Proof. The assertion follows from [Reference de Fernex and Hacon7, Lemma 2.7] and [Reference de Fernex and Hacon7, Remark 2.13].

Proof. This is an immediate application of Lemma 2.9.

Proof. Take a log resolution $f: Y \to X$ of $(X, \Delta +D, \mathfrak {a} I)$ separating the components of D, and let F and G be Cartier divisors on Y, such that $\mathcal {O}_Y(-F)= \mathfrak {a} \mathcal {O}_Y$ and $\mathcal {O}_Y(-G)=I \mathcal {O}_Y$ . For all integers $m \geqslant 1$ , such that $m\Delta $ is an integral Weil divisor, we define the $\mathbb {R}$ -Weil divisors $(\Delta +D)_{Y}^+$ and $(\Delta +D)_{m,Y}^+$ on Y as

$$ \begin{align*} (\Delta+D)_{Y}^+ &: = -f^{*}(-(K_X+\Delta+D)) - K_Y=-\sum_E a_E^+(X, \Delta+D)E,\\ (\Delta+D)_{m,Y}^+ &: = -\frac{f^{\natural}(-m(K_X+\Delta+D))}{m} - K_Y=-\sum_E a_{m,E}^+(X, \Delta+D)E, \end{align*} $$

where E runs through all prime divisors on Y.

To prove the “only if” part, it suffices to show by Lemma 2.9 that there exists a real number $\varepsilon>0$ , such that

$$\begin{align*}\operatorname{ord}_E((\Delta+D)_{m,Y}^{+} -f^{-1}_*D + \lambda F + \varepsilon G ) <1 \end{align*}$$

for every integer $m \geqslant 1$ with $m\Delta $ an integral Weil divisor and for every prime divisor E on Y. Since $(X, \Delta +D, \mathfrak {a}^\lambda )$ is valuatively plt along D,

$$\begin{align*}\operatorname{ord}_E((\Delta+D)_Y^{+} -f^{-1}_*D+ \lambda F )<1 \end{align*}$$

for every prime divisor E on Y. Therefore, there exists $\varepsilon> 0$ , such that

$$\begin{align*}\operatorname{ord}_E((\Delta+D)_Y^{+} -f^{-1}_*D + \lambda F + \varepsilon G)<1 \end{align*}$$

for all prime divisors E on Y. Then we have

$$\begin{align*}\operatorname{ord}_E((\Delta+D)_{m,Y}^{+} -f^{-1}_*D + \lambda F + \varepsilon G ) \leqslant \operatorname{ord}_E((\Delta+D)_Y^{+} -f^{-1}_*D+ \lambda F + \varepsilon G)<1. \end{align*}$$

For the “if” part, we fix a prime divisor E on Y. It is enough to show by Lemma 2.9 that

$$\begin{align*}\operatorname{ord}_E((\Delta+D)_{Y}^{+} -f^{-1}_*D + \lambda F ) <1. \end{align*}$$

If $K_X+\Delta +D$ is $\mathbb {Q}$ -Cartier at the center of E, then this inequality follows from Remark 2.8. Therefore, we may assume that $K_X+\Delta +D$ is not $\mathbb {Q}$ -Cartier at the center of E. Then by the definition of I, the center of E is contained in the zero locus of I, which implies that $\operatorname {ord}_E(G)>0$ . Since $(X, \Delta +D, \mathfrak {a}^\lambda I^\varepsilon )$ is m-weakly valuatively plt along D for all $m \geqslant 1$ , such that $m\Delta $ is an integral Weil divisor,

$$\begin{align*}\operatorname{ord}_E((\Delta+D)_{m,Y}^{+} -f^{-1}_*D + \lambda F ) <1 - \varepsilon \operatorname{ord}_E(G). \end{align*}$$

Taking the supremum over all such m, we have

$$\begin{align*}\operatorname{ord}_E((\Delta+D)_{Y}^{+} -f^{-1}_*D + \lambda F ) \leqslant 1 - \varepsilon \operatorname{ord}_E(G) < 1.\\[-34pt] \end{align*}$$

Proof. Since the pullback of a $\mathbb {Q}$ -Weil divisor on a surface, defined in Definition 2.5, coincides with Mumford’s numerical pullback, the pair $(X,\Delta )$ is numerically lc at x (see [Reference Kollár and Mori21, Section 4.1] for the definition of numerically lc pairs). The assertion then follows from [Reference Kollár and Mori21, Proposition 4.11 (2)].

Proof. We write

$$\begin{align*}D= \sum_{i=1}^n a_i E_i - \sum_j^m b_j E_j, \end{align*}$$

where $E_i$ and $E_j$ are prime divisors on X whose generic points are regular points of X and $a_i$ and $b_j$ are positive integers. Let $\mathcal {G}$ be the coherent sheaf

$$\begin{align*}\mathcal{H}om_{X}\left( \bigotimes_i \mathcal{I}_{E_i}^{\otimes a_i}, \Big(\bigotimes_j \mathcal{I}_{E_j}^{\otimes b_j}\Big)^{**}\right), \end{align*}$$

where $\mathcal {I}_{E_i}$ (respectively, $\mathcal {I}_{E_j}$ ) is the ideal sheaf of $E_i$ (respectively, $E_j$ ) and $(-)^{**}$ denotes the reflexive hull. Note by [Reference Schwede29, Corollary 2.9] that $\mathcal {G}$ is reflexive.Footnote ⁴

Let $j: V \hookrightarrow U$ be the open immersion from an open subset $V \subseteq U$ containing all codimension one points of X, such that $E_i|_V$ (respectively, $E_j|_V\kern-1pt)$ is Cartier for all i (respectively, j). Since $\mathcal {G}|_V \cong \mathcal {O}_V(E_U |_V)$ , it follows from Lemma 2.15 that

$$\begin{align*}\mathcal{G} \cong (i \circ j)_* (\mathcal{G}|_V) \cong i_* j_* \mathcal{O}_V(E_U|_V) \cong i_* \mathcal{O}_U(E_U) = \mathcal{O}_X(D).\\[-34pt] \end{align*}$$

Proof. The proof is very similar to the argument in [Reference Schwede29, Section 2] and [Reference Stacks31, Section 0AUY], which can be traced back to [Reference Hartshorne12] and [Reference Hartshorne13].

Proof. It follows from Lemmas A.16 and A.17.

Setting 2.19. Let $(Y, W, W', i, \mu , f, \Delta )$ be a tuple satisfying the following conditions.

1. 1. Y is an excellent reduced $(S_2)$ and $(G_1)$ scheme over S admitting a canonical divisor $K_Y \in \mathrm {WDiv}^*(Y)$ associated to $\omega _Y^{\bullet }:= \pi _Y^! \omega _S^{\bullet }$ , where $\pi _Y:Y \to S$ is the structure morphism.

2. 2. $i: W \hookrightarrow Y$ is the closed immersion from a reduced closed subscheme W whose generic points are codimension one regular points of Y. In particular, $W \in \mathrm {WDiv}^*(Y)$ .

3. 3. $\mu : W' \to W$ is a finite birational morphism from a reduced $(S_2)$ and $(G_1)$ scheme $W'$ and $f: = i \circ \mu : W' \to Y$ is the composite of i and $\mu $ .

   

4. 4. $\Delta \in \mathrm {WDiv}^*_{\mathbb {Q}}(Y)$ is a $\mathbb {Q}$ -Weil divisor on Y, such that the support of $\Delta $ has no common components with W and $K_Y+ \Delta +W \in \mathrm {WDiv}_{\mathbb {Q}}^*(Y)$ is $\mathbb {Q}$ -Cartier at every codimension one point w of W.

5. 5. For each codimension one singular point $w'$ of $W'$ , there exists an open neighborhood $U \subseteq Y$ of $f(w') \in Y$ , such that $\Delta |_U=0$ and $W|_U$ is Cartier, that is, $W|_U$ is contained in the image of the natural injection $\mathrm {Div}^*(U) \to \mathrm {WDiv}^*(U)$ .

Proof. This is a special case of Lemma A.10.

Proof. We write $\mathcal {A}' : =(Y^n, W", W', j, \pi , g , \nu ^*\Delta +C_Y)$ , where $g: W' \to Y^n$ is the morphism induced by f, $W" \subseteq Y^n$ is the reduced image of g, and j, $\pi $ , and $\rho $ are natural morphisms, such that the following diagram commutes:

It is clear that $\mathcal {A}'$ satisfies the conditions (1)–(3) in Setting 2.19. The tuple $\mathcal {A}'$ also satisfies (4), because $\nu $ is an isomorphism over the generic points of W, and therefore, $\nu ^*W=W"$ . By the assumption that Y is normal at the image of every codimension one singular point $w'$ of $W'$ , the conductor divisor $C_Y$ is trivial near $g(w')$ , which implies that $\mathcal {A}'$ satisfies the condition (5) too. Then the assertion is a special case of Lemma A.11.

Proof. This is just a reformulation of Lemma A.8.

Setting 2.26. Suppose that k is an algebraically closed field of characteristic zero, X is a reduced $(S_2)$ and $(G_1)$ scheme of finite type over k, T is an irreducible scheme over k with generic point $\eta $ , and $t \in T$ is a closed point. Let $(\mathcal {X},i)$ be a deformation of X over T with reference point t, such that $\mathcal {X}$ is a reduced $(S_2)$ and $(G_1)$ scheme. Let $\mathcal {D} \in \mathrm {WDiv}^*_{\mathbb {Q}}(\mathcal {X})$ be an effective $\mathbb {Q}$ -Weil divisor on $\mathcal {X}$ whose support does not contain any generic points of the closed fiber X nor any singular codimension one points of X. Let $\mathfrak {a} \subseteq \mathcal {O}_{\mathcal {X}}$ be a coherent ideal sheaf, such that $\mathfrak {a} \mathcal {O}_X$ is nonzero, and let $\lambda> 0$ be a real number.

Proof. The proof is straightforward.

Proof. The proof is similar to that of [Reference Fujino9, Lemma 2.7].

Proof. All the assertions immediately follow from Lemma 4.2.

Proof. We use the notation established in Theorem 4.4.

(1) Since $f_* W_Y = W$ , $f_* \Delta _Y = \Delta $ , and $f_* F =G$ , one has

$$\begin{align*}f_* (\Theta^{W_Y}(\Delta_Y + \lambda F )) = \Theta^W(\Delta + G), \end{align*}$$

which implies the first inclusion ${\mathcal {I}^{W}}(X, \Delta , \mathfrak {a}^\lambda ) \subseteq \mathcal {O}_X(- \lfloor \Theta ^W(\Delta + \lambda G) \rfloor )$ . The second inclusion is shown similarly.

(2) First note that the fractional ideals ${\mathcal {I}^{W}}(X, \Delta , \mathfrak {a}^\lambda )$ and ${\mathcal {I}_{D}^{W}}(X, \Delta , \mathfrak {a}^\lambda )$ are ideals in $\mathcal {O}_X$ by (1) and the assumption that W is contained in the support of $\Delta +G$ . Therefore, (a) (respectively, (b)) holds if and only if $\lfloor \Theta ^{W_Y}(\Delta _Y + \lambda F ) \rfloor \leqslant 0$ (respectively, $\lfloor \Theta ^{W_Y}_0(\Delta _Y + \lambda F) \rfloor \leqslant 0$ ). It is easy to see that these inequalities are equivalent to (c).

Proof. (1) The assertion follows from arguments similar to the proof of [Reference Sato and Takagi28, Proposition 4.1] (1) by replacing [Reference Lazarsfeld23, Proposition 9.2.31] with Lemma 4.7 (2).

(2) Assume to the contrary that there exists a log resolution $f: Y \to X$ of $(X, A+W+\Delta , \mathfrak {a},\mathcal {O}_X(-mB))$ and a prime divisor E on Y, such that $a_{m,E}^+(X, \Delta , \mathfrak {a}^\lambda )<-1$ . We write $\mathfrak {a} \mathcal {O}_Y= \mathcal {O}_Y(-F_a)$ and $\mathcal {O}_X(-mB) \mathcal {O}_Y = \mathcal {O}_Y(-F_b)$ . Since $K_X+A$ is Cartier, we have

$$ \begin{align*} \mathcal{O}_X(m(K_X+\Delta)) \mathcal{O}_Y &= \mathcal{O}_X(m(K_X+A)-m B) \mathcal{O}_Y\\ &=\mathcal{O}_Y(m(K_Y+A_Y) -F_b), \end{align*} $$

where $A_Y : = f^*(K_X+A)- K_Y$ . Then

$$ \begin{align*} a_{m, E}^+ (X, \Delta, \mathfrak{a}^{\lambda}) &= \operatorname{ord}_E ( K_Y + \frac{1}{m}(-m(K_Y+A_Y)+ F_b) -\lambda F_a)\\ &= \operatorname{ord}_E ( -A_Y - \lambda F_a + \frac{1}{m}F_b)\\ &= \operatorname{ord}_E(- A_Y - \lambda F_a - \frac{m-1}{m} F_b) + \operatorname{ord}_E(F_b). \end{align*} $$

By the assumption that $a_{m,E}^+(X, \Delta , \mathfrak {a}^{\lambda }) <-1$ , one has

$$\begin{align*}\operatorname{ord}_E(\lfloor \Theta^{W_Y}(A_Y+ \lambda F + \frac{m-1}{m} F_b) \rfloor)> \operatorname{ord}_E(F_b), \end{align*}$$

where $W_Y$ is the reduced divisor on Y whose support is the union of the strict transform $f^{-1}_*W$ of W and the exceptional locus $\mathrm {Exc}(f)$ of f. Therefore,

$$ \begin{align*} {\mathcal{I}^{W}}(X, A, \mathfrak{a}^{\lambda} \mathcal{O}_X(-m B)^{1-1/m}) &= f_* \mathcal{O}_Y(- \lfloor \Theta^{W_Y}(A_Y+ \lambda F_a + \frac{m-1}{m} F_b) \rfloor)\\ &\subsetneq f_* \mathcal{O}_Y(-F_b)\\ &= \mathcal{O}_X(-m B), \end{align*} $$

where the strict containment on the second line follows from the fact that $\mathcal {O}_Y(-F_b)$ is f-free. This is a contradiction.

(3) First note that $mB$ is Cartier by assumption, and we set $\mathfrak {q} : = \mathfrak {b} \otimes \mathcal {O}_X(m B)$ . It then follows from Lemma 4.7 (2) that the inclusion in (3) is equivalent to the inclusion $\mathfrak {q} \subseteq {\mathcal {I}_{0}^{W}} (X, \Delta , \mathfrak {a}^{\lambda } \mathfrak {q}^{1-1/m})$ . Take a log resolution $f: Y \to X$ of $(X, W, \mathfrak {a} \mathfrak {q})$ with $\mathfrak {a} \mathcal {O}_Y=\mathcal {O}_Y(-F_a)$ and $\mathfrak {q} \mathcal {O}_Y=\mathcal {O}_Y(-F_q)$ . Since $(X, \Delta , \mathfrak {a}^{\lambda })$ is lc, all the coefficients of $ \Delta _Y+ \lambda F_a$ are less than or equal to one, where $\Delta _Y:= f^*(K_X+\Delta )-K_Y$ . Noting that $F_q$ is an effective integral divisor on Y, we have

$$\begin{align*}\lfloor \Theta^{W_Y}_0( \Delta_Y+ \lambda F_a + \frac{m-1}{m} F_q) \rfloor \leqslant F_q, \end{align*}$$

where $W_Y$ is the reduced divisor on Y whose support is the union of the strict transform $f^{-1}_*W$ of W and the exceptional locus $\mathrm {Exc}(f)$ of f. Therefore,

$$\begin{align*}\mathfrak{q} \subseteq f_* \mathcal{O}_Y(-F_q) \subseteq {\mathcal{I}_{0}^{W}} (X, \Delta, \mathfrak{a}^{\lambda} \mathfrak{q}^{1-1/m}). \end{align*}$$

Proof. Let V be the complement of U in Z. Take a log resolution $f : Y \to X$ of $(X, \mathfrak {a}, \mathfrak {b})$ with $\mathfrak {a} \mathcal {O}_Y= \mathcal {O}_Y(-F_a)$ and $\mathfrak {b} \mathcal {O}_Y= \mathcal {O}_Y(-F_b)$ , such that $f^{-1}(V)$ is a closed subset of pure codimension one in Y and that the union of $f^{-1}(\operatorname {Supp} A)$ , $f^{-1}(Z)$ , $f^{-1}(W)$ , $f^{-1}(V)$ , the support of the divisor $F_a+F_b$ , and the exceptional locus $\mathrm {Exc}(f)$ of f is a simple normal crossing divisor on Y. Let g and $g'$ denote the induced morphisms $g: \widetilde {Z} \to Z^n$ and $g': \widetilde {Z} \to Z$ , respectively, where $\widetilde {Z}$ is the strict transform of Z on Y. Let $W_{Z^n}$ be the union of the support of $D+\mathrm {Diff}_{Z^n}(A)$ and all the codimension one irreducible components of the closed subscheme of $Z^n$ defined by $\mathfrak {a} \mathfrak {b} S^n$ . After replacing Y by its blowing up along $g^{-1}(W_{Z^n}) \subseteq Y$ , we may assume that $g^{-1}(W_{Z^n}) \subseteq \operatorname {Exc}(f)$ . Then $g: \widetilde {Z} \to Z^n$ is a log resolution of $(Z^n, \mathrm {Diff}_{Z^n}(A)+W_{Z^n}, (\mathfrak {a} S^n) (\mathfrak {b} S^n))$ with $(\mathfrak {a} S^n) \mathcal {O}_{\widetilde {Z}}= \mathcal {O}_{\widetilde {Z}}(-F_a|_{\widetilde {Z}})$ and $(\mathfrak {b} S^n) \mathcal {O}_{\widetilde {Z}}= \mathcal {O}_{\widetilde {Z}}(-F_b|_{\widetilde {Z}})$ .

Let $W_Y$ (respectively, $W_{\widetilde {Z}}$ ) be the reduced divisor on Y (respectively, $\widetilde {Z}$ ) whose support is the union of the strict transform $f^{-1}_* W$ (respectively, $g^{-1}_* W_{Z^n}$ ) and the exceptional locus of f (respectively, $g'$ ). Since $g^{-1}(W_{Z^n})$ is contained in the exceptional locus of f, we have $W_Y|_{\widetilde {Z}} \geqslant W_{\widetilde {Z}}$ . We decompose $W_Y=W_Y^1 + W_Y^2$ as follows:

1. (a) $f(W_Y^1) \subset V$ ,

2. (b) no irreducible components of $W_Y^2$ are mapped into V by f.

Set $\Gamma : = f^*(K_X+A+Z)- K_Y- \widetilde {Z}$ . Since $\Gamma |_{\widetilde {Z}} = g^*( K_{Z^n}+ \mathrm {Diff}_{Z^n}(A)) - K_{\widetilde {Z}}$ (see [Reference Kollár19, Paragraph 4.7]),

$$ \begin{align*} {\mathcal{I}_{D}}(Z^n, \mathrm{Diff}_{Z^n}(A), (\mathfrak{a} S^n)^\lambda (\mathfrak{b} S^n)^{\lambda'}) &={\mathcal{I}_{D}^{W_{Z^n}}}(Z^n, \mathrm{Diff}_{Z^n}(A), (\mathfrak{a} S^n)^\lambda (\mathfrak{b} S^n)^{\lambda'})\\ &= g_* \mathcal{O}_{\widetilde{Z}} ( - \lfloor \Theta^{W_{\widetilde{Z}}}_{g^*D}(\Gamma|_{\widetilde{Z}} + \lambda F_a|_{\widetilde{Z}} + \lambda' F_b|_{\widetilde{Z}}) \rfloor)\\ & \subseteq g_* \mathcal{O}_{\widetilde{Z}} ( - \lfloor \Theta^{W_Y|_{\widetilde{Z}}}_{g^*D}(\Gamma|_{\widetilde{Z}} + \lambda F_a|_{\widetilde{Z}} + \lambda' F_b|_{\widetilde{Z}}) \rfloor)\\ & \subseteq g_* \mathcal{O}_{\widetilde{Z}} ( - \lfloor \Theta^{W_Y^1|_{\widetilde{Z}}}_{g^*D}(\Theta^{W_Y^2|_{\widetilde{Z}}}(\Gamma|_{\widetilde{Z}} + \lambda F_a|_{\widetilde{Z}} + \lambda' F_b|_{\widetilde{Z}})) \rfloor), \end{align*} $$

where the containment on the third line follows from Lemma 4.2 (3) and the last containment does from Lemma 4.2 (1), (4), and (6). Setting $\Lambda : = \Theta ^{W_Y^2}(\Gamma + \lambda F_a + \lambda ' F_b)$ and noting that the union of the supports of $\widetilde {Z}$ , $W_Y^2$ and $\Gamma +\lambda F_a + \lambda ' F_b$ is a simple normal crossing divisor on Y, one has

$$\begin{align*}\Lambda|_{\widetilde{Z}} = \Theta^{W_Y^2|_{\widetilde{Z}}}(\Gamma|_{\widetilde{Z}} + \lambda F_a|_{\widetilde{Z}} + \lambda' F_b|_{\widetilde{Z}}), \end{align*}$$

and therefore,

$$\begin{align*}{\mathcal{I}_{D}}(Z^n, \mathrm{Diff}_{Z^n}(A), (\mathfrak{a} S^n)^\lambda (\mathfrak{b} S^n)^{\lambda'}) \subseteq g_* \mathcal{O}_{\widetilde{Z}} ( - \lfloor \Theta^{W_Y^1|_{\widetilde{Z}}}_{g^*D}(\Lambda|_{\widetilde{Z}}) \rfloor). \end{align*}$$

Proof of Claim.

It is enough to show that $J S^n \subseteq g_* \mathcal {O}_{\widetilde {Z}} (- \lfloor \Lambda |_{\widetilde {Z}} \rfloor )$ . Take a connected component C of $Z^n$ , and let $\widetilde {C}$ denote the corresponding component of $\widetilde {Z}$ . By the assumption (i), for any nonzero element $r \in H^0(C, J S^n|_C)$ ,

$$\begin{align*}\operatorname{div}_{\widetilde{C}}(r) \geqslant \lfloor \Theta^{W_Y^1|_{\widetilde{Z}}}_{g^*D}(\Lambda|_{\widetilde{Z}}) \rfloor|_{\widetilde{C}} \; \; \mathrm{and} \; \; \operatorname{div}_{\widetilde{C}}(r) \geqslant g^* D|_{\widetilde{C}}. \end{align*}$$

Fix any prime divisor E on $\widetilde {C}$ . If $\operatorname {ord}_E( \Lambda |_{\widetilde {Z}}) = \operatorname {ord}_E(g^*D) + 1$ and E is contained in $W_Y^1|_{\widetilde {Z}}$ , then $g'( E) \subseteq V$ by the definition of $W_Y^1$ , and it therefore follows from the assumption (ii) that

$$\begin{align*}\operatorname{ord}_E(r) \geqslant \operatorname{ord}_E(g^*D) + 1 = \operatorname{ord}_E( \Lambda|_{\widetilde{Z}}). \end{align*}$$

If $\operatorname {ord}_E( \Lambda |_{\widetilde {Z}}) \neq \operatorname {ord}_E(g^*D) + 1$ or $E \not \subseteq W_Y^1|_{\widetilde {Z}}$ , then

$$\begin{align*}\operatorname{ord}_E( \Theta^{W_Y^1|_{\widetilde{Z}}}_{g^*D}(\Lambda|_{\widetilde{Z}}))=\operatorname{ord}_E( \Lambda|_{\widetilde{Z}}). \end{align*}$$

Thus, we obtain the inequality $\operatorname {div}_{\widetilde {C}}(r) \geqslant \lfloor \Lambda |_{\widetilde {Z}} \rfloor |_{\widetilde {C}}$ , which implies $J S^n \subseteq g_* \mathcal {O}_{\widetilde {Z}} (- \lfloor \Lambda |_{\widetilde {Z}} \rfloor )$ .

By the above claim, we have the following commutative diagram:

Noting that ${\mathcal {I}^{W+Z}}(X,A+ Z, \mathfrak {a}^\lambda \mathfrak {b}^{\lambda '}) = f_* \mathcal {O}_Y(- \lfloor \Theta ^{W_Y^1}(\Lambda ) \rfloor )$ by Lemma 4.2 (6), we have

$$\begin{align*}\mathrm{Im}\; \beta \subseteq \mathrm{Im}\; \alpha = {\mathcal{I}^{W+Z}}(X,A+ Z, \mathfrak{a}^\lambda \mathfrak{b}^{\lambda'}) S. \end{align*}$$

Take any element $r \in J$ . In order to prove the assertion of this theorem, it suffices to prove that the morphism $\delta : J \hookrightarrow g^{\prime }_* \mathcal {O}_Y(- \lfloor \Lambda |_{\widetilde {Z}} \rfloor ) \to \mathrm {Coker}\; \beta $ sends r to zero. Since $(f_* W_Y^1)|_U=0$ , one has an inclusion $J|_U \subseteq {\mathcal {I}^{W+Z}}(X,A + Z, \mathfrak {a}^\lambda \mathfrak {b}^{\lambda '}) S|_U=\mathrm {Im}\; \beta |_U$ by the assumption (iii), which implies that the support of $\delta (r) \in \mathrm {Coker}\; \beta $ is contained in V.

On the other hand, by pushing forward the short exact sequence

$$\begin{align*}0 \to \mathcal{O}_Y(- \lfloor \Lambda \rfloor -\widetilde{Z}) \to \mathcal{O}_Y(- \lfloor \Lambda \rfloor) \to \mathcal{O}_{\widetilde{Z}}(- \lfloor \Lambda|_{\widetilde{Z}} \rfloor) \to 0, \end{align*}$$

we obtain an inclusion $\mathrm {Coker}\; \beta \subseteq R^1f_*\mathcal {O}_Y(- \lfloor \Lambda \rfloor -\widetilde {Z})$ . Let H be the f-semiample $\mathbb {R}$ -divisor $- (K_Y +\Gamma + \widetilde {Z} + \lambda F_a + \lambda ' F_b) $ on Y, let $\Delta $ be the fractional part of the $\mathbb {R}$ -divisor $\Gamma + \lambda F_a + \lambda ' F_b$ , and let B be the reduced divisor on Y whose support is the union of all prime divisors E, such that $E \subseteq W_Y^2$ and $E \not \subseteq \operatorname {Supp} \Delta $ . Since $B+\Delta $ has simple normal crossing support and

$$\begin{align*}-\lfloor \Lambda \rfloor -\widetilde{Z} = (K_Y+ B+ \Delta)+H, \end{align*}$$

it follows from [Reference Ambro2, Theorem 3.2] (see also [Reference Fujino8, Theorem 1.1]) that if

$$\begin{align*}\delta(r) \in R^1f_* \mathcal{O}_Y(-\lfloor \Lambda \rfloor -\widetilde{Z}) \end{align*}$$

is a nonzero element, then the support of $\delta (r)$ contains $f(T)$ , where T is a stratum of the simple normal crossing pair $(Y, B)$ . Taking into account that B and $f^{-1}(V)$ have no common components and $B+f^{-1}(V)$ has simple normal crossings, we have $f(T) \not \subseteq V$ , which contradicts the fact that the support of $\delta (r)$ is contained in V. Therefore, we conclude that $\delta (r)=0$ as desired.

Setting 4.11. Let $(R,\mathfrak {m})$ be an equidimensional local ring essentially of finite type over an algebraically closed field of characteristic zero, let $h \in R$ be a nonzero divisor, let $\lambda>0$ be a real number, and let $\mathfrak {a} \subseteq R$ be an ideal with the following properties:

1. (1) $S : = R/(h)$ is reduced and satisfies $(S_2)$ and $(G_1)$ . Therefore, so is R.

2. (2) $\mathfrak {a}$ is nonzero at any generic point of $X: = \operatorname {Spec} R$ and is trivial at any generic point of $Z: =\operatorname {Spec} S$ .

3. (3) Any generic point of Z is a regular point of X.

Moreover, let $\Delta $ be an effective $\mathbb {Q}$ -Weil divisor on X contained in $\mathrm {WDiv}^*_{\mathbb {Q}}(X)$ , and let $K_X$ and $K_Z$ be canonical divisors contained in $\mathrm {WDiv}^*(X)$ and $\mathrm {WDiv}^*(Z)$ , respectively, which exist by Proposition 2.16 and Example A.15. We further assume that

1. (4) neither any generic points of Z nor any codimension one singular points of Z are contained in the support of $\Delta $ , and

2. (5) $K_Z + \Delta |_Z$ is $\mathbb {Q}$ -Cartier, where $\Delta |_Z$ denotes the $\mathbb {Q}$ -Weil divisor $\mathrm {Diff}_Z(\Delta )$ (see Lemma 2.24).

Proof. We only consider the slc case, as the lc case follows essentially the same arguments.

First note that since the morphism $Z' \to Z$ is finite and birational, $Z'$ is reduced and the normalization $Z^n=\operatorname {Spec} S^n$ of Z is isomorphic to that of $Z'$ . We consider the following diagram:

By prime avoidance, we can take an effective Weil divisor $A \in \mathrm {WDiv}^*(X)$ , linearly equivalent to $-K_X$ , whose support contains neither any generic points of Z nor any codimension one singular points of Z. We may also assume that $B:= A -\Delta $ is effective. Fix an integer $m \geqslant 1$ , such that $m \Delta \in \mathrm {WDiv}^*_{\mathbb {Q}}(X)$ is an integral Weil divisor and $m(K_Z+\Delta |_Z) \in \mathrm {WDiv}^*_{\mathbb {Q}}(Z)$ is a Cartier divisor. It then follows from Lemma 2.24 (2) that $m B|_Z = m A|_Z - m \Delta |_Z \sim -m(K_Z +\Delta |_Z)$ is Cartier.

We define the $\mathbb {Q}$ -Weil divisors $\Delta _{X^n}$ , $A_{X^n}$ , and $B_{X^n}$ on $X^n$ as

$$\begin{align*}\Delta_{X^n} : = \nu^* \Delta + C_X, \ \ A_{X^n} : = \nu^* A + C_X\ \mathrm{and}\ B_{X^n} : = A_{X^n}- \Delta_{X^n}, \end{align*}$$

where $C_X$ is the conductor divisor of $\nu $ on $X^n$ . Let W be the reduced Weil divisor on $X^n$ whose support coincides with the union of the support of $\Delta _{X^n}$ and all the codimension one irreducible components of the closed subscheme of $X^n$ defined by $\mathfrak {a} R^n$ . Since $A_{X^n} \sim -K_{X^n}$ , it suffices to show by Proposition 4.9 (2) that

(⋆) $$ \begin{align} \mathfrak{b} &= {\mathcal{I}^{W+Z'}}(X^n, A_{X^n}, \mathcal{O}_{X^n}(-Z')^1 (\mathfrak{a} R^n)^\lambda \mathfrak{b}^{1-1/m}) \nonumber\\ &= {\mathcal{I}^{W+Z'}}(X^n, A_{X^n} +Z', (\mathfrak{a} R^n)^\lambda \mathfrak{b}^{1-1/m}) \end{align} $$

where $\mathfrak {b} : = \mathcal {O}_{X^n}(-mB_{X^n})$ .

As an intermediate step to prove (⋆), we show the inclusion

$$\begin{align*}\mathfrak{b} S' \subseteq {\mathcal{I}^{W+Z'}}(X^n, A_{X^n} +Z', (\mathfrak{a} R^n)^\lambda \mathfrak{b}^{1-1/m}) S'. \end{align*}$$

By Proposition 4.10, it is enough to show that if we set $\lambda ' : = (m-1)/m$ , $J : = \mathfrak {b} S'$ , and $D : =\rho ^* (m B|_Z)$ and if $U \subseteq Z'$ denotes the locus where $J=L$ , then the assumptions (i), (ii), and (iii) in Proposition 4.10 are satisfied. We define $\mathbb {Q}$ -Weil divisors $\Delta _{Z^n}$ and $A_{Z^n}$ on $Z^n$ as

$$\begin{align*}\Delta_{Z^n} : = \rho^* \Delta|_Z + C_Z\ \mathrm{and}\ A_{Z^n} : = \rho^* A|_Z + C_Z, \end{align*}$$

where $C_Z$ is the conductor divisor of $\rho $ on $Z^n$ . The assumption (i) is an immediate consequence of Claim 1 and Proposition 4.9 (3), because $\mathrm {Diff}_{Z^n}(A_{X^n}) =A_{Z^n} \sim - K_{Z^n}$ by Lemmas 2.21, 2.22, and 2.24 and $D = m(A_{Z^n} - \Delta _{Z^n})$ . Since L is a principal ideal, $J_x \subseteq \mathfrak {m}_{Z', x} L_x$ for all points $x \in Z' \setminus U$ , from which the assumption (ii) follows. In order to verify the assumption (iii), we need the following claim.

Let $\widetilde {U} : = V \cap Z'$ and $\widetilde {U}^n : = \pi ^{-1}(\widetilde {U}) \subseteq Z^n$ . By Lemma 4.7, we have

$$ \begin{align*} &{\mathcal{I}^{W+Z'}}(X^n,A_{X^n} +Z', (\mathfrak{a} R^n)^{\lambda} \mathfrak{b}^{\lambda'})|_V \\ =& {\mathcal{I}^{W|_V+ \widetilde{U}}}(V, A_{X^n}|_V +\widetilde{U}, (\mathfrak{a} R^n)|_V^{\lambda} \mathfrak{b}|_V^{\lambda'}) \\ =& {\mathcal{I}^{W|_V+ \widetilde{U}}}(V, A_{X^n}|_V+\widetilde{U} + \frac{m-1}{m}(mB_{X^n}|_V) , (\mathfrak{a} R^n)|_V^{\lambda}) \\ =& {\mathcal{I}^{W|_V+ \widetilde{U}}}(V, \Delta_{X^n}|_V+ \widetilde{U} , (\mathfrak{a} R^n)|_V^{\lambda}) \otimes_{\mathcal{O}_V} \mathcal{O}_V(-mB_{X^n}|_V), \end{align*} $$

where the second equality follows from the fact that $mB_{X^n}|_V$ is Cartier. Since the triple $(Z, \Delta |_Z, (\mathfrak {a} S)^\lambda )$ is slc and

$$\begin{align*}\Delta_{Z^n}|_{\widetilde{U}^n} = \mathrm{Diff}_{Z^n}(\Delta_{X^n})|_{\widetilde{U}^n}= \mathrm{Diff}_{\widetilde{U}^n}(\Delta_{X^n}|_V) \end{align*}$$

by Lemmas 2.21, 2.22, and 2.24, the triple $(\widetilde {U}^n, \mathrm {Diff}_{\widetilde {U}^n}(\Delta _{X^n}|_V), (\mathfrak {a} \mathcal {O}_{\widetilde {U}^n} )^\lambda )$ is lc. Noting that $m(K_V + \Delta _{X^n}|_V +\widetilde {U})$ is Cartier, we use inversion of adjunction for lc singularities [Reference Kawakita17]Footnote ⁶ to deduce that there exists an open subscheme $\widetilde {V} \subseteq V$ containing $\widetilde {U}$ , such that $(\widetilde {V}, \Delta _{X^n}|_{\widetilde {V}} +\widetilde {U}, \mathfrak {a}|_{\widetilde {V}}^{\lambda })$ is lc, which is equivalent by Lemma 4.8 (2) to saying that ${\mathcal {I}^{W|_{\widetilde {V}} + \widetilde {U}}}(\widetilde {V}, \Delta _{X^n}|_{\widetilde {V}} +\widetilde {U}, \mathfrak {a}|_{\widetilde {V}}^{\lambda }) = \mathcal {O}_{\widetilde {V}}$ . Therefore,

$$\begin{align*}{\mathcal{I}^{W+Z'}}(X^n,A_{X^n}+Z', (\mathfrak{a} R^n)^{\lambda} \mathfrak{b}^{\lambda'})|_{\widetilde{V}} = \mathcal{O}_{\widetilde{V}}(-mB_{X^n}|_{\widetilde{V}}), \end{align*}$$

and it follows from Claim 2 that the assumption (iii) of Theorem 4.10 is satisfied. Thus, we obtain the inclusion

$$\begin{align*}\mathfrak{b} S' \subseteq {\mathcal{I}^{W+Z'}}(X^n, A_{X^n}+Z', (\mathfrak{a} R^n)^\lambda \mathfrak{b}^{1-1/m}) S'. \end{align*}$$

Finally, combining this inclusion with Proposition 4.9 (1) yields that

$$\begin{align*}\mathfrak{b} \subseteq {\mathcal{I}^{W+Z'}}(X^n, A_{X^n}+Z', (\mathfrak{a} R^n)^\lambda \mathfrak{b}^{1-1/m}) + \mathfrak{b} \cap (h). \end{align*}$$

Since $B_{X^n}$ has no common component with $Z'$ , the ideal $\mathfrak {b} \cap (h)$ is contained in $h \mathfrak {b}$ . By Nakayama’s lemma, one has the desired inclusion (⋆), that is,

$$\begin{align*}\mathfrak{b}={\mathcal{I}^{W+Z'}}(X^n, A_{X^n}+Z', (\mathfrak{a} R^n)^\lambda \mathfrak{b}^{1-1/m}).\\[-34pt] \end{align*}$$

Proof. This is an immediate consequence of Theorem 4.12. Since $X^n \cong X$ , the assumption (6) in Theorem 4.12 is clearly satisfied.

Proof. As in the proof of Theorem 4.12, we consider the following diagram.

Since $\mathrm {Diff}_{Z^n}(\Theta ) \leqslant \mathrm {Diff}_{Z^n}(\nu ^* \Delta +C_X) =\rho ^*(\Delta |_Z) +C_Z$ by Lemmas 2.21, 2.22, and 2.24, the pair $(Z^n, \mathrm {Diff}_{Z^n}(\Theta ))$ is lc. We use inversion of adjunction for lc singularities [Reference Kawakita17] to deduce that $(X^n, \Theta + Z')$ is lc near $Z'$ . It then follows from [Reference Alexeev1, Theorem 3.4] that $Z'$ satisfies $(S_2)$ . Now we apply Theorem 4.12 to obtain the result.

Proof. It follows from Theorem 4.12, Corollary 4.13, and Corollary 4.14 that $(\mathcal {X}, \mathcal {D}, \mathfrak {a}^\lambda )$ is valuatively (s)lc at x. Since y is a generalization of x, the triple $(\mathcal {X}, \mathcal {D}, \mathfrak {a}^\lambda )$ is valuatively (s)lc at y by Remark 2.18, which completes the proof.

Proof. Since the structure map $\mathcal {X} \to T$ is a closed map, it follows from an argument similar to the proof of Corollary 4.15 that $(\mathcal {X}, \mathcal {D}, \mathfrak {a}^\lambda )$ is valuatively (s)lc near $\mathcal {X}_{\eta }$ , which implies the assertion.

Proof. In both cases, it suffices to show that the generic fiber $(X_{\eta }, D_{\eta }, (\mathfrak {a} \mathcal {O}_{X_\eta })^{\lambda })$ is lc. In (1), since $K_{X_{\eta }}+D_{\eta }$ is $\mathbb {Q}$ -Cartier by [Reference Sato and Takagi28, Remark 2.15], it follows from Corollary 4.17 and Remark 2.8 that $(X_{\eta }, D_{\eta }, (\mathfrak {a} \mathcal {O}_{X_\eta })^{\lambda })$ is lc. In (2), we deduce from Corollary 4.17 that $(X_{\eta }, D_{\eta }, (\mathfrak {a} \mathcal {O}_{X_\eta })^{\lambda })$ is two-dimensional valuatively lc, which implies by Lemma 2.12 that $(X_{\eta }, D_{\eta }, (\mathfrak {a} \mathcal {O}_{X_\eta })^{\lambda })$ is lc.