Proof. (1). The open subset $X \subset \overline {X}$ is the locus $t \neq 0$ , so the complement V is the divisor defined by t. Note that $\mathscr {O}_{\overline {X}}(V) = \mathscr {O}_{\overline {X}}(1)$ , hence V is relatively ample over S.

We have $V = {\mathrm {Proj}_S} \mathscr {A}$ where $\mathscr {A}$ is given the $\mathbb {N}$ -grading with $\deg (\mathscr {A}_m) = \langle \xi , m \rangle $ , and X (resp., $\overline {X}$ ) is the affine cone (resp., projective cone), relatively over S, for $\mathscr {O}_V(1) = \mathscr {O}_{\overline {X}}(1)|_V$ . Thus $V \subset \overline {X}$ is Cartier on the locus where $\mathscr {O}_{V}(1)$ is invertible; in particular, $V \subset \overline {X}$ is Cartier in codimension $2$ .

(2). First assume S is the spectrum of a DVR, with the closed point $s \in S$ . Then $(X, \Delta + X_s)$ is plt by [Reference Kollár15, Thm. 4.54], hence $(V, \Delta _V + V_s)$ is plt, where $\Delta _V = \mathrm {Diff}_{V}(\overline {\Delta }) = \overline {\Delta }|_V$ (see [Reference Kollár14, Lem. 3.1]). Then $(\overline {X}, \overline {\Delta } + V + \overline {X}_s)$ is dlt. Thus the conclusion holds in this case by [Reference Kollár15, Thm. 4.54] again. Note that the closed fiber $\overline {\Delta }_s$ of $\overline {\Delta }$ is the closure of $\Delta _s$ .

In general, the formation of $\overline {X}$ commutes with base change. By the DVR case above, the formation of $\overline {\Delta }$ commutes with specialization, hence $\bar {\pi } \colon (\overline {X}, \overline {\Delta } + V) \to S$ is a well-defined family of pairs by [Reference Kollár15, Thm-Def. 4.3]. Then it is a locally stable family of plt pairs since its base change to every DVR is so by [Reference Kollár15, Def-Thm. 4.7].

(3). Let

(2.56) $$ \begin{align} Y = {\mathrm{Spec}_V} \bigoplus_{n \geq 0} \mathscr{O}_V(n), \end{align} $$

with the canonical morphisms $p \colon Y \to V$ and $\mu \colon Y \to X$ , and $E = \mu ^{-1}(x(S)) \simeq V$ . Then

(2.57) $$ \begin{align} \mu \colon (Y, E) \to (X, \Delta) \end{align} $$

is a family of birational models, and we can write

(2.58) $$ \begin{align} \mu^{*}(K_{X/S} + \Delta) = K_{Y/S} + \mu^{-1}_{*}\Delta + E' \end{align} $$

for some $E'$ on Y supported on E. For each fiber we have $E^{\prime }_s = (1 - A_{X_s,\Delta _s}(E_s))E_s$ , and $E_s$ is a prime divisor. Thus the function $s \mapsto A_{X_s, \Delta _s}(E_s)$ is locally constant. If S is connected, then it is constant, say with value a. Note that $a> 0$ since $(X_s, \Delta _s)$ is klt for all $s \in S$ . Then

(2.59) $$ \begin{align} K_{\overline{X}_s} + \overline{\Delta}_s + (1 + a)V_s \sim_{\mathbb{Q}} 0 \end{align} $$

for all $s \in S$ by [Reference Zhuang38, Lem. 3.3]. Hence $K_{\overline {X}/S} + \overline {\Delta } + (1+a)V \sim _{S, \mathbb {Q}} 0$ .

Proof. Write $n = \dim (X_s)$ for all $s \in S$ .

(1). By definition,

(2.61) $$ \begin{align} \mathrm{vol}_{X_s}(\mathrm{wt}_{\xi}) = \lim_{\lambda \to \infty} \frac{n!}{\lambda^n} \sum_{\langle \xi, m \rangle < \lambda} \dim_{\kappa(s)}(\mathscr{A}_{m} \otimes_{\mathscr{O}_S} \kappa(s)), \end{align} $$

where the summation ranges over all $m \in M$ with $\langle \xi , m \rangle < \lambda $ . Since $\mathscr {A}_m$ is a flat coherent $\mathscr {O}_S$ -module,

(2.62) $$ \begin{align} s \mapsto \dim_{\kappa(s)}(\mathscr{A}_{m} \otimes_{\mathscr{O}_S} \kappa(s)) \end{align} $$

is locally constant for all $m \in M$ (see [26, 05P2]). Hence $s \mapsto \mathrm {vol}_{X_s}(\mathrm {wt}_{\xi })$ is locally constant.

(2). Fix $s \in S$ , then $\xi \mapsto A_{X_s,\Delta _s}(\mathrm {wt}_{\xi })$ is a linear map on $N_{\mathbb {R}}$ by [Reference Li and Xu22, Lem. 2.18]. Hence it suffices to show $s \mapsto A_{X_s,\Delta _s}(\mathrm {wt}_{\xi })$ is locally constant when $\xi \in N$ . We may assume that S is connected. Let

(2.63) $$ \begin{align} \bar{\pi} \colon (\overline{X}, \overline{\Delta} + V) \to S \end{align} $$

be the family constructed in Lemma 2.20 using $\xi $ . Then

(2.64) $$ \begin{align} K_{\overline{X}/S} + \overline{\Delta} + (1 + a)V \sim_{S, \mathbb{Q}} 0 \end{align} $$

for some $a> 0$ . Restricting to each fiber, we get $a = A_{X_s, \Delta _s}(\mathrm {wt}_{\xi })$ for all $s \in S$ by [Reference Zhuang38, Lem. 3.3].

(3). By definition,

(2.65) $$ \begin{align} \widehat{\mathrm{vol}}_{X_s,\Delta_s}(\mathrm{wt}_{\xi}) = A_{X_s,\Delta_s}(\mathrm{wt}_{\xi})^n \cdot \mathrm{vol}_{X_s}(\mathrm{wt}_{\xi}). \end{align} $$

Hence it is a locally constant function on S by (1) and (2).

If $\pi \colon (X, \Delta ) \to S$ is a family of K-semistable log Fano cone singularities with the polarization $\xi $ , then

(2.66) $$ \begin{align} \widehat{\mathrm{vol}}(x_s; X_s,\Delta_s) = \widehat{\mathrm{vol}}_{X_s,\Delta_s}(\mathrm{wt}_{\xi}) \end{align} $$

by [Reference Li and Xu22, Thm. 3.5]. Hence $s \mapsto \widehat {\mathrm {vol}}(x_s; X_s,\Delta _s)$ is locally constant by (3).

Proof. By Lemma 3.2, we may assume that $-(K_Y + f^{-1}_{*}\Delta + E)$ is ample. Since Y and $Y'$ have the same exceptional divisors over X, the birational map $\phi ^{-1} \colon Y' \dashrightarrow Y$ is an isomorphism in codimension $1$ , so it is the relative ample model for $-(K_{Y'} + f^{\prime -1}_{*}\Delta + E')$ . Since $-(K_{Y'} + f^{\prime -1}_{*}\Delta + E')$ is $f'$ -nef, $\phi ^{-1} \colon Y' \to Y$ is a morphism. Note that Y is $\mathbb {Q}$ -factorial at the generic points of strata of $(Y, E)$ , so $\phi \colon Y \dashrightarrow Y'$ is a local isomorphism at these points, as otherwise $\phi ^{-1}$ would have an exceptional divisor over there. Thus $f' \colon (Y', E') \to (X, \Delta )$ is a model of qdlt Fano type by [Reference Xu and Zhuang33, Prop. 3.6].

Proof. (i) $\Rightarrow $ (ii). See [Reference Xu and Zhuang33, Thm. 3.14] or [Reference Xu31, Thm. 5.26].

(ii) $\Rightarrow $ (i). By Lemma 3.2, we may assume $(Y, f^{-1}_{*}\Delta + E)$ is qdlt and $-(K_Y + f^{-1}_{*}\Delta + E)$ is ample. Choose a general $\mathbb {Q}$ -divisor $G \in |{-(K_Y + f^{-1}_{*}\Delta + E)}|_{\mathbb {Q}}$ such that $(Y, f^{-1}_{*}\Delta + G + E)$ is qdlt. Note that $f_{*}(G + E) \sim _{\mathbb {Q}} -(K_X + \Delta )$ , where $f_{*}E$ consists of components of E that are not f-exceptional. Let

(3.7) $$ \begin{align} \mu \colon (Z, F) \to (Y, f^{-1}_{*}\Delta + G + E) \end{align} $$

be a log resolution such that $\mu $ is a local isomorphism over the simple-toroidal locus of $(Y, E)$ , and there is a $\mu $ -ample divisor $-A$ on Z with $A \geq 0$ and $\mathrm {Supp}(A) = \mathrm {Ex}(\mu )$ . Then we have

(3.8) $$ \begin{align} 0 \sim_{\mathbb{Q}} \mu^{*}(K_Y + f^{-1}_{*}\Delta + G + E) = K_Z + \mu^{-1}_{*}f^{-1}_{*}\Delta + (\mu^{*}G - \epsilon A) + \mu^{-1}_{*}E + F' + \epsilon A. \end{align} $$

where $F' = \sum _{i} (1 - A_{Y, f^{-1}_{*}\Delta + E}(F_i)) \cdot F_i$ and $F_i$ ranges over all $\mu $ -exceptional divisors. By our choice of $\mu $ , we have $\lfloor F' \rfloor \leq 0$ . Thus $\lfloor F' + \epsilon A \rfloor \leq 0$ , and $\mu ^{*}G - \epsilon A$ is ample for $0 < \epsilon \ll 1$ . Now

(3.9) $$ \begin{align} \mathrm{LCP}(Z, \mu^{-1}_{*}f^{-1}_{*}\Delta + \mu^{-1}_{*}E + F' + \epsilon A) = \mathrm{QM}(Z, \mu^{-1}_{*}E) = \mathrm{QM}(Y, E). \end{align} $$

Choose a general $H \in |\mu ^{*}G - \epsilon A|_{\mathbb {Q}}$ , such that $(Z, \mu ^{-1}_{*}f^{-1}_{*}\Delta + H + \mu ^{-1}_{*}E + F' + \epsilon A)$ is sub-lc and

(3.10) $$ \begin{align} \mathrm{LCP}(Z, \mu^{-1}_{*}f^{-1}_{*}\Delta + H + \mu^{-1}_{*}E + F' + \epsilon A) = \mathrm{LCP}(Z, \mu^{-1}_{*}f^{-1}_{*}\Delta + \mu^{-1}_{*}E + F' + \epsilon A). \end{align} $$

Let $\Gamma = f_{*}\mu _{*}H + f_{*}E \sim _{\mathbb {Q}} f_{*}(G + E) \sim _{\mathbb {Q}} -(K_X + \Delta )$ , then $\mu ^{-1}_{*}f^{-1}_{*}\Gamma = H$ , and

(3.11) $$ \begin{align} \mu^{*}f^{*}(K_X + \Delta + \Gamma) = K_Z + \mu^{-1}_{*}f^{-1}_{*}\Delta + H + \mu^{-1}_{*}E + F' + \epsilon A. \end{align} $$

Hence $\Gamma $ is a special $\mathbb {Q}$ -complement with respect to $(Z, F) \to (X, \Delta )$ , and $\mathrm {LCP}(X, \Delta + \Gamma ) = \sigma $ .

Proof. (i), (ii), and (iii) are equivalent by [Reference Xu and Zhuang33, Thm. 4.1].

(i-iii) $\Rightarrow $ (iv). The valuation v induces a canonical valuation $\mathrm {wt}_v$ on $X_v$ , such that $\mathrm {wt}_v(D_v) = v(D)$ for every effective $\mathbb {Q}$ -Cartier $\mathbb {Q}$ -divisor D on X. By the Izumi type inequality [Reference Zhuang37, Lem. 3.5],

(3.13) $$ \begin{align} \mathrm{lct}(X_v, \Delta_v; D_v) \geq c_0\frac{\widehat{\mathrm{vol}}(x_v; X_v, \Delta_v)}{\widehat{\mathrm{vol}}_{X_v, \Delta_v}(\mathrm{wt}_v)} \cdot \frac{A_{X_v, \Delta_v}(\mathrm{wt}_v)}{\mathrm{wt}_v(D_v)}, \end{align} $$

where $x_v \in X_v$ is the vertex defined by the irrelevant ideal of $R_v$ , and $c_0$ is a constant only depending on $\dim _x(X) = \dim (X_v)$ . Thus, if

(3.14)

then $(X_v, \Delta _v + D_v)$ is klt. Since we know (iii) $\Rightarrow $ (i), this implies that v is special for $(X, \Delta + D)$ .

(iv) $\Rightarrow $ (v). It suffices to show that if v is special for $(X, \Delta )$ , then v computes the lc threshold of $\mathfrak {a}_{\bullet }$ . Suppose v is special, then $v \in \mathrm {LCP}(X, \Delta + \Gamma )$ for some $\mathbb {Q}$ -complement $\Gamma $ . Hence we get the conclusion by [Reference Jonsson and Mustaţă12, Thm. 7.8]Footnote ⁴ . We also give a proof here: For $w \in \mathrm {Val}_{X,x}$ , we have

(3.15) $$ \begin{align} A_{X,\Delta}(w) - w(\Gamma) = A_{X,\Delta+\Gamma}(w) \geq 0 \end{align} $$

since $(X, \Delta + \Gamma )$ is lc, and $w(\mathfrak {a}_{\bullet }) \leq w(\Gamma )/v(\Gamma )$ . Meanwhile, $A_{X,\Delta }(v) = v(\Gamma )$ since $v \in \mathrm {LCP}(X, \Delta + \Gamma )$ . Thus

(3.16) $$ \begin{align} \frac{A_{X,\Delta}(w)}{w(\mathfrak{a}_{\bullet})} \geq \frac{A_{X,\Delta}(w)}{w(\Gamma)} v(\Gamma) \geq v(\Gamma) = A_{X, \Delta}(v) = \frac{A_{X,\Delta}(v)}{v(\mathfrak{a}_{\bullet})}. \end{align} $$

Taking infimum over all $w \in \mathrm {Val}_{X,x}$ , we get $\mathrm {lct}(X, \Delta; \mathfrak {a}_{\bullet }) = A_{X,\Delta }(v)/v(\mathfrak {a}_{\bullet })$ .

(v) $\Rightarrow $ (vi). Let $\epsilon < \delta /v(D)$ , so $(X, \Delta + \epsilon D)$ is klt and v computes the lc threshold $\mathrm {lct}(X, \Delta + \epsilon D; \mathfrak {a}_{\bullet })$ by (v). Then the conclusion follows from the fact that a quasi-monomial valuation that computes the lc threshold of an ideal sequence is an lc place of a $\mathbb {Q}$ -complement; see [Reference Li and Xu22, Rmk. 4.4]. This fact is also the special case of Lemma 4.2 when the base S is a point.

(vi) $\Rightarrow $ (i). The proof is the same as [Reference Xu and Zhuang33, Lem. 3.4]. Let $f \colon (Y, E) \to (X, \Delta )$ be a simple-toroidal model, and $y \in (Y, E)$ be a generic point of a stratum, such that $v \in \mathrm {QM}_y(Y, E)$ , and there is an f-ample divisor $-A$ with $A \geq 0$ and $\mathrm {Supp}(A) = \mathrm {Ex}(f)$ . Then $-A$ is ample since X is affine. Let $H \in |{-A}|_{\mathbb {Q}}$ be a general $\mathbb {Q}$ -divisor whose support does not contain any stratum of $(Y, \mathrm {Supp}(f^{-1}_{*}\Delta + E + \mathrm {Ex}(f)))$ . Let $D = f_{*}H$ , then there is a $\mathbb {Q}$ -complement $\Gamma \geq \epsilon D$ of $(X, \Delta )$ such that $v \in \mathrm {LCP}(X, \Delta + \Gamma )$ by (vi). By construction, $f^{-1}_{*}\Gamma \geq \epsilon H$ . That is, $\Gamma $ is a special $\mathbb {Q}$ -complement with respect to $f \colon (Y, E) \to (X, \Delta )$ .

Proof. We know v computes the lc threshold of $\mathfrak {a}_{\bullet }$ by Lemma 3.10. Suppose $w \in \mathrm {Val}_{X,x}$ such that

(3.18) $$ \begin{align} \frac{A_{X,\Delta}(v)}{v(\mathfrak{a}_{\bullet})} = \mathrm{lct}(X, \Delta; \mathfrak{a}_{\bullet}) = \frac{A_{X,\Delta}(w)}{w(\mathfrak{a}_{\bullet})}. \end{align} $$

By rescaling, we may assume that $A_{X,\Delta }(v) = A_{X,\Delta }(w) = 1$ . Hence

(3.19) $$ \begin{align} \inf_{\lambda> 0} \frac{w(\mathfrak{a}_{\lambda}(v))}{\lambda} = w(\mathfrak{a}_{\bullet}) = v(\mathfrak{a}_{\bullet}) = 1. \end{align} $$

Thus $w(f) \geq v(f)$ for every $f \in \mathscr {O}_{X}$ . Assume that $w(f)> v(f)$ for some f, then $f \in \mathfrak {m}_x \smallsetminus \{0\}$ , where $\mathfrak {m}_x \subset \mathscr {O}_{X}$ is the maximal ideal of x. Let $D = \mathrm {div}(f)$ . Then, for $\epsilon> 0$ ,

(3.20) $$ \begin{align} \frac{A_{X, \Delta + \epsilon D}(w)}{w(\mathfrak{a}_{\bullet})} = 1 - \epsilon w(f) < 1 - \epsilon v(f) = \frac{A_{X, \Delta + \epsilon D}(v)}{v(\mathfrak{a}_{\bullet})}. \end{align} $$

This contradicts that v computes $\mathrm {lct}(X, \Delta + \epsilon D; \mathfrak {a}_{\bullet })$ for $0 < \epsilon \ll 1$ by Lemma 3.10.

Proof. Let $I_a = Rt_1^{a_1} \cdots t_r^{a_r} \subset R$ for each $a = (a_1, \ldots , a_r) \in \mathbb {N}^r$ . Then

(3.24) $$ \begin{align} I_a \otimes_R N \simeq I_aN = \bigoplus_{m \in \mathbb{Z}^r} M_{a+m}t_1^{-m_1} \cdots t_r^{-m_r} \subset N. \end{align} $$

Note that $I_a \cap I_b = I_{\max (a, b)}$ , where $\max (a, b) = (\max (a_1, b_1), \ldots , \max (a_r,b_r))$ . Since N is flat over R, we have $I_aN \cap I_bN = I_{\max (a, b)}N$ , that is,

(3.25) $$ \begin{align} M_{k} \cap M_{\ell} = M_{\max(k,\ell)} \end{align} $$

for all $k, \ell \in \mathbb {Z}^r$ . Since $M_0 = M$ , we have $M_m = M_{\max (m,0)}$ for all $m \in \mathbb {Z}^r$ . Hence it suffices to prove

(3.26) $$ \begin{align} \frac{\sum_{\langle \xi, a \rangle \geq \lambda} M_a}{\sum_{\langle \xi, a \rangle> \lambda} M_a} \simeq \bigoplus_{\langle \xi, a \rangle = \lambda} M_a/M_{>a}, \end{align} $$

where instead of $m \in \mathbb {Z}^r$ , the summations only range over $a \in \mathbb {N}^r$ .

Note that there is a canonical isomorphism

(3.27) $$ \begin{align} \frac{\sum_{\langle \xi, a \rangle \geq \lambda} I_a}{\sum_{\langle \xi, a \rangle> \lambda} I_a} \simeq \bigoplus_{\langle \xi, a \rangle = \lambda} At_1^{a_1} \cdots t_r^{a_r} \simeq \bigoplus_{\langle \xi, a \rangle = \lambda} I_a/I_{>a}. \end{align} $$

Since N is flat over R, we get

(3.28) $$ \begin{align} \frac{\sum_{\langle \xi, a \rangle \geq \lambda} I_aN}{\sum_{\langle \xi, a \rangle> \lambda} I_aN} \simeq \bigoplus_{\langle \xi, a \rangle = \lambda} I_aN/I_{>a}N. \end{align} $$

Taking the degree $0$ component, we get the desired conclusion.

Proof. By Lemma 3.2, we may assume $f \colon (Y, E) \to (X, \Delta )$ is the Kollár model, that is, $(Y, f^{-1}_{*}\Delta + E)$ is qdlt and $-(K_Y + f^{-1}_{*}\Delta + E)$ is ample. For each $m = (m_1, \ldots , m_r) \in \mathbb {Z}^r$ , let

(3.29) $$ \begin{align} I_m = f_{*}\mathscr{O}_Y\left(- \sum_{i=1}^{r} m_iE_i \right) \subset \mathscr{O}_X. \end{align} $$

Let $R_m = I_m/I_{>m}$ and $R = \bigoplus _{m \in \mathbb {N}^r} R_m$ be the associated graded ring. By [Reference Xu and Zhuang33, §4.2], R is a finitely generated integral domain over $R_0 = \kappa (x)$ , and $\mathrm {Spec}(R)$ is the equivariant degeneration of $x \in (X, \Delta )$ induced by the Kollár model $(Y, E)$ . Let $\overline {W} \subset \mathbb {R}_{\geq 0}^r$ be the closed convex cone generated by all $m \in \mathbb {N}^r$ such that $R_m \neq 0$ , and let $W = \mathrm {int}(\overline {W})$ be its interior. Note that for all $a \in \overline {W} \cap \mathbb {Q}^r$ , we have $R_{ka} \neq 0$ for all sufficiently divisible integers k since R is finitely generated and integral. We claim $\dim (\overline {W}) = r$ , hence W is a nonempty open subset in $\mathbb {R}_{>0}^r$ .

For $\xi \in \mathbb {R}_{\geq 0}^r$ , let $v_{\xi } \in \mathrm {QM}(Y, E) \subset \mathrm {Val}_{X,x}$ be the corresponding quasi-monomial valuation. Then its ideal sequence is

(3.30) $$ \begin{align} \mathfrak{a}_{\lambda}(v_{\xi}) = \sum_{\langle \xi, m \rangle \geq \lambda} I_m \subset \mathscr{O}_X \end{align} $$

by [Reference Xu and Zhuang33, Cor. 4.10]. Since the extended Rees algebra $\bigoplus _{m \in \mathbb {Z}^r} I_mt_1^{-m_1} \cdots t_r^{-m_r}$ is flat over $\Bbbk [t_1, \ldots , t_r]$ by [Reference Xu and Zhuang33, §4.2], where $\Bbbk $ is the base field, we have

(3.31) $$ \begin{align} \mathfrak{a}_{\lambda}(v_{\xi})/\mathfrak{a}_{>\lambda}(v_{\xi}) \simeq \bigoplus_{\langle \xi, m \rangle = \lambda} R_m \end{align} $$

by Lemma 3.14. If we choose $\xi $ such that $\xi _1, \ldots , \xi _r$ are linearly independent over $\mathbb {Q}$ , then $\mathrm {rat.rank}(v_{\xi }) = r$ ; on the other hand, the value group $\Gamma _{v_{\xi }}$ is generated by all $\langle \xi , m \rangle $ for $R_m \neq 0$ . Hence we conclude that the linear subspace spanned by all such m has dimension r.

Suppose $a = (a_1, \ldots , a_r) \in W$ . Since Y is of Fano type over X, we can take the relative ample model $\phi \colon Y \dashrightarrow Y'$ for $-\sum _{i=1}^{r} a_iE_i$ after a small $\mathbb {Q}$ -factorial modification. Then we need to show $\phi $ satisfies all the desired conditions. For some $a' \in W \cap \mathbb {Q}^r$ near a, the divisor $-\sum _{i=1}^{r} a_i'E_i$ yields the same ample model. Hence we may assume that $a \in W \cap \mathbb {Q}^r$ .

Note that

(3.32) $$ \begin{align} -(K_Y + f^{-1}_{*}\Delta + E) \sim_{X, \mathbb{Q}} -\sum_{i=1}^{r} A_{X,\Delta}(E_i) \cdot E_i \end{align} $$

is ample. Then we can choose a general $\mathbb {Q}$ -divisor $F \sim _{\mathbb {Q}} -\sum _{i=1}^{r} A_{X,\Delta }(E_i) \cdot E_i$ that does not contain any stratum of $(Y, E)$ . Since W is open, $a' = (a_i - \delta A_{X,\Delta }(E_i))_i \in W$ for some rational $0 < \delta \ll 1$ . Then $R_{ka'} \neq 0$ for all sufficiently divisible k since R is an integral domain. Choose $g \in I_{ka'}$ such that its image in $R_{ka'}$ is nonzero. Let $\Gamma = \mathrm {div}_X(g)$ , then

(3.33) $$ \begin{align} 0 \sim f^{*}\Gamma = f^{-1}_{*}\Gamma + \sum_{i=1}^{r} ka_i'E_i. \end{align} $$

For $\xi \in \mathbb {R}_{\geq 0}^r$ , we have $v_{\xi }(g) = \langle \xi , ka' \rangle $ since the image of g in $R_{ka'}$ is nonzero; but we also have

(3.34) $$ \begin{align} v_{\xi}(g) = v_{\xi}(f^{-1}_{*}\Gamma) + \sum_{i=1}^{r} ka_i' v_{\xi}(E_i) = v_{\xi}(f^{-1}_{*}\Gamma) + \sum_{i=1}^{r} ka_i'\xi_i. \end{align} $$

Thus $v_{\xi }(f^{-1}_{*}\Gamma ) = 0$ . It follows that $f^{-1}_{*}\Gamma $ does not contain any stratum of $(Y, E)$ . Let $G = \frac {1}{k}f^{-1}\Gamma $ . Then

(3.35) $$ \begin{align} G + \delta F \sim_{X, \mathbb{Q}} -\sum_{i=1}^{r} a_i E_i, \end{align} $$

and $\mathrm {Supp}(G + \delta F)$ does not contain any stratum of $(Y, E)$ . By [Reference Xu and Zhuang33, Lem. 2.3], for some $0 < \epsilon \ll 1$ , there exists an effective $\mathbb {Q}$ -divisor H on Y such that $(Y, f^{-1}_{*}\Delta + E + \epsilon (G + \delta F) + H)$ is qdlt,

(3.36) $$ \begin{align} \lfloor f^{-1}_{*}\Delta + E + \epsilon(G + \delta F) + H \rfloor = E \end{align} $$

and $-(K_Y + f^{-1}_{*}\Delta + E + \epsilon (G + \delta F) + H)$ is ample. Add a general ample effective $\mathbb {Q}$ -divisor, we may instead assume that

(3.37) $$ \begin{align} K_Y + f^{-1}_{*}\Delta + E + \epsilon(G + \delta F) + H \sim_{X, \mathbb{Q}} 0. \end{align} $$

In other words, let $\Delta ' = \Delta + f_{*}H$ , then

(3.38) $$ \begin{align} K_Y + f^{-1}_{*}\Delta' + E = K_Y + f^{-1}_{*}\Delta + E + H \sim_{X, \mathbb{Q}} -\epsilon(G + \delta F) \sim_{X, \mathbb{Q}} \sum_{i=1}^{r} \epsilon a_i E_i. \end{align} $$

So $(X, \Delta ')$ is klt since it is crepant to $(Y, f^{-1}_{*}\Delta ' + \sum _{i=1}^{r} (1-\epsilon a_i)E_i)$ . Moreover,

(3.39) $$ \begin{align} -(K_Y + f^{-1}_{*}\Delta' + E + \epsilon G) \sim_{X, \mathbb{Q}} \epsilon \delta F \end{align} $$

is ample, and $(Y, f^{-1}_{*}\Delta ' + E + \epsilon G)$ is qdlt. Thus $f \colon (Y, E) \to (X, \Delta ')$ is also a model of qdlt Fano type. Then the ample model $(Y', E' = \phi _{*}E)$ for $-\sum _{i=1}^{r} \epsilon a_i E_i$ is the corresponding Kollár model for $(X, \Delta ')$ by [Reference Xu and Zhuang33, Prop. 3.9]. Hence $(Y', E')$ is also a model of qdlt Fano type for $(X, \Delta )$ .

Proof. We will apply the ACC of lc thresholds in [Reference Hacon, McKernan and Xu10, Thm. 1.1]. Let $\mathrm {Coeff}(\Delta ) \subset \mathbb {R}$ denote the set of coefficients of $\Delta $ . Let $\mathrm {LCT} \subset \mathbb {R}_{\geq 0}$ be the set of all numbers $\mathrm {lct}(W, D_W; M)$ where $(W, D_W)$ is an slc pair with $\dim (W) = \dim (X)$ , with coefficients of $D_W$ in $\mathrm {Coeff}(\Delta ) \cup \{1\}$ , and M is an effective nonzero $\mathbb {Q}$ -Cartier $\mathbb {Z}$ -divisor on W. Then $\mathrm {LCT}$ satisfies the ACC by [Reference Hacon, McKernan and Xu10, Thm. 1.1]; note that we may allow slc pairs $(W, D_W)$ by [Reference Kollár14, Thm. 5.38]. Since $\mathrm {LCT} \subset [0,\, 1]$ and $1 \in \mathrm {LCT}$ , there exists $\epsilon> 0$ such that

(4.1) $$ \begin{align} \mathrm{LCT} \cap [1 - \epsilon,\, 1] = \{1\} \end{align} $$

by the ACC. In our case, $(Y, f^{-1}_{*}\Delta + (1-\epsilon )E + f^{*}\pi ^{*}H)$ is slc by [Reference Kollár15, Thm. 4.54], that is,

(4.2) $$ \begin{align} 1 - \epsilon \leq \mathrm{lct}(Y, f^{-1}_{*}\Delta + f^{*}\pi^{*}H; E) \in \mathrm{LCT}. \end{align} $$

Hence $\mathrm {lct}(Y, f^{-1}_{*}\Delta + f^{*}\pi ^{*}H; E) = 1$ , that is, $(Y, f^{-1}_{*}\Delta + E + f^{*}\pi ^{*}H)$ is slc. So $(Y, f^{-1}_{*}\Delta + E) \to S$ is a locally stable family by [Reference Kollár15, Thm. 4.54].

Proof. By rescaling the index of $I_{\bullet }$ , we may assume that $\mathrm {lct}(X_{s}, \Delta _{s}; I_{s, \bullet }) = \mathrm {lct}(X_{\eta }, \Delta _{\eta }; I_{\eta , \bullet }) = 1$ . The assumption that $v_0$ computes the lc threshold means

(4.5) $$ \begin{align} \mathrm{lct}(X_{\eta}, \Delta_{\eta}; I_{\eta, \bullet}) = \frac{A_{X_{\eta}, \Delta_{\eta}}(v_0)}{v_0(I_{\eta, \bullet})}. \end{align} $$

By rescaling $v_0$ , we may assume $A_{X_{\eta }, \Delta _{\eta }}(v_0) = 1$ . Hence $v_0(I_{\eta , \bullet }) = 1$ . Since $X_{\eta } \subset X$ is the generic fiber, we identify $\mathrm {Val}_{X_{\eta }, x_{\eta }} = \mathrm {Val}_{X,x}$ , so that $A_{X, \Delta }(v_0) = v_0(I_{\bullet }) = 1$ . By inversion of adjunction, $(X, \Delta + \pi ^{*}H)$ is dlt, and

(4.6) $$ \begin{align} \mathrm{lct}(X, \Delta + \pi^{*}H; I_{\bullet}) = \mathrm{lct}(X_s, \Delta_s; I_{s, \bullet}) = 1. \end{align} $$

It follows that $\mathrm {lct}(X, \Delta; I_{\bullet }) = 1$ , and $v_0$ computes the lc threshold of $I_{\bullet }$ on $(X, \Delta )$ .

Fix an isomorphism $\sigma \simeq \mathbb {R}_{\geq 0}^r$ . Note that the function

(4.7)

is non-negative, homogeneous, and convex on $\sigma \simeq (\mathbb {R}_{\geq 0})^r$ . Choose a norm $\| \cdot \|$ on $\mathbb {R}^r$ , and write $\mathrm {d}(\cdot , \cdot )$ for the induced metric on $\sigma $ . Then $A_{X, \Delta +I_{\bullet }}$ is a Lipschitz function on $\sigma $ .Footnote ⁵ Hence there exists $C> 0$ such that

(4.8) $$ \begin{align} A_{X, \Delta + I_{\bullet}}(v) \leq A_{X, \Delta + I_{\bullet}}(v_0) + C \cdot \mathrm{d}(v_0, v) = C \cdot \mathrm{d}(v_0, v). \end{align} $$

Fix $0 < \epsilon < 1$ as in Lemma 4.1 for $\pi \colon (X, \Delta ) \to S$ . By Diophantine approximation (see [Reference Li and Xu22, Lem. 2.7] or [Reference Xu31, Lem. 4.47]), there exist $v_1, \ldots , v_r \in \sigma $ and $q_1, \ldots , q_r \in \mathbb {Z}_{>0}$ such that:

1. (1) $v_0$ is in the convex cone $\tau $ spanned by $v_1, \ldots , v_r$ ;

2. (2) $q_iv_i = c_i {\mathrm {ord}_{E_i}}$ , where $E_i$ is a prime divisor over X and $c_i \in \mathbb {Z}_{>0}$ , for all $i = 1, \ldots , r$ ;

3. (3) $\mathrm {d}(v_0, v_i) < \frac {\epsilon }{2Cq_i}$ for all $i = 1, \ldots , r$ .

Thus

(4.9) $$ \begin{align} 0 \leq A_{X, \Delta + I_{\bullet}}(E_i) = \frac{q_i}{c_i} A_{X, \Delta + I_{\bullet}}(v_i) < \frac{\epsilon}{2} \end{align} $$

for all $i = 1, \ldots , r$ .

Choose $\epsilon '> 0$ such that $2\epsilon ' {\mathrm {ord}_{E_i}}(I_{\bullet }) < \epsilon $ for all $i = 1, \ldots , r$ . Since

(4.10) $$ \begin{align} \sup_{\lambda> 0} \lambda \cdot \mathrm{lct}(X_s, \Delta_s; I_{s,\lambda}) = \lim_{\lambda \to \infty} \lambda \cdot \mathrm{lct}(X_s, \Delta_s; I_{s,\lambda}) = \mathrm{lct}(X_s, \Delta_s; I_{s,\bullet}) = 1, \end{align} $$

we have

(4.11) $$ \begin{align} (1 - \epsilon')/\lambda < \mathrm{lct}(X_s, \Delta_s; I_{s, \lambda}) \leq 1/\lambda \end{align} $$

for $\lambda \gg 0$ . Write $c = (1 - \epsilon ')/\lambda $ , then $\mathrm {lct}(X, \Delta; I_{\lambda }^c) \geq \mathrm {lct}(X_s, \Delta _s; I_{s, \lambda }^c)> 1$ , and

(4.12) $$ \begin{align} \begin{aligned} A_{X, \Delta + I_{\lambda}^{c}}(E_i) &\leq A_{X, \Delta + I_{\bullet}}(E_i) + ({\mathrm{ord}_{E_i}}(I_{\bullet}) - c{\mathrm{ord}_{E_i}}(I_{\lambda})) \\ &< \frac{\epsilon}{2} + (1 - \epsilon')\left( {\mathrm{ord}_{E_i}}(I_{\bullet}) - \frac{1}{\lambda}{\mathrm{ord}_{E_i}}(I_{\lambda}) \right) + \epsilon' {\mathrm{ord}_{E_i}}(I_{\bullet}) \\ &< \frac{\epsilon}{2} + 0 + \frac{\epsilon}{2} = \epsilon \end{aligned} \end{align} $$

where we used the general fact that $v(I_{\bullet }) \leq v(I_{\lambda })/\lambda $ for every valuation v.

Write . By Lemma 2.4, there exists a model

(4.13) $$ \begin{align} f \colon (Y, E) \to (X, \Delta) \end{align} $$

such that $E = E_1 + \cdots + E_r$ is the sum of all exceptional divisors of f, and Y is $\mathbb {Q}$ -factorial. Since

(4.14) $$ \begin{align} \mathrm{lct}(X, \Delta + \pi^{*}H; I_{\lambda}^c) = \mathrm{lct}(X_s, \Delta_s; I_{s, \lambda}^c)> 1, \end{align} $$

$(Y, f^{-1}_{*}\Delta + \sum _{i=1}^{r} (1-a_i)E_{i} + f^{*}\pi ^{*}H)$ is lc, that is, $\pi \circ f \colon (Y, f^{-1}_{*}\Delta + \sum _{i=1}^{r} (1-a_i)E_{i}) \to S$ is locally stable. Thus $\pi \circ f \colon (Y, f^{-1}_{*}\Delta + E) \to S$ is locally stable since $a_i < \epsilon $ for all i, by Lemma 4.1.

Since Y is of Fano type over X, we can run a $-(K_Y + f^{-1}_{*}\Delta + E)$ -MMP over X and get a $\mathbb {Q}$ -factorial good minimal model

(4.15)

where $-(K_{Y'} + f^{\prime -1}_{*}\Delta + \phi _{*}E)$ is $f'$ -semi-ample. As above, $(Y', f^{\prime -1}_{*}\Delta + \sum _{i=1}^{r} (1-a_i)\phi _{*}E_{i} + f^{\prime *}\pi ^{*}H)$ is also lc, hence $\pi \circ f' \colon (Y', f^{\prime -1}_{*}\Delta + \phi _{*}E) \to S$ is a locally stable family by Lemma 4.1. Applying [Reference Kollár and Mori16, Lem. 3.38] in each step of the $-(K_Y + f^{-1}_{*}\Delta + E)$ -MMP, we have

(4.16) $$ \begin{align} A_{Y, f^{-1}_{*}\Delta + E + f^{*}\pi^{*}H}(F) \geq A_{Y', f^{\prime -1}_{*}\Delta + \phi_{*}E + f^{\prime *}\pi^{*}H}(F) \end{align} $$

for every prime divisor F, and the equality holds if and only if $\phi $ is a local isomorphism at $\mathrm {center}_Y(F)$ . Thus $\phi $ is a local isomorphism at every lc center of $(Y, f^{-1}_{*}\Delta + E + f^{*}\pi ^{*}H)$ . In particular, $\phi $ does not contract any component $E_i$ of E. Thus $f' \colon (Y', E' = \phi _{*}E) \to (X, \Delta )$ satisfies (1), (2), and (4).

It remains to show (3). Since $-(K_{Y} + f^{-1}_{*}\Delta + E)$ is f-semi-ample and X is affine, we can choose a general $\mathbb {Q}$ -divisor

(4.17) $$ \begin{align} G \in |{-(K_{Y} + f^{-1}_{*}\Delta + E)}|_{\mathbb{Q}} \end{align} $$

such that $(Y, f^{-1}_{*}\Delta + G + E + f^{*}\pi ^{*}H)$ is lc. Let $\Gamma = f_{*}G \sim _{\mathbb {Q}} -(K_X + \Delta )$ , then

(4.18) $$ \begin{align} f^{*}(K_X + \Delta + \Gamma + \pi^{*}H) = K_Y + f^{-1}_{*}\Delta + G + E + f^{*}\pi^{*}H, \end{align} $$

so $\pi \colon (X, \Delta + \Gamma ) \to S$ is locally stable, and each $E_i$ is an lc place of $(X, \Delta + \Gamma )$ . Since the function

(4.19) $$ \begin{align} v \mapsto A_{X,\Delta + \Gamma}(v) = A_{X, \Delta}(v) - v(\Gamma) \end{align} $$

is non-negative, homogeneous, and convex on $\tau $ , while vanishing at $\mathrm {ord}_{E_i}$ , we get $A_{X,\Delta + \Gamma }(v) = 0$ for all $v \in \tau $ , that is, $\tau \subset \mathrm {LCP}(X, \Delta + \Gamma )$ .