Proof. By Remark 2.2, we can take the Stein factorization, and then we may assume that $\pi $ has connected fibers. When $\dim Y=0$ , we have $U=W=Y$ and there is nothing to prove. When $\dim Y=1$ , since we have replaced $\pi $ with its Stein factorization, Y is a smooth curve and $\pi $ has smooth connected fibers on $U-W$ . Then $\pi $ is flat over Y, and $\pi $ has irreducible fibers on $U-W$ . Pick a point $P\in W$ . Since $\pi $ is flat, we have $\pi ^*Q\cdot D=\pi ^*P\cdot D\geq 0$ for each $Q\in Y$ . Since the fiber $\pi ^{-1}(Q)$ is irreducible for each $Q\in U-W$ , D is $\pi $ -nef over $U-W$ . Since D is also $\pi $ -nef over W, D is $\pi $ -nef over U. When $\dim Y=2$ , since $\pi $ has connected fibers, $\pi $ is isomorphic over $U-W$ . Then D is clearly $\pi $ -nef over U.

Proof. This is obvious by [Reference Fujino6, Theorems 4.6.1 and 4.6.2]. Since $\operatorname {dim}\operatorname {Nlc}(X, \Delta )=0$ , we have $\overline {\text {NE}}(X/Y; W)_{\operatorname {Nlc}(X, \Delta )}=0$ .

Proof. By Lemma 3.1, we have $[C]\in R$ . Since $\operatorname {dim}Y\geq 1$ , we have $\operatorname {dim}Z\geq 1$ . When $\operatorname {dim}Z=1$ , then Z is a smooth curve. If $\operatorname {Supp}\varphi _R^*(\varphi _R(C))$ is irreducible, C is proportional to the pullback of a point in Z. In particular, C is $\mathbb {Q}$ -Cartier. Suppose now that $\operatorname {Supp}\varphi _R^*(\varphi _R(C))$ is reducible. (Actually, it turns out that this situation does not occur. See the proof of the next lemma.) In this case, we have $C^2<0$ in the sense of Mumford (see, e.g., [Reference Sakai14, Section 1]), because $\varphi _R$ has connected fibers. On the other hand, when $\operatorname {dim}Z=2$ , we also have $C^2<0$ in the sense of Mumford. Then there exists a bimeromorphic contraction morphism $\psi \colon X\rightarrow X'$ such that $\operatorname {Exc}(\psi )=C$ , by the Grauert–Sakai contraction theorem (cf. [Reference Sakai14, Theorem 1.2]). By Lemma 2.11, $(X', \psi _*\Delta )$ is numerically log canonical (for the definition of numerical log canonicity, see [Reference Kollár and Mori11, Notation 4.1]). This implies that $K_{X'}+\psi _*\Delta $ is $\mathbb {R}$ -Cartier. Thus, we can write $K_X+\Delta =\psi ^*(K_{X'}+\psi _*\Delta )+aC$ , where $a\in \mathbb {R}_{>0}$ . In particular, C is $\mathbb {Q}$ -Cartier.

Proof. By Lemma 3.1, we have a contraction morphism $\phi _R$ over some open neighborhood of W. We divide the argument according to the dimensions of Y and $\operatorname {Im} \phi _R$ .

Case 1. When $\operatorname {dim}Y=0$ , we have $U=W=Y$ . Then this lemma is obvious by Lemma 3.1.

We note that when $\operatorname {dim}Y\geq 1$ , every exceptional curve of $\phi _R$ is $\mathbb {Q}$ -Cartier. Indeed, it is clear in Cases (A) and (B), and follows from Lemma 3.2 in Case (C). In Case (D), any divisor on an open subset of X is $\mathbb {Q}$ -Cartier (cf. [Reference Ishii10, Theorem 7.3.2]).

In Cases 2 and 3, we assume that $\dim Y=1$ . In these cases, taking the Stein factorization of $\pi $ , we may assume that the fiber $\pi ^{-1}(P)$ is smooth and irreducible for each $P\in Y-W$ . Let C be a curve on X mapped into a point on W such that the numerical equivalence class $[C]\in \overline {\text {NE}}(X/Y; W)$ generates R.

Case 2. When $\dim Y=1$ and $\dim (\operatorname {Im} \phi _R)=1$ , we can write $\phi _R^*(\phi _R(C))=E=\sum a_iE_i$ , where $a_i>0$ and $E_i$ ’s are irreducible curves. As noted above, each $E_i$ is a $\mathbb {Q}$ -Cartier divisor. If $\operatorname {Supp}E\neq \operatorname {Supp}C$ , then some $E_i$ satisfies $E_i\cdot C>0$ because $\phi _R$ has connected fibers. By the property (i) in Lemma 3.1, we have $E^2>0$ . This is a contradiction, by Lemma 2.12. Then, $\operatorname {Supp}\phi _R^*(\phi _R(C))=\operatorname {Supp}C$ . By Lemma 2.12, we have $C^2=0$ . Then, C is a support function of R. Moreover, since C is proportional to a pullback of a point in Z, $mC$ is $\pi $ -free over Y for some positive integer m. We obtain a morphism $X\rightarrow \mathbb {P}_Y(\pi _*\mathcal {O}_X(mC))$ . Taking the Stein factorization of this morphism, we obtain a contraction morphism $\varphi _R\colon X\rightarrow Z$ over Y. This $\varphi _R$ coincides with $\phi _R$ on the open subset of X on which $\phi _R$ is defined. This morphism satisfies the properties (i), (ii), and (iv) of this lemma.

To show that $\varphi _R$ satisfies (iii)’, take a line bundle $\mathcal {L}$ on X such that $\mathcal {L}\cdot C=0$ . We take a general fiber F of $\varphi _R$ . We may assume that F is smooth, by Sard’s theorem (cf. [Reference Ueno16, Corollary 1.8]). Since $-(K_X+\Delta )$ is $\varphi _R$ -ample, $-(K_X+\Delta )|_F=-K_F-\Delta |_F$ is ample. Then, $-K_F$ is ample. This implies that the genus of the smooth curve F is 0, that is, $F\cong \mathbb {P}^1$ . Since $\varphi _R$ is flat, $\mathcal {L}\cdot F=0$ . So, $\mathcal {L}|_F\cong \mathcal {O}_{\mathbb {P}^1}$ . Then, for a general point $z\in Z$ , we have

$$ \begin{align*} h^0(X_z, \mathcal{L}|_{X_z})=1,\quad\text{and}\quad h^0(X_z, \mathcal{L}^{-1}|_{X_z})=1. \end{align*} $$

By the upper semi-continuity (cf. [Reference Ueno16, Theorem 1.4]), for every point $z\in Z$ , we have

$$ \begin{align*} h^0(X_z, \mathcal{L}|_{X_z})\geq1,\quad\text{and}\quad h^0(X_z, \mathcal{L}^{-1}|_{X_z})\geq1. \end{align*} $$

This implies that $\mathcal {L}|_{X_z}\cong \mathcal {O}_{X_z}$ for every point $z\in Z$ . In particular, $\mathcal {L}$ is $\varphi _R$ -free over Z. Since $\mathcal {L}$ is $\varphi _R$ -numerically trivial, it is a pull-back of a line bundle on Z. Then, $\varphi _R$ satisfies (iii)’.

Case 3. When $\dim Y=1$ and $\dim (\operatorname {Im}\phi _R)=2$ , we have $C^2<0$ by Lemma 2.11. Take a $\pi $ -ample line bundle $\mathcal {A}$ on X. Then there is a positive rational number $\lambda $ such that $C+\lambda \mathcal {A}$ is a support function of R. Put $H:=C+\lambda \mathcal {A}$ . We can take some positive integer a such that $aH-(K_X+\Delta )$ is $\pi $ -ample over W. By the basepoint-free theorem (cf. [Reference Fujino6, Theorem 1.1.11]), $mH$ is $\pi $ -free over a neighborhood of W for every sufficiently large positive integer m. By the definition of H, for a point $P\in Y-W$ , we have $H\cdot \pi ^*P>0$ . Then $m_1(H\cdot \pi ^*P)\geq 2g+1$ for every sufficiently large positive integer $m_1$ , where g is the genus of the smooth curve $\pi ^{-1}(P)$ . Then $m_1H|_{\pi ^{-1}(P)}$ is very ample. Since $\pi \colon X\rightarrow Y$ is flat, $H\cdot \pi ^*Q=H\cdot \pi ^*P$ and the genus of $\pi ^{-1}(Q)$ is equal to the genus g of $\pi ^{-1}(P)$ for every point $Q\in Y-W$ . Then $m_1H|_{\pi ^{-1}(Q)}$ is very ample for every point $Q\in Y-W$ . This implies that $mH$ is $\pi $ -very ample over $Y-W$ for every sufficiently large positive integer m. Then, $mH$ is $\pi $ -free over Y for a positive integer m, and we obtain a morphism $\varphi _R\colon X\rightarrow Z$ over Y as in Case 2. This $\varphi _R$ coincides with $\phi _R$ on the open subset of X on which $\phi _R$ is defined. This morphism satisfies the properties (i), (ii), and (iv) of this lemma.

To show that $\varphi _R$ satisfies (iii)’, we take a line bundle $\mathcal {L}$ on X such that $\mathcal {L}\cdot C=0$ . Then $\mathcal {L}-(K_X+\Delta )$ is $\varphi _R$ -ample over $\pi _Z^{-1}W$ . By the basepoint-free theorem (cf. [Reference Fujino6, Theorem 1.1.11]), $m\mathcal {L}$ is $\varphi _R$ -free over a neighborhood of $\pi _Z^{-1}W$ for every sufficiently large positive integer m.

We will check that $m\mathcal {L}$ is $\varphi _R$ -free over $Z-\pi _Z^{-1}W$ for every sufficiently large positive integer m. Since $\dim Z=2$ , let E be a divisor on X such that $\operatorname {Supp}E$ corresponds to the exceptional locus of $\varphi _R$ . By the property (i), the numerical equivalence class $[E_i]\in \overline {\text {NE}}(X/Y; W)$ of each irreducible component $E_i$ of E is included in R. Since $E_i\cdot H=0$ , we have $E_i\subset \operatorname {Supp}C$ . Then, the exceptional locus of $\varphi _R$ corresponds to $\operatorname {Supp}C$ . Since $\varphi _R$ is isomorphic over $Z-\pi _Z^{-1}W$ , $m\mathcal {L}$ is $\varphi _R$ -free over $Z-\pi _Z^{-1}W$ for any positive integer m.

We have checked that $m\mathcal {L}$ is $\varphi _R$ -free over Z for every sufficiently large positive integer m. Then, there is some positive integer $m_0$ such that both $m_0\mathcal {L}$ and $(m_0+1)\mathcal {L}$ are $\varphi _R$ -free over Z. Since both $m_0\mathcal {L}$ and $(m_0+1)\mathcal {L}$ are $\varphi _R$ -numerically trivial, they are pull-backs of line bundles on Z. Therefore, $\mathcal {L}$ is also a pull-back of a line bundle on Z. Then, $\varphi _R$ satisfies (iii)’.

Case 4. When $\dim Y=2$ , taking the Stein factorization of $\pi $ , we may assume that $\pi \colon X\rightarrow Y$ is isomorphic over $Y-W$ . Then any line bundle on X is $\pi $ -free over $Y-W$ . We can show this case in the same way as in Case 3.

Proof. We note that the contraction morphism $\varphi $ preserves normality by (ii) of Lemma 3.3. We also note that $\rho (X/Y; W)-\rho (X'/Y; W)=1$ , because we have the following exact sequence

$$ \begin{align*}0\rightarrow A^1(X'/Y; W)\xrightarrow{D\mapsto\varphi^*D} A^1(X/Y; W)\xrightarrow{D\mapsto(D\cdot C)} \mathbb{Z}, \end{align*} $$

where C is a curve on X such that its numerical equivalence class $[C]\in \overline {\text {NE}}(X/Y; W)$ generates the extremal ray associated with $\varphi $ , by (iii)’ of Lemma 3.3.

In order to show that $(X', Y, W, \pi ', \Delta ')$ satisfies the condition ( $\bigstar $ ), we only have to show that $K_{X'}+\Delta '$ is $\mathbb {R}$ -Cartier because $(X, Y, W, \pi , \Delta )$ satisfies the condition ( $\bigstar $ ) and $\varphi \colon X\rightarrow X'$ is a bimeromorphic contraction morphism.

In Case (A), let E be a divisor on X such that its support $\operatorname {Supp}E$ corresponds to the exceptional locus of $\varphi $ . Let $E_i$ be an irreducible component of E. By assumption, each $E_i$ is $\mathbb {Q}$ -Cartier. Since the contraction morphism $\varphi $ satisfies the conditions in Lemma 3.3, the numerical equivalence class $[E_i]\in \overline {\text {NE}}(X/Y; W)$ is proportional to $[C]\in \overline {\text {NE}}(X/Y; W)$ . By assumption, C is also $\mathbb {Q}$ -Cartier. Since $C^2<0$ implies $C\cdot E_i<0$ , we have $\operatorname {Supp}E_i=\operatorname {Supp}C$ . Then $\operatorname {Supp}E=\operatorname {Supp}C$ and in particular the exceptional locus of $\varphi $ is irreducible. Pick a $\mathbb {Q}$ -Cartier $\mathbb {Q}$ -divisor $\Gamma $ on X. Since $\rho (X/Y; W)-\rho (X'/Y; W)=1$ and $E\cdot E<0$ , there is a rational number $\lambda $ such that $(\Gamma +\lambda E)\cdot E=0$ . Then $\Gamma +\lambda E$ is both $\mathbb {Q}$ -Cartier and $\varphi $ -numerically trivial. This implies that $\varphi _*(\Gamma +\lambda E)=\varphi _*\Gamma $ is $\mathbb {Q}$ -Cartier, by (iii)’ of Lemma 3.3. Since $K_X+\Delta $ is $\mathbb {R}$ -Cartier, we can write $K_X+\Delta =\sum _{i=1}^na_i(K_X+\Delta _i)$ on some open neighborhood V of $\operatorname {Supp}E$ on X, where every $\Delta _i$ is a boundary $\mathbb {Q}$ -divisor such that $K_X+\Delta _i$ is $\mathbb {Q}$ -Cartier and every $a_i$ is a positive real number such that $\sum a_i=1$ . Since $\operatorname {Supp}E$ corresponds to the exceptional locus of $\varphi $ , $\varphi (V)\subset X'$ is open and $V=\varphi ^{-1}(\varphi (V))$ . Substituting $\Gamma =K_X+\Delta _i$ , we see that $\varphi _*(K_X+\Delta _i)$ is $\mathbb {Q}$ -Cartier for every i. Then, $K_{X'}+\Delta '=\sum _{i=1}^na_i(K_{X'}+\varphi _*\Delta _i)$ is $\mathbb {R}$ -Cartier on $\varphi (V)$ . Since $\varphi $ is isomorphic on $X-\operatorname {Supp}E$ , $K_{X'}+\Delta '$ is $\mathbb {R}$ -Cartier on $X'$ . Substituting $\Gamma =K_X$ , $K_{X'}$ is $\mathbb {Q}$ -Cartier. Pick a prime divisor $D'$ on $X'$ which is mapped into W. Let D be the strict transform of $D'$ . Then $\pi (D)\subset W$ implies that D is $\mathbb {Q}$ -Cartier. Then, substituting $\Gamma =D$ , $\varphi _*D=D'$ is $\mathbb {Q}$ -Cartier. Therefore, $(X', Y, W, \pi ', \Delta ')$ satisfies the condition ( $\bigstar $ ) and $K_{X'}$ is $\mathbb {Q}$ -Cartier and every prime divisor on $X'$ which is mapped into W is $\mathbb {Q}$ -Cartier.

In Case (B), as above, the exceptional locus of $\varphi $ is irreducible. In order to prove that $X'$ is $\mathbb {Q}$ -factorial over W, pick a prime divisor $D'$ defined on an open neighborhood U of $(\pi ')^{-1}(W)$ . Let D be the strict transform of $D'$ . Then D is $\mathbb {Q}$ -Cartier over a neighborhood of W by the assumption of (B). Then as above, $\varphi _*D=D'$ is $\mathbb {Q}$ -Cartier over a neighborhood of W. Then $X'$ is $\mathbb {Q}$ -factorial over W. We can show that $K_{X'}+\Delta '$ is $\mathbb {R}$ -Cartier in the same way as above. Therefore, $(X', Y, W, \pi ', \Delta ')$ satisfies the condition ( $\bigstar $ ) and $X'$ is $\mathbb {Q}$ -factorial over W.

In Case (C), by Lemma 2.11, $(X', \Delta ')$ is numerically log canonical (for the definition of numerical log canonicity, see [Reference Kollár and Mori11, Notation 4.1]). This implies that $K_{X'}+\Delta '$ is $\mathbb {R}$ -Cartier. Therefore, $(X', Y, W, \pi ', \Delta ')$ satisfies the condition ( $\bigstar $ ) and $(X', \Delta ')$ is log canonical.

In Case (D), $X'$ also has only rational singularities by the vanishing theorem (cf. [Reference Fujino6, Theorem 3.3.1]), because the same argument as [Reference Fujino4, Proposition 3.7] is available, and we have

$$ \begin{align*}\mathcal{R}^j\varphi_*\mathcal{O}_X=0 \end{align*} $$

for all $j>0$ . In particular, any divisor on an open subset of X is $\mathbb {Q}$ -Cartier (cf. [Reference Ishii10, Theorem 7.3.2]). This implies that $K_{X'}+\Delta '$ is $\mathbb {R}$ -Cartier. Therefore, $(X', Y, W, \pi ', \Delta ')$ satisfies the condition ( $\bigstar $ ) and $X'$ also has only rational singularities.

Proof. We consider the following step-by-step procedure.

1. Step 1. We start with $(X, \Delta )$ .

2. Step 2. If $K_X+\Delta $ is nef over W, we stop.

3. Step 3. If $K_X+\Delta $ is not nef, then there is a rational $(K_X+\Delta )$ -negative extremal ray R of $\overline {\text {NE}}(X/Y; W)$ and we have the corresponding contraction morphism $\varphi _R\colon X\rightarrow X'$ over Y in Lemma 3.3. We note that $X'$ is normal, and that $\rho (X/Y; W)-\rho (X'/Y; W)=1$ as shown in the proof of Lemma 3.4.

4. Step 4. If $\varphi _R\colon X\rightarrow X'$ is not bimeromorphic, then it is a Mori fiber space and we stop.

5. Step 5. Otherwise, $\varphi _R\colon X\rightarrow X'$ is bimeromorphic. Let $\pi '\colon X'\rightarrow Y$ be the induced morphism and let $\Delta ':={\varphi _R}_*\Delta $ . By Lemma 3.4, $(X', Y, W, \pi ', \Delta ')$ also satisfies the condition ( $\bigstar $ ). Moreover, $K_{X'}$ is $\mathbb {Q}$ -Cartier and every prime divisor on $X'$ which is mapped into W is $\mathbb {Q}$ -Cartier in Case (A), $X'$ is $\mathbb {Q}$ -factorial over W in Case (B), $(X', \Delta ')$ is log canonical in Case (C), and $X'$ has only rational singularities in Case (D). Then, we can replace $(X, \Delta )$ with $(X', {\varphi _R}_*\Delta )$ and go back to the first procedure.

We may repeat this process. Since $\rho (X_i/Y; W)-\rho (X_{i+1}/Y; W)=1$ , we obtain a sequence of at most $\rho (X/Y;W)-1$ contractions

$$ \begin{align*}(X, \Delta)=(X_0, \Delta_0)\stackrel{\varphi_0}{\rightarrow}(X_1, \Delta_1)\stackrel{\varphi_1}{\rightarrow}\cdots\stackrel{\varphi_{k-1}}{\rightarrow}(X_k, \Delta_k)=(X^*, \Delta^*) \end{align*} $$

over Y such that either $K_{X^*}+\Delta ^*$ is nef over W or $(X^*, \Delta ^*)$ is a Mori fiber space.

By Lemma 2.11, we can write

$$ \begin{align*}K_X+\Delta=\Phi^*(K_{X^*}+\Delta^*)+E, \end{align*} $$

where $\Phi \colon X\rightarrow X^*$ is the composition of the $\varphi _i$ ’s, and E is an effective $\Phi $ -exceptional divisor on X whose support corresponds to the exceptional locus of $\Phi $ . If $K_{X^*}+\Delta ^*$ is nef over W, then it is nef over Y by Lemma 2.6. Then $K_X+\Delta $ is $\pi $ -pseudo-effective. Conversely, if $K_X+\Delta $ is $\pi $ -pseudo-effective, then so is $K_{X^*}+\Delta ^*$ , because $\Phi \colon X\rightarrow X^*$ is bimeromorphic. In order to show that $(X^*, \Delta ^*)$ is a minimal model, we suppose that $\varphi _k\colon (X_k, \Delta _k)=(X^*, \Delta ^*)\rightarrow Z$ is a Mori fiber space over Y. In this case, $-(K_{X_k}+\Delta _k)$ is $\varphi _k$ -ample and $\operatorname {dim}Z<2$ . For every point $P\in X_k$ , there exists a curve C on $X_k$ passing through P which is contracted by $\varphi _k$ , which satisfies that $(K_{X_k}+\Delta _k)\cdot C<0$ . This contradicts the fact that $(K_{X_k}+\Delta _k)\cdot C\geq 0$ for a very general curve C on $X_k$ . Therefore, $(X^*, \Delta ^*)$ is a minimal model when $K_X+\Delta $ is $\pi $ -pseudo-effective and it is a Mori fiber space when $K_X+\Delta $ is not $\pi $ -pseudo-effective.

Proof of Lemma 4.1.

The corresponding statement for algebraic surfaces holds by [Reference Fujino4, Lemma 4.4]. Using the technique from [Reference Ramis, Ruget and Verdier13], the analogous statement in the analytic setting also holds.

Proof of Theorem 4.2.

By Lemma 2.6, we may assume that $K_X+\Delta $ is nef over Y.

Step 1. We may assume that $\pi $ has connected fibers.

Proof of Step 1.

By Remark 2.2, we can take the Stein factorization $X\xrightarrow {\pi '}Y'\xrightarrow {\nu }Y$ of $\pi \colon X\rightarrow Y$ . Since $\nu $ is finite, $K_X+\Delta $ is $\pi '$ -nef over a neighborhood of $\nu ^{-1}W$ and

$$ \begin{align*}\nu^*\nu_*\pi^{\prime}_*\mathcal{O}_X(m(K_X+\Delta))\rightarrow\pi^{\prime}_*\mathcal{O}_X(m(K_X+\Delta)) \end{align*} $$

is surjective for any positive divisible integer m. Since $\pi ^{\prime *}$ is a right exact functor,

$$ \begin{align*} \pi^*\pi_*\mathcal{O}_X(m(K_X+\Delta))&=\pi^{\prime*}\nu^*\nu_*\pi^{\prime}_*{\mathcal{O}_X(m(K_X+\Delta))}\\ &\rightarrow\pi^{\prime*}\pi^{\prime}_*{\mathcal{O}_X(m(K_X+\Delta))} \end{align*} $$

is also surjective. Then if $K_X+\Delta $ is $\pi '$ -semi-ample over a neighborhood of $\nu ^{-1}W$ , it is $\pi $ -semi-ample over a neighborhood of W. Therefore, we can replace $\pi $ by $\pi '$ and so we may assume that $\pi $ has connected fibers.

Step 2. Taking a relatively compact open neighborhood of W, we may assume that $\operatorname {Supp}\Delta $ has only finitely many irreducible components. Let $\llcorner \Delta \lrcorner =\sum _iC_i$ be the irreducible decomposition. Shrinking Y around W, we may assume that if $C_i$ is mapped into a point in Y, then it is mapped into W. We put

$$ \begin{align*} A&:=\sum_{C_i\cdot(K_X+\Delta)=0\text{ and }\pi(C_i)\text{ is a point}}C_i, \text{ and}\\ B&:=\sum_{C_i\cdot(K_X+\Delta)>0\text{ or }\pi(C_i)\text{ is a curve}}C_i. \end{align*} $$

Then $\llcorner \Delta \lrcorner =A+B$ . If $C_i$ is a component of A, then we may assume that $C_i^2<0$ .

From Step 3 to Step 5, we will check that there exists a positive integer m such that the subset

$$ \begin{align*}T:=\operatorname{Supp}\operatorname{Coker}(\pi^*\pi_*\mathcal{O}_X(m(K_X+\Delta))\rightarrow\mathcal{O}_X(m(K_X+\Delta)))\subset X \end{align*} $$

contains no non-klt centers of $(X, \Delta )$ , after shrinking Y around W suitably.

Step 3. Let Q be a zero-dimensional non-klt center of $(X, \Delta )$ such that $Q\notin \operatorname {Supp}A$ . Then $Q \notin T$ .

Proof of Step 3.

Let $f\colon \widetilde {X}\rightarrow X$ be a resolution such that $K_{\widetilde {X}}+\Delta _{\widetilde {X}}=f^*(K_X+\Delta )$ . We may assume that

1. (1) $f^{-1}(A)$ has simple normal crossing support,

2. (2) $\operatorname {Supp} f_*^{-1} \Delta \cup \operatorname {Exc}(f)$ is a simple normal crossing divisor on $\widetilde {X}$ , and

3. (3) $f^{-1}(Q)$ has simple normal crossing support.

Let $W_1$ be the union of the irreducible components of $\Delta _{\widetilde {X}}^{=1}$ which are mapped into $A \cup Q$ by f. We put $\Delta _{\widetilde {X}}^{=1}=W_1+W_2$ . Then

$$ \begin{align*}-W_1-\llcorner\Delta_{\widetilde{X}}^{>1}\lrcorner+\ulcorner-(\Delta_{\widetilde{X}}^{<1})\urcorner-(K_{\widetilde{X}}+\{\Delta_{\widetilde{X}}\}+W_2) \sim_{\mathbb{Q}}-f^*(K_X+\Delta). \end{align*} $$

We put

$$ \begin{align*}\mathcal{J}_1=f_* \mathcal{O}_{\widetilde{X}}(-W_1-\llcorner\Delta_{\widetilde{X}}^{>1}\lrcorner+\ulcorner-(\Delta_{\widetilde{X}}^{<1})\urcorner)\subset \mathcal{O}_X. \end{align*} $$

Then we can easily check that

$$ \begin{align*}0 \rightarrow \mathcal{J}_1 \rightarrow \mathcal{O}_X(-A) \rightarrow \delta \rightarrow 0 \end{align*} $$

is exact, where $\delta $ is a skyscraper sheaf. We will check that the conditions in Lemma 2.14 hold. In other words, we will check that

• • $\mathcal {L}:=\mathcal {O}_{\widetilde {X}}(m(K_{\widetilde {X}}+\Delta _{\widetilde {X}})-W_1-\llcorner \Delta _{\widetilde {X}}^{>1}\lrcorner +\ulcorner -(\Delta _{\widetilde {X}}^{<1})\urcorner )$ is a line bundle on $\widetilde {X}$ for some divisible positive integer m,

• • $(\widetilde {X}, \Delta _{\widetilde {X}}-W_1-\llcorner \Delta _{\widetilde {X}}^{>1}\lrcorner +\ulcorner -(\Delta _{\widetilde {X}}^{<1})\urcorner )$ is a pair of smooth complex analytic variety and a boundary $\mathbb {R}$ -divisor on it which has a simple normal crossing support,

• • $f\colon \widetilde {X}\rightarrow X$ and $\pi \colon X\rightarrow Y$ are projective,

• • $\mathcal {L}-\mathcal {O}_{\widetilde {X}}(K_{\widetilde {X}}+\Delta _{\widetilde {X}}-W_1-\llcorner \Delta _{\widetilde {X}}^{>1}\lrcorner +\ulcorner -(\Delta _{\widetilde {X}}^{<1})\urcorner ) =\mathcal {O}_{\widetilde {X}}((m-1)(K_{\widetilde {X}}+\Delta _{\widetilde {X}}))=f^*\mathcal {O}_X((m-1)(K_X+\Delta ))$ , and

• • $\mathcal {O}_X((m-1)(K_X+\Delta ))$ is nef and log big over Y with respect to $f\colon (\widetilde {X}, \Delta _{\widetilde {X}}-W_1-\llcorner \Delta _{\widetilde {X}}^{>1}\lrcorner +\ulcorner -(\Delta _{\widetilde {X}}^{<1})\urcorner ) \rightarrow X$ .

We only have to show the last condition. We note that an $\mathbb {R}$ -line bundle $\mathcal {H}$ on X is log big over Y with respect to $f\colon (\widetilde {X}, \Gamma :=\Delta _{\widetilde {X}}-W_1-\llcorner \Delta _{\widetilde {X}}^{>1}\lrcorner +\ulcorner -(\Delta _{\widetilde {X}}^{<1})\urcorner )\rightarrow X$ if $\mathcal {H}$ is $\pi $ -big and $\mathcal {H}|_{f(S)}$ can be written as a finite $\mathbb {R}_{>0}$ -linear combination of $\pi $ -big line bundles on $f(S)$ for every lc center S of $(\widetilde {X}, \Gamma )$ . By assumption, $K_X+\Delta $ is $\pi $ -big. An lc center S of $(\widetilde {X}, \Gamma )$ is zero-dimensional or a component of $W_2$ . If $f(S)$ is a point, $(K_X+\Delta )|_{f(S)}$ is clearly $\pi $ -big. If $f(S)$ is a curve and it is not contracted by $\pi $ , then $\pi |_{f(S)}$ is finite. This implies that $(K_X+\Delta )|_{f(S)}$ is $\pi $ -big. If $f(S)$ is a curve and it is contracted by $\pi $ , then $(K_X+\Delta )\cdot f(S)>0$ by the definitions of B and $\Gamma $ . This implies that $(K_X+\Delta )|_{f(S)}$ is $\pi $ -big. Therefore, $\mathcal {O}_X((m-1)(K_X+\Delta ))$ is log big over Y with respect to $f\colon (\widetilde {X}, \Gamma )\rightarrow X$ . Then we can apply Lemma 2.14 and we have

$$ \begin{align*}\mathcal{R}^p\pi_*(\mathcal{O}_X(m(K_X+\Delta)) \otimes \mathcal{J}_1)=0 \end{align*} $$

for every $p>0$ by Lemma 2.14, where m is some divisible positive integer. This implies that $\pi _*\mathcal {O}_X(m(K_X+\Delta )) \rightarrow \pi _*(\mathcal {O}_X(m(K_X+\Delta ))\otimes \mathcal {O}_X/\mathcal {J}_1) $ is surjective. We consider the commutative diagram

We note that $\pi ^*\pi _*\mathcal {O}_X(m(K_X+\Delta )) \rightarrow \pi ^*\pi _*(\mathcal {O}_X(m(K_X+\Delta ))\otimes \mathcal {O}_X/\mathcal {J}_1) $ is surjective because $\pi ^*$ is a right exact functor. By the definition of $\mathcal {J}_1$ , the point Q is isolated in $\operatorname {Supp}(\mathcal {O}_X/\mathcal {J}_1)$ . Then $\pi ^*\pi _* (\mathcal {O}_X(m(K_X+\Delta ))\otimes \mathcal {O}_X/\mathcal {J}_1) \rightarrow (\mathcal {O}_X(m(K_X+\Delta ))\otimes \mathcal {O}_X/\mathcal {J}_1)$ is surjective at Q. Therefore, $Q\notin T$ .

Step 4. $T\cap A= \emptyset $ .

Proof of Step 4.

Let $A=\sum _jA_j$ be the connected decomposition. Now, $\pi (A_j)=P$ for each j. Let $C_i$ be an irreducible component of $A_j$ . Then $[C_i]\in N_1(X/Y; W)$ generates an extremal ray of $\overline {\text {NE}}(X/Y; W)$ because $C_i^2<0$ . By assumption, $C_i$ and $\Delta $ are $\mathbb {Q}$ -Cartier. If $A_j$ is reducible, or it is not isolated in $\operatorname {Supp}\Delta $ , then $C_i\cdot (\Delta -C_i)>0$ . So,

$$ \begin{align*}C_i\cdot(K_X+C_i)<C_i\cdot(K_X+\Delta)=0. \end{align*} $$

We note that $\rho (X/Y; W)<\infty $ (cf. Nakayama’s finiteness, [Reference Fujino5, Theorem 4.7]). By Lemma 3.1, after shrinking Y around W, there exists a contraction morphism $\varphi _{C_i}\colon X\rightarrow X'$ over Y. Let $\pi '\colon X'\rightarrow Y$ be the induced morphism and $\Delta '={\varphi _{C_i}}_*\Delta $ . Since $(X, Y, W, \pi , \Delta )$ satisfies the condition ( $\bigstar $ ), so does $(X', Y, W, \pi ', \Delta ')$ by Lemma 3.4. Moreover, $K_{X'}$ is $\mathbb {Q}$ -Cartier and every prime divisor on $X'$ which is mapped into W is $\mathbb {Q}$ -Cartier by Lemma 3.4.

We will check that we can replace $(X, Y, W, \pi , \Delta )$ by $(X', Y, W, \pi ', \Delta ')$ in order to prove that $T\cap A= \emptyset $ . Since $K_X+\Delta $ is $\varphi _{C_i}$ -trivial, we have $K_X+\Delta =\varphi _{C_i}^*(K_{X'}+\Delta ')$ . Then $K_{X'}+\Delta '$ is nef over Y. Let

$$ \begin{align*}T'=\operatorname{Supp}\operatorname{Coker}(\pi^{\prime*}\pi^{\prime}_*\mathcal{O}_{X'}(m(K_{X'}+\Delta'))\rightarrow\mathcal{O}_{X'}(m(K_{X'}+\Delta')))\subset X'. \end{align*} $$

Then, on $X'-T'$ , we have the exact sequence

$$ \begin{align*}\pi^{\prime*}\pi^{\prime}_*\mathcal{O}_{X'}(m(K_{X'}+\Delta'))\rightarrow\mathcal{O}_{X'}(m(K_{X'}+\Delta'))\rightarrow0. \end{align*} $$

Taking pull-back by $\varphi _{C_i}\colon X\rightarrow X'$ , on $\varphi _{C_i}^{-1}(X'-T')=X-\varphi _{C_i}^{-1}T'$ , we have the exact sequence

$$ \begin{align*}\varphi_{C_i}^*\pi^{\prime*}\pi^{\prime}_*\mathcal{O}_{X'}(m(K_{X'}+\Delta'))\rightarrow\varphi_{C_i}^*\mathcal{O}_{X'}(m(K_{X'}+\Delta'))\rightarrow0. \end{align*} $$

Since ${\varphi _{C_i}}_*\varphi _{C_i}^*\mathcal {O}_{X'}(m(K_{X'}+\Delta '))=\mathcal {O}_{X'}(m(K_{X'}+\Delta '))$ , on $X-\varphi _{C_i}^{-1}T'$ , we have the exact sequence

$$ \begin{align*}\varphi_{C_i}^*\pi^{\prime*}\pi^{\prime}_*{\varphi_{C_i}}_*\varphi_{C_i}^*\mathcal{O}_{X'}(m(K_{X'}+\Delta'))\rightarrow\varphi_{C_i}^*\mathcal{O}_{X'}(m(K_{X'}+\Delta'))\rightarrow0. \end{align*} $$

Then, on $X-\varphi _{C_i}^{-1}T'$ , we have the exact sequence

$$ \begin{align*}\pi^*\pi_*\mathcal{O}_X(m(K_X+\Delta))\rightarrow\mathcal{O}_X(m(K_X+\Delta))\rightarrow0. \end{align*} $$

Then $T\subset \varphi _{C_i}^{-1}T'$ . This implies that we can replace $(X, Y, W, \pi , \Delta )$ by $(X', Y, W, \pi ', \Delta ')$ in order to prove that $T\cap A= \emptyset $ .

We put

$$ \begin{align*} A^\sharp&:=\sum_{A_j \cap (\operatorname{Supp}\Delta-A_j)=\emptyset}A_j, \text{ and}\\ A^\flat&:=\sum_{A_j \cap (\operatorname{Supp}\Delta-A_j)\neq\emptyset}A_j. \end{align*} $$

Therefore, by replacing X with its contraction repeatedly as far as possible, we may assume that $A_j$ is an irreducible and isolated curve in $\operatorname {Supp}\Delta $ when $A_j\subset A^\sharp $ . Moreover, by replacing X with its contraction, $A_j$ is contracted to a point Q in $\operatorname {Supp}\Delta $ when $A_j\subset A^\flat $ . This point Q is a zero-dimensional non-klt center of $(X, \Delta )$ such that $Q\notin A$ and it satisfies $Q \notin T$ by Step 3. Then we could assume that $A=A^\sharp $ and each $A_j$ is irreducible from the beginning of the proof of Step 4.

We note that since $A_j$ is contracted by a projective morphism, we can treat $A_j$ as an algebraic curve by Serre’s GAGA principle. Pick a positive integer m such that $m(K_X+\Delta )$ is Cartier. If $A_j$ is $\mathbb {P}^1$ , then $\mathcal {O}_{A_j}(m(K_X+\Delta )) \cong \mathcal {O}_{A_j}$ because $A_j \cdot \left (K_X+\Delta \right )=0$ and a line bundle on $\mathbb {P}^1$ is trivial if and only if it has degree zero. If $A_j \ncong \mathbb {P}^1$ , then we obtain $h^1(A_j, \mathcal {O}_{A_j})=p_a(A_j) \neq 0$ . Since $A_j$ is projective, it has the dualizing sheaf $\omega _{A_j}$ . By Serre duality, we obtain $H^0(A_j, \omega _{A_j}) \neq 0$ . We note that

$$ \begin{align*}0 \rightarrow \mathcal{T} \rightarrow \mathcal{O}_X\left(K_X+A_j\right) \otimes \mathcal{O}_{A_j} \rightarrow \omega_{A_j} \rightarrow 0 \end{align*} $$

is exact, where the support of $\mathcal {T}$ is of dimension zero by Lemma 4.1. Now, we have the exact sequence

$$ \begin{align*}H^0(A_j, \mathcal{O}_X(K_X+A_j) \otimes \mathcal{O}_{A_j})\rightarrow H^0(A_j, \omega_{A_j})\rightarrow H^1(A_j, \mathcal{T})=0. \end{align*} $$

Since $H^0(A_j, \omega _{A_j})\neq 0$ , $H^0(A_j, \mathcal {O}_X(K_X+A_j) \otimes \mathcal {O}_{A_j})\neq 0$ . Furthermore, as $A_j$ is isolated,

$$ \begin{align*}H^0(A_j, \mathcal{O}_X(K_X+\Delta) \otimes \mathcal{O}_{A_j})=H^0(A_j, \mathcal{O}_X(K_X+A_j) \otimes \mathcal{O}_{A_j}). \end{align*} $$

Then we see that $\mathcal {O}_{A_j}(m(K_X+\Delta ))\cong \mathcal {O}_{A_j}$ because $A_j \cdot \left (K_X+\Delta \right )=0$ . Therefore, we obtain $\mathcal {O}_{A}(m(K_X+\Delta )) \cong \mathcal {O}_{A}$ .

Let $f\colon \widetilde {X}\rightarrow X$ be a resolution such that $K_{\widetilde {X}}+\Delta _{\widetilde {X}}=f^*(K_X+\Delta )$ . We may assume that

1. (1) $f^{-1}(A)$ has simple normal crossing support, and

2. (2) $\operatorname {Supp} f_*^{-1} \Delta \cup \operatorname {Exc}(f)$ is a simple normal crossing divisor on Y.

Let $W_3$ be the union of the irreducible components of $\Delta _{\widetilde {X}}^{=1}$ which are mapped into A by f. We write $\Delta _{\widetilde {X}}^{=1}=W_3+W_4$ . Then

$$ \begin{align*}-W_3-\llcorner\Delta_{\widetilde{X}}^{>1}\lrcorner+\ulcorner-(\Delta_{\widetilde{X}}^{<1})\urcorner-(K_{\widetilde{X}}+\{\Delta_{\widetilde{X}}\}+W_4) \sim_{\mathbb{Q}}-f^*(K_X+\Delta). \end{align*} $$

We put

$$ \begin{align*}\mathcal{J}_2=f_*\mathcal{O}_{\widetilde{X}}(-W_3-\llcorner\Delta_{\widetilde{X}}^{>1}\lrcorner+\ulcorner-(\Delta_{\widetilde{X}}^{<1})\urcorner) \subset \mathcal{O}_X. \end{align*} $$

Then we can easily check that

$$ \begin{align*}0 \rightarrow \mathcal{J}_2 \rightarrow \mathcal{O}_X(-A) \rightarrow \delta \rightarrow 0 \end{align*} $$

is exact, where $\delta $ is a skyscraper sheaf. As in the proof of Step 3, we have

$$ \begin{align*}\mathcal{R}^p\pi_*(\mathcal{O}_X(m(K_X+\Delta)) \otimes \mathcal{J}_2)=0 \end{align*} $$

for every $p>0$ , where m is some divisible positive integer. By the above exact sequence, we obtain

$$ \begin{align*}\mathcal{R}^p\pi_*(\mathcal{O}_X(m(K_X+\Delta)) \otimes \mathcal{O}_X(-A))=0 \end{align*} $$

for every $p>0$ . Then $\pi _*\mathcal {O}_X(m(K_X+\Delta )) \rightarrow \pi _*\mathcal {O}_A(m(K_X+\Delta )) $ is surjective. We consider the commutative diagram

We note that $\pi ^*\pi _*\mathcal {O}_X(m(K_X+\Delta )) \rightarrow \pi ^*\pi _*\mathcal {O}_A(m(K_X+\Delta )) $ is surjective because $\pi ^*$ is a right exact functor, and $\pi ^*\pi _*\mathcal {O}_A(m(K_X+\Delta )) \rightarrow \mathcal {O}_A(m(K_X+\Delta ))$ is surjective because $\mathcal {O}_A(m(K_X+\Delta )) \simeq \mathcal {O}_A$ . Therefore, we have $T\cap A=\emptyset $

Step 5. Let $E_i$ be an irreducible component of B. Then $E_i \not \subset T$ .

Proof of Step 5.

We may assume that $E_i \cap A=\emptyset $ by Step 4. When $\pi (E_i)$ is a point, $E_i\cdot (K_X+\Delta )>0$ and so $\mathcal {O}_{E_i}(m(K_X+\Delta ))$ is ample. Then $\pi ^*\pi _*\mathcal {O}_{E_i}(m(K_X+\Delta ))\rightarrow \mathcal {O}_{E_i}(m(K_X+\Delta ))$ is surjective. When $\pi (E_i)$ is a curve, $\pi |_{E_i}$ is a finite morphism. Then $\pi ^*\pi _*\mathcal {O}_{E_i}(m(K_X+\Delta ))\rightarrow \mathcal {O}_{E_i}(m(K_X+\Delta ))$ is surjective.

Let $f\colon \widetilde {X}\rightarrow X$ be the resolution as in the proof of Step 4. We can further assume that (4) $f^{-1}\left (E_i\right )$ has simple normal crossing support. Let $W_5$ be the union of the irreducible components of $\Delta _{\widetilde {X}}^{=1}$ which are mapped into $A \amalg E_i$ by f. We put $\Delta _{\widetilde {X}}^{=1}=W_5+W_6$ . Then

$$ \begin{align*}-W_5-\llcorner\Delta_{\widetilde{X}}^{>1}\lrcorner+\ulcorner-(\Delta_{\widetilde{X}}^{<1})\urcorner-(K_{\widetilde{X}}+\{\Delta_{\widetilde{X}}\}+W_6) \sim_{\mathbb{Q}}-f^*(K_X+\Delta). \end{align*} $$

We put

$$ \begin{align*}\mathcal{J}_3=f_* \mathcal{O}_{\widetilde{X}}(-W_5-\llcorner\Delta_{\widetilde{X}}^{>1}\lrcorner+\ulcorner-(\Delta_{\widetilde{X}}^{<1})\urcorner) \subset \mathcal{O}_X. \end{align*} $$

Then, as in the proof of Step 3, we have

$$ \begin{align*}R^p\pi_*(\mathcal{O}_X(m(K_X+\Delta)) \otimes \mathcal{J}_3)=0 \end{align*} $$

for every $p>0$ by Lemma 2.14, where m is some divisible positive integer. We note that there exists a short exact sequence

$$ \begin{align*}0 \rightarrow \mathcal{J}_3 \rightarrow \mathcal{O}_X\left(-A-E_i\right) \rightarrow \delta^{\prime} \rightarrow 0, \end{align*} $$

where $\delta ^{\prime }$ is a skyscraper sheaf on X. Thus,

$$ \begin{align*}R^p\pi_*(\mathcal{O}_X(m(K_X+\Delta)) \otimes \mathcal{O}_X(-A-E_i))=0 \end{align*} $$

for every $p>0$ . This implies that

$$ \begin{align*}\pi_*\mathcal{O}_X(m(K_X+\Delta)) \rightarrow \pi_*\mathcal{O}_{E_i}(m(K_X+\Delta)) \end{align*} $$

is surjective since $\operatorname {Supp} E_i \cap \operatorname {Supp} A=\emptyset $ . Therefore, $E_i \not \subset T$ for every irreducible component $E_i$ of B.

We have checked that T contains no non-klt centers of $(X, \Delta )$ , after shrinking Y around W suitably. In fact, it contains no non-klt centers which are components of $\llcorner \Delta \lrcorner $ by Steps 4 and 5, and it contains no non-klt centers of dimension zero by Steps 3 and 4.

Finally, we will prove that $K_X+\Delta $ is semi-ample over a neighborhood of W.

Step 6. If $T\cap \pi ^{-1}(P)=\emptyset $ , then there is nothing to prove because $T=\emptyset $ over a neighborhood of W. So, we assume that $T\cap \pi ^{-1}(P)\neq \emptyset $ .

Since $W=\{P\}$ , we only have to prove the result in the case where Y is a Stein space. In this case, we have

$$ \begin{align*} T&:=\operatorname{Supp}\operatorname{Coker}(\pi^*\pi_*\mathcal{O}_X(m(K_X+\Delta))\rightarrow\mathcal{O}_X(m(K_X+\Delta)))\\ &=\bigcap_{m(K_X+\Delta)\sim E\geq0}\operatorname{Supp}E \end{align*} $$

set-theoretically by [Reference Enokizono and Hashizume3, Lemma 2.7]. By the argument from Step 3 to Step 5, after shrinking Y around W suitably, T is an analytic subset of X and contains no non-klt centers of $(X,\Delta )$ . We take general members $\Xi _1, \Xi _2, \Xi _3 \in \{E\geq 0|m(K_X+\Delta )\sim E\}$ and put $\Theta =\Xi _1+\Xi _2+\Xi _3$ . Then $\Theta $ contains no non-klt centers of $(X, \Delta )$ and $K_X+\Delta +\Theta $ is not log canonical at $\operatorname {Supp}\Xi _1\cap \operatorname {Supp}\Xi _2\cap \operatorname {Supp}\Xi _3$ by [Reference Fujino6, Lemma 2.1.3]. Since $T\cap \pi ^{-1}(P)\neq \emptyset $ , $K_X+\Delta +\Theta $ is not log canonical at $\pi ^{-1}(P)$ . We put

$$ \begin{align*}c=\max \{t\in\mathbb{R}|K_X+\Delta+t\Theta \text { is log canonical at } \pi^{-1}(P)\backslash\operatorname{Nlc}(X, \Delta)\}. \end{align*} $$

Then we can easily check that $c\in \mathbb {Q}$ and $0<c<1$ . In this case,

$$ \begin{align*}K_X+\Delta+c\Theta \sim_{\mathbb{Q}}(1+c m)\left(K_X+\Delta\right) \end{align*} $$

and there exists an lc center $\mathcal {C}$ of $(X, \Delta +c \Theta )$ which is contained in $\cap _{m(K_X+\Delta )\sim E\geq 0}\operatorname {Supp}E$ , and satisfies $\mathcal {C}\cap \pi ^{-1}(P)\neq \emptyset $ . We take positive integers l and n such that

$$ \begin{align*}l(K_X+\Delta+c\Theta) \sim n m(K_X+\Delta) \end{align*} $$

Replace $m(K_X+\Delta )$ with $l(K_X+\Delta +c\Theta )$ and apply the previous arguments. Then, after shrinking Y around W, $\cap _{kl(K_X+\Delta +c\Theta )\sim E\geq 0}\operatorname {Supp}E$ contains no non-klt centers of $(X, \Delta +c\Theta )$ for some positive integer k. Since $\mathcal {C}\cap \pi ^{-1}(P)\neq \emptyset $ , $\mathcal {C}$ is still a non-empty subset of X. We obtain $\mathcal {C}\not \subset \cap _{kl(K_X+\Delta +c\Theta )\sim E\geq 0}\operatorname {Supp}E$ . Therefore,

$$ \begin{align*}\bigcap_{knm(K_X+\Delta)\sim E\geq0}\operatorname{Supp}E \subsetneq \bigcap_{m(K_X+\Delta)\sim E\geq0}\operatorname{Supp}E. \end{align*} $$

This is because there is an lc center $\mathcal {C}$ of $(X, \Delta +c \Theta )$ such that $\mathcal {C} \subset \cap _{m(K_X+\Delta )\sim E\geq 0}\operatorname {Supp}E$ , and $l(K_X+\Delta +c\Theta )\sim nm(K_X+\Delta )$ . By noetherian induction, we deduce that $K_X+\Delta $ is semi-ample over a neighborhood of W.

The proof of Theorem 4.2 is finished.

Proof. In Case (C), we replace X with its minimal resolution. In Case (D), any divisor on an open subset of X is $\mathbb {Q}$ -Cartier (cf. [Reference Ishii10, Theorem 7.3.2]). So, in these two cases, we may always assume that X is $\mathbb {Q}$ -factorial over W. So, we only consider Case (A) and Case (B). In Case (B), every prime divisor on X which is mapped into W is $\mathbb {Q}$ -Cartier and, shrinking Y around W, $\Delta $ is $\mathbb {Q}$ -Cartier. By assumption of ( $\bigstar $ ), $K_X$ is also $\mathbb {Q}$ -Cartier. So, we only consider Case (A). In this case, for any point $P\in W$ , every prime divisor on X which is included in $\pi ^{-1}(P)$ is $\mathbb {Q}$ -Cartier.

As in the proof of Theorem 4.2, we may assume that $K_X+\Delta $ is nef over Y. When $\dim Y=0$ , the statement is obvious by Serre’s GAGA principle and [Reference Fujino4, Theorem 7.2]. Then we may also assume that $\dim Y>0$ . If $\dim Y$ =2, then $K_X+\Delta $ is always $\pi $ -big. In order to prove that $K_X+\Delta $ is semi-ample over a neighborhood of W, we can replace W with P and Y with a Stein open neighborhood of P. Then by Theorem 4.2, $K_X+\Delta $ is semi-ample over a neighborhood of W. If $\operatorname {dim}Y$ =1 and $K_X+\Delta $ is $\pi $ -big, by Theorem 4.2, $K_X+\Delta $ is semi-ample over a neighborhood of W. We may assume that $\dim Y$ =1 and $K_X+\Delta $ is not $\pi $ -big. Taking the Stein factorization of $\pi \colon X\rightarrow Y$ , we may assume that Y is smooth and that $\pi $ has connected fibers.

Let $y \in Y$ be any point, $X_y$ the fiber of $\pi \colon X\rightarrow Y$ over y. Sard’s theorem implies that $X_y$ is smooth for an analytically sufficiently general fiber $X_y$ (cf. [Reference Ueno16, Corollary 1.8]). Since $X_y$ is connected for each point $y \in Y$ , $X_y$ is a smooth irreducible curve for an analytically sufficiently general fiber $X_y$ . Replacing Y with a relatively compact open neighborhood of W, we may assume that $\Delta $ has only finitely many irreducible components. Then $X_y$ and $\Delta $ have no common components for an analytically sufficiently general fiber $X_y$ .

Then $(K_X+\Delta )|_{X_y}=K_{X_y}+\Delta |_{X_y}$ . Let $\Delta _{X_y}=\Delta |_{X_y}$ . Then $\Delta _{X_y}\geq 0$ by assumption. Since $X_y$ is a smooth irreducible curve,

$$ \begin{align*} &h^0(X_y, \mathcal{O}_X(m(K_X+\Delta))|_{X_y})-h^1(X_y, \mathcal{O}_X(m(K_X+\Delta))|_{X_y})\\ &\quad =h^0(X_y, \mathcal{O}_{X_y})-h^1(X_y, \mathcal{O}_{X_y})\\ &\quad =1-h^1(X_y, \mathcal{O}_{X_y}), \end{align*} $$

by $\operatorname {deg}(K_X+\Delta )|_{X_y}=0$ . When $h^1(X_y, \mathcal {O}_{X_y})=0$ , we have

$$ \begin{align*}H^0(X_y, \mathcal{O}_X(m(K_X+\Delta))|_{X_y})\neq0. \end{align*} $$

We may assume that $h^1(X_y, \mathcal {O}_{X_y})\geq 1$ . By Serre duality,

$$ \begin{align*}1\leq h^1(X_y, \mathcal{O}_{X_y})=h^0(X_y, \mathcal{O}_{X_y}(K_{X_y})). \end{align*} $$

Since $\Delta _{X_y}\geq 0$ , there is an effective $\mathbb {Q}$ -divisor $D\neq 0$ which is $\mathbb {Q}$ -linearly equivalent to $K_{X_y}+\Delta _{X_y}$ . Then, for a divisible positive integer m, we have

$$ \begin{align*}1\leq h^0(X_y, \mathcal{O}_{X_y}(m(K_{X_y}+\Delta_{X_y})))=h^0(X_y, \mathcal{O}_X(m(K_X+\Delta))|_{X_y}). \end{align*} $$

This implies that $\pi _*\mathcal {O}_X(m(K_X+\Delta ))\neq 0$ . Since Y is Stein, we have

$$ \begin{align*}H^0(Y, \pi_*\mathcal{O}_X(m(K_X+\Delta)))=H^0(X, \mathcal{O}_X(m(K_X+\Delta)))\neq0. \end{align*} $$

Taking the minimal resolution of X, we may assume that X is smooth. We note that $\Delta $ is effective, but it is not necessarily boundary, after replacing X by its minimal resolution.

Since $H^0(X, \mathcal {O}_X(m(K_X+\Delta )))\neq 0$ , there is an effective Cartier divisor D on X such that

$$ \begin{align*}\mathcal{O}_X(m(K_X+\Delta))\cong\mathcal{O}_X(D). \end{align*} $$

It suffices to prove that D is semi-ample over Y. Since we have already replaced Y with a relatively compact open neighborhood of W, D has only finitely many irreducible components. We can write $D=D_{\text {horiz}}+D_{\text {vert}}$ , where $D_{\text {horiz}}$ is the horizontal part of D and $D_{\text {vert}}$ is the vertical part of D.

Assume that $D_{\text {horiz}}>0$ . Then $D_{\text {horiz}}|_{X_y}$ is ample for any irreducible fiber $X_y$ of $\pi \colon X\rightarrow Y$ . This implies that $(K_X+\Delta )|_{X_y}$ is ample for general fibers $X_y$ . In this case, $K_X+\Delta $ is $\pi $ -big. This is a contradiction.

So, we may assume that $D_{\text {horiz}}=0$ . We have

$$ \begin{align*}0\leq(K_X+\Delta)\cdot D_{\text{vert}}=D_{\text{vert}}^2\leq0, \end{align*} $$

by Lemma 2.12. Since $D_{\text {vert}}^2=0$ , by Lemma 2.12, $D_{\text {vert}}$ is proportional to a pull-back of an effective $\mathbb {Q}$ -divisor on Y. In this case, $D_{\text {vert}}=D$ is $\pi $ -semi-ample.

Proof. As in the proof of Corollary 4.4, it suffices to consider only Case (A).

Step 1. Taking a relatively compact open neighborhood of W, we may assume that $\operatorname {Supp}\Delta $ has only finitely many irreducible components. We put $F=\operatorname {Supp}\Delta $ and consider the real vector space $V=\bigoplus _k\mathbb {R}F_k$ , where $F=\sum _kF_k$ is the irreducible decomposition. We put $\mathcal {P}=\{D\in V|D\text { is boundary and }K_X+D\text { is }\mathbb {R}\text {-Cartier at }\pi ^{-1}W\}$ . Since $\pi ^{-1}W$ is a compact subset, $\mathcal {P}$ is a rational polytope. Let $\{R_\lambda \}_{\lambda \in \Lambda }$ be the set of all extremal rays of $\overline {\text {NE}}(X/Y; W)$ spanned by curves. We put $\mathcal {N}=\{D\in \mathcal {P}|(K_X+D)\cdot R_\lambda \geq 0\text { for every }\lambda \in \Lambda \}.$ We note that every $(K_X+\Delta )$ -negative extremal ray is spanned by a rational curve C with $0<-(K_X+\Delta )\cdot C\leq 4$ (cf. [Reference Fujino6, Theorem 1.1.6]).

From Step 2 to Step 4, we will check that $\mathcal {N}$ is a rational polytope in $\mathcal {P}$ . The proof of Steps 2–4 is essentially the same as the proof of [Reference Birkar2, Proposition 3.2].

Step 2. There is a real number $\alpha>0$ such that for any extremal curve C of $\overline {\text {NE}}(X/Y; W)$ , if $(K_X+\Delta )\cdot C>0$ , then $(K_X+\Delta )\cdot C>\alpha $ .

Step 3. There is a real number $\delta>0$ such that if $D\in \mathcal {P}$ , $\lVert D-\Delta \rVert <\delta $ and $(K_X+D)\cdot R\leq 0$ for an extremal ray R of $\overline {\text {NE}}(X/Y; W)$ , then $(K_X+\Delta )\cdot R\leq 0$ .

Proof of Step 3.

If the statement is not true, then there is an infinite sequence of $D_\lambda \in \mathcal {P}$ and extremal rays $R_\lambda $ of $\overline {\text {NE}}(X/Y; W)$ such that for each $\lambda $ we have $(K_X+D_\lambda )\cdot R_\lambda \leq 0$ , $(K_X+\Delta )\cdot R_\lambda>0$ , and $\lVert D_\lambda -\Delta \rVert $ converges to 0. For each $\lambda $ , there are nonnegative real numbers $a_{1, \lambda }$ , $\dots $ , $a_{r, \lambda }$ such that $D_\lambda =\sum _ia_{i, \lambda }\Delta _i$ and $\sum _ia_{i, \lambda }=1$ . The representation of $\Delta =\sum _ia_i\Delta _i$ and $D_\lambda =\sum _ia_{i, \lambda }\Delta _i$ is not necessarily unique now. Taking an infinite subsequence of $\{D_\lambda \}$ , we may assume that there is some simplex $\mathcal {S}$ which is spanned by some vertices of $\mathcal {P}$ such that every $D_\lambda $ is in the inner of $\mathcal {S}$ . We can write $D_\lambda =\sum _ia_{i, \lambda }\Delta _i$ , where $a_{i, \lambda }>0$ if and only if $\Delta _i$ is a vertex of $\mathcal {S}$ , and this representation is unique. Since $\lVert D_\lambda -\Delta \rVert $ converges to 0, the limits $\lim _{\lambda \to \infty }a_{i, \lambda }$ exist and we have $\Delta =\sum _i(\lim _{\lambda \to \infty }a_{i, \lambda })\Delta _i$ . We assume that $a_i=\lim _{\lambda \to \infty }a_{i, \lambda }$ . After replacing the sequence with an infinite subsequence, we can assume that the sign of $(K_X+\Delta _i)\cdot R_\lambda $ is independent of $\lambda $ , and that for each $\lambda ,$ we have an extremal curve $C_\lambda $ for $R_\lambda $ . Now, if $(K_X+\Delta _i)\cdot C_\lambda \leq 0$ , then it is bounded from below, hence there are only finitely many possibilities for this number and we could assume that it is independent of $\lambda $ . On the other hand, if $a_i\neq 0$ , then $(K_X+\Delta _i)\cdot C_\lambda $ is bounded from below and above because $(K_X+D_\lambda )\cdot C_\lambda =\sum _ia_{i, \lambda }(K_X+\Delta _i)\cdot C_\lambda \leq 0$ , hence there are only finitely many possibilities for $(K_X+\Delta _i)\cdot C_\lambda $ and we could assume that it is independent of $\lambda $ . Assume that $a_i\neq 0$ for $1\leq i\leq l$ but $a_i=0$ for $i>l$ . Then, it is clear that

$$ \begin{align*} (K_X+D_\lambda)\cdot C_\lambda=(K_X+\Delta)\cdot C_\lambda&+\sum_{i\leq l}(a_{i, \lambda}-a_i)(K_X+\Delta_i)\cdot C_\lambda\\ &+\sum_{i>l}a_{i, \lambda}(K_X+\Delta_i)\cdot C_\lambda \end{align*} $$

would be positive by Step 2 if $\lambda \gg 0$ , which gives a contradiction.

Step 4. The set $\mathcal {N}$ is a rational polytope.

We have checked that $\mathcal {N}$ is a rational polytope in $\mathcal {P}$ .

Finally, we will prove that $K_X+\Delta $ is semi-ample over a neighborhood of W.

Step 5. We note that $\mathcal {N}=\{D\in \mathcal {P}|K_X+D\text { is nef}\}$ . By the above construction, we have $\Delta \in \mathcal {N}$ . Let $\mathcal {F}$ be the minimal face of $\mathcal {N}$ containing $\Delta $ . Then we can take $\mathbb {Q}$ -divisors $\Delta _1,\dots ,\Delta _q$ on X and positive real numbers $a_1,\ldots ,a_q$ such that $\Delta _i$ is in the relative interior of $\mathcal {F}$ for every i, $K_X+\Delta =\sum _ia_i(K_X+\Delta _i)$ and $\sum _ia_i=1$ . By Corollary 4.4, $K_X+\Delta _i$ is semi-ample over a neighborhood of W for every i since $K_X+\Delta _i$ is nef over W. Therefore, $K_X+\Delta $ is semi-ample over a neighborhood of W.

Proof of Theorem 1.1.

Theorems 3.6 and 5.1 imply Theorem 1.1.

Proof of Corollary 1.5.

We may assume that $\pi _*\mathcal {O}_X(\llcorner m(K_X+\Delta )\lrcorner )\neq 0$ for some $m\geq 0$ . In this case, $K_X+\Delta $ is $\pi $ -pseudo-effective. By Theorem 1.1, shrinking Y around W, we may assume that $K_X+\Delta $ is $\pi $ -semi-ample. Then Corollary 1.5 follows from [Reference Fujino5, Lemma 2.36].

Proof of Corollary 1.6.

In Case (C), we replace X with its minimal resolution. So, we may always assume that X has only rational singularities. Then we only have to prove the statement in Case (D). In this case, any divisor on an open subset of X is $\mathbb {Q}$ -Cartier (cf. [Reference Ishii10, Theorem 7.3.2]). Then, for each point $y\in Y$ , X is $\mathbb {Q}$ -factorial over $W=\{y\}$ . We note that $\rho (X/Y; W)<\infty $ (cf. Nakayama’s finiteness, [Reference Fujino5, Theorem 4.7]). By Theorem 1.1, if $K_X+\Delta $ is $\pi $ -pseudo-effective, we may assume that $K_X+\Delta $ is $\pi $ -semi-ample over some open neighborhood of y. By [Reference Fujino5, 2.36], the log canonical ring $R(X,\Delta )$ is locally finitely generated over some open neighborhood of y. Then it is locally finitely generated on Y.