Proof. The argument is similar to [Reference Birkar, Cascini, Hacon and McKernanBCHM10, Proof of Lemma 3.5.6]. We fix elements $D_{1},\cdots ,\,D_{p}$ of $|D/Z|_{\mathbb {R}}$ such that $\bigcap _{i=1}^{p}\mathrm {Supp}\,D_{i}=\mathrm {Bs}|D/Z|_{\mathbb {R}}$ . We prove Lemma 2.4 by induction on the sum of the numbers of components of the divisors $D_{1},\cdots ,\,D_{p}$ . For each $2 \leq i \leq p$ , we consider the set

$$ \begin{align*}\mathcal{W}_{i}:=\left\{(E_{1}, E_{i})\in \mathrm{WDiv}_{\mathbb{R}}(X)\times \mathrm{WDiv}_{\mathbb{R}}(X)\,\middle|\begin{array}{l} E_{1} \geq 0, E_{i} \geq0, E_{1}\sim_{\mathbb{R},\,Z}E_{i},\\ \mathrm{Supp}\,E_{1}=\mathrm{Supp}\,D_{1}\text{, and}\\ \mathrm{Supp}\,E_{i}=\mathrm{Supp}\,D_{i} \end{array}\right\}.\end{align*} $$

By the argument from convex geometry, we can find a rational polytope $\mathcal {C}_{i} \subset \mathcal {W}_{i}$ that contains $(D_{1},D_{i})$ . Then there are effective $\mathbb {Q}$ -divisors $N_{1},\cdots ,\, N_{p}$ on X such that $(N_{1},N_{i}) \in \mathcal {C}_{i}$ for every $2 \leq i \leq p$ . Then $\mathrm {Supp}\,N_{i}=\mathrm {Supp}\,D_{i}$ for all $1 \leq i \leq p$ , and it follows that $N_{1}\sim _{\mathbb {Q},\,Z}N_{i}$ for all $2 \leq i \leq p$ . Since we have

$$ \begin{align*}\mathrm{Bs}|N_{1}/Z|_{\mathbb{Q}}\subset \bigcap_{i=1}^{p}\mathrm{Supp}\,N_{i}= \bigcap_{i=1}^{p}\mathrm{Supp}\,D_{i}=\mathrm{Bs}|D/Z|_{\mathbb{R}},\end{align*} $$

it follows that $N_{1}$ is movable over Z and $\mathrm {Bs}|N_{1}/Z|_{\mathbb {Q}} \subset \mathrm {Bs}|D/Z|_{\mathbb {R}}$ . We set

$$ \begin{align*}M_{1}:=N_{1} \quad \mathrm{and} \quad r_{1}:=\mathrm{sup}\{t \in \mathbb{R}_{\geq 0}\,|\, D_{i}-t N_{i}\geq 0\text{ for every }1\leq i \leq p\}.\end{align*} $$

Putting $D^{\prime }_{i}=D_{i}-r_{1} N_{i}$ for each $1\leq i \leq p$ and $D':=D-r_{1}M_{1}$ , then the sum of the numbers of components of $D^{\prime }_{1},\cdots ,\,D^{\prime }_{p}$ is strictly less than the sum of the numbers of components of $D_{1},\cdots ,\,D_{p}$ . By construction, the relation $D'\sim _{\mathbb {R},\,Z} D^{\prime }_{i}$ holds for every $1 \leq i \leq p$ , and

$$ \begin{align*}\mathrm{Bs}|D'/Z|_{\mathbb{R}} \subset \bigcap_{i=1}^{p}\mathrm{Supp}\,D^{\prime}_{i}\subset \bigcap_{i=1}^{p}\mathrm{Supp}\,D_{i}=\mathrm{Bs}|D/Z|_{\mathbb{R}}.\end{align*} $$

Thus $D'$ is movable over Z. We apply the induction hypothesis to $D'$ . Then we can find positive real numbers $r^{\prime }_{1},\cdots ,\,r^{\prime }_{q}$ and effective $\mathbb {Q}$ -divisors $N^{\prime }_{1},\cdots ,\,N^{\prime }_{q}$ on X such that $D' \sim _{\mathbb {R},\,Z}\sum _{j=1}^{q}r^{\prime }_{j}N^{\prime }_{j}$ and $\mathrm { Bs}|N^{\prime }_{j}/Z|_{\mathbb {Q}} \subset \mathrm {Bs}|D'/Z|_{\mathbb {R}}$ for every $1 \leq j \leq q$ . Since we have $D'=D-r_{1}N_{1}$ and $\mathrm {Bs}|D'/Z|_{\mathbb {R}} \subset \mathrm {Bs}|D/Z|_{\mathbb {R}}$ , by renaming $r^{\prime }_{1},\cdots ,\,r^{\prime }_{q}$ and $N^{\prime }_{1},\cdots ,\,N^{\prime }_{q}$ we obtain positive real numbers $r_{1},\cdots ,\,r_{l}$ and effective $\mathbb {Q}$ -divisors $M_{1},\cdots ,\,M_{l}$ on X satisfying the conditions of Lemma 2.4.

Proof. Let $D_{1},\cdots ,\,D_{k}$ be elements of $|D/Z|_{\mathbb {R}}$ such that $\bigcap _{i=1}^{k}\mathrm {Supp}\,D_{i}=\mathrm {Bs}|D/Z|_{\mathbb {R}}$ . For any prime divisor P on X, we define $r_{P}:=\underset {1\leq i \leq k}{\mathrm {min}}\, \mathrm {coeff}_{P}(D_{i})$ , and we set

$$ \begin{align*}F:=\sum_{P}r_{P}P,\end{align*} $$

where P runs over prime divisors on X. By construction, F is well defined as an effective $\mathbb {R}$ -divisor on X and every component of F is an irreducible component of $\mathrm {Bs}|D/Z|_{\mathbb {R}}$ . We put

$$ \begin{align*}D':=D-F.\end{align*} $$

Then $D' \sim _{\mathbb {R},\,Z}D_{i}-F$ for all $1\leq i \leq k$ , and construction of F implies that any prime divisor is not contained in $\bigcap _{i=1}^{k}\mathrm {Supp}\,(D_{i}-F)$ . Hence, $D'$ is movable over Z. We also have

$$ \begin{align*}\mathrm{Bs}|D'/Z|_{\mathbb{R}}\subset \bigcap_{i=1}^{k}\mathrm{Supp}\,(D_{i}-F) \subset \bigcap_{i=1}^{k}\mathrm{Supp}\,D_{i}=\mathrm{Bs}|D/Z|_{\mathbb{R}}.\end{align*} $$

By applying Lemma 2.4 to $D'$ , we get positive real numbers $r_{1},\cdots ,\,r_{l}$ and effective $\mathbb {Q}$ -divisors $M_{1},\cdots ,\,M_{l}$ on X such that

• • $D' \sim _{\mathbb {R},\,Z}\sum _{i=1}^{l}r_{i}M_{i}$ , and

• • $M_{i}$ is movable over Z and $\mathrm {Bs}|M_{i}/Z|_{\mathbb {Q}} \subset \mathrm {Bs}|D'/Z|_{\mathbb {R}}$ for every $1 \leq i \leq l$ .

Putting $M=\sum _{i=1}^{l}r_{i}M_{i}$ , then M and F satisfy the conditions of Lemma 2.5.

Proof. Replacing X by a resolution, we may assume that X is smooth. By Lemma 2.5, there exist effective $\mathbb {R}$ -divisors M and F on X satisfying the following.

• • $D \sim _{\mathbb {R},Z}M+F$ ,

• • every component of F is an irreducible component of $\mathrm {Bs}|D/Z|_{\mathbb {R}}$ , and

• • we may write $M=\sum _{i=1}^{l}r_{i}M_{i}$ for some positive real numbers $r_{1},\cdots ,\,r_{l}$ and effective $\mathbb {Q}$ -divisors $M_{1},\cdots ,\,M_{l}$ on X such that every $M_{i}$ is movable over Z and $\mathrm { Bs}|M_{i}/Z|_{\mathbb {Q}} \subset \mathrm {Bs}|D/Z|_{\mathbb {R}}$ for every $1\leq i \leq l$ .

We take a positive integer k such that $kM_{i}$ is Cartier for every $1 \leq i \leq l$ . Since Z is quasi-projective, choosing k sufficiently large and divisible, we may assume

$$ \begin{align*}\mathrm{Bs}|M_{i}/Z|_{\mathbb{Q}}=\mathrm{Supp}\big(\mathrm{Coker}(\pi^{*}\pi_{*}\mathcal{O}_{X}(kM_{i})\otimes_{\mathcal{O}_{X}} \mathcal{O}_{X}(-kM_{i}) \longrightarrow \mathcal{O}_{X})\big)\end{align*} $$

as Zariski closed subsets of X. Define $\mathcal {I}_{i}:=\mathrm {Im}(\pi ^{*}\pi _{*}\mathcal {O}_{X}(kM_{i})\otimes _{\mathcal {O}_{X}} \mathcal {O}_{X}(-kM_{i}) \to \mathcal {O}_{X})$ for each $1 \leq i \leq l$ . We take a log resolution $f \colon Y \to X$ of $(X,\Delta +F)$ such that $\mathcal {I}_{i}\cdot \mathcal {O}_{Y}$ is an invertible sheaf on Y for every $1 \leq i \leq l$ . We can write $\mathcal {I}_{i}\cdot \mathcal {O}_{Y}=\mathcal {O}_{Y}(-E_{i})$ for some effective Cartier divisor $E_{i}$ on Y. Since $\mathrm {Bs}|M_{i}/Z|_{\mathbb {Q}} \subset \mathrm {Bs}|D/Z|_{\mathbb {R}}$ , every component of $E_{i}$ is an irreducible component of $\mathrm {Bs}|f^{*}D/Z|_{\mathbb {R}}$ . By taking some blow-ups if necessary, we may assume that $\mathrm {Supp}(f_{*}^{-1}(\Delta +F)+\sum _{i=1}^{l}E_{i})\cup \mathrm {Ex}(f)$ is a simple normal crossing divisor on Y. Now we put

$$ \begin{align*}M_{Y}:=f^{*}M-\frac{1}{k}\sum_{i=1}^{l}r_{i}E_{i} \quad \mathrm{and} \quad F_{Y}:=f^{*}F+\frac{1}{k}\sum_{i=1}^{l}r_{i}E_{i}.\end{align*} $$

Then $M_{Y}=\frac {1}{k}\sum _{i=1}^{l}r_{i}\big ( f^{*}(kM_{i})-E_{i}\big )$ . Hence $M_{Y}$ is semi-ample over Z. By construction, $f \colon Y \to X$ , $M_{Y}$ , and $F_{Y}$ satisfy the conditions of Theorem 2.6.

Proof. Let $f \colon Y \to X$ be a resolution of X such that $f^{-1}(C)$ is pure codimension one. Let A be an ample Cartier divisor on Y. Since $C \not \subset {\mathrm {NNef}}(D/Z)$ , by the same argument as in [Reference NakayamaN04, Proof of III, 1.7 Lemma], for any $k \in \mathbb {Z}_{>0}$ we see that $\mathrm {Bs}|f^{*}D+\frac {1}{k}A/Z|_{\mathbb {R}}$ does not contain any irreducible component of $f^{-1}(C)$ mapping onto C. We can find a curve $C' \subset Y$ such that $f(C')=C$ and $C' \not \subset \mathrm {Bs}|f^{*}D+\frac {1}{k}A/Z|_{\mathbb {R}}$ for any $k \in \mathbb {Z}_{>0}$ . Then $(f^{*}D+\frac {1}{k}A)\cdot C' \geq 0$ for all $k \in \mathbb {Z}_{>0}$ , and therefore $(f^{*}D\,\cdot \, C') \geq 0$ . Thus $(D \,\cdot \, C) \geq 0$ .

Proof. By Remark 2.12, it is enough to prove ${\mathrm {NNef}}(D+A/Z)\supset f({\mathrm {NNef}}(f^{*}(D+A)/Z))$ for any $\pi $ -ample divisor A on X. In particular, we may assume that D is $\pi $ -big.

We will prove $X\setminus {\mathrm {NNef}}(D/Z)\subset X \setminus f({\mathrm {NNef}}(f^{*}D/Z))$ . We fix $x \in X\setminus {\mathrm {NNef}}(D/Z)$ . We pick ${y \in f^{-1}(x)}$ and an arbitrary prime divisor P over Y such that $c_{Y}(P)\ni y$ . We will prove that $\sigma _{P}(f^{*}D/Z)=0$ . Let $g \colon X' \to X$ be a resolution of X such that $g^{-1}(f(c_{Y}(P)))$ is pure codimension one in $X'$ , and let $h \colon Y' \to Y$ be a resolution of Y such that P appears as a prime divisor on $Y'$ and the map $f' \colon Y' \dashrightarrow X'$ is a morphism. Since D is $\pi $ -big and Z is quasi-projective, by Definition 2.8, the equality

$$ \begin{align*}\sigma_{P}(f^{*}D/Z)=\mathrm{inf}\left\{\mathrm{coeff}_{P}(D')\,\middle|\begin{array}{l}D'\geq 0\text{ and }\mathbb{R}\text{-Cartier,}\, D'\sim_{\mathbb{Q}}h^{*}f^{*}D+h^{*}f^{*}\pi^{*}H_{Z}\\ \text{for some }\mathbb{Q}\text{-Cartier }\mathbb{Q}\text{-divisor }H_{Z}\text{ on }Z\end{array}\right\} \end{align*} $$

holds. Let $g^{-1}(f(c_{Y}(P)))=\sum _{i=1}^{l}Q_{i}$ be the prime decomposition. As above, we have

$$ \begin{align*}\sigma_{Q_{i}}(D/Z)=\mathrm{inf}\left\{\mathrm{coeff}_{P}(E')\,\middle|\begin{array}{l}E'\geq 0\text{ and }\mathbb{R}\text{-Cartier,}\, E'\sim_{\mathbb{Q}}g^{*}D+g^{*}\pi^{*}H^{\prime}_{Z}\\ \text{for some }\mathbb{Q}\text{-Cartier }\mathbb{Q}\text{-divisor }H^{\prime}_{Z}\text{ on }Z\end{array}\right\} \end{align*} $$

for all i. Since $x \in X\setminus {\mathrm {NNef}}(D/Z)$ , we have $\sigma _{Q_{i}}(D/Z)=0$ for all i such that $g(Q_{i})\ni x$ . For any $\epsilon \in \mathbb {R}_{>0}$ , by taking a general effective $\mathbb {R}$ -Cartier $\mathbb {R}$ -divisor $E_{\epsilon }$ on $X'$ such that $E_{\epsilon }\sim _{\mathbb {Q}}g^{*}D+g^{*}\pi ^{*}H^{\prime }_{Z}$ for some $\mathbb {Q}$ -Cartier $\mathbb {Q}$ -divisor $H^{\prime }_{Z}$ on Z, we have $\mathrm {coeff}_{Q_{i}}(E_{\epsilon }) \leq \epsilon $ for all i such that $g(Q_{i})\ni x$ . By construction, P is a component of $f^{\prime *}(\sum _{j}Q_{j})$ , where j runs over the indices such that $g(Q_{j})\ni x$ . Therefore, for any $\epsilon \in \mathbb {R}_{>0}$ , we can find $D_{\epsilon } \geq 0$ such that $D_{\epsilon }\sim _{\mathbb {Q}}h^{*}f^{*}D+h^{*}f^{*}\pi ^{*}H_{Z}$ for some $\mathbb {Q}$ -Cartier $\mathbb {Q}$ -divisor $H_{Z}$ on Z and $\mathrm {coeff}_{P}(D_{\epsilon }) \leq \epsilon $ . This implies $\sigma _{P}(f^{*}D/Z)=0$ . Therefore, $y \not \in {\mathrm {NNef}}(f^{*}D/Z)$ for any $y \in f^{-1}(x)$ . Then $f^{-1}(y) \cap {\mathrm {NNef}}(f^{*}D/Z) =\emptyset $ . Hence $x \in X \setminus f({\mathrm {NNef}}(f^{*}D/Z))$ . We finish the proof.

Proof. We have

$$ \begin{align*}\Delta=\{\Delta\}+\Delta^{=1}+\lfloor \Delta^{<1}\rfloor+\lfloor \Delta^{>1}\rfloor=\{\Delta\}+\Delta^{=1}-D.\end{align*} $$

Since $K_{Y}+\Delta $ and $K_{Y}+\{\Delta \}+\Delta ^{=1}$ are $\mathbb {R}$ -Cartier, D is $\mathbb {Q}$ -Cartier.

We will prove that $G-\lfloor g^{*}D \rfloor $ is effective and g-exceptional. Clearly we have $g_{*}\Gamma =\Delta $ , $g_{*}\{\Gamma \}= \{\Delta \}$ , and $g_{*}\Gamma ^{=1}=\Delta ^{=1}$ . Thus $g_{*}G=D$ . It is also clear that $g_{*}^{-1}D$ is a Weil divisor, and therefore $\lfloor g^{*}D\rfloor -g^{*}D$ is g-exceptional. From these facts, it follows that $G-\lfloor g^{*}D\rfloor $ is g-exceptional. We also have

$$ \begin{align*} K_{W}+\{\Gamma\}+\Gamma^{=1}+\lfloor \Gamma^{<1}\rfloor+\lfloor \Gamma^{>1} \rfloor=g^{*}(K_{Y}+\{\Delta\}+\Delta^{=1})+g^{*}(\lfloor \Delta^{<1}\rfloor+\lfloor \Delta^{>1}\rfloor). \end{align*} $$

By the definitions of D and G, we obtain

$$ \begin{align*}G-g^{*}D=K_{W}+\{\Gamma\}+\Gamma^{=1}-g^{*}(K_{Y}+\{\Delta\}+\Delta^{=1}).\end{align*} $$

Since all g-exceptional prime divisors P on W satisfy $a(P,Y,\{\Delta \}+\Delta ^{=1})>-1$ , we have $\lceil (K_{W}+\{\Gamma \}+\Gamma ^{=1}-g^{*}(K_{Y}+\{\Delta \}+\Delta ^{=1}))\rceil \geq 0$ . Therefore, we obtain

$$ \begin{align*} \lceil G \rceil+\lceil -g^{*}D\rceil\geq \lceil(G-g^{*}D)\rceil=\lceil (K_{W}+\{\Gamma\}+\Gamma^{=1}-g^{*}(K_{Y}+\{\Delta\}+\Delta^{=1}))\rceil\geq0. \end{align*} $$

Since G is a Weil divisor, it follows that $G-\lfloor g^{*}D\rfloor \geq 0$ .

By the above arguments, D is $\mathbb {Q}$ -Cartier and $G-\lfloor g^{*}D\rfloor $ is effective and g-exceptional. Thus, Lemma 2.19 holds.

Proof. As in [Reference Fujino, Schwede and TakagiFST11, Lemma 5.2], the sheaf $g_{*}\mathcal {O}_{W}(\lceil -(\Gamma ^{<1})\rceil -\lfloor \Gamma ^{>1} \rfloor )$ does not depend on $g \colon W \to Y$ . From this fact, we may freely replace g without loss of generality.

Since the problem is local, we may shrink Y, and therefore we may assume that Y is quasi-projective. Since $(Y,\Delta ^{<1}+\mathrm {Supp}\,\Delta ^{\geq 1})$ is dlt, there is a small $\mathbb {Q}$ -factorialization $h \colon Y' \to Y$ . The quasi-projectivity of Y was used to construct this $h \colon Y' \to Y$ . Put $\Delta '=h_{*}^{-1}\Delta $ . Since h is small, h is an isomorphism over the smooth locus of Y (see, e.g., [Reference Kollár and MoriKM98, Corollary 2.63]). Thus, the dlt property ([Reference Kollár and MoriKM98, Definition 2.37]) of $(Y,\Delta ^{<1}+\mathrm {Supp}\,\Delta ^{\geq 1})$ is preserved after we replace $(Y,\Delta )$ by $(Y',\Delta ')$ . Since h is a small projective birational morphism, we have

$$ \begin{align*}h_{*}\mathcal{O}_{Y'}(-\lfloor \Delta^{\prime>1} \rfloor)= \mathcal{O}_{Y}(-\lfloor \Delta^{>1}\rfloor).\end{align*} $$

From these facts, we may replace $(Y,\Delta )$ and $g \colon W\to Y$ by $(Y',\Delta ')$ and some log resolution of $(Y',\Delta ')$ , respectively. Therefore, we may assume that Y is $\mathbb {Q}$ -factorial.

Since $(Y,\Delta ^{<1}+\mathrm {Supp}\,\Delta ^{\geq 1})$ is a $\mathbb {Q}$ -factorial dlt pair, $K_{Y}+\{\Delta \}+\Delta ^{=1}$ is an $\mathbb {R}$ -Cartier $\mathbb {R}$ -divisor and there is a log resolution $g' \colon W' \to Y$ of $(Y,\Delta )$ such that all $g'$ -exceptional prime divisors P on $W'$ satisfy $a(P,Y,\Delta ^{<1}+\mathrm {Supp}\,\Delta ^{\geq 1})>-1$ . It is easy to check the relation $\{\Delta \}+\Delta ^{=1} \leq \Delta ^{<1}+\mathrm {Supp}\,\Delta ^{\geq 1}$ , and thus all the $g'$ -exceptional prime divisors P on $W'$ satisfy $a(P,Y,\{\Delta \}+\Delta ^{=1})>-1$ . By replacing g by $g'$ , we may assume that all g-exceptional prime divisors P on W satisfy $a(P,Y,\{\Delta \}+\Delta ^{=1})>-1$ . We put

$$ \begin{align*}D=-\lfloor \Delta^{>1}\rfloor \qquad \mathrm{and} \qquad G=\lceil -(\Gamma^{<1})\rceil-\lfloor \Gamma^{>1} \rfloor.\end{align*} $$

By Lemma 2.19, D is $\mathbb {Q}$ -Cartier and $G-\lfloor g^{*}D\rfloor $ is effective and g-exceptional. Then

$$ \begin{align*}\mathcal{O}_{Y}(-\lfloor \Delta^{>1}\rfloor) = \mathcal{O}_{Y}(D) \cong g_{*}\mathcal{O}_{W}(G) = g_{*}\mathcal{O}_{W}(\lceil -(\Gamma^{<1}) \rceil-\lfloor \Gamma^{>1} \rfloor)\end{align*} $$

by the standard argument of divisorial sheaves.

Proof. We prove the theorem by induction on ${\mathrm {dim}}\,Y$ . The theorem is clear if ${\mathrm {dim}}\,Y=1$ . From now on, we assume that ${\mathrm {dim}}\,Y>1$ . Since S is not contained in $\mathrm {Supp}\,\Delta ^{>1}$ , it follows that S is an lc center of $(Y,\Delta ^{<1}+\mathrm {Supp}\,\Delta ^{\geq 1})$ . Then we can find a component T of $\Delta ^{=1}$ containing S. Since T is not contained in $\mathrm {Supp}\,\Delta ^{>1}$ , the divisor

$$ \begin{align*}G_{T}:=(\Delta^{>1}-\mathrm{Supp}\,\Delta^{>1})|_{T}=(K_{Y}+\Delta-(K_{Y}+\Delta^{<1}+\mathrm{Supp}\,\Delta^{\geq 1}))|_{T}\end{align*} $$

is well defined as an effective $\mathbb {R}$ -Cartier $\mathbb {R}$ -divisor on T. Let $(T,B_{T})$ be a dlt pair defined by adjunction $K_{T}+B_{T}=(K_{Y}+\Delta ^{<1}+\mathrm {Supp}\,\Delta ^{\geq 1})|_{T}$ . We set

$$ \begin{align*}\Delta_{T}:=B_{T}+G_{T}.\end{align*} $$

We will prove that $\Delta _{T}$ satisfies all the properties of Theorem 2.21. We have

$$ \begin{align*}K_{T}+\Delta_{T}=(K_{Y}+\Delta^{<1}+\mathrm{Supp}\,\Delta^{\geq 1})|_{T}+(\Delta^{>1}-\mathrm{Supp}\,\Delta^{>1})|_{T}=(K_{Y}+\Delta)|_{T}.\end{align*} $$

Hence $K_{T}+\Delta _{T}$ is $\mathbb {R}$ -Cartier and we see that $(T,\Delta _{T})$ is a pair. Therefore $\Delta _{T}$ satisfies the first property of Theorem 2.21. We may write

$$ \begin{align*}\Delta^{\geq 1}=T+\sum_{i}D_{i}+\sum_{j}(1+\gamma_{j})D^{\prime}_{j}\end{align*} $$

for some prime divisors $D_{i}$ and $D^{\prime }_{j}$ and positive real numbers $\gamma _{j}$ . By the dlt property of $(Y,\Delta ^{<1}+\mathrm {Supp}\,\Delta ^{\geq 1})$ , we see that $(Y,\Delta )$ is log smooth at the generic point of $T\cap D_{i}$ and $T\cap D_{j}$ for all i and j. By considering the locus on which $(Y,\Delta )$ is log smooth, we see that any component P of $G_{T}=(\Delta ^{>1}-\mathrm {Supp}\,\Delta ^{>1})|_{T}$ is an irreducible component of $T \cap D^{\prime }_{j}$ for some j. This fact implies that any component P of $G_{T}$ satisfies

$$ \begin{align*}\mathrm{coeff}_{P}(\Delta_{T})=1+\gamma_{j}>1, \quad \mathrm{coeff}_{P}(B_{T})=1, \quad \mathrm{and} \quad \mathrm{coeff}_{P}(\Delta_{T}^{<1}+\mathrm{Supp}\,\Delta^{\geq 1}_{T})=1.\end{align*} $$

Therefore, any component P of $G_{T}$ satisfies

$$ \begin{align*}\mathrm{coeff}_{P}(B_{T})=\mathrm{coeff}_{P}(\Delta_{T}^{<1}+\mathrm{Supp}\,\Delta^{\geq 1}_{T}).\end{align*} $$

Since $\Delta _{T}=B_{T}+G_{T}$ , if a prime divisor Q is not a component of $G_{T}$ , then we have

$$ \begin{align*}\mathrm{coeff}_{Q}(\Delta_{T})=\mathrm{coeff}_{Q}(B_{T}) \leq 1,\end{align*} $$

where the inequality follows from the dlt property of $(T,B_{T})$ . This shows that Q is not a component of $\Delta _{T}^{>1}$ . Hence we have $\mathrm {coeff}_{Q}(\Delta _{T})=\mathrm {coeff}_{Q}(\Delta _{T}^{<1}+\mathrm {Supp}\,\Delta ^{\geq 1}_{T}).$ Then

$$ \begin{align*}\mathrm{coeff}_{Q}(B_{T})=\mathrm{coeff}_{Q}(\Delta_{T}^{<1}+\mathrm{Supp}\,\Delta^{\geq 1}_{T}).\end{align*} $$

By the discussion, we obtain

$$ \begin{align*}B_{T}=\Delta_{T}^{<1}+\mathrm{Supp}\,\Delta^{\geq 1}_{T},\end{align*} $$

which is the second property of Theorem 2.21.

In this paragraph, we will prove the third property of Theorem 2.21 for T and $\Delta _{T}^{>1}$ . Since $(Y,\Delta ^{<1}+\mathrm {Supp}\,\Delta ^{\geq 1})$ is dlt, there is a log resolution $g \colon W \to Y$ of $(Y,\Delta )$ such that all g-exceptional prime divisor E on W satisfy $a(E,Y,\Delta ^{<1}+\mathrm { Supp}\,\Delta ^{\geq 1})>-1$ . We put $T'=g_{*}^{-1}T$ and $g_{T'}=g|_{T'}\colon T' \to T$ , and we define an $\mathbb {R}$ -divisor $\Gamma $ on W by

$$ \begin{align*}K_{W}+T'+\Gamma=g^{*}(K_{Y}+\Delta).\end{align*} $$

By the first property of Theorem 2.21 for $(T,\Delta _{T})$ , we have $K_{T'}+\Gamma |_{T'}=g_{T'}^{*}(K_{T}+\Delta _{T})$ . By the exact sequence

$$ \begin{align*} \begin{aligned} 0 \longrightarrow \mathcal{O}_{W}(\lceil -(\Gamma^{<1})\rceil-\lfloor \Gamma^{>1} \rfloor-T')&\longrightarrow \mathcal{O}_{W}(\lceil -(\Gamma^{<1})\rceil-\lfloor \Gamma^{>1} \rfloor)\\ & \longrightarrow \mathcal{O}_{T'}(\lceil -(\Gamma^{<1})|_{T'}\rceil-\lfloor \Gamma^{>1}|_{T'} \rfloor) \longrightarrow 0, \end{aligned} \end{align*} $$

we get the exact sequence

$$ \begin{align*} \begin{aligned} 0 \longrightarrow g_{*}\mathcal{O}_{W}(\lceil -(\Gamma^{<1})\rceil-\lfloor \Gamma^{>1} \rfloor-T')&\longrightarrow g_{*}\mathcal{O}_{W}(\lceil -(\Gamma^{<1})\rceil-\lfloor \Gamma^{>1} \rfloor)\\ & \longrightarrow g_{T'*}\mathcal{O}_{T'}(\lceil -(\Gamma^{<1})|_{T'}\rceil-\lfloor \Gamma^{>1}|_{T'} \rfloor). \end{aligned} \end{align*} $$

By Corollary 2.20, the isomorphism $\mathcal {O}_{Y} \to g_{*}\mathcal {O}_{W}(\lceil -(\Gamma ^{<1})\rceil )$ induces an isomorphism $\mathcal {O}_{Y}(- \lfloor \Delta ^{>1} \rfloor ) \cong g_{*}\mathcal {O}_{W}(\lceil -(\Gamma ^{<1})\rceil -\lfloor \Gamma ^{>1} \rfloor )$ . This isomorphism and $T'=g_{*}^{-1}T$ imply that $\mathcal {O}_{Y}(- \lfloor \Delta ^{>1} \rfloor -T) \to g_{*}\mathcal {O}_{W}(\lceil -(\Gamma ^{<1})\rceil -\lfloor \Gamma ^{>1} \rfloor -T')$ is an isomorphism. Now $(T',\Gamma |_{T'})$ is log smooth by construction, and $(T, \Delta ^{<1}_{T}+\mathrm {Supp}\,\Delta ^{\geq 1}_{T})$ is dlt by the second property of Theorem 2.21 for T. Since $g_{T'} \colon T' \to T$ is birational and $K_{T'}+\Gamma |_{T'}=g_{T'}^{*}(K_{T}+\Delta _{T})$ , Corollary 2.20 implies that the isomorphism $\mathcal {O}_{T} \to g_{T'*}\mathcal {O}_{T'}(\lceil -(\Gamma ^{<1})|_{T'}\rceil )$ induces an isomorphism $\mathcal {O}_{T}( - \lfloor \Delta _{T}^{>1} \rfloor ) \cong g_{T'*}\mathcal {O}_{T'}(\lceil -(\Gamma ^{<1})|_{T'}\rceil -\lfloor \Gamma ^{>1}|_{T'} \rfloor )$ . By these isomorphisms, we get the exact sequence

$$ \begin{align*}0 \longrightarrow \mathcal{O}_{Y}(- \lfloor \Delta^{>1} \rfloor-T)\longrightarrow \mathcal{O}_{Y}(- \lfloor \Delta^{>1} \rfloor) \longrightarrow \mathcal{O}_{T}( - \lfloor \Delta_{T}^{>1} \rfloor)\end{align*} $$

such that the morphism $\mathcal {O}_{Y}(- \lfloor \Delta ^{>1} \rfloor ) \to \mathcal {O}_{T}( - \lfloor \Delta _{T}^{>1} \rfloor )$ is induced by $\mathcal {O}_{Y} \to \mathcal {O}_{T}$ . This shows that T and $\Delta _{T}^{>1}$ satisfy the third property of Theorem 2.21.

The arguments in the previous paragraphs show that $\Delta _{T}$ satisfies all the properties of Theorem 2.21. We recall that T is a component of $\Delta ^{=1}$ containing S. Therefore S is an lc center of $(T, \Delta _{T})$ . By the second property of Theorem 2.21 for $\Delta _{T}$ , we obtain

$$ \begin{align*}B_{T}=\Delta_{T}^{<1}+\mathrm{Supp}\,\Delta^{\geq 1}_{T} \qquad \mathrm{and} \qquad G_{T}=\Delta_{T}^{>1}-\mathrm{Supp}\,\Delta_{T}^{>1}.\end{align*} $$

Therefore, we have

$$ \begin{align*} \begin{aligned} &G_{S}=\big((\Delta^{>1}-\mathrm{Supp}\,\Delta^{>1})|_{T}\big)|_{S}=(\Delta_{T}^{>1}-\mathrm{Supp}\,\Delta_{T}^{>1})|_{S},\quad \mathrm{and}\\ &K_{S}+B_{S}=\big((K_{Y}+\Delta^{<1}+\mathrm{Supp}\,\Delta^{\geq 1})|_{T}\big)|_{S} =(K_{T}+\Delta_{T}^{<1}+\mathrm{Supp}\,\Delta^{\geq 1}_{T})|_{S}. \\ \end{aligned} \end{align*} $$

By the induction hypothesis of Theorem 2.21, the divisor $\Delta _{S}$ satisfies the first and the second properties of Theorem 2.21, and moreover we obtain the morphism

$$ \begin{align*}\mathcal{O}_{Y}(- \lfloor \Delta^{>1} \rfloor) \longrightarrow \mathcal{O}_{T}( - \lfloor \Delta_{T}^{>1} \rfloor) \longrightarrow \mathcal{O}_{S}( - \lfloor \Delta_{S}^{>1} \rfloor)\end{align*} $$

induced by $\mathcal {O}_{Y} \to \mathcal {O}_{T} \to \mathcal {O}_{S}$ . From this, $\Delta _{S}$ satisfies all properties of Theorem 2.21. We finish the proof.

Proof. We may assume that Z is affine. By using [Reference FujinoF25, Lemma 4.25] and the argument from convex geometry, we can find positive real numbers $r_{1},\,\cdots , \,r_{m}$ , $\mathbb {Q}$ -line bundles $\omega _{1},\,\cdots ,\,\omega _{m}$ on X, and $\pi $ -ample $\mathbb {Q}$ -divisors $A_{1},\,\cdots ,\,A_{m}$ on X such that

• • $\sum _{i=1}^{m}r_{i}=1$ , $\sum _{i=1}^{m}r_{i}\omega _{i}=\omega $ , and $\sum _{i=1}^{m}r_{i}A_{i}=A$ ,

• • for every $1\leq i \leq m$ , there is the structure of a quasi-log scheme $[X,\omega _{i}]$ that has the same non-qlc locus as $\mathrm {Nqlc}(X, \omega )$ , and

• • all $(\omega _{i}+A_{i})|_{\mathrm {Nqlc}(X, \omega )}$ are semi-ample over Z.

For each $1\leq i \leq m$ , the cone theorem [Reference FujinoF17, Theorem 6.7.4] implies that we may write

$$ \begin{align*}\overline{\mathrm{NE}}(X/Z)=\overline{\mathrm{NE}}(X/Z)_{(\omega_{i}+A_{i}) \geq 0}+\overline{\mathrm{NE}}(X/Z)^{(i)}_{-\infty}+\sum_{j=1}^{p_{i}}\mathbb{R}_{\geq 0}[C_{i,j}]\end{align*} $$

for some $p_{i} \in \mathbb {Z}_{>0}$ and curves $C_{i,j} \subset X$ contained in a fiber of $\pi $ , where

$$ \begin{align*}\overline{\mathrm{NE}}(X/Z)^{(i)}_{-\infty}:=\mathrm{Im}(\overline{\mathrm{NE}}(\mathrm{Nqlc}(X, \omega_{i})/Z) \to \overline{\mathrm{NE}}(X/Z))\qquad (\text{[}{\rm F17}\text{, Definition 6.7.1]}).\end{align*} $$

Since $(\omega _{i}+A_{i})|_{\mathrm {Nqlc}(X, \omega )}$ is semi-ample over Z, we have $\overline {\mathrm {NE}}(X/Z)^{(i)}_{-\infty } \subset \overline {\mathrm {NE}}(X/Z)_{(\omega _{i}+A_{i}) \geq 0}$ for any $1 \leq i \leq m$ . For each $1 \leq i \leq m$ and $1 \leq j \leq p_{i}$ , we consider the set

$$ \begin{align*}\mathcal{H}_{i,j}:=\left\{(t_{1},\cdots,\,t_{m})\in (\mathbb{R}_{\geq 0})^{m}\,\middle|\,\sum_{k=1}^{m}t_{k}=1, \, \Bigl(\sum_{k=1}^{m}t_{k} (\omega_{k}+A_{k})\Bigr)\cdot C_{i,j}\geq 0\right\}.\end{align*} $$

Since all $\omega _{k}+A_{k}$ are $\mathbb {Q}$ -Cartier and all $\mathcal {H}_{i,j}$ contain $(r_{1}, \cdots ,\,r_{m})$ , we see that $\bigcap _{i,j}\mathcal {H}_{i,j}$ is a rational polytope that contains $(r_{1}, \cdots ,\,r_{m})$ . By the standard argument from convex geometry, we can find positive real numbers $r^{\prime }_{1},\,\cdots , \,r^{\prime }_{n}$ , $\mathbb {Q}$ -line bundles $\omega ^{\prime }_{1},\,\cdots ,\,\omega ^{\prime }_{n}$ on X, and $\pi $ -ample $\mathbb {Q}$ -divisors $A^{\prime }_{1},\,\cdots ,\,A^{\prime }_{n}$ on X such that

• • $\sum _{l=1}^{n}r^{\prime }_{l}=1$ , $\sum _{l=1}^{n}r^{\prime }_{l}\omega ^{\prime }_{l}=\omega $ , and $\sum _{l=1}^{n}r^{\prime }_{l}A^{\prime }_{l}=A$ ,

• • every $\omega ^{\prime }_{l}+A^{\prime }_{l}$ is a convex linear combination of $\omega _{1}+A_{1}, \,\cdots ,\,\omega _{m}+A_{m}$ ,

• • for every $1\leq l \leq n$ , there is the structure of a quasi-log scheme $[X,\omega ^{\prime }_{l}]$ that has the same non-qlc locus as $\mathrm {Nqlc}(X, \omega )$ ,

• • all $(\omega ^{\prime }_{l}+A^{\prime }_{l})|_{\mathrm {Nqlc}(X, \omega )}$ are semi-ample over Z, and

• • $(\omega ^{\prime }_{l}+A^{\prime }_{l})\cdot C_{i,j} \geq 0$ for all l, i, and j.

Then $\omega ^{\prime }_{l}+A^{\prime }_{l}$ is $\pi $ -nef for all l. Indeed, if $\omega ^{\prime }_{l}+A^{\prime }_{l}$ is not $\pi $ -nef for some l, then there exists an $(\omega ^{\prime }_{l}+A^{\prime }_{l})$ -negative extremal ray R of $\overline {\mathrm {NE}}(X/Z)$ . By construction, there is an index i such that R is an $(\omega _{i}+A_{i})$ -negative extremal ray of $\overline {\mathrm {NE}}(X/Z)$ . Then R is generated by $C_{i,j}$ for some j. However, it contradicts the condition $(\omega ^{\prime }_{l}+A^{\prime }_{l})\cdot C_{i,j} \geq 0$ . Therefore, $\omega ^{\prime }_{l}+A^{\prime }_{l}$ is $\pi $ -nef for all l. By the base point free theorem [Reference FujinoF17, Theorem 6.5.1], every $\omega ^{\prime }_{l}+A^{\prime }_{l}$ is $\pi $ -semi-ample. Therefore, $\omega +A=\sum _{l=1}^{n}r^{\prime }_{l}(\omega ^{\prime }_{l}+A^{\prime }_{l})$ is also $\pi $ -semi-ample.

Proof. This is a direct consequence of [Reference FujinoF17, Proposition 6.3.1].

Proof. Let $\tau \colon T \to X$ and $f_{S}\colon S \to T$ be as in the theorem. To define the structure of a quasi-log scheme induced by a normal pair $f_{S} \colon (S,\Delta _{S}) \to [T,\omega _{T}]$ , we need to check $f_{S}(\mathrm {Nlc}(S,\Delta _{S})) \neq T$ . By Theorem 2.21, the pair $(S,\Delta _{S}^{<1}+\mathrm {Supp}\,\Delta _{S}^{\geq 1})$ is a dlt pair and the natural morphism $\mathcal {O}_{Y} \to \mathcal {O}_{S}$ induces a morphism $\mathcal {O}_{Y}(- \lfloor \Delta ^{>1} \rfloor )\to \mathcal {O}_{S}( - \lfloor \Delta _{S}^{>1} \rfloor )$ . By Corollary 2.20, the support of $\mathrm {Nlc}(S,\Delta _{S})$ is contained in $\mathrm {Nlc}(Y,\Delta )\cap S$ . Then $f(\mathrm {Nlc}(S,\Delta _{S})) \subset \mathrm {Nqlc}(X, \omega ) \cap f(S)$ . This fact and the hypothesis $f(S)\not \subset \mathrm {Nqlc}(X, \omega )$ imply $f_{S}(\mathrm {Nlc}(S,\Delta _{S})) \neq T$ . Therefore, we may define the structure of a quasi-log scheme induced by a normal pair $f_{S} \colon (S,\Delta _{S}) \to [T,\omega _{T}]$ .

Since $(Y,\Delta ^{<1}+\mathrm {Supp}\,\Delta ^{\geq 1})$ is a dlt pair and S is an lc center of $(Y,\Delta )$ , any lc center of $(S,\Delta _{S})$ is an lc center of $(Y,\Delta )$ contained in S. From this, the image $\tau (C)$ of any qlc center C of $[T ,\omega _{T}]$ is a qlc center of $[X,\omega ]$ .

By Corollary 2.20, the defining ideal sheaf $\mathcal {I}_{X}$ of $\mathrm {Nqlc}(X, \omega )$ is $f_{*}\mathcal {O}_{X}(-\lfloor \Delta ^{>1}\rfloor )$ , and the defining ideal sheaf $\mathcal {I}_{T}$ of $\mathrm {Nqlc}(T, \omega _{T})$ is $f_{S*}\mathcal {O}_{S}(-\lfloor \Delta ^{>1}_{S}\rfloor )$ . By Theorem 2.21, the natural morphism $\mathcal {O}_{Y} \to \mathcal {O}_{S}$ induces a morphism $\mathcal {I}_{X} \to \tau _{*}\mathcal {I}_{T}$ . By the following diagram

there exists a natural morphism $\mathcal {O}_{\mathrm {Nqlc}(X, \omega )}\to \tau _{*}\mathcal {O}_{\mathrm {Nqlc}(T, \omega _{T})}$ . This induces the desired morphism $\mathrm {Nqlc}(T, \omega _{T}) \to \mathrm {Nqlc}(X, \omega )$ as a morphism between closed subschemes.

Proof. By the definitions of the discriminant $\mathbb {R}$ -b-divisors and the moduli $\mathbb {R}$ -b-divisors as in [Reference Fujino and HashizumeFH23, Corollary 5.2] (see [Reference AmbroA04] for the case of $\mathbb {Q}$ -divisors), we obtain a generalized pair $(X,B+M)$ as in [Reference Birkar and ZhangBZ16] such that $\omega \sim _{\mathbb {R},\,Z} K_{X}+B+M$ and the generalized non-klt locus of $(X,B+M)$ is equal to $\mathrm { Nqklt}(X,\omega )$ (cf. [Reference AmbroA04, Lemma 3.2]). Let $g \colon X' \to X$ be a log resolution of $(X,B)$ such that the moduli part $M'$ on $X'$ is nef over Z. We define an $\mathbb {R}$ -divisor $B'$ on $X'$ by $K_{X'}+B'+M'=g^{*}(K_{X}+B+M)$ . Since $g^{*}A+M'$ is nef and big over Z, we can find an effective $\mathbb {R}$ -divisor $E'$ on $X'$ and a general $(\pi \circ g)$ -ample $\mathbb {R}$ -divisor $H'$ on $X'$ such that $E'+H' \sim _{\mathbb {R},\,Z}g^{*}A+M'$ and $\lfloor B' \rfloor = \lfloor (B'+E'+H') \rfloor $ . By replacing $X'$ with a log resolution of $(X',B'+E')$ , we may assume that $(X',B'+E'+H')$ is log smooth. Put $G:=g_{*}(B'+E'+H')$ . By construction, we have $K_{X}+G\sim _{\mathbb {R},\,Z}\omega +A$ and $\mathrm {Nklt}(X,G)$ coincides with the generalized non-klt locus of $(X,B+M)$ , and therefore we have $\mathrm {Nklt}(X,G)=\mathrm {Nqklt}(X,\omega )$ as closed subschemes of X.

Proof. We may assume that Z is affine. Let $U \supset X'$ be an open subset of X on which L is Cartier. Let $g \colon W \to Y$ be a log resolution of $(Y,\Delta )$ , and we define $\Gamma $ by

$$ \begin{align*}K_{W}+\Gamma=g^{*}(K_{Y}+\Delta).\end{align*} $$

Then $g^{*}f^{*}L=\lfloor g^{*}f^{*}L \rfloor + \{g^{*}f^{*}L\}$ and $\{g^{*}f^{*}L\}$ is an effective g-exceptional $\mathbb {Q}$ -divisor such that $\{g^{*}f^{*}L\}|_{(f \circ g)^{-1}(U)}=0$ . By taking some blow-ups on W if necessary, we may assume that the union of all strata of $(W,\mathrm {Supp}\,\Gamma ^{\geq 1})$ mapping into $X'$ is $\mathrm {Supp}\,(\Gamma ^{>1}+S)$ for some union S of components of $\Gamma ^{= 1}$ . We put

$$ \begin{align*}T:=S+\lfloor \Gamma^{>1}\rfloor \qquad \mathrm{and}\qquad B:= \Gamma^{=1}+\{\Gamma\}-S.\end{align*} $$

Then B is an snc boundary $\mathbb {R}$ -divisor, and $B^{=1}$ and T have no common components. Moreover, $\Gamma =T+B-\lceil -(\Gamma ^{<1})\rceil $ and $\mathcal {I}_{X'}=(f\circ g)_{*}\mathcal {O}_{W}(-T+\lceil -(\Gamma ^{<1})\rceil )$ . We have

$$ \begin{align*} \begin{aligned} \lfloor g^{*}f^{*}L \rfloor=&g^{*}f^{*}(L-\omega)+g^{*}f^{*}\omega-\{g^{*}f^{*}L\}\\ \sim_{\mathbb{R}}&g^{*}f^{*}(L-\omega)+K_{W}+\Gamma-\{g^{*}f^{*}L\}\\ \sim_{\mathbb{R}}&g^{*}f^{*}(L-\omega)+K_{W}+T+B-\lceil-(\Gamma^{<1})\rceil-\{g^{*}f^{*}L\}. \end{aligned} \end{align*} $$

We take a reduced divisor E on W such that if we define

$$ \begin{align*}B':=B-\{g^{*}f^{*}L\}+E,\end{align*} $$

then $B'$ is a boundary $\mathbb {R}$ -divisor and $B^{\prime =1}\leq B^{=1}$ . More precisely, for each prime divisor P on W, the coefficient of P in E is defined by

$$ \begin{align*} \mathrm{coeff}_{P}(E)=\left\{ \begin{array}{l}{0 \qquad (\mathrm{coeff}_{P}(B-\{g^{*}f^{*}L\})\geq 0)} \\ {1 \qquad (\mathrm{coeff}_{P}(B-\{g^{*}f^{*}L\})< 0)} \end{array}\right. \end{align*} $$

We note that E is g-exceptional and $E|_{(f \circ g)^{-1}(U)}=0$ because any component of E is a component of $\{g^{*}f^{*}L\}$ . By the above relation, we have

(♠) $$ \begin{align} \lfloor g^{*}f^{*}L \rfloor+E+\lceil-(\Gamma^{<1})\rceil-T\sim_{\mathbb{R}}g^{*}f^{*}(L-\omega)+K_{W}+B'. \end{align} $$

We put

$$ \begin{align*}A:=\lfloor g^{*}f^{*}L \rfloor+E+\lceil-(\Gamma^{<1})\rceil\sim_{\mathbb{R}}g^{*}f^{*}(L-\omega)+K_{W}+T+B',\end{align*} $$

and we consider the exact sequence

$$ \begin{align*}0 \longrightarrow \mathcal{O}_{W}(A-T) \longrightarrow \mathcal{O}_{W}(A) \longrightarrow \mathcal{O}_{T}(A|_{T}) \longrightarrow 0.\end{align*} $$

Now T is an snc Weil divisor, and $B^{=1}$ and T have no common components. By this fact and the log smoothness of $(W,\mathrm {Supp}\,\Gamma )$ , any stratum of $(W,B^{=1})$ is not contained in $\mathrm {Supp}\,T$ . By these facts, we can check that the image of any lc center of $(W,B)$ by $f \circ g$ is not contained in $X'$ . Indeed, if some lc center C of $(W,B)$ is mapped into $X'$ , then C is a stratum of $(W,\mathrm {Supp}\,\Gamma ^{\geq 1})$ by $B^{=1} \leq \mathrm {Supp}\,\Gamma ^{\geq 1}$ . Then $C \subset \mathrm {Supp}\,B^{=1}\cap \mathrm {Supp}\,T$ since the union of all strata of $(W,\mathrm {Supp}\,\Gamma ^{\geq 1})$ mapping into $X'$ is $\mathrm {Supp}\,T$ , which follows from the definitions of T and S. This contradicts the log smoothness of $(W,\mathrm {Supp}\,\Gamma )$ and the fact that $B^{=1}$ and T have no common components. Thus, the image of any lc center of $(W,B)$ is not mapped into $X'$ . Since $B^{\prime =1}\leq B^{=1}$ , any lc center of $(W,B')$ is not mapped into $X'$ . By the torsion-free theorem [Reference FujinoF17, Theorem 5.6.2 (i)], the connecting morphism $(f\circ g)_{*}\mathcal {O}_{T}(A|_{T}) \to R^{1}(f\circ g)_{*}\mathcal {O}_{W}(A-T)$ is the zero morphism. Therefore,

$$ \begin{align*}0 \longrightarrow (f\circ g)_{*}\mathcal{O}_{W}(A-T) \longrightarrow (f\circ g)_{*}\mathcal{O}_{W}(A) \longrightarrow (f\circ g)_{*}\mathcal{O}_{T}(A|_{T}) \longrightarrow 0\end{align*} $$

is exact.

Now we have the exact sequence

$$ \begin{align*}0 \longrightarrow \mathcal{I}_{X'} \longrightarrow \mathcal{O}_{X} \longrightarrow \mathcal{O}_{X'} \longrightarrow 0.\end{align*} $$

We recall that $L|_{U}$ is Cartier and $U \supset X'$ . From this, we have the exact sequence

$$ \begin{align*}0 \longrightarrow \mathcal{I}_{X'}\otimes_{\mathcal{O}_{X}} \mathcal{O}_{X}(L) \longrightarrow \mathcal{O}_{X}(L) \longrightarrow \mathcal{O}_{X'}(L|_{X'}) \longrightarrow 0.\end{align*} $$

We recall that $f^{*}L$ is a Weil divisor on Y, which is the hypothesis of Theorem 2.29, and $A=\lfloor g^{*}f^{*}L \rfloor +E+\lceil -(\Gamma ^{<1})\rceil $ such that $E+\lceil -(\Gamma ^{<1})\rceil $ is an effective g-exceptional divisor. Thus, we have

$$ \begin{align*}(f\circ g)_{*}\mathcal{O}_{W}(A) \cong f_{*}\mathcal{O}_{Y}(f^{*}L)\cong \mathcal{O}_{X}(L)\end{align*} $$

via the natural morphism $\mathcal {O}_{X}(L) \to (f\circ g)_{*}\mathcal {O}_{W}(A)$ (cf. [Reference NakayamaN04, II, 2.11. Lemma]).

Recall that $\mathcal {I}_{X'}=(f\circ g)_{*}\mathcal {O}_{W}(-T+\lceil -(\Gamma ^{<1})\rceil )$ . Since $U \supset X'=(f \circ g)(\mathrm {Supp}\,T)$ , we have $\mathrm {Supp}\,T \subset (f \circ g)^{-1}(U)$ . By recalling the fact $E|_{(f \circ g)^{-1}(U)}=0$ , we have

$$ \begin{align*} \begin{aligned} (f\circ g)_{*}\mathcal{O}_{T}(A|_{T}) &=(f\circ g)_{*}\mathcal{O}_{T}\big((\lfloor g^{*}f^{*}L \rfloor+E+\lceil-(\Gamma^{<1})\rceil)|_{T}\big)\\&\cong(f\circ g)_{*}\mathcal{O}_{T}\big(\lceil-(\Gamma^{<1})\rceil|_{T}\big)\otimes_{\mathcal{O}_{X}} \mathcal{O}_{X}(L). \end{aligned} \end{align*} $$

Note that this isomorphism is induced by the restriction of $\mathcal {O}_{X}(L) \to (f\circ g)_{*}\mathcal {O}_{W}(A)$ to $X'$ . By the definitions of T and B, we also have

$$ \begin{align*}\lceil-(\Gamma^{<1})\rceil-T\sim_{\mathbb{R}}-g^{*} f^{*}\omega+K_{W}+\Gamma+\lceil-(\Gamma^{<1})\rceil-T\sim_{\mathbb{R},\,X}K_{W}+B.\end{align*} $$

In the second paragraph of this proof, we have checked that any lc center of $(W,B)$ is not mapped into $X'$ . By the torsion-free theorem [Reference FujinoF17, Theorem 5.6.2 (i)] and applying $(f \circ g)_{*}$ to the exact sequence

$$ \begin{align*}0 \longrightarrow \mathcal{O}_{W}(\lceil-(\Gamma^{<1})\rceil-T) \longrightarrow \mathcal{O}_{W}(\lceil-(\Gamma^{<1})\rceil) \longrightarrow \mathcal{O}_{T}(\lceil-(\Gamma^{<1})\rceil|_{T}) \longrightarrow 0,\end{align*} $$

we have the exact sequence

$$ \begin{align*} 0 \longrightarrow \mathcal{I}_{X'} \longrightarrow \mathcal{O}_{X} \longrightarrow (f \circ g)_{*}\mathcal{O}_{T}(\lceil-(\Gamma^{<1})\rceil|_{T}) \longrightarrow 0. \end{align*} $$

This shows $(f \circ g)_{*}\mathcal {O}_{T}(\lceil -(\Gamma ^{<1})\rceil |_{T})= \mathcal {O}_{X'}$ . Therefore we have

$$ \begin{align*}(f\circ g)_{*}\mathcal{O}_{T}(A|_{T}) \cong (f\circ g)_{*}\mathcal{O}_{T}\big(\lceil-(\Gamma^{<1})\rceil|_{T}\big)\otimes_{\mathcal{O}_{X}} \mathcal{O}_{X}(L) \cong \mathcal{O}_{X'}\otimes_{\mathcal{O}_{X}} \mathcal{O}_{X}(L) \cong \mathcal{O}_{X'}(L|_{X'}).\end{align*} $$

Here, the final isomorphism follows from the Cartier property of L around $X'$ .

From the above argument, we obtain the diagram

From this, we have $(f\circ g)_{*}\mathcal {O}_{W}(A-T) \cong \mathcal {I}_{X'}\otimes _{\mathcal {O}_{X}} \mathcal {O}_{X}(L)$ . By (♠) and the definition of A, we have $A-T \sim _{\mathbb {R}}g^{*}f^{*}(L-\omega )+K_{W}+B'.$ By construction of $B'$ , any stratum of $B^{\prime =1}$ is a stratum of $\Gamma ^{=1}$ . Since $L-\omega $ is $\pi $ -nef and $\pi $ -log big with respect to $[X,\omega ]$ (see [Reference FujinoF16, Definition 3.7]), which is the hypothesis of Theorem 2.29, the vanishing theorem for simple normal crossing varieties [Reference FujinoF17, Theorem 5.7.3 (ii)] implies

$$ \begin{align*}R^{i}\pi_{*}(\mathcal{I}_{X'}\otimes_{\mathcal{O}_{X}} \mathcal{O}_{X}(L)) \cong R^{i}\pi_{*}\big((f\circ g)_{*}\mathcal{O}_{W}(A-T)\big)=0\end{align*} $$

for all $i>0$ . This is what we wanted to prove.

Proof. We will prove the case of minimal model because the case of good minimal model can be proved similarly.

Let $(X",\Delta ")$ be a minimal model of $(X',\Delta ')$ over Z. For any prime divisor P on X, we have

$$ \begin{align*}a(P,X,\Delta)=a(P,X', \Delta') \leq a(P,X",\Delta").\end{align*} $$

Moreover, if P is exceptional over $X"$ then $f_{*}^{-1}P$ is exceptional over $X"$ . Hence

$$ \begin{align*}a(P,X,\Delta)< a(P,X",\Delta").\end{align*} $$

Therefore, the first condition of a minimal model in Definition 3.1 holds. For any prime divisor $Q"$ on $X"$ , we have

$$ \begin{align*}\mathrm{coeff}_{Q"}(\Delta")=-a(Q",X',\Delta')=-a(Q",X,\Delta).\end{align*} $$

Suppose that $Q"$ is exceptional over X. If $Q"$ is not exceptional over $X'$ , then the birational transform $Q'$ of $Q"$ on $X'$ is f-exceptional, and therefore

$$ \begin{align*}a(Q",X,\Delta)=a(Q',X,\Delta)\leq -1.\end{align*} $$

If $Q"$ is exceptional over $X'$ , then

$$ \begin{align*}a(Q",X,\Delta)=a(Q",X',\Delta')\leq -1.\end{align*} $$

In any case, we have $a(Q",X,\Delta )\leq -1$ . Thus the second condition of a minimal model in Definition 3.1 holds. By definition, $K_{X"}+\Delta "$ is nef over Z, and thus the third condition of a minimal model in Definition 3.1 holds. Therefore, $(X",\Delta ")$ is a minimal model of $(X,\Delta )$ over Z. From this, we see that Lemma 3.2 holds.

Proof. The proof is the same as the lc case. Let $(X",\Delta ")$ be a good minimal model of $(X,\Delta )$ over Z. Take a common resolution $f \colon X' \to X$ and $g \colon X' \to X"$ of $X \dashrightarrow X"$ . By Definition 3.1, we can write

$$ \begin{align*}f^{*}(K_{X}+\Delta)=g^{*}(K_{X"}+\Delta")+E\end{align*} $$

for some effective g-exceptional $\mathbb {R}$ -divisor E on $X'$ . By [Reference Liu and XieLX23a, Lemma 3.4], we have

$$ \begin{align*}\sigma_{P}(f^{*}(K_{X}+\Delta)/Z)=\sigma_{P}(g^{*}(K_{X"}+\Delta")+E/Z)=\mathrm{coeff}_{P}(E)\end{align*} $$

for any prime divisor P on $X'$ , where the final equality follows from the semi-ampleness of $K_{X"}+\Delta "$ . By Definition 2.9, we have $N_{\sigma }(f^{*}(K_{X}+\Delta ); X'/Z)=E$ , and therefore we have $P_{\sigma }(f^{*}(K_{X}+\Delta ); X'/Z)=g^{*}(K_{X"}+\Delta ")$ , which is semi-ample over Z.

Proof. We have

$$ \begin{align*}\mathrm{Bs}|K_{Y}+\Gamma+B/Z|_{\mathbb{R}} \subset \mathrm{Bs}|K_{Y}+\Gamma-E/Z|_{\mathbb{R}}\cup\mathrm{Supp}\,(B+E).\end{align*} $$

Since $K_{Y}+\Gamma =f^{*}(K_{X}+\Delta )+E$ and $\Gamma \geq 0$ and $E \geq 0$ have no common components, any component $\tilde {E}$ of E satisfies $a(\tilde {E},X,\Delta )>0$ . Then $\mathrm {Supp}\,E \subset \mathrm {Bs}|f^{*}(K_{X}+\Delta )/Z|_{\mathbb {R}}$ by the assumption of $f \colon Y \to X$ . Since $\mathrm {Supp}\,B \subset \mathrm {Bs}|f^{*}(K_{X}+\Delta )/Z|_{\mathbb {R}}$ , we obtain

(*) $$\begin{align} \mathrm{Supp}\,(B+E) \subset \mathrm{Bs}|f^{*}(K_{X}+\Delta)/Z|_{\mathbb{R}}. \end{align} $$

From this fact and the relation $K_{Y}+\Gamma -E=f^{*}(K_{X}+\Delta )$ , we have

$$ \begin{align*}\mathrm{Bs}|K_{Y}+\Gamma+B/Z|_{\mathbb{R}} \subset \mathrm{Bs}|K_{Y}+\Gamma-E/Z|_{\mathbb{R}}\cup\mathrm{Supp}\,(B+E) \subset \mathrm{Bs}|f^{*}(K_{X}+\Delta)/Z|_{\mathbb{R}}.\end{align*} $$

Therefore, any prime divisor contracted by $\phi $ is contained in $\mathrm {Bs}|f^{*}(K_{X}+\Delta )/Z|_{\mathbb {R}}$ . We put $\Gamma '=\phi _{*}\Gamma $ and $B'=\phi _{*}B$ . Let $g \colon W \to Y$ and $g' \colon W \to Y'$ be a common resolution of $\phi $ . We may write

(**) $$\begin{align} g^{*}f^{*}(K_{X}+\Delta)+F=g^{\prime *}(K_{Y'}+\Gamma'+B')+F' \end{align} $$

for some effective $\mathbb {R}$ -divisors F and $F'$ on W that have no common components. Then $F'$ is $g'$ -exceptional since $g^{\prime }_{*}(F-F')=B'+\phi _{*}E$ and $g^{\prime }_{*}F$ and $g^{\prime }_{*}F'$ have no common components.

In this paragraph, we will prove that $F=0$ . We assume that F is $(f \circ g)$ -exceptional. Then the semi-ampleness of $K_{Y'}+\Gamma '+B'$ over Z implies

$$ \begin{align*} \begin{aligned} \mathrm{Supp}\,F'=\mathrm{Bs}|g^{\prime *}(K_{Y'}+\Gamma'+B')+F'/Z|_{\mathbb{R}} &=\mathrm{Bs}|g^{*}f^{*}(K_{X}+\Delta)+F/Z|_{\mathbb{R}} \\&= \mathrm{Bs}|g^{*}f^{*}(K_{X}+\Delta)/Z|_{\mathbb{R}}\cup\mathrm{Supp}\,F. \end{aligned} \end{align*} $$

From this relation and the fact that F and $F'$ have no common components, we have $F=0$ . Therefore, it is sufficient to prove that F is $(f \circ g)$ -exceptional. By definition, we have $g^{\prime }_{*}F=B'+\phi _{*}E$ . Hence every component of $g_{*}F$ is $\phi $ -exceptional or a component of $B+E$ . By the inclusion (*) and the fact that $\phi $ only contracts divisors contained in $\mathrm {Bs}|f^{*}(K_{X}+\Delta )/Z|_{\mathbb {R}}$ , we have $\mathrm {Supp}\,g_{*}F\subset \mathrm { Bs}|f^{*}(K_{X}+\Delta )/Z|_{\mathbb {R}}$ . Thus

$$ \begin{align*}\mathrm{Supp}\,(f \circ g)_{*}F\subset \mathrm{Bs}|K_{X}+\Delta/Z|_{\mathbb{R}}.\end{align*} $$

From now on, we suppose that $(f \circ g)_{*}F \neq 0$ and we will get a contradiction. By (**) and the semi-ampleness of $K_{Y'}+\Gamma '+B'$ over Z, any component of $(f \circ g)_{*}F$ is not an irreducible component of $\mathrm {Bs}|K_{X}+\Delta +(f \circ g)_{*}F/Z|_{\mathbb {R}}$ . Note that $K_{X}+\Delta +(f \circ g)_{*}F$ may not be $\mathbb {R}$ -Cartier. We pick $D \in \mathrm {Bs}|K_{X}+\Delta /Z|_{\mathbb {R}}$ , and let $D=\sum _{i=1}^{l}a_{i}D_{i}$ be the prime decomposition, where $a_{1},\cdots ,\, a_{l}$ are positive real numbers. Since every component of $\mathrm {Supp}\,(f \circ g)_{*}F$ is contained in $\mathrm {Bs}|K_{X}+\Delta /Z|_{\mathbb {R}}$ , we can write $(f \circ g)_{*}F=\sum _{i=1}^{l}b_{i}D_{i}$ for some nonnegative real numbers $b_{1},\cdots ,\, b_{l}$ . Relabeling the indices, we may assume $\frac {b_{1}}{a_{1}}\geq \frac {b_{i}}{a_{i}}$ for all $2 \leq i \leq l$ . Then $b_{1}>0$ because otherwise the relation $\frac {b_{1}}{a_{1}}\geq \frac {b_{i}}{a_{i}}$ implies $b_{i}=0$ for all $2 \leq i \leq l$ , which shows $(f \circ g)_{*}F=0$ . Moreover