The stability threshold and Diophantine approximation

Abstract

The purpose of this paper is to use the filtration that appeared in Ru and Vojta [Amer. J. Math. 142 (2020), pp. 957-991] to extend the result of Blum-Jonsson [Adv. Math. 365 (2020), p. 57], as well as to explore some connections between the notion of the $K$-stability and Diophantine approximation, especially the $\beta$-constant and the Ru-Vojta’s theorem.

1. Introduction

The notion of the K-stability of Fano varieties is an algebro-geometric stability condition originally motivated by studies of Kähler metrics. When the base field is the complex number field, it was recently established that the existence of positive scalar curvature Kähler-Einstein metric is actually equivalent to the K-stability condition, by the works of Tia97, Don02, B16, and others, including the recent celebrated result CDS15a, CDS15c. This equivalence had been known before as the Yau-Tian-Donaldson conjecture (for the case of Fano varieties).

The original notion of K-stability in Tia97, Don02 is defined in terms of the sign of the generalised Futaki invariant on all test configurations or at least on some special test configurations (see LX14). Recently, there has been tremendous progress in reinterpreting K-stability in terms of invariants associated to valuations rather than test configurations. More specifically, in BHJ17, the data of a test configuration was translated into the data of a filtration and it was shown that a nontrivial special test configuration yields a divisorial valuation. In 2016, K. Fujita Fuj16 introduced divisorial stability: Let $X$ be a ${\mathbb{Q}}$-fano variety, i.e., a projective variety over the complex number field which has at worst klt singularities such that the anticanonical divisor $-K_X$ of $X$ is ample (as ${\mathbb{Q}}$-divisor). The pair $(X, -K_X)$ is said to be divisorially stable (resp. semi-) if the value

$$\begin{equation*} \eta (D): =\operatorname {Vol}(-K_X)-\int _0^{\infty } \operatorname {Vol}(-K_X-tD) dt \end{equation*}$$

satisfies $\eta (D)>0$ (resp. $\eta (D)\ge 0$) for any nonzero divisor $D$ on $X$. Fujita Fuj16 showed that if $X$ is $K$-(semi) stable, then it is divisorially (semi) stable. Later, based on the work of LX14, K. Fujita Fuj19 and C. Li Li17 independently proved that the $K$-(semi) stability and divisorial (semi) stability are indeed equivalent if one goes to the birational model and modifies the constant to

$$\begin{equation*} \tilde{\eta }(E): =A_X(E)\operatorname {Vol}(-K_X)-\int _0^{\infty } \operatorname {Vol}(-K_X-t E) dt \end{equation*}$$

for any prime divisors $E$ over $X$, i.e. they are prime divisors on a birational model $\pi : Y\rightarrow X$, where $A_X(E)\coloneq 1+\operatorname {ord}_E(K_{Y/X})$ is the log discrepancy. Namely, For ${\mathbb{Q}}$-Fano $X$, $X$ is $K$-stable (resp. semi-) if and only if $\tilde{\eta }(E)>0$ (resp. $\tilde{\eta }(E)\ge 0$) for any prime divisors $E$ over $X$. If we denote, for a line bundle $L$ and a Cartier divisor $D$,

$$\begin{equation} \beta (L, D)\coloneq {1\over \operatorname {Vol}(L)}\int _0^{\infty } \operatorname {Vol}(L-tD)dt, {\tag{1}} \end{equation}$$

then it states: For ${\mathbb{Q}}$-Fano $X$, $X$ is $K$-stable (resp. semi-) if and only if ${A_X(E)\over \beta (-K_X, E)}> 1$ (resp. $\geq 1$) for any prime divisors $E$ over $X$. In BJ20 (see also FO18), Blum-Jonsson introduced the stability threshold $\delta (L)=\inf _E {A_X(E)\over \beta (L, E)}$ $=\lim _{m\rightarrow \infty } \delta _m(L)$, where $\delta _m(L)\coloneq \inf \{\text{lct}(D)\nobreakspace |\nobreakspace D\sim _{{\mathbb{Q}}} L\nobreakspace \text{of m-basis type}\}$. The result is re-formulated as follows: For ${\mathbb{Q}}$-Fano $X$, $X$ is $K$-stable (resp. semi-) if and only if $\delta (-K_X)> 1$ (resp. $\delta (-K_X)\geq 1$).

The $\beta$-constant $\beta (L, D)$ defined in 1 played an important role in Diophantine approximation (see MR15, RV20). In particular, Ru-Vojta RV20 proved the following result, which is viewed as an extension of Schmidt’s subspace theorem (for notations, see RV20).

Theorem A (RV20).

Let $X$ be a projective variety, and $D_1$, …, $D_q$ be effective Cartier divisors, both defined over a number field $k$. Assume that $D_1$, …, $D_q$ intersect properly on $X$. Let $S\subset M_k$ be a finite set of places on $k$. Let $L$ be a big line sheaf on $X$. Then, for every $\epsilon >0$, there is a proper Zariski-closed subset $Z$ of $X$ such that the inequality

$$\begin{equation} \sum _{j=1}^q \beta (L,D_j) m_S(x,D_j) \leq (1+\epsilon ) h_{L}(x) \tag{2} \end{equation}$$

holds for all $x\in X(k)$ outside of $Z$.

The proof of Theorem A uses the m-basis type divisor chosen from a filtration which is similar to, but more sophisticated than, the filtration used in the paper of Blum-Jonsson BJ20. This filtration used in RV20 is multi-variable which allows us to deal with the divisors $D_1+\cdots +D_q$, where $D_1$, …, $D_q$ are in general position, rather than a single divisor in the case of Blum-Jonsson BJ20. The purpose of this paper is to use this filtration to extend the result of Blum-Jonsson BJ20, as well as to explore some connections among these areas. In the last section, we explore the relation between the constant $\beta (L, D)$, the Seshadri constant $\epsilon (L, D)$ and the pseudo-effective constant $T(L, D)$, and use the relation to derive some corollaries of Theorem A.

2. The Okounkov body and the $\beta$-constant

We work on the field ${\mathbb{C}}$ although the results hold for any algebraically closed field with characteristic zero. Throughout the paper, we use $X$ to denote a normal projective variety of dimension $n$.

The $\beta$-constant

Let $L$ be a big line bundle (we also regard $L$ as a line sheaf or a Cartier divisor) and let $D$ be a nonzero effective Cartier divisor on $X$. In RV20 (see also MR15), the following constant was introduced

$$\begin{equation} \beta ( L, D) = \liminf _{m\to \infty } \frac{\sum _{j\ge 1} \dim H^0(X, mL-jD)}{m \dim H^0(X, mL)}. {\tag{3}} \end{equation}$$

The $\beta$-constant appeared in Theorem A above.

The volume function

The volume of $L$ is defined by

$$\begin{equation*} \operatorname {Vol}(L)=\limsup _{m\rightarrow \infty } {\dim H^0(X, mL)\over m^n/n!}. \end{equation*}$$

Notice that $\operatorname {Vol}(kL) = k^n\operatorname {Vol}(L)$ so the volume function can be extended to ${\mathbb{Q}}$-divisors. Also note that the volume function depends only on the numerical class of $L$, so it is defined on $\mathrm{NS}(X): = \mathrm{Div}(X)/\mathrm{Num}(X)$ and extends to a continuous function on $\mathrm{NS}(X)_{{\mathbb{R}}}$. By using the theory of Okounkov bodies described below, one can prove (see Theorem 2.5) that the $\liminf$ in 3 is indeed a limit when $L$ is big, and that $\beta (L, D)$ can be expressed through the notion of volume function as in 1.

Okounkov bodies of a graded linear series of $L$

An Okounkov body $\Delta (L)\subset {\mathbb{R}}^n$ (where $n=\dim X$) is a compact convex set designed to study the asymptotic behavior of $H^0(X, mL)$, as $m\rightarrow \infty$. They have the crucial property that $\operatorname {Vol} (\Delta )=\lim _{m\rightarrow \infty } {\dim H^0(X, mL)\over m^n}= {\text{Vol(L)}\over n!}$. More generally, one can also attach to a graded linear series of $L$, i.e. $V_{\bullet }=\bigoplus _{m} V_{m} \subset \bigoplus _{m} H^0(X, mL)$, a convex body $\Delta \left(V_{\bullet }\right) \subset {\mathbb{R}}^n$ such that

$$\begin{equation*} \operatorname {Vol} \left(\Delta \left(V_{\bullet }\right) \right)=\lim _{m\rightarrow \infty }{\dim V_m\over m^n}. \end{equation*}$$

Here is the detailed description. Let $L$ be a big line bundle on $X$. Fix a system $z=(z_1, \dots , z_n)$ of parameters centered at a regular closed point $\xi$ of $X$. It gives a rank-$n$ valuation

$$\begin{equation*} \operatorname {ord}_{z} : \mathcal{O}_{X, \xi } \backslash \{0\} \rightarrow \mathbb{N}^{n} \end{equation*}$$

centered at $\xi$ as follows: expand $f \in {\mathcal{O}}_{X, \xi }$ as a power series

$$\begin{equation*} f=\sum _{\alpha \in {\mathbb{N}}^n} a_{\alpha } z^{\alpha } \end{equation*}$$

and set

$$\begin{equation*} \mathrm{ord}_z(f)=\min _{lex}\{\alpha \in {\mathbb{N}}^n\nobreakspace |\nobreakspace a_{\alpha }\not =0\}, \end{equation*}$$

where the minimum is taken with respect to the lexicographic order on ${\mathbb{N}}^n$. This extends to holomorphic section $s\in H^0(X, L)$ with the basic property that each graded piece has

$$\begin{equation} \dim \left(\{s\in W, \mathrm{ord}_z(s)\ge _{lex} \alpha \}/\{s\in W, \mathrm{ord}_z(s)>_{lex} \alpha \}\right)\leq 1 {\tag{4}} \end{equation}$$

for each subspace $W\subset H^0(X, L)$. Indeed, given $s_1, s_2$ with $\operatorname {ord}_z(s_1)=\operatorname {ord}_z(s_2)=\alpha$, we have $s_j=c_jz^{\alpha }+$ (high order terms), and it immediately follows that $s_1, s_2$ are linearly independent modulo $\{\mathrm{ord}_z>\alpha \}$ (see also Lemma 1.3 in LM09). Note that 4 implies in particular that $\#(\mathrm{ord}_z(W\setminus \{0\}))=\dim W$.

Let $V_{\bullet }=\bigoplus _{m} V_{m} \subset \bigoplus _{m} H^0(X, mL)$ be a nonzero graded linear series. For $m\in \mathbb{N}$, by 4, the subset $\Gamma _m=\Gamma _m(V_{\bullet })\coloneq \mathrm{ord}_z(V_m\setminus \{0\})$ has cardinality $\dim V_m$. One associates to $V_{\bullet }$ a semigroup

$$\begin{equation*} \Gamma \left(V_{\bullet }\right) \coloneq \left\{(m, \alpha ) \in \mathbb{N}^{n+1} |\nobreakspace \alpha \in \Gamma _m\right\}. \end{equation*}$$

Let $\Sigma =\Sigma \left(V_{\bullet }\right) \subset \mathbb{R}^{n+1}$ be the closed convex cone generated by $\Gamma (V_{\bullet })$. The Okounkov body of $V$ with respect to $z$ is given by

$$\begin{equation*} \Delta =\Delta _{z}\left(V_{\bullet }\right)=\left\{\alpha \in \mathbb{R}^{n} |(1, \alpha ) \in \Sigma \right\}. \end{equation*}$$

This is a compact convex subset of ${\mathbb{R}}^n$.

Remark.

The Okounkov body of $V$ depends on the choice of the system of parameters $z$. But the properties we are concerned about are independent of $z$.

For $m\ge 1$, let $\rho _m$ be the atomic positive measure (called the Duistermaat-Heckman measure) on $\Delta$ given by

$$\begin{equation*} \rho _{m}=m^{-n} \sum _{\alpha \in \Gamma _m}\delta _{m^{-1} \alpha } \end{equation*}$$

The following result is a special case of Theorem 1.12 in Bo14.

Theorem 2.1 (Bo14, Theorem 1.12).

Assume that $V_{\bullet }$ contains an ample series, i.e. $L=A+E$ (as ${\mathbb{Q}}$-divisor) with $A$ being ${\mathbb{Q}}$-ample and $E$ being effective such that $H^0(X, mA)\subset V_m\subset H^0(X, mL)$. Then its Okounkov body $\Delta \subset {\mathbb{R}}^n$ has nonempty interior, and we have $\lim _{m\rightarrow \infty } \rho _m=\rho$ in the weak topology of measures, where $\rho$ denotes the Lebesgue measure on $\Delta \subset {\mathbb{R}}^n$. In particular, the limit

$$\begin{equation} \operatorname {Vol}\left(V_{\bullet }\right): =\lim _{m \rightarrow \infty } \frac{n !}{m^{n}} \dim V_{m} \in (0, \operatorname {Vol}(L)] {\tag{5}} \end{equation}$$

exists, and equals $n!\operatorname {Vol}(\Delta )$.

Filtrations

We apply the above results to a special graded linear series $V_{\bullet }$ which is associated to a filtration ${\mathcal{F}}$. By a filtration ${\mathcal{F}}$ on $R(X, L)\coloneq \bigoplus _{m} R_{m}$ we mean a family $\mathcal{F}^{\lambda } R_{m} \subset R_{m}$ of ${\mathbb{C}}$-vector subspaces of $R_m$ for $m\in {\mathbb{N}}$ and $\lambda \in {\mathbb{R}}^+$, satisfying

(F1)

$\mathcal{F}^{\lambda } R_{m} \subset \mathcal{F}^{\lambda ^{\prime }} R_{m}$ when $\lambda \ge \lambda '$;

(F2)

$\mathcal{F}^{\lambda } R_{m}=\bigcap _{\lambda ^{\prime }<\lambda } \mathcal{F}^{\lambda ^{\prime }} R_{m}$ for $\lambda >0$;

(F3)

$\mathcal{F}^{0} R_{m}=R_{m}$ and $\mathcal{F}^{\lambda } R_{m}= 0$ for $\lambda \gg 0$;

(F4)

$\mathcal{F}^{\lambda } R_{m} \cdot \mathcal{F}^{\lambda ^{\prime }} R_{m^{\prime }} \subset \mathcal{F}^{\lambda +\lambda ^{\prime }} R_{m+m^{\prime }}$.

A simple example of a filtration is given by

$$\begin{equation} \mathcal{F}^{\lambda } R_{m}: =H^0(mL-\lambda D), \tag{6} \end{equation}$$

where $D$ is an effective Cartier divisor on $X$. Here we use the following convention: for $\lambda \in {\mathbb{R}}^+$ and $j\in {\mathbb{N}}$ with $j \leq \lambda < j+1$, we set $H^0(mL-\lambda D)=H^0(mL-jD)$.

A filtration ${\mathcal{F}}$ on $R(X, L)$ defines a family

$$\begin{equation} V_{\bullet }^{t}=V_{\bullet }^{\mathcal{F}, t}=\bigoplus _{m} V_{m}^{t} \tag{7} \end{equation}$$

of graded linear series of $L$, indexed by $t$, given by $V_{m}^{t} \coloneq \mathcal{F}^{m t} R_{m}$ for $m\in {\mathbb{N}}$.

Set

$$\begin{equation} T_{m} \coloneq T_{m}(\mathcal{F}) \coloneq \sup \left\{t \geq 0\nobreakspace |\nobreakspace V_{m}^{t} \neq 0\right\} {\tag{8}} \end{equation}$$

with the convention $T_m = 0$ if $R_m = 0$. By (F4) above,

$$\begin{equation*} T_{m+m^{\prime }} \geq \frac{m}{m+m^{\prime }}T_{m}+\frac{m^{\prime }}{m+m^{\prime }} T_{m^{\prime }}. \end{equation*}$$

so Fekete’s Lemma implies that the limit

$$\begin{equation} T(\mathcal{F}) \coloneq \lim _{m \rightarrow \infty } T_{m}(\mathcal{F}) \in [0,+\infty ] {\tag{9}} \end{equation}$$

exists, and equals $\sup _m T_m({\mathcal{F}})$. Hence

$$\begin{equation} T(\mathcal{F})=\sup \left\{t \geq 0 | \operatorname {Vol}\left(V_{\bullet }^{t}\right)>0\right\}, {\tag{10}} \end{equation}$$

because $V^t_{\bullet }$ contains an ample linear series for any $t < T({\mathcal{F}})$ (see [BC11, Lemma 1.6]), and hence, by using Theorem 2.1, $\operatorname {Vol}\left(V^t_{\bullet }\right)=\lim _{m \rightarrow \infty } \frac{n !}{m^{n}} \operatorname {dim}V_m^t$. We say that the filtration $\mathcal{F}$ is linearly bounded if $T(\mathcal{F})<\infty$.

Let $\Delta = \Delta (L)\subset {\mathbb{R}}^n$ be the Okounkov body of $R(X, L)$. The filtration ${\mathcal{F}}$ of $(R, L)$ induces a concave transform

$$\begin{equation} G=G^{\mathcal{F}} : \Delta \rightarrow \mathbb{R}_{+} \tag{11} \end{equation}$$

which is given by $G(\alpha )=\sup \left\{t \in \mathbb{R}_{+} | \alpha \in \Delta ^{t}\right\}$, where $\Delta ^t= \Delta (V_{\bullet }^{t})\subset {\mathbb{R}}^n$ is the Okounkov body associated to the graded linear series $V_{\bullet }^{t} \subset R(X, L)$. Note that, for $t'>t\ge 0$, we have $\Delta ^{t} \supset \Delta ^{t^{\prime }}$, and $\Delta ^0=\Delta$ and $\Delta ^t=\emptyset$ for $t>T({\mathcal{F}})$. It is easy to see that $\{G\ge t\}= \Delta ^t$ for $0<t<T({\mathcal{F}})$. Thus $G$ is a concave, upper semicontinuous function on $\Delta$ with values in $[0, T({\mathcal{F}})]$. As noted in the proof of [BKMS15, Lemma 2.22], the Brunn-Minkowski inequality implies:

Proposition 2.2.

The function $t \mapsto \operatorname {Vol}(V_{\bullet }^{t})$ is non-increasing and concave on $[0, T({\mathcal{F}})]$. As a consequence, it is continuous on ${\mathbb{R}}_+$, except possibly at $t = T({\mathcal{F}})$.

We define the limit measure $\mu =\mu ^{{\mathcal{F}}}$ of the filtration ${\mathcal{F}}$ as the pushforward

$$\begin{equation*} \mu =G_*\rho . \end{equation*}$$

Thus $\mu$ is a positive measure on ${\mathbb{R}}_+$ of mass $\operatorname {Vol}(\Delta ) ={1\over n!} \operatorname {Vol}(L)$ with support in $[0, T({\mathcal{F}})]$. Theorem 2.1 thus gives Corollary 2.3.

Corollary 2.3 (Corollary 2.4 in BJ20).

The limit measure $\mu$ satisfies

$$\begin{equation} \mu =-\frac{1}{n !} \frac{d}{d t} \operatorname {Vol}\left(V_{\bullet }^{t}\right)=-\frac{d}{d t} \operatorname {Vol}\left(\Delta ^{t}\right) \tag{12} \end{equation}$$

and is absolutely continuous with respect to Lebesgue measure, except possibly at $t = T({\mathcal{F}})$, where $\mu \{T(\mathcal{F})\}=\lim _{t \rightarrow T(\mathcal{F})-} \operatorname {Vol}\left(V_{\bullet }^{t}\right)$.

Let $N_m\coloneq \dim H^0(X, mL)$, and let $M(L)$ be the set of $m\in {\mathbb{N}}$ for which $N_m>0$. Given a filtration ${\mathcal{F}}$, consider the jumping numbers

$$\begin{equation*} 0\leq a_{m, 1}\leq \cdots \leq a_{m, N_m}=mT_m({\mathcal{F}}), \end{equation*}$$

defined by, for $m\in M(L)$,

$$\begin{equation*} a_{m, j}=a^{\mathcal{F}}_{m, j}=\inf \{\lambda \in {\mathbb{R}}_+\nobreakspace |\nobreakspace \text{codim} {\mathcal{F}}^{\lambda } R_m\ge j\} \end{equation*}$$

for $1\leq j\leq N_m$. Note that the non-increasing step functions $t\mapsto \dim {\mathcal{F}}^t R_m$ satisfy the condition that $\operatorname {codim} {\mathcal{F}}^t R_m=j$ if and only if $t\in (a_{m, j-1}, a_{m, j}]$. In particular, we have that

$$\begin{equation*} {d\over dt} \dim {\mathcal{F}}^t R_m = -\sum _{j=1}^{N_m} \delta _{a_{m, j}}. \end{equation*}$$

Define a positive measure $\mu _m=\mu _m^{\mathcal{F}}$ on ${\mathbb{R}}_+$ by

$$\begin{equation*} \mu _{m}={1\over m^n} \sum _{j=1}^{N_m} \delta _{m^{-1} a_{m, j}}=- {1 \over m^n} \frac{d}{d t} \dim {\mathcal{F}}^{mt} R_m. \end{equation*}$$

The following result is [BC11, Theorem 1.11].

Theorem 2.4 (BC11, Theorem 1.11).

If ${\mathcal{F}}$ is linearly bounded, i.e. $T({\mathcal{F}}) < +\infty$, then we have

$$\begin{equation*} \lim _{m\rightarrow +\infty } \mu _m=\mu \end{equation*}$$

in the weak sense of measures on ${\mathbb{R}}_+$.

Let $D$ be an effective Cartier divisor on $X$, and consider $\mathcal{F}^{\lambda } R_{m}: =H^0(mL-\lambda D)$. Then,

$$\begin{eqnarray*} \beta _m(L, D):&=& {1\over mN_m} \sum _{j\ge 1 } \dim H^0(X, mL-jD) = {1\over mN_m} \sum _{j\ge 1 } \dim {\mathcal{F}}^{j} R_m \\ &=& {1\over mN_m} \sum _{j\ge 0} j\left(\dim {\mathcal{F}}^{j} R_m - \dim {\mathcal{F}}^{j+1} R_m\right)\\ & =& {1\over mN_m} \sum _j a_{j, m} ={m^n\over N_m}\int _0^{\infty } t d\mu _m(t). \end{eqnarray*}$$

Then Theorem 2.4 implies that the limit in the definition of 3 exists, i.e. the limit $\lim _{m\rightarrow \infty } \beta _m(L, D)$ exists when $L$ is big. Moreover,

$$\begin{align*} \beta (L, D)&= \lim _{m\rightarrow \infty } \beta _m(L, D) =\frac{n !}{\operatorname {Vol}(L)} \int _{0}^{\infty } t d \mu (t)\\ &= \frac{1}{\operatorname {Vol}(L)} \int _{0}^{\infty } \operatorname {Vol}\left(V_{\bullet }^{t}\right) d t \\ &={1\over \operatorname {Vol}(L)}\int _0^{\infty } \operatorname {Vol}(L-tD)dt =\frac{1}{\operatorname {Vol}(\Delta )} \int _{\Delta } G d \rho . \end{align*}$$

We thus derive the main result of this section as follows:

Theorem 2.5.

Let $X$ be a normal projective variety of dimension $n$ and $L$ be a big line bundle on $X$. Let $D$ be an effective Cartier divisor on $X$. Then

$$\begin{eqnarray*} \beta (L, D) &=& \lim _{m\rightarrow \infty } {1\over mN_m} \sum _{j\ge 1 } \dim H^0(X, mL-jD) ={1\over \operatorname {Vol}(L)}\int _0^{\infty } \operatorname {Vol}(L-tD)dt \\ &=&\frac{1}{ \operatorname {Vol}(\Delta )} \int _{\Delta } G d \rho . \end{eqnarray*}$$

3. The stability threshold introduced by Blum-Jonsson

The log canonical threshold of $L$

Tian Tia87 in 1987 introduced $\alpha (L)$, the log canonical threshold of $L$, as follows: Let $h=e^{-\phi }$ be a singular metric of $L$ with $\Theta _{L, h}\ge 0$, where $\Theta _{L, h}= {\sqrt {-1}\over \pi }\partial \bar{\partial }\log \phi$. Let $p\in X$ and define $c_p(h)=\sup \{c\nobreakspace |\nobreakspace e^{-2c\phi }$ is locally integrable at $p$ $\}$. Define

$$\begin{equation*} \alpha (L)=\inf _{h: \Theta _{L, h}\ge 0} \inf _{p\in X} c_p(h). \end{equation*}$$

Tian Tia87 proved that, for ${\mathbb{Q}}$-Fano $X$, if $\alpha (-K_X)> {n\over n+1}$, then $X$ is $K$-stable.

Let $D$ be an effective Cartier divisor on $X$ and $[D]$ be its associated line bundle over $X$. Then the canonical section $s_D$ of $[D]$ gives a singular metric on $[D]$ with $\phi \coloneq \log |s_D|$. With the singular metric $h\coloneq e^{-\phi _D}$, we denote $\operatorname {lct}_p(D): = c_p(h)$ and $\operatorname {lct}(D): = \inf _{p\in X} c_p(h)$. $\operatorname {lct}(D)$ is called the log canonical threshold of $D$. According to Demailly (see CS08, Appendix A),

$$\begin{equation*} \alpha (L) =\inf \{\operatorname {lct}(D)\nobreakspace |\nobreakspace D\nobreakspace \text{is effective}, D\sim _{{\mathbb{Q}}}L\}, \end{equation*}$$

where $D\sim _{{\mathbb{Q}}}L$ means that $D$ is an effective ${\mathbb{Q}}$-divisor ${\mathbb{Q}}$-linearly equivalent to $L$.

We also have an alternative (algebraic geometry) definition for lct$(D)$ (see BJ20): Recall that a prime divisor $E$ over $X$ is a prime divisor on $Y$, where $\pi : Y\rightarrow X$ is a proper birational morphism and $Y$ is normal. Then $E$ defines a valuation $\mathbb{C}(X)^*\to \mathbb{Z}$ given by order of vanishing at the generic point of $E$, where $\mathbb{C}(X)^*$ is the set of nontrivial rational functions on $X$. The definition extends to $\operatorname {ord}_E(D)$, where $D$ is an effective $\mathbb{Q}$-divisor: Pick $m\geq 1$ such that $mD$ is Cartier and set $\operatorname {ord}_E(D)\coloneq m^{-1}\operatorname {ord}_E(f)$, where $f$ is a local equation of $mD$. Equivalently, $\operatorname {ord}_E(D)=m^{-1}\operatorname {ord}_E(s)$, where $s$ is the canonical section of $[mD]$. We define (see BJ20):

$$\begin{equation} \operatorname {lct}(D)=\min _{E \text{ over }X}\frac{A_X(E)}{\operatorname {ord}_E(D)}, {\tag{13}} \end{equation}$$

where $A_X(E)\coloneq 1+\operatorname {ord}_E(K_{Y/X})$ is the log discrepancy. We say $X$ has at worst klt singularities if $A_X(E)>0$ for all prime divisors $E$ over $X$.

Blum-Jonsson’s stability threshold

In BJ20 (see also FO18), Blum-Jonsson introduced the stability threshold $\delta (L)$ to replace $\alpha (L)$ by replacing $D$ with only the m-basis type divisors. Recall that an effective ${\mathbb{Q}}$-divisor $D\sim _{\mathbb{Q}} L$ on $X$ is of m-basis type (with respect to the line bundle $L$) if there is a basis $s_1$, …, $s_{N_m}$ of $H^0(X, mL)$ such that

$$\begin{equation*} D={1\over mN_m} (\{s_1=0\}+\cdots +\{s_{N_m}=0\}). \end{equation*}$$

Define

$$\begin{equation} \delta _m(L)\coloneq \inf \{\text{lct}(D)\nobreakspace |\nobreakspace D\sim _{{\mathbb{Q}}} L\nobreakspace \text{of m-basis type}\}, {\tag{14}} \end{equation}$$

and

$$\begin{equation} \delta (L)=\inf _{E} {A_X(E)\over \beta (L, E)}. {\tag{15}} \end{equation}$$

The result of Blum-Jonsson BJ20 is as follows:

Theorem 3.1 (Blum-Jonsson BJ20).

Let $X$ be a normal complex projective variety of dimension $n$ with at worst klt singularities, and let $L$ be a big line bundle on $X$. Then

(a)

$\lim _{m\rightarrow \infty } \delta _m(L)$ exists and is equal to $\delta (L)$;

(b)

$\alpha (L)\leq \delta (L)\leq (n+1)\alpha (L)$;

(c)

For ${\mathbb{Q}}$-Fano $X$, $X$ is $K$-semistable ($K$-stable) iff $\delta (-K_X)\ge 1$ $(\delta (-K_X)>1)$.

The proof of (b) depends on the relationship of the three constants described in the next section, and (c) directly follows from the recent result of Fujita Fuj19 and C. Li Li17. So here we only outline the proof of (a).

Let $E$ be a prime divisor over $X$. Denote by

$$\begin{equation} \beta _m(L, E): = {1\over mN_m}\sum _j a_{m, j},{\tag{16}} \end{equation}$$

where $a_{m, j}$ are the jumping numbers of the filtration $\mathcal{F}^{\lambda } R_{m}: =H^0(m\pi ^* L-\lambda E)$.

Lemma 3.2.

$$\begin{equation*} \beta _m(L, E)=\max _{\mathcal{B}} {1\over mN_m} \sum _{j=1}^{N_m} \operatorname {ord}_E (s_j), \end{equation*}$$

where the maximum is over all bases $\mathcal{B}=\{s_1, \dots , s_{N_m}\}$ of $H^0(X, mL)$.

Proof.

First consider any basis $s_1$, …, $s_{N_m}$ of $H^0(X, mL)$. We may assume $\operatorname {ord}_E(s_1)\leq \operatorname {ord}_E(s_2)\leq \cdots \leq \operatorname {ord}_E(s_{N_m})$. Then $\operatorname {ord}_E(s_j)\leq a_{m, j}$, where $a_{m, j}$ is the $j$-th jumping number of the filtration $\mathcal{F}^{\lambda } R_{m}: =H^0(m\pi ^* L-\lambda E)$. This implies that

$$\begin{equation*} {1\over mN_m} \sum _{j=1}^{N_m} \operatorname {ord}_E (s_j)\leq {1\over mN_m} \sum _{j=1}^{N_m} a_{m, j}= \beta _m(L, E). \end{equation*}$$

On the other hand, we can pick the basis such that $\operatorname {ord}_E(s_j) = a_{m, j}$, and then

$$\begin{equation*} {1\over mN_m} \sum _{j=1}^{N_m} \operatorname {ord}_E(s_j)= \beta _m(L, E). \end{equation*}$$

This proves the lemma.

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Proposition 3.3.

For $m\in M(L)$, we have

$$\begin{equation*} \delta _m(L) =\inf _E {A(E)\over \beta _m(L, E)}, \end{equation*}$$

where the inf runs through prime divisors $E$ over $X$.

Proof.

Note that

$$\begin{equation*} \delta _m(L) = \inf _{\text{D of m-basis type}} \left(\inf _E {A(E)\over \operatorname {ord}_E(D)}\right), \end{equation*}$$

where the inner infimum runs through the prime divisors over $X$. Switching the order of the two infimums and applying Lemma 3.2 above yield the desired equality.

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Proof of (a) in Theorem 3.1.

From 16 and Theorem 2.5, we have

$$\begin{equation*} \lim _{m\rightarrow \infty } \beta _m(L, E)=\beta (L, E). \end{equation*}$$

This, together with Proposition 3.3, gives

$$\begin{equation} \delta (L)= \limsup _{m\to \infty } \delta _m(L) \leq \inf _E {A(E)\over \beta (L, E)}.{\tag{17}} \end{equation}$$

On the other hand, given $\epsilon >0$, there exists $m_0$ such that $\beta _m(L, E)\leq (1+\epsilon ) \beta (L, E)$ for all the prime divisors $E$ over $X$. Thus

$$\begin{equation*} \delta (L) = \limsup _{m\to \infty } \delta _m(L) = \limsup _{m\to \infty } \inf _E {A(E)\over \beta _m(L, E)} \ge (1+\epsilon )^{-1} \inf _E {A(E)\over \beta (L, E)}. \end{equation*}$$

Letting $\epsilon \rightarrow 0$ and combining this inequality with 17 completes the proof.

An upper bound of $\delta (L)$

We now derive an upper bound for $\delta (L)$ in terms of $\operatorname {lct}(D)$, where $D$ is an effective Cartier divisor on $X$, not necessarily in $|L|$. Take a basis $B$ of the filtration $\mathcal{F}^{\lambda } R_{m}: =H^0(mL-\lambda D)$. Notice that, for any $s\in W_t\coloneq H^0(X, mL-tD)$, $\operatorname {ord}_E(s)\ge t \operatorname {ord}_E(D)$, so, from Lemma 3.2,

$$\begin{align*} \beta _m(L, E)&\ge {1\over mN_m} \sum _{s\in B} \operatorname {ord}_E(s) \ge {1\over m N_m} \left( \sum _{t=0}^{\infty } t (\dim W_t-\dim W_{t+1})\right)\operatorname {ord}_E(D)\\ &={1\over mN_m} \left(\sum _{t=1}^{\infty } \dim W_t\right)\operatorname {ord}_E(D). \end{align*}$$

Thus we have, from 3, 13, and Proposition 3.3, we get

$$\begin{equation*} \delta (L) \leq {1 \over \beta (L, D)}\operatorname {lct}(D). \end{equation*}$$

Thus we proved the following result.

Theorem 3.4.

Let $X$ be a normal complex projective variety of dimension $n$ with at worst klt singularities, and let $L$ a big line bundle on $X$. Then for any effective Cartier divisor $D$ on $X$, we have

$$\begin{equation*} \delta (L) \leq {1 \over \beta (L, D)}\operatorname {lct}(D). \end{equation*}$$

The role of the $m$-basis in Ru-Vojta’s result RV20

Note that the concept of $m$-basis is also used in the proof of the main Diophantine result in RV20. In particular, the following result, which is a re-formulation of Schmidt’s subspace theorem, involves the $m$-basis. To keep the notation to a minimum, we don’t recall the notation here. Instead, we refer to RV20.

Theorem 3.5 (Theorem 2.10 in RV20).

Let $k$ be a number field, let $S$ be a finite set of places of $k$ containing all archimedean places, let $X$ be a complete variety over $k$, and let $L$ be a line bundle on $X$. Let ${\mathcal{B}}$ be a finite set of the divisors $D$ which are of $m$-basis type with respect to $L$. Then, for any $\epsilon >0$, there is a proper Zariski-closed subset $Z$ of $X$ such that

$$\begin{equation} \sum _{\upsilon \in S}\max _{D\in {\mathcal{B}}} \lambda _{D,\upsilon }(x) \le (1+\epsilon )h_L(x) {\tag{18}} \end{equation}$$

holds for all $x\in (X\setminus Z)(k)$, where $\lambda _{D,\upsilon }$ is a local height function and $h$ is a logarithmic height function.

The main result of this section

With the more sophisticated multi-dimensional filtration in RV20 (see also Aut11), we extend Theorem 3.4 by proving the following more general result (which can be viewed as a counter-part of the arithmetic general theorem of Ru-Vojta RV20).

Let $D_1$, …, $D_q$ be effective Cartier divisors on $X$. We say that $D_1$, …, $D_q$ lie in general position if for any $I\subset \{1,\dots ,q\}$, we have $\dim (\bigcap _{i\in I}\operatorname {Supp} D_i) = n-\#I$ if $\# I \leq n$, and $\bigcap _{i\in I}\operatorname {Supp} D_i = \emptyset$ if $\#I>n$. We say that $D_1$, …, $D_q$ intersect properly if for any subset $I\subset \{1,\dots ,q\}$ and any $x\in \bigcap _{i\in I}\operatorname {Supp} D_i$, the sequence $(\phi _i)_{i\in I}$ is a regular sequence in the local ring $\mathscr{O}_{X,x}$, where $\phi _i$ are the local defining functions of $D_i$, $1\leq i\leq q$. It is known (see RV20) if $D_1$, …, $D_q$ intersect properly, then they lie in general position. The converse holds if $X$ is Cohen-Macaulay (this is true if $X$ is nonsingular).

Theorem 3.6.

Let $X$ be a normal complex projective variety of dimension $n$ with at worst klt singularities, and let $L$ a big line bundle on $X$. Then

$$\begin{equation*} \delta (L)\leq {1\over \max _{1\leq i \leq q} \beta (L,D_i)} \operatorname {lct}(D), \end{equation*}$$

for any divisor $D=D_1+\cdots +D_q$ with $D_1$, …, $D_q$ intersecting properly on $X$.

Proof.

Let $\Sigma = \biggl \{\sigma \subseteq \{1,\dots ,q\} \bigm | \bigcap _{j\in \sigma } \operatorname {Supp} D_j\ne \emptyset \biggr \}$. Since $D_1$, …, $D_q$ intersect properly on $X$, $\#\sigma \leq n$ for $\forall \sigma \in \Sigma$. Fix an integer $b>0$. For $\sigma \in \Sigma$, let

$$\begin{equation*} \bigtriangleup _{\sigma }=\left\{{\mathbf{a}}=(a_i)\in {\mathbb{N}}^{\#\sigma }\nobreakspace |\nobreakspace \sum _{i\in \sigma } a_i=b\right\}. \end{equation*}$$

For ${\mathbf{a}}\in \bigtriangleup _{\sigma }$ and $x\in {\mathbb{R}}_+$, one defines the ideal ${\mathcal{I}}_{\mathbf{a}}(x)$ of ${\mathscr{O}}_X$ by

$$\begin{equation*} \mathcal{I}_{\mathbf{a}}(x) = \sum _{{\mathbf{b}}} {\mathscr{O}}_X\left(-\sum _{i\in \sigma } b_iD_{i}\right), \end{equation*}$$

where the sum is taken for ${\mathbf{b}}\in {\mathbb{N}}^{\#\sigma }$ with $\sum _{i\in \sigma } a_ib_i\ge bx$. Write $L$ as the line sheaf $\mathscr{L}$ and consider the filtration ${\mathcal{F}}(\sigma ; {\mathbf{a}})_x = H^0(X, \mathscr{L}^m \otimes {\mathcal{I}_{\mathbf{a}}}(x))$, which are regarded as subspaces of $H^0(X,\mathscr{L}^m)$, and let

$$\begin{equation*} F(\sigma ; {\mathbf{a}}) = {1\over h^0(\mathscr{L}^m)}\int _0^{+\infty }(\dim {\mathcal{F}}(\sigma ; {\mathbf{a}})_x)\nobreakspace dx. \end{equation*}$$

The key result from Ru-Vojta (see Proposition 6.7 in RV20) is that

$$\begin{equation*} F(\sigma ; {\mathbf{a}})\ge \min _{1\leq i \leq q}\left({1\over h^0(\mathscr{L}^m)}\sum _{m\ge 1} h^0(\mathscr{L}^m(-kD_i))\right). \end{equation*}$$

It then gives (see Remark 6.6 in RV20), for any basis $\mathcal{B}_{\sigma ; {\mathbf{a}}}$ of $H^0(X, \mathscr{L}^m)$ adapted to the above filtration,

$$\begin{equation} \sum _{s\in \mathcal{B}_{\sigma ; {\mathbf{a}}}} \mu _{\mathbf{a}}(s) \ge \min _{1\leq i\leq q} \sum _{m\ge 1} h^0(\mathscr{L}^m(-kD_i)), {\tag{19}} \end{equation}$$

where for any $s\in H^0(X,\mathscr{L}^m)$, $\mu _{\mathbf{a}}(s) = \sup \{x\in \mathbb{R}^{+} : s\in \mathcal{F}(\sigma ; \mathbf{a})_x\}$.

Note that there are only finitely many ordered pairs $(\sigma , \mathbf{a})$ for $\sigma \in \Sigma , \mathbf{a}\in \bigtriangleup _{\sigma }$. We also note that, for any prime divisor $E$ over $X$ and $s\in H^0(X, \mathscr{L}^m \otimes {\mathcal{I}}_{\mathbf{a}}(\mu _{\mathbf{a}}(s)))$,

$$\begin{equation*} \text{ord}_E(s) \ge \min _{\mathbf{b}\in K} \sum _{i\in \sigma } b_i\nobreakspace \text{ord}_E(D_i), \end{equation*}$$

where $K=K_{\sigma ,\mathbf{a},s}$ is the set of minimal elements of $\{\mathbf{b}\in \mathbb{N}^{\#\sigma }:\sum _{i\in \sigma } a_ib_i\ge b\mu _{\mathbf{a}}(s)\}$ relative to the coordinatewise partial order on $\mathbb{N}^{\#\sigma }$. The set $K$ is a finite set. Let $t_i\coloneq {\text{ord}_E(D_i)\over \text{ord}_E(D)}$, then $\sum _{i\in \sigma } t_i=1$. Therefore, using $\#\sigma \le n$, $b \le \sum _{i\in \sigma } \lfloor (b+n)t_i \rfloor \le b+n$, and we may choose $\mathbf{a} = (a_i) \in \bigtriangleup _\sigma$ such that $t_i \ge \frac{a_i}{b+n}$ for all $i\in \sigma$. Thus, for any $s\in \mathcal{B}_{\sigma ; \mathbf{a}}$,

$$\begin{equation*} \begin{split} \text{ord}_E(s) &\ge \min _{\mathbf{b}\in K} \sum _{i\in \sigma } b_i \text{ord}_E(D_i) = \left( \min _{\mathbf{b}\in K} \sum _{i\in \sigma } b_i t_i \right) \text{ord}_E(D) \\ & \ge \left( \min _{\mathbf{b}\in K} \sum _{i\in \sigma } \frac{a_i b_i}{b+n} \right) \text{ord}_E(D) \ge \left( \frac{b}{b+n} \right) \mu _{\mathbf{a}}(s) \text{ord}_E(D). \end{split} \end{equation*}$$

Hence, by Lemma 3.2 as well as using 19,

$$\begin{equation*} \beta _m(L,E)\geq \sum _{s\in \mathcal{B}_{\sigma ;\mathbf{a}}}\frac{1}{mN_m}\operatorname {ord}_E(s)\geq \frac{b}{b+n}\operatorname {ord}_E(D) \left( \min _{1\leq i\leq q}\frac{\sum _{k\geq 1}h^0(\mathscr{L}^m(-kD_i) )}{mh^0(\mathscr{L}^m)} \right). \end{equation*}$$

Thus, by Proposition 3.3 and Theorem 2.5,

$$\begin{equation*} \delta _m(L) \leq \frac{A_X(E)}{\beta _m(L,E)} \leq \left( \frac{b}{b+n} \right) {1\over \max _{1\leq i \leq q} \beta (L,D_i)} {A_X (\text{ord}_E)\over \text{ord}_E(D)}. \end{equation*}$$

By letting $b\rightarrow \infty$ and $m\rightarrow \infty$, we get

$$\begin{equation*} \delta (L)\leq {1\over \max _{1\leq i \leq q} \beta (L,D_i)} \text{lct}(D). \end{equation*}$$ ■

4. Three important constants

Definition 4.1.

Let $L$ be an ample line bundle over $X$, we define the Seshadri constant $\epsilon (L, D)$ by $\epsilon (L, D) = \sup \{\gamma \in {\mathbb{Q}}: L-\gamma D \text{ is nef }\}$. We also define the pseudo-effective constant as $T(L, D) = \sup \{\gamma \in {\mathbb{Q}}: L-\gamma D \text{ is pseudo-effective}\}$.

Theorem 4.2.

We have ${1\over n+1} T(L, D) \leq \beta (L, D) \leq T(L, D)$.

Proof.

Given a filtration ${\mathcal{F}}$, we show that

$$\begin{equation} {1\over n+1} T(\mathcal{F}) \leq \beta (\mathcal{F}) \leq T(\mathcal{F}), {\tag{20}} \end{equation}$$

where $T(\mathcal{F})$ is given in 9 and $\beta (\mathcal{F})$ is given by

$$\begin{equation*} \beta (\mathcal{F}): = {1\over \operatorname {Vol}(L)}\int _0^{\infty } \operatorname {Vol}\left(V_{\bullet }^{t}\right) dt. \end{equation*}$$

The second inequality is clear since $\operatorname {Vol}\left(V_{\bullet }^{t}\right) \leq \operatorname {Vol}(L)$ and $\operatorname {Vol}\left(V_{\bullet }^{t}\right) =0$ for $t> T(\mathcal{F})$. The first follows from the Proposition of 2.2 which states that concavity of $t\mapsto \mathrm{vol}\left(V_{\bullet }^{t}\right)^{1/n}$ thus yields $\operatorname {Vol}\left(V_{\bullet }^{t}\right) \ge \operatorname {Vol}(L) \left(1- {t\over T(\mathcal{F})} \right)^{n}$. Therefore 20 is proved. The theorem follows from 20 by taking the filtration $\mathcal{F}^{\lambda } R_{m}: =H^0(mL-\lambda D)$ and by noticing 10.

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Combining Theorem 4.2 with Theorem A gives the following result.

Theorem 4.3.

Let $X$ be a projective variety, and $D_1$, …, $D_q$ be effective Cartier divisors, both defined over a number field $k$. Assume that $D_1$, …, $D_q$ intersect properly on $X$. Let $S\subset M_k$ be a finite set of places on $k$. Let $L$ be an ample line sheaf on $X$. Then, for every $\epsilon >0$, there is a proper Zariski-closed subset $Z$ of $X$ such that the inequality

$$\begin{equation} \sum _{j=1}^q T(L,D_j) m_S(x,D_j) \leq (n+1+\epsilon ) h_{L}(x) \tag{21} \end{equation}$$

holds for all $x\in X(k)$ outside of $Z$.

Theorem 4.3 is reminiscent of Theorem 3.3 in MR16, since both use the pseudo-effective constant to get bounds on the quality of Diophantine approximations.

Theorem 4.3 implies, noticing it is trivial from the definition that $\epsilon (L, D) \leq T(L, D)$, the following recent result of Levin-Heier HL20.

Theorem 4.4.

Let $X$ be a projective variety, and $D_1$, …, $D_q$ be effective Cartier divisors, both defined over a number field $k$. Assume that $D_1$, …, $D_q$ intersect properly on $X$. Let $S\subset M_k$ be a finite set of places on $k$. Let $L$ be an ample line sheaf on $X$. Then, for every $\epsilon >0$, there is a proper Zariski-closed subset $Z$ of $X$ such that the inequality

$$\begin{equation} \sum _{j=1}^q \epsilon (L,D_j) m_S(x,D_j) \leq (n+1+\epsilon ) h_{L}(x) \tag{22} \end{equation}$$

holds for all $x\in X(k)$ outside of $Z$.