Proof. Let us quickly recall the b-fold cyclic cover $\pi \colon \widetilde X \to X$ . We fix a rational function $\phi $ on X such that $b\Delta = \operatorname {\mathrm {div}}(\phi ).$ As usual, we can define an $\mathcal O_X$ -algebra structure of $\bigoplus _{i=0}^{b-1}\mathcal O_X(\lfloor i\Delta \rfloor )$ by $b\Delta ={\operatorname {\mathrm {div}}}(\phi )$ . We note that

$$ \begin{align*}\mathcal O_X(\lfloor i\Delta\rfloor)\times \mathcal O_X(\lfloor j\Delta\rfloor) \to \mathcal O_X(\lfloor (i+j)\Delta\rfloor) \end{align*} $$

is well defined for $0\leq i, j \leq b-1$ by $\lfloor i\Delta \rfloor +\lfloor j\Delta \rfloor \leq \lfloor (i+j)\Delta \rfloor $ and that

$$ \begin{align*}\mathcal O_X(\lfloor (i+j)\Delta\rfloor)\simeq \mathcal O_X(\lfloor (i+j-b)\Delta\rfloor) \end{align*} $$

for $i+j\geq b$ defined by the multiplication with $\phi ^{-1}$ . We put

$$ \begin{align*}\pi\colon \widetilde X=\operatorname{\mathrm{Spec}} _X \bigoplus _{i=0}^{b-1} \mathcal O_X(\lfloor i\Delta\rfloor) \end{align*} $$

and call it a b-fold cyclic cover associated with $b\Delta \sim 0$ . By construction, $\pi \colon \widetilde X\to X$ is étale outside $\operatorname {\mathrm {Supp}} \{\Delta \}$ . We note that $\widetilde X$ is normal over a neighborhood of the generic point of every irreducible component of $\operatorname {\mathrm {Supp}} \{\Delta \}$ . We also note that $(\widetilde X, B_{\widetilde X})$ is simple normal crossing in codimension one. Throughout this proof, we freely use the following commutative diagram:

Step 1. Let U and Z be affine open neighborhood of $x\in X$ and $y=f(x)\in Y$ , respectively. Without loss of generality, we may assume that U is a simple normal crossing divisor on a smooth affine variety W since $(X, B)$ is a simple normal crossing pair. By shrinking W, U, and Z suitably, we get the following commutative diagram:

where $\iota $ is the natural closed embedding $U\hookrightarrow W$ . From now on, we repeatedly shrink W, U, and Z suitably without mentioning it explicitly. Since every stratum of X is smooth over Y, we may assume that p is a smooth morphism between smooth affine varieties.

Step 2. Since $p\colon W\to Z$ is a smooth morphism, there exists a commutative diagram:

where g is étale and $p_1$ is the first projection (see, e.g., [Reference Altman and Kleiman1, Chap. VII, Def. 1.1 and Th. 1.8]). By choosing a coordinate system $(z_1, \ldots , z_n)$ of ${\mathbb C}^n$ suitably and shrinking U and W if necessary, we may further assume that U is defined by a monomial

$$ \begin{align*}x_1\cdots x_p=0 \end{align*} $$

on W, where $x_i=g^*z_i$ for $1\leq i\leq p$ , and

$$ \begin{align*}B|_{U}=\sum _{i=1}^r\alpha _i (y_i=0)|_U \quad \text{with} \quad \alpha _i \in {\mathbb Q} \end{align*} $$

holds, where $y_i=g^*z_{p+i}$ for $1\leq i\leq r$ . Here, we used the hypothesis that every stratum of $(X, \operatorname {\mathrm {Supp}} B)$ is smooth over Y.

Step 3. We put $L=(z_1\cdots z_p=0)$ in ${\mathbb C}^n$ . Then we have the following commutative diagram.

Note that $g|_{U}$ is étale because it is the base change of g by $L\hookrightarrow {\mathbb C}^n$ . We put

$$ \begin{align*}D=\sum _{i=1}^r \alpha_i (z_{p+i}=0)\end{align*} $$

on ${\mathbb C}^n$ . Let $p_2\colon Z\times {\mathbb C}^n\to {\mathbb C}^n$ be the second projection. Then $B|_{U}=g^*p^*_2D|_U$ holds.

Step 4. Without loss of generality, we may assume that $K_Z\sim 0$ by shrinking Z suitably. Then $K_U\sim 0$ holds. Hence, by using the second projection $p_2\colon Z\times {\mathbb C}^n \to {\mathbb C}^n$ , we have

$$ \begin{align*}0\sim b\Delta|_U =b(K_U+B|_U)\sim bg|^*_U\bigl(p_2^*D|_{Z\times L}\bigr). \end{align*} $$

Since $g|_{U}$ is étale, we see that all the coefficients of $bp_{2}^{*}D|_{Z\times L}$ are integers. Since $p_{2}$ is the second projection and $D+L$ is a simple normal crossing divisor on $\mathbb {C}^{n}$ , all the coefficients of $bD$ are integers. Therefore, we have $bD\sim 0$ . We fix a rational function $\sigma $ on $\mathbb {C}^{n}$ such that $bD=\operatorname {\mathrm {div}}(\sigma )$ . We consider the b-fold cyclic cover $\alpha \colon M\to {\mathbb C}^n$ associated with $bD=\operatorname {\mathrm {div}}(\sigma )$ . We put $N=\alpha ^{-1}L$ . We define $B_N$ by $K_N+B_N=(\alpha |_{N})^*(K_L+D|_L)$ and put $B_{Z\times N}=p^*_2B_N$ , where $p_2\colon Z\times N\to N$ is the second projection. Then we get the following commutative diagram:

where $g'\colon U'\to Z\times N$ is the base change of $g|_U\colon U\to Z\times L$ by $\mathrm {id}_Z\times (\alpha |_N)$ . We put $B_{U'}=g^{\prime *}B_{Z\times N}$ . Then $K_{U'}+B_{U'}$ is equal to the pullback of $K_{U}+B|_{U}$ to $U'$ .

Step 5. Since $\alpha \colon M\to {\mathbb C}^n$ is the b-fold cyclic cover associated with $bD=\operatorname {\mathrm {div}}(\sigma )$ , we see that

$$ \begin{align*}M=\operatorname{\mathrm{Spec}} _{{\mathbb C}^n} \bigoplus _{i=0}^{b-1} \mathcal O_{{\mathbb C}^n}(\lfloor iD\rfloor).\end{align*} $$

Since $p_{2}\circ g\colon W\to Z\times {\mathbb C}^n\to {\mathbb C}^{n}$ is the composition of an étale morphism and the second projection, we have $g^{*}p_{2}^{*}\lfloor iD\rfloor |_{U} =\lfloor ig^{*}p_{2}^{*}D\rfloor |_{U}=\lfloor iB\rfloor |_{U} =\lfloor iB|_{U}\rfloor $ , where the last equality follows from that $(U,B|_{U})$ is a simple normal crossing pair. Let $\sigma _{U}$ be a rational function on U which is the pullback of $\sigma $ . Then $bB|_{U}=\operatorname {\mathrm {div}}(\sigma _{U})$ because we have $bD=\operatorname {\mathrm {div}}(\sigma )$ . By the construction of $U'\to U$ , we see that

$$ \begin{align*}U'=\operatorname{\mathrm{Spec}} _{U} \bigoplus _{i=0}^{b-1} \mathcal O_U(\lfloor iB|_{U}\rfloor)\end{align*} $$

and $U'\to U$ is the b-fold cyclic cover associated with $bB|_{U}=\operatorname {\mathrm {div}}(\sigma _{U})$ .

We recall that $\Delta =K_{X}+B$ and $\widetilde X\to X$ is the b-fold cyclic cover associated with $b\Delta =\operatorname {\mathrm {div}}(\phi )$ . We put $\phi _{U}$ as the restriction of $\phi $ to U. Then, the morphism $\pi ^{-1}(U)\to U$ is the b-fold cyclic cover associated with $b\Delta |_{U}=\operatorname {\mathrm {div}}(\phi _{U})$ . Now, $\Delta |_{U}-B|_{U}$ is a Cartier divisor on U and $b(\Delta |_{U}-B|_{U})= \operatorname {\mathrm {div}}(\phi _{U}\cdot \sigma _{U}^{-1})$ . With this relation, we construct a b-fold cyclic cover $\tau \colon \overline {U}\to U$ . Then $\tau $ is étale, $\tau ^{*}(\Delta |_{U}-B|_{U})$ is Cartier, and $\tau ^{*}(\Delta |_{U}-B|_{U})\sim 0$ . So there exists a rational function $\xi $ on $\overline {U}$ such that $\xi ^{b} =\tau ^*(\phi _U\cdot \sigma ^{-1}_U)$ , equivalently, $\tau ^{*}(\Delta |_{U}-B|_{U}) =\operatorname {\mathrm {div}}(\xi )$ . From this, the b-fold cyclic cover $U^{\dagger } _1\to \overline U$ associated with $b\tau ^{*}\Delta |_{U}= \operatorname {\mathrm {div}}(\tau ^{*}\phi _{U})$ is isomorphic to the b-fold cyclic cover $U^{\dagger }_2\to \overline U$ associated with $b\tau ^{*}B|_{U} =\operatorname {\mathrm {div}}(\tau ^{*}\sigma _{U})= \operatorname {\mathrm {div}}(\tau ^{*}\phi _{U}\cdot \xi ^{-b})$ . Since $\tau \colon \overline {U}\to U$ is étale, the construction of $U^{\dagger }_2$ shows that $U_2^{\dagger } \to \overline {U}$ is the base change of $U'\to U$ by $\overline {U}\to U$ . Similarly, we see that $U^{\dagger }_1 \to \overline {U}$ is the base change of $\pi ^{-1}(U)\to U$ by $\overline {U}\to U$ .

We put $a_1\colon U^{\dagger }_1\to \pi ^{-1}(U)$ and $a_2\colon U^{\dagger }_2\to U'$ . By construction, $a_1$ and $a_2$ are étale. We see that the composition $U^{\dagger }_1 \to \pi ^{-1}(U)\to U$ is isomorphic to the composition $U^{\dagger }_2 \to U'\to U$ by construction. By this isomorphism, we obtain that $a_1^*(B_{\widetilde X}|_{\pi ^{-1}(U)})$ is isomorphic to $a_2^*B_{U'}$ .

In this way, there exist étale morphisms $a\colon U^{\dagger }\to \pi ^{-1}(U)$ and $a'\colon U^{\dagger }\to U'$ over Z such that $U^{\dagger }_1\simeq U^{\dagger }\simeq U^{\dagger }_2$ with the following commutative diagram:

such that $a^*(B_{\widetilde X}|_{\pi ^{-1}(U)})=a^{\prime *}B_{U'}$ .

Step 7. Recall that $B_{Z\times N}=p^*_2B_N$ , where $p_2\colon Z\times N \to N$ , and $B_{U'}=g^{\prime *}B_{Z\times N}$ . Recall also the relation $a^*(B_{\widetilde X}|_{\pi ^{-1}(U)})=a^{\prime *}B_{U'}$ . We apply [Reference Bierstone and Vera Pacheco2, Th. 1.4] (see Theorem 4.2) to N and $B_N$ , and we obtain a morphism $\beta \colon N'\to N$ given by a composition of blowups. We apply [Reference Bierstone and Vera Pacheco2, Th. 1.4] again to $Z\times N$ and $B_{Z \times N}$ . Then we get a morphism $\mathrm {id}_Z\times \beta \colon Z\times N'\to Z\times N$ by the functoriality of [Reference Bierstone and Vera Pacheco2, Th. 1.4] (see Remark 4.3). We put $\widehat {V}=d^{-1}(\pi ^{-1}(U)) \subset V$ , and we apply [Reference Bierstone and Vera Pacheco2, Th. 1.4] to the pair of $U^{\dagger }$ and $a^*(B_{\widetilde X}|_{\pi ^{-1}(U)})$ , and the pair of $U'$ and $B_{U'}$ . Then we obtain morphisms $V^{\dagger } \to U^{\dagger }$ and $V'\to U'$ .

We check that we may apply the functoriality (see Remark 4.3) to the morphisms

$$ \begin{align*} g'\colon U' \to Z\times N, a'\colon U^{\dagger} \to U', \text{ and } a\colon U^{\dagger} \to \pi^{-1}(U) \end{align*} $$

(see the diagram in the next paragraph) and divisors

$$ \begin{align*} B_{Z\times N} \text{ on } Z\times N, B_{U'} \text{ on } U', \text{ and } B_{\widetilde X}|_{\pi^{-1}(U)} \text{ on } \pi^{-1}(U) \text{ and their pullbacks.}\end{align*} $$

We only check the second condition of Remark 4.3 for schemes because the case of divisors can be proved by the same way. By construction, $g'$ is the base change of $g|_{U}\colon U \to Z\times L$ by the morphism $Z\times N \to Z \times L$ . Because $Z \times L$ is a simple normal crossing divisor on $Z\times {\mathbb C}^n$ and $g|_{U}$ is étale, by arguing locally, we see that $g|_{U}$ satisfies the second condition of Remark 4.3. Then so does $g'$ since $g'$ is constructed by the base change of $g|_{U}$ . Similarly, $a'$ (resp. a) is constructed with the base change of $\tau \colon \overline {U}\to U$ by $U' \to U$ (resp. $\pi ^{-1}(U)\to U$ ), and U is a simple normal crossing divisor on W. Thus, the same argument as above implies that $a'$ and a satisfy the second condition of Remark 4.3. Thus, we may apply the functoriality (see Remark 4.3) to the above morphisms and divisors.

Applying the functoriality (see Remark 4.3), we have the following diagram:

where each square is the fiber product. By construction, all the upper horizontal morphisms are étale. Let $B_{V^{\dagger }}$ (resp. $B_{\widehat {V}}$ ) be the sum of the birational transform of $a^{\prime *}B_{U'}$ (resp. $B_{\widetilde X}|_{\pi ^{-1}(U)}$ ) and the exceptional locus of $V^{\dagger } \to U^{\dagger }$ (resp. $\widehat {V}\to \pi ^{-1}(U)$ ). Then, every stratum of $(V^{\dagger }, \operatorname {\mathrm {Supp}} B_{V^{\dagger }})$ is smooth over Z. Since each irreducible component of $V^{\dagger }$ is smooth over Z and $V^{\dagger } \to \widehat {V}$ is étale, we see that each irreducible component of $\widehat {V}$ is smooth over Z. By a similar argument, we see that every stratum of $(\widehat {V}, \operatorname {\mathrm {Supp}} B_{\widehat {V}})$ is smooth over Z. This implies that $d\colon V\to \widetilde X$ satisfies (iii).

We finish the proof of Lemma 4.4.

Proof of Theorem 4.1

Here, we only explain how to modify the proof of [Reference Fujino6, Th. 1.2] by using Lemma 4.4.

By taking a completion as in [Reference Fujino6, Lem. 4.12], we may further assume that Y is projective. By Lemma 4.4, we can construct a commutative diagram (6.4) in [Reference Fujino6, §6] satisfying (a)–(g) such that $\Sigma _Y=\Sigma $ holds without taking birational modifications of Y. Here, we do not require the condition $\operatorname {\mathrm {Supp}} M_Y\subset \operatorname {\mathrm {Supp}} \Sigma _Y$ in part (d) in [Reference Fujino6, §6] (see Remark 3.8). We also do not require the condition that the general fiber of $h\colon V\to Y$ is connected (see Remark 4.5). The covering arguments and [Reference Fujino6, Prop. 6.3] work without any modifications. We note that Y is a smooth projective variety. In what follows, we apply the proof of [Reference Fujino6, Th. 8.1]. Let $\gamma \colon Y' \to Y$ be a projective birational morphism from a normal variety $Y'$ . By replacing $Y'$ with a higher model if necessary, we may assume that $Y'$ is smooth and that $\gamma ^{-1}\Sigma _Y$ is a simple normal crossing divisor on $Y'$ . With [Reference Fujino6, Lem. 7.3], we construct $\tau \colon \overline {Y}\to Y$ a unipotent reduction of the local monodromies around $\Sigma _{Y}$ . Then the induced fibration over $\overline {Y}$ satisfies [Reference Fujino6, Prop. 6.3(iv) and (v)]. As in the proof of [Reference Fujino6, Th. 8.1], we get a diagram:

such that $\tau '$ is finite and the induced fibration over $\overline {Y}'$ satisfies [Reference Fujino6, Prop. 6.3(iv) and (v)]. By [Reference Fujino6, Th. 3.1], we see that ${\mathbf M}_{\overline {Y}}$ is a nef Cartier divisor and $\gamma ^{\prime *}{\mathbf M}_{\overline {Y}}={\mathbf M}_{\overline {Y}'}$ . Moreover, we have $\tau ^{*}{\mathbf M}_{Y} ={\mathbf M}_{\overline {Y}}$ and $\tau ^{\prime *}{\mathbf M}_{Y'}={\mathbf M}_{\overline {Y}'}$ because $\tau $ and $\tau '$ are both finite (see [Reference Fujino6, Lem. 4.10]). Thus, we have that ${\mathbf M}_{Y}$ is a nef $\mathbb {Q}$ -divisor and $\gamma ^{*}{\mathbf M}_{Y}={\mathbf M}_{Y'}$ . This is Theorem 4.1 (ii). Theorem 4.1(i) immediately follows from Theorem 4.1(ii). So we are done.

Proof. The proof of [Reference Fujino6, Lem. 11.1] works with some suitable minor modifications. Therefore, we can take $B_i$ , $D_i$ , and $r_i$ for $1\leq i\leq k$ satisfying (1)–(6). By (2), $B_i=B^{\leq 1}_i$ holds over the generic point of Y for every i. By (2) again, $\operatorname {\mathrm {rank}} f_*\mathcal O_X(\lceil -(B^{<1}_i)\rceil ) =\operatorname {\mathrm {rank}} f_*\mathcal O_X(\lceil -(B^{<1})\rceil )=1$ . Hence, $f\colon (X, B_i)\to Y$ is a basic ${\mathbb Q}$ -slc-trivial fibration with $K_X+B_i\sim _{{\mathbb Q}} f^*D_i$ for every i. We put $\widetilde B=\sum _{i=1}^k t_i B_i$ . Then $\widetilde B=\widetilde B^{\leq 1}$ holds over the generic point of Y by (2). By (2) again, we see that $\lceil -(\widetilde B^{<1})\rceil =\lceil -(B^{<1})\rceil $ holds. Therefore, $f\colon (X, \widetilde B)\to Y$ is a basic ${\mathbb R}$ -slc-trivial fibration.

Proof. Since ${\mathbf B}$ is an ${\mathbb R}$ -b-divisor, it is sufficient to prove the lemma for a resolution of $Y'\to Y$ and the induced basic slc-trivial fibrations $f'\colon (X',B_{X'})\to Y$ and $f'\colon (X',(B_{i})_{X'})\to Y'$ . Moreover, by Definition 3.5 and taking the normalization of X, we may assume that X is a disjoint union of smooth varieties. Therefore, by replacing X, Y, B, and $B_{i}$ , we may assume that Y is smooth and there are simple normal crossing divisors $\Sigma _{X}$ on X and $\Sigma _{Y}$ on Y such that:

• • $\operatorname {\mathrm {Supp}} B\subset \Sigma _{X}$ and $\operatorname {\mathrm {Supp}} B_{i}\subset \Sigma _{X}$ for every i,

• • $\Sigma _{X}^{v}\subset f^{-1} \Sigma _{Y}\subset \Sigma _{X}$ , where $\Sigma _{X}^{v}$ is the vertical part of $\Sigma _{X}$ ,

• • f is smooth over $Y\setminus \Sigma _{Y}$ , and

• • $\Sigma _{X}$ is relatively simple normal crossing over $Y\setminus \Sigma _{Y}$ .

Then it is clear that $\operatorname {\mathrm {Supp}} {\mathbf B}_{Y}\subset \Sigma _{Y}$ and $\operatorname {\mathrm {Supp}} {\mathbf B}^{\Delta }_{Y} \subset \Sigma _{Y}$ for all $\Delta \in \mathcal {P}$ . We consider a rational convex polytope

$$ \begin{align*} \mathcal C=\left\{\boldsymbol{v}=(v_1, \ldots, v_k)\in [0, 1]^k\, \left|\, \sum _jv_j=1\right.\right\}\subset [0, 1]^k. \end{align*} $$

Then we may identify $\mathcal C$ with $\mathcal P$ by putting $\Delta _{\boldsymbol {v}}= \sum _i v_i B_i\in \mathcal P$ for $\boldsymbol {v}=(v_1, \ldots , v_k)\in \mathcal C$ . We define $\boldsymbol {v}_0 \in \mathcal C$ to be the point such that $\Delta _{\boldsymbol {v}_0}=B$ .

Fix a prime divisor Q on Y which is a component of $\Sigma _{Y}$ . We shrink Y near the generic point of Q so that all components of $f^{*}Q$ dominate Q. We can write $f^{*}Q=\sum _{i}m_{P_{i}}P_{i}$ , where $P_{i}$ are components of $\Sigma _{X}$ such that $f(P_{i})=Q$ , and $m_{P_{i}}=\operatorname {\mathrm {coeff}}_{P_{i}}(f^{*}Q)$ . We fix a component $P_{(B,Q)}$ of $f^{*}Q$ such that

$$ \begin{align*} \frac{1-\operatorname{\mathrm{coeff}}_{P_{(B,Q)}}(B)}{m_{P_{(B,Q)}}}= \underset{P_{i}}\min\left\{\frac{1- \operatorname{\mathrm{coeff}}_{P_{i}}(B)}{m_{P_{i}}}\right\}. \end{align*} $$

Note that $\frac {1-\operatorname {\mathrm {coeff}}_{P_{(B,Q)}}(B)}{m_{P_{(B,Q)}}}$ is the log canonical threshold of $(X,B)$ with respect to $f^*Q$ over the generic point of Q because $(X,B+\mu f^{*}Q)$ is sub log canonical over the generic point of Q if and only if $\operatorname {\mathrm {coeff}}_{P_{i}}(B)+\mu m_{P_{i}}\leq 1$ for all $P_{i}$ . For every component $P_{i}$ of $f^{*}Q$ , we can define a function

$$ \begin{align*} H^{(P_{i})}(\boldsymbol{v}):= \frac{1-\operatorname{\mathrm{coeff}}_{P_{(B,Q)}}(\Delta_{\boldsymbol{v}})} {m_{P_{(B,Q)}}}-\frac{1-\operatorname{\mathrm{coeff}}_{P_{i}}(\Delta_{\boldsymbol{v}})}{m_{P_{i}}} \end{align*} $$

and the half-space

$$ \begin{align*} H^{(P_{i})}_{\leq0}:=\{\boldsymbol{v} \in \mathcal{C}\;|\; H^{(P_{i})}(\boldsymbol{v})\leq 0\}. \end{align*} $$

It is easy to check that $H^{(P_{i})}$ are rational affine functions and the half spaces $H^{(P_{i})}_{\leq 0}$ contain $\boldsymbol {v}_{0}$ since $\boldsymbol {v}_{0}$ is the point such that $\Delta _{\boldsymbol {v}_{0}}=B$ . Therefore, the set

$$ \begin{align*}\mathcal{C}_{Q}:=\mathcal{C}\cap \left(\bigcap_{P_{i}}H^{(P_{i})}_{\leq0}\right)\end{align*} $$

is a rational polytope in $\mathcal {C}$ containing $\boldsymbol {v}_{0}$ , where $P_{i}$ runs over components of $f^{*}Q$ . We put

$$ \begin{align*} \begin{aligned} t(\Delta_{\boldsymbol{v}},Q)&:=1-\operatorname{\mathrm{coeff}}_{Q} \left({\mathbf B}^{\Delta_{\boldsymbol{v}}}_{Y}\right)\\ &=\sup\{\mu\in \mathbb{R}\,|\, (X,\Delta_{\boldsymbol{v}}+ \mu f^{*}Q) \text{ is sub log canonical over the generic point of } Q\}. \end{aligned} \end{align*} $$

Then, by the definitions of $H^{(P_{i})}_{\leq 0}$ , every $\boldsymbol {v}\in \mathcal {C}_{Q}$ satisfies

(1) $$ \begin{align} t(\Delta_{\boldsymbol{v}},Q)=\underset{P_{i}} \min\left\{\frac{1-\operatorname{\mathrm{coeff}}_{P_{i}}(\Delta_{\boldsymbol{v}})}{m_{P_{i}}}\right\} =\frac{1-\operatorname{\mathrm{coeff}}_{P_{(B,Q)}}(\Delta_{\boldsymbol{v}})}{m_{P_{(B,Q)}}}. \end{align} $$

Here, to prove the first equality, we used the fact that $(X,\Delta _{\boldsymbol {v}}+\mu f^{*}Q)$ is sub log canonical over the generic point of Q if and only if $\operatorname {\mathrm {coeff}}_{P_{i}}(\Delta _{\boldsymbol {v}})+\mu m_{P_{i}}\leq 1$ for all $P_{i}$ .

Finally, we define

$$ \begin{align*}\mathcal{C'}:=\bigcap_{Q}\mathcal{C}_{Q},\end{align*} $$

where Q runs over all irreducible components of $\Sigma _{Y}$ . It is easy to see that $\mathcal {C'}$ is a rational polytope in $\mathcal {C}$ and $\mathcal {C'}$ contains $\boldsymbol {v}_{0}$ . Thus, we can find rational points $\boldsymbol {v}_{1},\ldots , \boldsymbol {v}_{l}$ and positive real numbers $s_{1},\ldots , s_{l}$ such that $\sum _{j=1}^{l}s_{j}=1$ and $\sum _{j=1}^{l}s_{j} \boldsymbol {v}_{j} =\boldsymbol {v}_{0}$ . We put $\Delta _{j}=\Delta _{\boldsymbol {v}_{j}}$ for each $1\leq j \leq l$ . Then $B=\Delta _{\boldsymbol {v}_{0}}=\sum _{j=1}^{l}s_{j}\Delta _{j}$ . For every component Q of $\Sigma _{Y}$ , the equation (1) implies that

$$ \begin{align*} \begin{array}{llll} t(B,Q)&=&\dfrac{1-\operatorname{\mathrm{coeff}}_{P_{(B,Q)}}(B)} {m_{P_{(B,Q)}}} &\qquad\qquad(\text{see (1)})\\\\ &=&\dfrac{1-\operatorname{\mathrm{coeff}}_{P_{(B,Q)}} \left(\sum_{j=1}^{l}s_{j} \Delta_{j}\right)}{m_{P_{(B,Q)}}}&\qquad \qquad(B=\sum_{j=1}^{l}s_{j}\Delta_{j})\\\\ &=&\sum_{j=1}^{l}s_{j}\cdot \dfrac{1-\operatorname{\mathrm{coeff}}_{P_{(B,Q)}}(\Delta_{j})} {m_{P_{(B,Q)}}}&\qquad\qquad(\sum_{j=1}^{l}s_{j}=1)\\\\ &=& \sum_{j=1}^{l}s_{j} \cdot t(\Delta_{j},Q) &\qquad\qquad(\text{see (1)}). \end{array} \end{align*} $$

Since $t(\Delta _{j},Q)=1-\operatorname {\mathrm {coeff}}_{Q} \left ({\mathbf B}^{\Delta _{j}}_{Y}\right )$ for every $1\leq j \leq l$ and every irreducible component Q of $\Sigma _{Y}$ , we see that ${\mathbf B}_{Y}=\sum _{j=1}^{l}s_{j}{\mathbf B}^{\Delta _{j}}_{Y}$ .

Proof of Theorem 5.1

Fix an arbitrary projective birational morphism $\sigma \colon Y' \to Y$ from a normal quasi-projective variety $Y'$ , and let

be the induced basic ${\mathbb R}$ -slc-trivial fibration (see Definition 3.4). It is sufficient to show that $\sigma ^{*}(K_{Y}+{\mathbf B}_Y) =K_{Y'}+{\mathbf B}_{Y'}$ and ${\mathbf M}_Y$ is a potentially nef ${\mathbb R}$ -divisor on Y with $\sigma ^{*}{\mathbf M}_{Y}={\mathbf M}_{Y'}$ .

We pick $\mathbb {Q}$ -divisors $B_{1},\dots , B_{k}$ on X, $\mathbb {Q}$ -divisors $D_{1},\dots , D_{k}$ on Y and positive real numbers $r_{1},\dots , r_{k}$ as in Lemma 5.4. Then, the following properties hold:

• • $\sum _{i=1}^k r_i=1$ with $\sum _{i=1}^k r_i B_i=B$ and $\sum _{i=1}^k r_i D_i=D$ ,

• • $\operatorname {\mathrm {Supp}} B=\operatorname {\mathrm {Supp}} B_i$ and $\operatorname {\mathrm {Supp}} D=\operatorname {\mathrm {Supp}} D_i$ hold for every i, and

• • $K_X+B_i\sim _{{\mathbb Q}} f^*D_i$ holds for every i.

We put $D^{\prime }_{i}=\sigma ^{*}D_{i}$ and we define $B^{\prime }_{i}$ by $K_{X'}+B^{\prime }_{i}=\mu ^{*}(K_{X}+B_{i})$ for any $1\leq i\leq k$ . Then $f'\colon (X',B^{\prime }_{i})\to Y'$ are basic ${\mathbb Q}$ -slc-trivial fibrations with $K_{X'}+B^{\prime }_{i}\sim _{\mathbb {Q}}f^{\prime *}D^{\prime }_{i}$ . As in Lemma 5.5, we put

$$ \begin{align*} \mathcal P'=\left\{\left.\sum _{i=1}^{k} t_i B^{\prime}_i \, \right|\, 0\leq t_i\leq 1 \text{ for every } i \text{ with } \sum _{i=1}^{k}t_i=1\right\}. \end{align*} $$

We may assume that $f'\colon (X', \Delta )\to Y'$ is a basic ${\mathbb R}$ -slc-trivial fibration for every $\Delta \in \mathcal P'$ . We define $\mathcal P^{\prime }_{\mathbb {Q}}$ as

$$ \begin{align*} \mathcal P^{\prime}_{\mathbb{Q}}:=\left\{\left.\sum _{i=1}^{k} t_i B^{\prime}_i \,\right|\, t_i\in {\mathbb Q} \text{ and } 0\leq t_i\leq 1 \text{ for every } i \text{ with } \sum _{i=1}^{k}t_i=1\right\}. \end{align*} $$

Note that $B_{X'}\in \mathcal P'$ .

Pick any $\Delta =\sum _{i=1}^{k} t_i B^{\prime }_i\in \mathcal P^{\prime }_{\mathbb {Q}}$ . Since $\mu _{*}B^{\prime }_{i}=B_{i}$ , we have $\mu _{*}\Delta =\sum _{i=1}^{k} t_i B_i$ such that $t_{i}\in \mathbb {Q}$ . Therefore, the morphism $f\colon (X,\mu _{*}\Delta )\to Y$ is a basic ${\mathbb Q}$ -slc-trivial fibration such that $K_{X}+\mu _{*}\Delta \sim _{\mathbb {Q}}f^{*} \bigl (\sum _{i=1}^{k} t_i D_{i}\bigr )$ . Let ${\mathbf B}^{\Delta }$ and ${\mathbf M}^{\Delta }$ be the discriminant ${\mathbb Q}$ -b-divisor and the moduli ${\mathbb Q}$ -b-divisor of the basic ${\mathbb Q}$ -slc-trivial fibration $f\colon (X,\mu _{*}\Delta )\to Y$ , respectively. Because we have $\operatorname {\mathrm {Supp}} \bigl (\sum _{i=1}^{k} t_i D_{i}\bigr )\subset \operatorname {\mathrm {Supp}} D$ and $\operatorname {\mathrm {Supp}} \mu _{*}\Delta \subset \operatorname {\mathrm {Supp}} B$ , we may apply Theorem 4.1. Therefore, for every $\Delta \in \mathcal P^{\prime }_{\mathbb {Q}}$ , it follows that $\sigma ^{*}(K_{Y}+{\mathbf B}^{\Delta }_{Y}) =K_{Y'}+{\mathbf B}^{\Delta }_{Y'}$ and ${\mathbf M}^{\Delta }_Y$ is a potentially nef ${\mathbb Q}$ -divisor on Y with $\sigma ^{*}{\mathbf M}^{\Delta }_{Y}={\mathbf M}^{\Delta }_{Y'}$ . It also follows from the construction that $f'\colon (X',\Delta )\to Y'$ is the basic ${\mathbb Q}$ -slc-trivial fibration induced from $f\colon (X,\mu _{*}\Delta )\to Y$ such that $K_{X'}+\Delta \sim _{\mathbb {Q}}f^{\prime *}\bigl (\sum _{i=1}^{k} t_i D^{\prime }_{i}\bigr )$ . It is because $K_{X'}+\Delta =\mu ^{*}(K_{X}+\mu _{*}\Delta )$ by construction.

We apply Lemma 5.5 to $f'\colon (X',B_{X'})\to Y'$ and $\mathcal {P}'$ . Then, we can find $\Delta _{1},\dots , \Delta _{l}\in \mathcal P^{\prime }_{\mathbb {Q}}$ and positive real numbers $s_{1},\dots ,s_{l}$ such that:

• • $\sum _{j=1}^l s_j=1$ and $\sum _{j=1}^l s_j \Delta _j=B_{X'}$ , and

• • ${\mathbf B}_{Y'}=\sum _{j=1}^{l}s_{j}{\mathbf B}^{\Delta _{j}}_{Y'}$ .

Since ${\mathbf B}$ and ${\mathbf B}^{\Delta _{j}}$ are ${\mathbb R}$ -b-divisors, we have ${\mathbf B}_{Y}= \sum _{j=1}^{l}s_{j}{\mathbf B}^{\Delta _{j}}_{Y}$ . Then

$$ \begin{align*} \begin{aligned} \sigma^{*}(K_{Y}+{\mathbf B}_Y)&= \sigma^{*}\left(K_{Y}+\sum_{j=1}^{l}s_{j}{\mathbf B}^{\Delta_{j}}_{Y}\right) =\sum_{j=1}^{l}s_{j}\sigma^{*}(K_{Y}+{\mathbf B}^{\Delta_{j}}_{Y})\\ &=\sum_{j=1}^{l}s_{j}(K_{Y'}+{\mathbf B}^{\Delta_{j}}_{Y'}) =K_{Y'}+\sum_{j=1}^{l}s_{j}{\mathbf B}^{\Delta_{j}}_{Y'}\\ &=K_{Y'}+{\mathbf B}_{Y'}. \end{aligned} \end{align*} $$

Therefore, we have $\sigma ^{*}(K_{Y}+{\mathbf B}_Y) =K_{Y'}+{\mathbf B}_{Y'}$ . Recalling that $\sigma \colon Y' \to Y$ is an arbitrary projective birational morphism, we see that part (i) of Theorem 5.1 holds, that is,

$$ \begin{align*} {{\mathbf K}} + {{\mathbf B}} = \overline {{{\mathbf K}}_Y + {{\mathbf B}}_Y}. \end{align*} $$

As in the third paragraph, for each j, we define $D^{\prime }_{\Delta _{j}}$ to be the $\mathbb {Q}$ -divisor on $Y'$ associated with the basic ${\mathbb Q}$ -slc-trivial fibration $f'\colon (X',\Delta _{j})\to Y'$ . Note that $K_{X'}+\Delta _{j} \sim _{\mathbb {Q}}f^{\prime *}D^{\prime }_{\Delta _{j}}$ for all j. Since $\sum _{j=1}^l s_j=1$ and $\sum _{j=1}^l s_j \Delta _j=B_{X'}$ , we have $\sigma ^{*}D=\sum _{j=1}^{l}s_{j}D^{\prime }_{\Delta _{j}}$ . By the relation ${\mathbf B}_{Y'}= \sum _{j=1}^{l}s_{j}{\mathbf B}^{\Delta _{j}}_{Y'}$ and the definition of the moduli ${\mathbb R}$ -b-divisors (see Definition 3.5), we have

$$ \begin{align*}{\mathbf M}_{Y'}=\sum_{j=1}^{l}s_{j} {\mathbf M}^{\Delta_{j}}_{Y'}\qquad \text{and} \qquad {\mathbf M}_{Y}=\sum_{j=1}^{l}s_{j}{\mathbf M}^{\Delta_{j}}_{Y}.\end{align*} $$

Then ${\mathbf M}_Y$ is a potentially nef ${\mathbb R}$ -divisor on Y and

$$ \begin{align*} \begin{aligned} \sigma^{*}{\mathbf M}_{Y}&= \sigma^{*}\left(\sum_{j=1}^{l}s_{j} {\mathbf M}^{\Delta_{j}}_{Y}\right)=\sum_{j=1}^{l}s_{j} {\mathbf M}^{\Delta_{j}}_{Y'}={\mathbf M}_{Y'}. \end{aligned} \end{align*} $$

Here, we used $\sigma ^{*}{\mathbf M}^{\Delta _{j}}_{Y} ={\mathbf M}^{\Delta _{j}}_{Y'}$ for every j, which follows from the third paragraph. We complete the proof.

Sketch of Proof

It can be proved by Theorem 5.1 and Lemmas 5.4 and 5.5. We only outline the proof.

We note that the properties of Theorem 5.6 except the last two properties correspond to parts (1)–(6) of Lemma 5.4, respectively. By Lemma 5.4, we can find $\mathbb {Q}$ -divisors $\widetilde {B}_{1},\dots , \widetilde {B}_{k}$ on X, $\mathbb {Q}$ -Cartier ${\mathbb Q}$ -divisors $\widetilde {D}_{1},\dots , \widetilde {D}_{k}$ on Y, and positive real numbers $\widetilde {r}_{1},\dots , \widetilde {r}_{k}$ satisfying parts (1)–(6) of Lemma 5.4. Then $\widetilde {B}_{i}$ , $\widetilde {D}_{i}$ , and $\widetilde {r}_{i}$ satisfy all the properties of Theorem 5.6 except the last two properties. More specifically, $\widetilde {B}_{i}$ , $\widetilde {D}_{i}$ , and $\widetilde {r}_{i}$ satisfy:

• • $\sum _{i=1}^k \widetilde {r}_i=1$ with $\sum _{i=1}^k \widetilde {r}_i \widetilde {B}_i=B$ and $\sum _{i=1}^k \widetilde {r}_i \widetilde {D}_i =D$ (see part (1) of Lemma 5.4),

• • $\operatorname {\mathrm {Supp}} B=\operatorname {\mathrm {Supp}} \widetilde {B}_i$ and $\operatorname {\mathrm {Supp}} D=\operatorname {\mathrm {Supp}} \widetilde {D}_i$ hold for every i, and

• • $K_X+\widetilde {B}_i\sim _{{\mathbb Q}} f^*\widetilde {D}_i$ holds for every i (see part (6) of Lemma 5.4),

and parts (2)–(5) in Lemma 5.4. We take a smooth higher model $\sigma \colon Y'\to Y$ so that the induced basic $\mathbb {R}$ -slc-trivial fibration $f'\colon (X',B')\to Y'$ satisfies the property that there exists a simple normal crossing divisor $\Sigma '$ on $Y'$ such that $\operatorname {\mathrm {Supp}} \sigma ^{*}D\subset \Sigma '$ and that every stratum of $(X', \operatorname {\mathrm {Supp}} B')$ is smooth over $Y'\setminus \Sigma '$ . The morphism $X'\to X$ is denoted by $\mu $ . For each $1\leq i\leq k$ , let $\widetilde {B}^{\prime }_i$ be a $\mathbb {Q}$ -divisor on $X'$ defined by $K_{X'}+\widetilde {B}^{\prime }_i=\mu ^{*}(K_{X}+\widetilde {B}_i)$ . Note that $K_{X'}+\widetilde {B}^{\prime }_i\sim _{\mathbb {Q}}f^{\prime *}\sigma ^{*}\widetilde {D}_{i}$ . We may assume that $\operatorname {\mathrm {Supp}} \sigma ^{*}\widetilde {D}_{i}\subset \Sigma '$ and that every stratum of $(X', \operatorname {\mathrm {Supp}} \widetilde {B}^{\prime }_i)$ is smooth over $Y'\setminus \Sigma '$ for every i by taking $\sigma \colon Y'\to Y$ suitably. We define

$$ \begin{align*} \mathcal P= \left\{\left.\sum _{i=1}^{k} t_i \widetilde{B}^{\prime}_i \, \right|\, 0\leq t_i\leq 1 \text{ for every } i \text{ with } \sum _{i=1}^{k}t_i=1\right\}. \end{align*} $$

By Lemma 5.5, we can find $B^{\prime }_{1},\dots , B^{\prime }_{l}\in \mathcal P$ , which are $\mathbb {Q}_{\geq 0}$ -linear combinations of $\widetilde {B}^{\prime }_{1},\dots , \widetilde {B}^{\prime }_{l}$ and positive real numbers $r_{1},\dots ,r_{l}$ such that:

• • $\sum _{j=1}^l r_j=1$ and $\sum _{j=1}^l r_j B^{\prime }_j=B'$ , and

• • ${\mathbf B}_{Y'}=\sum _{j=1}^{l}r_{j}{\mathbf B}_{jY'}$ .

Here, ${\mathbf B}_j$ is the discriminant $\mathbb {Q}$ -b-divisor associated with $f'\colon (X',B^{\prime }_{j})\to Y'$ . By Theorem 5.1, we have ${\mathbf K}+{\mathbf B}= \overline {{\mathbf K}_{Y'}+{\mathbf B}_{Y'}}$ and ${\mathbf K}+ {\mathbf B}_j=\overline {{\mathbf K}_{Y'}+{\mathbf B}_{jY'}}$ . We put $B_j=\mu _{*}B^{\prime }_j$ for each $1\leq j \leq l$ . Then we can find $\mathbb {Q}$ -divisors $D_{1},\dots , D_{l}$ on Y such that $K_{X}+B_{j}\sim _{\mathbb {Q}}f^{*}D_{j}$ and $\sum _{j=1}^{l}r_{j}D_{j}=D$ . By construction, we can easily see that $B_1\ldots , B_l$ , $D_1, \ldots , D_l$ , and $r_1, \ldots , r_l$ constructed above satisfy the desired properties.

Proof of Theorem 1.2

From Step 1 to Step 3, we define a natural quasi-log scheme structure on Z. This part is essentially contained in [Reference Fujino5, Chap. 6] and [Reference Fujino7].

Step 1. In this step, we give a natural quasi-log scheme structure on $W':=W\cup \operatorname {\mathrm {Nlc}}(X, \Delta )$ . This step is essentially the adjunction for quasi-log schemes (see [Reference Fujino5, Th. 6.3.5(i)]).

We put $W':=W\cup \operatorname {\mathrm {Nlc}}(X, \Delta )$ as above. We sketch how to define a natural quasi-log scheme structure on $W'$ . Let $f\colon Y\to X$ be a projective birational morphism from a smooth quasi-projective variety Y such that $K_Y+\Delta _Y=f^*(K_X+\Delta )$ and that $\operatorname {\mathrm {Supp}} \Delta _Y$ is a simple normal crossing divisor on Y. By taking some more blowups, we may assume that the union of all log canonical centers of $(Y, \Delta _Y)$ mapped to $W'$ by f, which is denoted by $V'$ , is a union of some irreducible components of $\Delta ^{=1}_Y$ . As usual, we put $A=\lceil -(\Delta ^{<1}_Y)\rceil $ and $N=\lfloor \Delta ^{>1}_Y\rfloor $ and consider the following short exact sequence:

$$ \begin{align*}0\to \mathcal O_Y(A-N-V')\to \mathcal O_Y(A-N)\to \mathcal O_{V'}(A-N)\to 0. \end{align*} $$

By taking $R^if_*$ , we obtain

$$ \begin{align*} \begin{aligned} 0&\longrightarrow f_*\mathcal O_Y(A-N-V')\longrightarrow f_*\mathcal O_Y(A-N)\longrightarrow f_*\mathcal O_{V'}(A-N)\\ &\overset{\delta}{\longrightarrow} R^1f_*\mathcal O_Y(A-N-V')\longrightarrow \cdots. \end{aligned} \end{align*} $$

The connecting homomorphism $\delta $ is zero since no associated prime of $R^1f_*\mathcal O_Y(A-N-V')$ is contained in $W'=f(V')$ (see [Reference Fujino4, Th. 6.3 (i)] and [Reference Fujino5, Th. 5.6.2(i)]). Hence, we have

$$ \begin{align*}0\to f_*\mathcal O_Y(A-N-V')\to f_*\mathcal O_Y(A-N)\to f_*\mathcal O_{V'}(A-N)\to 0. \end{align*} $$

Note that $\mathcal J_{\operatorname {\mathrm {NLC}}}(X, \Delta )=f_*\mathcal O_Y(A-N)$ by definition. We put $\mathcal I_{W'}=f_*\mathcal O_Y(A-N-V')$ and $\mathcal I_{W^{\prime }_{-\infty }}=f_*\mathcal O_{V'}(A-N)$ . We define $\Delta _{V'}$ by $(K_Y+\Delta _Y)|_{V'}=K_{V'}+ \Delta _{V'}$ . Then

$$ \begin{align*}\left(W', (K_X+\Delta)|_{W'}, f\colon (V', \Delta_{V'})\to W'\right) \end{align*} $$

is a quasi-log scheme. By construction, $\operatorname {\mathrm {Nqlc}}(W', (K_X+\Delta )|_{W'})=\operatorname {\mathrm {Nlc}}(X, \Delta )$ holds. By construction again, a subset $C\subset X$ is a qlc stratum of $[W', (K_X+\Delta )|_{W'}]$ if and only if C is a log canonical center of $(X, \Delta )$ included in W. We note that the above construction is independent of the choice of $f\colon Y\to X$ by [Reference Fujino5, Prop. 6.3.1].

Step 2. In this step, we give a natural quasi-log scheme structure on $[W, (K_X+\Delta )|_W]$ . This step is essentially [Reference Fujino7, Lem. 4.19].

In Step 1, we may further assume that the union of all strata of $(V', \Delta _{V'})$ mapped to $W\cap \operatorname {\mathrm {Nlc}}(X, \Delta )$ is also a union of some irreducible components of $V'$ . Let $\widehat V$ be the union of the irreducible components of $V'$ mapped to W by f. We put $\Delta _{\widehat V}$ by $(K_Y+\Delta _Y)|_{\widehat V} =K_{\widehat V}+\Delta _{\widehat V}$ . Then, by the proof of [Reference Fujino7, Lems. 4.18 and 4.19],

$$ \begin{align*}\left(W, (K_X+\Delta)|_W, f\colon (\widehat V, \Delta_{\widehat V}) \to W\right) \end{align*} $$

is a quasi-log scheme. By [Reference Fujino7, Lem. 4.19], we obtain that $\mathcal I_{W_{-\infty }}=\mathcal I_{W^{\prime }_{-\infty }}$ holds and that a subset $C\subset X$ is a qlc stratum of $[W', (K_X+\Delta )|_{W'}]$ if and only if C is a qlc stratum of $[W, (K_X+\Delta )|_W]$ . Hence, $W\cap \operatorname {\mathrm {Nlc}}(X, \Delta )=W_{-\infty }$ and

$$ \begin{align*}W\cap \left(\operatorname{\mathrm{Nlc}}(X, \Delta)\cup \bigcup _{W\not\subset W^{\dagger}}W^{\dagger}\right) =\operatorname{\mathrm{Nqklt}}(W, (K_X+\Delta)|_W)\end{align*} $$

hold set-theoretically, where $W^{\dagger }$ runs over log canonical centers of $(X, \Delta )$ which do not contain W.

Step 3. In this step, we give a natural quasi-log scheme structure on Z. This step is nothing but [Reference Fujino7, Th. 1.9].

In Step 2, we may further assume that the union of all strata of $(\widehat V, \Delta _{\widehat V})$ mapped to $\operatorname {\mathrm {Nqklt}}(W, (K_X+\Delta )|_W)$ is a union of some irreducible components of $\widehat V$ . Let V be the union of the irreducible components of $\widehat V$ which are dominant onto W. Then, by the proof of [Reference Fujino7, Th. 1.9], $f\colon V\to W$ factors through Z and

$$ \begin{align*}\left(Z, \nu^*(K_X+\Delta), f\colon (V, \Delta_V)\to Z\right) \end{align*} $$

becomes a quasi-log scheme, where $\Delta _V$ is defined by $(K_Y+\Delta _Y)|_V =K_V+\Delta _V$ . By construction, we have $\nu _*\mathcal I_{\operatorname {\mathrm {Nqklt}}(Z, \nu ^*(K_X+\Delta ))}= \mathcal I_{\operatorname {\mathrm {Nqklt}}(W, (K_X+\Delta )|_W)}$ . Hence,

$$ \begin{align*} \operatorname{\mathrm{Nqklt}}(Z, \nu^*(K_X+\Delta))=\nu^{-1}\operatorname{\mathrm{Nqklt}}(W, (K_X+\Delta)|_W) \end{align*} $$

holds.

Step 6. (see [Reference Fujino and Hashizume11, Th. 5.4]).

In this final step, we prove (iv). This step is essentially [Reference Fujino and Hashizume11, Th. 5.4]. We explain it here for the reader’s convenience.

Without loss of generality, we may assume that X is affine by taking a finite affine open cover of X. Let $g_{dlt}\colon X_{dlt}\to X$ be a good dlt blowup of $(X, \Delta )$ such that $K_{X_{dlt}}+\Delta _{X_{dlt}}= g^*_{dlt}(K_X+\Delta )$ (see [Reference Fujino and Hashizume11, Lem. 3.5]). We may assume that there is an irreducible component S of $\Delta ^{=1}_{X_{dlt}}$ with $g_{dlt}(S)=W$ . We put

$$ \begin{align*} D=\Delta^{\geq 1}_{X_{dlt}}-\operatorname{\mathrm{Supp}} \Delta^{\geq 1}_{X_{dlt}}= \Delta^{> 1}_{X_{dlt}}-\operatorname{\mathrm{Supp}} \Delta^{>1}_{X_{dlt}}. \end{align*} $$

Then $-D$ is semi-ample over X and $\operatorname {\mathrm {Supp}} D=\operatorname {\mathrm {Nlc}}(X_{dlt}, \Delta _{X_{dlt}})$ holds set-theoretically (see [Reference Fujino and Hashizume11, Lem. 3.5]). By taking the contraction morphism $\varphi \colon X_{dlt}\to X_{lc}$ associated with $-D$ over X, we get a log canonical modification $g_{lc}\colon X_{lc}\to X$ with $K_{X_{lc}}+ \Delta _{X_{lc}}=g^*_{lc}(K_X+\Delta )$ (see [Reference Fujino and Hashizume11, Th. 1.3]).

We put $D'=\varphi _*D$ . Then $-D'$ is ample over X, and

$$ \begin{align*} g^{-1}_{lc}\operatorname{\mathrm{Nlc}}(X, \Delta) =\operatorname{\mathrm{Nlc}}(X_{lc}, \Delta_{X_{lc}})=\operatorname{\mathrm{Supp}} D' \end{align*} $$

holds set-theoretically. We note that

$$ \begin{align*} \operatorname{\mathrm{Nlc}}(X_{dlt}, \Delta_{X_{dlt}}) =\varphi^{-1}\operatorname{\mathrm{Nlc}}(X_{lc}, \Delta_{X_{lc}})=g^{-1}_{dlt}\operatorname{\mathrm{Nlc}}(X, \Delta) \end{align*} $$

holds set-theoretically. Let $\tilde {W}$ be the strict transform of W on $X_{lc}$ . Let $\tilde \nu \colon \tilde Z \to \tilde W$ be the normalization. Then we can easily see that

$$ \begin{align*} \operatorname{\mathrm{Supp}} {\mathbf B}^{>1}_{\tilde Z} =\widetilde \nu ^*D'=\tilde{\nu}^{-1}\left(\operatorname{\mathrm{Nlc}} (X_{lc}, \Delta_{X_{lc}}) \cap\tilde W\right) = (g_{lc}\circ \tilde \nu)^{-1}\left(\operatorname{\mathrm{Nlc}}(X, \Delta)\cap W\right) \end{align*} $$

holds set-theoretically. We note that ${\mathbf B}^{>1}=0$ over $X\setminus \operatorname {\mathrm {Nlc}}(X, \Delta )$ by construction. Hence, we obtain $\nu \circ p({\mathbf B}^{>1}_{Z'}) =W\cap \operatorname {\mathrm {Nlc}}(X, \Delta )$ set-theoretically.

We finish the proof of Theorem 1.2.

Proof of Theorem 1.1

Here, we use the same notation as in Theorem 1.2. We put $B_Z={\mathbf B}_Z$ and $M_Z={\mathbf M}_Z$ in Theorem 1.2. We note that ${\mathbf M}_{Z'}$ is a finite ${\mathbb R}_{>0}$ -linear combination of potentially nef Cartier divisors on $Z'$ with $p_*{\mathbf M}_{Z'}=M_Z$ . Hence, the desired statement follows from Theorem 1.2.

Proof of Corollary 1.4

By the definition of ${\mathbf B}$ in Theorem 1.2 (see the proof of Theorem 1.2 and Definition 1.3), we can easily check that $B_Z$ is nothing but Shokurov’s different (see [Reference Fujino4, §14]) and $\nu ^*(K_X+\Delta )=K_Z+B_Z$ holds, where $\nu \colon Z\to W$ is the normalization of W. In particular, we have $M_Z=0$ . By part (A) in Theorem 1.1, we obtain that $(X, \Delta )$ is log canonical in a neighborhood of W if and only if $(Z, B_Z)$ is log canonical in the usual sense. It recovers Kawakita’s inversion of adjunction (see [Reference Kawakita15, Th.]). By part (B), we see that $(Z, B_Z)$ is Kawamata log terminal if and only if $(X, \Delta )$ is log canonical in a neighborhood of W and W is a minimal log canonical center of $(X, \Delta )$ (see [Reference Fujino4, Th. 9.1] and [Reference Fujino5, Th. 6.3.11]). Note that $(X, \Delta )$ is purely log terminal in a neighborhood of W if and only if $(X, \Delta )$ is log canonical in a neighborhood of W and W is a minimal log canonical center of $(X, \Delta )$ .