alpha H+\beta
a(K_{X}+\Delta))|_{\operatorname{Nlc}(X,\Delta)}$ is
$\pi|_{\operatorname{Nlc}(X,\Delta)}$-ample over $W$ because
$(H+r(K_{X}+\Delta))|_{\operatorname{Nlc}(X,\Delta)}$ is
$\pi|_{\operatorname{Nlc}(X,\Delta)}$-ample over $W$. We take any point $w\in
W$. Then it is sufficient to prove that
$L_{0}\cap\operatorname{Nlc}(X,\Delta)=\emptyset$ holds over some open
neighborhood of $w$. From now on, we will freely shrink $Y$ around $w$ without
mentioning it explicitly. We take a sufficiently large and divisible positive
integer $m^{\prime}$ such that

$m^{\prime}(\alpha H+\beta a(K_{X}+\Delta))|_{\operatorname{Nlc}(X,\Delta)}$

is $\pi|_{\operatorname{Nlc}(X,\Delta)}$-very ample. We put
$(p_{0},q_{0}):=(m^{\prime}\alpha,m^{\prime}\beta)$ and apply the argument in
the proof of Claim 1. Thus, if $0<aq-rp<1$ and $(p,q)\in\mathbb{Z}^{2}$ is
sufficiently large, then we can write

$M(p,q)=mM(\alpha,\beta)+(M(p,q)-mM(\alpha,\beta))$

such that $M(p,q)-mM(\alpha,\beta)$ is $\pi$-very ample and that

$m(\alpha H+\beta a(K_{X}+\Delta))|_{\operatorname{Nlc}(X,\Delta)}$

is also $\pi|_{\operatorname{Nlc}(X,\Delta)}$-very ample. Hence,
$\mathcal{O}_{\operatorname{Nlc}(X,\Delta)}(M(p,q))$ is $\pi$-very ample. We
note that

$M(p,q)-(K_{X}+\Delta)=pH+(qa-1)(K_{X}+\Delta)$

is $\pi$-ample over some open neighborhood of $w$ because $(p,q)$ is
sufficiently large and $aq-rp<1$. Thus, by the vanishing theorem: Theorem 6.1,
the restriction map

$\pi_{*}\mathcal{O}_{X}(M(p,q))\to\pi_{*}\mathcal{O}_{\operatorname{Nlc}(X,\Delta)}(M(p,q))$

is surjective. Therefore, $L(p,q)\cap\operatorname{Nlc}(X,\Delta)=\emptyset$
holds over some open neighborhood of $w$. By Claim 1, we have
$L_{0}\cap\operatorname{Nlc}(X,\Delta)=\emptyset$ over some open neighborhood
of $w$. Since $w$ is an arbitrary point of $W$,
$L_{0}\cap\operatorname{Nlc}(X,\Delta)=\emptyset$ holds over some open
neighborhood of $W$. This is what we wanted. ∎

######  Claim 3.

We assume that $r$ is not rational or that $r$ is rational and has denominator
$>a(d+1)$ in reduced form. Then, for $(p,q)$ sufficiently large and $0<aq-
rp<1$, $\mathcal{O}_{X}(M(p,q))$ is $\pi$-generated at general points of every
log canonical center of $(X,\Delta)$.

We will explain the proof of Claim 3 in detail because we have to change the
proof of Claim 3 in the proof of [Fu2, Theorem 15.1] slightly.

###### Proof of Claim 3.

After shrinking $Y$ around $W$ suitably, it is sufficient to consider minimal
log canonical centers $C$ of $(X,\Delta)$ such that $\pi(C)\cap
W\neq\emptyset$. By Claim 2, we may assume that
$C\cap\operatorname{Nlc}(X,\Delta)=\emptyset$ holds. We take a point
$w\in\pi(C)\cap W$. It is sufficient to consider everything over some small
open enighborhood of $w$ in $Y$. We take an analytically sufficiently general
fiber $F$ of $C\to\pi(C)$. Then we may assume that $(H+r(K_{X}+\Delta))|_{F}$
and $(H+r(K_{X}+\Delta))|_{\pi^{-1}(\pi(F))}$ are both nef by Lemma 3.5 (see
also Remark 3.7). We note that

$\begin{split}M(p,q)-(K_{X}+\Delta)&=pH+(qa-1)(K_{X}+\Delta)\\\
&=\left(p-\frac{qa-1}{r}\right)H+\frac{(qa-1)}{r}(H+r(K_{X}+\Delta))\end{split}$

holds. Therefore, if $aq-rp<1$ and $(p,q)$ is sufficiently large, then we see
that

$\left(M(p,q)-(K_{X}+\Delta)\right)|_{\pi^{-1}(\pi(F))}$

is ample. We note that

$P_{F}(p,q):=\chi(F,\mathcal{O}_{F}(M(p,q)))$

is a non-zero polynomial of degree at most $\dim F\leq d$ by Lemma 10.3. We
also note that $F$ is an analytically sufficiently general fiber of
$C\to\pi(C)$. By Lemma 10.2, there exists $(p,q)$ such that $P_{F}(p,q)\neq
0$, $(p,q)$ sufficiently large, and $0<aq-rp<1$. By the $\pi$-ampleness of
$M(p,q)-(K_{X}+\Delta)$ over some open neighborhood of $\pi(F)$,

$P_{F}(p,q)=\chi(F,\mathcal{O}_{F}(M(p,q)))=\dim_{\mathbb{C}}H^{0}(F,\mathcal{O}_{F}(M(p,q)))$

and

$\pi_{*}\mathcal{O}_{X}(M(p,q))\to\pi_{*}\mathcal{O}_{C}(M(p,q))$

is surjective over some open neighborhood of $\pi(F)$ by Theorem 6.1 (see also
2.14). We note that $\pi_{*}\mathcal{O}_{C}(M(p,q))\neq 0$ by $P_{F}(p,q)\neq
0$ and that $C\cap\operatorname{Nlc}(X,\Delta)=\emptyset$ by assumption.
Therefore, $\mathcal{O}_{X}(M(p,q))$ is $\pi$-generated at general points of
$C$. By combining this fact with Claim 1, $\mathcal{O}_{X}(M(p,q))$ is
$\pi$-generated at general points of every log canonical center of
$(X,\Delta)$ if $(p,q)$ is sufficiently large with $0<aq-rp<1$. Hence we
obtain Claim 3. ∎

Note that $\mathcal{O}_{X}(M(p,q))$ is not $\pi$-generated for $(p,q)\in I$
because $M(p,q)$ is not $\pi$-nef over $W$. Therefore, $L_{0}\neq\emptyset$
with $\pi(L_{0})\cap W\neq\emptyset$. We take a point $w\in\pi(L_{0})\cap W$
and replace $Y$ with a relatively compact Stein open neighborhood of $w$. From
now on, we will freely shrink $Y$ around $w$ suitably. Let
$D_{1},\ldots,D_{n+1}$ be general members of

$\pi_{*}\mathcal{O}_{X}(M(p_{0},q_{0}))=H^{0}(X,\mathcal{O}_{X}(M(p_{0},q_{0})))$

with $(p_{0},q_{0})\in I$. We put $D:=\sum_{i=1}^{n+1}D_{i}$. Let $x\in X$ be
any point of $L_{0}$. Then, by Lemma 2.3, $K_{X}+\Delta+D$ is not log
canonical at $x$. On the other hand, we may assume that $K_{X}+\Delta+D$ is
log canonical outside $L_{0}\cup\operatorname{Nlc}(X,\Delta)$ since $D_{i}$ is
a general member of $|M(p_{0},q_{0})|$ for every $i$. We put

$c:=\sup\left\\{t\in\mathbb{R}\,\mid\text{$(X,\Delta+tD)$ is log canonical at
$\pi^{-1}(w)\cap\left(X\setminus\operatorname{Nlc}(X,\Delta)\right)$}\right\\}.$

Then we can check that $0<c<1$ by Claim 3. We note that $w\in\pi(L_{0})\cap
W$. Thus, the pair $(X,\Delta+cD)$ has some log canonical centers contained in
$L_{0}$ and intersecting $\pi^{-1}(w)$. By shrinking $Y$ around $w$, we may
assume that $(X,\Delta+cD)$ is log canonical outside
$\operatorname{Nlc}(X,\Delta)$. Let $C$ be a log canonical center contained in
$L_{0}$ and intersecting $\pi^{-1}(w)$. We note that
$\mathcal{J}_{\operatorname{NLC}}(X,\Delta+cD)=\mathcal{J}_{\operatorname{NLC}}(X,\Delta)$
by Lemma 4.3 and its proof and that
$C\cap\operatorname{Nlc}(X,\Delta+cD)=C\cap\operatorname{Nlc}(X,\Delta)=\emptyset$.
We consider

$K_{X}+\Delta+cD=c(n+1)p_{0}H+(1+c(n+1)q_{0}a)(K_{X}+\Delta).$

Thus we have

$\displaystyle pH+qa(K_{X}+\Delta)-(K_{X}+\Delta+cD)$
$\displaystyle=(p-c(n+1)p_{0})H+(qa-(1+c(n+1)q_{0}a))(K_{X}+\Delta).$

If $p$ and $q$ are large enough and $0<aq-rp\leq aq_{0}-rp_{0}$, then

$pH+qa(K_{X}+\Delta)-(K_{X}+\Delta+cD)$

is $\pi$-ample over $w$. It is because

$\displaystyle(p-c(n+1)p_{0})H+(qa-(1+c(n+1)q_{0}a))(K_{X}+\Delta)$
$\displaystyle=(p-(1+c(n+1))p_{0})H+(qa-(1+c(n+1))q_{0}a)(K_{X}+\Delta)$
$\displaystyle\ +p_{0}H+(q_{0}a-1)(K_{X}+\Delta).$

By shrinking $Y$ around $w$ suitably, we may further assume that it is
$\pi$-ample over $Y$. We consider an analytically sufficiently general fiber
$F$ of $C\to\pi(C)$ as in the proof of Claim 3. We note that
$\left(H+r(K_{X}+\Delta)\right)|_{\pi^{-1}(\pi(F))}$ is nef by the choice of
$F$.

Suppose that $r$ is not rational. There exists an arbitrarily large
$(p,q)\in\mathbb{Z}^{2}$ such that $0<aq-rp<\varepsilon=aq_{0}-rp_{0}$ and
$\chi(F,\mathcal{O}_{F}(M(p,q)))\neq 0$ by Lemma 10.2 because
$P_{F}(p,q)=\chi(F,\mathcal{O}_{F}(M(p,q)))$ is a non-trivial polynomial of
degree at most $\dim F\leq d$ by Lemma 10.3. Since

$\left(M(p,q)-(K_{X}+\Delta+cD)\right)|_{\pi^{-1}(\pi(F))}$

is ample by $0<aq-rp<aq_{0}-rp_{0}$, we have

$\dim_{\mathbb{C}}H^{0}(F,\mathcal{O}_{F}(M(p,q)))=\chi(F,\mathcal{O}_{F}(M(p,q)))\neq
0$

by the vanishing theorem: Theorem 6.1 (see also 2.14). By the vanishing
theorem: Theorem 6.1,

$\pi_{*}\mathcal{O}_{X}(M(p,q))\to\pi_{*}\mathcal{O}_{C}(M(p,q))$

is surjective over a neighborhood of $\pi(F)$ because
$\left(M(p,q)-(K_{X}+\Delta+cD)\right)|_{\pi^{-1}(\pi(F))}$ is ample. We note
that $C\cap\operatorname{Nlc}(X,\Delta+cD)=\emptyset$. Thus $C$ is not
contained in $L(p,q)$. Therefore, $L(p,q)$ is a proper subset of
$L(p_{0},q_{0})=L_{0}$, which gives the desired contradiction. Hence we know
that $r$ is rational.

We next suppose that the assertion of the theorem concerning the denominator
of $r$ is false. We choose $(p_{0},q_{0})\in I$ such that $aq_{0}-rp_{0}$ is
the maximum, say it is equal to $e/v$. If $0<aq-rp\leq e/v$ and $(p,q)$ is
sufficiently large, then

$\chi(F,\mathcal{O}_{F}(M(p,q)))=\dim_{\mathbb{C}}H^{0}(F,\mathcal{O}_{F}(M(p,q)))$

since $\left(M(p,q)-(K_{X}+\Delta+cD)\right)|_{\pi^{-1}(\pi(F))}$ is ample.
There exists sufficiently large $(p,q)\in\mathbb{Z}^{2}$ in the strip $0<aq-
rp<1$ with $\varepsilon=1$ for which

$\dim_{\mathbb{C}}H^{0}(F,\mathcal{O}_{F}(M(p,q)))=\chi(F,\mathcal{O}_{F}(M(p,q)))\neq
0$

by Lemma 10.2 since $P_{F}(p,q)=\chi(F,\mathcal{O}_{F}(M(p,q)))$ is a non-
trivial polynomial of degree at most $\dim F\leq d$ by Lemma 10.3. Note that
$aq-rp\leq e/v=aq_{0}-rp_{0}$ holds automatically for $(p,q)\in I$. Since

$\pi_{*}\mathcal{O}_{X}(M(p,q))\to\pi_{*}\mathcal{O}_{C}(M(p,q))$

is surjective over some open neighborhood of $\pi(F)$ by the ampleness of

$\left(M(p,q)-(K_{X}+\Delta+cD)\right)|_{\pi^{-1}(\pi(F))},$

we obtain the desired contradiction by the same reason as above.

Thus, we finish the proof of the rationality theorem. ∎

We close this section with an easy remark.

######  Remark 10.4.

The proof of Theorem 10.1 shows that Theorem 10.1 holds true under the
assumption that $H$ is a $\pi$-ample line bundle.

##  11\. Kleiman–Mori cones

In this section, we will define Kleiman–Mori cones for projective morphisms
between complex analytic spaces under some suitable assumption.

######  11.1.

Throughout this section, let $\pi\colon X\to Y$ be a projective morphism of
complex analytic spaces and let $W$ be a compact subset of $Y$. Let
$Z_{1}(X/Y;W)$ be the free abelian group generated by the projective integral
curves $C$ on $X$ such that $\pi(C)$ is a point of $W$. Let $U$ be any open
neighborhood of $W$. Then we can consider the following intersection pairing

$\cdot:\operatorname{Pic}\\!\left(\pi^{-1}(U)\right)\times
Z_{1}(X/Y;W)\to\mathbb{Z}$

given by $\mathcal{L}\cdot C\in\mathbb{Z}$ for
$\mathcal{L}\in\operatorname{Pic}\\!\left(\pi^{-1}(U)\right)$ and $C\in
Z_{1}(X/Y;W)$. We say that $\mathcal{L}$ is $\pi$-numerically trivial over $W$
when $\mathcal{L}\cdot C=0$ for every $C\in Z_{1}(X/Y;W)$. We take
$\mathcal{L}_{1},\mathcal{L}_{2}\in\operatorname{Pic}\\!\left(\pi^{-1}(U)\right)$.
If $\mathcal{L}_{1}\otimes\mathcal{L}_{2}^{-1}$ is $\pi$-numerically trivial
over $W$, then we write $\mathcal{L}_{1}\equiv_{W}\mathcal{L}_{2}$ and say
that $\mathcal{L}_{1}$ is numerically equivalent to $\mathcal{L}_{2}$ over
$W$. We put

$\widetilde{A}(U,W):=\operatorname{Pic}\\!\left(\pi^{-1}(U)\right)/{\equiv_{W}}$

and define

$A^{1}(X/Y;W):=\underset{W\subset U}{\varinjlim}\widetilde{A}(U,W),$

where $U$ runs through all the open neighborhoods of $W$.

######  11.2.

We assume that $A^{1}(X/Y;W)$ is a finitely generated abelian group. Then we
can define the relative Picard number $\rho(X/Y;W)$ to be the rank of
$A^{1}(X/Y;W)$. We put

$N^{1}(X/Y;W):=A^{1}(X/Y;W)\otimes_{\mathbb{Z}}\mathbb{R}.$

Let $A_{1}(X/Y;W)$ be the image of

$Z_{1}(X/Y;W)\to\operatorname{Hom}_{\mathbb{Z}}\left(A^{1}(X/Y;W),\mathbb{Z}\right)$

given by the above intersection pairing. Then we set

$N_{1}(X/Y;W):=A_{1}(X/Y;W)\otimes_{\mathbb{Z}}\mathbb{R}.$

As usual, we can define the Kleiman–Mori cone

$\operatorname{\overline{NE}}(X/Y;W)$

of $\pi\colon X\to Y$ over $W$, that is, $\operatorname{\overline{NE}}(X/Y;W)$
is the closure of the convex cone in $N_{1}(X/Y;W)$ spanned by the projective
integral curves $C$ on $X$ such that $\pi(C)$ is a point of $W$. An element
$\zeta\in N^{1}(X/Y;W)$ is called $\pi$-nef over $W$ or nef over $W$ if
$\zeta\geq 0$ on $\operatorname{\overline{NE}}(X/Y;W)$, equivalently,
$\zeta|_{\pi^{-1}(w)}$ is nef in the usual sense for every $w\in W$.

######  Remark 11.3.

We assume that $\pi\colon X\to Y$ decomposes as

$\textstyle{\pi\colon
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{Z}}$$\textstyle{Y,}$

where $\pi_{Z}\colon Z\to Y$ is a projective morphism of complex analytic
spaces. Then $\varphi$ is always projective and $\pi^{-1}_{Z}(W)$ is a compact
subset of $Z$. Therefore, we can define
$A^{1}\left(X/Z;\pi^{-1}_{Z}(W)\right)$ and
$N^{1}\left(X/Z;\pi^{-1}_{Z}(W)\right)$ as above. By definition,
$N^{1}\left(X/Z;\pi^{-1}_{Z}(W)\right)$ is a quotient vector space of
$N^{1}\left(X/Y;W\right)$. Hence, if $\dim_{\mathbb{R}}N^{1}(X/Y;W)<\infty$,
then we see that
$\dim_{\mathbb{R}}N^{1}\left(X/Z;\pi^{-1}_{Z}(W)\right)<\infty$ holds.

######  Lemma 11.4 ([Na1, Proposition 4.7 (2)]).

$\operatorname{\overline{NE}}(X/Y;W)$ contains no lines of $N_{1}(X/Y;W)$.

###### Proof.

Suppose that $\operatorname{\overline{NE}}(X/Y;W)$ contains a line of
$N_{1}(X/Y;W)$. Then we can take
$\Gamma\in\operatorname{\overline{NE}}(X/Y;W)$ such that
$\Gamma,-\Gamma\in\operatorname{\overline{NE}}(X/Y;W)$. We take a $\pi$-ample
$\mathbb{R}$-line bundle $\mathcal{A}$ on $X$. By definition, $\mathcal{A}$ is
$\pi$-nef over $W$. Therefore, we obtain $\mathcal{A}\cdot\Gamma\geq 0$ and
$-\mathcal{A}\cdot\Gamma\geq 0$. This means that $\mathcal{A}\cdot\Gamma=0$.
On the other hand, after shrinking $Y$ around $W$ suitably, we can take a line
bundle $\mathcal{M}$ on $X$ such that $\Gamma\cdot\mathcal{M}>0$ since
$\Gamma\neq 0$ in $N_{1}(X/Y;W)$. Since $\mathcal{A}$ is $\pi$-ample,
$m\mathcal{A}-\mathcal{M}$ is also $\pi$-ample over some open neighborhood of
$W$, where $m$ is a large positive integer. This implies that
$(m\mathcal{A}-\mathcal{M})\cdot\Gamma\geq 0$ since
$\Gamma\in\operatorname{\overline{NE}}(X/Y;W)$. Thus,
$m\mathcal{A}\cdot\Gamma\geq\mathcal{M}\cdot\Gamma>0$ holds. Hence we obtain
$\mathcal{A}\cdot\Gamma>0$. This is a contradiction. Therefore, there are no
lines in $\operatorname{\overline{NE}}(X/Y;W)$. ∎

The following theorem is Kleiman’s ampleness criterion for projective
morphisms between complex analytic spaces (see [Na1, Proposition 4.7]).

######  Theorem 11.5 (Kleiman’s ampleness criterion).

Let $\pi\colon X\to Y$ be a projective morphism between complex analytic
spaces and let $W$ be a compact subset of $Y$ such that the dimension of
$N^{1}(X/Y;W)$ is finite. Let $\mathcal{L}$ be an $\mathbb{R}$-line bundle on
$X$. Then the following conditions are equivalent.

  * (i)

$\mathcal{L}$ is $\pi$-ample over $W$.

  * (ii)

$\mathcal{L}$ is $\pi$-ample over some open neighborhood $U$ of $W$.

  * (iii)

$\mathcal{L}$ is positive on
$\operatorname{\overline{NE}}(X/Y;W)\setminus\\{0\\}$.

###### Proof.

We have already proved the equivalence of (i) and (ii) in Lemma 3.2 without
assuming $\dim_{\mathbb{R}}N^{1}(X/Y;W)<\infty$.

######  Step 1.

In this step, we will prove that (i) follows from (iii).

We assume that $\mathcal{L}$ is positive on
$\operatorname{\overline{NE}}(X/Y;W)\setminus\\{0\\}$. Then we can take a
$\pi$-ample $\mathbb{R}$-line bundle $\mathcal{A}$ on $X$ such that
$\mathcal{N}:=\mathcal{L}-\mathcal{A}$ is non-negative on
$\operatorname{\overline{NE}}(X/Y;W)$. This means that
$\mathcal{N}|_{\pi^{-1}(w)}$ is nef for every $w\in W$. Since
$\mathcal{L}|_{\pi^{-1}(w)}=\mathcal{N}|_{\pi^{-1}(w)}+\mathcal{A}|_{\pi^{-1}(w)}$,
$\mathcal{N}|_{\pi^{-1}(w)}$ is nef, and $\mathcal{A}|_{\pi^{-1}(w)}$ is
ample, $\mathcal{L}|_{\pi^{-1}(w)}$ is ample by the usual Kleiman’s ampleness
criterion. Hence (i) follows from (iii).

######  Step 2.

In this step, we will prove that (iii) follows from (ii).

We assume that $\mathcal{L}$ is $\pi$-ample over some open neighborhood $U$ of
$W$. By replacing $Y$ with $U$, we may assume that $Y=U$. In the proof of
Lemma 11.4, we have already checked that $\mathcal{L}\cdot\Gamma>0$ for every
$\Gamma\in\operatorname{\overline{NE}}(X/Y;W)\setminus\\{0\\}$. This means
that (iii) follows from (ii).

We finish the proof. ∎

From now on, we always assume that the dimension of $N^{1}(X/Y;W)$ is finite.
In order to formulate the cone and contraction theorem, we need the following
definitions.

######  Definition 11.6.

Let $\pi\colon X\to Y$ be a projective morphism of complex analytic spaces and
let $W$ be a compact subset of $Y$. Let $X$ be a normal complex variety and
let $\Delta$ be an effective $\mathbb{R}$-divisor on $X$ such that
$K_{X}+\Delta$ is $\mathbb{R}$-Cartier. We assume that the dimension of
$N^{1}(X/Y;W)$ is finite. Then we define a subcone

$\operatorname{\overline{NE}}(X/Y;W)_{\operatorname{Nlc}(X,\Delta)}$

of $\operatorname{\overline{NE}}(X/Y;W)$ as the closure of the convex cone
spanned by the projective integral curves $C$ on
$\operatorname{Nlc}(X,\Delta)$ such that $\pi(C)$ is a point of $W$. Let $D$
be an element of $N^{1}(X/Y;W)$. We define

$D_{\geq 0}:=\\{z\in N_{1}(X/Y;W)\,|\,D\cdot z\geq 0\\}.$

Similarly, we can define $D_{>0}$, $D_{\leq 0}$, and $D_{<0}$. We also define

$D^{\perp}:=\\{z\in N_{1}(X/Y;W)\,|\,D\cdot z=0\\}.$

We use the following notation

$\operatorname{\overline{NE}}(X/Y;W)_{D\geq
0}:=\operatorname{\overline{NE}}(X/Y;W)\cap D_{\geq 0},$

and similarly for $>0$, $\leq 0$, and $<0$.

######  Definition 11.7.

An extremal face of the Kleiman–Mori cone
$\operatorname{\overline{NE}}(X/Y;W)$ is a non-zero subcone
$F\subset\operatorname{\overline{NE}}(X/Y;W)$ such that
$z,z^{\prime}\in\operatorname{\overline{NE}}(X/Y;W)$ and $z+z^{\prime}\in F$
imply that $z,z^{\prime}\in F$. Equivalently,
$F=\operatorname{\overline{NE}}(X/Y;W)\cap\mathcal{H}^{\perp}$ for some
$\mathbb{R}$-line bundle $\mathcal{H}$ which is defined on some open
neighborhood of $\pi^{-1}(W)$ and is $\pi$-nef over $W$. We call $\mathcal{H}$
a support function of $F$. An extremal ray is a one-dimensional extremal face.

  * (1)

An extremal face $F$ is called $(K_{X}+\Delta)$-negative if

$F\cap\operatorname{\overline{NE}}(X/Y;W)_{(K_{X}+\Delta)\geq 0}=\\{0\\}.$

  * (2)

An extremal face $F$ is called rational if we can choose a $\mathbb{Q}$-line
bundle $\mathcal{H}$, which is defined on some open neighborhood of
$\pi^{-1}(W)$ and is $\pi$-nef over $W$, as a support function of $F$.

  * (3)

An extremal face $F$ is called relatively ample at
$\operatorname{Nlc}(X,\Delta)$ if

$F\cap\operatorname{\overline{NE}}(X/Y;W)_{\operatorname{Nlc}(X,\Delta)}=\\{0\\}.$

Equivalently, $\mathcal{H}|_{\operatorname{Nlc}(X,\Delta)}$ is
$\pi|_{\operatorname{Nlc}(X,\Delta)}$-ample over $W$ for every support
function $\mathcal{H}$ of $F$.

  * (4)

An extremal face $F$ is called contractible at $\operatorname{Nlc}(X,\Delta)$
if it has a rational support function $\mathcal{H}$ such that
$\mathcal{H}|_{\operatorname{Nlc}(X,\Delta)}$ is
$\pi|_{\operatorname{Nlc}(X,\Delta)}$-semiample over some open neighborhood of
$W$.

We make a remark on (3) in Definition 11.7.

######  Remark 11.8.

In (3) in Definition 11.7, the condition that $F$ is relatively ample at
$\operatorname{Nlc}(X,\Delta)$ implies that the support function $\mathcal{H}$
of $F$ is positive on
$\operatorname{\overline{NE}}(X/Y;W)_{\operatorname{Nlc}(X,\Delta)}\setminus\\{0\\}$.
Let $\mathcal{A}$ be a $\pi$-ample $\mathbb{R}$-line bundle on $X$. Then
$\mathcal{N}:=\mathcal{H}-\varepsilon\mathcal{A}$ is positive on
$\operatorname{\overline{NE}}(X/Y;W)_{\operatorname{Nlc}(X,\Delta)}\setminus\\{0\\}$
for some $0<\varepsilon\ll 1$. Thus, it is easy to see that
$\mathcal{N}|_{\operatorname{Nlc}(X,\Delta)}$ is
$\pi|_{\operatorname{Nlc}(X,\Delta)}$-nef over $W$. Note that
$\mathcal{A}|_{\operatorname{Nlc}(X,\Delta)}$ is obviously
$\pi|_{\operatorname{Nlc}(X,\Delta)}$-ample. Hence
$\mathcal{H}|_{\operatorname{Nlc}(X,\Delta)}$ is
$\pi|_{\operatorname{Nlc}(X,\Delta)}$-ample over $W$.

###  11.1. Nakayama’s finiteness

In this subsection, we quickly recall Nakayama’s finiteness. As we saw above,
for the cone and contraction theorem in this paper, we need the assumption
that the dimension of $N^{1}(X/Y;W)$ is finite. In general, the dimension of
$N^{1}(X/Y;W)$ may be infinite. The author learned the following example from
Noboru Nakayama.

######  Example 11.9 (Nakayama).

Let $\pi\colon X\to Y$ be a projective surjective morphism of complex analytic
spaces such that $X$ is a normal complex variety with $\dim X\geq 2$ and that
$Y=\\{z\in\mathbb{C}\,|\,|z|<2\\}$. We put

$W:=\left\\{\left.\frac{1}{n}\,\right|\,n\in\mathbb{Z}_{>0}\right\\}\cup\\{0\\}.$

Then $W$ is a compact subset of $Y$. It is obvious that $W$ has infinitely
many connected components. In this case, we can see that the abelian group
$A^{1}(X/Y;W)$ is not finitely generated. Hence we have
$\dim_{\mathbb{R}}N^{1}(X/Y;W)=\infty$.

The following theorem gives an important and useful sufficient condition for
the finite-dimensionality of $N^{1}(X/Y;W)$. We state it here for the reader’s
convenience.

######  Theorem 11.10 (Nakayama’s finiteness, see [Na2, Chapter II. 5.19.
Lemma]).

Let $\pi\colon X\to Y$ be a projective surjective morphism of complex analytic
spaces such that $W$ is a compact subset of $Y$. We assume that $W\cap V$ has
only finitely many connected components for every analytic subset $V$ defined
over an open neighborhood of $W$. Then $A^{1}(X/Y;W)$ is a finitely generated
abelian group.

###### Proof.

For the details, see [Fu8, 4.1. Nakayama’s finiteness]. ∎