Proof. By [Reference Xu and ZhuangXZ20, Theorem 4.3], we already have $\mu ({\mathcal {F}})\ge \mathbf {L}^{\mathrm {NA}}({\mathcal {F}})$ , thus it suffices to show $\mu ({\mathcal {F}})\le \mathbf {L}^{\mathrm {NA}}({\mathcal {F}})$ . By [Reference Jonsson and MustaţăJM12, Theorem 7.3], $\mathrm { lct}(X_{\mathbb {A}^1},\Delta _{\mathbb {A}^1}+{\mathcal {I}}_{\bullet }^{\frac {1}{r}}; (t))=\mathbf {L}^{\mathrm {NA}}({\mathcal {F}})+1$ is computed by some $\mathbb {G}_m$ -invariant valuation $w\in \mathrm {Val}_{X\times \mathbb {A}^1}^{\circ }$ (the $\mathbb {G}_m$ -equivariant version is not proved in [Reference Jonsson and MustaţăJM12] but is not hard to achieve from the proof). By [Reference Boucksom, Hisamoto and JonssonBHJ17, Lemma 4.2], up to rescaling w is the Gauss extension of a valuation $v\in \mathrm { Val}_X^{\circ }\cup \{v_{\mathrm {triv}}\}$ , that is, $w(ft^i)=v(f)+i$ for any $0\neq f\in K(X)$ and $i\in \mathbb {Z}$ . Since w computes the lct and $w(t)=1$ , we have

$$\begin{align*}\mathbf{L}^{\mathrm{NA}}({\mathcal{F}})+1=A_{X_{\mathbb{A}^1},\Delta_{\mathbb{A}^1}}(w)-w({\mathcal{I}}_{\bullet}^{\frac{1}{r}})=A_{X,\Delta}(v)+1-w({\mathcal{I}}_{\bullet}^{\frac{1}{r}}). \end{align*}$$

Thus, $A_{X,\Delta }(v)-\mathbf {L}^{\mathrm {NA}}({\mathcal {F}})=w({\mathcal {I}}_{\bullet }^{\frac {1}{r}})\le w({\mathcal {I}}_m^{\frac {1}{mr}})\le \frac {v(I_{m,i})-i}{mr}$ for all $m\in \mathbb {N}$ and $i\in \mathbb {Z}$ . It follows that

$$\begin{align*}r^{-1}v(I_{\bullet}^{(\lambda)})\ge \lambda+A_{X,\Delta}(v)-\mathbf{L}^{\mathrm{NA}}({\mathcal{F}}) \end{align*}$$

for all $\lambda \in \mathbb {R}$ . In particular, $\mathrm {lct}(X,\Delta ;I_{\bullet }^{(\lambda )})<r^{-1}$ for any $\lambda>\mathbf {L}^{\mathrm {NA}}({\mathcal {F}})$ . By the definition of log canonical slope, this implies $\mu ({\mathcal {F}})\le \mathbf {L}^{\mathrm {NA}}({\mathcal {F}})$ and proves the first part of the lemma.

If ${\mathcal {F}}$ is finitely generated, then ${\mathcal {I}}_{pm}={\mathcal {I}}_m^p$ for any sufficiently divisible $m,p\in \mathbb {N}$ and thus w can be chosen as a divisorial valuation. Since $\mathbf {L}^{\mathrm {NA}}({\mathcal {F}})<\lambda _{\max }({\mathcal {F}})$ , the valuation v cannot be the trivial one, otherwise equation (3.4) becomes $\lambda \le \mathbf {L}^{\mathrm {NA}}({\mathcal {F}})$ for all $\lambda < \lambda _{\max }({\mathcal {F}})$ and therefore $\lambda _{\max }({\mathcal {F}})\le \mathbf {L}^{\mathrm {NA}}({\mathcal {F}})$ , a contradiction. By [Reference Boucksom, Hisamoto and JonssonBHJ17, Lemma 4.1], we know that v is divisorial. Let ${\mathfrak {a}}_m=I_{m,\mu mr}$ . Again, ${\mathfrak {a}}_{pm}={\mathfrak {a}}_m^p$ for any sufficiently divisible $m,p\in \mathbb {N}$ as ${\mathcal {F}}$ is finitely generated. By equation (3.4), we have $r^{-1}v({\mathfrak {a}}_{\bullet })\ge A_{X,\Delta }(v)$ ; thus, from the definition of log canonical slope, we see that v necessarily computes

$$\begin{align*}\mathrm{lct}({\mathfrak{a}}_{\bullet})=m\cdot \mathrm{lct}({\mathfrak{a}}_m)=m\cdot \mathrm{ lct}(X,\Delta;\{s=0\}) \end{align*}$$

for sufficiently divisible m and general $s\in {\mathcal {F}}^{\mu mr} R_m$ . As $K_X+\Delta +\frac {1}{mr}\{s=0\}\sim _{\mathbb {Q}} 0$ , this easily implies that v is an lc place of some ${\mathbb {Q}}$ -complement.

Proof. It is not hard to see from the definition that $\mu ({\mathcal {F}}_v)\le A_{X,\Delta }(v)$ (c.f. [Reference Xu and ZhuangXZ20, Proposition 4.2]), thus the first inequality follows from Lemma 3.8. The second inequality follows from the first and the definition of $\mathbf {H}^{\mathrm {NA}}$ and $\tilde {\beta }$ .

Proof. It is not hard to see that the statement is unaffected by translation of the filtrations. Thus, by Lemma 3.8, we may assume that, after shifting the filtrations, there exists valuations $v_0, v_1$ on X with $A_{X,\Delta }(v_i)<\infty $ such that $\mathbf {L}^{\mathrm {NA}}({\mathcal {F}}_i)=\mu ({\mathcal {F}}_i)=A_{X,\Delta }(v_i)$ and $v_i(I_{{\mathcal {F}}_i,\bullet }^{(\lambda )})\ge r\lambda $ for all $\lambda \in \mathbb {R}$ and $i=0,1$ . In particular, ${\mathcal {F}}_i^{\lambda } R\subseteq {\mathcal {F}}_{v_i}^{\lambda } R$ for all $\lambda \in \mathbb {R}$ .

Let $(Y=\mathrm {Spec}(R),\Gamma )$ denote the affine cone over $(X,\Delta )$ with respect to the polarization $L=-r(K_X+\Delta )$ . Let $w_i$ be the $\mathbb {G}_m$ -invariant valuation on Y given by $w_i(s)=mr+v_i(s)$ for $s\in R_m$ (informally $w_i=r\cdot \mathrm {ord}_o+v_i$ ). Let ${\mathfrak {b}}_{t,\bullet }$ be the graded sequence of ideals on Y defined by

$$\begin{align*}{\mathfrak{b}}_{t,m}={\mathfrak{a}}_{\bullet}((1-t)w_0)\boxplus {\mathfrak{a}}_{\bullet}(tw_1):=\sum_{i=0}^m {\mathfrak{a}}_{m-i}((1-t)w_0)\cap {\mathfrak{a}}_i(tw_1). \end{align*}$$

In other words, ${\mathfrak {b}}_{t,m}$ is generated by those $s\in R$ with $\lfloor (1-t)w_0(s) \rfloor + \lfloor tw_1(s) \rfloor \ge m$ . For any $k\in \mathbb {Z}$ and any $s\in {\mathcal {F}}_t^{k+2} R_m$ , we have $(1-t)w_0(s)+tw_1(s)\ge mr+k+2$ by equation (3.1). It follows that s is a section of ${\mathfrak {b}}_{t,mr+k}$ (for if $a+b\ge k+2$ then $\lfloor a \rfloor +\lfloor b \rfloor \ge k$ ). Therefore, elements in ${\mathcal {F}}_t^{k+2} R_m$ yield sections of ${\mathfrak {b}}_{t,mr+k}$ on Y for any $k\in \mathbb {Z}$ . By [Reference Xu and ZhuangXZ21, Theorem 3.11], we have

$$ \begin{align*} \mathrm{lct}({\mathfrak{b}}_{t,\bullet}) & \le \mathrm{lct}({\mathfrak{a}}_{\bullet}((1-t)w_0))+\mathrm{lct}({\mathfrak{a}}_{\bullet}(tw_1)) \le (1-t)A_{Y,\Gamma}(w_0)+tA_{Y,\Gamma}(w_1) \\ & = 1+(1-t)A_{X,\Delta}(v_0)+tA_{X,\Delta}(v_1) = 1+(1-t) \mathbf{L}^{\mathrm{NA}}({\mathcal{F}}_0) + t \mathbf{L}^{\mathrm{NA}}({\mathcal{F}}_1). \end{align*} $$

Thus, for any rational $c>(1-t) \mathbf {L}^{\mathrm {NA}}({\mathcal {F}}_0) + t \mathbf {L}^{\mathrm {NA}}({\mathcal {F}}_1)$ and any $s\in {\mathcal {F}}^{cmr+2}_t R_m$ (where m is sufficiently divisible), as it yields a section of ${\mathfrak {b}}_{t,(1+c)mr}$ on Y, the pair $(Y,\Gamma +\frac {1}{mr}\{s=0\})$ is not lc. Using [Reference KollárKol13, Lemma 3.1(5)], it follows that the base $(X,\Delta +\frac {1}{mr}\{s=0\})$ is not lc for any $m\in \mathbb {N}$ and any $s\in {\mathcal {F}}^{cmr+2}_t R_m$ . By the definition of log canonical slope, this implies that $\mu ({\mathcal {F}}_t)\le c$ . By Lemma 3.8 and the fact that $c>(1-t) \mathbf {L}^{\mathrm {NA}}({\mathcal {F}}_0) + t \mathbf {L}^{\mathrm {NA}}({\mathcal {F}}_1)$ was arbitrary, the result follows.

Proof. Set $\boldsymbol {\nu }:= \mathrm {DH}_{{\mathcal {F}}_0,{\mathcal {F}}_1}$ . We compute

(3.5) $$ \begin{align} \mathbf{E}^{\mathrm{NA}}({\mathcal{F}}_t) = \int_{\mathbb{R}} \lambda \, \mathrm{DH}_{{\mathcal{F}}_t}(\mathrm{d} \lambda) =\int_{\mathbb{R}^2} \left( (1-t) x+ ty \right) \, \mathrm{d} \boldsymbol{\nu} = (1-t) \int_{\mathbb{R}^2} x \, \mathrm{d}\boldsymbol{\nu} + t \int_{\mathbb{R}^2} y \, \mathrm{d} \boldsymbol{\nu}, \end{align} $$

where the second equality is by Proposition 3.6. From this, the result follows.

Proof. Set $\boldsymbol {\nu }: = \mathrm {DH}_{{\mathcal {F}}_0,{\mathcal {F}}_1}$ , $f(x,y): = e^{-x}$ , and $g(x,y) := e^{-y}$ . For $t\in [0,1]$ ,

$$\begin{align*}\widetilde{S}({\mathcal{F}}_t) = -\log \int_{\mathbb{R}^2} e^{-\lambda } \, \mathrm{DH}_{{\mathcal{F}}_t}(\mathrm{d}\lambda) = -\log \int_{\mathbb{R}^2} e^{-(1-t)x - ty} \, \mathrm{d}\boldsymbol{\nu} = - \log \lVert f^{1-t} g^{t} \rVert_{1,\boldsymbol{\nu}} , \end{align*}$$

where the second equality is by Proposition 3.6. Hölder’s inequality implies

$$ \begin{align*} -\log \lVert f^{1-t} g^{t} \rVert_{1,\boldsymbol{\nu}} \geq - \log \left( \lVert f \rVert_{1,\boldsymbol{\nu}}^{1-t} \lVert g \rVert_{1,\boldsymbol{\nu}}^t \right) &= - (1-t) \log \lVert f \rVert_{1,\boldsymbol{\nu}} - t \log \lVert g \rVert_{1,\boldsymbol{\nu}}\\ & = (1-t) \widetilde{S}({\mathcal{F}}_0) + t \widetilde{S}({\mathcal{F}}_1) \end{align*} $$

Furthermore, the inequality is strict unless (i) $f=0$ or $g=0 \ \boldsymbol {\nu }$ -a.e. or (ii) there exists $c>0$ so that $f- c g =0 \ \boldsymbol {\nu }$ -a.e.

Condition (i) cannot occur since f and g are $>0$ . Condition (ii) is equivalent to saying $x-y-d=0 \ \boldsymbol {\nu }$ -a.e., where $d: = -\ln (c)$ . Now, if we write ${\mathcal {G}}$ for the filtration of R defined by ${\mathcal {G}}^{\lambda } R_m : = {\mathcal {F}}_1^{\lambda -mrd}R_m$ , then

$$\begin{align*}\lVert x-y-d \rVert_{1,\boldsymbol{\nu}} = \int_{\mathbb{R}^2} \left| x-y-d \right| \, \mathrm{d}\boldsymbol{\nu} = \int_{\mathbb{R}} \left| \lambda \right| \, \mathrm{RLM}_{{\mathcal{F}}_0,{\mathcal{G}}} (\mathrm{d}\lambda) = d_1({\mathcal{F}},{\mathcal{G}}) , \end{align*}$$

where the second is by Proposition 3.6. Therefore, (ii) holds iff $d_{1}({\mathcal {F}},{\mathcal {G}})=0$ .

Proof of Theorem 3.7.

The result follows immediately from Propositions 3.12, 3.13 and 3.14.

Proof. Consider the geodesic $({\mathcal {F}}_t)_{t\in [0,1]}$ connecting ${\mathcal {F}}_0 : = {\mathcal {F}}_v$ and ${\mathcal {F}}_1 :={\mathcal {F}}_w$ . For $t\in (0,1)$ ,

$$\begin{align*}\mathbf{H}^{\mathrm{NA}}({\mathcal{F}}_t) \leq (1-t) \mathbf{H}^{\mathrm{NA}}({\mathcal{F}}_0) + t \mathbf{H}^{\mathrm{NA}}({\mathcal{F}}_1) \leq (1-t) \tilde{\beta}_{X,\Delta}(v) + t \tilde{\beta}_{X,\Delta}(w) = h(X,\Delta), \end{align*}$$

where first inequality is Theorem 3.7 and the second Corollary 3.11. Since $h(X,\Delta ) \leq \mathbf {H}^{\mathrm {NA}}({\mathcal {F}}_t)$ , the first inequality cannot be strict. Therefore, Theorem 3.7 further implies there exists $d\in \mathbb {R}$ so that $d_{1}({\mathcal {F}}_0,{\mathcal {G}})=0$ , where ${\mathcal {G}}^{\lambda } R_m : = {\mathcal {F}}_1^{\lambda -mrd} R_m$ .

Next, note that $d=0$ , since

$$\begin{align*}0= \lambda_{\min}({\mathcal{F}}_0) = \lambda_{\min}({\mathcal{G}}) = \lambda_{\min}({\mathcal{F}}_1) +d = d, \end{align*}$$

where first and last inequality is by Lemma 2.4 and the second by Proposition 3.1. Therefore, $d_{1}({\mathcal {F}}_0,{\mathcal {F}}_1)=0$ . By Lemma 3.16, we conclude $v=w$ .

Proof. It is enough to show $v(f)=w(f)$ for all $f\in R_m$ and $m \in \mathbb {N}$ . Indeed, for any $\lambda \in \mathbb {R}$ , we may choose some integer $m\gg 0$ such that $ {\mathcal {O}}_X(mL)\otimes {\mathfrak {a}}_{\lambda } (v)$ is globally generated, where ${\mathfrak {a}}_{\lambda } (v):=\{f\in {\mathcal {O}}_X\,\vert \,v(f)\ge \lambda \}$ denotes the valuation ideal. If $v(f)=w(f)$ for all $f\in H^0(X, {\mathcal {O}}_X(mL) \otimes {\mathfrak {a}}_{\lambda } (v))\subseteq R_m$ , then ${\mathfrak {a}}_{\lambda } (v)\subseteq {\mathfrak {a}}_{\lambda } (w)$ . Switching the role of v and w gives the reverse containment. Thus, ${\mathfrak {a}}_{\lambda } (v) = {\mathfrak {a}}_{\lambda } (w)$ for all $\lambda \in \mathbb {R}$ , and, hence, $v=w$ .

Suppose now that $a=v(f)\neq w(f)=b$ for some $f\in R_m$ . Without loss of generality, we may assume $a>b$ . Let $\lambda =\lambda _{\max }({\mathcal {F}}_w)$ . Fix some $\varepsilon \in (0,a-b)$ , and let p be a sufficiently large integer such that $\lambda r\le (a-b-\varepsilon )p$ . Consider the subspace $V_k:=f^{kp} \cdot R_k\subseteq {\mathcal {F}}_v^{akp} R_{k(mp+1)}$ . For any $g\in V_k$ , we have $w(g)\le bkp+k\lambda r\le (a-\varepsilon )kp$ by our choice of p. Thus, for any basis $(s_1,\cdots ,s_N)$ of $R_{k(mp+1)}$ that is compatible with both ${\mathcal {F}}_v$ and ${\mathcal {F}}_w$ , the part that spans ${\mathcal {F}}_v^{akp} R_{k(mp+1)}$ contains at least $\dim V_k = \dim R_k$ elements $s_i$ with $w(s_i)\le (a-\varepsilon )kp$ . In particular for these $s_i$ , we have $v(s_i)-w(s_i)\ge \varepsilon kp$ . It follows that

$$\begin{align*}\int_{\mathbb{R}} |\lambda| \mathrm{d} \nu_{k(mp+1)}^{{\mathcal{F}}_v,{\mathcal{F}}_w}\ge \frac{\frac{\varepsilon p}{(mp+1)r}\cdot \dim R_k}{\dim R_{k(mp+1)}} \end{align*}$$

for all $k\gg 0$ . Letting $k\to \infty $ , we get $d_1({\mathcal {F}}_v,{\mathcal {F}}_w)>0$ , contradicting our assumption. Thus, we must have $v(f)=w(f)$ for all $m\in \mathbb {N}$ and all $f\in R_m$ and therefore $v=w$ .

Proof. By [Reference Han and LiHL20b, Corollary 4.9], there is a quasi-monomial valuation $v\in \mathrm {Val}_X^{\circ } \cup \{v_{\mathrm {triv}}\}$ computing $h(X,\Delta )$ . The uniqueness is by Theorem 3.15.

Proof. Let r be the rational rank of $v_0$ . Since $v_0$ is quasi-monomial, there exists a log resolution $\pi \colon Y\to X$ , a regular system of parameters $y_1,\ldots , y_r$ at a point $\eta \in Y$ , and $\alpha \in \mathbb {R}^r_+$ such that $v_0=v_{\alpha }$ . Let $W=C_Y(v_0)$ . Possibly after further blowups, we may choose a flag $W_{\bullet }\colon W=W_0\supseteq \cdots \supseteq W_{n-r}=\{\text {point}\}$ of smooth subvarieties such that each $W_{i+1}$ is a divisor in $W_i$ . Let $\nu =\nu _{W_{\bullet }}\colon K(W)^{\times } \to \mathbb {Z}^{n-r}$ be the induced valuation as in [Reference Lazarsfeld and MustaţăLM09]. Any nonzero $f\in {\mathcal {O}}_{Y,W}$ can be written as $f=c_{\beta } y^{\beta } +f_1$ for some (uniquely determined) $\beta \in \mathbb {N}^r$ and $c_{\beta },f_1\in {\mathcal {O}}_{Y,W}$ such that $\langle \alpha ,\beta \rangle =v_0(f)$ and $v_0(f_1)>v_0(f)$ . Moreover, $0\neq \bar {c}_{\beta } \in K(W)$ is well defined and does not depend on the choice of $c_{\beta }$ . Now, consider the valuation $w_0\colon K(X)^{\times } \to \mathbb {Z}^n$ given by setting $w_0(f)=(\beta ,\nu (\bar {c}_{\beta }))$ for $f\in {\mathcal {O}}_{Y,W}$ . It is not hard to check from the construction that $w_0$ is faithful and has one-dimensional leaves. Clearly, $v_0(f)=\langle (\alpha ,0,\cdots ,0),w_0(f)\rangle $ . Thus, $w_0$ is the good valuation we want.

Proof of Lemma 4.2.

Since the argument is very similar to those in [Reference Blum and JonssonBJ20], we only sketch the proof. By Lemma 4.4, there exists some good valuation $w_0\colon K(X)\to \mathbb {Z}^n$ and some $u_0\in \mathbb {R}^n_{\ge 0}$ such that $v_0(f)=\langle u_0,w_0(f) \rangle $ . Let $\Sigma \subseteq \mathbb {R}^n$ be the corresponding Okounkov body (see [Reference Kaveh and KhovanskiiKK12]), that is, the closed convex hull of $\bigcup _{m\geq 1}\{\frac {w_0(s)}{mr}\,\vert \,s\in R_m\setminus \{0\}\}$ . We regard the function g also as a positive function on $\Sigma $ by $g(\alpha )=g(\langle u_0,\alpha \rangle )$ . Let $\rho $ denote the Lebesgue measure on $\Sigma $ and $\rho _m$ the atomic probability measure supported on $\Sigma \cap \frac {1}{m} \mathbb {Z}^n$ as defined in [Reference Blum and JonssonBJ20, Section 2.2]. Note that $\lim _{m\to \infty }\rho _m=\rho $ in the weak topology of measures (see [Reference Blum and JonssonBJ20, Theorem 2.1]). Using the argument of [Reference Blum and JonssonBJ20, Lemma 2.2] and the uniform continuity of g on $\Sigma $ , we see that for each $\varepsilon>0$ there exists $m_0=m_0(\varepsilon )$ such that

$$\begin{align*}\int_{\Sigma} fg \mathrm{d} \rho_m \le \int_{\Sigma} fg \mathrm{d} \rho + \varepsilon \end{align*}$$

for every $m\ge m_0$ and every concave function $f\colon \Sigma \to \mathbb {R}$ satisfying $0\le f\le 1$ . We may then apply the proof of [Reference Blum and JonssonBJ20, Corollary 2.10] to the concave transform of ${\mathcal {F}}$ and conclude that (after possibly enlarging $m_0$ ) $S_{g,m}(v_0;{\mathcal {F}})\le (1+\varepsilon )S_g(v_0;{\mathcal {F}})$ for all linearly bounded filtrations ${\mathcal {F}}$ and all $m\ge m_0$ .

Proof. It is not hard to check from the definition that

(4.1) $$ \begin{align} S_{g,m}(v_0;v) = \max \{ v(D) \, \vert \, D \text{ is a } g\text{-weighted } (m,v_0)\text{-basis type divisor} \} \end{align} $$

and in fact $S_{g,m}(v_0;v)=v(D)$ for any g-weighted $(m,v_0)$ -basis type divisor D that’s also compatible with v. Moreover, when $v_0,v\in \mathrm {Val}_X^{\mathbb {T},\circ }$ , such a divisor D can be chosen to be $\mathbb {T}$ -invariant by choosing compatible basis in each component of the weight decomposition under the torus action. Hence,

(4.2) $$ \begin{align} \delta_{g,\mathbb{T},m}(X,\Delta,v_0) =\inf_{v\in\mathrm{Val}_X^{\mathbb{T},\circ}} \frac{A_{X,\Delta}(v)}{S_{g,m}(v_0;v)}. \end{align} $$

Combined with Lemma 4.2, the argument in the proof of [Reference Blum and JonssonBJ20, Theorem 4.4] then yields $\delta _{g,\mathbb {T}}(X,\Delta ,v_0)=\lim _{m\to \infty }\delta _{g,\mathbb {T},m}(X,\Delta ,v_0)$ .

Proof. Let ${\mathcal {F}}$ be the trivial filtration of R. By Lemma 3.8 and [Reference Xu and ZhuangXZ20, Lemma A.6], we have $\mathbf {L}^{\mathrm {NA}}({\mathcal {F}})=\mathbf {L}^{\mathrm {NA}}({\mathcal {F}}_{\eta })$ for any $\eta \in N_{\mathbb {R}}$ . Thus, $\mathbf {D}^{\mathrm {NA}}_g({\mathcal {F}}_{\eta })=\mathbf {D}^{\mathrm {NA}}_g({\mathcal {F}})+\mathrm {Fut}_g(\eta )=\mathrm { Fut}_g(\eta )$ . By assumption, $\mathbf {D}^{\mathrm {NA}}_g({\mathcal {F}}_{\eta })\ge 0$ for any $\eta \in N$ . By linearity, $\mathrm {Fut}_g\equiv 0$ on N and, hence, the same holds on $N_{\mathbb {R}}$ .

Proof. When $v_0=\mathrm {wt}_{\xi }$ this is treated in [Reference Han and LiHL20, Section 7] using [Reference Li and XuLX14]. Here, we present a proof that is independent of [Reference Li and XuLX14]. It is clear that (1) implies (2). By [Reference Blum, Liu and XuBLX22, Theorem A.2], (2) implies (3). Thus, it remains to show (3) implies (1). To see this, let ${\mathcal {F}}$ be a finitely generated $\mathbb {T}$ -equivariant $\mathbb {Z}$ -filtration of R. If $\mathbf {L}^{\mathrm {NA}}({\mathcal {F}})\ge \lambda _{\max }({\mathcal {F}})$ , then we already have $\mathbf {D}^{\mathrm {NA}}_g({\mathcal {F}})\ge \mathbf {J}^{\mathrm {NA}}_g({\mathcal {F}})\ge \mathbf {J}^{\mathrm {NA}}_{g,\mathbb {T}}({\mathcal {F}})$ . Thus, to prove (1) we may assume that $\mathbf {L}^{\mathrm {NA}}({\mathcal {F}})<\lambda _{\max }({\mathcal {F}})$ . By the second part of Lemma 3.8 and Remark 3.9, there exists some divisorial lc place of a ${\mathbb {Q}}$ -complement $v\in \mathrm {Val}_X^{\mathbb {T},\circ }$ such that

$$\begin{align*}r^{-1}v(I^{(\lambda)}_{\bullet}) - A_{X,\Delta}(v)\ge \lambda-\mathbf{L}^{\mathrm{NA}}({\mathcal{F}}) \end{align*}$$

for all $\lambda \in \mathbb {R}$ . Since $\mathbf {D}^{\mathrm {NA}}_g({\mathcal {F}})$ and $\mathbf {J}^{\mathrm {NA}}_{g,\mathbb {T}}({\mathcal {F}})$ are both translation invariant, we may shift ${\mathcal {F}}$ so that $\mathbf {L}^{\mathrm {NA}}({\mathcal {F}})=A_{X,\Delta }(v)$ . The above inequality then becomes $v(I^{(\lambda )}_{\bullet })\ge \lambda r$ and therefore ${\mathcal {F}}^{\lambda } R\subseteq {\mathcal {F}}_v^{\lambda } R$ for all $\lambda \in \mathbb {R}$ .

Let $\eta \in N_{\mathbb {R}}$ , and let ${\mathcal {G}}$ be the $\eta $ -twist of ${\mathcal {F}}_v$ . Then we also have ${\mathcal {F}}_{\eta }^{\lambda } R\subseteq {\mathcal {G}}^{\lambda } R$ for all $\lambda $ , and this clearly implies

$$\begin{align*}\mathbf{E}^{\mathrm{NA}}_g({\mathcal{F}}_{\eta})\le \mathbf{E}^{\mathrm{NA}}_g({\mathcal{G}})\quad \text{and}\quad \lambda_{\max}({\mathcal{F}}_{\eta})\le \lambda_{\max}({\mathcal{G}}). \end{align*}$$

Since $\mathbf {L}^{\mathrm {NA}}({\mathcal {F}})=A_{X,\Delta }(v)\ge \mathbf {L}^{\mathrm {NA}}({\mathcal {F}}_v)$ by Corollary 3.11, we also get $\mathbf {L}^{\mathrm {NA}}({\mathcal {F}}_{\eta })\ge \mathbf {L}^{\mathrm {NA}}({\mathcal {G}})$ by Lemma 3.8 and [Reference Xu and ZhuangXZ20, Lemma A.6]. Note that by Lemma 4.8, [Reference Blum, Liu and XuBLX22, Theorem A.2] and (3) we have $\mathrm {Fut}_g\equiv 0$ on $N_{\mathbb {R}}$ , thus $\mathbf {D}^{\mathrm {NA}}_g({\mathcal {F}}_{\eta })=\mathbf {D}^{\mathrm {NA}}_g({\mathcal {F}})$ and $\mathbf {D}^{\mathrm {NA}}_g({\mathcal {G}})=\mathbf {D}^{\mathrm {NA}}_g({\mathcal {F}}_v)$ .

Since

$$ \begin{align*} \mathbf{D}^{\mathrm{NA}}_g({\mathcal{F}})-c\cdot \mathbf{J}^{\mathrm{NA}}_g({\mathcal{F}}) = \mathbf{L}^{\mathrm{NA}}({\mathcal{F}}) - (1-c)\mathbf{E}^{\mathrm{NA}}_g({\mathcal{F}})-c\cdot \lambda_{\max}({\mathcal{F}}), \end{align*} $$

we then obtain

$$ \begin{align*} \mathbf{D}^{\mathrm{NA}}_g({\mathcal{F}}) - c\cdot \mathbf{J}^{\mathrm{NA}}_{g,\mathbb{T}}({\mathcal{F}}) & \ge \mathbf{D}^{\mathrm{NA}}_g({\mathcal{F}}_{\eta}) - c \cdot \mathbf{J}^{\mathrm{NA}}_g({\mathcal{F}}_{\eta}) \\ & \ge \mathbf{D}^{\mathrm{NA}}_g({\mathcal{G}}) - c \cdot \mathbf{J}^{\mathrm{NA}}_g({\mathcal{G}}) = \mathbf{D}^{\mathrm{NA}}_g({\mathcal{F}}_v) - c \cdot \mathbf{J}^{\mathrm{NA}}_g({\mathcal{G}}). \end{align*} $$

As $\eta \in N_{\mathbb {R}}$ is arbitrary, this gives

$$\begin{align*}\mathbf{D}^{\mathrm{NA}}_g({\mathcal{F}}) - c \cdot \mathbf{J}^{\mathrm{NA}}_{g,\mathbb{T}}({\mathcal{F}}) \ge \mathbf{D}^{\mathrm{NA}}_g({\mathcal{F}}_v) - c \cdot \mathbf{J}^{\mathrm{NA}}_{g,\mathbb{T}}({\mathcal{F}}_v). \end{align*}$$

By (3), the right-hand side is $\ge 0$ . Thus, the same is true for the left-hand side. Since ${\mathcal {F}}$ is arbitrary, this proves that (3) implies (1).

Proof. Assume that $(X,\Delta ,v_0)$ is $\mathbb {T}$ -equivariantly g-Ding semistable. If $\delta :=\delta _{g,\mathbb {T}}(X,\Delta ,v_0) < 1$ , then we may choose some $\varepsilon>0$ such that $(1+\varepsilon )^2 \delta <1$ . Let $m\gg 0$ be such that $\delta _m:=\delta _{g,\mathbb {T},m}(X,\Delta ,v_0)<(1+\varepsilon )\delta $ . By Lemma 4.2, we may also assume that $S_{g,m}(v_0;v)\le (1+\varepsilon )S_g(v_0;v)$ for any $v\in \mathrm {Val}_X^{\circ }$ . By definition, $\delta _m=\mathrm {lct}(X,\Delta ;D)$ for some $\mathbb {T}$ -invariant g-weighted $(m,v_0)$ -basis type divisor D. Thus, if w is a $\mathbb {T}$ -invariant divisorial valuation that computes the lct, we would have

$$ \begin{align*}A_{X,\Delta}(w)=\delta_m S_{g,m}(v_0;w)<(1+\varepsilon)^2 \delta\cdot S_g(v_0;w)<S_g(v_0;w).\end{align*} $$

On the other hand, since $\delta _m<1$ , we know that $-(K_X+\Delta +\delta _m D)$ is ample, thus $\mathrm {gr}_w R$ is finitely generated by [Reference Birkar, Cascini, Hacon and McKernanBCHM10, Corollary 1.4.3]. In other words, w induces a $\mathbb {T}$ -equivariant test configuration of $(X,\Delta )$ . As $(X,\Delta ,v_0)$ is $\mathbb {T}$ -equivariantly g-Ding semistable, Corollary 3.11 gives $A_{X,\Delta }(w) - S_g(v_0;w)\ge \mathbf {D}^{\mathrm {NA}}_{g}({\mathcal {F}}_w) \geq 0$ , a contradiction. Therefore, $\delta _{g,\mathbb {T}}(X,\Delta ,v_0)\ge 1$ when $(X,\Delta ,v_0)$ is $\mathbb {T}$ -equivariantly g-Ding semistable.

Conversely, if $\delta _{g,\mathbb {T}}(X,\Delta ,v_0)\ge 1$ , then for any $v\in \mathrm { Val}_X^{\mathbb {T},\circ }$ we have $A_{X,\Delta }(v)\ge S_g(v_0;v)$ . If v is an lc place of a ${\mathbb {Q}}$ -complement, then we also have $\mathbf {L}^{\mathrm {NA}}({\mathcal {F}}_v)=A_{X,\Delta }(v)$ (c.f. [Reference Xu and ZhuangXZ20, Proposition 4.2]). Thus, $\mathbf {D}^{\mathrm {NA}}_g({\mathcal {F}}_v)\ge 0$ for any $v\in \mathrm {Val}_X^{\mathbb {T},\circ }$ that is an lc place of a ${\mathbb {Q}}$ -complement. By Lemma 4.12 (with $c=0$ ), this implies that $(X,\Delta ,v_0)$ is $\mathbb {T}$ -equivariantly g-Ding semistable.

Proof. The argument is very similar to those in [Reference Xu and ZhuangXZ20, Appendix], so we only give a sketch. First, note that the assumption remains true if we enlarge the torus $\mathbb {T}$ , and clearly if the conclusion holds for a maximal torus containing $\mathbb {T}$ , then it also holds for $\mathbb {T}$ . Thus, we may assume that $\mathbb {T}$ is a maximal torus.

By Lemma 4.12 and [Reference Blum, Liu and XuBLX22, Theorem 3.5], we know that there exist some integer $N>0$ and a sequence of divisorial lc places of N-complements $v_i\in \mathrm { Val}_X^{\mathbb {T},\circ }$ (not of the form $\mathrm {wt}_{\eta }$ ) such that

$$\begin{align*}\lim_{i\to\infty} \frac{\mathbf{D}^{\mathrm{NA}}_g({\mathcal{F}}_{v_i})}{\mathbf{J}^{\mathrm{NA}}_{g,\mathbb{T}}({\mathcal{F}}_{v_i})} = 0. \end{align*}$$

By the constructibility result [Reference Xu and ZhuangXZ20, Lemma A.11] and arguing as in the proof of [Reference Xu and ZhuangXZ20, Theorem A.5], we may assume that the $v_i$ ’s are lc places of the same ${\mathbb {Q}}$ -complement and after rescaling $v=\lim _i v_i\in \mathrm {Val}_X^{\mathbb {T},\circ }$ exists and $v\neq \mathrm {wt}_{\eta }$ for any $\eta \in N_{\mathbb {R}}$ . By the following Lemma 4.15, we have $\lim _i S_g(v_0;v_i) = S_g(v_0;v)$ . Since $v_i$ are lc places of ${\mathbb {Q}}$ -complements, it follows from Lemma 3.8 that $\mathbf {L}^{\mathrm {NA}}({\mathcal {F}}_{v_i})=A_{X,\Delta }(v_i)$ and thus $\lim _i \mathbf {D}^{\mathrm {NA}}_g({\mathcal {F}}_{v_i}) = \mathbf {D}^{\mathrm {NA}}_g({\mathcal {F}}_{v})$ . Since the function $v\mapsto \lambda _{\max }({\mathcal {F}}_v)=T_{X,\Delta }(v)$ is also continuous on $\mathrm {QM}(Y,E)$ by [Reference Blum, Liu and XuBLX22, Proposition 2.4], we get $\lim _i \mathbf {J}^{\mathrm {NA}}_g({\mathcal {F}}_{v_i}) = \mathbf {J}^{\mathrm {NA}}_g({\mathcal {F}}_{v})$ as well. Thus,

$$\begin{align*}\frac{\mathbf{D}^{\mathrm{NA}}_g({\mathcal{F}}_v)}{\mathbf{J}^{\mathrm{NA}}_{g,\mathbb{T}}({\mathcal{F}}_v)} = \lim_{i\to\infty} \frac{\mathbf{D}^{\mathrm{NA}}_g({\mathcal{F}}_{v_i})}{\mathbf{J}^{\mathrm{NA}}_{g,\mathbb{T}}({\mathcal{F}}_{v_i})}= 0, \end{align*}$$

which implies $\mathbf {D}^{\mathrm {NA}}_g({\mathcal {F}}_v)=A_{X,\Delta }(v)-S_g(v_0;v)=0$ . Since $\delta _g(X,\Delta ,\xi )\ge 1$ by Corollary 4.13, we see that $\delta _g(X,\Delta ,\xi ) = 1$ and v computes $\delta _g (X,\Delta ,\xi )$ .

Proof. Fix a point $\eta \in \mathrm {Supp}(E)$ and local coordinates $y_1,\ldots , y_r \in {\mathcal {O}}_{Y,\eta }$ so that each $y_j$ cuts out an irreducible component of E at $\eta $ . For $\alpha =( \alpha _1,\ldots , \alpha _r)\in \mathbb {R}_{\geq 0}^{r}$ , write $v_{\alpha }\in \mathrm {Val}_X$ for the associated quasi-monomial valuation satisfying $v_{\alpha }(y_j) = \alpha _j$ .

To prove the continuity of $S_g(v_0;\, \cdot \, )$ on $\mathbb {R}_{\geq 0}^r$ , fix a convergent sequence $\alpha ^{i}$ in $\mathbb {R}_{\geq 0}^{r}$ and set $ \alpha : = \lim _i \alpha ^{i}$ . We aim to show $\lim _i S_g(v_0;v_i) = S_g(v_0;v)$ , where $v_i: = v_{\alpha ^i}$ and $v : = v_{\alpha }$ .

First, note that $S_g(v_0;c w)= cS_g(v_0;w) $ for all $w\in \mathrm {Val}_X^{\circ }$ and $c\in \mathbb {R}_{>0}$ . Therefore, after rescaling the $\alpha ^i$ and removing finitely many terms, we may assume the sequence $(\alpha ^i)$ is nonincreasing. Hence, $v_i \geq v_{i+1} \geq v$ for all i. Next, consider the Okounkov body $\Sigma \subset \mathbb {R}^n$ induced by the good valuation form Lemma 4.4. Write $G_i$ and G for the concave functions $\Sigma \to \mathbb {R}_{\geq 0}$ induced by the filtrations ${\mathcal {F}}_{v}$ and ${\mathcal {F}}_{v_i}$ (see [Reference Blum and JonssonBJ20, Section 2.5]) and set $\mathrm {vol}_g(\Sigma )= \int _{\Sigma } g\, \mathrm {d} \rho $ . Note that

(4.3) $$ \begin{align} |S_g(v_0;v_i) - S_g(v_0;v) | = \left| \frac{1}{ \mathrm{vol}_g (\Sigma) } \int_{\Sigma }g G_i \, \mathrm{d} \rho - \frac{1}{ \mathrm{vol}_g (\Sigma) } \int_{\Sigma } g G \, \mathrm{d} \rho \right | = \frac{1}{\mathrm{vol}_g(\Sigma)} \int_{\Sigma} g |G_i-G| \, \mathrm{d} \rho ,\end{align} $$

where the second equality uses that $G_i\leq G$ . Using that $\lim _i S_1(v_0; v_i) = S_{1}(v_0;v)$ by [Reference Blum, Liu and XuBLX22, Proposition 2.4] and equation (4.3), we see $(G_i)$ converges to G a.e. Therefore, the dominated convergence theorem implies

$$\begin{align*}\lim_{i\to \infty} |S_g(v_0;v_i) - S_g(v_0;v) | = \lim_{i \to \infty } \frac{1}{\mathrm{vol}_g(\Sigma)} \int_{\Sigma} g |G_i-G| \, \mathrm{d} \rho =0 ,\end{align*}$$

which completes the proof.

Proof. We first show the ‘if’ part. It suffices to show that $\tilde {\beta }_{X,\Delta }(v) \geq \tilde {\beta }_{X,\Delta }(v_0)$ for any valuation $v\in \mathrm {Val}_X^{\circ }$ . Let ${\mathcal {F}}_0:={\mathcal {F}}_{v_0}$ and ${\mathcal {F}}_1:={\mathcal {F}}_{v}$ . Let $\boldsymbol {\nu }$ denote the compatible measure of ${\mathcal {F}}_0$ and ${\mathcal {F}}_1$ from Section 3.1.3. From the definitions, we know that

(5.1) $$ \begin{align} \widetilde{S}(v_0;v_0)=\frac{\int_{\mathbb{R}^2} x e^{-x} d\boldsymbol{\nu}}{\int_{\mathbb{R}^2} e^{-x} d\boldsymbol{\nu}}, \quad \textrm{and}\quad \widetilde{S}(v_0;v)=\frac{\int_{\mathbb{R}^2} y e^{-x} d\boldsymbol{\nu}}{\int_{\mathbb{R}^2} e^{-x} d\boldsymbol{\nu}}. \end{align} $$

Consider the following function $f:[0,1]\to \mathbb {R}$ given by

$$\begin{align*}f(t):= (1-t)A_{X,\Delta}(v_0) +tA_{X,\Delta}(v) +\log \int_{\mathbb{R}^2} e^{-(1-t)x-ty} d\boldsymbol{\nu}. \end{align*}$$

It is clear that $f(0)=\tilde {\beta }_{X,\Delta }(v_0)$ and $f(1)=\tilde {\beta }_{X,\Delta }(v)$ . By Hölder’s inequality as in the proof of Proposition 3.14, we know that $f(t)$ is convex in t. Moreover, we have

$$ \begin{align*} f'(0) & = A_{X,\Delta}(v) - A_{X,\Delta}(v_0) + \frac{\int_{\mathbb{R}^2} (x-y)e^{-x} d\boldsymbol{\nu}}{\int_{\mathbb{R}^2} e^{-x} d\boldsymbol{\nu}}\\ & = (A_{X,\Delta}(v)- \widetilde{S}(v_0;v)) - (A_{X,\Delta}(v_0)-\widetilde{S}(v_0;v_0)) \geq 0. \end{align*} $$

Thus, $f(1)\geq f(0)$ and the ‘if’ part is proved.

Next, we show the ‘only if’ part. Let $({\mathcal {F}}_t)_{t\in [0,1]}$ be the geodesic of filtrations connecting ${\mathcal {F}}_0$ and ${\mathcal {F}}_1$ . Since $v_0$ computes $h(X,\Delta )$ , we know that $\mathbf {H}^{\mathrm {NA}}({\mathcal {F}}_t)\geq \mathbf {H}^{\mathrm {NA}}({\mathcal {F}}_0)=f(0)$ for any $t\in [0,1]$ . Recall that

$$\begin{align*}\mathbf{H}^{\mathrm{NA}}({\mathcal{F}}_t)= \mathbf{L}^{\mathrm{NA}}({\mathcal{F}}_t)-\widetilde{S}({\mathcal{F}}_t)=\mathbf{L}^{\mathrm{NA}}({\mathcal{F}}_t)+ \log \int_{\mathbb{R}^2} e^{-(1-t)x-ty} d\boldsymbol{\nu}. \end{align*}$$

By Proposition 3.12 and Corollary 3.11, we obtain

$$\begin{align*}\mathbf{L}^{\mathrm{NA}}({\mathcal{F}}_t) \leq (1-t)\mathbf{L}^{\mathrm{NA}}({\mathcal{F}}_0)+t\mathbf{L}^{\mathrm{NA}}({\mathcal{F}}_1)\leq (1-t) A_{X,\Delta}(v_0)+t A_{X,\Delta}(v). \end{align*}$$

Hence, we have $\mathbf {H}^{\mathrm {NA}}({\mathcal {F}}_t)\leq f(t)$ which implies that $f(t)\ge f(0)$ for all $t\in [0,1]$ and thus $f'(0)\geq 0$ , that is,

$$\begin{align*}A_{X,\Delta}(v)- \widetilde{S}(v_0;v) \geq A_{X,\Delta}(v_0)-\widetilde{S}(v_0;v_0). \end{align*}$$

Since $v\in \mathrm {Val}_{X}^{\circ }$ is arbitrary, the above inequality remains true if we replace v by $\lambda v$ for any $\lambda \in \mathbb {R}_{>0}$ , that is,

$$\begin{align*}\lambda (A_{X,\Delta}(v) - \widetilde{S}(v_0;v))\geq A_{X,\Delta}(v_0)-\widetilde{S}(v_0;v_0). \end{align*}$$

Thus, we have $A_{X,\Delta }(v) \geq \widetilde {S}(v_0;v)$ for any $v\in \mathrm {Val}_X^{\circ }$ and $A_{X,\Delta }(v_0)\leq \widetilde {S}(v_0;v_0)$ , which implies that $A_{X,\Delta }(v_0)= \widetilde {S}(v_0;v_0)$ . This finishes the proof.

Question 5.3. Assume that $\delta _{g,\mathbb {T}}(X,\Delta ,v_0)\le 1$ . Let $v\in \mathrm {Val}_X^{\mathbb {T},\circ }$ be a valuation that computes $\delta _{g,\mathbb {T}}(X,\Delta ,v_0)$ . Is the associated graded ring $\mathrm {gr}_v R:=\mathrm {gr}_{{\mathcal {F}}_v} R$ finitely generated?

Proof. We only sketch the proof since the argument is almost the same as in [Reference Liu, Xu and ZhuangLXZ22, Lemma 3.1]. After rescaling, we assume that $A_{X,\Delta }(v)=1$ . Since v is quasi-monomial, we have $v\in \mathrm {QM}(Y,E)$ for some log smooth model $(Y,E)\to (X,\Delta )$ . Let ${\mathfrak {a}}_m = {\mathfrak {a}}_m(v)$ ( $m\in \mathbb {N}$ ) be the valuation ideals. By the proof of [Reference Liu, Xu and ZhuangLXZ22, (3.1)] (which only uses the fact that v is quasi-monomial), we know that, for any $\varepsilon \in (0,1)$ , there exists $\varepsilon _0>0$ and divisorial valuations $v_i=\mathrm { ord}_{F_i}\in \mathrm {QM}(Y,E)$ ( $i=1,\cdots ,r$ ) such that v is in the convex hull of $v_1,\cdots ,v_r$ and $A_{X,\Delta +{\mathfrak {a}}_{\bullet }^{1-\varepsilon _0}}(F_i)<\varepsilon $ for all i. By assumption, we have $v(D_m)>1-\varepsilon _0$ for sufficiently large m and for such m we obtain

(5.2) $$ \begin{align} A_{X,\Delta+D_m}(F_i)\le A_{X,\Delta+{\mathfrak{a}}_{\bullet}^{1-\varepsilon_0}}(F_i)<\varepsilon. \end{align} $$

By [Reference Birkar, Cascini, Hacon and McKernanBCHM10, Corollary 1.4.3], we get a ${\mathbb {Q}}$ -factorial birational model $p\colon \widetilde {X}\to X$ that extracts exactly the divisors $F_i$ . By assumption, all $(X,\Delta +D_m)$ ( $m\in \mathbb {N}$ ) have ${\mathbb {Q}}$ -complements. Together with equation (5.2), this implies that $(\widetilde {X},p_*^{-1}\Delta +(1-\varepsilon )\sum _{i=1}^r F_i)$ has ${\mathbb {Q}}$ -complements as well. Using [Reference Liu, Xu and ZhuangLXZ22, Lemma 3.2], we conclude that ${\mathbb {Q}}$ -complements also exist for $(\widetilde {X},p_*^{-1}\Delta +\sum _{i=1}^r F_i)$ as long as $\varepsilon $ is sufficiently small. Since v is in the complex hull of $\mathrm {ord}_{F_i}$ , this yields a ${\mathbb {Q}}$ -complement $\Gamma $ of $(X,\Delta )$ that has v as an lc place.

Proof. Let $T=T_{X,\Delta }(v_0)<\infty $ , and let

$$\begin{align*}c_0=\frac{\inf_{x\in[0,T]} g(x)}{\sup_{x\in[0,T]} g(x)}>0. \end{align*}$$

Let $\boldsymbol {\nu }$ denote the compatible DH measure associated to ${\mathcal {F}}_{v_0}$ and ${\mathcal {F}}_G$ as in Section 3.1.3. Then $\boldsymbol {\nu }$ is supported in $[0,T]\times \mathbb {R}$ and we have

$$\begin{align*}S_g(v_0;G)=\frac{\int_{\mathbb{R}^2} yg(x)\mathrm{d} \boldsymbol{\nu}}{\int_{\mathbb{R}^2} g(x)\mathrm{d} \boldsymbol{\nu}}\ge c_0\cdot \frac{\int_{\mathbb{R}^2} y\mathrm{d} \boldsymbol{\nu}}{\int_{\mathbb{R}^2} \mathrm{d} \boldsymbol{\nu}} = c_0\cdot S_{X,\Delta}(G)=\frac{c_0}{n+1}, \end{align*}$$

where the last equality is by [Reference Liu, Xu and ZhuangLXZ22, Lemma 2.20]. Thus, we may take, for example, $c=\frac {c_0}{3n}$ .

Proof of Theorem 5.4.

The plan is to use Lemma 5.5 to show that v is a monomial lc place of a special complement (in the sense of [Reference Liu, Xu and ZhuangLXZ22, Definition 3.3]) and then apply [Reference Liu, Xu and ZhuangLXZ22, Theorem 4.2] to get the finite generation. To this end, let $\pi :(Y,E=\sum _{i=1}^r E_i)\to (X,\Delta )$ be a $\mathbb {T}$ -equivariant log smooth model such that $\mathrm {QM}(Y,E)$ is a simplicial cone whose interior contains v, $C_Y(v)=\cap _{i=1}^r E_i$ , and there is a $\pi $ -exceptional and $\pi $ -ample ${\mathbb {Q}}$ -divisor $-F$ on Y.

Let $G_Y$ be a $\mathbb {T}$ -invariant ${\mathbb {Q}}$ -divisor in the ample ${\mathbb {Q}}$ -linear system $|-\pi ^*(K_X+\Delta )-\varepsilon F|_{{\mathbb {Q}}}$ ( $0<\varepsilon \ll 1$ ) whose support does not contain $C_Y(v_0)$ (such $G_Y$ exists because there is some $\mathbb {T}$ -invariant element $\Gamma $ in $H^0(Y,m(-\pi ^*(K_X+\Delta )-\varepsilon F))$ for sufficiently divisible m with $v_0(\Gamma )\neq 0$ ). Let $G:=\pi _*G_Y$ . For any $m\in \mathbb {N}$ , let $D_m\in |-K_X-\Delta |_{\mathbb {R}}$ be a $\mathbb {T}$ -invariant g-weighted $(m,v_0)$ -basis type divisor that is also compatible with both v and G. Such divisors exist because:

• • Both v and G are $\mathbb {T}$ -invariant (so we can choose compatible basis in each individual piece in the weight decomposition), and

• • By our assumption, any $\mathbb {T}$ -invariant basis that is compatible with both v and G is automatically compatible with $v_0$ .

By Lemma 5.6, there exists some $c\in {\mathbb {Q}}_+$ such that $D_m\ge cG$ for all $m\gg 0$ . Let $\delta _m=\min \{\delta _{g,\mathbb {T},m}(X,\Delta ,v_0),1\}$ . Then $(X,\Delta +\delta _m D_m)$ is lc and $\lim _{m\to \infty } \delta _m = \delta _{g,\mathbb {T}}(X,\Delta ,v_0)$ by Lemma 4.5 and the assumption that $\delta _{g,\mathbb {T}}(X,\Delta ,v_0)\le 1$ .

Since v computes $\delta _{g,\mathbb {T}}(X,\Delta ,v_0)$ , we also have

$$\begin{align*}\lim_{m\to\infty} \delta_m v(D_m) = \delta_{g,\mathbb{T}}(X,\Delta,v_0) S_g(v_0;v) = A_{X,\Delta}(v). \end{align*}$$

Note that by construction $D_m=\sum \lambda _i D_m^{(i)}$ for some $\lambda \in \mathbb {R}_+$ and some effective ${\mathbb {Q}}$ -divisors $D_m^{(i)}\sim _{\mathbb {Q}} -(K_X+\Delta )$ . Thus, by perturbing the coefficients $\lambda _i$ , for each $m\gg 0$ we get a ${\mathbb {Q}}$ -divisor $D^{\prime }_m=\sum \lambda ^{\prime }_i D_m^{(i)}$ such that $\delta _m D_m> D^{\prime }_m\ge \frac {1}{2}cG$ and $\lim _{m\to \infty } v(D^{\prime }_m)=A_{X,\Delta }(v)$ . It follows that $(X,\Delta +cG+\widetilde {D}_m)$ is lc and $-(K_X+\Delta +cG+\widetilde {D}_m)$ is ample where $\widetilde {D}_m=D^{\prime }_m-\frac {1}{2}cG$ . By Lemma 5.5, we see that v is an lc place of $(X,\Delta +\frac {1}{2}cG+\Gamma )$ for some ${\mathbb {Q}}$ -complement $\Gamma $ of $(X,\Delta +\frac {1}{2}cG)$ . Recall the $\pi _*^{-1}G$ is ample and does not contain $C_Y(v)$ . By [Reference Liu, Xu and ZhuangLXZ22, Theorem 4.2], this implies that $\mathrm {gr}_v R$ is finitely generated.

Proof. By [Reference Han and LiHL20b, Theorem 1.5], $v_0$ is quasi-monomial. So the result follows immediately from Corollary 5.2 and Theorem 5.4 (with $g(x)=e^{-x}$ and $\mathbb {T}=\{1\}$ ).

Proof of Theorem 1.2.

By Corollary 5.7, we know that v yields a special $\mathbb {R}$ -test configuration in the sense of [Reference Han and LiHL20b, Definition 2.8]. Thus, by [Reference Han and LiHL20b, Theorem 1.6], we know that $(X_0,\Delta _0,\xi _v)$ is a K-semistable triple.

Proof. In view of Theorem 5.4, it suffices to show that $\delta _g(X,\Delta ,\xi )\le 1$ . Indeed, we will prove a stronger statement:

(5.3) $$ \begin{align} \delta_{g,\mathbb{T},m} (X,\Delta,v_0)\le 1 \end{align} $$

for all $v_0\in \mathrm {Val}_X^{\mathbb {T},\circ }$ , $m\in \mathbb {N}$ and all positive functions $g\in C^0(\mathbb {R})$ , as long as $\mathbb {T}\neq \{1\}$ . To see this, let D be a $\mathbb {T}$ -invariant g-weighted $(m,v_0)$ -basis type divisor. If $(X,\Delta +D)$ is klt, then after perturbing the coefficients of D as in the proof of Theorem 5.4, we get a $\mathbb {T}$ -invariant ${\mathbb {Q}}$ -divisor $\widetilde {D}>D$ proportional to $-(K_X+\Delta )$ such that $(X,\Delta +\widetilde {D})$ is still klt and $K_X+\Delta +\widetilde {D}$ is ample. So such pairs have finite automorphism groups and this is a contradiction as $\mathbb {T}<\operatorname {Aut}(X,\Delta +\widetilde {D})$ by construction. Thus, $(X,\Delta +D)$ is not klt and $\mathrm {lct}(X,\Delta ;D)\le 1$ . This proves equation (5.3).

Proof. The proof is very similar to that of [Reference Liu, Xu and ZhuangLXZ22, Theorem 5.2]. Let $\mathbb {T}<\operatorname {Aut}(X,\Delta ,\xi )$ be a maximal torus such that $\xi \in N_{\mathbb {R}}$ . Assume to the contrary that $(X,\Delta ,\xi )$ is g-Ding polystable but not reduced uniformly g-Ding stable. Then by Proposition 4.14, we know that $\delta _g(X,\Delta ,\xi )=1$ is computed by some quasi-monomial valuation $v\in \mathrm {Val}_X^{\mathbb {T},\circ }$ that is not of the form $\mathrm {wt}_{\eta }$ . By Corollary 5.8, the associated graded ring $\mathrm {gr}_v R$ is finitely generated. Let $\pi \colon (Y,E)\to (X,\Delta )$ be a $\mathbb {T}$ -equivariant log smooth model such that $\mathrm {QM}(Y,E)$ is a simplicial cone containing v and that its dimension is the same as the rational rank of v. As in [Reference XuXu21, Claim 3.10], this implies that in a neighbourhood of v in $\mathrm {QM}(Y,E)$ , the function $w\mapsto S_g(v_0;w)$ is linear, and we have $\mathrm {gr}_wR\cong \mathrm {gr}_vR$ .

Thus, $\delta _g(X,\Delta ,\xi )$ is also computed by some $\mathbb {T}$ -invariant divisorial valuation $w\in \mathrm {QM}(Y,E)$ that is sufficiently close to v. In particular, $\mathbf {D}^{\mathrm {NA}}_g({\mathcal {F}}_w)\le A_{X,\Delta }(w)-S_g(v_0;w)=0$ (the first inequality is by Corollary 3.11) and $w\neq \mathrm {wt}_{\eta }$ for any $\eta \in N_{\mathbb {R}}$ . It induces a nonproduct type $\mathbb {T}$ -equivariant test configuration $({\mathcal {X}},\Delta _{{\mathcal {X}}},{\mathcal {L}})$ such that $\mathbf {D}^{\mathrm {NA}}_g({\mathcal {X}},\Delta _{{\mathcal {X}}},{\mathcal {L}})\le 0$ . This contradicts the g-Ding polystability of $(X,\Delta ,\xi )$ and proves the first part of the corollary. The remaining part follows from Theorem 4.9.

Proof of Theorem 1.3.

By [Reference Berman and NyströmBWN14, Theorem 1.5], $(X,\Delta ,\xi )$ is K-polystable if it admits a Kähler–Ricci soliton. Thus, the result follows immediately from Theorem 4.9 and Corollary 5.9 by setting $g(x)=e^{-x}$ .

Proof of Theorem 1.1.

It is a combination of Theorem 1.2, [Reference Han and LiHL20b, Theorem 1.3] and Theorem 1.3.

Proof. By Corollary 3.17, there is a unique valuation $v\in \mathrm {Val}_X$ which computes $h(X,\Delta )$ . By the uniqueness, v is $\mathbb {T}$ -invariant, that is, $v=\mathrm {wt}_{\xi }$ for some $\xi \in N_{\mathbb {R}}$ . Therefore, the K-semistable triple produced in Theorem 1.2 is $(X,\Delta , \xi )$ .

Since $(X,\Delta ,\xi )$ is K-semistable and toric, it is reduced uniformly Ding stable by Theorem 4.12. Indeed, condition (3) of the theorem holds trivially, since any $w\in \mathrm { Val}_{X}^{\mathbb {T},\circ }$ is of the form $w=\mathrm {wt}_{\eta }$ for some $\eta \in N_{\mathbb {R}}$ and, hence, satisfies $\mathbf {J}^{\mathrm {NA}}_{g,\mathbb {T}}(w) = 0$ where $g=e^{-x}$ . Therefore, $(X,\Delta ,\xi )$ admits a Kähler–Ricci soliton by [Reference Han and LiHL20].