Proof. The $\mathbb {Q}$ -divisor class $D|_F$ is defined in [Reference Han, Liu and Shokurov26, Definition A.2 and A.4] by localising at every codimension $1$ point of F, and the isomorphism is established by [Reference Han, Liu and Shokurov26, Lemma A.3].

Proof. By [Reference Lazarsfeld and Mustaţă33, Lemma 4.20] as in the previous remark, the semigroup $\Gamma (W_{\vec {\bullet }})$ generates $\mathbb {Z}^{r+\ell }$ as a group. Let $\Gamma \subseteq \Gamma (W_{\vec {\bullet }})$ be a finitely generated subsemigroup that still generates $\mathbb {Z}^{r+\ell }$ and such that $\gamma \in \mathrm {int}(\Sigma )$ , where $\Sigma \subseteq \Sigma (W_{\vec {\bullet }})$ is the subcone generated by $\Gamma $ . By [Reference Khovanskiĭ29, Proposition 3], there exists some $\gamma _0\in \Gamma $ such that

$$\begin{align*}(\Sigma+\gamma_0)\cap \mathbb{N}^{r+\ell} \subseteq \Gamma \subseteq \Gamma(W_{\vec{\bullet}}). \end{align*}$$

As $\gamma \in \mathrm {int}(\Sigma )$ , we have $k\gamma \in \Sigma +\gamma _0$ when $k\gg 0$ , and thus the lemma follows.

Proof. The argument is almost identical to those in [Reference Blum and Jonsson2] (which is in turn based on [Reference Boucksom and Chen8]). Using the filtration $\mathcal {F}$ , we define a family $W_{\vec {\bullet }}^t$ of multigraded linear series on X (indexed by $t\in \mathbb {R}$ ) where $W^t_{m,\vec {a}}=\mathcal {F}^{{\kern1pt}mt} W_{m,\vec {a}}$ . Set

$$\begin{align*}T_m(W_{\vec{\bullet}};\mathcal{F}{\kern1pt})=\max\{j\in\mathbb{N}\,|\,\mathcal{F}^j W_{m,\vec{a}}\neq 0 \mbox{ for some } \vec{a}\}. \end{align*}$$

It is easy to see that the sequence $T_m(W_{\vec {\bullet }};\mathcal {F}{\kern1pt})$ is super-additive, and we set

$$\begin{align*}T(W_{\vec{\bullet}};\mathcal{F}{\kern1pt})=\lim_{m\to \infty} \frac{T_m(W_{\vec{\bullet}};\mathcal{F}{\kern1pt})}{m} = \sup_{m\in \mathbb{N}} \frac{T_m(W_{\vec{\bullet}};\mathcal{F}{\kern1pt})}{m}. \end{align*}$$

One can check as in [Reference Boucksom and Chen8, Lemma 1.6] that for any $t< T(W_{\vec {\bullet }};\mathcal {F}{\kern1pt})$ , the multigraded linear series $W_{\vec {\bullet }}^t$ contains an ample series. Therefore, for any fixed admissible flag $Y_{\bullet }$ of length $n=\dim X$ centered at a general point of X, we have the associated Okounkov bodies $\Delta ^t=\Delta _{Y_{\bullet }}(W_{\vec {\bullet }}^t)$ ( $t\in \mathbb {R}$ ). The result is now simply a consequence of properties of Okounkov bodies. More precisely, consider the function $G\colon \Delta :=\Delta ^0\to [0,T(W_{\vec {\bullet }};\mathcal {F}{\kern1pt})]$ given by

$$\begin{align*}G(\gamma)=\sup\{t\in\mathbb{R}\,|\,\gamma\in \Delta^t\}. \end{align*}$$

It is straightforward to check that G is concave and hence continuous in the interior of $\Delta $ . By the exact same proof of [Reference Blum and Jonsson2, Lemma 2.9] (using [Reference Boucksom and Chen8, Theorem 1.11]), we get the equality (where $\rho $ is the Lebesgue measure on $\Delta ^0$ )

$$\begin{align*}S(W_{\vec{\bullet}};\mathcal{F}{\kern1pt})=\frac{1}{\mathrm{vol}(\Delta)}\int_{\Delta}G \mathrm{d} \rho = \lim_{m\to \infty} S_m(W_{\vec{\bullet}};\mathcal{F}{\kern1pt}) \end{align*}$$

and an estimate

$$\begin{align*}S_m(W_{\vec{\bullet}};\mathcal{F}{\kern1pt})\le \frac{m^{n+r}}{h^0(W_{m,\vec{\bullet}})} \int_{\Delta}G\mathrm{d} \rho_m, \end{align*}$$

where $\rho _m$ is as in equation (2.4) (note that $\Delta =\Delta (W_{\vec {\bullet }})$ ). Applied to $\mathcal {F}=\mathcal {F}_v$ , the argument of [Reference Blum and Jonsson2, Lemma 2.2 and Corollary 2.10] then implies that for any $\epsilon>0$ , there exists some $m_0=m_0(\epsilon )$ such that $S_m(W_{\vec {\bullet }};v)\le (1+\epsilon )S(W_{\vec {\bullet }};v)$ for any valuation $v\in \mathrm {Val}_X^*$ with $A_{X,\Delta }(v)<\infty $ and any $m\ge m_0$ (the key point is that $m_0$ doesn’t depend on v). The remaining equalities in the lemma now follow from the exact same proof of [Reference Blum and Jonsson2, Theorem 4.4].

Proof. We already have $S(W_{\vec {\bullet }};\mathcal {F}{\kern1pt})=\frac {1}{\mathrm {vol}(\Delta )}\int _{\Delta }G \mathrm {d} \rho $ . It is not hard to see that $\int _{\Delta }G \mathrm {d} \rho = \int _0^{\infty } \mathrm {vol}(\Delta ^t)\mathrm {d} t$ . Since $\mathrm {vol}(W_{\vec {\bullet }})=(n+r)!\cdot \mathrm {vol}(\Delta )$ and $\mathrm {vol}(W_{\vec {\bullet }}^t)=(n+r)!\cdot \mathrm {vol}(\Delta ^t)$ for all $t\ge 0$ (see Remark 2.3), the result follows.

Proof. Let $\mathrm {pr} \colon \mathbb {R}^{r+1+n}\to \mathbb {R}^{r+1}$ be the projection to the first $r+1$ coordinates that induces a map $p\colon \Delta ^t \to \Delta ^t_{\mathrm {supp}}$ . By [Reference Lazarsfeld and Mustaţă33, Theorem 2.13 and 4.21], we know that $\mathrm {vol}(W_{\vec {\bullet }}^t)=(n+r)!\cdot \mathrm {vol}(\Delta ^t)$ and $\mathrm {vol}_{W_{\vec {\bullet }}^t}(\gamma )=n!\cdot \mathrm {vol}(p^{-1}(\gamma ))$ for all $\gamma \in \mathrm {int}(\Delta ^t_{\mathrm {supp}})$ . The lemma then follows from the obvious identity $\mathrm {vol}(\Delta ^t)=\int _{\Delta ^t_{\mathrm {supp}}} \mathrm {vol}(p^{-1}(\gamma )) \mathrm {d} \gamma $ .

Proof. We use the same notation as in the proof of Lemma 2.9 and let $\Delta '$ , $G'$ , and so on be the counterparts on $W_{\vec {\bullet }}'$ . Let $f\colon \mathbb {R}^{r+n}\to \mathbb {R}^{r+n}$ be given by

$$ \begin{align*} (x_1,\cdots,x_{r+n})\mapsto (kx_1,\cdots,kx_r,x_{r+1},\cdots,x_{r+n}). \end{align*} $$

We claim that

(2.6) $$ \begin{align} \Sigma(W_{\vec{\bullet}})=f(\Sigma(W_{\vec{\bullet}}')). \end{align} $$

Indeed, it is clear from the construction that $f(\Gamma (W_{\vec {\bullet }}'))\subseteq \Gamma (W_{\vec {\bullet }})$ , hence $f(\Sigma (W_{\vec {\bullet }}'))\subseteq \Sigma (W_{\vec {\bullet }})$ . On the other hand, from the proof of Lemma 2.4, we know that there exists some $\gamma _0\in \Gamma (W_{\vec {\bullet }})$ such that

$$\begin{align*}(\Sigma(W_{\vec{\bullet}})+\gamma_0)\cap \mathbb{N}^{r+n} \subseteq \Gamma(W_{\vec{\bullet}}), \end{align*}$$

hence as $f(\Gamma (W_{\vec {\bullet }}'))=f(\mathbb {N}^{r+n})\cap \Gamma (W_{\vec {\bullet }})$ , we have $(\Sigma (W_{\vec {\bullet }})+\gamma _0)\cap f(\mathbb {N}^{r+n})\subseteq f(\Gamma (W_{\vec {\bullet }}'))$ and therefore $\Sigma (W_{\vec {\bullet }})\subseteq f(\Sigma (W_{\vec {\bullet }}'))$ , which proves the claim.

It follows from equation (2.6) that $\Delta (W_{\vec {\bullet }})=\frac {1}{k}f(\Delta (W_{\vec {\bullet }}'))$ (recall that we identify $\Delta (W_{\vec {\bullet }})$ as a subset of $\{1\}\times \mathbb {R}^{r-1+n}$ ). Replace $W_{\vec {\bullet }}$ with $W_{\vec {\bullet }}^{t/k}$ , noting that $W^{\prime t}_{m,\vec {a}}:=\mathcal {F}^{{\kern1pt}mt}W^{\prime }_{m,\vec {a}}=W^{t/k}_{km,k\vec {a}}$ , and we deduce $\Delta ^{t/k}=\frac {1}{k}f(\Delta ^{\prime t})$ . Hence $\Delta =\frac {1}{k}f(\Delta ')$ and

(2.7) $$ \begin{align} G\left(\frac{f(\gamma)}{k}\right)=\frac{G'(\gamma)}{k} \end{align} $$

for any $\gamma \in \Delta '$ . Substitute it into the equality $S(W_{\vec {\bullet }};v)=\frac {1}{\mathrm {vol}(\Delta )}\int _{\Delta }G \mathrm {d} \rho $ from the proof of Lemma 2.9, we obtain $S(W_{\vec {\bullet }};v)=\frac {1}{k}S(W_{\vec {\bullet }}';v)$ . The remaining parts of the lemma now follow immediately from Lemma 2.9.

Proof. Let $\vec {L}=(L_1,\cdots ,L_r)$ . In the notation of Definitions 2.9 and 2.11, we have

$$\begin{align*}\frac{c_1(W_{m,\vec{\bullet}})}{m\cdot h^0(W_{m,\vec{\bullet}})} = L_0 + \frac{\sum_{\vec{a}\in \mathbb{N}^r} h^0(W_{m,\vec{a}})\cdot (\vec{a}\cdot \vec{L})}{m\cdot h^0(W_{m,\vec{\bullet}})}. \end{align*}$$

Thus for $c_1(W_{\vec {\bullet }})$ , it suffices to show that $\lim _{m\to \infty } \frac {\sum _{\vec {a}\in \mathbb {N}^r} h^0(W_{m,\vec {a}})\cdot a_i}{m\cdot h^0(W_{m,\vec {\bullet }})}$ exists for each $1\le i\le r$ . In the notation of Remark 2.3, we have

$$\begin{align*}\frac{\sum_{\vec{a}\in \mathbb{N}^r} h^0(W_{m,\vec{a}})\cdot a_i}{m\cdot h^0(W_{m,\vec{\bullet}})} = \frac{\int x_i \mathrm{d} \rho_m}{\int \mathrm{d} \rho_m}, \end{align*}$$

where $x_i$ denotes the ith entry of an element of $\mathbb {R}^{r+n}$ . Hence by [Reference Boucksom7, Théorème 1.12], the limit exists and equals $\frac {1}{\mathrm {vol}(\Delta )}\int _{\Delta } x_i \mathrm {d} \rho $ , where $\Delta = \Delta (W_{\vec {\bullet }})$ .

For $F(W_{\vec {\bullet }})$ , it suffices to show that $\lim _{m\to \infty }\mathrm {ord}_D(F_m)$ exists for any prime divisor D. First note that since $W_{\vec {\bullet }}$ has bounded support, there exists some constant $C_1>0$ such that $|\vec {a}|\le C_1 m$ for any $\vec {a}\in \mathbb {N}^{r}$ with $W_{m,\vec {a}}\neq 0$ . Thus as $mL_0+\vec {a}\cdot \vec {L}-F_{m,\vec {a}}$ is effective, we further deduce $\mathrm {ord}_D(F_{m,\vec {a}})\le Tm$ for some absolute constant T. Let $\Delta _0:=\mathrm {int}(\mathrm {Supp}(W_{\vec {\bullet }}))\cap (\{1\}\times \mathbb {R}^r)\subseteq \mathbb {R}^r$ . Since $W_{\vec {a}}\cdot W_{\vec {a}'}\subseteq W_{\vec {a}+\vec {a}'}$ , we have $F_{\vec {a}}+F_{\vec {a}'}\ge F_{\vec {a}+\vec {a}'}$ (whenever $W_{\vec {a}},W_{\vec {a}'}\neq 0$ ); thus if we let

$$\begin{align*}f_{W_{\vec{\bullet}},D}(\gamma):=\inf_{m} \frac{\mathrm{ord}_D(F_{m,m\vec{\gamma}})}{m} = \lim_{m\to\infty} \frac{\mathrm{ord}_D(F_{m,m\vec{\gamma}})}{m} \end{align*}$$

for $\gamma \in \Delta _0 \cap \mathbb {Q}^r$ , where the infimum and limit are taken over sufficiently divisible integers m, then $f_{W_{\vec {\bullet }},D}(t\gamma _1+(1-t)\gamma _2)\le tf_{W_{\vec {\bullet }},D}(\gamma _1)+(1-t)f_{W_{\vec {\bullet }},D}(\gamma _2)$ for any $\gamma _1,\gamma _2\in \Delta _0$ . Therefore it naturally extends to a convex (and hence continuous) function $f_{W_{\vec {\bullet }},D}$ on $\Delta _0$ . For simplicity, we denote $f_{W_{\vec {\bullet }},D}$ by f. By the previous discussion, $f(\gamma )\le T$ for all $\gamma \in \Delta _0$ .

We claim that $f(\gamma )=\lim _{m\to \infty } f_m(\gamma )$ for any $\gamma \in \Delta _0$ where

$$\begin{align*}f_m(\gamma):= \begin{cases} \frac{1}{m}\mathrm{ord}_D (F_{m,\lfloor m\vec{\gamma}\rfloor}) & \text{if } W_{m,\lfloor m\vec{\gamma}\rfloor}\neq 0 \\ T & \text{if } W_{m,\lfloor m\vec{\gamma}\rfloor} = 0. \end{cases} \end{align*}$$

Indeed, as $f_m(\gamma )\ge f(\frac {\lfloor m\vec {\gamma }\rfloor }{m})$ ( $m\gg 0$ ) by definition, we have

$$\begin{align*}\liminf_{m\to \infty} f_m(\gamma)\ge \lim_{m\to\infty} f(\frac{\lfloor m\vec{\gamma}\rfloor}{m}) = f(\gamma). \end{align*}$$

To get the reverse direction, let $\varepsilon>0$ and choose $\gamma _i\in \Delta _0\cap \mathbb {Q}^r$ ( $i=0,\cdots ,r$ ) that are sufficiently close to $\gamma $ such that their convex hull contains $\gamma $ in the interior and $f(\gamma _i) < f(\gamma )+\varepsilon $ . Then we may choose some sufficiently divisible $m_0\in \mathbb {N}$ such that $f_{m_0}(\gamma _i)<f(\gamma )+\varepsilon $ . Let $\Pi \subseteq \mathbb {R}^{r+1}$ be the cone spanned by all the $\gamma _i$ s. From the proof of Lemma 2.4, we know that there exists some $\vec {a}_0\in \mathbb {N}^{r+1}$ such that $W_{\vec {a}}\neq 0$ for all $\vec {a}\in (\Pi +\vec {a}_0)\cap \mathbb {N}^{r+1}$ (consider the semigroup $\{\vec {a}\,\vert \,W_{\vec {a}}\neq 0\}\subseteq \mathbb {N}^{r+1}$ , choose a finitely generated subsemigroup that generates $\mathbb {Z}^{r+1}$ such that the cone it spans contains $\Pi $ , and apply [Reference Khovanskiĭ29, Proposition 3]). Then one can verify that there exists some constant $C>0$ such that for all $m\gg 0$ , we have

$$\begin{align*}(m,\lfloor m\vec{\gamma}\rfloor) = \vec{a} + \sum_{i=0}^r k_i (m_0,m_0\gamma_i) \end{align*}$$

for some $k_i\in \mathbb {N}$ and some $\vec {a}\in \mathbb {N}^{r+1}$ satisfying $W_{\vec {a}}\neq 0$ and $|\vec {a}|\le C$ . In particular, $|m-m_0\sum k_i|\le C$ and $\mathrm {ord}_D(W_{\vec {a}})\le CT$ . It follows that

$$ \begin{align*} \mathrm{ord}_D (F_{m,\lfloor m\vec{\gamma}\rfloor}) & \le \mathrm{ord}_D (W_{\vec{a}})+\sum_{i=0}^r k_i \mathrm{ord}_D(F_{m_0,m_0\gamma_i}) = \mathrm{ord}_D (W_{\vec{a}})+\sum_{i=0}^r k_i m_0 f_{m_0}(\gamma_i) \\ & \le \mathrm{ord}_D (W_{\vec{a}})+(m-C)(f(\gamma)+\varepsilon) \le CT+ (m-C)(f(\gamma)+\varepsilon). \end{align*} $$

Hence $\limsup _{m\to \infty } f_m(\gamma )\le f(\gamma )+\varepsilon $ . Since $\varepsilon>0$ is arbitrary, we get $\limsup _{m\to \infty } f_m(\gamma )\le f(\gamma )$ , and this proves the claim. Note that the argument also shows $W_{m,\lfloor m\vec {\gamma }\rfloor }\neq 0$ for $m\gg 0$ .

It is clear that

$$\begin{align*}\mathrm{ord}_D(F_m) = \frac{\int (f_m\circ p) \mathrm{d} \rho_m}{\int \mathrm{d} \rho_m}, \end{align*}$$

where $p\colon \Delta =\Delta (W_{\vec {\bullet }})\to \Delta _0$ is the natural projection. By dominated convergence and the above claim, the latter limit exists and equals $\frac {1}{\mathrm {vol}(\Delta )}\int _{\Delta } (f\circ p) \mathrm {d} \rho $ .

Proof. The above proof gives $\mathrm {ord}_D(F(W_{\vec {\bullet }}))=\frac {1}{\mathrm {vol}(\Delta )}\int _{\Delta } (f\circ p) \mathrm {d} \rho $ . We have $\mathrm {vol}(W_{\vec {\bullet }})=(n+r)!\cdot \mathrm {vol}(\Delta )$ , $p(\Delta )=\Delta _{\mathrm {supp}}$ and $\mathrm {vol}_{W_{\vec {\bullet }}}(\gamma )=n!\cdot \mathrm {vol}(p^{-1}(\gamma ))$ for any $\gamma \in \mathrm {int}(\Delta _{\mathrm {supp}})$ . These together imply the given formula.

Proof. Let $M(\gamma ):=\lim _{m\to \infty } \frac {1}{m} c_1(M_{m,\lfloor m\vec {\gamma }\rfloor })\in \mathrm {Cl}(X)_{\mathbb {R}}$ and $F(\gamma )=\lim _{m\to \infty } \frac {1}{m} F_{m,\lfloor m\vec {\gamma }\rfloor }$ for $\gamma \in \mathrm {int}(\Delta _{\mathrm {supp}})$ . As in the previous proof, the limit exists: $M(\gamma )=\vec {\gamma }\cdot \vec {L}-\sum _D f_{W_{\vec {\bullet }},D}(\gamma )\cdot D$ and $F(\gamma )=\sum _D f_{W_{\vec {\bullet }},D}(\gamma )\cdot D$ in the notation of Corollary 2.15. Moreover, M is continuous, and we have

$$\begin{align*}c_1(M_{\vec{\bullet}})=\frac{1}{\mathrm{vol}(\Delta)}\int_{\Delta} (M\circ p) \mathrm{d} \rho, \end{align*}$$

where $\Delta =\Delta (W_{\vec {\bullet }})$ and $p\colon \Delta \to \Delta _{\mathrm {supp}}$ is the natural projection. Since $W_{\vec {\bullet }}$ is almost complete, we see that $M(\gamma )$ is $\mathbb {R}$ -Cartier and $M(\gamma ) \equiv g(\gamma )L$ for some $g(\gamma )\in \mathbb {R}$ . It follows that $c_1(M_{\vec {\bullet }})$ is also $\mathbb {R}$ -Cartier and $c_1(M_{\vec {\bullet }})=\mu L$ in $\mathrm {NS}(X)_{\mathbb {R}}$ , where

$$\begin{align*}\mu=\frac{1}{\mathrm{vol}(\Delta)}\int_{\Delta} (g\circ p) \mathrm{d} \rho=\frac{1}{\mathrm{vol}(\Delta)}\int_{\Delta_{\mathrm{supp}}} \mathrm{vol}(p^{-1}(\gamma))\cdot g(\gamma) \mathrm{d} \gamma. \end{align*}$$

Since $F(W_{\vec {\bullet }})\sim _{\mathbb {R}} c_1(W_{\vec {\bullet }})-c_1(M_{\vec {\bullet }})$ , we also see that $F(W_{\vec {\bullet }})$ is $\mathbb {R}$ -Cartier. It remains to prove equation (2.8).

As $F(\gamma )\sim _{\mathbb {R}} \vec {\gamma }\cdot \vec {L}-M(\gamma )$ is also $\mathbb {R}$ -Cartier, we may define $h(\gamma )=v(F(\gamma ))$ ; and as in the proof of Corollary 2.15, we have

$$\begin{align*}v(F(W_{\vec{\bullet}}))=\frac{1}{\mathrm{vol}(\Delta)}\int_{\Delta} (h\circ p) \mathrm{d} \rho. \end{align*}$$

We claim that

(2.9) $$ \begin{align} \mathrm{vol}_{W^t_{\vec{\bullet}}}(\gamma)=\mathrm{vol}(g(\gamma)L;v\ge t-h(\gamma)) \end{align} $$

in the notation of Corollary 2.13 and Lemma 2.10. For this, we may assume that $\gamma \in \mathbb {Q}^r$ . Let $\mathcal {F}$ be the filtration induced by v, and let m be a sufficiently divisible integer. From the exact sequence

$$\begin{align*}0\to \mathcal{F}^{\lambda}M_{m,m\vec{\gamma}}\to \mathcal{F}^{\lambda}H^0(X,L_{m,\vec{\gamma}})\to H^0(X,L_{m,\vec{\gamma}})/M_{m,m\vec{\gamma}} \end{align*}$$

and the obvious equality

$$\begin{align*}|\mathcal{F}^{{\kern1pt}mt}W_{m,m\vec{\gamma}}|=|\mathcal{F}^{{\kern1pt}mt-v(F_{m,m\vec{\gamma}})}M_{m,m\vec{\gamma}}|+F_{m,m\vec{\gamma}}, \end{align*}$$

we deduce that

(2.10) $$ \begin{align} \left| \dim (\mathcal{F}^{{\kern1pt}mt}W_{m,m\vec{\gamma}}) - \dim (\mathcal{F}^{{\kern1pt}mt-v(F_{m,m\vec{\gamma}})}H^0(X,L_{m,\vec{\gamma}})) \right| \le h^0(X,L_{m,\vec{\gamma}})-h^0(M_{m,m\vec{\gamma}}). \end{align} $$

By [Reference Lazarsfeld32, Lemma 2.2.42], there exists a fixed effective divisor N on X such that $N\pm (L_{m,\vec {\gamma }} - \ell _{m,\vec {\gamma }}L)$ is effective. In particular, we have the inclusions

$$\begin{align*}H^0(X,\ell_{m,\vec{\gamma}}L-N)\hookrightarrow H^0(X,L_{m,\vec{\gamma}})\hookrightarrow H^0(X,\ell_{m,\vec{\gamma}}L+N), \end{align*}$$

$$\begin{align*}H^0(X,\ell_{m,\vec{\gamma}}L-N)\hookrightarrow H^0(X,\ell_{m,\vec{\gamma}}L)\hookrightarrow H^0(X,\ell_{m,\vec{\gamma}}L+N), \end{align*}$$

which implies

$$\begin{align*}\lim_{m\to\infty} \frac{\dim (\mathcal{F}^{{\kern1pt}mt-v(F_{m,m\vec{\gamma}})}H^0(X,L_{m,\vec{\gamma}}))}{m^n/n!} = \lim_{m\to\infty} \frac{\dim (\mathcal{F}^{{\kern1pt}mt-v(F_{m,m\vec{\gamma}})}H^0(X,\ell_{m,\vec{\gamma}}L))}{m^n/n!}. \end{align*}$$

Thus as we divide equation (2.10) by $m^n/n!$ , and letting $m\to \infty $ , the right side of the inequality becomes $0$ by the definition of almost completeness, and the equality equation (2.9) follows as $g(\gamma )=\lim _{m\to \infty }\frac {1}{m}\ell _{m,\vec {\gamma }}$ .

By Corollary 2.13 and Lemma 2.10, we have

$$\begin{align*}S(W_{\vec{\bullet}};v) = \frac{1}{n!\mathrm{vol}(\Delta)} \iint_{\Delta_{\mathrm{supp}}\times \mathbb{R}_+} \mathrm{vol}_{W^t_{\vec{\bullet}}}(\gamma) \mathrm{d} t \mathrm{d} \gamma. \end{align*}$$

Combined with equation (2.9), we then obtain

$$ \begin{align*} S(W_{\vec{\bullet}};v) & = \frac{1}{n!\mathrm{vol}(\Delta)} \iint_{\Delta_{\mathrm{supp}}\times \mathbb{R}_+} \mathrm{vol}(g(\gamma)L; v\ge t-h(\gamma)) \mathrm{d} t \mathrm{d} \gamma \\ & = \frac{1}{n!\mathrm{vol}(\Delta)} \int_{\Delta_{\mathrm{supp}}}\left(\int_0^{h(\gamma)}+\int_{h(\gamma)}^{\infty}\right)\mathrm{vol}(g(\gamma)L; v\ge t-h(\gamma)) \mathrm{d} t \mathrm{d} \gamma \\ & = \frac{1}{n!\mathrm{vol}(\Delta)} \int_{\Delta_{\mathrm{supp}}}\left(h(\gamma)\cdot g(\gamma)^n\mathrm{vol}(L) + \int_0^{\infty} g(\gamma)^{n+1} \mathrm{vol}(L; v\ge t) \mathrm{d} t\right) \mathrm{d} \gamma. \end{align*} $$

Notice that $\mathrm {vol}_{W_{\vec {\bullet }}}(\gamma ) = \mathrm {vol}_{W_{\vec {\bullet }}^0}(\gamma )=g(\gamma )^n\mathrm {vol}(L)$ by equation (2.9); thus we deduce that

$$ \begin{align*} S(W_{\vec{\bullet}};v) & = \frac{1}{n!\mathrm{vol}(\Delta)} \int_{\Delta_{\mathrm{supp}}} \mathrm{vol}_{W_{\vec{\bullet}}}(\gamma)\cdot (h(\gamma)+g(\gamma)S(L;v)) \mathrm{d} \gamma\\ & = \frac{1}{\mathrm{vol}(\Delta)} \int_{\Delta_{\mathrm{supp}}} \mathrm{vol}(p^{-1}(\gamma))\cdot (h(\gamma)+g(\gamma)S(L;v)) \mathrm{d} \gamma\\ & = v(F(W_{\vec{\bullet}}))+\mu\cdot S(L;v). \end{align*} $$

This finishes the proof.

Proof. We have $S(L;P)=\frac {1}{2}\deg L$ for any ample line bundle L and any closed point P on C. Combining with Lemma 2.13, we see that $S(W_{\vec {\bullet }};P)=S(c_1(W_{\vec {\bullet }})-F(W_{\vec {\bullet }});P)+\mathrm {mult}_P F(W_{\vec {\bullet }}) = \frac {1}{2}\deg (c_1(W_{\vec {\bullet }})-F(W_{\vec {\bullet }}))+\mathrm {mult}_P F(W_{\vec {\bullet }})$ . Since $\delta _P(C;W_{\vec {\bullet }})=\frac {1}{S(W_{\vec {\bullet }};P)}$ and $\delta (C;W_{\vec {\bullet }})=\inf _{P\in C}\delta _P(C;W_{\vec {\bullet }})$ by definition, the result follows.

Proof. We only prove equation (3.2), since the proof for equation (3.3) is almost identical. By Proposition 3.1, we have $\delta _Z(V_{\vec {\bullet }})=\delta _Z(V_{\vec {\bullet }},\mathcal {F}{\kern1pt})$ (where $\mathcal {F}$ is the filtration on $V_{\vec {\bullet }}$ induced by F); thus it suffices to show that

(3.4) $$ \begin{align} \delta_{Z,m}(V_{\vec{\bullet}},\mathcal{F}{\kern1pt})\ge \min\left\{\frac{A_{X,\Delta}(F)}{S_m(V_{\vec{\bullet}};F)},\inf_{\pi(Z')=Z_0}\delta_{Z',m}(F,\Delta_F;W_{\vec{\bullet}})\right\} \end{align} $$

for all $m\in M(V_{\vec {\bullet }})$ . Letting $m\to \infty $ , we obtain equation (3.2). Let D be an m-basis type $\mathbb {Q}$ -divisor of $V_{\vec {\bullet }}$ that’s compatible with F. From the discussion before, we have

(3.5) $$ \begin{align} \pi^*D = S_m(V_{\vec{\bullet}};F)\cdot F + \Gamma, \end{align} $$

where $\Gamma =D_Y$ is the strict transform of D on Y and $\Gamma |_F$ is an m-basis type $\mathbb {Q}$ -divisor of $W_{\vec {\bullet }}$ . Let $\lambda _m$ (respectively, $\lambda $ ) be the right-hand side of equation (3.4) (respectively, equation (3.2)). Then we have $\pi ^*(K_X+\Delta +\lambda _m D)=K_Y+\Delta _Y+a_m F+\lambda _m \Gamma $ , where $a_m=1-A_{X,\Delta }(F)+\lambda _m S_m(V_{\vec {\bullet }};F)\le 1$ . In addition, the non-lc centre of $(F,\Delta _F+\lambda _m \Gamma |_F)$ doesn’t contain $Z_0\subseteq Z$ in its image (under the morphism $\pi $ ) by the definition of stability thresholds, and hence by inversion of adjunction the same is true for $(Y,\Delta _Y+F+\lambda _m \Gamma )$ . It follows that $(X,\Delta +\lambda _m D)$ is lc at the generic point of Z and indeed

$$\begin{align*}A_{X,\Delta}(v)\ge \lambda_m v(D) + (1-a_m)\cdot v(F) \end{align*}$$

for all valuations v on X whose centre contains Z (when $Z\not \subseteq C_X(F)$ . The value of $a_m$ doesn’t matter to us since $v(F)=0$ ; this is the main difference between the proof of equation (3.2) and equation (3.3)). Since D is arbitrary, we get $\delta _{Z,m}(V_{\vec {\bullet }},\mathcal {F}{\kern1pt})\ge \lambda _m$ , which proves equation (3.4), and

$$\begin{align*}A_{X,\Delta}(v)\ge \lambda_m S_m(V_{\vec{\bullet}},\mathcal{F};v) + (A_{X,\Delta}(F) - \lambda_m S_m(V_{\vec{\bullet}};F))\cdot v(F). \end{align*}$$

As in the proof of Lemma 2.9, we have $\lim _{m\to \infty } \lambda _m = \lambda $ . Thus, letting $m\to \infty $ and noting that $S(V_{\vec {\bullet }};v)=S(V_{\vec {\bullet }},\mathcal {F};v)$ by Proposition 3.1, we obtain equation (3.2) as well as the following inequality:

(3.6) $$ \begin{align} A_{X,\Delta}(v)\ge \lambda \cdot S(V_{\vec{\bullet}};v) + (A_{X,\Delta}(F) - \lambda \cdot S(V_{\vec{\bullet}};F))\cdot v(F). \end{align} $$

Now assume that equality holds in equation (3.2) and $\delta _Z(V_{\vec {\bullet }})$ is computed by some valuation $v\in \mathrm {Val}_X^*$ : that is, $Z\subseteq C_X(v)$ and $A_{X,\Delta }(v) = \lambda \cdot S(V_{\vec {\bullet }};v)$ . By equation (3.6), we see that either $A_{X,\Delta }(F) = \lambda \cdot S(V_{\vec {\bullet }};F)$ , in which case F computes $\delta _Z(V_{\vec {\bullet }})$ and we are done, or

(3.7) $$ \begin{align} \lambda = \inf_{\pi(Z')=Z_0}\delta_{Z'}(W_{\vec{\bullet}}) < \frac{A_{X,\Delta}(F)}{S(V_{\vec{\bullet}};F)} \end{align} $$

and $v(F)=0$ : that is, $C_Y(v)\not \subseteq F$ . Now assume that we are in the latter case, and let S be an irreducible component of $C_Y(v)\cap F$ with $Z_0\subseteq \pi (S)$ . After rescaling the valuation v, we may also assume that $A_{Y,\Delta _Y}(v)=A_{X,\Delta }(v)=1$ . Let $\mathfrak {a}_{\bullet }(v)\subseteq \mathcal {O}_Y$ be the valuation ideals, and let $\mathfrak {b}_{\bullet }=\mathfrak {a}_{\bullet }(v)|_F$ . Clearly $\mathrm {lct}_x(Y,\Delta _Y;\mathfrak {a}_{\bullet }(v))\le \frac {A_{Y,\Delta _Y}(v)}{v(\mathfrak {a}_{\bullet }(v))}\le 1$ for any $x\in C_Y(v)$ , hence by inversion of adjunction, we have $\mathrm {lct}(F,\Delta _F;\mathfrak {b}_{\bullet })\le 1$ at the generic point of S. By [Reference Jonsson and Mustaţă27, Theorem A], there exists some valuation $v_0$ on F with centre S such that

(3.8) $$ \begin{align} \frac{A_{F,\Delta_F}(v_0)}{v_0(\mathfrak{b}_{\bullet})}\le 1. \end{align} $$

To finish the proof, it suffices to show that this valuation computes $\delta _{Z'}(W_{\vec {\bullet }})$ for any subvarieties $Z'\subseteq S$ with $\pi (Z')=Z_0$ and that $\delta _{Z'}(W_{\vec {\bullet }})=\lambda $ . To see this, let D be an m-basis type $\mathbb {Q}$ -divisor of $V_{\vec {\bullet }}$ that’s compatible with both F and v (which exists by Lemma 3.1), and let $D_Y$ be its strict transform on Y. Then as before, we have $v(D_Y)=v(D)=S_m(V_{\vec {\bullet }};v)$ (here we use the fact that $C_Y(v)\not \subseteq F$ ) and $D_Y|_F$ is an m-basis type $\mathbb {Q}$ -divisor of $W_{\vec {\bullet }}$ . Using equation (3.8), we further see that

$$\begin{align*}A_{F,\Delta_F}(v_0)\le \frac{v_0(D_Y|_F)}{v(D_Y)}=\frac{v_0(D_Y|_F)}{S_m(V_{\vec{\bullet}};v)}, \end{align*}$$

hence $S_m(W_{\vec {\bullet }};v_0)\ge v_0(D_Y|_F)\ge S_m(V_{\vec {\bullet }};v)\cdot A_{F,\Delta _F}(v_0)$ . Letting $m\to \infty $ , we obtain

$$\begin{align*}\delta_{Z'}(W_{\vec{\bullet}})\le \frac{A_{F,\Delta_F}(v_0)}{S(W_{\vec{\bullet}};v_0)}\le \frac{1}{S(V_{\vec{\bullet}};v)}=\frac{A_{X,\Delta}(v)}{S(V_{\vec{\bullet}};v)}=\lambda. \end{align*}$$

Combined with equation (3.7), this implies $\delta _{Z'}(W_{\vec {\bullet }})=\lambda $ , and it’s computed by $v_0$ .

Proof. This is immediate from Theorem 3.2 and Lemma 2.13.

Proof. By Proposition 3.1, we have $\delta _Z(V_{\vec {\bullet }})=\delta _Z(V_{\vec {\bullet }},\mathcal {F}{\kern1pt})=\delta _{Z\cap Y_0}(Y_0,\Delta _0;W_{\vec {\bullet }}^{(0)},\mathcal {F}{\kern1pt})$ . Thus it suffices to prove that

(3.10) $$ \begin{align} \delta_{Z\cap Y_i}(Y_i,\Delta_i;W_{\vec{\bullet}}^{(i)},\mathcal{F}{\kern1pt}) \ge \min\left\{ \frac{A_{Y_i,\Delta_i}(Y_{i+1})}{S(W_{\vec{\bullet}}^{(i)};Y_{i+1})}, \delta_{Z\cap Y_{i+1}}(Y_{i+1},\Delta_{i+1};W_{\vec{\bullet}}^{(i+1)},\mathcal{F}{\kern1pt}) \right\} \end{align} $$

for all $0\le i\le j-1$ ; equation (3.9) then follows by induction.

As in the proof of Theorem 3.2, let D be an m-basis type $\mathbb {Q}$ -divisor of $W_{\vec {\bullet }}^{(i)}$ that’s compatible with $\mathcal {F}$ . Then in particular it is compatible with $Y_{i+1}$ , and we may write

$$\begin{align*}D = S_m(W_{\vec{\bullet}}^{(i)};Y_{i+1})\cdot Y_{i+1} + \Gamma, \end{align*}$$

where $\Gamma $ doesn’t contain $Y_{i+1}$ in its support and $\Gamma |_{Y_{i+1}}$ is an m-basis type $\mathbb {Q}$ -divisor of $W_{\vec {\bullet }}^{(i+1)}$ (since this is the same as the refinement of $W_{\vec {\bullet }}^{(i)}$ by $Y_{i+1}$ ) that is compatible with $\mathcal {F}$ (since the same is true for D and the filtration $\mathcal {F}$ on $W_{\vec {\bullet }}^{(i)}$ is a refinement of the filtration induced by $Y_{i+1}$ ). Thus by inversion of adjunction as in the proof of Theorem 3.2, we get

$$\begin{align*}\delta_{Z\cap Y_i,m}(Y_i,\Delta_i;W_{\vec{\bullet}}^{(i)},\mathcal{F}{\kern1pt}) \ge \min\left\{ \frac{A_{Y_i,\Delta_i}(Y_{i+1})}{S_m(W_{\vec{\bullet}}^{(i)};Y_{i+1})}, \delta_{Z\cap Y_{i+1},m}(Y_{i+1},\Delta_{i+1};W_{\vec{\bullet}}^{(i+1)},\mathcal{F}{\kern1pt}) \right\}. \end{align*}$$

Letting $m\to \infty $ , we obtain equation (3.10), and this finishes the proof.

Proof. Let r be a sufficiently large integer such that $rL$ is very ample, and let $H\in |rL|$ be a general member. Let $V_{\vec {\bullet }}$ be the complete linear series associated to L, and let $\mathcal {F}$ be the filtration on $V_{\vec {\bullet }}$ induced by H. By Proposition 3.1, we have $S(L;v)=S(V_{\vec {\bullet }};v)=S(V_{\vec {\bullet }},\mathcal {F};v)$ . Let D be an m-basis type $\mathbb {Q}$ -divisor of L that’s compatible with $\mathcal {F}$ . By the same discussion as at the beginning of Section 3.1, we have

$$\begin{align*}D=S_m(L;H)\cdot H + \Gamma \end{align*}$$

for some effective $\mathbb {Q}$ -divisor $\Gamma $ whose support doesn’t contain H. Since H is general, we have $C_X(v)\not \subseteq H$ . Thus $v(D)=v(\Gamma )\le T(L-S_m(L;H)\cdot H; v))$ and $S_m(V_{\vec {\bullet }},\mathcal {F};v)\le T(L-S_m(L;H)\cdot H; v))$ . Letting $m\to \infty $ , we see that

$$\begin{align*}S(L;v)\le T(L-S(L;H)\cdot H; v)). \end{align*}$$

By direct calculation for any irreducible divisor $H\in |rL|$ , we have

(4.1) $$ \begin{align} S(L;H)=\int_0^{1/r}(1-rx)^n \mathrm{d} x=\frac{1}{r(n+1)}; \end{align} $$

putting it into the previous inequality we get $S(L;v)\le \frac {n}{n+1}T(L;v)$ as desired.

Proof of Theorem 4.1

We only prove the K-stability part since the argument for K-semistability is similar (and simpler). Let $r>0$ be an integer such that $-r(K_X+\Delta )$ is Cartier. Following [Reference Fujita23], we say that a divisor E over X is dreamy if the double graded algebra $\bigoplus _{k,j\in \mathbb {N}} H^0(Y,-kr\pi ^*(K_X+\Delta )-jE)$ is finitely generated (where $\pi \colon Y\to X$ is a proper birational morphism such that the centre of E is a prime divisor on Y). For such E, there exists some effective $\mathbb {Q}$ -divisor $D\sim _{\mathbb {Q}} -(K_X+\Delta )$ such that $T(-K_X-\Delta ;E)=\mathrm {ord}_E(D)$ . By assumption, $(X,\Delta +\frac {n}{n+1}D)$ is klt, hence $\frac {n}{n+1}T(-K_X-\Delta ;E)<A_{X,\Delta }(E)$ and by Lemma 4.1, we have $\beta _{X,\Delta }(E)=A_{X,\Delta }(E)-S(-K_X-\Delta ;E)>0$ . Since this holds for any dreamy divisor E over X, $(X,\Delta )$ is K-stable by [Reference Fujita23, Theorem 1.6 and §6].

Proof. We may assume that $T(L;v)>\eta (L;v)$ , otherwise the statement follows from Lemma 4.1. We claim that there exists a unique irreducible $\mathbb {Q}$ -divisor $G\sim _{\mathbb {Q}} L$ such that $v(G)>\eta $ . The uniqueness simply follows from the definition of movable threshold. To see the existence, let $\widetilde {G}\sim _{\mathbb {Q}} L$ be an effective $\mathbb {Q}$ -divisor on X such that $v(\widetilde {G})>\eta $ (such $\widetilde {G}$ exists by the definition of pseudo-effective thresholds). Since X is $\mathbb {Q}$ -factorial and has Picard number one, every divisor on X is $\mathbb {Q}$ -linearly equivalent to a rational multiple of L. In particular, we may write $\widetilde {G}=\sum \lambda _i G_i$ , where $\sum \lambda _i = 1$ and each $G_i\sim _{\mathbb {Q}} L$ is irreducible. As $v(\widetilde {G})>\eta $ , we have $v(G_i)>\eta $ for some i that proves the claim. Note that by the definition of pseudo-effective threshold, we then necessarily have $v(G)=T(L;v)$ . Write $G=\lambda G_0$ , where $G_0$ is a prime divisor on X.

As in the proof of Lemma 4.1, let $r\in \mathbb {Z}_+$ be such that $rL$ is very ample, and let $H\in |rL|$ be a general member. Let $V_{\vec {\bullet }}$ be the complete linear series associated to L, and let $\mathcal {F}$ be the filtration on $V_{\vec {\bullet }}$ induced by H. By Proposition 3.1, we have $S(L;v)=S(V_{\vec {\bullet }},\mathcal {F};v)$ . Let D be an m-basis type $\mathbb {Q}$ -divisor of L that’s compatible with both $\mathcal {F}$ and $\mathcal {F}_v$ (which exists by Lemma 3.1). We have

$$\begin{align*}D=S_m(L;H)\cdot H + \Gamma \end{align*}$$

for some effective $\mathbb {Q}$ -divisor $\Gamma $ whose support doesn’t contain H. We further decompose $\Gamma =\mu G_0 + \Gamma _0$ , where the support of $\Gamma _0$ doesn’t contain $G_0$ . Note that $v(\Gamma _0)\le \eta (\Gamma _0;v)$ by our choice of $G_0$ . As H is general and D is of m-basis type, we have $\mu =\mathrm {ord}_{G_0}(\Gamma )=\mathrm {ord}_{G_0}(D)\le S_m(L;G_0)$ ; thus

$$ \begin{align*} S_m(V_{\vec{\bullet}},\mathcal{F};v) & = v(D) = v(\Gamma) = \mu\cdot v(G_0) + v(\Gamma_0) \\ & \le S_m(L;G_0)\cdot v(G_0) + \eta(\Gamma - S_m(L;G_0)\cdot G_0;v) \\ & = T(S_m(L;G_0)\cdot G_0;v) + \eta(L- S_m(L;H)\cdot H - S_m(L;G_0)\cdot G_0;v). \end{align*} $$

Since $\rho (X)=1$ , for any prime divisor F on X, we have $S(L;F)\cdot F \sim _{\mathbb {Q}} \frac {1}{n+1}L$ as in the proof of Lemma 4.1, hence letting $m\to \infty $ in the above inequality, we obtain

$$\begin{align*}S(L;v)=\lim_{m\to\infty} S_m(V_{\vec{\bullet}},\mathcal{F};v)\le \frac{1}{n+1}T(L;v)+\frac{n-1}{n+1}\eta(L;v) \end{align*}$$

as desired.

Proof of Theorem 4.2

As in Theorem 4.1, we only prove the K-stability part. Let E be a dreamy divisor over X. If the centre of E is a prime divisor on X, then we have $-(K_X+\Delta )\sim _{\mathbb {Q}} \lambda E$ for some $\lambda>0$ as X has Picard number one. By assumption $(X,\Delta +\frac {\lambda }{n+1} E)$ is klt, hence $\beta _{X,\Delta }(E) = A_{X,\Delta }(E)-S(-K_X-\Delta ;E) = A_{X,\Delta }(E) - \frac {\lambda }{n+1}> 0$ . If the centre of E has codimension at least two on X, then since E is dreamy there are effective $\mathbb {Q}$ -divisor $D\sim _{\mathbb {Q}} -(K_X+\Delta )$ and movable boundary $M\sim _{\mathbb {Q}} -(K_X+\Delta )$ such that $\mathrm {ord}_E(D)=T(-K_X-\Delta ;E)$ and $\mathrm {ord}_E(M)=\eta (-K_X-\Delta ;E)$ . By assumption $(X,\Delta +\frac {1}{n+1}D+\frac {n-1}{n+1}M)$ is klt, thus

$$ \begin{align*} A_{X,\Delta}(E) &> \frac{1}{n+1}\mathrm{ord}_E(D)+\frac{n-1}{n+1}\mathrm{ord}_E(M) \\ & = \frac{1}{n+1}T(-K_X-\Delta;E)+\frac{n-1}{n+1}\eta(-K_X-\Delta;E) \\ & \ge S(-K_X-\Delta;E), \end{align*} $$

where the last inequality follows from Lemma 4.2. Therefore $\beta _{X,\Delta }(E)>0$ for all dreamy divisors E over X, and $(X,\Delta )$ is K-stable by [Reference Fujita23, Theorem 1.6 and §6].

Proof. By [Reference Cheltsov and Shramov12, Theorem 1.26], X is $\mathbb {Q}$ -factorial of Picard number one and $(X,M)$ has canonical singularities (in particular it is klt) for every movable boundary $M\sim _{\mathbb {Q}} -K_X$ . Let $D\sim _{\mathbb {Q}} -K_X$ be an effective $\mathbb {Q}$ -divisor. By assumption, $(X,\frac {1}{2}D)$ is lc. As $\frac {1}{n+1}D+\frac {n-1}{n+1}M=\frac {2}{n+1}\cdot \frac {1}{2}D+\frac {n-1}{n+1}M$ is a convex linear combination of $\frac {1}{2}D$ and M, we see that the conditions of Theorem 4.2 are satisfied and therefore X is K-stable.

Proof. Let $V_{\vec {\bullet }}$ be the complete linear series associated to L, and let $W_{\vec {\bullet }}$ be its refinement by H. By Example 2.12, $W_{\vec {\bullet }}$ is almost complete and $F(W_{\vec {\bullet }})=0$ . By equation (4.1), we have $S(L;H)=\frac {1}{n+1}$ . As discussed in Section 3, any m-basis type $\mathbb {Q}$ -divisor $D\sim _{\mathbb {Q}} L$ that is compatible with H can be written as $D=S_m(L;H)\cdot H + \Gamma $ , where $\Gamma |_H$ is an m-basis type $\mathbb {Q}$ -divisor of $W_{\vec {\bullet }}$ ; thus, letting $m\to \infty $ , we see that

$$\begin{align*}c_1(W_{\vec{\bullet}})\sim_{\mathbb{Q}} L|_H- S(L;H)\cdot H|_H\sim_{\mathbb{Q}} \frac{n}{n+1}L|_H \end{align*}$$

and $\delta _x(c_1(W_{\vec {\bullet }}))=\frac {n+1}{n}\delta _x(L|_H)$ . The result now follows directly from Corollary 3.3 with $F=H$ .

Proof. First assume that $n=1$ : that is, X is a curve that is smooth at x (in this case, the statement should be well-known to experts). By direct calculation, we have $S(L;x)=\frac {1}{2}\deg L$ . Hence $\delta _x(L) = \frac {2}{\deg L}$ as desired.

Assume now that the statement has been proved in smaller dimensions. Let $H\in |L|$ be a general divisor containing x. By (*), H is smooth at x and $L|_H$ also satisfies (*). By induction hypothesis, we have $\delta _x(L|_H)\ge \frac {n}{(L^{n-1}\cdot H)}=\frac {n}{(L^n)}$ , hence by Lemma 4.3, we see that $\delta _x(L)\ge \frac {n+1}{(L^n)}$ . Suppose that equality holds, $(L^n)>1$ , and let v be a valuation on X that computes $\delta _x(L)$ . Then by Lemma 4.3, we see that the centre $C_X(v)$ of v is not contained in H, $\delta _x(L|_H)=\frac {n}{(L^n)}$ , and it is computed by some valuation $v_0$ on H with centre $Z\subseteq C_X(v)\cap H$ . But by induction hypothesis, $v_0$ is divisorial and its centre Z is a prime divisor on H, hence $C_X(v)$ has to be a divisor on X. It follows that v is divisorial as well and is induced by a divisor on X.

Proof. By Lemma 4.4, we have $\delta _x(L)\ge \frac {n+1}{(L^n)}$ at every $x\in X$ , hence $\delta (L)\ge \frac {n+1}{(L^n)}$ . By (1), we then obtain $\delta (-K_X)\ge \frac {n+1}{r\cdot (L^n)}\ge 1$ and X is K-semistable. Assume that X is not uniformly K-stable: that is, $\delta (-K_X)=1$ . Then equality holds in (1) and $\delta (L) = \frac {n+1}{(L^n)}$ . By Lemma 4.4, either $(L^n)=1$ or $\delta (L)$ is computed by some prime divisor E on X. In the latter case, there is nothing left to prove. In the former case, we have $r=\frac {n+1}{(L^n)}=n+1$ , hence $X\cong \mathbb {P}^n$ by [Reference Kobayashi and Ochiai30] and $\beta _X(H)=0$ for any hyperplane H on X.

Proof. Note that (3) is a special case of (4) with $m=1$ . We will also treat (5) separately. In each remaining case, we will find an ample line bundle L on the Fano variety that satisfies the assumptions of Corollary 4.4. Indeed, for del Pezzo surfaces X of degree $2$ or $3$ (respectively, degree $1$ ), we take $L=-K_X$ (respectively, $L=-2K_X$ ). We also set $L=-K_X$ for hypersurfaces $X\subseteq \mathbb {P}^{n+1}$ of degree $n+1$ . In case (4), we choose $L=\pi ^*H$ . It is straightforward to verify that they all satisfy the assumptions of Corollary 4.4, hence by Corollary 4.4, all Fano manifolds X in (1)–(4) are K-semistable. Moreover, del Pezzo surface of degree $1$ or $2$ are uniformly K-stable since $(L^n)<\frac {n+1}{r}$ for our choice of L. It remains to check that there are no divisors E on X with $\beta _X(E)=0$ in the other cases.