In this paper, we do not need Theorem 11.10 except in the proof of Corollary
1.5. We only need the assumption that the dimension of $N^{1}(X/Y;W)$ is
finite.

######  Remark 11.11.

Let $\pi\colon X\to Y$ be a smooth projective surjective morphism between
smooth irreducible complex analytic spaces. Let $W$ be a compact subset of
$Y$. Assume that $W$ is connected. Then we can easily check that the dimension
of $N^{1}(X/Y;W)$ is finite. However, $W\cap V$ may have infinitely many
connected components for some analytic subset $V$ defined over an open
neighborhood of $W$.

We close this subsection with some remarks on Nakayama’s fundamental paper
[Na1].

######  Remark 11.12.

Example 11.9 shows that [Na1, Proposition 4.3] is not correct. In [Fu1,
Section 4], we gave an alternative simple proof of [Na1, Theorem 5.5] (see
also [Fu1, 5.3]).

##  12\. Cone theorem

In this section, we will explain the cone and contraction theorem of normal
pairs for projective morphisms between complex analytic spaces. The proof
given in this section is essentially the same as that in [Fu2] for algebraic
varieties. The main ingredients of this section is the basepoint-free theorem
(see Theorem 9.1) and the rationality theorem (see Theorem 10.1).

We first treat the contraction theorem, which is a direct consequence of the
basepoint-free theorem: Theorem 9.1. We will use it in the proof of the cone
theorem: Theorem 12.2.

######  Theorem 12.1 (Contraction theorem).

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
$Y$ such 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. Let $\mathcal{H}$ be a line bundle which is defined on
some open neighborhood of $\pi^{-1}(W)$ and is $\pi$-nef over $W$ such that
the extremal face

$F=\mathcal{H}^{\perp}\cap\operatorname{\overline{NE}}(X/Y;W)$

is $(K_{X}+\Delta)$-negative and contractible at
$\operatorname{Nlc}(X,\Delta)$. Then, after shrinking $Y$ around $W$ suitably,
there exists a projective morphism $\varphi_{F}\colon X\to Z$ over $Y$ with
the following properties.

  * $(1)$

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 $F$.

  * $(2)$

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

  * $(3)$

Let $\mathcal{L}$ be a line bundle on $X$ such that $\mathcal{L}\cdot C=0$ for
every curve $C$ with $[C]\in F$. Assume that $\mathcal{L}^{\otimes
m}|_{\operatorname{Nlc}(X,\Delta)}$ is
$\varphi_{F}|_{\operatorname{Nlc}(X,\Delta)}$-generated for every $m\gg 0$.
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.

As we mentioned above, Theorem 12.1 easily follows from the basepoint-free
theorem: Theorem 9.1.

###### Proof of Theorem 12.1.

Since $F$ is contractible at $\operatorname{Nlc}(X,\Delta)$ by assumption, we
may assume that $\mathcal{H}|_{\operatorname{Nlc}(X,\Delta)}$ is
$\pi|_{\operatorname{Nlc}(X,\Delta)}$-semiample over some open neighborhood of
$W$. Since $F$ is $(K_{X}+\Delta)$-negative by assumption, we can take some
positive integer $a$ such that $a\mathcal{H}-(K_{X}+\Delta)$ is $\pi$-ample
over $W$. By the basepoint-free theorem (see Theorem 9.1), after shrinking $Y$
around $W$ suitably, $\mathcal{H}^{\otimes m}$ is $\pi$-generated for some
positive integer $m$. We take the Stein factorization of the associated
morphism. Then we can obtain a contraction morphism $\varphi_{F}\colon X\to Z$
over $Y$ satisfying the properties (1) and (2). We consider $\varphi_{F}\colon
X\to Z$ and $\operatorname{\overline{NE}}(X/Z;\pi_{Z}^{-1}(W))$, where
$\pi_{Z}\colon Z\to Y$ is the structure morphism. Then
$\operatorname{\overline{NE}}(X/Z;\pi_{Z}^{-1}(W))=F$ holds by construction,
$\mathcal{L}$ is numerically trivial over $\pi_{Z}^{-1}(W)$, and
$-(K_{X}+\Delta)$ is $\varphi_{F}$-ample over $\pi_{Z}^{-1}(W)$. We use the
basepoint-free theorem over $Z$ (see Theorem 9.1). Then, after shrinking $Z$
around $\pi^{-1}_{Z}(W)$ suitably, both $\mathcal{L}^{\otimes m}$ and
$\mathcal{L}^{\otimes(m+1)}$ are pull-backs of line bundles on $Z$. Their
difference gives a line bundle $\mathcal{L}_{Z}$ on $Z$ such that
$\mathcal{L}\simeq\varphi^{*}_{F}\mathcal{L}_{Z}$ holds. We finish the proof
of Theorem 12.1. ∎

The following theorem is the main result of this section, which is the cone
theorem of normal pairs for projective morphisms between complex analytic
spaces.

######  Theorem 12.2 (Cone theorem, see [Fu2, Theorem 16.6]).

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
$Y$ such 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 the following properties.

  * $(1)$

We can write

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

where $R_{j}$’s are the $(K_{X}+\Delta)$-negative extremal rays of
$\operatorname{\overline{NE}}(X/Y;W)$ that are rational and relatively ample
at $\operatorname{Nlc}(X,\Delta)$. In particular, each $R_{j}$ is spanned by
an integral curve $C_{j}$ on $X$ such that $\pi(C_{j})$ is a point of $W$.

  * $(2)$

Let $\mathcal{A}$ be a $\pi$-ample $\mathbb{R}$-line bundle defined on some
open neighborhood of $\pi^{-1}(W)$. 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
$\operatorname{\overline{NE}}(X/Y;W)_{(K_{X}+\Delta)<0}$.

  * $(3)$

Let $F$ be a $(K_{X}+\Delta)$-negative extremal face of
$\operatorname{\overline{NE}}(X/Y;W)$ that is relatively ample at
$\operatorname{Nlc}(X,\Delta)$. Then $F$ is a rational face. In particular,
$F$ is contractible at $\operatorname{Nlc}(X,\Delta)$.

By combining Theorem 12.2 with Theorem 12.1, we obtain the cone and
contraction theorem of normal pairs for projective morphisms between complex
analytic spaces.

###### Proof of Theorem 12.2.

Without loss of generality, we can freely shrink $Y$ around $W$ suitably
throughout this proof. From Step 1 to Step 5, we will prove Theorem 12.2 under
the extra assumption that $K_{X}+\Delta$ is $\mathbb{Q}$-Cartier. Then, in
Step 6, we will treat the general case. We note that we may assume that
$\dim_{\mathbb{R}}N_{1}(X/Y;W)\geq 2$ and $K_{X}+\Delta$ is not $\pi$-nef over
$W$. Otherwise, the theorem is obvious.

######  Step 1.

In this step, we will prove:

######  Claim 1.

When $K_{X}+\Delta$ is $\mathbb{Q}$-Cartier, the following equality

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

holds, where $F$’s vary among all rational proper $(K_{X}+\Delta)$-negative
extremal faces that are relatively ample at $\operatorname{Nlc}(X,\Delta)$.

We note that in Claim 1 —– denotes the closure with respect to the real
topology.

###### Proof of Claim 1.

We put

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

The inclusion $\operatorname{\overline{NE}}(X/Y;W)\supset\mathfrak{B}$
obviously holds by definition. We note that each $F$ is spanned by curves on
$X$ mapped to points in $W$ by Theorem 12.1 (1). From now on, we suppose
$\operatorname{\overline{NE}}(X/Y;W)\neq\mathfrak{B}$. Then we will derive a
contradiction. We can take a separating function $M$ which is a line bundle on
some open neighborhood of $\pi^{-1}(W)$ and is not a multiple of
$K_{X}+\Delta$ in $N^{1}(X/Y;W)$ such that $M>0$ on
$\mathfrak{B}\setminus\\{0\\}$ and $M\cdot z_{0}<0$ for some
$z_{0}\in\operatorname{\overline{NE}}(X/Y;W)$. Let $\mathcal{C}$ be the dual
cone of $\operatorname{\overline{NE}}(X/Y;W)_{(K_{X}+\Delta)\geq 0}$, that is,

$\mathcal{C}=\\{D\in N^{1}(X/Y;W)\,|\,D\cdot z\geq 0\ \text{for}\
z\in\operatorname{\overline{NE}}(X/Y;W)_{(K_{X}+\Delta)\geq 0}\\}.$

Then $\mathcal{C}$ is generated by $K_{X}+\Delta$ and $\mathbb{R}$-line
bundles on $X$ which are $\pi$-nef over $W$. Since $M>0$ on
$\operatorname{\overline{NE}}(X/Y;W)_{(K_{X}+\Delta)\geq 0}\setminus\\{0\\}$,
$M$ is in the interior of $\mathcal{C}$. Hence there exists a $\pi$-ample
$\mathbb{R}$-line bundle $A$ such that

$M-A=L^{\prime}+p(K_{X}+\Delta)$

in $N^{1}(X/Y;W)$, where $L^{\prime}$ is an $\mathbb{R}$-line bundle on some
open neighborhood of $\pi^{-1}(W)$ which is $\pi$-nef over $W$, and $p$ is a
non-negative rational number. Therefore, $M$ is expressed in the form

$M=H+p(K_{X}+\Delta)$

in $N^{1}(X/Y;W)$, where $H=A+L^{\prime}$ is a $\mathbb{Q}$-line bundle on $X$
which is $\pi$-ample over $W$. The rationality theorem (see Theorem 10.1)
implies that there exists a positive rational number $r<p$ such that

$L=H+r(K_{X}+\Delta)$

is $\pi$-nef over $W$ but not $\pi$-ample over $W$, and
$L|_{\operatorname{Nlc}(X,\Delta)}$ is
$\pi|_{\operatorname{Nlc}(X,\Delta)}$-ample over $W$. We note that $L\neq 0$
in $N^{1}(X/Y;W)$ since $M$ is not a multiple of $K_{X}+\Delta$. Thus the
extremal face $F_{L}$ associated to the support function $L$ is contained in
$\mathfrak{B}$, which implies $M>0$ on $F_{L}$. Therefore, $p<r$. It is a
contradiction. This completes the proof of Claim 1. ∎

######  Step 2.

In this step, we will prove:

######  Claim 2.

In the equality in Claim 1, we can assume that every extremal face $F$ is one-
dimensional.

###### Proof of Claim 2.

Let $F$ be a rational proper $(K_{X}+\Delta)$-negative extremal face that is
relatively ample at $\operatorname{Nlc}(X,\Delta)$. We assume that $\dim F\geq
2$. After shrinking $Y$ around $W$ suitably, we can take the contraction
morphism $\varphi_{F}:X\to Z$ over $Y$ associated to $F$ (see Theorem 12.1).
We note that $F=\operatorname{\overline{NE}}(X/Z;\pi^{-1}_{Z}(W))$, where
$\pi_{Z}\colon Z\to Y$ is the structure morphism, and that $-(K_{X}+\Delta)$
is $\varphi_{F}$-ample over $\pi^{-1}_{Z}(W)$ by construction. By Claim 1 in
Step 1, we obtain

(12.1)
$F=\operatorname{\overline{NE}}(X/Z;\pi^{-1}_{Z}(W))=\overline{\sum_{G}G},$

where the $G$’s in (12.1) are the rational proper $(K_{X}+\Delta)$-negative
extremal faces of $\operatorname{\overline{NE}}(X/Z;\pi^{-1}_{Z}(W))$. We note
that
$\operatorname{\overline{NE}}(X/Z;\pi^{-1}_{Z}(W))_{\operatorname{Nlc}(X,\Delta)}=0$
holds because $\varphi_{F}$ embeds $\operatorname{Nlc}(X,\Delta)$ into $Z$.
The $G$’s are also $(K_{X}+\Delta)$-negative extremal faces of
$\operatorname{\overline{NE}}(X/Y;W)$ that are ample at
$\operatorname{Nlc}(X,\Delta)$ with $\dim G<\dim F$. By induction, we finally
obtain

(12.2)
$\operatorname{\overline{NE}}(X/Y;W)=\overline{\operatorname{\overline{NE}}(X/Y;W)_{(K_{X}+\Delta)\geq
0}+\operatorname{\overline{NE}}(X/Y;W)_{\operatorname{Nlc}(X,\Delta)}+\sum
R_{j}},$

where the $R_{j}$’s are $(K_{X}+\Delta)$-negative rational extremal rays. Note
that each $R_{j}$ does not intersect
$\operatorname{\overline{NE}}(X/Y;W)_{\operatorname{Nlc}(X,\Delta)}$. We
finish the proof of Claim 2. ∎

######  Step 3.

In this step, we still assume that $K_{X}+\Delta$ is $\mathbb{Q}$-Cartier. We
will finish the proof of (1) when $K_{X}+\Delta$ is $\mathbb{Q}$-Cartier.

The contraction theorem (see Theorem 12.1) guarantees that for each extremal
ray $R_{j}$, which is $(K_{X}+\Delta)$-negative, rational, and relatively
ample at $\operatorname{Nlc}(X,\Delta)$, there exists a projective integral
curve $C_{j}$ on $X$ such that $[C_{j}]\in R_{j}$. Let $\psi_{j}:X\to Z_{j}$
be the contraction morphism of $R_{j}$ over $Y$ after shrinking $Y$ around $W$
suitably, and let $A$ be a $\pi$-ample line bundle on $X$. Let
$\pi_{Z_{j}}\colon Z_{j}\to Y$ be the structure morphism. We set

$r_{j}=-\frac{A\cdot C_{j}}{(K_{X}+\Delta)\cdot C_{j}}.$

Then $A+r_{j}(K_{X}+\Delta)$ is $\psi_{j}$-nef over $\pi^{-1}_{Z_{j}}(W)$ but
not $\psi_{j}$-ample over $\pi^{-1}_{Z_{j}}(W)$, and

$(A+r_{j}(K_{X}+\Delta))|_{\operatorname{Nlc}(X,\Delta)}$

is $\psi_{j}|_{\operatorname{Nlc}(X,\Delta)}$-ample over
$\pi^{-1}_{Z_{j}}(W)$. By the rationality theorem (see Theorem 10.1),
expressing $r_{j}=u_{j}/v_{j}$ with $u_{j},v_{j}\in\mathbb{Z}_{>0}$ and
$(u_{j},v_{j})=1$, we have the inequality $v_{j}\leq a(\dim X+1)$. After
shrinking $Y$ around $W$ suitably, we take $\pi$-ample line bundles
$H_{1},H_{2},\ldots,H_{\rho-1}$ on $X$ such that $K_{X}+\Delta$ and the
$H_{i}$’s form a basis of $N^{1}(X/Y;W)$, where
$\rho=\dim_{\mathbb{R}}N^{1}(X/Y;W)<\infty$. As we saw above, the intersection
of the extremal rays $R_{j}$ with the hyperplane

$\\{z\in N_{1}(X/Y;W)\,|\,a(K_{X}+\Delta)\cdot z=-1\\}$

in $N_{1}(X/Y;W)$ lie on the lattice

$\Lambda=\\{z\in N_{1}(X/Y;W)\,|\,a(K_{X}+\Delta)\cdot z=-1,H_{i}\cdot
z\in(a(a(\dim X+1))!)^{-1}\mathbb{Z}\\}.$

This implies that the extremal rays are discrete in the half space

$\\{z\in N_{1}(X/Y;W)\,|\,(K_{X}+\Delta)\cdot z<0\\}.$

Thus we can omit the closure sign —– from the formula (12.2) and this
completes the proof of (1) when $K_{X}+\Delta$ is $\mathbb{Q}$-Cartier.

######  Step 4.

In this step, we will prove (2) under the assumption that $K_{X}+\Delta$ is
$\mathbb{Q}$-Cartier.

Let $\mathcal{A}$ be a $\pi$-ample $\mathbb{R}$-line bundle on $X$. We choose
$0<\varepsilon_{i}\ll 1$ for $1\leq i\leq\rho-1$ such that
$\mathcal{A}-\sum_{i=1}^{\rho-1}\varepsilon_{i}H_{i}$ is still $\pi$-ample.
Then the $R_{j}$’s included in $(K_{X}+\Delta+\mathcal{A})_{<0}$ correspond to
some elements of the above lattice $\Lambda$ in Step 3 for which
$\sum_{i=1}^{\rho-1}\varepsilon_{i}H_{i}\cdot z<1/a$. Therefore, we obtain (2)
when $K_{X}+\Delta$ is $\mathbb{Q}$-Cartier.

######  Step 5.

In this step, we will prove (3) under the extra assumption that $K_{X}+\Delta$
is $\mathbb{Q}$-Cartier.

Let $F$ be a $(K_{X}+\Delta)$-negative extremal face as in (3). The vector
space $V=F^{\perp}\subset N^{1}(X/Y;W)$ is defined over $\mathbb{Q}$ because
$F$ is generated by some of the $R_{j}$’s. There exists a $\pi$-ample
$\mathbb{R}$-line bundle $\mathcal{A}$ such that $F$ is contained in
$(K_{X}+\Delta+\mathcal{A})_{<0}$. Let $\langle F\rangle$ be the vector space
spanned by $F$. We put

$\mathcal{C}_{F}:=\operatorname{\overline{NE}}(X/Y;W)_{(K_{X}+\Delta+\mathcal{A})\geq
0}+\operatorname{\overline{NE}}(X/Y;W)_{\operatorname{Nlc}(X,\Delta)}+\sum_{R_{j}\not\subset
F}R_{j}.$

Then $\mathcal{C}_{F}$ is a closed cone,

$\operatorname{\overline{NE}}(X/Y;W)=\mathcal{C}_{F}+F,$

and

$\mathcal{C}_{F}\cap\langle F\rangle=\\{0\\}.$

The support functions of $F$ are the elements of $V$ that are positive on
$\mathcal{C}_{F}\setminus\\{0\\}$. This is a non-empty open subset of $V$ and
thus it contains a rational element that, after scaling, gives a line bundle
$L$ defined over some open neighborhood of $W$ such that $L$ is $\pi$-nef over
$W$ and that $F=L^{\perp}\cap\operatorname{\overline{NE}}(X/Y;W)$. Therefore,
$F$ is rational. Hence, we obtain (3) when $K_{X}+\Delta$ is
$\mathbb{Q}$-Cartier.

We finish the proof of Theorem 12.2 under the extra assumption that
$K_{X}+\Delta$ is $\mathbb{Q}$-Cartier. Therefore, from now on, we can freely
use Theorem 12.2 when $K_{X}+\Delta$ is $\mathbb{Q}$-Cartier.

######  Step 6.

In this final step, we will treat the general case. This means that we will
treat the case where $K_{X}+\Delta$ is $\mathbb{R}$-Cartier.

Let $\mathcal{A}$ be a $\pi$-ample $\mathbb{R}$-line bundle on $X$. First we
will prove (2). By Lemma 4.4, after shrinking $Y$ around $W$ suitably, we can
take 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

$K_{X}+\Delta=\sum_{i=1}^{k}r_{i}(K_{X}+\Delta_{i})$

and that
$\mathcal{J}_{\operatorname{NLC}}(X,\Delta_{i})=\mathcal{J}_{\operatorname{NLC}}(X,\Delta)$
holds for every $i$. Since $K_{X}+\Delta_{i}$ is $\mathbb{Q}$-Cartier, there
are only finitely many $(K_{X}+\Delta_{i}+\mathcal{A})$-negarive extremal rays
of $\operatorname{\overline{NE}}(X/Y;W)$ which are rational and relatively
ample at $\operatorname{Nlc}(X,\Delta_{i})=\operatorname{Nlc}(X,\Delta)$ for
every $i$. Therefore, since

$K_{X}+\Delta+\mathcal{A}=\sum_{i=1}^{k}r_{i}(K_{X}+\Delta_{i}+\mathcal{A})$

holds, there exist only finitely many $(K_{X}+\Delta+\mathcal{A})$-negative
extremal rays of $\operatorname{\overline{NE}}(X/Y;W)$ which are rational and
relatively ample at $\operatorname{Nlc}(X,\Delta)$. Thus we obtain (2) in full
generality. The statement (1) is a direct and formal consequence of (2). For
the details, see, for example, the proof of [Ko, Chapter III. 1.2 Theorem].
Finally, we will prove (3). Let $F$ be a $(K_{X}+\Delta)$-negative extremal
face of $\operatorname{\overline{NE}}(X/Y;W)$ as in (3). By using Lemma 4.4,
after shrinking $Y$ around $W$ suitably, we can take an effective
$\mathbb{Q}$-divisor $\Delta^{\dagger}$ on $X$, which is sufficiently close to
$\Delta$, such that $K_{X}+\Delta^{\dagger}$ is $\mathbb{Q}$-Cartier,
$\mathcal{J}_{\operatorname{NLC}}(X,\Delta^{\dagger})=\mathcal{J}_{\operatorname{NLC}}(X,\Delta)$
holds, and $F$ is $(K_{X}+\Delta^{\dagger})$-negative. Therefore, we see that
$F$ is a rational face of $\operatorname{\overline{NE}}(X/Y;W)$. This is what
we wanted.

We finish the proof of the cone theorem. ∎

###  12.1. Proof of Theorem 1.4 and Corollary 1.5

In this subsection, we will prove Theorem 1.4 as an application of the
vanishing theorem for projective quasi-log schemes (see [Fu3, Theorem 6.3.5
(ii)]). Note that Corollary 1.5 is an easy consequence of Theorem 1.4. For the
details of the framework of quasi-log schemes, see [Fu3, Chapter 6], [Fu4],
[Fu6], and [Fu7]. Let us start with an easy lemma.

######  Lemma 12.3 (see [FM1, Lemma 4.2]).

Let $[V,\omega]$ be an irreducible positive-dimensional projective quasi-log
scheme with $\dim\operatorname{Nqlc}(V,\omega)=0$ or
$\operatorname{Nqlc}(V,\omega)=\emptyset$ and let $\mathcal{M}$ be an ample
line bundle on $V$. Assume that $\omega+r\mathcal{M}$ is numerically trivial
for some real number $r$. Then $r\leq\dim V+1$ holds.

###### Proof.

If $r\leq 0$, then $r\leq\dim V+1$ is obvious. Hence we may assume that $r$ is
positive. We consider the following short exact sequence:

(12.3)
$0\to\mathcal{I}_{\operatorname{Nqlc}(V,\omega)}\to\mathcal{O}_{V}\to\mathcal{O}_{\operatorname{Nqlc}(V,\omega)}\to
0,$

where $\mathcal{I}_{\operatorname{Nqlc}(V,\omega)}$ is the defining ideal
sheaf of $\operatorname{Nqlc}(V,\omega)$ on $V$. Since $l\mathcal{M}-\omega$
is ample for $l>-r$, we have

$H^{i}(V,\mathcal{I}_{\operatorname{Nqlc}(V,\omega)}\otimes\mathcal{M}^{\otimes
l})=0$

for every $i\neq 0$ and $l>-r$ by the vanishing theorem for quasi-log schemes
(see [Fu3, Theorem 6.3.5 (ii)]). Since $\dim\operatorname{Nqlc}(V,\omega)=0$
or $\operatorname{Nqlc}(V,\omega)=\emptyset$,

$H^{i}(V,\mathcal{O}_{\operatorname{Nqlc}(V,\omega)}\otimes\mathcal{M}^{\otimes
l})=0$

for every $i\neq 0$ and every $l$. Therefore, we obtain
$H^{i}(V,\mathcal{M}^{\otimes l})=0$ for every $i\neq 0$ and $l>-r$ by (12.3).
Let $V^{\prime}$ be the unique maximal (with respect to the inclusion) qlc
stratum of $[V,\omega]$. Then we have the following short exact sequence:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ker}\alpha\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathcal{O}_{V^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

such that $\dim\operatorname{Supp}\operatorname{Ker}\alpha\leq 0$ since
$\dim\operatorname{Nqlc}(V,\omega)=0$ or
$\operatorname{Nqlc}(V,\omega)=\emptyset$. Hence we have
$H^{i}(V^{\prime},\mathcal{M}^{\otimes l}|_{V^{\prime}})=0$ for every $i\neq
0$ and $l>-r$. Since $\dim V^{\prime}=\dim V>0$, it is obvious that
$H^{0}(V^{\prime},\mathcal{M}^{\otimes l}|_{V^{\prime}})=0$ holds for every
$l<0$. We consider

$\chi(t):=\sum_{i=0}^{\dim
V^{\prime}}(-1)^{i}\dim_{\mathbb{C}}H^{i}(V^{\prime},\mathcal{M}^{\otimes
t}|_{V^{\prime}}).$

Then it is well known that $\chi(t)$ is a non-trivial polynomial of
$\deg\chi(t)=\dim V^{\prime}=\dim V$. By the above observation, $\chi(l)=0$
for $l\in\mathbb{Z}$ with $-r<l<0$. This implies that $r\leq\dim V+1$. We
finish the proof. ∎

Let us start the proof of Theorem 1.4.

###### Proof of Theorem 1.4.

We put $\mathcal{L}:=f^{*}\mathcal{A}_{Y^{\flat}}$. Without loss of
generality, we may assume that $\dim X\geq 1$. Since $\mathcal{L}$ is
$\pi$-nef over $W$, $R$ is a $(K_{X}+\Delta)$-negative extremal ray of
$\operatorname{\overline{NE}}(X/Y;W)$. Therefore, by Theorem 12.2 (3) and
Theorem 12.1, after shrinking $Y$ around $W$ suitably, we obtain a contraction
morphism $\varphi_{R}\colon X\to Z$ over $Y$ associated to $R$. It is
sufficient to prove that $R\cdot\mathcal{L}=0$ holds.

######  Step 1.

In this step, we will treat the case where $\dim Z=0$.

From now on, we assume that $\dim Z=0$ holds. Then $X$ is projective with
$\rho(X)=1$ and $\mathcal{L}$ is a nef line bundle on $X$ in the usual sense.
Suppose that $\mathcal{L}$ is ample. Then we obtain that
$K_{X}+\Delta+r\mathcal{L}$ is numerically trivial for some $r>\dim X+1$ since
$(K_{X}+\Delta+(\dim X+1)\mathcal{L})\cdot R<0$ and $\rho(X)=1$. We note that
$[X,K_{X}+\Delta]$ naturally becomes an irreducible projective quasi-log
scheme with $\operatorname{Nqlc}(X,K_{X}+\Delta)=\emptyset$ (see, for example,
[Fu3, 6.4.1]). Therefore, we get a contradiction by Lemma 12.3. This implies
that $\mathcal{L}$ is numerically trivial, that is, $R\cdot
f^{*}\mathcal{A}_{Y^{\flat}}=R\cdot\mathcal{L}=0$. This is what we wanted.

######  Step 2.

In this step, we will treat the case where $\dim Z\geq 1$.

From now on, we assume that $\dim Z\geq 1$ holds. Then we can always take a
point $P\in Z$ such that $\dim\varphi^{-1}_{R}(P)\geq 1$. We shrink $Z$ around
$P$ and assume that $Z$ is Stein. Then we can take an effective
$\mathbb{R}$-Cartier divisor $B$ on $Z$ such that
$(X,\Delta+\varphi^{*}_{R}B)$ is log canonical outside $\varphi^{-1}_{R}(P)$,
there exists a positive-dimensional log canonical center $C$ of
$(X,\Delta+\varphi^{*}_{R}B)$ with $\varphi_{R}(C)=P$, and
$\dim\operatorname{Nlc}(X,\Delta+\varphi^{*}_{R}B)=0$ or
$\operatorname{Nlc}(X,\Delta+\varphi^{*}_{R}B)=\emptyset$. After shrinking $Z$
around $P$ suitably again, we can take a projective bimeromorphic morphism
$f\colon Y\to X$ from a smooth complex variety $Y$ such that $f^{-1}(C)$ and
the exceptional locus $\operatorname{Exc}(f)$ of $f$ are both simple normal
crossing divisors on $Y$ and that the union of $f^{-1}(C)$,
$\operatorname{Exc}(f)$, and
$\operatorname{Supp}\left(f^{-1}_{*}(\Delta+\varphi^{*}_{R}B)\right)$ is a
simple normal crossing divisor on $Y$ (see [BM2, Theorem 13.2]). We define
$B_{Y}$ by the formula $K_{Y}+B_{Y}=f^{*}(K_{X}+\Delta+\varphi^{*}B)$. Then we
see that $\operatorname{Supp}B_{Y}$ is a simple normal crossing divisor on
$Y$. By shrinking $X$ around $C$, we assume that $(X\setminus
C)\cap\operatorname{Nlc}(X,\Delta+\varphi^{*}_{R}B)=\emptyset$. Let $T$ be the
union of the irreducible components of $B^{=1}_{Y}$ that are mapped to $C$ by
$f$. We define $B_{T}$ by adjunction: $K_{T}+B_{T}=(K_{Y}+B_{Y})|_{T}$. We
consider the following short exact sequence:

$\begin{split}0&\to\mathcal{O}_{Y}(\lceil-(B^{<1}_{Y})\rceil-\lfloor
B^{>1}_{Y}\rfloor-T)\to\mathcal{O}_{Y}(\lceil-(B^{<1}_{Y})\rceil-\lfloor
B^{>1}_{Y}\rfloor)\\\ &\to\mathcal{O}_{T}(\lceil-(B^{<1}_{T})\rceil-\lfloor
B^{>1}_{T}\rfloor)\to 0.\end{split}$

We note that

(12.4) $\lceil-(B^{<1}_{Y})\rceil-\lfloor
B^{>1}_{Y}\rfloor-T-(K_{Y}+\\{B_{Y}\\}+B^{=1}_{Y}-T)=-f^{*}(K_{X}+\Delta+\varphi^{*}B).$

By taking $R^{i}f_{*}$, we have a long exact sequence:

(12.5) $\begin{split}0&\longrightarrow
f_{*}\mathcal{O}_{Y}(\lceil-(B^{<1}_{Y})\rceil-\lfloor
B^{>1}_{Y}\rfloor-T)\longrightarrow\mathcal{J}_{\operatorname{NLC}}(X,\Delta+\varphi^{*}B)\\\
&\longrightarrow f_{*}\mathcal{O}_{T}(\lceil-(B^{<1}_{T})\rceil-\lfloor
B^{>1}_{T}\rfloor)\overset{\delta}{\longrightarrow}R^{1}f_{*}\mathcal{O}_{Y}(\lceil-(B^{<1}_{Y})\rceil-\lfloor
B^{>1}_{Y}\rfloor-T)\longrightarrow\cdots.\end{split}$

The support of $f_{*}\mathcal{O}_{T}(\lceil-(B^{<1}_{T})\rceil-\lfloor
B^{>1}_{T}\rfloor)$ is contained in $C$ since $f(T)=C$. On the other hand, any
associated subvarieties of
$R^{1}f_{*}\mathcal{O}_{Y}(\lceil-(B^{<1}_{Y})\rceil-\lfloor
B^{>1}_{Y}\rfloor-T)$ are not contained in $C$ by (12.4) and Theorem 5.5 (i).
Hence, the connecting homomorphism $\delta$ in (12.5) is zero. We put

$\mathcal{J}:=f_{*}\mathcal{O}_{Y}(\lceil-(B^{<1}_{Y})\rceil-\lfloor
B^{>1}_{Y}\rfloor-T).$

Then it is an ideal sheaf contained in
$\mathcal{J}_{\operatorname{NLC}}(X,\Delta+\varphi^{*}B)$. Let $X^{\prime}$
denote the closed analytic subspace of $X$ defined by $\mathcal{J}$. By
applying the snake lemma to the following commutative diagram:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{J}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{J}_{\operatorname{NLC}}(X,\Delta+\varphi^{*}B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{f_{*}\mathcal{O}_{T}(\lceil-(B^{<1}_{T})\rceil-\lfloor
B^{>1}_{T}\rfloor)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

we obtain the short exact sequence:

$0\to f_{*}\mathcal{O}_{T}(\lceil-(B^{<1}_{T})\rceil-\lfloor
B^{>1}_{T}\rfloor)\to\mathcal{O}_{X^{\prime}}\to\mathcal{O}_{\operatorname{Nlc}(X,\Delta+\varphi^{*}B)}\to
0.$

Since $C$ is projective and $T$ is projective over $C$ by construction,

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

is a projective quasi-log scheme with
$\operatorname{Nqlc}(X^{\prime},(K_{X}+\Delta)|_{X^{\prime}})=\operatorname{Nlc}(X,\Delta+\varphi^{*}_{R}B)$
by [Fu4, Theorem 4.9]. In particular,
$\dim\operatorname{Nqlc}(X^{\prime},(K_{X}+\Delta)|_{X^{\prime}})=0$ or
$\operatorname{Nqlc}(X^{\prime},(K_{X}+\Delta)|_{X^{\prime}})=\emptyset$
holds. We note that $X^{\prime}=C$ holds set theoretically by construction. We
put $\omega:=(K_{X}+\Delta)|_{X^{\prime}}$. Then $-\omega$ is ample since
$\varphi_{R}(X^{\prime})=P$.

Suppose that $R\cdot\mathcal{L}>0$ holds. Then
$\mathcal{L}^{\prime}:=\mathcal{L}|_{X^{\prime}}$ is ample and
$\omega+r\mathcal{L}^{\prime}$ is numerically trivial on $X^{\prime}$ for some
positive real number $r$ with $r>\dim X+1>\dim X^{\prime}+1$. This is a
contradiction by Lemma 12.3. Hence we obtain $R\cdot\mathcal{L}=0$. Therefore,
$R$ is a $(K_{X}+\Delta)$-negative extremal ray of
$\operatorname{\overline{NE}}(X/{Y^{\flat}};g^{-1}(W))$.

We finish the proof. ∎

###### Proof of Corollary 1.5.

Let $P\in Y$ be any point. We put $W:=\\{P\\}$. Then the dimension of
$N^{1}(X/Y;W)$ is finite by Theorem 11.10. Suppose that $K_{X}+\Delta+(\dim
X+1)\mathcal{A}$ is not $\pi$-nef over $W$. Then there exists a
$(K_{X}+\Delta+(\dim X+1)\mathcal{A})$-negative extremal ray $R$ of
$\operatorname{\overline{NE}}(X/Y;W)$. We put $Y^{\flat}:=Y$,
$\mathcal{A}_{Y^{\flat}}:=\mathcal{A}$, and $f:=id_{Y}$. Then we use Theorem
1.4. Thus we obtain $R\cdot\mathcal{A}=0$. This is a contradiction since
$\mathcal{A}$ is $\pi$-ample over $W$. Therefore, $K_{X}+\Delta+(\dim
X+1)\mathcal{A}$ is $\pi$-nef over $W$. Since $P$ is any point of $Y$, this
means that $K_{X}+\Delta+(\dim X+1)\mathcal{A}$ is nef over $Y$. ∎

We close this section with a remark on Theorem 1.4 and Corollary 1.5.

######  Remark 12.4.

In Theorem 1.4 and Corollary 1.5, we can replace $(\dim X+1)$ with $\dim X$
when $\pi(X)$ is not a point. We can check it easily by the proof of Theorem
1.4.

##  13\. Lengths of extremal rational curves

In this section, we will quickly explain that every extremal ray is spanned by
a rational curve. Our result in this section generalizes Kawamata’s famous
result in [Ka]. We first prove the following theorem as an application of
[Fu6, Theorem 1.12].

######  Theorem 13.1.

Let $\varphi\colon X\to Z$ be a projective morphism of complex analytic spaces
such that $X$ is a normal complex variety and let $\Delta$ be an effective
$\mathbb{R}$-divisor on $X$ such that $K_{X}+\Delta$ is $\mathbb{R}$-Cartier.
Assume that $-(K_{X}+\Delta)$ is $\varphi$-ample. Let $P$ be an arbitrary
point of $Z$. Let $E$ be any positive-dimensional irreducible component of
$\varphi^{-1}(P)$ such that $E\not\subset\operatorname{Nlc}(X,\Delta)$. Then
$E$ is covered by possibly singular rational curves $\ell$ with

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

In particular, $E$ is uniruled.

In the proof of Theorem 13.1, we will use the theory of quasi-log schemes (see
[Fu3, Chapter 6], [Fu4], [Fu6], and [Fu7]).

###### Proof of Theorem 13.1.

If $\varphi(X)=P$, then $E=X$ obviously holds. In this case, the statement
follows from [Fu6, Theorem 1.12] since we can see $[X,K_{X}+\Delta]$ as a
projective quasi-log scheme (see, for example, [Fu3, 6.4.1]). Therefore, from
now on, we may assume that $\varphi(X)\neq P$. We shrink $Z$ around $P$ and
may assume that $Z$ is Stein. Then we can take an effective
$\mathbb{R}$-Cartier divisor $B$ on $Z$ such that $E$ is a log canonical
center of $(X,\Delta+\varphi^{*}B)$. After shrinking $Z$ around $P$ suitably
again, we can take a projective bimeromorphic morphism $f\colon Y\to X$ from a
smooth complex variety $Y$ such that $f^{-1}(E)$ is a simple normal crossing
divisor on $Y$,

$K_{Y}+B_{Y}:=f^{*}(K_{X}+\Delta+\varphi^{*}B),$

and $\operatorname{Supp}B_{Y}$ is a simple normal crossing divisor on $Y$ (see
[BM2, Theorem 13.2]). We may further assume that the support of the union of
$f^{-1}(E)$ and $\operatorname{Supp}B_{Y}$ is also a simple normal crossing
divisor on $Y$. Let $T$ be the union of the irreducible components of
$B^{=1}_{Y}$ that are mapped to $E$ by $f$. We put
$A:=\lceil-(B^{<1}_{Y})\rceil$ and $N:=\lfloor B^{>1}_{Y}\rfloor$ and consider
the following short exact sequence:

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

We note that

(13.1)
$A-N-T-(K_{Y}+\\{B_{Y}\\}+B^{=1}_{Y}-T)=-f^{*}(K_{X}+\Delta+\varphi^{*}B).$

By taking $R^{i}f_{*}$, we have a long exact sequence:

(13.2) $\begin{split}0&\longrightarrow
f_{*}\mathcal{O}_{Y}(A-N-T)\longrightarrow\mathcal{J}_{\operatorname{NLC}}(X,\Delta+\varphi^{*}B)\longrightarrow
f_{*}\mathcal{O}_{T}(A-N)\\\
&\overset{\delta}{\longrightarrow}R^{1}f_{*}\mathcal{O}_{Y}(A-N-T)\longrightarrow\cdots.\end{split}$

The support of $f_{*}\mathcal{O}_{T}(A-N)$ is contained in $E$ since $f(T)=E$.
On the other hand, any associated subvarieties of
$R^{1}f_{*}\mathcal{O}_{Y}(A-N-T)$ are not contained in $E$ by (13.1) and
Theorem 5.5 (i). Hence, the connecting homomorphism $\delta$ in (13.2) is
zero. We put $\mathcal{J}:=f_{*}\mathcal{O}_{Y}(A-N-T)$. Then it is an ideal
sheaf contained in $\mathcal{J}_{\operatorname{NLC}}(X,\Delta+\varphi^{*}B)$.
Let $X^{\prime}$ denote the closed analytic subspace of $X$ defined by
$\mathcal{J}$. Thus we obtain the following big commutative diagram.

---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{J}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{J}_{\operatorname{NLC}}(X,\Delta+\varphi^{*}B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{f_{*}\mathcal{O}_{T}(A-N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\mathcal{O}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{f_{*}\mathcal{O}_{T}(A-N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{X^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{\operatorname{Nlc}(X,\Delta+\varphi^{*}B)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$

We note that $X^{\prime}=E\cup\operatorname{Nlc}(X,\Delta+\varphi^{*}B)$ holds
set theoretically. On $X^{\prime}$, by the above big commutative diagram, we
see that $\operatorname{Nlc}(X,\Delta+\varphi^{*}B)$ is defined by the ideal
sheaf $f_{*}\mathcal{O}_{T}(A-N)$. We write $T=T^{\prime}+T^{\prime\prime}$,
where $T^{\prime\prime}$ is the union of the irreducible components of $T$
mapped to $\operatorname{Nlc}(X,\Delta+\varphi^{*}B)$ by $f$. We put
$\mathcal{I}:=f_{*}\mathcal{O}_{T^{\prime\prime}}(A-N-T^{\prime})$. Then

$\mathcal{I}=f_{*}\mathcal{O}_{T^{\prime\prime}}(A-N-T^{\prime})\subset
f_{*}\mathcal{O}_{T}(A-N)\subset\mathcal{O}_{X^{\prime}}.$

Since $\mathcal{I}\subset f_{*}\mathcal{O}_{T}(A-N)$, $\mathcal{I}$ is zero
when it is restricted to $\operatorname{Nlc}(X,\Delta+\varphi^{*}B)$. Since
$f(T^{\prime\prime})\subset\operatorname{Nlc}(X,\Delta+\varphi^{*}B)$,
$\mathcal{I}$ is zero on
$X^{\prime}\setminus\operatorname{Nlc}(X,\Delta+\varphi^{*}B)$. Thus, we have
$\mathcal{I}=\\{0\\}$. By construction, we see that $E$ is a closed analytic
subvariety of $X^{\prime}$. Let $\mathcal{I}_{E}$ be the defining ideal sheaf
of $E$ on $X^{\prime}$. Then we obtain

$\mathcal{I}_{E}\cap f_{*}\mathcal{O}_{T}(A-N)\subset
f_{*}\mathcal{O}_{T^{\prime\prime}}(A-N-T^{\prime})=\mathcal{I}=\\{0\\}.$

Hence, we can see $f_{*}\mathcal{O}_{T}(A-N)$ as an ideal sheaf on $E$. Since
$E$ is projective and $T$ is projective over $E$ by construction,

$\left(E,(K_{X}+\Delta)|_{E},f\colon(T,B_{T})\to E\right)$

is a projective quasi-log scheme, where $K_{T}+B_{T}:=(K_{Y}+B_{Y})|_{T}$, by
[Fu4, Theorem 4.9]. Since $\varphi(E)=P$, $-(K_{X}+\Delta)|_{E}$ is ample.
Thus, by [Fu6, Theorem 1.12], $E$ is covered by possibly singular rational
curves $\ell$ with $0<-(K_{X}+\Delta)\cdot\ell\leq 2\dim E$. In particular,
this implies that $E$ is uniruled. ∎

Theorem 13.2 is an easy consequence of Theorem 13.1. It seems to be
indispensable for the minimal model program with scaling.

######  Theorem 13.2 (Lengths of extremal rational curves).

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
$Y$ such 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. If $R$ is a $(K_{X}+\Delta)$-negative extremal ray of
$\operatorname{\overline{NE}}(X/Y;W)$ which is relatively ample at
$\operatorname{Nlc}(X,\Delta)$, then there exists a rational curve $\ell$
spanning $R$ with

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

###### Proof.

By the cone and contraction theorem (see Theorems 12.1 and 12.2), after
shrinking $Y$ around $W$ suitably, we obtain a contraction morphism
$\varphi\colon X\to Z$ over $Y$ associated to $R$. We note that
$-(K_{X}+\Delta)$ is $\varphi$-ample and
$\varphi\colon\operatorname{Nlc}(X,\Delta)\to\varphi(\operatorname{Nlc}(X,\Delta))$
is finite by construction. Therefore, we can find a rational curve $\ell$ in a
fiber of $\varphi$ with $0<-(K_{X}+\Delta)\cdot\ell\leq 2\dim X$ by Theorem
13.1. This $\ell$ is a desired rational curve spanning $R$. ∎

##  14\. On Shokurov’s polytopes

In this section, we will discuss Shokurov’s polytopes for projective morphisms
of complex analytic spaces. Here, we will follow the presentation in [Fu3,
Section 4.7]. Let us recall the definition of extremal curves.

######  Definition 14.1 (Extremal curves).

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
$Y$ such that the dimension of $N^{1}(X/Y;W)$ is finite. A curve $\Gamma$ on
$X$ is called extremal over $W$ if the following properties hold.

  * (i)

$\Gamma$ generates an extremal ray $R$ of
$\operatorname{\overline{NE}}(X/Y;W)$.

  * (ii)

There exists a $\pi$-ample line bundle $\mathcal{H}$ over some open
neighborhood of $W$ such that

$\mathcal{H}\cdot\Gamma=\min_{\ell}\\{\mathcal{H}\cdot\ell\\},$

where $\ell$ ranges over curves generating $R$.

By Lemma 13.2, we have:

######  Lemma 14.2.

Let $\pi\colon X\to Y$ be a projective morphism of complex analytic spaces and
let $(X,\Delta)$ be a log canonical pair. Let $W$ be a compact subset of $Y$
such that the dimension of $N^{1}(X/Y;W)$ is finite. Let $R$ be a
$(K_{X}+\Delta)$-negative extremal ray of
$\operatorname{\overline{NE}}(X/Y;W)$. If $\Gamma$ is an extremal curve over
$W$ generating $R$, then

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

holds.

###### Proof.

By Theorem 13.2, we can take a rational curve $\ell$ spanning $R$ such that

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

Let $\mathcal{H}$ be a line bundle as in Definition 14.1. Then

$\frac{-(K_{X}+\Delta)\cdot\Gamma}{\mathcal{H}\cdot\Gamma}=\frac{-(K_{X}+\Delta)\cdot\ell}{\mathcal{H}\cdot\ell}.$

holds. Hence we obtain

$0<-(K_{X}+\Delta)\cdot\Gamma=\left(-(K_{X}+\Delta)\cdot\ell\right)\cdot\frac{\mathcal{H}\cdot\Gamma}{\mathcal{H}\cdot\ell}\leq
2\dim X.$

This is what we wanted. ∎

One of the main purposes of this section is to explain the following theorem,
which is very well known and has already played an important role when
$\pi\colon X\to Y$ is algebraic.

######  Theorem 14.3.

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
$Y$ such that the dimension of $N^{1}(X/Y;W)$ is finite. Let $V$ be a finite-
dimensional affine subspace of $\operatorname{WDiv}_{\mathbb{R}}(X)$, which is
defined over the rationals. We fix an $\mathbb{R}$-divisor
$\Delta\in\mathcal{L}(V;\pi^{-1}(W))$, that is, $\Delta\in V$ and $(X,\Delta)$
is log canonical at $\pi^{-1}(W)$. Then we can find positive real numbers
$\alpha$ and $\delta$, which depend on $(X,\Delta)$ and $V$, with the
following properties.

  * (1)

If $\Gamma$ is any extremal curve over $W$ and $(K_{X}+\Delta)\cdot\Gamma>0$,
then $(K_{X}+\Delta)\cdot\Gamma>\alpha$.

  * (2)

If $D\in\mathcal{L}(V;\pi^{-1}(W))$, $|\\!|D-\Delta|\\!|<\delta$, and
$(K_{X}+D)\cdot\Gamma\leq 0$ for an extremal curve $\Gamma$ over $W$, then
$(K_{X}+\Delta)\cdot\Gamma\leq 0$.

  * (3)

Let $\\{R_{t}\\}_{t\in T}$ be any set of extremal rays of
$\operatorname{\overline{NE}}(X/Y;W)$. Then

$\mathcal{N}_{T}:=\\{D\in\mathcal{L}(V;\pi^{-1}(W))\,|\,{\text{$(K_{X}+D)\cdot
R_{t}\geq 0$ for every $t\in T$}}\\}$

is a rational polytope in $V$. In particular,

$\mathcal{N}^{\sharp}_{\pi}(V;W):=\\{\Delta\in\mathcal{L}(V;\pi^{-1}(W))\,|\,{\text{$K_{X}+\Delta$
is nef over $W$}}\\}$

is a rational polytope.

###### Proof of Theorem 14.3.

Throughout this proof, we can freely shrink $Y$ around $W$ suitably. We first
note that $\mathcal{L}(V;\pi^{-1}(W))$ is a rational polytope in $V$ (see
2.10).

(1) If $\Delta$ is a $\mathbb{Q}$-divisor, then we may assume that
$m(K_{X}+\Delta)$ is Cartier for some positive integer $m$ by shrinking $Y$
around $W$ suitably. Therefore, the statement is obvious even if $\Gamma$ is
not extremal. Hence, from now on, we assume that $\Delta$ is not a
$\mathbb{Q}$-divisor. Then we can write
$K_{X}+\Delta=\sum_{j}a_{j}(K_{X}+D_{j})$ as in Lemma 4.4. This means that
$a_{j}$ is a positive real number for every $j$ with $\sum_{j}a_{j}=1$ and
that $D_{j}\in\mathcal{L}(V;\pi^{-1}(W))$ is a $\mathbb{Q}$-divisor for every
$j$. Thus we have
$(K_{X}+\Delta)\cdot\Gamma=\sum_{j}a_{j}(K_{X}+D_{j})\cdot\Gamma$. If
$(K_{X}+\Delta)\cdot\Gamma<1$, then

$\displaystyle-2\dim X\leq(K_{X}+D_{j_{0}})\cdot\Gamma$
$\displaystyle<\frac{1}{a_{j_{0}}}\left\\{-\sum_{j\neq
j_{0}}a_{j}(K_{X}+D_{j})\cdot\Gamma+1\right\\}$ $\displaystyle\leq\frac{2\dim
X+1}{a_{j_{0}}}$

for $a_{j_{0}}\neq 0$. This is because $(K_{X}+D_{j})\cdot\Gamma\geq-2\dim X$
holds for every $j$ by Lemma 14.2. Thus there are only finitely many
possibilities of the intersection numbers $(K_{X}+D_{j})\cdot\Gamma$ for
$a_{j}\neq 0$ when $(K_{X}+\Delta)\cdot\Gamma<1$. Therefore, the existence of
$\alpha$ is obvious.

(2) If we take $\delta$ sufficiently small, then, for every
$D\in\mathcal{L}(V;\pi^{-1}(W))$ with $|\\!|D-\Delta|\\!|<\delta$, we can
always find $D^{\prime}\in\mathcal{L}(V;\pi^{-1}(W))$ such that

$K_{X}+D=(1-s)(K_{X}+\Delta)+s(K_{X}+D^{\prime})$

with

$0\leq s\leq\frac{\alpha}{\alpha+2\dim X}.$

Since $\Gamma$ is extremal, we have $(K_{X}+D^{\prime})\cdot\Gamma\geq-2\dim
X$ for every $D^{\prime}\in\mathcal{L}(V;\pi^{-1}(W))$ by Lemma 14.2. We
assume that $(K_{X}+\Delta)\cdot\Gamma>0$. Then
$(K_{X}+\Delta)\cdot\Gamma>\alpha$ by (1). Therefore,

$\displaystyle(K_{X}+D)\cdot\Gamma$
$\displaystyle=(1-s)(K_{X}+\Delta)\cdot\Gamma+s(K_{X}+D^{\prime})\cdot\Gamma$
$\displaystyle>(1-s)\alpha+s(-2\dim X)\geq 0.$

This is a contradiction. Hence, we obtain $(K_{X}+\Delta)\cdot\Gamma\leq 0$.
We complete the proof of (2).

(3) For every $t\in T$, we may assume that there is some
$D(t)\in\mathcal{L}(V;\pi^{-1}(W))$ such that $(K_{X}+D(t))\cdot R_{t}<0$. Let
$B_{1},\ldots,B_{r}$ be the vertices of $\mathcal{L}(V;\pi^{-1}(E))$. We note
that $(K_{X}+D)\cdot R_{t}<0$ for some $D\in\mathcal{L}(V;\pi^{-1}(W))$
implies $(K_{X}+B_{j})\cdot R_{t}<0$ for some $j$. Therefore, we may assume
that $T$ is contained in $\mathbb{N}$. This is because there are only
countably many $(K_{X}+B_{j})$-negative extremal rays for every $j$ by the
cone theorem (see Theorem 12.2). We note that $\mathcal{N}_{T}$ is a closed
convex subset of $\mathcal{L}(V;\pi^{-1}(W))$ by definition. If $T$ is a
finite set, then the claim is obvious. Thus, we may assume that
$T=\mathbb{N}$. By (2) and by the compactness of $\mathcal{N}_{T}$, we can
take $\Delta_{1},\ldots,\Delta_{n}\in\mathcal{N}_{T}$ and
$\delta_{1},\ldots,\delta_{n}>0$ such that $\mathcal{N}_{T}$ is covered by

$\mathcal{B}_{i}=\\{D\in\mathcal{L}(V;\pi^{-1}(W))\,|\,|\\!|D-\Delta_{i}|\\!|<\delta_{i}\\}$

and that if $D\in\mathcal{B}_{i}$ with $(K_{X}+D)\cdot R_{t}<0$ for some $t$,
then $(K_{X}+\Delta_{i})\cdot R_{t}=0$. If we put

$T_{i}=\\{t\in T\,|\,(K_{X}+D)\cdot R_{t}<0\ \text{for some}\
D\in\mathcal{B}_{i}\\},$

then $(K_{X}+\Delta_{i})\cdot R_{t}=0$ for every $t\in T_{i}$ by the above
construction. Since $\\{\mathcal{B}_{i}\\}_{i=1}^{n}$ gives an open covering
of $\mathcal{N}_{T}$, we have $\mathcal{N}_{T}=\bigcap_{1\leq i\leq
n}\mathcal{N}_{T_{i}}$ by the following claim.

######  Claim 1.

$\mathcal{N}_{T}=\bigcap_{1\leq i\leq n}\mathcal{N}_{T_{i}}$.

###### Proof of Claim 1.

We note that $\mathcal{N}_{T}\subset\bigcap_{1\leq i\leq
n}\mathcal{N}_{T_{i}}$ is obvious. Suppose that
$\mathcal{N}_{T}\subsetneq\bigcap_{1\leq i\leq n}\mathcal{N}_{T_{i}}$ holds.
We take $D\in\bigcap_{1\leq i\leq
n}\mathcal{N}_{T_{i}}\setminus\mathcal{N}_{T}$ which is very close to
$\mathcal{N}_{T}$. Since $\mathcal{N}_{T}$ is covered by
$\\{\mathcal{B}_{i}\\}_{i=1}^{n}$, there is some $i_{0}$ such that
$D\in\mathcal{B}_{i_{0}}$. Since $D\not\in\mathcal{N}_{T}$, there is some
$t_{0}\in T$ such that $(K_{X}+D)\cdot R_{t_{0}}<0$. Thus, $t_{0}\in
T_{i_{0}}$. This is a contradiction because $D\in\mathcal{N}_{T_{i_{0}}}$.
Therefore, we obtain the desired equality $\mathcal{N}_{T}=\bigcap_{1\leq
i\leq n}\mathcal{N}_{T_{i}}$. ∎

Therefore, it is sufficient to see that each $\mathcal{N}_{T_{i}}$ is a
rational polytope in $V$. By replacing $T$ with $T_{i}$, we may assume that
there is some $D\in\mathcal{N}_{T}$ such that $(K_{X}+D)\cdot R_{t}=0$ for
every $t\in T$.

######  Claim 2.

If $\dim_{\mathbb{R}}\mathcal{L}(V;\pi^{-1}(W))=1$, then $\mathcal{N}_{T}$ is
a rational polytope in $V$.

###### Proof of Claim 2.

As we explained above, we can take some $D\in\mathcal{N}_{T}$ such that
$(K_{X}+D)\cdot R_{t}=0$ for every $t\in T$. Since
$\dim_{\mathbb{R}}\mathcal{L}(V;\pi^{-1}(W))=1$, we can write

$K_{X}+D=b_{1}(K_{X}+D_{1})+b_{2}(K_{X}+D_{2})$

such that $K_{X}+D_{i}\in\mathcal{L}(V;\pi^{-1}(W))$, $D_{i}$ is a
$\mathbb{Q}$-divisor, and $0\leq b_{i}\leq 1$ for $i=1,2$, and
$b_{1}+b_{2}=1$. By $(K_{X}+D)\cdot R_{t}=0$, we see that $b_{1}$ and $b_{2}$
are rational numbers. This implies that $D$ is a $\mathbb{Q}$-divisor.
Therefore, $\mathcal{N}_{T}$ is a rational polytope in $V$. ∎

Hence we assume $\dim_{\mathbb{R}}\mathcal{L}(V;\pi^{-1}(W))>1$. Let
$\mathcal{L}^{1},\ldots,\mathcal{L}^{p}$ be the proper faces of
$\mathcal{L}(V;\pi^{-1}(W))$. Then
$\mathcal{N}_{T}^{i}=\mathcal{N}_{T}\cap\mathcal{L}^{i}$ is a rational
polytope by induction on dimension. Moreover, for each
$D^{\prime\prime}\in\mathcal{N}_{T}$ which is not $D$, there is $D^{\prime}$
on some proper face of $\mathcal{L}(V;\pi^{-1}(W))$ such that
$D^{\prime\prime}$ is on the line segment determined by $D$ and $D^{\prime}$.
Note that $(K_{X}+D)\cdot R_{t}=0$ for every $t\in T$. Therefore, if
$D^{\prime}\in\mathcal{L}^{i}$, then $D^{\prime}\in\mathcal{N}_{T}^{i}$. Thus,
$\mathcal{N}_{T}$ is the convex hull of $D$ and all the $\mathcal{N}_{T}^{i}$.
There is a finite subset $T^{\prime}\subset T$ such that

$\bigcup_{i}\mathcal{N}_{T}^{i}=\mathcal{N}_{T^{\prime}}\cap(\bigcup_{i}\mathcal{L}^{i}).$

Therefore, the convex hull of $D$ and $\bigcup_{i}\mathcal{N}_{T}^{i}$ is just
$\mathcal{N}_{T^{\prime}}$. We complete the proof of (3). ∎

As an application of Theorem 14.3, we have:

######  Theorem 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
$Y$ such that the dimension of $N^{1}(X/Y;W)$ is finite. Let $(X,\Delta)$ be a
log canonical pair and let $H$ be an effective $\mathbb{R}$-Cartier
$\mathbb{R}$-divisor on $X$ such that $(X,\Delta+H)$ is log canonical and that
$K_{X}+\Delta+H$ is nef over $W$. Then, either $K_{X}+\Delta$ is nef over $W$
or there is 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

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

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

###### Proof.

We assume that $K_{X}+\Delta$ is not $\pi$-nef over $W$. Let $\\{R_{j}\\}$ be
the set of the $(K_{X}+\Delta)$-negative extremal rays of
$\operatorname{\overline{NE}}(X/Y;W)$. Let $C_{j}$ be an extremal curve over
$W$ spanning $R_{j}$ for every $j$. We put
$\mu=\underset{j}{\sup}\\{\mu_{j}\\}$, where

$\mu_{j}=\frac{-(K_{X}+\Delta)\cdot C_{j}}{H\cdot C_{j}}.$

By definition, it is obvious that $\lambda=\mu$ and $0<\mu\leq 1$ hold. Hence
it is sufficient to prove that $\mu=\mu_{j_{0}}$ for some $j_{0}$. By Lemma
4.4, after shrinking $Y$ around $W$ suitably, we can find effective
$\mathbb{Q}$-divisors $\Delta_{1},\ldots,\Delta_{k}$ and positive real numbers
$r_{1},\ldots,r_{k}$ with $\sum_{i=1}^{k}r_{i}=1$ such that
$m(K_{X}+\Delta_{i})$ is Cartier for every $i$,
$\Delta=\sum_{i=1}^{k}r_{i}\Delta_{i}$ holds, and $(X,\Delta_{i})$ is log
canonical for every $i$. Therefore, by Lemma 14.2, we can write

$-(K_{X}+\Delta)\cdot C_{j}=\sum_{i=1}^{l}\frac{r_{i}n_{ij}}{m}>0,$

where $n_{ij}$ is an integer with $n_{ij}\leq 2m\dim X$ for every $i$ and $j$
since $C_{j}$ is extremal over $W$. If $(K_{X}+\Delta+H)\cdot R_{j_{0}}=0$ for
some $j_{0}$, then there are nothing to prove since $\lambda=1$ and
$(K_{X}+\Delta+H)\cdot R=0$ with $R=R_{j_{0}}$. Thus, we assume that
$(K_{X}+\Delta+H)\cdot R_{j}>0$ for every $j$. We put
$F=\operatorname{Supp}(\Delta+H)$. Let $F=\sum_{k}F_{k}$ be the irreducible
decomposition. We put $V=\bigoplus_{k}\mathbb{R}F_{k}$,

$\mathcal{L}(V;\pi^{-1}(W)):=\\{D\in V\,|\,(X,D)\ \text{is log canonical at
$\pi^{-1}(W)$}\\},$

and

$\mathcal{N}:=\\{D\in\mathcal{L}(V;\pi^{-1}(W))\,|\,(K_{X}+D)\cdot R_{j}\geq
0\ \text{for every $j$}\\}.$

Then $\mathcal{N}$ is a rational polytope in $V$ by Theorem 14.3 (3) and
$\Delta+H$ is in the relative interior of $\mathcal{N}$ by the above
assumption. Therefore, after shrinking $Y$ around $W$ suitably again, we can
write

$K_{X}+\Delta+H=\sum_{p=1}^{q}r^{\prime}_{p}(K_{X}+D_{p}),$

where $r^{\prime}_{1},\ldots,r^{\prime}_{q}$ are positive real numbers such
that $\sum_{p}r^{\prime}_{p}=1$, $(X,D_{p})$ is log canonical for every $p$,
$m^{\prime}(K_{X}+D_{p})$ is Cartier for some positive integer $m^{\prime}$
and every $p$, and $(K_{X}+D_{p})\cdot C_{j}>0$ for every $p$ and $j$. So, we
obtain

$(K_{X}+\Delta+H)\cdot
C_{j}=\sum_{p=1}^{q}\frac{r^{\prime}_{p}n^{\prime}_{pj}}{m^{\prime}}$

with $0<n^{\prime}_{pj}=m^{\prime}(K_{X}+D_{p})\cdot C_{j}\in\mathbb{Z}$. Note
that $m^{\prime}$ and $r^{\prime}_{p}$ are independent of $j$ for every $p$.
We also note that

$\displaystyle\frac{1}{\mu_{j}}=\frac{H\cdot C_{j}}{-(K_{X}+\Delta)\cdot
C_{j}}$ $\displaystyle=\frac{(K_{X}+\Delta+H)\cdot C_{j}}{-(K_{X}+\Delta)\cdot
C_{j}}+1$
$\displaystyle=\frac{m\sum_{p=1}^{q}r^{\prime}_{p}n^{\prime}_{pj}}{m^{\prime}\sum_{i=1}^{l}r_{i}n_{ij}}+1.$

Since

$\sum_{i=1}^{l}\frac{r_{i}n_{ij}}{m}>0$

for every $j$ and $n_{ij}\leq 2m\dim X$ with $n_{ij}\in\mathbb{Z}$ for every
$i$ and $j$, the number of the set $\\{n_{ij}\\}_{i,j}$ is finite. Thus,

$\inf_{j}\left\\{\frac{1}{\mu_{j}}\right\\}=\frac{1}{\mu_{j_{0}}}$

for some $j_{0}$. Therefore, we obtain $\mu=\mu_{j_{0}}$. We finish the proof.
∎

##  15\. Basepoint-free theorem for $\mathbb{R}$-divisors

In this section, we will discuss the basepoint-free theorem for
$\mathbb{R}$-divisors, although we do not need it in this paper. The proof of
Theorem 15.1 needs the cone theorem (see Theorem 12.2). Hence Theorem 15.1
looks much deeper than the basepoint-free theorem for Cartier divisors (see
Theorem 9.1).

######  Theorem 15.1 (Basepoint-free theorem for $\mathbb{R}$-divisors).

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
$Y$ such that the dimension of $N^{1}(X/Y;W)$ is finite. Let $\Delta$ be an
effective $\mathbb{R}$-divisor on $X$ such that $(X,\Delta)$ is log canonical.
Let $D$ be an $\mathbb{R}$-Cartier $\mathbb{R}$-divisor defined on some open
neighborhood of $\pi^{-1}(W)$ such that $D$ is $\pi$-nef over $W$. Assume that
$aD-(K_{X}+\Delta)$ is $\pi$-ample over $W$ for some positive real number $a$.
Then there exists an open neighborhood $U$ of $W$ such that $D$ is semiample
over $U$.

Theorem 15.1 is an application of the cone theorem (see Theorem 12.2) and the
basepoint-free theorem for Cartier divisors (see Theorem 9.1).

###### Proof of Theorem 15.1.

Without loss of generality, by replacing $D$ with $aD$, we may assume that
$a=1$ holds. By replacing $Y$ with a relatively compact open neighborhood of
$W$, we may further assume that $\operatorname{Supp}D$ has only finitely many
irreducible components and that $D$ is a globally $\mathbb{R}$-Cartier
$\mathbb{R}$-divisor on $X$. We consider

$F=\\{z\in\operatorname{\overline{NE}}(X/Y;W)\,|\,D\cdot z=0\\}.$

Then $F$ is a face of $\operatorname{\overline{NE}}(X/Y;W)$ and
$(K_{X}+\Delta)\cdot z<0$ holds for $z\in F$. We take an ample
$\mathbb{R}$-line bundle $\mathcal{A}$ on $X$ such that
$D-(K_{X}+\Delta+\mathcal{A})$ is still $\pi$-ample over $W$. Let $R$ be a
$(K_{X}+\Delta)$-negative extremal ray of
$\operatorname{\overline{NE}}(X/Y;W)$ such that $R\subset F$. Then $R$ is
automatically a $(K_{X}+\Delta+\mathcal{A})$-negative extremal ray of
$\operatorname{\overline{NE}}(X/Y;W)$ since $D\cdot R=0$ and
$D-(K_{X}+\Delta+\mathcal{A})$ is $\pi$-ample over $W$. Therefore, $F$
contains only finitely many $(K_{X}+\Delta)$-negative extremal rays
$R_{1},\ldots,R_{k}$ of $\operatorname{\overline{NE}}(X/Y;W)$. Thus, $F$ is
spanned by the extremal rays $R_{1},\ldots,R_{k}$. Let
$\operatorname{Supp}D=\sum_{j}D_{j}$ be the irreducible decomposition of
$\operatorname{Supp}D$. Then we consider the finite-dimensional real vector
space $V=\underset{j}{\bigoplus}\mathbb{R}D_{j}$. In this situation, we can
easily check that

$\mathcal{R}:=\\{B\in V\,|\,\text{$B$ is a globally $\mathbb{R}$-Cartier
$\mathbb{R}$-divisor and $B\cdot z=0$ for every $z\in F$}\\}$

is a rational affine subspace of $V$ with $D\in\mathcal{R}$. As in Step 5 in
the proof of Theorem 12.2, we put

$\mathcal{C}_{F}:=\operatorname{\overline{NE}}(X/Y;W)_{(K_{X}+\Delta+\mathcal{A})\geq
0}+\sum_{R_{j}\not\subset F}R_{j}.$

and

$\mathcal{R}^{+}:=\\{B\in\mathcal{R}\,|\,\text{$B$ is positive on
$\mathcal{C}_{F}\setminus\\{0\\}$}\\}.$

We note that
$\operatorname{\overline{NE}}(X/Y;W)_{\operatorname{Nlc}(X,\Delta)}=\emptyset$
since $(X,\Delta)$ is log canonical. Then $\mathcal{R}^{+}$ is a non-empty
open subset of $\mathcal{R}$ with $D\in\mathcal{R}^{+}$. Hence we can find
positive real numbers $r_{1},r_{2},\ldots,r_{m}$ and globally
$\mathbb{Q}$-Cartier $\mathbb{Q}$-divisors
$B_{1},B_{2},\ldots,B_{m}\in\mathcal{R}^{+}$ such that
$D=\sum_{i=1}^{m}r_{i}B_{i}$ and $B_{i}-(K_{X}+\Delta)$ is $\pi$-ample over
$W$ for every $i$. We note that $B_{i}$ is automatically $\pi$-nef over $W$
for every $i$ since $B_{i}\in\mathcal{R}^{+}$. By the basepoint-free theorem
for Cartier divisors (see Theorem 9.1), there exists a relatively compact open
neighborhood $U$ of $W$ such that $B_{i}$ is $\pi$-semiample over $U$ for
every $i$. Therefore, $D=\sum_{i=1}^{m}r_{i}B_{i}$ is $\pi$-semiample over
$U$. This is what we wanted. ∎

Theorem 15.1 will play an important role in the study of minimal models of
complex analytic spaces.

##  16\. Proof of Main theorem

In this final section, we will prove Theorem 1.2, which is the main theorem of
this paper.

###### Proof of Theorem 1.2.

By Theorem 12.2 (1) and (2), we obtain the following equality

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

satisfying (1), (2), and (3) in Theorem 1.2. By Theorem 12.2 (3), $F$ in
Theorem 1.2 (4) is rational and hence contractible at
$\operatorname{Nlc}(X,\Delta)$. Thus, by the contraction theorem (see Theorem
12.1), we obtain the desired contraction morphism $\varphi_{F}\colon X\to Z$
over $Y$ after shrinking $Y$ around $W$ suitably. By Theorem 13.2, we see that
(5) holds. We note that (6) is nothing but Theorem 14.4. Finally, we will
prove (7). We may assume that $K_{X}+\Delta$ is not $\pi$-nef over $W$. Then
we can take a small positive real number $\varepsilon$ such that
$K_{X}+\Delta+\varepsilon\mathcal{H}$ is not $\pi$-nef over $W$. By the cone
theorem (see Theorem 12.2 (2)), there exist only finitely many
$(K_{X}+\Delta+\varepsilon\mathcal{H})$-negative extremal rays
$R_{1},\ldots,R_{k}$ of $\operatorname{\overline{NE}}(X/Y;W)$. We put

$\mu_{i}:=\frac{-(K_{X}+\Delta)\cdot R_{i}}{\mathcal{H}\cdot R_{i}}$

for every $i$. Then it is obvious that $\lambda=\max_{1\leq i\leq k}\mu_{i}$.
If $\lambda=\mu_{i_{0}}$ holds for $1\leq i_{0}\leq k$, then
$(K_{X}+\Delta+\lambda\mathcal{H})\cdot R_{i_{0}}=0$. By construction,
$K_{X}+\Delta+\lambda\mathcal{H}$ is $\pi$-nef over $W$. We finish the proof
of Theorem 1.2. ∎

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