and $M_{X^{\prime}}$ are obviously $\mathbb{Q}$-divisors by construction.

######  Step 4.

In this step, we treat the general case. We first use Lemma 4.25 and get
positive real numbers $r_{i}$ and $\left(X,\omega_{i},f\colon(Y,D_{i})\to
X\right)$ for $1\leq i\leq k$ with the properties in Lemma 4.25. Then we apply
the argument in Step 3 to

$\left(X,\omega_{i},f\colon(Y,D_{i})\to X\right)$

for every $i$. By Theorem 6.10, we can take a projective birational morphism
$p\colon X^{\prime}\to X$ from a smooth quasi-projective variety $X^{\prime}$
which works for

$\left(X,\omega_{i},f\colon(Y,D_{i})\to X\right)$

for every $i$. By summing them up with weight $r_{i}$, we get
$\mathbb{R}$-divisors $B_{X^{\prime}}$ and $M_{X^{\prime}}$ with the desired
properties.

We finish the proof of Theorem 7.1. ∎

##  8\. Proof of Theorem 1.10

In this section, we prove Theorem 1.10 as an application of Theorem 7.1. Then,
by using Theorem 1.10, we prove Corollary 8.2 and Lemma 8.3, which will play
an important role in Section 9. Let us start with the following elementary
lemma for the proof of Theorem 1.10.

######  Lemma 8.1.

Let $X$ be a quasi-projective variety and let $H$ be an ample
$\mathbb{R}$-divisor on $X$. Let $p\colon X^{\prime}\to X$ be a projective
birational morphism from a smooth quasi-projective variety $X^{\prime}$ and
let $F$ be an effective $p$-exceptional Cartier divisor on $X^{\prime}$ such
that $-F$ is $p$-ample. Let $M^{\prime}$ be a potentially nef
$\mathbb{R}$-divisor on $X^{\prime}$. Then $p^{*}H-\varepsilon F+M^{\prime}$
is ample for any $0<\varepsilon\ll 1$.

###### Proof.

We can write $H=\sum_{i=0}^{k}a_{i}H_{i}$ such that $a_{i}$ is a positive real
number and $H_{i}$ is an ample Cartier divisor on $X$ for every $i$. If
$0<\varepsilon\ll 1$, then we can take $\varepsilon_{i}$ such that
$p^{*}H_{i}-\varepsilon_{i}F$ is ample for every $i$ with
$\sum_{i=0}^{k}a_{i}\varepsilon_{i}=\varepsilon$. Since $M^{\prime}$ is a
potentially nef $\mathbb{R}$-divisor on $X^{\prime}$, we can construct a
smooth projective completion $X^{\dagger}$ of $X^{\prime}$ and a nef
$\mathbb{R}$-divisor $M^{\dagger}$ on $X^{\dagger}$ such that
$M^{\dagger}|_{X^{\prime}}=X^{\prime}$. By taking a suitable birational
modification of $X^{\dagger}$, we may further assume that there is an ample
$\mathbb{R}$-divisor $A$ on $X^{\dagger}$ such that
$A|_{X^{\prime}}=p^{*}H_{0}-\varepsilon_{0}F$ holds. Hence
$a_{0}p^{*}H_{0}-a_{0}\varepsilon_{0}F+M^{\prime}$ is ample. Therefore,
$p^{*}H-\varepsilon F+M^{\prime}$ is ample for any $0<\varepsilon\ll 1$. This
is what we wanted. ∎

Let us start the proof of Theorem 1.10.

###### Proof of Theorem 1.10.

By Theorem 7.1, there is a projective birational morphism $p\colon
X^{\prime}\to X$ from a smooth quasi-projective variety $X^{\prime}$ such that

$K_{X^{\prime}}+B_{X^{\prime}}+M_{X^{\prime}}=p^{*}\omega,$

where $B_{X^{\prime}}$ is an $\mathbb{R}$-divisor on $X^{\prime}$ whose
support is a simple normal crossing divisor, $B^{<0}_{X^{\prime}}$ is
$p$-exceptional, $M_{X^{\prime}}$ is a potentially nef $\mathbb{R}$-divisor on
$X^{\prime}$, and $p(B^{\geq 1}_{X^{\prime}})=\operatorname{Nqklt}(X,\omega)$.
By taking some more blow-ups, we may further assume that there is an effective
$p$-exceptional Cartier divisor $F$ on $X^{\prime}$ such that $-F$ is
$p$-ample and that $\operatorname{Supp}F\cup\operatorname{Supp}B_{X^{\prime}}$
is contained in a simple normal crossing divisor on $X^{\prime}$. Then
$p^{*}H-\varepsilon F+M_{X^{\prime}}$ is ample for any $0<\varepsilon\ll 1$ by
Lemma 8.1. We take a general effective $\mathbb{R}$-divisor $G$ on
$X^{\prime}$ such that $G\sim_{\mathbb{R}}p^{*}H-\varepsilon F+M_{X^{\prime}}$
with $0<\varepsilon\ll 1$,
$\operatorname{Supp}G\cup\operatorname{Supp}B_{X^{\prime}}\cup\operatorname{Supp}F$
is contained in a simple normal crossing divisor on $X^{\prime}$, and
$(B_{X^{\prime}}+\varepsilon F+G)^{\geq 1}=B^{\geq 1}_{X^{\prime}}$ holds set
theoretically. Then we have

$\begin{split}K_{X^{\prime}}+B_{X^{\prime}}+M_{X^{\prime}}+p^{*}H&=K_{X^{\prime}}+B_{X^{\prime}}+\varepsilon
F+p^{*}H-\varepsilon F+M_{X^{\prime}}\\\
&\sim_{\mathbb{R}}K_{X^{\prime}}+B_{X^{\prime}}+\varepsilon F+G.\end{split}$

We put $\Delta:=p_{*}(B_{X^{\prime}}+\varepsilon F+G)$. By construction,
$K_{X}+\Delta\sim_{\mathbb{R}}\omega+H$. By construction again, we have

$\operatorname{Nklt}(X,\Delta)=p\left((B_{X^{\prime}}+\varepsilon F+G)^{\geq
1}\right)=p\left(B^{\geq
1}_{X^{\prime}}\right)=\operatorname{Nqklt}(X,\omega)$

set theoretically.

When $[X,\omega]$ has a $\mathbb{Q}$-structure, we can make $B_{X^{\prime}}$
and $M_{X^{\prime}}$ $\mathbb{Q}$-divisors by Theorem 7.1. Then it is easy to
see that we can make $\Delta$ a $\mathbb{Q}$-divisor on $X$ such that
$K_{X}+\Delta\sim_{\mathbb{Q}}\omega+H$ when $H$ is an ample
$\mathbb{Q}$-divisor and $[X,\omega]$ has a $\mathbb{Q}$-structure by the
above construction of $\Delta$.

Finally, if $X$ is a curve in the above argument, then $p\colon X^{\prime}\to
X$ is an isomorphism and $M_{X^{\prime}}$ is semi-ample (see Theorem 7.1).
Hence we can take $\Delta$ such that

$K_{X}+\Delta\sim_{\mathbb{R}}\omega$

with the desired properties. ∎

For some related results, see [FG1], [F15], and so on. By applying Theorem
1.10 to normal pairs, we have the following useful result.

######  Corollary 8.2.

Let $X$ be a normal variety and let $\Delta$ be an effective
$\mathbb{R}$-divisor on $X$ such that $K_{X}+\Delta$ is $\mathbb{R}$-Cartier.
Let $C$ be a log canonical center of $(X,\Delta)$ such that $C$ is a smooth
curve. Then

$(K_{X}+\Delta)|_{C}\sim_{\mathbb{R}}K_{C}+\Delta_{C}$

holds for some effective $\mathbb{R}$-divisor $\Delta_{C}$ such that

$\operatorname{Supp}\Delta^{\geq
1}_{C}=C\cap\left(\operatorname{Nlc}(X,\Delta)\cup\bigcup_{C\not\subset
W}W\right),$

where $W$ runs over lc centers of $(X,\Delta)$ which do not contain $C$, holds
set theoretically. When $K_{X}+\Delta$ is $\mathbb{Q}$-Cartier, we can make
$\Delta_{C}$ a $\mathbb{Q}$-divisor such that

$(K_{X}+\Delta)|_{C}\sim_{\mathbb{Q}}K_{C}+\Delta_{C}$

in the above statement.

###### Proof.

As we saw in Example 4.11, $[X,K_{X}+\Delta]$ naturally becomes a quasi-log
scheme. By construction,
$\operatorname{Nqlc}(X,K_{X}+\Delta)=\operatorname{Nlc}(X,\Delta)$, $W$ is a
qlc center of $[X,K_{X}+\Delta]$ if and only if $W$ is a log canonical center
of $(X,\Delta)$. Hence we can see that $C$ is a qlc center of
$[X,K_{X}+\Delta]$. Therefore, by adjunction (see Theorem 4.6 (i) and [F11,
Theorem 6.3.5 (i)]), $[C^{\prime},(K_{X}+\Delta)|_{C^{\prime}}]$ is a quasi-
log scheme, where $C^{\prime}=C\cup\operatorname{Nlc}(X,\Delta)$. By Lemma
4.20, we see that $[C,(K_{X}+\Delta)|_{C}]$ is also a quasi-log scheme such
that

$\operatorname{Nqklt}(C,(K_{X}+\Delta)|_{C})=\operatorname{Nqklt}(C^{\prime},(K_{X}+\Delta)|_{C^{\prime}})\cap
C$

holds set theoretically. By construction, we can easily see that

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

where $W$ runs over lc centers of $(X,\Delta)$ which do not contain $C$, holds
set theoretically (see Theorem 4.10 and Corollary 4.14). By applying Theorem
1.10 to $[C,(K_{X}+\Delta)|_{C}]$, we can find an effective
$\mathbb{R}$-divisor $\Delta_{C}$ on $C$ such that

$(K_{X}+\Delta)|_{C}\sim_{\mathbb{R}}K_{C}+\Delta_{C}$

with

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

Of course, if $K_{X}+\Delta$ is $\mathbb{Q}$-Cartier, then we can make
$\Delta_{C}$ a $\mathbb{Q}$-divisor such that

$(K_{X}+\Delta)|_{C}\sim_{\mathbb{Q}}K_{C}+\Delta_{C}$

in the above statement. ∎

We will use the following lemma in Section 9.

######  Lemma 8.3.

Let $\varphi\colon X\to Y$ be a proper surjective morphism between normal
varieties such that $R^{1}\varphi_{*}\mathcal{O}_{X}=0$ and that
$\dim\varphi^{-1}(y)\leq 1$ holds for every closed point $y\in Y$. Let $C$ be
a projective curve on $X$ such that $\varphi(C)$ is a point. Then

$C\simeq\mathbb{P}^{1}.$

Let $\Delta$ be an effective $\mathbb{R}$-divisor on $X$ such that
$K_{X}+\Delta$ is $\mathbb{R}$-Cartier. If
$C\not\subset\operatorname{Nlc}(X,\Delta)$ and

$C\cap\left(\operatorname{Nlc}(X,\Delta)\cup\bigcup_{C\not\subset
W}W\right)\neq\emptyset,$

where $W$ runs over lc centers of $(X,\Delta)$ which do not contain $C$, then
the following inequality

$-(K_{X}+\Delta)\cdot C\leq 1$

holds.

###### Proof.

In Step 1, we will prove that $C\simeq\mathbb{P}^{1}$ holds. In Step 2, we
will prove that $-(K_{X}+\Delta)\cdot C\leq 1$ by Corollary 8.2.

######  Step 1.

Although the argument in this step is well known, we will explain it in detail
for the reader’s convenience. Let us consider the following short exact
sequence

$0\to\mathcal{I}_{C}\to\mathcal{O}_{X}\to\mathcal{O}_{C}\to 0,$

where $\mathcal{I}_{C}$ is the defining ideal sheaf of $C$ on $X$. Since
$\dim\varphi^{-1}(y)\leq 1$ for every $y\in Y$ by assumption,
$R^{2}\varphi_{*}\mathcal{I}_{C}=0$ holds. Therefore, we get the following
surjection

$R^{1}\varphi_{*}\mathcal{O}_{X}\to R^{1}\varphi_{*}\mathcal{O}_{C}\to 0.$

By assumption, $R^{1}\varphi_{*}\mathcal{O}_{X}=0$. Hence
$R^{1}\varphi_{*}\mathcal{O}_{C}=0$ holds. Since $\varphi(C)$ is a point by
assumption, $H^{1}(C,\mathcal{O}_{C})=0$ holds. This means that
$C\simeq\mathbb{P}^{1}$.

######  Step 2.

By shrinking $Y$ around $\varphi(C)$, we may assume that $Y$ is quasi-
projective. Let $B_{1},\ldots,B_{n+1}$ be general very ample Cartier divisors
on $Y$ passing through $\varphi(C)$ with $n=\dim X$. Then it is well known
that

$\left(X,\Delta+\sum_{i=1}^{n+1}\varphi^{*}B_{i}\right)$

is not log canonical at any point of $C$ (see, for example, [F6, Lemma 13.2])
such that

$\operatorname{Nklt}\left(X,\Delta+(1-\varepsilon)\sum_{i=1}^{n+1}\varphi^{*}B_{i}\right)=\operatorname{Nklt}(X,\Delta)$

holds outside $\varphi^{-1}(\varphi(C))$ for every $0<\varepsilon\leq 1$.
Hence we can take $0\leq c<1$ such that $C$ is a log canonical center of
$(X,\Delta+\varphi^{*}B)$, where $B=c\sum_{i=1}^{n+1}B_{i}$. Since $B$ is
effective, we see that

$C\cap\left(\operatorname{Nlc}(X,\Delta+\varphi^{*}B)\cup\bigcup_{C\not\subset
W}W\right)\neq\emptyset,$

where $W$ runs over lc centers of $(X,\Delta+\varphi^{*}B)$ which do not
contain $C$. By Corollary 8.2, we can take an effective $\mathbb{R}$-divisor
$\Delta_{C}$ on $C$ such that

$(K_{X}+\Delta)|_{C}\sim_{\mathbb{R}}(K_{X}+\Delta+\varphi^{*}B)|_{C}\sim_{\mathbb{R}}K_{C}+\Delta_{C}$

and that

$\operatorname{Supp}\Delta^{\geq
1}_{C}=C\cap\left(\operatorname{Nlc}(X,\Delta+\varphi^{*}B)\cup\bigcup_{C\not\subset
W}W\right)\neq\emptyset$

holds. This implies that

$-(K_{X}+\Delta)\cdot C=-\deg(K_{C}+\Delta_{C})=2-\deg\Delta_{C}\leq 1.$

We finish the proof of Lemma 8.3. ∎

##  9\. Proof of Theorem 1.8

In this section, we prove Theorem 1.8. Let us start with the following
proposition, which is a consequence of the cone and contraction theorem for
normal pairs (see [F6, Theorem 1.1]) with the aid of Lemma 8.3. This is
essentially due to [S, Proposition 5.2].

######  Proposition 9.1 ([S, Proposition 5.2] and [F15, Proposition 7.1]).

Let $\pi\colon X\to S$ be a projective morphism from a normal
$\mathbb{Q}$-factorial variety $X$ onto a scheme $S$. Let
$\Delta=\sum_{i}d_{i}\Delta_{i}$ be an effective $\mathbb{R}$-divisor on $X$,
where the $\Delta_{i}$’s are the distinct prime components of $\Delta$ for all
$i$, such that

$\left(X,\Delta^{\prime}:=\sum_{d_{i}<1}d_{i}\Delta_{i}+\sum_{d_{i}\geq
1}\Delta_{i}\right)$

is dlt. Assume that $(K_{X}+\Delta)|_{\operatorname{Nklt}(X,\Delta)}$ is nef
over $S$. Then $K_{X}+\Delta$ is nef over $S$ or there exists a non-constant
morphism

$f\colon\mathbb{A}^{1}\longrightarrow X\setminus\operatorname{Nklt}(X,\Delta)$

such that $\pi\circ f(\mathbb{A}^{1})$ is a point. More precisely, the curve
$C$, the closure of $f(\mathbb{A}^{1})$ in $X$, is a (possibly singular)
rational curve with

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

Moreover, if $C\cap\operatorname{Nklt}(X,\Delta)\neq\emptyset$, then we can
make $C$ satisfy a sharper estimate

$0<-(K_{X}+\Delta)\cdot C\leq 1.$

###### Proof.

We note that $\operatorname{Nklt}(X,\Delta)$ coincides with
$(\Delta^{\prime})^{=1}=\lfloor\Delta^{\prime}\rfloor$, $\Delta^{\geq 1}$, and
$\lfloor\Delta\rfloor$ set theoretically because $(X,\Delta^{\prime})$ is dlt
by assumption. It is sufficient to construct a non-constant morphism

$f\colon\mathbb{A}^{1}\longrightarrow X\setminus\operatorname{Nklt}(X,\Delta)$

such that $\pi\circ f(\mathbb{A}^{1})$ is a point with the desired properties
when $K_{X}+\Delta$ is not nef over $S$. When $(X,\Delta)$ is kawamata log
terminal, that is, $\lfloor\Delta\rfloor=0$, the statement is well known (see,
for example, [F6, Theorem 1.1], Theorem 1.12, or Corollary 12.3 below).
Therefore, we may assume that $(X,\Delta)$ is not kawamata log terminal. By
shrinking $S$ suitably, we may assume that $S$ and $X$ are both quasi-
projective. By the cone and contraction theorem for normal pairs (see [F6,
Theorem 1.1]), we can take a $(K_{X}+\Delta)$-negative extremal ray $R$ of
$\operatorname{\overline{NE}}(X/S)$ and the associated extremal contraction
morphism $\varphi:=\varphi_{R}\colon X\to Y$ over $S$ since
$(K_{X}+\Delta)|_{\operatorname{Nklt}(X,\Delta)}$ is nef over $S$. Note that
$(K_{X}+\Delta^{<1})\cdot R<0$ and $(K_{X}+\Delta^{\prime})\cdot R<0$ hold
because $(K_{X}+\Delta)|_{\operatorname{Nklt}(X,\Delta)}$ is nef over $S$.
Since $(X,\Delta^{<1})$ is kawamata log terminal and $-(K_{X}+\Delta^{<1})$ is
$\varphi$-ample, we get $R^{i}\varphi_{*}\mathcal{O}_{X}=0$ for every $i>0$ by
the relative Kawamata–Viehweg vanishing theorem (see [F11, Corollary 5.7.7]).
By construction,
$\varphi\colon\operatorname{Nklt}(X,\Delta)\to\varphi(\operatorname{Nklt}(X,\Delta))$
is finite. We have the following short exact sequence

$0\to\mathcal{O}_{X}(-\lfloor\Delta^{\prime}\rfloor)\to\mathcal{O}_{X}\to\mathcal{O}_{\lfloor\Delta^{\prime}\rfloor}\to
0.$

Since
$-\lfloor\Delta^{\prime}\rfloor-(K_{X}+\\{\Delta^{\prime}\\})=-(K_{X}+\Delta^{\prime})$
is $\varphi$-ample and $(X,\\{\Delta^{\prime}\\})$ is kawamata log terminal,
$R^{i}\varphi_{*}\mathcal{O}_{X}(-\lfloor\Delta^{\prime}\rfloor)=0$ holds for
every $i>0$ by the relative Kawamata–Viehweg vanishing theorem again (see
[F11, Corollary 5.7.7]). Therefore,

$0\to\varphi_{*}\mathcal{O}_{X}(-\lfloor\Delta^{\prime}\rfloor)\to\mathcal{O}_{Y}\to\varphi_{*}\mathcal{O}_{\lfloor\Delta^{\prime}\rfloor}\to
0$

is exact. This implies that
$\operatorname{Supp}\lfloor\Delta^{\prime}\rfloor=\operatorname{Supp}\Delta^{\geq
1}$ is connected in a neighborhood of any fiber of $\varphi$.

######  Case 1.

Assume that $\varphi$ is a Fano contraction, that is, $\dim Y<\dim X$. Then we
see that $\Delta^{\geq 1}$ is $\varphi$-ample and that $\dim Y=\dim X-1$. Note
that $\operatorname{Supp}\Delta^{\geq 1}$ is finite over $Y$ since no curves
in $\operatorname{Supp}\Delta^{\geq 1}$ are contracted by $\varphi$.

Assume that there exists a closed subvariety $\Sigma$ on $X$ with
$\dim\Sigma\geq 2$ such that $\varphi(\Sigma)$ is a point. Then

$\dim\left(\Sigma\cap\operatorname{Supp}\Delta^{\geq 1}\right)\geq 1$

holds since $\Delta^{\geq 1}$ is $\varphi$-ample. This is a contradiction
because $\operatorname{Supp}\Delta^{\geq 1}$ is finite over $Y$. Hence we
obtain that $\dim\varphi^{-1}(y)=1$ for every closed point $y\in Y$.

Let $C$ be any projective curve on $X$ such that $\varphi(C)$ is a point. Then
$(X,\Delta)$ is log canonical at the generic point of $C$, equivalently,
$C\not\subset\operatorname{Nlc}(X,\Delta)$, since
$\operatorname{Supp}\Delta^{\geq 1}$ is finite over $Y$. More precisely, since
$\operatorname{Supp}\Delta^{\geq 1}=\operatorname{Nklt}(X,\Delta)$ is finite
over $Y$, $C\not\subset\operatorname{Nklt}(X,\Delta)$ and

$\operatorname{Supp}\Delta^{\geq
1}=\operatorname{Nklt}(X,\Delta)=\left(\operatorname{Nlc}(X,\Delta)\cup\bigcup_{C\not\subset
W}W\right)$

holds, where $W$ runs over lc centers of $(X,\Delta)$ which do not contain
$C$. On the other hand,

$C\cap\operatorname{Supp}\Delta^{\geq 1}\neq\emptyset$

because $\Delta^{\geq 1}$ is $\varphi$-ample. Hence, by Lemma 8.3, we obtain
that $C\simeq\mathbb{P}^{1}$ and that $-(K_{X}+\Delta)\cdot C\leq 1$.

By the connectedness of $\operatorname{Supp}\Delta^{\geq 1}$ discussed above,
$C\cap\operatorname{Supp}\Delta^{\geq 1}$ is a point. Therefore, we can find a
non-constant morphism

$f\colon\mathbb{A}^{1}\longrightarrow X\setminus\operatorname{Nklt}(X,\Delta)$

such that $\pi\circ f(\mathbb{A}^{1})$ is a point and that
$0<-(K_{X}+\Delta)\cdot C\leq 1$ holds, where $C$ is the closure of
$f(\mathbb{A}^{1})$ in $X$.

######  Case 2.

Assume that $\varphi$ is a birational contraction and that the exceptional
locus $\operatorname{Exc}(\varphi)$ of $\varphi$ is disjoint from
$\operatorname{Nklt}(X,\Delta)$. In this situation, we can find a rational
curve $C$ in a fiber of $\varphi$ with $0<-(K_{X}+\Delta)\cdot C\leq 2\dim X$
by the cone theorem for kawamata log terminal pairs (see [F6, Theorem 1.1],
Theorem 1.12, or Corollary 12.3 below). It is obviously disjoint from
$\operatorname{Nklt}(X,\Delta)$. Therefore, we can take a non-constant
morphism

$f\colon\mathbb{A}^{1}\longrightarrow X\setminus\operatorname{Nklt}(X,\Delta)$

such that the closure of $f(\mathbb{A}^{1})$ is $C$.

######  Case 3.

Assume that $\varphi$ is a birational contraction and that
$\operatorname{Exc}(\varphi)\cap\operatorname{Nklt}(X,\Delta)\neq\emptyset$.
In this situation, as in Case 1, we see that $\Delta^{\geq 1}$ is
$\varphi$-ample and that $\dim\varphi^{-1}(y)\leq 1$ for every $y\in Y$. Let
$C$ be any projective curve $C$ on $X$ such that $\varphi(C)$ is a point.
Then, $C\cap\operatorname{Supp}\Delta^{\geq 1}\neq\emptyset$ holds since
$\Delta^{\geq 1}$ is $\varphi$-ample, and
$C\cap\operatorname{Supp}\Delta^{\geq 1}$ is a point by the connectedness of
$\operatorname{Supp}\Delta^{\geq 1}$ discussed above. In particular, we obtain
$C\not\subset\operatorname{Nklt}(X,\Delta)$ and

$C\cap\operatorname{Supp}\Delta^{\geq 1}\neq\emptyset,$

and

$\operatorname{Supp}\Delta^{\geq
1}=\operatorname{Nklt}(X,\Delta)=\left(\operatorname{Nlc}(X,\Delta)\cup\bigcup_{C\not\subset
W}W\right),$

where $W$ runs over lc centers of $(X,\Delta)$ which do not contain $C$.
Hence, by Lemma 8.3, $C\simeq\mathbb{P}^{1}$ with $-(K_{X}+\Delta)\cdot C\leq
1$. Since $C\cap\operatorname{Supp}\Delta^{\geq 1}$ is a point, we get a non-
constant morphism

$f\colon\mathbb{A}^{1}\longrightarrow X\setminus\operatorname{Nklt}(X,\Delta)$

such that $f(\mathbb{A}^{1})=C\cap(X\setminus\operatorname{Nklt}(X,\Delta))$.

Therefore, we get the desired statement. ∎

Let us prove Theorem 1.8 as an application of Proposition 9.1.

###### Proof of Theorem 1.8.

By shrinking $S$ suitably, we may assume that $X$ and $S$ are both quasi-
projective. By Lemma 3.10, we can construct a projective birational morphism
$g\colon Y\to X$ from a normal $\mathbb{Q}$-factorial variety $Y$ satisfying
(i), (ii), and (iv) in Lemma 3.10. Let us consider $\pi\circ g\colon Y\to S$.
Note that $K_{Y}+\Delta_{Y}$ is not nef over $S$ since
$K_{Y}+\Delta_{Y}=g^{*}(K_{X}+\Delta)$ holds. It is obvious that
$(K_{Y}+\Delta_{Y})|_{\operatorname{Nklt}(Y,\Delta_{Y})}$ is nef over $S$ by
(iv) because so is $(K_{X}+\Delta)|_{\operatorname{Nklt}(X,\Delta)}$.
Therefore, by Proposition 9.1, we have a non-constant morphism

$h\colon\mathbb{A}^{1}\longrightarrow
Y\setminus\operatorname{Nklt}(Y,\Delta_{Y})$

such that $(\pi\circ g)\circ h(\mathbb{A}^{1})$ is a point and that

$0<-(K_{Y}+\Delta_{Y})\cdot C_{Y}\leq 2\dim Y=2\dim X$

holds, where $C_{Y}$ is the closure of $h(\mathbb{A}^{1})$ in $Y$. Since
$K_{Y}+\Delta_{Y}=h^{*}(K_{X}+\Delta)$ holds, $g$ does not contract $C_{Y}$ to
a pont. This implies that

$f:=g\circ h\colon\mathbb{A}^{1}\longrightarrow
X\setminus\operatorname{Nklt}(X,\Delta)$

is a desired non-constant morphism such that $\pi\circ f(\mathbb{A}^{1})$ is a
point by (iv). ∎

For the proof of Theorem 1.6, we prepare the following somewhat artificial
statement as an application of Theorem 1.8.

######  Theorem 9.2.

Let $\pi\colon X\to S$ be a proper surjective morphism from a normal quasi-
projective variety $X$ onto a scheme $S$. Let $\mathcal{P}$ be an
$\mathbb{R}$-Cartier divisor on $X$ and let $H$ be an ample Cartier divisor on
$X$. Let $\Sigma$ be a closed subset of $X$. Assume that $\pi$ is not finite,
$-\mathcal{P}$ is $\pi$-ample, and $\pi\colon\Sigma\to\pi(\Sigma)$ is finite.
We further assume

  * •

$\\{\varepsilon_{i}\\}_{i=1}^{\infty}$ is a set of positive real numbers with
$\varepsilon_{i}\searrow 0$ for $i\nearrow\infty$, and

  * •

for every $i$, there exists an effective $\mathbb{R}$-divisor $\Delta_{i}$ on
$X$ such that

$\mathcal{P}+\varepsilon_{i}H\sim_{\mathbb{R}}K_{X}+\Delta_{i}$

and that

$\Sigma=\operatorname{Nklt}(X,\Delta_{i})$

holds set theoretically.

Then there exists a non-constant morphism

$f\colon\mathbb{A}^{1}\longrightarrow X\setminus\Sigma$

such that $\pi\circ f(\mathbb{A}^{1})$ is a point and that the curve $C$, the
closure of $f(\mathbb{A}^{1})$ in $X$, is a rational curve with

$0<-\mathcal{P}\cdot C\leq 2\dim X.$

###### Proof.

We take an ample $\mathbb{Q}$-divisor $A$ on $X$ such that $-(\mathcal{P}+A)$
is $\pi$-ample. Without loss of generality, we may assume that
$-(\mathcal{P}+A+\varepsilon_{i}H)$ is $\pi$-ample for every $i$ because
$\varepsilon_{i}\searrow 0$ for $i\nearrow\infty$. By assumption,

$\mathcal{P}+\varepsilon_{i}H\sim_{\mathbb{R}}K_{X}+\Delta_{i}$

with

$\operatorname{Nklt}(X,\Delta_{i})=\Sigma$

for every $i$. Hence, by Theorem 1.8, there is a non-constant morphism

$f_{i}\colon\mathbb{A}^{1}\longrightarrow X\setminus\Sigma$

such that $\pi\circ f_{i}(\mathbb{A}^{1})$ is a point and that

$0<-(K_{X}+\Delta_{i})\cdot C_{i}=-(\mathcal{P}+\varepsilon_{i}H)\cdot
C_{i}\leq 2\dim X,$

where $C_{i}$ is the closure of $f_{i}(\mathbb{A}^{1})$ in $X$. We note that

$0<A\cdot
C_{i}=\left((\mathcal{P}+\varepsilon_{i}H+A)-(\mathcal{P}+\varepsilon_{i}H)\right)\cdot
C_{i}<2\dim X.$

It follows that the curves $C_{i}$ belong to a bounded family. Thus, possibly
passing to a subsequence, we may assume that $f_{i}$ and $C_{i}$ are constant,
that is, there is a non-constant morphism

$f\colon\mathbb{A}^{1}\longrightarrow X\setminus\Sigma$

such that $C_{i}=C$ for every $i$, where $C$ is the closure of
$f(\mathbb{A}^{1})$ in $X$. Therefore, we get

$0<-\mathcal{P}\cdot C=\lim_{i\to\infty}-(\mathcal{P}+\varepsilon_{i}H)\cdot
C=\lim_{i\to\infty}-(\mathcal{P}+\varepsilon_{i}H)\cdot C_{i}\leq 2\dim X.$

We finish the proof of Theorem 9.2. ∎

##  10\. Proof of Theorems 1.4, 1.5, and 1.6

In this section, we prove Theorems 1.4, 1.5 and 1.6. Since Theorem 1.4 is an
easy consequence of Theorem 1.5 and Theorem 1.5 can be seen as a very special
case of Theorem 1.6 by Example 4.11, it is sufficient to prove Theorem 1.6.
Let us start with the proof of Theorem 1.6.

###### Proof of Theorem 1.6.

We note that (i) and (ii) were already established in [F11, Theorem 6.7.4].
Therefore, it is sufficient to prove (iii). From Step 1 to Step 4, we will
reduce the problem to the case where $X$ is a normal variety. Then, in Step 5,
we will obtain a desired non-constant morphism from $\mathbb{A}^{1}$ by
Theorem 9.2.

######  Step 1.

Let $\varphi_{R_{j}}\colon X\to Y$ be the extremal contraction associated to
$R_{j}$ (see Theorem 4.18 and [F11, Theorems 6.7.3 and 6.7.4]). We note that

$\varphi_{R_{j}}\colon\operatorname{Nqlc}(X,\omega)\to\varphi_{R_{j}}\left(\operatorname{Nqlc}(X,\omega)\right)$

is finite. By replacing $\pi\colon X\to S$ with $\varphi_{R_{j}}\colon X\to
Y$, we may assume that $-\omega$ is $\pi$-ample and that
$\operatorname{\overline{NE}}(X/S)_{-\infty}=\\{0\\}$.

######  Step 2.

We take a qlc stratum $W$ of $[X,\omega]$ such that $\pi\colon W\to\pi(W)$ is
not finite and that $\pi\colon W^{\dagger}\to\pi(W^{\dagger})$ is finite for
every qlc center $W^{\dagger}$ of $[X,\omega]$ with $W^{\dagger}\subsetneq W$.
We put $W^{\prime}=W\cup\operatorname{Nqlc}(X,\omega)$. Then, by adjunction
(see Theorem 4.6 (i) and [F11, Theorem 6.3.5 (i)]),
$[W^{\prime},\omega|_{W^{\prime}}]$ naturally becomes a quasi-log scheme. By
replacing $[X,\omega]$ with $[W^{\prime},\omega|_{W^{\prime}}]$, we may
further assume that $X\setminus X_{-\infty}$ is irreducible and that

$\pi\colon\operatorname{Nqklt}(X,\omega)\to\pi\left(\operatorname{Nqklt}(X,\omega)\right)$

is finite.

######  Step 3.

By Lemma 4.20, we may replace $X$ with $\overline{X\setminus X_{-\infty}}$ and
assume that $X$ is a variety. We note that the finiteness of

$\pi\colon\operatorname{Nqklt}(X,\omega)\to\pi\left(\operatorname{Nqklt}(X,\omega)\right)$

still holds.

######  Step 4.

Let $\nu\colon Z\to X$ be the normalization. Then $[Z,\nu^{*}\omega]$
naturally becomes a quasi-log scheme by Theorem 1.9. Since
$\operatorname{Nqklt}(Z,\nu^{*}\omega)=\nu^{-1}\operatorname{Nqklt}(X,\omega)$
by Theorem 1.9, we may assume that $X$ is normal by replacing $[X,\omega]$
with $[Z,\nu^{*}\omega]$.

######  Step 5.

By shrinking $S$ suitably, we may further assume that $X$ and $S$ are both
quasi-projective. Hence we have the following properties:

  * (a)

$\pi\colon X\to S$ is a projective morphism from a normal quasi-projective
variety $X$ to a scheme $S$,

  * (b)

$-\omega$ is $\pi$-ample, and

  * (c)

$\pi\colon\Sigma\to\pi(\Sigma)$ is finite, where
$\Sigma:=\operatorname{Nqklt}(X,\omega)$.

Let $H$ be an ample Cartier divisor on $X$ and let
$\\{\varepsilon_{i}\\}_{i=1}^{\infty}$ be a set of positive real numbers such
that $\varepsilon_{i}\searrow 0$ for $i\nearrow\infty$. Then, by Theorem 1.10,
we have:

  * (d)

there exists an effective $\mathbb{R}$-divisor $\Delta_{i}$ on $X$ such that

$K_{X}+\Delta_{i}\sim_{\mathbb{R}}\omega+\varepsilon_{i}H$

with

$\operatorname{Nklt}(X,\Delta_{i})=\Sigma$

for every $i$.

Thus, by Theorem 9.2, we have a desired non-constant morphism

$f\colon\mathbb{A}^{1}\longrightarrow
X\setminus\operatorname{Nqklt}(X,\omega).$

We finish the proof of Theorem 1.6. ∎

As we already mentioned above, Theorem 1.5 is a very special case of Theorem
1.6.

###### Proof of Theorem 1.5.

By Example 4.11, $[X,K_{X}+\Delta]$ naturally becomes a quasi-log scheme.
Then, by Theorem 1.6, the desired cone theorem holds for $(X,\Delta)$. ∎

Theorem 1.4 easily follows from Theorem 1.5.

###### Proof of Theorem 1.4.

Since $(X,\Delta)$ is Mori hyperbolic by assumption, there is no
$(K_{X}+\Delta)$-negative extremal ray of $\operatorname{\overline{NE}}(X)$
that is rational and relatively ample at infinity (see Theorem 1.5). By
assumption, $(K_{X}+\Delta)|_{\operatorname{Nlc}(X,\Delta)}$ is nef. Hence the
subcone $\operatorname{\overline{NE}}(X)_{-\infty}$ is included in
$\operatorname{\overline{NE}}(X)_{(K_{X}+\Delta)\geq 0}$. This implies that

$\operatorname{\overline{NE}}(X)=\operatorname{\overline{NE}}(X)_{(K_{X}+\Delta)\geq
0}$

holds by Theorem 1.5. Thus $K_{X}+\Delta$ is nef. ∎

The author thinks that the proof of Theorems 1.4, 1.5 and 1.6 shows that the
framework of quasi-log schemes established in [F11, Chapter 6] and [F14] is
very powerful and useful even for the study of normal pairs.

##  11\. Ampleness criterion for quasi-log schemes

The main purpose of this section is to establish the following ampleness
criterion for quasi-log schemes. Then we will see that Theorem 1.11 is a very
special case of Theorem 11.1.

######  Theorem 11.1 (Ampleness criterion for quasi-log schemes).

Let $[X,\omega]$ be a quasi-log scheme and let $\pi\colon X\to S$ be a
projective morphism between schemes. Assume that
$\omega|_{\operatorname{Nqlc}(X,\omega)}$ is ample over $S$ and that $\omega$
is log big over $S$ with respect to $[X,\omega]$. We further assume that there
is no non-constant morphism

$f\colon\mathbb{A}^{1}\longrightarrow U$

such that $\pi\circ f(\mathbb{A}^{1})$ is a point, where $U$ is any open qlc
stratum of $[X,\omega]$. Then $\omega$ is ample over $S$.

Let us treat a special case of Theorem 11.1.

######  Theorem 11.2.

Let $[X,\omega]$ be a quasi-log scheme such that $X$ is a normal variety. Let
$\pi\colon X\to S$ be a projective morphism onto a scheme $S$. Assume that
$\omega|_{\operatorname{Nqklt}(X,\omega)}$ is ample over $S$ and that there is
no non-constant morphism

$f\colon\mathbb{A}^{1}\longrightarrow
X\setminus\operatorname{Nqklt}(X,\omega)$

such that $\pi\circ f(\mathbb{A}^{1})$ is a point. We further assume that
$\omega$ is big over $S$. Then $\omega$ is ample over $S$.

###### Proof.

We divide the proof into several small steps.

######  Step 1.

By Lemma 4.25, we can obtain quasi-log schemes

$\left(X,\omega_{i},f\colon(Y,D_{i})\to X\right)$

for $1\leq i\leq k$ with the following properties:

  * (a)

$[X,\omega_{i}]$ has a $\mathbb{Q}$-structure for every $i$,

  * (b)

$\operatorname{Nqlc}(X,\omega_{i})=\operatorname{Nqlc}(X,\omega)$ holds for
every $i$,

  * (c)

$W$ is a qlc stratum of $[X,\omega]$ if and only if $W$ is a qlc stratum of
$[X,\omega_{i}]$ for every $i$, and

  * (d)

there exist positive real numbers $r_{i}$ for $1\leq i\leq k$ such that
$\omega=\sum_{i=1}^{k}r_{i}\omega_{i}$ with $\sum_{i=1}^{k}r_{i}=1$.

By construction, we can make $\omega_{i}$ sufficiently close to $\omega$ (see
the proof of Lemma 4.25). Therefore, we may assume that
$\omega_{i}|_{\operatorname{Nqklt}(X,\omega_{i})}$ is ample over $S$ for every
$i$ by (b) and (c). Thus $[X,\omega_{i}]$ satisfies all the assumptions for
$[X,\omega]$ in Theorem 11.2. Hence, by replacing $[X,\omega]$ with
$[X,\omega_{i}]$, it is sufficient to prove the ampleness of $\omega$ under
the extra assumption that $[X,\omega]$ has a $\mathbb{Q}$-structure by (a) and
(d).

######  Step 2.

By assumption and Theorem 1.6 (iii), $\omega$ is nef over $S$. Since
$\omega|_{\operatorname{Nqklt}(X,\omega)}$ is ample over $S$ by assumption,
$\omega$ is nef and log big over $S$ with respect to $[X,\omega]$. Therefore,
by [F10, Theorem 1.1], we obtain that $\omega$ is semi-ample over $S$. Hence
$m\omega$ gives a birational contraction morphism $\Phi\colon X\to Y$ between
normal varieties over $S$, where $m$ is a sufficiently large and divisible
positive integer.

######  Step 3.

In this step, we will get a contradiction under the assumption that $\Phi$ is
not an isomorphism.

By shrinking $S$, we may assume that $S$, $X$, and $Y$ are quasi-projective.
By construction,

$\Phi\colon\operatorname{Nqklt}(X,\omega)\to\Phi(\operatorname{Nqklt}(X,\omega))$

is finite. Since $\Phi$ is birational and $Y$ is quasi-projective, we can take
an effective Cartier divisor $G$ on $X$ such that $-G$ is $\Phi$-ample. By
Lemma 4.24, for $0<\varepsilon\ll 1$, $[X,\omega+\varepsilon G]$ is a quasi-
log scheme such that

$\operatorname{Nqklt}(X,\omega+\varepsilon G)=\operatorname{Nqklt}(X,\omega)$

holds. By the cone theorem (see Theorem 1.6 (iii)), we can find a non-constant
morphism

$f\colon\mathbb{A}^{1}\longrightarrow
X\setminus\operatorname{Nqklt}(X,\omega+\varepsilon
G)=X\setminus\operatorname{Nqklt}(X,\omega)$

such that $\pi\circ f(\mathbb{A}^{1})$ is a point and that
$0<-(\omega+\varepsilon G)\cdot C\leq 2\dim X$ holds, where $C$ is the closure
of $f(\mathbb{A}^{1})$ in $X$. This is a contradiction.

Hence $\Phi$ is an isomorphism. Therefore, we obtain that $\omega$ is ample
over $S$. This is what we wanted. ∎

Once we know Theorem 11.2, it is not difficult to prove Theorem 11.1.

###### Proof of Theorem 11.1.

By Theorem 1.6 (iii), $\omega$ is nef and log big over $S$ with respect to
$[X,\omega]$. We put

$[X_{0},\omega_{0}]:=[X,\omega]$

and

$[X_{i+1},\omega_{i+1}]:=[\operatorname{Nqklt}(X_{i},\omega_{i}),\omega_{i}|_{\operatorname{Nqklt}(X_{i},\omega_{i})}]$

for $i\geq 0$. Then there exists $k\geq 0$ such that

$\operatorname{Nqklt}(X_{k},\omega_{k})=\operatorname{Nqlc}(X_{k},\omega_{k})=\operatorname{Nqlc}(X,\omega).$

We note that $\operatorname{Nqlc}(X,\omega)$ may be empty. By assumption,
$\omega_{k}|_{\operatorname{Nqklt}(X_{k},\omega_{k})}$ is ample over $S$. We
want to show by inverse induction on $i$ that $\omega_{i}$ is ample over $S$.
Therefore, it is sufficient to prove the following claim.

######  Claim.

Let $[X,\omega]$ be a quasi-log scheme and let $\pi\colon X\to S$ be a
projective morphism between schemes such that
$\omega|_{\operatorname{Nqklt}(X,\omega)}$ is ample over $S$ and that $\omega$
is nef and log big over $S$ with respect to $[X,\omega]$. Assume that there is
no non-constant morphism

$f\colon\mathbb{A}^{1}\longrightarrow
X\setminus\operatorname{Nqklt}(X,\omega)$

such that $\pi\circ f(\mathbb{A}^{1})$ is a point. Then $\omega$ is ample over
$S$.

###### Proof of Claim.

By adjunction (see Theorem 4.6 (i) and [F11, Theorem 6.3.5 (i)]), we may
assume that $\overline{X\setminus X_{-\infty}}$ is irreducible. By Lemma 4.20,
we may further assume that $X$ is irreducible. Then, by Theorem 1.9, we can
reduce the problem to the case where $X$ is a normal variety. Hence $\omega$
is ample over $S$ by Theorem 11.2. ∎

As we have already mentioned above, by applying Claim inductively, we obtain
the desired relative ampleness of $\omega=\omega_{0}$. ∎

We close this section with the proof of Theorem 1.11.

###### Proof of Theorem 1.11.

By Example 4.11, $[X,K_{X}+\Delta]$ naturally becomes a quasi-log scheme. We
apply Theorem 11.1 to $[X,K_{X}+\Delta]$. Then we obtain that $K_{X}+\Delta$
is ample. This is what we wanted. ∎

The author knows no proof of Theorem 1.11 that does not use the framework of
quasi-log schemes. Note that a similar result for dlt pairs was already
established in [F7, Theorem 5.1], whose proof is much easier than that of
Theorem 1.11 and depends on the basepoint-free theorem of Reid–Fukuda type for
dlt pairs (see [F1, Theorem 0.1]). We recommend the interested reader to see
[F7, Theorem 5.1] and [F1, Theorem 0.1].

##  12\. Proof of Theorems 1.12 and 1.13

In this section, we prove Theorems 1.12 and 1.13, and explain an application
for normal pairs. For the basic properties of uniruled varieties, see [Ko1,
Chapter IV. 1]. Let us start with the following lemma, which is a
generalization of [Ka, Lemma].

######  Lemma 12.1.

Let $[X,\omega]$ be a quasi-log scheme and let $\varphi\colon X\to W$ be a
projective morphism between schemes. Let $P$ be an arbitrary closed point of
$W$. Let $E$ be a positive-dimensional irreducible component of
$\varphi^{-1}(P)$ such that $E\not\subset X_{-\infty}$ and let
$\nu\colon\overline{E}\to E$ be the normalization. Then, for every ample
$\mathbb{R}$-divisor $H$ on $\overline{E}$, there exists an effective
$\mathbb{R}$-divisor $\Delta_{\overline{E},H}$ on $\overline{E}$ such that

$\nu^{*}\omega+H\sim_{\mathbb{R}}K_{\overline{E}}+\Delta_{\overline{E},H}$

holds. Therefore,

$\mathcal{A}^{\dim E-1}\cdot\omega\cdot
E\geq\left(\nu^{*}\mathcal{A}\right)^{\dim E-1}\cdot K_{\overline{E}}$

holds for every $\varphi$-ample line bundle $\mathcal{A}$ on $X$.

In the above statement, if $[X,\omega]$ has a $\mathbb{Q}$-structure and $H$
is an ample $\mathbb{Q}$-divisor on $\overline{E}$, then we can make
$\Delta_{\overline{E},H}$ an effective $\mathbb{Q}$-divisor on $\overline{E}$
with

$\nu^{*}\omega+H\sim_{\mathbb{Q}}K_{\overline{E}}+\Delta_{\overline{E},H}.$

###### Proof.

Our approach is different from Kawamata’s in [Ka]. A key ingredient of this
proof is Theorem 1.10.

######  Step 1.

If $E$ is a qlc stratum of $[X,\omega]$, then we put $B=0$ and go to Step 3.

######  Step 2.

By Step 1, we may assume that $E$ is not a qlc stratum of $[X,\omega]$.
Without loss of generality, we may assume that $W$ is quasi-projective by
shrinking $W$ around $P$. Let $B_{1},\ldots,B_{n+1}$ be general very ample
Cartier divisors on $W$ passing through $P$ with $n=\dim X$. Let
$f\colon(Y,B_{Y})\to X$ be a proper morphism from a globally embedded simple
normal crossing pair $(Y,B_{Y})$ as in Definition 4.2. Let $X^{\prime}$ be the
union of $X_{-\infty}=\operatorname{Nqlc}(X,\omega)$ and all qlc strata of
$[X,\omega]$ mapped to $P$ by $\varphi$. By [F11, Proposition 6.3.1] and [Ko2,
Theorem 3.35], we may assume that the union of all strata of $(Y,B_{Y})$
mapped to $X^{\prime}$ by $f$, which is denoted by $Y^{\prime}$, is a union of
some irreducible components of $Y$. As usual, we put
$Y^{\prime\prime}=Y-Y^{\prime}$,
$K_{Y^{\prime\prime}}+B_{Y^{\prime\prime}}=(K_{Y}+B_{Y})|_{Y^{\prime\prime}}$,
and $f^{\prime\prime}=f|_{Y^{\prime\prime}}$. By [F11, Proposition 6.3.1] and
[Ko2, Theorem 3.35] again, we may further assume that

$\left(Y^{\prime\prime},(f^{\prime\prime})^{*}\varphi^{*}\sum_{i=1}^{n+1}B_{i}+\operatorname{Supp}B_{Y^{\prime\prime}}\right)$

is a globally embedded simple normal crossing pair. By [F11, Lemma 6.3.13], we
can take $0<c<1$ such that

$f^{\prime\prime}\left(\left(B_{Y^{\prime\prime}}+c(f^{\prime\prime})^{*}\varphi^{*}\sum_{i=1}^{n+1}B_{i}\right)^{>1}\right)\not\supset
E$

and that there exists an irreducible component $G$ of

$\left(B_{Y^{\prime\prime}}+c(f^{\prime\prime})^{*}\varphi^{*}\sum_{i=1}^{n+1}B_{i}\right)^{=1}$

with $f^{\prime\prime}(G)=E$. By Lemma 4.23, we obtain that

$\left(X,\omega+B,f^{\prime\prime}\colon(Y^{\prime\prime},B_{Y^{\prime\prime}}+(f^{\prime\prime})^{*}B)\to
X\right),$

where $B=\varphi^{*}\left(c\sum_{i=1}^{n+1}B\right)$, is a quasi-log scheme
such that $E$ is a qlc stratum of this quasi-log scheme.

######  Step 3.

We put $E^{\prime}=E\cup\operatorname{Nqlc}(X,\omega+B)$. Then, by adjunction
(see Theorem 4.6 (i) and [F11, Theorem 6.3.5 (i)]),
$[E^{\prime},(\omega+B)|_{E^{\prime}}]$ is a quasi-log scheme. By Lemma 4.20,
$[E,(\omega+B)|_{E}]$ is also a quasi-log scheme. We note that
$(\omega+B)|_{E}\sim_{\mathbb{R}}\omega|_{E}$ since $\varphi(E)=P$. Hence
$[E,\omega|_{E}]$ is a quasi-log scheme. By Theorem 1.9,
$[\overline{E},\nu^{*}\omega]$ naturally becomes a quasi-log scheme. By
Theorem 1.10, there exists an effective $\mathbb{R}$-divisor
$\Delta_{\overline{E},H}$ on $\overline{E}$ such that

$\nu^{*}\omega+H\sim_{\mathbb{R}}K_{\overline{E}}+\Delta_{\overline{E},H}.$

This implies that

$(\nu^{*}\mathcal{A})^{\dim
E-1}\cdot(\nu^{*}\omega+H)\cdot\overline{E}=(\nu^{*}\mathcal{A})^{\dim
E-1}(K_{\overline{E}}+\Delta_{\overline{E},H})\geq(\nu^{*}\mathcal{A})^{\dim
E-1}\cdot K_{\overline{E}}.$

Since the above inequality holds for every ample $\mathbb{R}$-divisor $H$ on
$\overline{E}$, we obtain

$\mathcal{A}^{\dim E-1}\cdot\omega\cdot E=(\nu^{*}\mathcal{A})^{\dim
E-1}\cdot\nu^{*}\omega\cdot\overline{E}\geq(\nu^{*}\mathcal{A})^{\dim
E-1}\cdot K_{\overline{E}}.$

This is what we wanted. By the above proof, it is easy to see that we can make
$\Delta_{\overline{E},H}$ an effective $\mathbb{Q}$-divisor on $\overline{E}$
if $[X,\omega]$ has a $\mathbb{Q}$-structure and $H$ is an ample
$\mathbb{Q}$-divisor on $\overline{E}$.

We finish the proof of Lemma 12.1. ∎

######  Remark 12.2.

In the proof of [Ka, Lemma], Kawamata uses a relative Kawamata–Viehweg
vanishing theorem for projective bimeromorphic morphisms between complex
analytic spaces. His argument does not work for quasi-log schemes.

Let us prove Theorem 1.12.

###### Proof of Theorem 1.12.

In this proof, we will freely use the notation of Lemma 12.1.

######  Case 1.

We will treat the case where $\dim E=1$.

We take an ample $\mathbb{Q}$-divisor $H$ on $\overline{E}$ such that
$-(\nu^{*}\omega+H)$ is still ample. Then, by Lemma 12.1, $-K_{\overline{E}}$
is ample since $\Delta_{\overline{E},H}$ is effective. This means that
$\overline{E}\simeq\mathbb{P}^{1}$. By Lemma 12.1 again, we have

$0<-\omega\cdot E\leq-\deg K_{\overline{E}}=2.$

######  Case 2.

We will treat the case where $\dim E\geq 2$.

We take a $\varphi$-ample line bundle $\mathcal{A}$ such that
$\nu^{*}\mathcal{A}$ is very ample. We put $C=D_{1}\cap\cdots\cap D_{\dim
E-1}$, where $D_{i}$ is a general member of $|\nu^{*}\mathcal{A}|$ for every
$i$. Then $C$ is a smooth irreducible curve on $\overline{E}$ such that $C$
lies in the smooth locus of $\overline{E}$. By Lemma 12.1, we obtain

$C\cdot K_{\overline{E}}\leq\mathcal{A}^{\dim E-1}\cdot\omega\cdot E<0$

because $-\omega$ is $\varphi$-ample. We note that

$\begin{split}0>\nu^{*}\omega\cdot
C&=\nu^{*}\omega\cdot(\nu^{*}\mathcal{A})^{\dim E-1}\cdot\overline{E}\\\
&=\omega\cdot\mathcal{A}^{\dim E-1}\cdot E\\\ &\geq(\nu^{*}\mathcal{A})^{\dim
E-1}\cdot K_{\overline{E}}\\\ &=C\cdot K_{\overline{E}}.\end{split}$

Therefore, for any given point $x\in C$, there exists a rational curve
$\Gamma$ on $\overline{E}$ passing through $x$ with

$\begin{split}0<-\nu^{*}\omega\cdot\Gamma&\leq
2\dim\overline{E}\cdot\frac{-\nu^{*}\omega\cdot C}{-K_{\overline{E}}\cdot
C}\\\ &\leq 2\dim\overline{E}.\end{split}$

This is essentially due to Miyaoka–Mori (see [MM]). We note that
$\overline{E}$ is not always smooth but it is smooth in a neighborhood of $C$.
Hence we can use the argument of [MM]. For the details, see [Ko1, Chapter II.
5.8 Theorem]. Thus, $E$ is covered by rational curves $\ell:=\nu_{*}\Gamma$
with

$0<-\omega\cdot\ell\leq 2\dim E.$

Hence, by [Ko1, Chapter IV. 1.4 Proposition–Definition], $E$ is uniruled. We
finish the proof of Theorem 1.12. ∎

We prove Theorem 1.13.

###### Proof of Theorem 1.13.

Since $\varphi_{R}$ is the contraction morphism associated to $R$,

$\varphi_{R}\colon\operatorname{Nqlc}(X,\omega)\to\varphi_{R}(\operatorname{Nqlc}(X,\omega))$

is finite. We apply Theorem 1.12 to $\varphi_{R}\colon X\to W$, we can take a
rational curve $\ell$ on $X$ such that $\varphi_{R}(\ell)$ is a point with

$0<-\omega\cdot\ell\leq 2d.$

We finish the proof of Theorem 1.13. ∎

We explain an application of Theorems 1.12 and 1.13 for normal pairs, which is
a generalization of [Ka, Theorem 1].

######  Corollary 12.3.

Let $X$ be a normal variety and let $\Delta$ be an effective
$\mathbb{R}$-divisor on $X$ such that $K_{X}+\Delta$ is $\mathbb{R}$-Cartier.
Let $\pi\colon X\to S$ be a projective morphism between schemes. Let $R$ be a
$(K_{X}+\Delta)$-negative extremal ray of $\operatorname{\overline{NE}}(X/S)$
that are rational and relatively ample at infinity. Let $\varphi_{R}\colon
X\to W$ be the contraction morphism over $S$ associated to $R$. We put

$d=\min_{E}\dim E,$

where $E$ runs over positive-dimensional irreducible components of
$\varphi^{-1}_{R}(P)$ for all $P\in W$. Then $R$ is spanned by a (possibly
singular) rational curve $\ell$ with

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

Furthermore, if $\varphi_{R}$ is birational and $(X,\Delta)$ is kawamata log
terminal, then $R$ is spanned by a (possibly singular) rational curve $\ell$
with

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

Let $V$ be an irreducible component of the degenerate locus

$\left\\{x\in X\,|\,\text{$\varphi_{R}$ is not an isomorphism at
$x$}\right\\}$

of $\varphi_{R}$. Then $V$ is uniruled.

###### Proof.

We divide the proof into three small steps.

######  Step 1.

By Example 4.11, $[X,K_{X}+\Delta]$ naturally becomes a quasi-log scheme. By
applying Theorem 1.13 to $[X,K_{X}+\Delta]$, we see that $R$ is spanned by a
rational curve $\ell$ with

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

######  Step 2.

When $(X,\Delta)$ is kawamata log terminal and $\varphi_{R}$ is a birational
contraction, we take a $d$-dimensional irreducible component $E$ of
$\varphi^{-1}_{R}(P)$ for some $P\in W$. By shrinking $W$ around $P$, we may
assume that $W$ is affine. Since $\varphi_{R}$ is birational, there exists an
effective $\mathbb{Q}$-divisor $G$ on $X$ such that $(X,\Delta+G)$ is kawamata
log terminal and that $-G$ is $\varphi_{R}$-ample. By applying Theorem 1.12 to
$[X,K_{X}+\Delta+G]$, we see that $E$ is covered by rational curves $\ell$
with

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

Since $-G$ is $\varphi_{R}$-ample, we have

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

This implies that $R$ is spanned by a rational curve $\ell$ with

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

when $(X,\Delta)$ is kawamata log terminal and $\varphi_{R}$ is birational.

######  Step 3.

From now on, we will check that $V$ is uniruled. We shrink $W$ around the
generic point of $\varphi_{R}(V)$ and assume that $W$ is quasi-projective.
Then we take a sufficiently ample Cartier divisor $H$ on $W$ such that
$-(K_{X}+\Delta)+\varphi^{*}_{R}H$ is ample. By Theorem 1.12,
$V\cap\varphi^{-1}_{R}(P)$ is covered by rational curves $\ell$ of
$-\left((K_{X}+\Delta)+\varphi^{*}_{R}H\right)$-degree at most $2\dim V$ for
every $P\in\varphi_{R}(V)\subset W$. We take a suitable projective completion
$\overline{X}$ of $X$ and apply [Ko1, Chapter IV. 1.4 Proposition–Definition].
Then we obtain that $V$ is uniruled.

We finish the proof of Corollary 12.3. ∎

##  13\. Proof of Theorem 1.14

In this section, we prove Theorem 1.14. Let us start with the following
definition.

######  Definition 13.1 ([F11, Definition 6.8.1]).

Let $[X,\omega]$ be a quasi-log scheme and let $\pi\colon X\to S$ be a
projective morphism between schemes. If $-\omega$ is ample over $S$, then
$[X,\omega]$ is called a relative quasi-log Fano scheme over $S$. When $S$ is
a point, we simply say that $[X,\omega]$ is a quasi-log Fano scheme.

We recall an easy consequence of the vanishing theorem (see Theorem 4.6 (ii)),
which is missing in [F11, Section 6.8].

######  Lemma 13.2.

Let $[X,\omega]$ be a quasi-log scheme and let $\pi\colon X\to S$ be a proper
morphism between schemes with $\pi_{*}\mathcal{O}_{X}\simeq\mathcal{O}_{S}$.
Assume that $-\omega$ is nef and log big over $S$ with respect to
$[X,\omega]$. Then $X_{-\infty}\cap\pi^{-1}(P)$ is connected for every closed
point $P\in S$.

###### Proof.

By Theorem 4.6 (ii), $R^{1}\pi_{*}\mathcal{I}_{X_{-\infty}}=0$. Therefore, the
restriction map

$\mathcal{O}_{S}\simeq\pi_{*}\mathcal{O}_{X}\to\pi_{*}\mathcal{O}_{X_{-\infty}}$

is surjective. This implies that $X_{-\infty}\cap\pi^{-1}(P)$ is connected for
every closed point $P\in S$. ∎

Lemma 13.2 should have been stated in [F11, Lemma 6.8.3], which plays an
important role in this section. The main ingredient of the proof of Theorem
1.14 is the following theorem.

######  Theorem 13.3 ([Z, Theorem 1], [HM], and [BP, Corollary 1.4]).

Let $X$ be a normal projective 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 ample. Then $X$ is rationally chain connected
modulo $\operatorname{Nklt}(X,\Delta)$.

###### Proof.

We take an effective $\mathbb{Q}$-divisor $\Delta^{\prime}$ on $X$ such that
$K_{X}+\Delta^{\prime}$ is $\mathbb{Q}$-Cartier, $-(K_{X}+\Delta^{\prime})$ is
ample, and
$\operatorname{Nklt}(X,\Delta^{\prime})=\operatorname{Nklt}(X,\Delta)$ holds.
If $\operatorname{Nklt}(X,\Delta^{\prime})=\emptyset$, that is,
$(X,\Delta^{\prime})$ is kawamata log terminal, then $X$ is rationally
connected by [Z, Theorem 1]. In particular, $X$ is rationally chain connected
by Lemma 2.12. When $\operatorname{Nklt}(X,\Delta^{\prime})\neq\emptyset$, by
applying [BP, Corollary 1.4] to $(X,\Delta^{\prime})$, we obtain that for any
general point $x$ of $X$ there exists a rational curve $R_{x}$ passing through
$x$ and intersecting $\operatorname{Nklt}(X,\Delta^{\prime})$. By [Ko1,
Chapter II. 2.4 Corollary], for every $x\in X$, we can find a chain of
rational curves $R_{x}$ such that $x\in R_{x}$ and
$R_{x}\cap\operatorname{Nklt}(X,\Delta^{\prime})\neq\emptyset$. Hence $X$ is
rationally chain connected modulo $\operatorname{Nklt}(X,\Delta)$. We note
that if $-(K_{X}+\Delta)$ is an ample $\mathbb{Q}$-divisor then the proof of
[BP, Theorem 1.2 and Corollary 1.4] becomes much simpler than the general
case. Hence we obtain that $X$ is rationally chain connected modulo
$\operatorname{Nklt}(X,\Delta)$. ∎

We prepare one useful lemma.

######  Lemma 13.4.

Let $[X,\omega]$ be a projective quasi-log canonical pair such that
$\operatorname{Nqklt}(X,\omega)=\emptyset$, $-\omega$ is ample, and $X$ is
connected. Then $X$ is rationally connected. Hence $X$ is rationally chain
connected.

###### Proof.

By Lemma 4.8 and [F11, Theorem 6.3.11 (i)], $X$ is a normal variety. By
Theorem 1.10, we can find an effective $\mathbb{R}$-divisor $\Delta$ on $X$
such that $-(K_{X}+\Delta)$ is ample with
$\operatorname{Nklt}(X,\Delta)=\emptyset$. Hence $X$ is rationally connected
by [Z, Theorem 1] (see the proof of Theorem 13.3). ∎

By the framework of quasi-log schemes, we can prove the following lemma as a
generalization of Theorem 13.3 without any difficulties. We note that if
$\operatorname{Nqlc}(X,\omega)=\emptyset$ in Lemma 13.5 then it is nothing but
[F15, Theorem 1.7]. For semi-log canonical Fano pairs, we recommend the reader
to see [FLw].

######  Lemma 13.5.

Let $[X,\omega]$ be a projective quasi-log scheme such that $X$ is connected.
Assume that $-\omega$ is ample. Then $X$ is rationally chain connected modulo
$X_{-\infty}$.

###### Proof.

As in the proof of Theorem 11.1, we put

$[X_{0},\omega_{0}]:=[X,\omega]$

and

$[X_{i+1},\omega_{i+1}]:=[\operatorname{Nqklt}(X_{i},\omega_{i}),\omega_{i}|_{\operatorname{Nqklt}(X_{i},\omega_{i})}]$

for $i\geq 0$. Then there exists $k\geq 0$ such that

$\operatorname{Nqklt}(X_{k},\omega_{k})=\operatorname{Nqlc}(X_{k},\omega_{k})=\operatorname{Nqlc}(X,\omega)=X_{-\infty}.$

We note that if $X_{-\infty}=\emptyset$, that is, $[X,\omega]$ is quasi-log
canonical, then $X_{k}$ is the unique minimal qlc stratum of $[X,\omega]$ by
[F11, Theorem 6.8.3 (ii)]. By applying Lemma 13.4 to $[X_{k},\omega_{k}]$, we
obtain that $X_{k}$ is rationally connected when $X_{-\infty}=\emptyset$. We
want to show by inverse induction on $i$ that $X_{i+1}$ is rationally chain
connected modulo $X_{-\infty}=\operatorname{Nqlc}(X,\omega)$. Note that we
want to show that $X$ is rationally chain connected modulo $X_{k}$ when
$X_{-\infty}=\emptyset$. We also note that $X_{i}$ is connected by Lemma 13.2
and [F11, Theorem 6.8.3]. Hence it is sufficient to prove the following claim.

######  Claim.

Let $[X,\omega]$ be a quasi-log scheme such that $X$ is connected,
$\operatorname{Nqklt}(X,\omega)\neq\emptyset$, and $-\omega$ is ample. Then
$X$ is rationally chain connected modulo $\operatorname{Nqklt}(X,\omega)$.

###### Proof of Claim.

By adjunction (see Theorem 4.6 (i) and [F11, Theorem 6.3.5 (i)]) and [F11,
Theorem 6.8.3], we may assume that $\overline{X\setminus X_{-\infty}}$ is
irreducible. We note that every qlc stratum of $[X,\omega]$ intersects with
$\operatorname{Nqklt}(X,\omega)$ (see [F11, Theorem 6.8.3]). By Lemma 4.20, we
may further assume that $X$ itself is irreducible. Then, by Theorem 1.9, we
can further reduce the problem to the case where $X$ is a normal variety.
Then, by Theorem 1.10, we can take an ample $\mathbb{R}$-divisor $H$ on $X$
such that $-(\omega+H)$ is still ample and that

$K_{X}+\Delta\sim_{\mathbb{R}}\omega+H$

holds for some effective $\mathbb{R}$-divisor $\Delta$ on $X$ with

$\operatorname{Nklt}(X,\Delta)=\operatorname{Nqklt}(X,\omega).$

By applying Theorem 13.3 to $(X,\Delta)$, we obtain that $X$ is rationally
chain connected modulo $\operatorname{Nqklt}(X,\omega)$. We finish the proof
of Claim. ∎

By using Claim inductively, we can check that $X$ is rationally chain
connected modulo $X_{-\infty}=\operatorname{Nqlc}(X,\omega)$. ∎

Let us prove Theorem 1.14.

###### Proof of Theorem 1.14.

When $\pi^{-1}(P)\cap X_{-\infty}=\emptyset$, we may assume that
$X_{-\infty}=\emptyset$ by shrinking $X$ around $\pi^{-1}(P)$. We divide the
proof into several steps.

######  Step 1.

Let $X_{0}$ be the union of $X_{-\infty}$ and all qlc strata of $[X,\omega]$
contained in $\pi^{-1}(P)$. By Lemma 13.2 and [F11, Theorem 6.8.3],
$X_{0}\cap\pi^{-1}(P)$ is connected.

######  Case 1.

If $X_{0}\neq X_{-\infty}$, then $[X_{0},\omega_{0}]$, where
$\omega_{0}=\omega|_{X_{0}}$, is a quasi-log scheme by adjunction (see Theorem
4.6 (i) and [F11, Theorem 6.3.5 (i)]). Let us consider
$X^{\dagger}_{0}=\overline{X_{0}\setminus\operatorname{Nqlc}(X_{0},\omega_{0})}$.
Then $[X^{\dagger}_{0},\omega^{\dagger}_{0}]$, where
$\omega^{\dagger}_{0}=\omega|_{X^{\dagger}_{0}}$, is a quasi-log scheme by
Lemma 4.20. By construction, $-\omega^{\dagger}_{0}$ is ample since
$\pi(X^{\dagger}_{0})=P$. Therefore, by Lemma 13.5, $X^{\dagger}_{0}$ is
rationally chain connected modulo
$\operatorname{Nqlc}(X^{\dagger}_{0},\omega^{\dagger}_{0})$. This means that
$X_{0}\cap\pi^{-1}(P)$ is rationally chain connected modulo $\pi^{-1}(P)\cap
X_{-\infty}$.

######  Case 2.

If $X_{0}=X_{-\infty}$, that is, there is no qlc stratum of $[X,\omega]$
contained in $\pi^{-1}(P)$, then $X_{0}\cap\pi^{-1}(P)$ is obviously
rationally chain connected modulo $\pi^{-1}(P)\cap X_{-\infty}$ because
$X_{0}\cap\pi^{-1}(P)=\pi^{-1}(P)\cap X_{-\infty}$. Note that
$X_{0}=X_{-\infty}=\emptyset$ may happen.

Hence $\pi^{-1}(P)$ is rationally chain connected modulo $\pi^{-1}(P)\cap
X_{-\infty}$ when $\pi^{-1}(P)\subset X_{0}$. Thus, from now on, we may assume
that $\pi^{-1}(P)\not\subset X_{0}$.

######  Step 2.

Without loss of generality, we may assume that $S$ is quasi-projective by
shrinking $S$ around $P$. We take general very ample Cartier divisors
$B_{1},\ldots,B_{n+1}$ passing through $P$ with $n=\dim X$. Let
$f\colon(Y,B_{Y})\to X$ be a proper morphism from a globally embedded simple
normal crossing pair $(Y,B_{Y})$ as in Definition 4.2. By [F11, Proposition
6.3.1] and [Ko2, Theorem 3.35], we may assume that the union of all strata of
$(Y,B_{Y})$ mapped to $X_{0}$ by $f$, which is denoted by $Y^{\prime}$, is a
union of some irreducible components of $Y$. As usual, we put
$Y^{\prime\prime}=Y-Y^{\prime}$,
$K_{Y^{\prime\prime}}+B_{Y^{\prime\prime}}=(K_{Y}+B_{Y})|_{Y^{\prime\prime}}$,
and $f^{\prime\prime}=f|_{Y^{\prime\prime}}$. By [F11, Proposition 6.3.1] and
[Ko2, Theorem 3.35] again, we may further assume that

$\left(Y^{\prime\prime},(f^{\prime\prime})^{*}\pi^{*}\sum_{i=1}^{n+1}B_{i}+\operatorname{Supp}B_{Y^{\prime\prime}}\right)$

is a globally embedded simple normal crossing pair. By [F11, Lemma 6.3.13], we
can take $0<c_{1}<1$ such that

$f^{\prime\prime}\left(\left(B_{Y^{\prime\prime}}+c_{1}(f^{\prime\prime})^{*}\pi^{*}\sum_{i=1}^{n+1}B_{i}\right)^{>1}\right)=X_{0}$

holds set theoretically and that there exists an irreducible component $G$ of

$\left(B_{Y^{\prime\prime}}+c_{1}(f^{\prime\prime})^{*}\pi^{*}\sum_{i=1}^{n+1}B_{i}\right)^{=1}$

with $f^{\prime\prime}(G)\not\subset X_{0}$. By Lemma 4.23,

$\left(X,\omega+c_{1}B,f^{\prime\prime}\colon(Y^{\prime\prime},B_{Y^{\prime\prime}}+c_{1}(f^{\prime\prime})^{*}B)\to
X\right),$

where $B=\pi^{*}\left(\sum_{i=1}^{n+1}B\right)$, is a quasi-log scheme.

Let $X_{1}$ be the union of $\operatorname{Nqlc}(X,\omega+c_{1}B)$ and all qlc
strata of $[X,\omega+c_{1}B]$ contained in $\pi^{-1}(P)$. By construction,
$\operatorname{Nqlc}(X,\omega+c_{1}B)=X_{0}$ holds set theoretically.
Therefore, by Case 1 in Step 1, $X_{1}\cap\pi^{-1}(P)$ is rationally chain
connected modulo $X_{0}\cap\pi^{-1}(P)$. We note that by Step 1
$X_{0}\cap\pi^{-1}(P)$ is rationally chain connected modulo $\pi^{-1}(P)\cap
X_{-\infty}$. This means that $X_{1}\cap\pi^{-1}(P)$ is rationally chain
connected modulo $\pi^{-1}(P)\cap X_{-\infty}$.

######  Step 3.

By repeating the argument in Step 2, we can construct a finite increasing
sequence of positive real numbers

$0<c_{1}<\cdots<c_{k}<1$

and closed subschemes

$X_{1}\subsetneq\cdots\subsetneq X_{k}$

of $X$ with the following properties:

  * (a)

$[X_{i},\omega_{i}]$ is a quasi-log scheme, where
$\omega_{i}=(\omega+c_{i}B)|_{X_{i}}$, for every $i$,

  * (b)

$\operatorname{Nqlc}(X_{i+1},\omega_{i+1})=X_{i}$ holds set theoretically for
every $i$,

  * (c)

$\pi^{-1}(P)\subset X_{k}$ holds, and

  * (d)

$X_{i+1}\cap\pi^{-1}(P)$ is rationally chain connected modulo
$X_{i}\cap\pi^{-1}(P)$ for every $i$.

Hence we obtain that $\pi^{-1}(P)=\pi^{-1}(P)\cap X_{k}$ is rationally chain
connected modulo $\pi^{-1}(P)\cap X_{-\infty}$.

We finish the proof of Theorem 1.14. ∎

##  14\. Towards Conjecture 1.15

In this final section, we treat several results related to Conjecture 1.15.
This section needs some deep results on the theory of minimal models for
higher-dimensional algebraic varieties. Let us start with the following
special case of the flip conjecture II.

######  Conjecture 14.1 (Termination of flips).

Let $(X,\Delta)$ be a $\mathbb{Q}$-factorial klt pair and let $\pi\colon X\to
S$ be a projective surjective morphism between normal quasi-projective
varieties such that $K_{X}+\Delta$ is not pseudo-effective over $S$. Let

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

be a sequence of flips over $S$ starting from $(X,\Delta)$. Then it terminates
after finitely many steps.

In this section, we establish the following theorem, which is a precise
version of Theorem 1.16.

######  Theorem 14.2 (see Theorem 1.16).

Assume that Conjecture 14.1 holds true in dimension at most $\dim\pi^{-1}(P)$.
Then Conjecture 1.15 holds true.

For the proof of Theorem 14.2, we prepare a variant of Theorem 1.8. We need
the termination of flips in this theorem.

######  Theorem 14.3.

Let $X$ be a normal variety and let $\Delta$ be an effective
$\mathbb{R}$-divisor on $X$ such that $K_{X}+\Delta$ is $\mathbb{R}$-Cartier.
Assume that Conjecture 14.1 holds true in $\dim X$. Let $\pi\colon X\to S$ be
a projective morphism onto a scheme $S$ such that $-(K_{X}+\Delta)$ is
$\pi$-ample with $\dim S<\dim X$. We assume that
$\operatorname{Nklt}(X,\Delta)$ is not empty such that

$\pi\colon\operatorname{Nklt}(X,\Delta)\to\pi(\operatorname{Nklt}(X,\Delta))$

is finite. Then there exists a non-constant morphism

$f\colon\mathbb{A}^{1}\longrightarrow X\setminus\operatorname{Nklt}(X,\Delta)$

such that $\pi\circ f(\mathbb{A}^{1})$ is a point and that the curve $C$, the
closure of $f(\mathbb{A}^{1})$ in $X$, is a (possibly singular) rational curve
satisfying $C\cap\operatorname{Nklt}(X,\Delta)\neq\emptyset$ with

$0<-(K_{X}+\Delta)\cdot C\leq 1.$

###### Proof.

By shrinking $S$ suitably, we may assume that $X$ and $S$ are both quasi-
projective. By Lemma 3.10, we can construct a projective birational morphism
$g\colon Y\to X$ from a normal $\mathbb{Q}$-factorial variety satisfying (i),
(ii), and (iv) in Lemma 3.10. Since $K_{Y}+\Delta_{Y}=g^{*}(K_{X}+\Delta)$,
$(K_{Y}+\Delta_{Y})|_{\operatorname{Nklt}(Y,\Delta_{Y})}$ is nef over $S$ by
Lemma 3.10 (iv). Let us consider $\pi\circ g\colon Y\to S$. By construction,
$(Y,\Delta^{<1}_{Y})$ is a $\mathbb{Q}$-factorial klt pair. Since
$-(K_{X}+\Delta)$ is $\pi$-ample by assumption, $K_{Y}+\Delta_{Y}$ is not
pseudo-effective over $S$. Hence $K_{Y}+\Delta^{<1}_{Y}$ is not pseudo-
effective over $S$. Since
$(K_{Y}+\Delta_{Y})|_{\operatorname{Nklt}(Y,\Delta_{Y})}$ is nef over $S$, the
cone theorem

$\operatorname{\overline{NE}}(Y/S)=\operatorname{\overline{NE}}(Y/S)_{(K_{Y}+\Delta_{Y})\geq
0}+\sum_{j}R_{j}$

holds by [F6, Theorem 1.1] (see also Theorem 1.5 (i)). Since
$K_{Y}+\Delta_{Y}$ is not pseudo-effective over $S$, $K_{Y}+\Delta_{Y}$ is not
nef over $S$. Hence we have a $(K_{Y}+\Delta_{Y})$-negative extremal ray $R$
of $\operatorname{\overline{NE}}(Y/S)$. Then we consider the contraction
morphism $\varphi_{R}\colon Y\to W$ over $S$ associated to $R$ (see [F6,
Theorem 1.1] and Theorem 4.18). We note that $-(K_{Y}+\Delta^{<1}_{Y})\cdot
R>0$ since $(K_{Y}+\Delta_{Y})|_{\operatorname{Nklt}(Y,\Delta_{Y})}$ is nef
over $S$. If $\varphi_{R}$ is an isomorphism in a neighborhood of
$\operatorname{Nklt}(Y,\Delta_{Y})$, then we can run a minimal model program
with respect to $K_{Y}+\Delta_{Y}$ over $S$ by [BCHM]. Thus we run a minimal
model program with respect to $K_{Y}+\Delta_{Y}$ over $S$. Then we have a
sequence of flips and divisorial contractions

$\textstyle{Y=:Y_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{0}}$$\textstyle{Y_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{1}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{i-1}}$$\textstyle{Y_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{i}}$$\textstyle{\cdots}$

over $S$. As usual, we put $(Y_{0},\Delta_{Y_{0}}):=(Y,\Delta_{Y})$ and
$\Delta_{Y_{i+1}}={\phi_{i}}_{*}\Delta_{Y_{i}}$ for every $i$.

######  Case 1.

We assume that $\phi_{i}$ is an isomorphism in a neighborhood of
$\operatorname{Nklt}(Y_{i},\Delta_{Y_{i}})$ for every $i$. Then this minimal
model program is a minimal model program with respect to
$K_{Y}+\Delta^{<1}_{Y}$. Hence, by Conjecture 14.1, we finally get the
following diagram

$\textstyle{Y=:Y_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{0}}$$\scriptstyle{\pi\circ
g}$$\textstyle{Y_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{1}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{k-1}}$$\textstyle{Y_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S}$

where $\phi_{i}$ is a flip or a divisorial contraction for every $i$ and
$p\colon Y_{k}\to Z$ is a Fano contraction over $S$. We note that
$(K_{Y_{k}}+\Delta_{Y_{k}})|_{\operatorname{Nklt}(Y_{k},\Delta_{Y_{k}})}$ is
nef over $S$. By Case 1 in the proof of Proposition 9.1, we can find a curve
$C_{Y_{k}}\simeq\mathbb{P}^{1}$ on $Y_{k}$ such that $p(C_{Y_{k}})$ is a
point, $C_{Y_{k}}\cap\operatorname{Nklt}(Y_{k},\Delta_{Y_{k}})$ is a point,
and $0<-(K_{Y_{k}}+\Delta_{Y_{k}})\cdot C_{Y_{k}}\leq 1$ holds. By using the
negativity lemma, we can check that

$-(K_{Y}+\Delta_{Y})\cdot C_{Y}\leq-(K_{Y_{k}}+\Delta_{Y_{k}})\cdot
C_{Y_{k}}\leq 1$

holds, where $C_{Y}$ is the strict transform of $C_{Y_{k}}$ on $Y$. Note that
$C_{Y}\cap\operatorname{Nklt}(Y,\Delta_{Y})$ is a point since $\phi_{i}$ is an
isomorphism in a neighborhood of $\operatorname{Nklt}(Y_{i},\Delta_{Y_{i}})$
for every $i$. Therefore, $C=g(C_{Y})$ is a curve on $X$ such that
$C\cap\operatorname{Nklt}(X,\Delta)$ is a point by Lemma 3.10 (iv) with
$0<-(K_{X}+\Delta)\cdot C\leq 1$. Hence we can construct a morphism

$f\colon\mathbb{A}^{1}\longrightarrow X\setminus\operatorname{Nklt}(X,\Delta)$

such that $f(\mathbb{A}^{1})=C\cap(X\setminus\operatorname{Nklt}(X,\Delta))$.
This is a desired morphism.

######  Case 2.

We assume that there exists $i_{0}$ such that $\phi_{i}$ is an isomorphism in
a neighborhood of $\operatorname{Nklt}(Y_{i},\Delta_{Y_{i}})$ for $0\leq
i<i_{0}$ and $\phi_{i_{0}}$ is not an isomorphism in a neighborhood of
$\operatorname{Nklt}(Y_{i_{0}},\Delta_{Y_{i_{0}}})$. Then, by using Case 3 in
the proof of Proposition 9.1, we can find a curve
$C_{Y_{i_{0}}}\simeq\mathbb{P}^{1}$ on $Y_{i_{0}}$ such that
$C_{Y_{i_{0}}}\cap\operatorname{Nklt}(Y_{i_{0}},\Delta_{Y_{i_{0}}})$ is a
point, $C_{Y_{i_{0}}}$ is mapped to a point on $S$, and
$0<-(K_{Y_{i_{0}}}+\Delta_{Y_{i_{0}}})\cdot C_{Y_{i_{0}}}\leq 1$ holds. By the
same argument as in Case 1 above, we get a desired morphism

$f\colon\mathbb{A}^{1}\longrightarrow
X\setminus\operatorname{Nklt}(X,\Delta).$

We finish the proof of Theorem 14.3. ∎

By Theorem 14.3, we have:

######  Theorem 14.4.

In Theorem 9.2, we further assume that $\dim S<\dim X$ and that
$\Sigma\neq\emptyset$. If Conjecture 14.1 holds true in $\dim X$, then there
exists a non-constant morphism

$f\colon\mathbb{A}^{1}\longrightarrow X\setminus\Sigma$

such that $\pi\circ f(\mathbb{A}^{1})$ is a point and that the curve $C$, the
closure of $f(\mathbb{A}^{1})$ in $X$, is a rational curve satisfying
$C\cap\Sigma\neq\emptyset$ with

$0<-\mathcal{P}\cdot C\leq 1.$

###### Proof.

We use Theorem 14.3 instead of Theorem 1.8. Then the proof of Theorem 9.2
implies the existence of

$f\colon\mathbb{A}^{1}\longrightarrow X\setminus\Sigma$

with the desired properties. ∎

For the proof of Theorem 14.2, we establish the following somewhat technical
lemma.

######  Lemma 14.5.

Let $\pi\colon X\to S$ be a projective surjective morphism between normal
quasi-projective varieties with $\pi_{*}\mathcal{O}_{X}\simeq\mathcal{O}_{S}$
and $\dim S>0$ and let $[X,\omega]$ be a quasi-log scheme such that

$\pi\colon\operatorname{Nqklt}(X,\omega)\to\pi(\operatorname{Nqklt}(X,\omega))$

is finite. Let $P$ be a closed point of $S$ such that $\dim\pi^{-1}(P)>0$.
Then there exists an effective $\mathbb{R}$-Cartier divisor $B$ on $S$ such
that $[X,\omega+\pi^{*}B]$ is a quasi-log scheme with the following
properties:

  * (i)

$\operatorname{Nqklt}(X,\omega)\subset\operatorname{Nqklt}(X,\omega+\pi^{*}B)$,

  * (ii)

$\operatorname{Nqklt}(X,\omega)=\operatorname{Nqklt}(X,\omega+\pi^{*}B)$ holds
outside $\pi^{-1}(P)$,

  * (iii)

$\pi\colon\operatorname{Nqlc}(X,\omega+\pi^{*}B)\to\pi(\operatorname{Nqlc}(X,\omega+\pi^{*}B))$
is finite, and

  * (iv)

there exists a positive-dimensional qlc center of $[X,\omega+\pi^{*}B]$ in
$\pi^{-1}(P)$.

We further assume that $-\omega$ is $\pi$-ample. Let $W$ be a positive-
dimensional qlc center of $[X,\omega+\pi^{*}B]$ with $\pi(W)=P$. Let
$\nu\colon W^{\nu}\to W$ be the normalization. Then $[W^{\nu},\nu^{*}\omega]$
naturally becomes a quasi-log Fano scheme such that

$\nu^{-1}\left(\operatorname{Nqklt}(X,\omega)\cap\pi^{-1}(P)\right)\subset\operatorname{Nqklt}(W^{\nu},\nu^{*}\omega)$

holds set theoretically.

###### Proof.

Let $B_{1},\ldots,B_{n+1}$ be general very ample Cartier divisors on $S$
passing through $P$ with $n=\dim X$. Let $f\colon(Y,B_{Y})\to X$ be a proper
morphism from a globally embedded simple normal crossing pair $(Y,B_{Y})$ as
in Definition 4.2. Let $X^{\prime}$ be the union of
$\operatorname{Nqlc}(X,\omega)$ and all qlc centers of $[X,\omega]$ mapped to
$P$ by $\pi$. By [F11, Proposition 6.3.1] and [Ko2, Theorem 3.35], we may
assume that the union of all strata of $(Y,B_{Y})$ mapped to $X^{\prime}$ by
$f$, which is denoted by $Y^{\prime}$, is a union of some irreducible
components of $Y$. As usual, we put $Y^{\prime\prime}=Y-Y^{\prime}$,
$K_{Y^{\prime\prime}}+B_{Y^{\prime\prime}}=(K_{Y}+B_{Y})|_{Y^{\prime\prime}}$,
and $f^{\prime\prime}=f|_{Y^{\prime\prime}}$. By [F11, Proposition 6.3.1] and
[Ko2, Theorem 3.35] again, we may further assume that

$\left(Y^{\prime\prime},(f^{\prime\prime})^{*}\pi^{*}\sum_{i=1}^{n+1}B_{i}+\operatorname{Supp}B_{Y^{\prime\prime}}\right)$

is a globally embedded simple normal crossing pair. By [F11, Lemma 6.3.13], we
can take $0<c<1$ such that

  * (a)

we have

$f^{\prime\prime}\left(\left(B_{Y^{\prime\prime}}+c(f^{\prime\prime})^{*}\pi^{*}\sum_{i=1}^{n+1}B_{i}\right)^{>1}\right)\cap\pi^{-1}(P)=\emptyset$

or

$\dim\left(f^{\prime\prime}\left(\left(B_{Y^{\prime\prime}}+c(f^{\prime\prime})^{*}\pi^{*}\sum_{i=1}^{n+1}B_{i}\right)^{>1}\right)\cap\pi^{-1}(P)\right)=0,$

and

  * (b)

the following inequality

$\dim\left(f^{\prime\prime}\left(\left(B_{Y^{\prime\prime}}+c(f^{\prime\prime})^{*}\pi^{*}\sum_{i=1}^{n+1}B_{i}\right)^{=1}\right)\cap\pi^{-1}(P)\right)\geq
1$

holds.

By Lemma 4.23, we obtain that

$\left(X,\omega+\pi^{*}B,f^{\prime\prime}\colon(Y^{\prime\prime},B_{Y^{\prime\prime}}+(f^{\prime\prime})^{*}\pi^{*}B)\to
X\right),$

where $B=c\sum_{i=1}^{n+1}B_{i}$, is a quasi-log scheme. By construction, we
see that (i) holds true and $\operatorname{Nqklt}(X,\omega+\pi^{*}B)$
coincides with $\operatorname{Nqklt}(X,\omega)$ outside $\pi^{-1}(P)$ since
$B_{1},\ldots,B_{n+1}$ are general very ample Cartier divisors on $S$. Hence
we have (ii). Therefore, (iii) and (iv) follow from (a) and (b), respectively.

From now on, we further assume that $-\omega$ is $\pi$-ample. As usual, we put

$W^{\prime}=W\cup\operatorname{Nqlc}(X,\omega+\pi^{*}B).$

By adjunction (see Theorem 4.6 (i) and [F11, Theorem 6.3.5 (i)]),
$[W^{\prime},(\omega+\pi^{*}B)|_{W^{\prime}}]$ is a quasi-log scheme. By Lemma
4.20, $[W,(\omega+\pi^{*}B)|_{W}]$ naturally becomes a quasi-log scheme. We
note that $(\pi^{*}B)|_{W}\sim_{\mathbb{R}}0$ since $\pi(W)=P$. Therefore, by
replacing $(\omega+\pi^{*}B)|_{W}$ with $\omega|_{W}$, we see that
$[W,\omega|_{W}]$ is a quasi-log scheme. By Theorem 1.9,
$[W^{\nu},\nu^{*}\omega]$ becomes a quasi-log scheme. Note that
$-\nu^{*}\omega$ is ample since $\pi\circ\nu(W^{\nu})=P$.

######  Claim.

We have

$\begin{split}&\operatorname{Nqklt}(X,\omega)\cap\pi^{-1}(P)\\\
&\subset\operatorname{Nqklt}(W^{\prime},(\omega+\pi^{*}B)|_{W^{\prime}})\cap\pi^{-1}(P)\\\
&=\operatorname{Nqklt}(W,(\omega+\pi^{*}B)|_{W})\\\
&=\operatorname{Nqklt}(W,\omega|_{W})\end{split}$

set theoretically.

###### Proof of Claim.

We divide the proof into several steps.

######  Step 1.

By (iii) and Lemma 13.2,
$\operatorname{Nqlc}(X,\omega+\pi^{*}B)\cap\pi^{-1}(P)$ is empty or a point.
By [F11, Theorem 6.8.3 (i)], every qlc center of $[X,\omega+\pi^{*}B]$ in
$\pi^{-1}(P)$ contains $\operatorname{Nqlc}(X,\omega+\pi^{*}B)\cap\pi^{-1}(P)$
when $\operatorname{Nqlc}(X,\omega+\pi^{*}B)\cap\pi^{-1}(P)\neq\emptyset$.
When $\operatorname{Nqlc}(X,\omega+\pi^{*}B)\cap\pi^{-1}(P)=\emptyset$, the
set of all qlc centers intersecting $\pi^{-1}(P)$ has a unique minimal element
with respect to the inclusion by [F11, Theorem 6.8.3 (ii)].

######  Step 2.

In this step, we will check that

$\operatorname{Nqklt}(X,\omega)\cap\pi^{-1}(P)\subset\operatorname{Nqklt}(W^{\prime},(\omega+\pi^{*}B)|_{W^{\prime}})\cap\pi^{-1}(P)$

holds set theoretically.

If $\operatorname{Nqklt}(X,\omega)\cap\pi^{-1}(P)=\emptyset$, then it is
obvious. Hence we may assume that
$\operatorname{Nqklt}(X,\omega)\cap\pi^{-1}(P)\neq\emptyset$. By assumption,
$\operatorname{Nqklt}(X,\omega)\cap\pi^{-1}(P)$ is zero-dimensional. We take
$Q\in\operatorname{Nqklt}(X,\omega)\cap\pi^{-1}(P)$. If $Q$ is a qlc center of
$[X,\omega]$ or $Q\in\operatorname{Nqlc}(X,\omega)$, then
$Q\in\operatorname{Nqlc}(X,\omega+\pi^{*}B)$ by the construction of the quasi-
log scheme structure of $[X,\omega+\pi^{*}B]$. Then we have

$Q\in\operatorname{Nqlc}(W^{\prime},(\omega+\pi^{*}B)|_{W^{\prime}})\subset\operatorname{Nqklt}(W^{\prime},(\omega+\pi^{*}B)|_{W^{\prime}}).$

Therefore, we have

$Q\in\operatorname{Nqklt}(W^{\prime},(\omega+\pi^{*}B)|_{W^{\prime}})\cap\pi^{-1}(P).$

From now on, we assume that $Q$ is not a qlc center of $[X,\omega]$ and that
$Q\not\in\operatorname{Nqlc}(X,\omega)$. Then, there exists a positive-
dimensional qlc center $V$ of $[X,\omega]$ such that $Q\in V\cap\pi^{-1}(P)$.
Since $\operatorname{Nqklt}(X,\omega)=\operatorname{Nqklt}(X,\omega+\pi^{*}B)$
holds outside $\pi^{-1}(P)$ (see (ii)), $V$ is also a qlc center of
$[X,\omega+\pi^{*}B]$. If
$\operatorname{Nqlc}(X,\omega+\pi^{*}B)\cap\pi^{-1}(P)\neq\emptyset$, then
$\operatorname{Nqlc}(X,\omega+\pi^{*}B)\cap\pi^{-1}(P)$ is a point by (iii)
and Lemma 13.2. In this case, by [F11, Theorem 6.8.3 (i)], we have $Q\in
V\cap\pi^{-1}(P)\cap\operatorname{Nqlc}(X,\omega+\pi^{*}B)$. Hence
$Q\in\operatorname{Nqlc}(W^{\prime},(\omega+\pi^{*}B)|_{W^{\prime}})\cap\pi^{-1}(P)$.
This implies that
$Q\in\operatorname{Nqklt}(W^{\prime},(\omega+\pi^{*}B)|_{W^{\prime}})\cap\pi^{-1}(P)$.
Thus we further assume that
$\operatorname{Nqlc}(X,\omega+\pi^{*}B)\cap\pi^{-1}(P)=\emptyset$. By
shrinking $X$ around $\pi^{-1}(P)$, we may assume that $[X,\omega+\pi^{*}B]$
is quasi-log canonical. Then $Q\in V\cap W\cap\pi^{-1}(P)$ by Step 1 (see also
[F11, Theorem 6.8.3 (ii)]). Hence
$Q\in\operatorname{Nqklt}(W^{\prime},(\omega+\pi^{*}B)|_{W^{\prime}})\cap\pi^{-1}(P)$
by [F11, Theorem 6.3.11 (i)]. More precisely, $Q$ is a qlc center of
$[W^{\prime},(\omega+\pi^{*}B)|_{W^{\prime}}]$.

In any case, we obtain

$\operatorname{Nqklt}(X,\omega)\cap\pi^{-1}(P)\subset\operatorname{Nqklt}(W^{\prime},(\omega+\pi^{*}B)|_{W^{\prime}})\cap\pi^{-1}(P)$

set theoretically.

######  Step 3.

By Step 1 and Lemma 4.20,

$\operatorname{Nqklt}(W^{\prime},(\omega+\pi^{*}B)|_{W^{\prime}})\cap\pi^{-1}(P)=\operatorname{Nqklt}(W,(\omega+\pi^{*}B)|_{W})$

holds set theoretically. By the definition of the quasi-log scheme structure
of $[W,\omega|_{W}]$,

$\operatorname{Nqklt}(W,(\omega+\pi^{*}B)|_{W})=\operatorname{Nqklt}(W,\omega|_{W})$

obviously holds.

We finish the proof of Lemma 14.5. ∎

Hence by Claim

$\nu^{-1}\left(\operatorname{Nqklt}(X,\omega)\cap\pi^{-1}(P)\right)\subset\operatorname{Nqklt}(W^{\nu},\nu^{*}\omega)$

holds set theoretically since
$\nu^{-1}\left(\operatorname{Nqklt}(W,\omega|_{W})\right)=\operatorname{Nqklt}(W^{\nu},\nu^{*}\omega)$
by Theorem 1.9. ∎

Let us prove Theorem 14.2, which is stronger than Theorem 1.16.

###### Proof of Theorem 14.2.

We first use the reduction as in Steps 2, 3, and 4 in the proof of Theorem
1.6. Let us explain it more precisely for the reader’s convenience.

######  Step 1.

We take an irreducible component $W$ of $X$ such that $C^{\dagger}\subset W$.
We put $X^{\prime}=W\cup\operatorname{Nqlc}(X,\omega)$. Then, by adjunction
(see Theorem 4.6 (i) and [F11, Theorem 6.3.5 (i)]),
$[X^{\prime},\omega^{\prime}=\omega|_{X^{\prime}}]$ is a quasi-log scheme. By
replacing $[X,\omega]$ with $[X^{\prime},\omega^{\prime}]$, we may assume that
$X\setminus X_{-\infty}$ is irreducible. By Lemma 4.20, we may replace $X$
with $\overline{X\setminus X_{-\infty}}$ and assume that $X$ is a variety.
Then, by taking the normalization, we may further assume that $X$ is a normal
variety (see Theorem 1.9).

######  Step 2.

By taking the Stein factorization, we may further assume that
$\pi_{*}\mathcal{O}_{X}\simeq\mathcal{O}_{S}$. We put
$\Sigma=\operatorname{Nqklt}(X,\omega)$. It is sufficient to find a non-
constant morphism

$f\colon\mathbb{A}^{1}\longrightarrow(X\setminus\Sigma)\cap\pi^{-1}(P)$

such that the curve $C$, the closure of $f(\mathbb{A}^{1})$ in $X$, is a
(possibly singular) rational curve satisfying $C\cap\Sigma\neq\emptyset$ with

$0<-\omega\cdot C\leq 1.$

Without loss of generality, we may assume that $X$ and $S$ are quasi-
projective by shrinking $S$ suitably.

######  Step 3.

By assumption, $\dim\pi^{-1}(P)>0$ and $\pi^{-1}(P)\cap\Sigma\neq\emptyset$.
When $\dim S>0$, by Lemma 14.5, we take an effective $\mathbb{R}$-Cartier
divisor $B$ on $S$ such that $[X,\omega+\pi^{*}B]$ is a quasi-log scheme
satisfying the properties (i), (ii), (iii), and (iv) in Lemma 14.5. Then we
take a positive-dimensional qlc center $W$ of $[X,\omega+\pi^{*}B]$ in
$\pi^{-1}(P)$ such that there is no positive-dimensional qlc center
$W^{\dagger}\subsetneq W$ of $[X,\omega+\pi^{*}B]$. By Lemma 14.5,
$[W^{\nu},\nu^{*}\omega]$ naturally becomes a quasi-log Fano scheme, where
$\nu\colon W^{\nu}\to W$ is the normalization. When $\dim S=0$, it is
sufficient to put $B=0$ and $W=X$. By construction,
$\operatorname{Nqklt}(W^{\nu},\nu^{*}\omega)$ is finite. On the other hand,
$\operatorname{Nqklt}(W^{\nu},\nu^{*}\omega)$ is connected (see Lemma 13.2 and
[F11, Theorem 6.8.3]). By Lemma 14.5, we obtain

$\emptyset\neq\nu^{-1}\left(\Sigma\cap\pi^{-1}(P)\right)\subset\operatorname{Nqklt}(W^{\nu},\nu^{*}\omega).$

Hence $\operatorname{Nqklt}(W^{\nu},\nu^{*}\omega)$ is a point such that
$\nu^{-1}\left(\Sigma\cap\pi^{-1}(P)\right)=\operatorname{Nqklt}(W^{\nu},\nu^{*}\omega)$
holds set theoretically. By applying Theorems 1.10 and 14.4 to
$[W^{\nu},\nu^{*}\omega]$ as in Step 5 in the proof of Theorem 1.6, we obtain
a non-constant morphism

$h\colon\mathbb{A}^{1}\longrightarrow
W^{\nu}\setminus\operatorname{Nqklt}(W^{\nu},\nu^{*}\omega)$

such that $C^{\prime}$, the closure of $h(\mathbb{A}^{1})$ in $W^{\nu}$, is a
(possibly singular) rational curve satisfying
$C^{\prime}\cap\operatorname{Nqklt}(W^{\nu},\nu^{*}\omega)\neq\emptyset$ with
$0<-\nu^{*}\omega\cdot C^{\prime}\leq 1$. Then

$f:=\iota\circ\nu\circ
h\colon\mathbb{A}^{1}\longrightarrow(X\setminus\Sigma)\cap\pi^{-1}(P),$

where $\iota\colon W\hookrightarrow X$ is a natural inclusion, is a desired
morphism.

We finish the proof of Theorem 14.2. ∎

For the proof of Theorem 1.20, we prepare the following theorem. The proof of
Theorem 14.6 uses a deep result on the existence problem of minimal models in
[H1].

######  Theorem 14.6.

Let $(X,\Delta)$ be a dlt pair and let $\pi\colon X\to S$ be a projective
morphism between normal varieties such that $-(K_{X}+\Delta)$ is $\pi$-ample.
We assume that $\operatorname{Nklt}(X,\Delta)$ is not empty such that
$\pi\colon\operatorname{Nklt}(X,\Delta)\to\pi(\operatorname{Nklt}(X,\Delta))$
is finite and that there exists a curve $C^{\dagger}$ on $X$ such that
$\pi(C^{\dagger})$ is a point with
$C^{\dagger}\cap\operatorname{Nklt}(X,\Delta)\neq\emptyset$. Then there exists
a non-constant morphism

$f\colon\mathbb{A}^{1}\longrightarrow X\setminus\operatorname{Nklt}(X,\Delta)$

such that $\pi\circ f(\mathbb{A}^{1})$ is a point and that the curve $C$, the
closure of $f(\mathbb{A}^{1})$ in $X$, is a (possibly singular) rational curve
satisfying $C\cap\operatorname{Nklt}(X,\Delta)\neq\emptyset$ with

$0<-(K_{X}+\Delta)\cdot C\leq 1.$

###### Proof.

By shrinking $S$ suitably, we may assume that $X$ and $S$ are both quasi-
projective. By Lemma 3.10, we can construct a projective birational morphism
$g\colon Y\to X$ from a normal $\mathbb{Q}$-factorial variety satisfying (i),
(ii), and (iv) in Lemma 3.10. Since $K_{Y}+\Delta_{Y}=g^{*}(K_{X}+\Delta)$,
$(K_{Y}+\Delta_{Y})|_{\operatorname{Nklt}(Y,\Delta_{Y})}$ is nef over $S$ by
Lemma 3.10 (iv). Let us consider $\pi\circ g\colon Y\to S$. By construction,
$(Y,\Delta_{Y})$ is a $\mathbb{Q}$-factorial dlt pair.

If $\dim S=\dim X$, then $K_{Y}+\Delta_{Y}$ is pseudo-effective over $S$. In
this case, we can take an effective $\mathbb{R}$-divisor $A$ on $Y$ such that
$K_{Y}+\Delta_{Y}+A\sim_{\mathbb{R},\pi\circ g}0$ and that $(Y,\Delta_{Y}+A)$
is dlt since $-(K_{Y}+\Delta_{Y})=-g^{*}(K_{X}+\Delta)$ is $(\pi\circ
g)$-semi-ample. Hence $(Y,\Delta_{Y})$ has a good minimal model over $S$ by
[H1, Theorem 1.1] and any $(K_{Y}+\Delta_{Y})$-minimal model program over $S$
with scaling of an ample divisor terminates (see [H1, Theorem 2.11]).

If $\dim S<\dim X$, then $K_{Y}+\Delta_{Y}$ is not pseudo-effective over $S$
since $-(K_{X}+\Delta)$ is ample over $S$. In this case, we have a
$(K_{Y}+\Delta_{Y})$-minimal model program which terminates at a Mori fiber
space by [BCHM].

Therefore, we have a finite sequence of flips and divisorial contractions

$\textstyle{Y=:Y_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{0}}$$\textstyle{Y_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{1}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{i-1}}$$\textstyle{Y_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{i}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{k-1}}$$\textstyle{Y_{k}}$

starting from $(Y_{0},\Delta_{Y_{0}}):=(Y,\Delta_{Y})$ over $S$ such that
$(Y_{k},\Delta_{Y_{k}})$ is a good minimal model of $(Y,\Delta_{Y})$ over $S$
or $p\colon Y_{k}\to Z$ is a Mori fiber space with respect to
$K_{Y_{k}}+\Delta_{Y_{k}}$ over $S$, where
$\Delta_{Y_{i+1}}={\phi_{i}}_{*}\Delta_{Y_{i}}$ for every $i$. By assumption,
we can take a curve $C^{\prime}$ on $Y$ such that $-(K_{Y}+\Delta_{Y})\cdot
C^{\prime}>0$ with
$C^{\prime}\cap\operatorname{Nklt}(Y,\Delta_{Y})\neq\emptyset$. If $\phi_{i}$
is an isomorphism in a neighborhood of
$\operatorname{Nklt}(Y_{i},\Delta_{Y_{i}})$ for $0\leq i<l$, then

(14.1) $0<-(K_{Y}+\Delta_{Y})\cdot
C^{\prime}\leq-(K_{Y_{l}}+\Delta_{Y_{l}})\cdot C^{\prime}_{Y_{l}}$

holds by the negativity lemma, where $C^{\prime}_{Y_{l}}$ is the strict
transform of $C^{\prime}$ on $Y_{l}$.

######  Case 1.

We assume that $\phi_{i}$ is an isomorphism in a neighborhood of
$\operatorname{Nklt}(Y_{i},\Delta_{Y_{i}})$ for every $i$. Then, by (14.1),
the final model $Y_{k}$ has a Mori fiber space structure $p\colon Y_{k}\to Z$
over $S$. We note that
$(K_{Y_{k}}+\Delta_{Y_{k}})|_{\operatorname{Nklt}(Y_{k},\Delta_{Y_{k}})}$ is
nef over $S$. Hence the argument in Case 1 in the proof of Theorem 14.3 works
without any changes. Then we get a non-constant morphism

$f\colon\mathbb{A}^{1}\longrightarrow X\setminus\operatorname{Nklt}(X,\Delta)$

with the desired properties.

######  Case 2.

By Case 1, we may assume that there exists $i_{0}$ such that $\phi_{i}$ is an
isomorphism in a neighborhood of $\operatorname{Nklt}(Y_{i},\Delta_{Y_{i}})$
for $0\leq i<i_{0}$ and $\phi_{i_{0}}$ is not an isomorphism in a neighborhood
of $\operatorname{Nklt}(Y_{i_{0}},\Delta_{Y_{i_{0}}})$. The argument in Case 2
in the proof of Theorem 14.3 works without any changes. Then we get a non-
constant morphism

$f\colon\mathbb{A}^{1}\longrightarrow X\setminus\operatorname{Nklt}(X,\Delta)$

with the desired properties.

We finish the proof of Theorem 14.6. ∎

We close this section with the proof of Theorem 1.20. Since adjunction works
well for dlt pairs, Theorem 1.20 directly follows from Theorem 14.6.

###### Proof of Theorem 1.20.

We put $W=\overline{U_{j}}$. Then $W$ is an lc stratum of $(X,\Delta)$. By
adjunction, it is well known that we have

$(K_{X}+\Delta)|_{W}=K_{W}+\Delta_{W}$

such that $(W,\Delta_{W})$ is dlt and that the lc centers of $(W,\Delta_{W})$
are exactly the lc centers of $(X,\Delta)$ that are strictly included in $W$
(see, for example, [F3, Proposition 3.9.2]). By replacing $\pi\colon X\to S$
with the Stein factorization of $\varphi_{R_{j}}\colon
W\to\varphi_{R_{j}}(W)$, we may assume that
$\pi\colon\operatorname{Nklt}(X,\Delta)\to\pi(\operatorname{Nklt}(X,\Delta))$
is finite and that there exists a curve $C^{\dagger}$ on $X$ such that
$\pi(C^{\dagger})$ is a point with
$C^{\dagger}\cap\operatorname{Nklt}(X,\Delta)\neq\emptyset$. By Theorem 14.6,
we obtain a desired non-constant morphism

$f\colon\mathbb{A}^{1}\to X\setminus\operatorname{Nklt}(X,\Delta)$

with the desired properties. ∎

As we have already mentioned, we will completely prove Conjecture 1.15 in a
joint paper with Kenta Hashizume (see [FH1]), where we use some deep results
on the minimal model program for log canonical pairs. We strongly recommend
the interested reader to see [FH1].

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