Delta-invariants for Fano varieties with large automorphism groups

TL;DR

This paper introduces a $G$-invariant delta-threshold for Fano varieties, linking it to $G$-equivariant K-stability and exploring its implications for varieties with large automorphism groups, including spherical Fano varieties.

Contribution

It defines a new $G$-invariant delta-threshold and proves its equivalence to $G$-equivariant K-stability for Fano varieties, extending stability analysis to varieties with large symmetry groups.

Findings

The $G$-invariant delta-threshold characterizes $G$-equivariant K-stability.

Application to spherical Fano varieties with large automorphism groups.

Analysis of $G$-equivariant stability for varieties with finite automorphism groups.

Abstract

For a variety $X$, a big $\mathbb{Q}$-divisor $L$ and a closed connected subgroup $G \subset \mathrm{Aut}(X, L)$ we define a $G$-invariant version of the $\delta$-threshold. We prove that for a Fano variety $(X, -K_X)$ and a connected subgroup $G \subset \mathrm{Aut}(X)$ this invariant characterizes $G$-equivariant uniform $K$-stability. We also use this invariant to investigate $G$-equivariant $K$-stability of some Fano varieties with large groups of symmetries, including spherical Fano varieties. We also consider the case of $G$ being a finite group.

Equations

$$S_L(v)=\frac{1}{\mathrm{Vol}(L)}{\int }_0^{\infty }\mathrm{Vol}({\phi }^{\ast }L-tE)dt$$

$${\delta }_G(X,L)=\underset{v\in {\mathrm{DivVal}}_X^G}{\inf }\frac{A_X(v)}{S_L(v)}.$$

$${\delta }_G(X)=\underset{\begin{array}{c}{\mathrm{ord}}_{D_i}\in \mathcal{V}\cap \mathcal{C}\\ \mathcal{C}\in {\mathbb{F}}_X\end{array}}{\min }\frac{a_{D_i}}{a_{D_i}-V\cdot ⟨2{\rho }_Q-{\mathrm{bar}}_{DH}({\Delta }^{+}),{\pi }^{-1}({\mathrm{ord}}_{D_i})⟩}.$$

$${\delta }_G(X)=\delta (Y,B)$$

$$\operatorname{\mathrm{lct}}(X,\Delta;D)=\sup\{t\in\mathbb{R}\mid\mbox{$(X,\Delta+tD)$ is log canonical}\}.$$

$$\alpha(X,\Delta)=\inf\{\operatorname{\mathrm{lct}}(X,\Delta;D)\mid D\operatorname{\sim_{\mathbb{Q}}}-(K_{X}+\Delta)\mbox{ and $D$ is an effective $\mathbb{Q}$-divisor}\}.$$

$$D=\frac{1}{m{N}_m}(D(s_1)+\dots +D(s_{N_m}))$$

$$\delta_{m}(X,\Delta)=\inf\{\operatorname{\mathrm{lct}}(X,\Delta;D)\mid\mbox{$D$ is of $m$-basis type}\}$$

$$\delta (X,\Delta )=\underset{m\in \mathbb{N}}{\mathrm{limsup}}{\delta }_m(X,\Delta ).$$

$$A_X:{\mathrm{Val}}_X\to \mathbb{R}\cup \{\infty \}$$

$$\mathrm{Vol}(D)=\underset{m\to \infty }{\mathrm{limsup}}\frac{n!}{k^n\cdot m^n}{h}^0(X,O_X(mkD)).$$

$$T_L(v)=\sup \{t\in \mathbb{R}|\mathrm{Vol}(f^{\ast }L-tE)>0\}.$$

$$S_L(v)=\frac{1}{\mathrm{Vol}(L)}{\int }_0^{\infty }\mathrm{Vol}(f^{\ast }L-tE)dt.$$

$$\alpha (X,L)=\underset{v\in {\mathrm{Val}}_X}{\inf }\frac{A_X(v)}{T_L(v)}=\underset{v\in {\mathrm{DivVal}}_X}{\inf }\frac{A_X(v)}{T_L(v)}$$

$$\delta (X,L)=\underset{v\in {\mathrm{Val}}_X}{\inf }\frac{A_X(v)}{S_L(v)}=\underset{v\in {\mathrm{DivVal}}_X}{\inf }\frac{A_X(v)}{S_L(v)}.$$

$${\phi }_{|mL|}:X\to P(H^0(X,O_X(mL){)}^{\ast })$$

$$U_{f,i}=\{\gamma \in G\mid v_i(\gamma \cdot f)={\bar{v}}_i(f)\}.$$

$${\bar{v}}_i(f)=v_i(\gamma \cdot f)$$

$${\alpha }_G(X,L)=\underset{v\in {\mathrm{Val}}_X^G}{\inf }\frac{A_X(v)}{T_L(v)}=\underset{v\in {\mathrm{DivVal}}_X^G}{\inf }\frac{A_X(v)}{T_L(v)}$$

$${\delta }_G(X,L)=\underset{v\in {\mathrm{Val}}_X^G}{\inf }\frac{A_X(v)}{S_L(v)}=\underset{v\in {\mathrm{DivVal}}_X^G}{\inf }\frac{A_X(v)}{S_L(v)}.$$

$${\alpha }_G(X,L)=\underset{m\in \mathbb{N}}{\inf }\underset{\begin{array}{c}|\Sigma |\subset |mL|\\ |\Sigma {|}^G=|\Sigma |\end{array}}{\inf }\mathrm{lct}(X,\frac{1}{m}|\Sigma |)$$

$${\alpha }_{G,m}(X,L)=\underset{\begin{array}{c}|\Sigma |\subset |mL|\\ |\Sigma {|}^G=|\Sigma |\end{array}}{\inf }\mathrm{lct}(X,\frac{1}{m}|\Sigma |).$$

$$\underset{m\in \mathbb{N}}{\sup }\underset{\begin{array}{c}|\Sigma |\subset |mL|\\ |\Sigma {|}^G=|\Sigma |\end{array}}{\sup }\frac{1}{m}v({\mathfrak{b}}_{\Sigma })=T_L(v).$$

$$|\Sigma |=\{D\in |mL|\mid v(D)⩾\lambda /m\}$$

$$v(\frac{1}{m}|\Sigma |)=\underset{D\in \mid \Sigma |}{\inf }v(D)⩾\lambda /m.$$

$$\underset{m\in \mathbb{N}}{\inf }\underset{\begin{array}{c}|\Sigma |\subset |mL|\\ |\Sigma {|}^G=|\Sigma |\end{array}}{\inf }\mathrm{lct}(X,\frac{1}{m}|\Sigma |)=\underset{m\in \mathbb{N}}{\inf }\underset{\begin{array}{c}|\Sigma |\subset |mL|\\ |\Sigma {|}^G=|\Sigma |\end{array}}{\inf }\underset{v\in {\mathrm{Val}}_X^G}{\inf }\frac{A_X(v)}{\frac{1}{m}v({\mathfrak{b}}_{\Sigma })}=\underset{v\in {\mathrm{Val}}_X^G}{\inf }\frac{A_X(v)}{T_L(v)}$$

$$0<{\alpha }_G(X,L)⩽{\delta }_G(X,L)⩽(n+1){\alpha }_G(X,L).$$

$${\alpha }_G(X,L)(1+\frac{1}{n})⩽{\delta }_G(X).$$

$${\lambda }_{\max }(X,L)=\frac{(p_1\circ \pi {)}^{\ast }{L}^n\cdot {\phi }^{\ast }\bar{L}}{L^n}\text{mbox}and{J}^{NA}(X,L)={\lambda }_{\max }(X,L)-\frac{{\bar{L}}^{n+1}}{(n+1)L^n}$$

$$\mathrm{DF}(X,L)=\frac{n}{n+1}\cdot \frac{{\bar{L}}^{n+1}}{(-K_X{)}^n}+\frac{{\bar{L}}^n\cdot K_{X/{\mathbb{P}}^1}}{(-K_X{)}^n}.$$

Full text

The author is partially supported by Laboratory of Mirror Symmetry NRU HSE, RF Government grant, ag. 14.641.31.0001

1. Introduction

The problem of constructing Kähler–Einstein metrics on Fano varieties (over the field $\mathbb{C}$ of complex numbers) has been extensively studied in recent years. In particular, for smooth Fano varieties (Fano manifolds) the existence of Kähler–Einstein metrics was shown by Chen, Donaldson and Sun [CDS15] and Tian [Ti15] to be equivalent to an algebro-geometric condition of $K$-polystability. Another approach to this problem is the variational one, developed in [BBGZ13, Ber16, BBEGZ19]. For a Fano variety with finite automorphism group the existence of a Kähler–Einstein metric is equivalent to a stronger property of uniform $K$-stability. This was shown in [BBJ18] for $X$ smooth and in [Ber16, LTW19] for a Fano variety with klt singularities.

In view of the above results, it is important to be able to check if a given Fano variety $X$ is $K$-polystable or uniformly $K$-stable. A priori this requires computing certain numerical invariants for all polarized one-parameter degenerations of $X$ (see Definition 3.8 below for the precise definition of $K$-stability). Ideally, one would like to have a numerical invariant, depending on the variety $X$ and an ample (or, more generally, big) polarization $L$ only, such that the $K$-stability of $(X,L)$ is detected by this invariant. The first example of such invariant was the $\alpha$-invariant (or its version $\alpha_{G}(X)$ for a compact group $G$ of symmetries of $X$) introduced by Tian [Ti87, p. 229] via analytic methods. Tian gave a sufficient condition for the existence of a Kähler–Einstein metric on a Fano manifold in terms of $\alpha_{G}(X)$.

Theorem 1.1** ([Ti87, Theorems 2.1 and 4.1]).**

Let $X$ be a Fano manifold of dimension $n$ and $G\subset\operatorname{\mathrm{Aut}}(X)$ a compact subgroup. If $\alpha_{G}(X)>\frac{n}{n+1}$ then $X$ admits a Kähler–Einstein metric.

In early 2000s it became evident to experts that Tian’s $\alpha$-invariant coincides with the global log canonical threshold of $X$ (see [C08] and [CS08, Appendix A]). An algebraic counterpart of Tian’s result was given by Odaka and Sano [OS12, Theorem 1.4]. They have shown by purely algebraic methods that a klt Fano variety $X$ satisfying $\alpha(X)>n/(n+1)$ is $K$-stable. Moreover, for a Fano variety $X$ and a closed subgroup $G\subset\operatorname{\mathrm{Aut}}(X)$ Odaka and Sano proved [OS12, Theorem 1.10] that the condition $\alpha_{G}(X)>n/(n+1)$ implies $K$-stability of $X$ along $G$-equivariant degenerations (so-called $G$-equivariant $K$-stability, see Definition  3.8).

Automorphism groups have been successfully used to establish the existence (or nonexistence) of Kähler–Einstein metrics for many particular examples of Fano varieties. To list a few examples, we should mention the obstructions for the existence of such metrics [Mat57, Fut83]. Also, for smooth del Pezzo surfaces the necessary and sufficient condition for being Kähler–Einstein is reductivity of $\operatorname{\mathrm{Aut}}(X)$ [Ti90]. For toric Fano varieties $K$-stability was studied in [WZ04, Don02, Don08]. The case of varieties with torus action of complexity one was considered in [Su13, IS17]. In fact, by [LWX18, Theorem 1.4] torus-equivariant $K$-polystability of a Fano variety $X$ with a torus action is equivalent to $K$-polystability of $X$. Tian’s result was applied in [Nad90, C08, CS09] to prove the Kähler–Einstein property for certain Fano threefolds, including those of types $V_{1},V_{5}$ (see e.g. [IP99, §12] for the classification of Fano threefolds). Equivariant $K$-polystability of Fano threefolds of type $V_{22}$ was studied in [Don08, Section 5], [CS12, Appendix  A] and in [CS18]. For a criterion of equivariant $K$-stability of spherical Fano varieties see [Del16]. Also an extremely important general result was proved by Datar and Székelyhidi.

Theorem 1.2** ([DS16, Theorem 1]).**

Let $X$ be a Fano manifold and $G\subset\operatorname{\mathrm{Aut}}(X)$ a reductive subgroup. If $X$ is $K$-polystable with respect to $G$-equivariant test configurations then $X$ is Kähler–Einstein.

These results show the significance of equivariant $K$-stability for the study of Fano varieties.

Another invariant, the so-called $\delta$-invariant, was defined for Fano varieties by Fujita and Odaka [FO18, Definition 0.2] using log canonical thresholds of basis-type divisors. The inequality $\delta(X)>1$ implies uniform $K$-stability of $X$ by [FO18, Theorem 0.3] and in fact turns out to be equivalent to it (see [BlJ20, Theorem B]).

Since uniform $K$-stability forces the automorphism group of $X$ to be finite [BBEGZ19, Theorem 5.4], we cannot use $\delta$-invariant to study Kähler–Einstein property of Fano varieties with $\operatorname{\mathrm{Aut}}(X)$ infinite. However, if we restrict to $G$-equivariant degenerations for a suitable subgroup $G\subset\operatorname{\mathrm{Aut}}(X)$ (which is sufficient by Theorem 1.2), then in some examples (see e. g. Example 3.16) uniform $K$-stability holds. Therefore it is reasonable to expect that there is a version of $\delta$-invariant for a variety $X$ with a reductive subgroup $G\subset\operatorname{\mathrm{Aut}}(X)$.

The main goal of the present paper is to define the $\delta_{G}$-invariant for a pair $(X,L)$ where $X$ is a projective variety, $L$ is a big $\mathbb{Q}$-divisor and $G\subset\operatorname{\mathrm{Aut}}(X,L)$ is a closed connected subgroup. To do so, we follow the approach to $\delta$-invariants and $K$-stability via valuations on the field of rational functions on $X$, developed in [Fuj19a, Fuj19b, BlJ20, BHJ17]. Note also that an alternative valuative criterion for $K$-semistability was proved in [Li18]. In Section 2 we give the necessary definitions. The main object we consider is the space $\operatorname{\mathrm{DivVal}}^{G}_{X}$ of $G$-invariant divisorial valuations on $X$. Up to a multiplicative constant, every such valuation $v$ is given by the order of vanishing at the generic point of a $G$-stable prime divisor $E$ on a birational model $\varphi\colon Y\to X$. We consider the log discrepancy $A_{X}(v)=1+\operatorname{\mathrm{ord}}_{E}(K_{Y/X})$ and the expected vanishing order

[TABLE]

of the valuation. Being inspired by [BlJ20, Theorem 4.4], we define (see Definition 2.20):

[TABLE]

Also in Section 2 we study basic properties of this invariant and compare it to $\alpha_{G}$-invariant of Tian. In Section $3$ we discuss equivariant $K$-stability and prove our main result.

Theorem 1.3** (see Theorem 3.10 below).**

Let $(X,-K_{X})$ be a klt $\mathbb{Q}$-Fano variety with the anticanonical polarization. Let $G\subset\operatorname{\mathrm{Aut}}(X)$ be a closed connected subgroup. Then $(X,-K_{X})$ is uniformly $K$-stable (resp. $K$-semistable) with respect to $G$-equivariant degenerations if and only if the $\delta_{G}$-invariant of $(X,-K_{X})$ is greater than one (resp. greater or equal to one).

It is of course desirable to generalize this theorem to any closed subgroup $G\subset\operatorname{\mathrm{Aut}}(X)$ (e. g. a finite group $G$). In Section $3$ we also establish a further connection of $\delta_{G}(X)$ with metric invariants of $X$, see Proposition 3.21 below.

In Section $4$ we investigate $\delta$-invariants of varieties with an action of a torus $T=(\mathbb{G}_{m})^{k}$. We prove that $\delta$-invariant can be computed using only $T$-invariant valuations (Proposition 4.1), generalizing a result of Blum and Jonsson, who considered the case of a toric variety $X$ and a maximal torus $T$.

Section $5$ is devoted to $\delta_{G}$-invariants of spherical Fano varieties; we refer to this section for basic definitions and notation from the theory of spherical varieties. If $X$ is a spherical Fano variety under the action of a connected reductive group $G$, we give a formula for $\delta_{G}(X)$. We choose a Borel subgroup $B\subset G$ and a maximal torus $T\subset B$. This formula uses the description of $\operatorname{\mathrm{Val}}^{G}_{X}$ as the cone $\mathcal{V}$ in a finite-dimensional vector space. The log discrepancy $A_{X}(v)$ identifies with a certain piecewise linear function $h$ on the cones $C\subset\mathbb{F}_{X}$ in the complete colored fan $\mathbb{F}_{X}$ of $X$. The function $S_{-K_{X}}(v)$ can be expressed, following [Del16], via the moment polytope $\Delta^{+}$, the Duistermaat–Heckman measure $\mathrm{DH}$ on $\Delta^{+}$ (see [Del16, Theorem 4.5]) and the vector $2\rho_{Q}$, determined by $\Delta^{+}$ and the root system of $(G;T)$.

Proposition 1.4** (see Proposition 5.4 below).**

Let $X$ be a Fano variety which is spherical under the action of a connected reductive group $G$. Then $\delta_{G}$-invariant of $X$ can be expressed as follows:

[TABLE]

Here $\mathrm{bar}_{DH}(\Delta^{+})$ is the barycenter of $\Delta^{+}$ with respect to the Duistermaat–Heckman measure $\mathrm{DH}$ and $V$ is a constant depending on $\Delta^{+}$ and $\mathrm{DH}$ only. The minimum is taken over a finite set $\mathrm{ord}_{D_{1}},\ldots,\mathrm{ord}_{D_{N}}$ of valuations corresponding to generators of one-dimensional subcones (edges) in $\mathcal{C}\cap\mathcal{V},\mathcal{C}\in\mathbb{F}_{X}$.

In Section $6$ we consider the case of a variety with an action of a finite group $G$; aiming for possible generalization of [Su13, Theorem 1.2], we give an alternative definition of $\delta_{G}$ using $G$-invariant divisors and prove the ramification formula.

Proposition 1.5** (see Proposition 6.4 below).**

Let $X$ be a variety with klt singularities and $-K_{X}$ big. Let $G\subset\operatorname{\mathrm{Aut}}(X)$ be a finite group. Denote by $Y=X/G$ the quotient variety and let $B=\sum_{i}(1-1/m_{i})B_{i}$ be the branch divisor on $Y$. Then we have

[TABLE]

where $\delta(Y,B)$ is the $\delta$-invariant of the klt pair $(Y,B)$.

We expect that there is a unified definition of $\delta_{G}$ for any closed subgroup $G\subset\operatorname{\mathrm{Aut}}(X,L)$. We also hope for a generalization of Theorem 3.10 to the case of a Fano variety $X$ with an arbitrary closed subgroup $G\subset\operatorname{\mathrm{Aut}}(X)$.

Acknowledgement. We would like to thank our advisor Constantin Shramov for his patience and support. We thank Harold Blum for careful reading of a preliminary version of this text and pointing out numerous inaccuracies. We thank Kento Fujita, Thibaut Delcroix and Yuji Odaka for many helpful remarks; we also thank the referee for careful reading of the paper. The author is partially supported by Laboratory of Mirror Symmetry NRU HSE, RF Government grant, ag. 14.641.31.0001.

Note added to the updated version Soon after the first version of this paper had appeared on arXiv, we have been informed that Ziwen Zhu [Zhu19] gave an independent proof of a $G$-equivariant valuative criterion for $G\subset\operatorname{\mathrm{Aut}}(X)$ arbitrary (not nesessarily connected). He replaced $\operatorname{\mathrm{DivVal}}^{G}_{X}$ by the space of finite-orbit divisorial valuations (invariant under the connected component $G^{0}$) and the functions $\beta_{X}(v)$ and $S_{-K_{X}}(v)$ by their $G$-analogues. Using this criterion, Yuchen Liu and Ziwen Zhu defined in [LZ20] the invariant $\delta_{G}$ and generalized our Theorem 3.10 for any algebraic subgroup $G\subset\operatorname{\mathrm{Aut}}(X)$.

Moreover, at the same time Chi Li [Li19] has given a criterion for $G$-uniform $K$-stability of a Fano variety $(X,-K_{X})$ for $G\subset\operatorname{\mathrm{Aut}}(X)$ connected. The notion of $G$-uniform $K$-stability was introduced by Hisamoto [His16]. It is weaker than the notion of $G$-equivariant uniform $K$-stability since it replaces the usual norm $||(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})||$ on test configurations by the norm $||(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})||_{\mathbb{T}}$ orthogonal to the identity component of the center $\mathbb{T}=C(G)^{0}\subset\operatorname{\mathrm{Aut}}^{0}(X)$. He also proved that for $G=\operatorname{\mathrm{Aut}}^{0}(X)$ or $G=T\subset\operatorname{\mathrm{Aut}}(X)$ a maximal torus this condition is equivalent to existence of a Kähler–Einstein metric on $X$.

2. General definitions

2.1. Notation and conventions

We work over the field $\mathbb{C}$ of complex numbers. A $\mathbb{Q}$-Fano variety is a projective variety such that $K_{X}$ is an ample $\mathbb{Q}$-Cartier divisor. We restrict ourselves to $\mathbb{Q}$-Fano varieties (or pairs) with Kawamata log terminal (klt) singularities. For all basic information regarding singularities we refer to [Kol97]. A $\mathbb{Q}$-line bundle $L$ is a reflexive sheaf of rank $1$ such that some tensor power of $L$ is locally free.

In this section we recall the definitions of $\alpha$ and $\delta$-invariants via log canonical thresholds and valuations. Then we study the space of $G$-invariant valuations and define $\delta_{G}$.

2.2. Log canonical thresholds

Definition 2.1**.**

Let $X$ be a normal projective variety and let $\Delta$ be an effective $\operatorname{\mathbb{Q}}$-divisor on $X$ such that $K_{X}+\Delta$ is $\operatorname{\mathbb{Q}}$-Cartier and the pair $(X,\Delta)$ has klt singularities. For an effective $\operatorname{\mathbb{Q}}$-Cartier $\operatorname{\mathbb{Q}}$-divisor $D$ on $X$ we define the log canonical threshold of $D$ with respect to $(X,\Delta)$ by the formula

[TABLE]

Definition 2.2**.**

Let $(X,\Delta)$ be a klt log Fano pair. We define the $\alpha$-invariant of $(X,\Delta)$ by

[TABLE]

We also define the $\delta$-invariant of the pair $(X,\Delta)$ as follows. For every $m\in\mathbb{N}$ such that $m(K_{X}+\Delta)$ is a Cartier divisor we look at the space $H^{0}(X,-m(K_{X}+\Delta))$; put $N_{m}=h^{0}(X,-m(K_{X}+\Delta))$. For every basis $(s_{1},\ldots,s_{N_{m}})$ of $H^{0}(X,-m(K_{X}+\Delta))$ we denote $D(s_{1}),\ldots,D(s_{N_{m}})$ the corresponding divisors. We call effective $\operatorname{\mathbb{Q}}$-divisors $\mathbb{Q}$-linearly equivalent to $-K_{X}$ which have the form

[TABLE]

(anticanonical) $\operatorname{\mathbb{Q}}$-divisors of $m$-basis type on $(X,\Delta)$. We define for $m\in\mathbb{N}$ the invariant $\delta_{m}(X,\Delta)$ by

[TABLE]

and finally we define the $\delta$-invariant by

[TABLE]

If $\Delta=0$ then we simply write $\alpha(X)$ for $\alpha(X,0)$ and analogously for $\delta(X)$.

2.3. The space of valuations

In this subsection we recollect some basic information about the space of valuations on the function field of a variety. We refer to [JM12, BdFFU15, BlJ20] for more details. In this subsection $X$ is a normal and $\operatorname{\mathbb{Q}}$-Gorenstein projective variety over $\mathbb{C}$.

Definition 2.3**.**

A valuation on $X$ is a real valuation $v\colon\mathbb{C}(X)^{*}\to\mathbb{R}$ on the function field of $X$ which is trivial on $\mathbb{C}$. We denote by $\operatorname{\mathrm{Val}}_{X}$ the set of all nontrivial valuations on $X$. The latter is endowed with the topology of pointwise convergence, i. e. the weakest topology with the property that all evaluation maps $\mathrm{ev}_{f}\colon\mathrm{Val}_{X}\to\mathbb{R},v\mapsto v(f)$ are continuous for every $f\in\mathbb{C}(X)^{*}$.

Definition 2.4**.**

A valuation on $X$ is called divisorial if it has the form $v=c\cdot\operatorname{\mathrm{ord}}_{E}(\cdot)$ where $E$ is a prime divisor on a birational model of $X$ and $c\in\mathbb{R}$. The set of divisorial valuations on $X$ is denoted by $\operatorname{\mathrm{DivVal}}_{X}$.

Proposition 2.5**.**

The set of divisorial valuations is dense in the space of all valuations on $\mathbb{C}(X)$ in the topology of pointwise convergence.

Remark 2.6**.**

The above Proposition 2.5 is proved using the theory of Berkovich spaces. We sketch the proof for reader’s convenience. We can associate to the variety $X$ its Berkovich analytification $X^{an}$. The set of points of $X^{an}$ is, by definition, the set of pairs $(x,|\cdot|_{x})$ where $x\in X$ is a scheme point and $|\cdot|_{x}\colon k(x)\to\mathbb{R}$ is a valuation on the residue field of $x$. Then the set $\operatorname{\mathrm{Val}}_{X}$ is identified with the preimage of the generic point of $X$ under the projection $\pi\colon X^{an}\to X$. The topology induced on $\operatorname{\mathrm{Val}}_{X}$ from $X^{an}$ is precisely the topology of pointwise convergence. Thus Proposition 2.5 follows from the density theorem for divisorial points in a Berkovich space (see e.g. [Gub98, Theorem 7.12]).

Definition 2.7**.**

Let $v=\operatorname{\mathrm{ord}}_{E}$ be a divisorial valuation on $X$ where $E\subset Y$ is a prime divisor on a birational model $f\colon Y\to X$. Let $K_{Y/X}=K_{Y}-f^{*}K_{X}$ be the relative canonical divisor. Then the log discrepancy of $v$ is defined by $A_{X}(v)=1+\operatorname{\mathrm{ord}}_{E}(K_{Y/X})$.

Proposition 2.8**.**

([BdFFU15, Theorem 3.1]) The log discrepancy extends to a function

[TABLE]

which is lower semicontinuous and homogeneous of order $1$, that is, $A_{X}(\lambda\cdot v)=\lambda\cdot A_{X}(v)$ for $\lambda\in\mathbb{R}_{\geqslant 0}$.

We recall the definition of the volume function for $\mathbb{Q}$-divisors (see e.g. [Laz04, Section 2.2.C]).

Definition 2.9**.**

Let $D$ be a $\mathbb{Q}$-divisor on a variety $X$ of dimension $n$. Let $k\in\mathbb{N}$ be such that $kD$ is an integral Cartier divisor. Then the volume of $D$ is defined by

[TABLE]

Definition 2.10**.**

Let $v=\operatorname{\mathrm{ord}}_{E}$ be a divisorial valuation on $X$ where $E\subset Y$ is a prime divisor on a birational model $f\colon Y\to X$. Let $L$ be a big $\mathbb{Q}$-divisor on $X$. We define the pseudoeffective threshold of $v$ (or the maximal vanishing order) with respect to $L$ by

[TABLE]

We also define the expected vanishing order of $v$ with respect to $L$ by

[TABLE]

Proposition 2.11**.**

[BlJ20, Proposition 3.13] The functions $T_{L}$ and $S_{L}$ can be uniquely extended to functions $\operatorname{\mathrm{Val}}_{X}\to\mathbb{R}$ which are lower semicontinuous and homogeneous of order $1$, that is, $T(\lambda\cdot v)=\lambda\cdot T(v)$ and $S(\lambda\cdot v)=\lambda\cdot S(v)$ for $\lambda\in\mathbb{R}_{\geqslant 0}$.

Now we give the valuative definition of the $\alpha$ and $\delta$-thresholds, following [BlJ20]. This definition is equivalent to Definition 2.2, as shown in [BlJ20, Theorem C].

Definition 2.12**.**

We define the $\alpha$- and $\delta$-thresholds of $(X,L)$ (or the $\alpha$- and $\delta$-invariants of $(X,L)$) by the formulas

[TABLE]

and analogously

[TABLE]

In the case of a variety with a boundary divisor $(X,\Delta;L)$ such that $(X,\Delta)$ is klt we define $\alpha(X,\Delta;L)$ and $\delta(X,\Delta;L)$ in the same way by introducing the log discrepancy $A_{X,\Delta}(v)$ with respect to the pair $(X,\Delta)$.

2.4. Automorphisms preserving a big divisor class

We consider a pair $(X,L)$ where $X$ is a normal complex projective variety and $L$ is a big $\operatorname{\mathbb{Q}}$-Cartier divisor. We denote by $\operatorname{\mathrm{Aut}}(X,L)\subset\operatorname{\mathrm{Aut}}(X)$ the subgroup of automorphisms of $X$ preserving the class $[L]\in\mathrm{Cl}(X)$. The following proposition is well-known; however we do not know a standard reference (see e. g. [KPS18, Lemma 3.1.2]).

Proposition 2.13**.**

Let $(X,L)$ be a variety with a big $\mathbb{Q}$-Cartier divisor and let $m\in\mathbb{N}$ be such that $mL$ is Cartier and the map

[TABLE]

is birational onto its image. Then the above map is $\operatorname{\mathrm{Aut}}(X,L)$-equivariant and the group $\operatorname{\mathrm{Aut}}(X,L)$ embeds into $\mathrm{PGL}(H^{0}(X,\operatorname{\mathscr{O}}_{X}(mL)^{*})$. Therefore $\operatorname{\mathrm{Aut}}(X,L)$ is a linear algebraic group.

We are mostly interested in the case when $X$ is a Fano variety and $L=-K_{X}$. Since the automorphism group $\operatorname{\mathrm{Aut}}(X)$ preserves the anticanonical class, we have the following corollary.

Corollary 2.14**.**

The automorphism group of a $\operatorname{\mathbb{Q}}$-Fano variety is a linear algebraic group.

We will also consider triples $(X,\Delta;L)$; here the group $\operatorname{\mathrm{Aut}}(X,\Delta;L)\subset\operatorname{\mathrm{Aut}}(X)$ is a stabilizer of the boundary divisor $\Delta$ and the class of $L$. By Proposition 2.13 the group $\operatorname{\mathrm{Aut}}(X,\Delta;L)$ is a linear algebraic group as well.

2.5. $G$-invariant valuations

In this subsection we work with $(X,L)$ as above and a closed connected subgroup $G\subset\operatorname{\mathrm{Aut}}(X,L)$; then $G$ is a linear algebraic group. The group $G$ acts on the function field $\mathbb{C}(X)$ by $f\mapsto\gamma\cdot f$ and therefore on $\operatorname{\mathrm{Val}}_{X}$. We describe the space of $G$-invariant valuations mainly following [Kn93, Tim11].

Definition 2.15**.**

We denote by $\operatorname{\mathrm{Val}}^{G}_{X}\subset\operatorname{\mathrm{Val}}_{X}$ the set of all valuations $v\colon\mathbb{C}(X)^{*}\to\mathbb{R}$ which are invariant under the action of $G$. That is, a nontrivial valuation $v$ belongs to $\operatorname{\mathrm{Val}}^{G}_{X}$ if and only if for all $\gamma\in G$ and for all $f\in\mathbb{C}(X)^{*}$ we have $v(\gamma\cdot f)=v(f).$ We also denote by $\operatorname{\mathrm{DivVal}}^{G}_{X}$ the set of all divisorial $G$-invariant valuations. It has a topology induced from $\operatorname{\mathrm{Val}}_{X}$.

Proposition 2.16**.**

(see e. g. [Tim11, Proposition 19.8]) Let $X$ be a $G$-variety with $G$ connected. Then every $G$-invariant divisorial valuation $v\in\operatorname{\mathrm{DivVal}}^{G}_{X}$ is proportional to a valuation $\operatorname{\mathrm{ord}}_{D}$ where $D$ is a $G$-invariant prime divisor on some birational $G$-model of $X$.

Remark 2.17**.**

Observe that the trivial valuation $v_{0}\colon\mathbb{C}(X)\to\mathbb{R}^{*}$ does not belong to $\operatorname{\mathrm{Val}}_{X}$ or to $\operatorname{\mathrm{Val}}^{G}_{X}$. Thus for any $v\in\operatorname{\mathrm{Val}}^{G}_{X}$ the center of $v$ on $X$ (which exists since $X$ is projective) is $G$-invariant; its closure is a proper $G$-invariant subvariety of $X$. Hence the space $\operatorname{\mathrm{Val}}^{G}_{X}$ can be empty, for example if $G$-action on $X$ is transitive (see Remark 2.22 below).

The following approximation theorem of Sumihiro is crucial for the study of $G$-invariant valuations.

Theorem 2.18** ([Sum74, Lemma 10]).**

Let $G$ be a connected algebraic group and let $X$ be an irreducible $G$-variety. Then for any divisorial valuation $v$ on $\mathbb{C}(X)$ there exists a $G$-invariant valuation $\bar{v}\in\operatorname{\mathrm{DivVal}}^{G}_{X}\cup\mathbb{R}_{\geqslant 0}\{v_{0}\}$ with the following property: for any $f\in\mathbb{C}(X)$ there exists a nonempty Zariski-open subset $U_{f}\subset G$ such that for any $\gamma\in U_{f}$ we have $v(\gamma\cdot f)=\bar{v}(f)$.

Proposition 2.19**.**

Let $X$ be a $G$-variety. The set $\operatorname{\mathrm{DivVal}}^{G}_{X}$ of $G$-invariant divisorial valuations is dense in the set $\operatorname{\mathrm{Val}}^{G}_{X}$ in the topology of pointwise convergence.

Proof.

We assume that $\operatorname{\mathrm{Val}}^{G}_{X}$ is nonempty; otherwise there is nothing to prove. Suppose that $v\in\operatorname{\mathrm{Val}}_{X}$ is $G$-invariant. By Proposition 2.5 we can find a sequence $\{v_{i}\}$ of divisorial valuations converging to $v$. For every $i\in\mathbb{N}$ let $\bar{v}_{i}$ be the $G$-invariant valuation associated to $v_{i}$ by Theorem 2.18. It suffices to prove that the sequence $\{\bar{v}_{i}\}$ converges to $v$; since $v$ is nontrivial, almost all $v_{i}$ have to be divisorial in this case. Take a rational function $f\in\mathbb{C}(X)$; then for every $i\in\mathbb{N}$ we have the subset

[TABLE]

Let $U_{f}=\cap_{i\in\mathbb{N}}U_{f,i}$ be the intersection of these subsets; it is nonempty since the field $\mathbb{C}$ is uncountable. Then for any $\gamma\in U_{f}$ we obtain that

[TABLE]

converge to $v(\gamma\cdot f)=v(f)$ by definition of $\bar{v}_{i}$ and $G$-invariance of $v$. Therefore for any $f\in\mathbb{C}(X)$ the sequence $\bar{v}_{i}(f)$ converges to $v(f)$, as desired. This proves density of $G$-invariant divisorial valuations in the space $\operatorname{\mathrm{Val}}^{G}_{X}$. ∎

Now let us introduce the definitions for the $\alpha_{G}$ and $\delta_{G}$-thresholds via $G$-invariant valuations.

Definition 2.20**.**

Let $X$ be a variety and let $L$ be a big $\mathbb{Q}$-Cartier divisor. Let also $G\subset\operatorname{\mathrm{Aut}}(X,L)$ be a closed connected subgroup; we define the $G$-invariant $\alpha$- and $\delta$-thresholds of $(X,L)$ (we will also call them the $\alpha$- and $\delta$-invariants of $(X,L)$ with the action of $G$) by the formulas

[TABLE]

and analogously

[TABLE]

For a variety $X$ with $L=-K_{X}$ big we write $\delta_{G}(X,-K_{X})=\delta_{G}(X)$ and the same for $\alpha_{G}$. In the case of a pair $(X,\Delta;L)$ and a subgroup $G\subset\operatorname{\mathrm{Aut}}(X,\Delta;L)$ we can define $\alpha_{G}(X,\Delta;L)$ and $\delta_{G}(X,\Delta;L)$ in the same way by introducing the log discrepancy $A_{X,\Delta}(v)$ with respect to the pair $(X,\Delta)$.

Remark 2.21**.**

The extension procedure in Propositions 2.8 and 2.11 uses density of divisorial valuations in $\operatorname{\mathrm{Val}}_{X}$. By Proposition 2.19 we can restrict the functions $A_{X}(v)$, $T_{L}(v)$ and $S_{L}(v)$ to $\operatorname{\mathrm{DivVal}}^{G}_{X}$ and extend to $\operatorname{\mathrm{Val}}^{G}_{X}$ by the same procedure; the resulting functions will coincide with the restrictions to $\operatorname{\mathrm{Val}}^{G}_{X}$ of the respective extensions of $A_{X}(v)$, $T_{L}(v)$ and $S_{L}(v)$, given by Propositions 2.8 and 2.11.

Remark 2.22**.**

If a variety $X$ is a homogeneous space under the action of an algebraic group $G$ then both $\alpha_{G}$ and $\delta_{G}$ are infinite. Indeed, since the action of $G$ is transitive, the space $\operatorname{\mathrm{Val}}^{G}_{X}$ is empty.

Remark 2.23**.**

In [CS08, Appendix A] the algebraic version of Tian’s invariant $\alpha_{G}(X)$ was defined by

[TABLE]

where the second infimum is taken over all $G$-invariant linear subsystems $|\Sigma|\subset|mL|$.

Proposition 2.24**.**

The definition of $\alpha_{G}(X,L)$ given in Remark 2.23 is equivalent to Definition 2.20.

Proof.

By definition from Remark 2.23, we have $\alpha_{G}(X,L)=\inf_{m\in\mathbb{N}}\alpha_{G,m}(X,L)$ where

[TABLE]

Note that $\operatorname{\mathrm{lct}}(X,|\Sigma|)$ is defined as the log canonical threshold of the base ideal $\mathfrak{b}_{\Sigma}$ of $|\Sigma|$. Since $|\Sigma|$ is $G$-invariant, the base ideal is also $G$-invariant. There exists a divisorial valuation computing $\operatorname{\mathrm{lct}}(X,\mathfrak{b}_{\Sigma})$ by [BlJ20, Section 1.7]; moreover, by $G$-equivariant log resolution (see e.g. [Kol07, Section 3.9.1]) and connectedness of $G$ this valuation can be chosen to be $G$-invariant. Note also that for $v\in\operatorname{\mathrm{Val}}^{G}_{X}$ we have

[TABLE]

Indeed, by definition, $T(v)=\sup_{m\in\mathbb{N}}\sup_{D\in|mL|}\frac{1}{m}v(D)$. For any $D_{0}\in|mL|$ with $v(D_{0})=\lambda$ the linear system

[TABLE]

is nonempty and $G$-invariant by invariance of $v$; moreover,

[TABLE]

The reverse inequality is obvious. Expanding the definitions and switching the order of infima we can write

[TABLE]

and thus we obtain the equivalence of two definitions. ∎

Proposition 2.25**.**

Let $(X,L)$ be a variety with a big $\mathbb{Q}$-divisor and let $G\subset\operatorname{\mathrm{Aut}}(X,L)$ be a closed connected subgroup. We have the following inequalities for $\alpha_{G}(X,L)$ and $\delta_{G}(X,L)$ where $X$ has dimension  $n$:

[TABLE]

If, moreover, $L$ is ample then we have a stronger inequality

[TABLE]

Proof.

The first inequalities follow from [BlJ20, Lemma 2.6]. The second inequality follows from the fact that $\frac{S_{L}(v)}{T_{L}(v)}\leqslant\frac{n}{n+1}$ for $L$ an ample $\mathbb{Q}$-divisor and all valuations $v\in\operatorname{\mathrm{Val}}_{X}$ [Fuj19b, Proposition 2.1]. ∎

3. Equivariant $K$-stability

In this section we collect basic information on $G$-equivariant $K$-stability for a variety $X$ with an ample polarization $L$. Then we prove the main Theorem 1.3 (see Theorem 3.10 below).

We call a pair $(X,L)$ with $X$ projective and $L$ an ample $\operatorname{\mathbb{Q}}$-divisor a polarized variety.

3.1. Basic definitions related to $K$-stability

The following definitions are well-known and can be found e.g. in [Ti97, Don02, BHJ17].

Definition 3.1**.**

A test configuration for a polarized variety $(X,L)$ with $L$ ample is a pair $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ consisting of a variety $\operatorname{\mathcal{X}}$ with a projective surjective morphism $\pi\colon\operatorname{\mathcal{X}}\to\mathbb{A}^{1}$ and a $\pi$-semiample $\operatorname{\mathbb{Q}}$-line bundle $\operatorname{\mathcal{L}}$ together with a $\mathbb{G}_{m}$-action on $\operatorname{\mathcal{X}}$ preserving the class of $\operatorname{\mathcal{L}}$ such that

• •

The morphism $\pi\colon\operatorname{\mathcal{X}}\to\mathbb{A}^{1}$ is $\mathbb{G}_{m}$-equivariant with respect to the given action on $\operatorname{\mathcal{X}}$ and the multiplicative action on $\mathbb{A}^{1}$;

• •

The pair $(\operatorname{\mathcal{X}}\setminus\operatorname{\mathcal{X}}_{0},\operatorname{\mathcal{L}}|_{\operatorname{\mathcal{X}}\setminus\operatorname{\mathcal{X}}_{0}})$ is $\mathbb{G}_{m}$-equivariantly isomorphic to $(X\times(\mathbb{A}^{1}\setminus\{0\}),p_{1}^{*}L)$.

Definition 3.2** (see e. g. [OS12, DS16]).**

Let $G$ be a closed subgroup of $\operatorname{\mathrm{Aut}}(X,L)$. A test configuration $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ is called $G$-equivariant if there exists an action of $G$ on the pair $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ which commutes with the $\mathbb{G}_{m}$-action and restricts to the given $G$-action on $(\operatorname{\mathcal{X}}_{t},\operatorname{\mathcal{L}}|_{\operatorname{\mathcal{X}}_{t}})\simeq(X,L)$ for $t\neq 0$.

Definition 3.3**.**

Let $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ be a test configuration for $(X,L)$. The compactification $(\bar{\operatorname{\mathcal{X}}},\bar{\operatorname{\mathcal{L}}})$ of $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ is the degeneration of $(X,L)$ over $\mathbb{P}^{1}$ defined by gluing $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ with $(X\times(\mathbb{P}^{1}\setminus\{0\}),p_{1}^{*}L)$ along the subset $X\times(\mathbb{P}^{1}\setminus\{0,1\})$. The compactification is $G$-equivariant if $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ is.

Definition 3.4** (see e. g. [Der16, LX14]).**

A test configuration $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ is trivial if the relative canonical model $(\operatorname{\mathcal{X}}^{\operatorname{\mathrm{can}}},\operatorname{\mathcal{L}}^{\operatorname{\mathrm{can}}})$ of the normalization of $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ is $\mathbb{G}_{m}$-equivariantly isomorphic to $(X\times\mathbb{A}^{1},p_{1}^{*}L+c\operatorname{\mathcal{X}}_{0})$ for some $c\in\mathbb{Q}$. We say that $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ is of product type if $\operatorname{\mathcal{X}}^{\operatorname{\mathrm{can}}}$ is isomorphic to $X\times\mathbb{A}^{1}$.

Definition 3.5** ([BHJ17, Der16, LX14]).**

Consider a test configuration $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ for $(X,L)$. Let us denote by $\mathcal{Z}$ the normalization of the graph of the map $\bar{\operatorname{\mathcal{X}}}\dasharrow X\times\mathbb{P}^{1}$; it has natural maps $\pi\colon\mathcal{Z}\to X\times\mathbb{P}^{1}$ and $\varphi\colon\mathcal{Z}\to\operatorname{\mathcal{X}}$. We define the following invariants:

[TABLE]

where the latter is called the norm of $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$. Also for a normal test configuration $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ for a $\mathbb{Q}$-Fano variety $(X,-K_{X})$ we define the Donaldson–Futaki invariant $\mathrm{DF}(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ by

[TABLE]

We also define the Ding invariant of $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ as follows. Let $D_{(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})}$ be the unique $\mathbb{Q}$-divisor defined by the conditions $\mathrm{Supp}(D_{(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})})\subset\operatorname{\mathcal{X}}_{0}$ and $D_{(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})}\sim_{\mathbb{Q}}-K_{\operatorname{\mathcal{X}}/\mathbb{P}^{1}}-\bar{\operatorname{\mathcal{L}}}$. The Ding invariant $\mathrm{Ding}(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ is defined by

[TABLE]

Remark 3.6**.**

The crucial property of the norm $J^{NA}$ is the following: this function is nonnegative, and it vanishes on $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ if and only if the normalization of $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ is a trivial test configuration (for a proof see e. g. [Der16, Theorem 1.3]).

Remark 3.7**.**

Examples of product-type test configurations which are not trivial are given by one-parameter subgroups $\mathbb{G}_{m}\subset\operatorname{\mathrm{Aut}}(X)$ acting diagonally on $\operatorname{\mathcal{X}}=X\times\mathbb{A}^{1}$.

Definition 3.8**.**

A $\mathbb{Q}$-Fano variety $(X,-K_{X})$ is called

• •

$K$-semistable if $\mathrm{DF}(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})\geqslant 0$ for every normal test configuration $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ for $(X,-K_{X})$;

• •

$K$-polystable if $\mathrm{DF}(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})\geqslant 0$ for every normal test configuration $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ and $\mathrm{DF}(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})=0$ only if $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ is of product type;

• •

$K$-stable if $\mathrm{DF}(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})\geqslant 0$ for every normal test configuration $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ and $\mathrm{DF}(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})=0$ only if $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ is trivial;

• •

uniformly $K$-stable if there exists $\varepsilon>0$ such that $\mathrm{DF}(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})\geqslant\varepsilon J^{NA}(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ for every normal test configuration $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ for $(X,-K_{X})$.

Analogous notions for Ding-stability are defined in the same way with $\mathrm{DF}(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ replaced by $\mathrm{Ding}(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$. For a closed subgroup $G\subset\operatorname{\mathrm{Aut}}(X)$ we say that $(X,-K_{X})$ is $G$-equivariantly $K$-stable (or $K$-stable along $G$-equivariant test configurations) if the corresponding inequalities hold for $G$-equivariant test configurations; the same for uniform $K$- or Ding-stability.

Remark 3.9**.**

We have the obvious implications:

[TABLE]

and the same ones for $G$-equivariant stability. Note also that $K$-stability implies that $\operatorname{\mathrm{Aut}}(X)$ is discrete, whereas $K$-polystability allows $\operatorname{\mathrm{Aut}}(X)$ to be infinite. As a consequence, uniform $K$-stability is not equivalent to existence of Kähler–Einstein metrics. In fact, $G$-equivariant uniform $K$-stability is also stronger than being Kähler–Einstein in general.

3.2. The main result

Let us now state and prove the main result of this work (Theorem 1.3 from the introduction).

Theorem 3.10**.**

Let $(X,L)=(X,-K_{X})$ be a $\operatorname{\mathbb{Q}}$-Fano variety with the anticanonical polarization and let $G\subset\operatorname{\mathrm{Aut}}(X)$ be a connected subgroup. Then the following conditions are equivalent:

1. (1)

The variety $X$ is uniformly $K$-stable (resp. $K$-semistable) along $G$-equivariant test configurations; 2. (2)

The variety $X$ is uniformly Ding-stable (resp. Ding-semistable) along $G$-equivariant test configurations; 3. (3)

We have the inequality $\delta_{G}(X)>1$ (resp. $\delta_{G}(X)\geqslant 1$).

The proof of Theorem 3.10 mainly follows [Fuj19a, Theorem 1.4] and [LX14]. We divide the proof into Propositions 3.11 and 3.13 below. Our goal is to check that the constructions of test configurations used in [Fuj19a, Theorem 4.1] and [LX14, Theorem 4] can be made $G$-equivariantly, for $G\subset\operatorname{\mathrm{Aut}}(X)$ a connected subgroup. The next proposition uses the “blowing up” techniques to check $K$-stability (see [Od13, OS12, Der16]) in $G$-equivariant setting.

Proposition 3.11**.**

Let $(X,L)=(X,-K_{X})$ be a $\operatorname{\mathbb{Q}}$-Fano variety and $G\subset\operatorname{\mathrm{Aut}}(X)$ a closed connected subgroup; assume that for all $G$-equivariant normal test configurations $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ for $(X,-K_{X})$ we have the inequality

[TABLE]

for some $\varepsilon>0$. Then we have for every $G$-invariant divisorial valuation on $\mathbb{C}(X)$ the inequality

[TABLE]

In particular, we have $\delta_{G}(X)>1$.

Proof.

Let us first outline the idea of the proof of [Fuj19a, Theorem 4.1]. Starting from a divisorial valuation $v=\operatorname{\mathrm{ord}}_{E}$, we construct a sequence of test configurations $(\operatorname{\mathcal{X}}^{r},\operatorname{\mathcal{L}}^{r})$ using flag ideals (introduced in [Od13]) associated to the valuation $v$. Then from the computations in [Fuj19a, Claim 4.4] and [Fuj19b, Claim 2.5] we conclude that uniform boundedness from below of the Ding invariants $\mathrm{Ding}(\operatorname{\mathcal{X}}^{r},\operatorname{\mathcal{L}}^{r})$ implies the required bound on $A_{X}(v)/S(v)$. We show that this construction can be made $G$-equivariantly provided that we start from a $G$-invariant divisorial valuation. We assume that $\operatorname{\mathrm{Val}}^{G}_{X}$ is nonempty; otherwise there is nothing to prove.

Let $r_{0}\in\mathbb{N}$ be the Cartier index of $K_{X}$. To a divisorial valuation $v=\operatorname{\mathrm{ord}}_{E}$ where $E$ is a $G$-stable prime divisor on a birational model $\varphi\colon Y\to X$ we associate the order filtration $\mathcal{F}_{v}$ on the graded algebra

[TABLE]

The filtration is defined by

[TABLE]

It is a filtration by $G$-invariant linear subspaces of $R(X,-r_{0}K_{X})$. To this filtration and to any given $t\in\mathbb{R},m\in\mathbb{N}$ we can associate a nontrivial $G$-invariant ideal

[TABLE]

By [Fuj19a, Claim 4.2], we have

[TABLE]

We now define for appropriate $e_{+},e_{-}\in\mathbb{Z}$ and $r\in\mathbb{N}$ large enough (as in [Fuj19a, Theorem 4.1]) the flag ideal $\operatorname{\mathscr{I}}_{r}\subset\operatorname{\mathscr{O}}_{X\times\mathbb{A}^{1}}$ by

[TABLE]

This ideal is invariant under the action of $G$ on $X\times\mathbb{A}^{1}$. We construct a test configuration by blowing up the flag ideal $\Phi_{r}\colon\mathrm{Bl}_{\operatorname{\mathscr{I}}_{r}}(X\times\mathbb{A}^{1})\to X\times\mathbb{A}^{1}$. Let $E_{r}$ be the exceptional divisor of the blow-up; we set

[TABLE]

Then the $G$-action lifts to the blow-up leaving $\operatorname{\mathcal{L}}^{r}$ invariant; therefore $(\operatorname{\mathcal{X}}^{r},\operatorname{\mathcal{L}}^{r})$ is a $G$-equivariant test configuration for $(X,L)$. By assumption, for the (still $G$-equivariant) normalization $(\operatorname{\mathcal{X}}^{r,\nu},\nu^{*}\operatorname{\mathcal{L}}^{r})$ of any test configuration $(\operatorname{\mathcal{X}}^{r},\operatorname{\mathcal{L}}^{r}),r\in\mathbb{N}$ we have the inequality

[TABLE]

for some fixed $\varepsilon>0$. By the proof of [Fuj19a, Theorem 4.1] and [Fuj19b, Claims 2.4, 2.5] we can express $\mathrm{Ding}(\operatorname{\mathcal{X}}^{r,\nu},\nu^{*}\operatorname{\mathcal{L}}^{r})$ and $J^{NA}(\operatorname{\mathcal{X}}^{r,\nu},\nu^{*}\operatorname{\mathcal{L}}^{r})$ via the invariants of the valuation $v=\operatorname{\mathrm{ord}}_{E}$ and obtain an inequality

[TABLE]

Since $v=\operatorname{\mathrm{ord}}_{E}\in\operatorname{\mathrm{DivVal}}^{G}_{X}$ was arbitrary, we obtain

[TABLE]

as it was to be shown. ∎

We recall here some important results of Li–Xu and Fujita. These results show that uniform $K$- or Ding-stability can be checked using only special test configurations. Moreover, the Donaldson-Futaki invariants of these test configurations can be expressed via invariants of valuations.

Proposition 3.12** ([LX14, Theorem 4] and [Fuj19a, Theorem 5.2]).**

Let $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ be a normal test configuration for a $\mathbb{Q}$-Fano variety $(X,-K_{X})$. Then there exist

• •

a finite base change $(\operatorname{\mathcal{X}}^{(d)},\operatorname{\mathcal{L}}^{(d)})\to(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$;

• •

a $\mathbb{G}_{m}$-equivariant birational map $\operatorname{\mathcal{X}}^{(d)}\to\operatorname{\mathcal{X}}^{s}$ obtained by running a relative MMP with scaling of an ample divisor $\mathcal{H}$

such that the resulting test configuration $(\operatorname{\mathcal{X}}^{s},\operatorname{\mathcal{L}}^{s})$ is special, that is, $\operatorname{\mathcal{X}}^{s}_{0}$ is irreducible and reduced. Moreover, for every special test configuration $(\operatorname{\mathcal{X}}^{s},\operatorname{\mathcal{L}}^{s})$ obtained from $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ the following properties hold true:

• (a)

There exists $d\in\mathbb{N}$ such that $\mathrm{DF}(\operatorname{\mathcal{X}}^{s},\operatorname{\mathcal{L}}^{s})\leqslant d\cdot\mathrm{DF}(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ and such that for any $\varepsilon\in[0,1]$ we have

[TABLE]

• (b)

The Donaldson–Futaki invariant $\mathrm{DF}(\operatorname{\mathcal{X}}^{s},\operatorname{\mathcal{L}}^{s})$ is equal to the Ding invariant of $(\operatorname{\mathcal{X}}^{s},\operatorname{\mathcal{L}}^{s})$ and can be expressed via the invariants of the divisorial valuation $v=v_{\operatorname{\mathcal{X}}^{s}_{0}}$ as follows:

[TABLE]

The next proposition says that one can pass from any $G$-equivariant test configuration to a $G$-equivariant special test configuration.

Note that for any test configuration $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ the $\mathbb{G}_{m}$-equivariant birational map $\operatorname{\mathcal{X}}\dasharrow X\times\mathbb{A}^{1}$ gives an isomorphism $\mathbb{C}(\operatorname{\mathcal{X}})=\mathbb{C}(X)(t)$. This isomorphism is $G$-equivariant if the test configuration $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ is. The projection $X\times\mathbb{A}^{1}\to X$ gives a $G$-equivariant embedding $\mathbb{C}(X)\subset\mathbb{C}(X)(t)$. If the central fiber $\operatorname{\mathcal{X}}_{0}$ is irreducible then the restriction of the divisorial valuation $v_{\operatorname{\mathcal{X}}_{0}}$ to $\mathbb{C}(X)$ is either divisorial or trivial (see e.g. [BHJ17, Lemma 4.1] or [Tim11, Proposition B.8]).

Proposition 3.13**.**

Let $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ be a normal test configuration for a $\mathbb{Q}$-Fano variety $(X,-K_{X})$. If $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ is $G$-equivariant for a closed connected subgroup $G\subset\operatorname{\mathrm{Aut}}(X)$ then every special test configuration $(\operatorname{\mathcal{X}}^{s},\operatorname{\mathcal{L}}^{s})$ constructed from $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ as in Proposition 3.12 is $G$-equivariant and satisfies the properties (a) and (b) from Proposition 3.12. In particular, if $(\operatorname{\mathcal{X}}^{s},\operatorname{\mathcal{L}}^{s})$ is non-trivial then the restriction of the divisorial valuation $v_{\operatorname{\mathcal{X}}^{s}_{0}}$ to the subfield

[TABLE]

is a $G$-invariant divisorial valuation.

Proof.

We start from a normal $G$-equivariant test configuration $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ and compactify it to $(\bar{\operatorname{\mathcal{X}}},\bar{\operatorname{\mathcal{L}}})$. Taking an equivariant log resolution of the pair $(\bar{\operatorname{\mathcal{X}}},\bar{\operatorname{\mathcal{X}}}_{0})$ (by [Kol07, Section 3.9.1]) and a $G$-equivariant finite base change $t\mapsto t^{d}$ (by [LX14, Lemma 5]) we can assume that the pair $(\bar{\operatorname{\mathcal{X}}},\bar{\operatorname{\mathcal{X}}}_{0})$ is log canonical.

Then by Proposition 3.12 a special test configuration can be obtained from $(\bar{\operatorname{\mathcal{X}}},\bar{\operatorname{\mathcal{X}}}_{0})$ by running a relative $K_{\bar{\operatorname{\mathcal{X}}}/\operatorname{\mathbb{P}}^{1}}$-MMP with scaling of an ample $\mathbb{Q}$-divisor $\mathcal{H}$. We can take $\mathcal{H}$ such that the class of $\mathcal{H}$ lies in   $\operatorname{\mathrm{Pic}}^{G}(\bar{\operatorname{\mathcal{X}}})$. We recall that $\overline{\mathrm{NE}}(\bar{\operatorname{\mathcal{X}}})=\overline{\mathrm{NE}}(\bar{\operatorname{\mathcal{X}}})^{G}$ by [And01, Lemma 1.5] since $G$ is connected. Therefore for every divisorial extremal ray of $\overline{\mathrm{NE}(X)}$ of the Mori cone the contraction is $G$-equivariant. For flips, the statement follows since the $\mathrm{Proj}$ of an algebra with a $G$-action has an induced structure of a $G$-variety.

Thus at the final step of the MMP we obtain a $G$-equivariant test configuration $(\bar{\operatorname{\mathcal{X}}}^{s},\bar{\operatorname{\mathcal{L}}}^{s})$ such that $\operatorname{\mathcal{X}}^{s}_{0}$ is a $G$-stable prime divisor. So Proposition 3.12 applies to $(\operatorname{\mathcal{X}}^{s},\operatorname{\mathcal{L}}^{s})$ and ensures that the properties (a) and (b) are satisfied. Moreover, the valuation $v_{\operatorname{\mathcal{X}}^{s}_{0}}$ on $\mathbb{C}(\operatorname{\mathcal{X}}^{s})$ is $G$-invariant. If $(\operatorname{\mathcal{X}}^{s},\operatorname{\mathcal{L}}^{s})$ is nontrivial then the restriction of $v_{\operatorname{\mathcal{X}}^{s}_{0}}$ to $\mathbb{C}(X)$ is divisorial by [Tim11, Proposition B.8] or [BHJ17, Lemma 4.1]and invariant by the induced $G$-action on $X$. ∎

Proof of Theorem 3.10.

The implication $(2)\Rightarrow(3)$ is precisely Proposition 3.11. To show $(3)\Rightarrow(1)$ we use Propositions 3.12 and 3.13 in order to pass from a $G$-equivariant test configuration $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ to a special test configuration $(\operatorname{\mathcal{X}}^{s},\operatorname{\mathcal{L}}^{s})$ such that the valuation $v_{\operatorname{\mathcal{X}}^{s}_{0}}$ is $G$-invariant. We have the inequality

[TABLE]

for some $\varepsilon\in(0;1)$ by assumption and thus

[TABLE]

Therefore $\mathrm{DF}(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})\geqslant\varepsilon J^{NA}(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ by Proposition 3.12. Finally, since the Donaldson–Futaki and Ding invariants of special test configurations coincide, we get the implication $(1)\Rightarrow(2)$ The same implications for semistability follow by the same argument with $\varepsilon=0$ in Propositions 3.11 and 3.12. ∎

Remark 3.14**.**

We stated and proved the above theorem for a $\mathbb{Q}$-Fano variety $X$ with the anticanonical polarization. However, it can easily be seen that Theorem 3.10 is still true if we rescale the polarization by a rational number $t$. Indeed, the Donaldson–Futaki invariant does not change under rescaling the polarization (see [BHJ17, Definition 3.6]). The norm function is rescaled by a multiple of $t$ by [Der16, Definition 2.5]. Therefore, the same argument as in the proof of Theorem 3.10 applies.

Combining Theorem 3.10 and Proposition 2.25 we can strengthen the results of Odaka and Sano [OS12, Theorem 1.10].

Corollary 3.15**.**

Let $X$ be a Fano variety of dimension $n$ and let $G\subset\operatorname{\mathrm{Aut}}(X)$ be a closed connected subgroup. If $\alpha_{G}(X)>\frac{n}{n+1}$ (resp. $\alpha_{G}(X)\geqslant\frac{n}{n+1}$) then $(X,-K_{X})$ is $G$-equivariantly uniformly $K$-stable (resp. $G$-equivariantly $K$-semistable).

Example 3.16**.**

Let $X$ be a Mukai–Umemura threefold or a $V_{5}$ threefold. Then $\operatorname{\mathrm{Aut}}^{0}(X)\simeq\mathrm{PGL}_{2}(\mathbb{C})$. Taking $G=\operatorname{\mathrm{Aut}}^{0}(X)$ we have $\alpha_{G}(X)=\frac{5}{6}$ by [Don08] and [CS09]. Therefore these Fano varieties are $G$-equivariantly uniformly $K$-stable by Corollary 3.15.

Remark 3.17**.**

Suppose that the space $\operatorname{\mathrm{Val}}^{G}_{X}$ is empty (e.g. $X$ is a $G$-homogeneous variety). Then for every $G$-equivariant special test configuration $(\operatorname{\mathcal{X}}^{s},\operatorname{\mathcal{L}}^{s})$ the corresponding valuation on $X$ is trivial. Therefore we have $J^{NA}(\operatorname{\mathcal{X}}^{s},\operatorname{\mathcal{L}}^{s})=0$ and therefore the test configuration $(\operatorname{\mathcal{X}}^{s},\operatorname{\mathcal{L}}^{s})$ is trivial by [Der16, Theorem 1.3]. Thus the $G$-equivariant $K$-stability condition is trivially satisfied. Note that the existence of Kähler–Einstein metrics on compact Kähler homogeneous manifolds was established by Matsushima in [Mat72, Theorem 3].

Proposition 3.18**.**

Let $X$ be a smooth Fano variety and suppose that $(X,-K_{X})$ is $G$-equivariantly uniformly $K$-stable for a connected reductive subgroup $G\subset\operatorname{\mathrm{Aut}}(X)$. Then the centralizer $C_{\operatorname{\mathrm{Aut}}(X)}(G)$ of $G$ inside $\operatorname{\mathrm{Aut}}(X)$ is finite; in particular, the groups $G$ and $\operatorname{\mathrm{Aut}}(X)$ are semisimple.

Proof.

By [DS16, Theorem 1] and Remark 3.9 the variety $X$ admits a Kähler-Einstein metric, therefore the groups $\operatorname{\mathrm{Aut}}(X)$ and $C_{\operatorname{\mathrm{Aut}}(X)}(G)$ are reductive by Matsushima’s theorem. The condition of $G$-equivariant uniform $K$-stability implies that every $G$-equivariant product-type test configuration is $G$-equivariantly trivial. This implies that there are no one-parameter subgroups $\mathbb{G}_{m}\subset C_{\operatorname{\mathrm{Aut}}(X)}(G)$. Since $C_{\operatorname{\mathrm{Aut}}(X)}(G)$ is a reductive algebraic group, its connected component of the identity is trivial, so $C_{\operatorname{\mathrm{Aut}}(X)}(G)$, and therefore $C(G)$, is finite. Finally, since any $\mathbb{G}_{m}\subset C(\operatorname{\mathrm{Aut}}(X))$ lies in $C_{\operatorname{\mathrm{Aut}}(X)}(G)$ for any $G\subset\operatorname{\mathrm{Aut}}(X)$ we obtain that the center of $\operatorname{\mathrm{Aut}}(X)$ is discrete. Using the fact that any reductive algebraic group with discrete center is semisimple (see e. g. [Mil17, Proposition 19.3]) we obtain the statement of the theorem. ∎

Remark 3.19**.**

The above proposition shows that a Kähler–Einstein Fano manifold $X$ may not be uniformly $G$-equivariantly $K$-stable for any connected subgroup $G\subset\operatorname{\mathrm{Aut}}(X)$. For example, a smooth del Pezzo surface $X$ of degree 6 is isomorphic to a blow-up of $\mathbb{P}^{2}$ at three points and $\operatorname{\mathrm{Aut}}^{0}(X)\simeq(\mathbb{C}^{\times})^{2}$. Since the latter group is reductive, by [Ti90] the surface $X$ admits a Kähler–Einstein metric.

3.3. The greatest Ricci lower bound

Let now $X$ be a smooth Fano variety and let $G\subset\operatorname{\mathrm{Aut}}(X)$ be a connected reductive subgroup. We denote by $K_{G}$ the compact real form of $G$ (see e.g. [OV90, Theorem 5.2.8]); then $K_{G}$ is a compact Lie group acting on $X$ by automorphisms. Our Theorem 3.10 together with [DS16, Theorem 1] shows that the inequality $\delta_{G}(X)>1$ guarantees the existence of a $K_{G}$-invariant Kähler–Einstein metric on $X$. In this subsection we show that our invariant $\delta_{G}(X,-K_{X})$ is related to metric geometry of $(X,-K_{X})$ even in the case when $\delta_{G}(X)\leqslant 1$. Namely, we consider the following invariant:

[TABLE]

This is a $K_{G}$-invariant version of the greatest Ricci lower bound $\beta(X,-K_{X})$ defined by Rubinstein in [Rub08, Equation (32)]. For the case $G=\{1\}$ we have the following result.

Theorem 3.20** (see [CRZ19, Theorem 5.7] and [BBJ18, Theorem 7.5 and Corollary 7.6]).**

Let $X$ be a smooth Fano variety. Then its delta-invariant $\delta(X,-K_{X})$ and the greatest Ricci lower bound $\beta(X,-K_{X})$ are related by the formula

[TABLE]

We prove an analogous result in $G$-equivariant setup and establish relations between the above invariants (with and without $G$-action).

Proposition 3.21**.**

Let $X$ be a smooth Fano variety and let $G\subset\operatorname{\mathrm{Aut}}(X)$ be a connected reductive subgroup. Then there is an equality

[TABLE]

We also have the following relation between $\delta$-invariants

[TABLE]

Proof.

Let us first prove equality (1). Note that by $K_{G}$-invariant continuity method (see e.g. [DS16]) we have an equality of the greatest Ricci lower bounds

[TABLE]

Therefore, we combine this with Theorem 3.20 and an obvious inequality $\delta(X,-K_{X})\leqslant\delta_{G}(X,-K_{X})$ to obtain

[TABLE]

Conversely, let us choose $t\in(0;1]\cap\mathbb{Q}$ such that $0<t<\delta_{G}(X,-K_{X})$. Then by [BlJ20, Remark  4.5], we have $\delta_{G}(X,-tK_{X})>1$. Therefore by Remark 3.14 the polarized variety $(X,-tK_{X})$ is uniformly $G$-equivariantly $K$-stable. Then by [DS16, Proposition 10] the equation

[TABLE]

has a $K_{G}$-invariant solution for some $K_{G}$-invariant Kähler form $\alpha\in(1-t)c_{1}(X)$. This shows that

[TABLE]

and proves equality (1).

The equality (2) follows from the previous one and Theorem 3.20 in the case when $\delta(X,-K_{X})\leqslant 1$; otherwise, by [BBEGZ19, Theorem 5.4], we have $G=\{1\}$ and equality (2) then holds tautologically. ∎

Remark 3.22**.**

We expect that an appropriate version of the Proposition 3.21 holds for klt Fano varieties and singular Kähler metrics. Also, it would be interesting to see a purely algebro-geometric proof of the equality $\delta(X)=\min\{1,\delta_{G}(X)\}$.

4. Varieties with torus action

In this section we consider the case of a variety $X$ with an action of a torus $T=(\mathbb{G}_{m})^{k}\subset\operatorname{\mathrm{Aut}}(X)$.

4.1. Varieties with an action of a torus

Toric varieties form a well-understood class of varieties with large groups of symmetries. The formulas for the $\alpha$-invariant of a toric variety in terms of the associated fan were given in [BS99], in [S05] using analytic methods, and in [CS08, Lemma 5.1]. The analogous formula for the $\delta$-invariant first appeared in [BlJ20, Corollary 7.16]. The underlying idea in these formulas is that it suffices to consider only torus-invariant divisors (or valuations) in the computation of $\alpha(X)$ and $\delta(X)$. We show that the same is true for a subtorus $T\subset\operatorname{\mathrm{Aut}}(X)$ of any dimension.

Proposition 4.1**.**

Let $(X,L)$ be a variety with an ample polarization and let $T\subset\operatorname{\mathrm{Aut}}(X,L)$ be a subtorus. Then the following equalities hold:

[TABLE]

Proof.

We apply the degeneration to the initial filtration argument from [BlJ20]. Let us give an outline of their argument. Attached to a valuation $v\in\operatorname{\mathrm{Val}}_{X}$ is the filtration $\mathcal{F}_{v}$ on $R(X,L)$. Using a construction from [KK12] we can associate to $\mathcal{F}_{v}$ the so-called initial filtration $\mathrm{in}(\mathcal{F}_{v})$. Its crucial property is that the sequence of base ideals

[TABLE]

consists of monomial ideals and therefore is invariant under the action of $T$. Moreover, by [BlJ20, Proposition 7.13] it is possible to degenerate $\mathcal{F}_{v}$ to $\mathrm{in}(\mathcal{F}_{v})$ in a one-parameter family in such a way that the log canonical threshold of the sequence defined by

[TABLE]

does not increase after passing to $\mathrm{in}(\mathcal{F}_{v})$, that is

[TABLE]

By [JM12, Proposition 8.1] we can associate to $\mathrm{in}(\mathcal{F}_{v})$ a $T$-invariant valuation $\bar{v}$ computing the log canonical threshold $\operatorname{\mathrm{lct}}(\mathfrak{b}_{\bullet}(\mathrm{in}(\mathcal{F}_{v})))$. Such valuation will be a monomial valuation on a certain $T$-model of $X$. The valuation $\bar{v}$ has the property that

[TABLE]

by [BlJ20, Lemma 1.1]. Moreover, we have $S_{L}(\bar{v})\geqslant S_{L}(v)$ by the argument in [BlJ20, Proposition 6.8]. Therefore, the infimum in the definition of $\delta(X,L)$ is at least the infimum over the subspace $\operatorname{\mathrm{Val}}^{T}_{X}$. The statement now follows from the definition of $\delta_{T}(X,L)$. The proof for $\alpha_{T}(X,L)$ is the same. ∎

Remark 4.2**.**

In the case of $\alpha(X)$ this result was proved in [OS12, Corollary 1.8] for any connected solvable group using the Borel fixed point theorem. We expect Proposition 4.1 to hold in the case of a connected solvable group $G$.

By combining the criterion for $T$-equivariant semistability from Theorem 3.10 and Proposition 4.1 we immediately obtain the following corollary.

Corollary 4.3**.**

A Fano variety $(X,-K_{X})$ is $K$-semistable if and only if it is $K$-semistable with respect to all $T$-equivariant test configurations.

This result shows that for varieties with an action of a torus $T=(\mathbb{G}_{m})^{k}$ it is necessary to consider additional symmetries in order to use Tian’s criterion (or Theorem 3.10). This method was first implemented in [BS99] for symmetric toric Fano manifolds. In [Su13] the method was applied to $T$-varieties of complexity one, and in [CS18] for Fano threefolds from the $V_{22}$ family having automorphism groups $\mathbb{G}_{m}\rtimes\mathbb{Z}/2\mathbb{Z}$ (cf. [DKK17] where the additional symmetries were not used).

5. Spherical Fano varieties

A natural generalization of toric varieties is the class of spherical varieties. A variety $X$ is spherical if it has an action of a connected reductive group $G$ such that a Borel subgroup $B\subset G$ acts on $X$ with an open orbit. For general information on spherical varieties we refer to [BLV86], see also [Kn91, Tim11]. Log canonical thresholds of spherical varieties were investigated in [Pas16, Smi17, Del15]. An extensive study of $K$-stability of spherical Fano varieites was undertaken by Delcroix in [Del16].

In case of a spherical $G$-variety the space of $G$-invariant valuations has a particularly nice description (see Theorem 5.1 below). For a Fano variety $X$, spherical under the action of a connected reductive group $G$ we give a formula for $\delta_{G}$ in terms of the combinatorial data defined by $X$. Moreover, we recover the combinatorial criterion for $G$-equivariant $K$-polystability of a spherical Fano variety given by Delcroix [Del16, Theorem A]. Our proof is different, though not entirely independent of the one in [Del16].

We fix the notation, mostly following [Del16]. Let $X$ be a projective variety, spherical under the action of a connected reductive group $G$. Let $B\subset G$ be a Borel subgroup and let $T\subset B$ be a maximal torus. We denote by $\mathfrak{X}(T)$ the group of algebraic characters of $T$. Let also $\Phi$ be the root system of $(G,B,T)$ and $\Phi^{+}$ be the set of positive roots determined by $B$. We also let $\mathfrak{N}(T)$ be the group of $1$-parameter subgroups of $T$.

We describe the set of $G$-invariant valuations on $X$. This set depends on the open $G$-orbit $U$ only. Let us fix a Borel subgroup $B\subset G$. Associated to $B$ is the space of $B$-eigenfunctions

[TABLE]

We denote by $N_{B}(U)$ the dual of $M_{B}(U)$; it is a free abelian group of finite rank. To every $v\in\operatorname{\mathrm{Val}}^{G}_{X}$ we can associate a vector $\rho_{v}\in N_{B}(U)\otimes\mathbb{R}$ by the rule $\rho_{v}(\chi)=v(f)$ where $f\in\mathbb{C}(X)^{*}$ is a rational function as in definition of $M_{B}(U)$. This is a well defined map exactly because $B$ has an open orbit. The space $N_{B}\otimes\mathbb{R}$ is a quotient of $\mathfrak{N}(T)\otimes\mathbb{R}$; we denote by $\pi\colon\mathfrak{N}(T)\otimes\mathbb{R}\to N_{B}\otimes\mathbb{R}$ the quotient map. The next result gives a very simple description of the set $\operatorname{\mathrm{Val}}^{G}_{X}$.

Theorem 5.1** (see e.g. [Kn91, Corollaries 1.8 and 5.3]).**

The map $\rho\colon\operatorname{\mathrm{Val}}^{G}_{X}\to N_{B}(U)\otimes\mathbb{R}$ is injective and identifies the set $\operatorname{\mathrm{Val}}^{G}_{X}$ with a polyhedral convex cone $\mathcal{V}$ in the finite-dimensional space $N_{B}(U)\otimes\mathbb{R}$.

In [Pas17] Pasquier described the restriction of the log discrepancy function $A_{X}$ to the space $\operatorname{\mathrm{Val}}^{G}_{X}$ using the identification of the latter space with the cone $\mathcal{V}$ given by the above Theorem 5.1. This description uses the so-called complete colored fan $\mathbb{F}_{X}$ associated to $X$, which is a finite set of finitely generated cones $\mathcal{C}\subset N_{B}\otimes\mathbb{R}$ with some additional data (see e.g. [Pas17, Definition 2.13] for details).

Proposition 5.2** (see [Pas17, Theorem 2.20 and Proposition 5.2]).**

Let $X$ be a $\mathbb{Q}$-Gorenstein spherical variety and let $\mathbb{F}_{X}$ be the complete colored fan associated to $X$. For every cone $\mathcal{C}\subset\mathbb{F}_{X}$ there exists a function $h_{\mathcal{C}}\colon N_{B}\otimes\mathbb{R}\to\mathbb{R}$ such that its restriction to the cone $\mathcal{C}\in\mathbb{F}_{X}$ is linear and for every $v\in\mathcal{V}\cap\mathcal{C}$ we have $A_{X}(v)=h_{\mathcal{C}}(v)$. If $v=\mathrm{ord}_{D}$ corresponds to a primitive generator of an edge of $\mathcal{C}$ then $h_{\mathcal{C}}(v)=a_{D}$ with $a_{D}=1$ or $a_{D}\geqslant 2$.

The coefficients $a_{D}$ can be expressed via the spherical roots and coroots of the triple $(G,B,T)$ (see [Pas17, Theorem 2.20] for details). Note also that by [Pas17, Proposition 5.6] a $\mathbb{Q}$-Gorenstein spherical variety $X$ always has klt singularities.

In [Del16] Delcroix investigated $G$-equivariant $K$-stability of spherical Fano varieties. He gave a construction of $G$-equivariant special test configurations $(\operatorname{\mathcal{X}},\operatorname{\mathcal{L}})$ corresponding to vectors in the valuation cone $\mathcal{V}$. Moreover, he computed the Donaldson–Futaki invariants of these test configurations in terms of the moment polytope $\Delta^{+}\subset\mathfrak{X}(T)$ (see [Del16, Definition 3.14]) and the so-called Duistermaat–Heckman measure $\mathrm{DH}$ on $\Delta^{+}$ (see [Del16, Theorem 4.5]). The moment polytope determines a subsystem $\Phi_{L}\subset\Phi$; we denote by $2\rho_{Q}$ the sum of elements in $\Phi^{+}\setminus\Phi_{L}$.

Theorem 5.3** ([Del16, Theorems B and C]).**

Let $(X,-K_{X})$ be a spherical Fano variety. To every rational vector $v\in\mathcal{V}$ corresponds a $G$-equivariant test configuration $(\operatorname{\mathcal{X}}^{v},\operatorname{\mathcal{L}}^{v})$ for $(X,-K_{X})$ with irreducible central fiber $\operatorname{\mathcal{X}}^{v}_{0}$. Moreover, there exists $m\in\mathbb{N}$ such that the test configuration constructed from $mv$ is special and, conversely, every $G$-equivariant special test configuration for $(X,-K_{X})$ can be constructed in this way. The Donaldson–Futaki invariant of a special test configuration corresponding to $v$ is given by the formula

[TABLE]

Here $\mathrm{bar}_{DH}(\Delta^{+})$ is the barycenter of the moment polytope with respect to the Duistermaat–Heckman measure $\mathrm{DH}$ and $V$ is a universal constant depending on $\Delta^{+}$ and $\mathrm{DH}$ only. The above expression does not depend on the choice of a preimage $\pi^{-1}(v)$ of $v$.

This computation allows to express the $\delta_{G}$-invariant of $X$ in terms of the combinatorial data associated to $X$ and $B\subset G$ and prove Proposition 1.4. Note that the formula below does not depend on the choices of a Borel subgroup $B$ and a maximal torus $T\subset B$.

Proposition 5.4**.**

Let $X$ be a Fano variety which is spherical under the action of $G$; let $B\subset G$ be a Borel subgroup and $T\subset B$ a maximal torus. In the above notation, the following formula holds for the $\delta_{G}$-invariant of $X$:

[TABLE]

Here the minimum is taken over a finite set $\mathrm{ord}_{D_{1}},\ldots,\mathrm{ord}_{D_{N}}$ of divisorial valuations corresponding to primitive generators of edges in $\mathcal{V}\cap\mathbb{F}_{X}$.

Proof.

By Theorem 5.1 we can identify $\operatorname{\mathrm{Val}}^{G}_{X}$ with the cone $\mathcal{V}$. Since the functions $A_{X}$ and $S_{-K_{X}}$ are homogeneous of order $1$ by Propositions 2.8 and 2.11, their ratio depends only on the line generated by $v\in\mathcal{V}$. Thus for any $v\in\mathcal{V}$ we can consider the special test configuration $(\operatorname{\mathcal{X}}^{v},\operatorname{\mathcal{L}}^{v})$ from Theorem 5.3. Using Proposition 3.12 we find

[TABLE]

By Proposition 5.2 the log discrepancy function identifies with the piecewise linear function on the cones $\mathcal{C}\in\mathbb{F}_{X}$. Therefore the $\delta_{G}$-invariant is the minimum over the cones in $\mathcal{V}\cap\mathbb{F}_{X}$ of two piecewise linear functions. Thus the minimum is attained on one of the primitive generators of edges in $\mathcal{C}\cap\mathcal{V},\mathcal{C}\in\mathbb{F}_{X}$ corresponding to one of the valuations $\mathrm{ord}_{D_{1}},\ldots,\mathrm{ord}_{D_{N}}$. By Proposition 5.2 we have $A_{X}(\mathrm{ord}_{D_{i}})=a_{D_{i}}$ where $a_{D_{i}}=1$ or $a_{D_{i}}\geqslant 2$. The formula now follows. ∎

Remark 5.5**.**

It should be possible to prove the formula for $S_{-K_{X}}(v)$ for $v\in\mathcal{V}$ directly, not relying on the computation of [Del16].

Example 5.6**.**

In the case when $X$ is a toric variety and $G=(\mathbb{G}_{m})^{n}$ the polytope $\Delta^{+}$ is the usual polytope $\Delta$ associated to $X$. The cone of $G$-invariant valuations is generated by the valuations $v_{D_{i}}$ corresponding to the torus-invariant divisors $D_{1},\ldots,D_{d}$. The log discrepancy function is linear on the cones of the fan of $X$. Therefore, the infumum is attained on one of the valuations $v_{D_{i}}$ (see [BlJ20, Corollary  7.4]). Thus, by Proposition 4.1 we recover the formula from [BlJ20, Corollary 7.16]:

[TABLE]

Example 5.7**.**

More generally, let $X$ be a horospherical variety, that is, a $G$-equivariant compactification of a homogeneous space $G/H$ where $H$ contains the unipotent radical of a Borel subgroup of $G$. These varieties admit a characterization in terms of the valuation cone $\mathcal{V}$. Namely, by [Kn91, Corollary 6.2] a spherical variety $X$ is horospherical if and only if $\mathcal{V}=N_{B}(U)\otimes\mathbb{R}$. Assume that the valuation cone of $X$ is non-trivial. Then by Proposition 5.4 for a horospherical Fano variety $X$ we have $\delta_{G}(X)\leqslant 1$. Moreover, if $X$ is smooth then by [Del16, Corollary  5.7] the condition $2\rho_{Q}=\mathrm{bar}_{DH}(\Delta^{+})$ (equivalent to $\delta_{G}(X)=1$) is a criterion for existence of a Kähler–Einstein metric on $X$.

On the other hand, we have some obvious examples of horospherical Fano varieties with $\delta_{G}(X)>1$.

Example 5.8**.**

Let $X$ be a spherical homogeneous space under the action of $G$, for example a Grassmannian  $\mathrm{Gr}(k,n)$ with $G=\mathrm{GL}_{n}(\mathbb{C})$. Then, as in Remark 2.22, we obtain that the space $\operatorname{\mathrm{DivVal}}^{G}_{X}$ is empty and therefore $\delta_{G}(X)=\infty$.

To produce more examples of spherical Fano varieties $X$ satisfying $\delta_{G}(X)>1$ we need to study the group $\operatorname{\mathrm{Aut}}_{G}(X)$ of $G$-equivariant automorphisms of $X$. If the spherical variety $X$ is an equivariant compactification of the quotient $G/H$ then this group can be described by

[TABLE]

where $N_{G}(H)$ is the normalizer of $H$ inside $G$ (see e. g. [Del16, Section 3.1.3] and references therein). It turns out that the dimension of $\operatorname{\mathrm{Aut}}_{G}(X)$ can be recovered from the valuation cone $\mathcal{V}$, namely, from its linear part $\mathcal{V}\cap(-\mathcal{V})$.

Proposition 5.9** (see e. g. [Kn91, Theorem 6.1]).**

Let $X$ be a spherical variety and let $\operatorname{\mathrm{Aut}}_{G}(X)$ be the group of $G$-equivariant automorphisms of $X$. Then the connected component $\operatorname{\mathrm{Aut}}_{G}^{0}(X)$ is a central torus in $G$. Moreover, the dimension of $\operatorname{\mathrm{Aut}}_{G}(X)$ is equal to the dimension of the linear part $\mathcal{V}\cap(-\mathcal{V})$ of the valuation cone.

Using Theorem 3.10, we can deduce the statement of [Del16, Theorem A] (in the non-twisted case) and establish an analogous criterion for $G$-equivariant uniform $K$-stability.

Proposition 5.10** (see [Del16, Theorem A]).**

Let $X$ be a $G$-spherical Fano variety. Then $(X,-K_{X})$ is $G$-equivariantly $K$-polystable if and only if the vector

[TABLE]

lies in the relative interior of the dual cone to the closure of $\pi^{-1}(-\mathcal{V})$. Analogously, the variety $(X,-K_{X})$ is $G$-equivariantly uniformly $K$-stable if and only if the vector $2\rho_{Q}-\mathrm{bar}_{DH}(\Delta^{+})$ belongs to the relative interior of the dual cone to the closure of $\pi^{-1}(-\mathcal{V})$ and, in addition, the linear part of the valuation cone $\mathcal{V}$ is trivial.

Proof.

By Theorem 3.10 it follows that $(X,-K_{X})$ is $G$-equivariantly $K$-semistable iff $\delta_{G}(X)\geqslant 1$. By our formula for $\delta_{G}(X)$ from Proposition 5.4 we obtain that the condition $\delta_{G}(X)\geqslant 1$ is equivalent to

[TABLE]

for all $D_{i}\in\mathcal{V}$. This means that the vector $\mathrm{bar}_{DH}(\Delta^{+})-2\rho_{Q}$ belongs to the dual cone to $\pi^{-1}(-\mathcal{V})$. Moreover, a vector $\xi\in\mathfrak{N}(T)\otimes\mathbb{Q}$ corresponds (up to a multiple) to a product-type $G$-equivariant test configuration if and only if $\xi$ lies on one of the hyperplanes defining the dual cone to $\overline{\pi^{-1}(-\mathcal{V})}$. The latter happens precisely when $\xi$ projects to the linear part of $\mathcal{V}$ (see [Del16, the proof of Theorem 5.3]). This implies the first statement of the proposition.

If $X$ is $G$-equivariantly uniformly $K$-stable then again by Theorem 3.10 and Proposition 5.4 we have

[TABLE]

for any $v\in\mathcal{V}\setminus\{0\}$. Thus the vector $2\rho_{Q}-\mathrm{bar}_{DH}(\Delta^{+})$ belongs to the dual cone to the closure of $\pi^{-1}(-\mathcal{V})$. Moreover, by linearity of the $\mathrm{DF}$ functional, the linear part $\mathcal{V}\cap-\mathcal{V}$ has to be trivial. Conversely, if the condition

[TABLE]

is satisfied for any $v\in\mathcal{V}\setminus\{0\}$, then the linear part $\mathcal{V}\cap(-\mathcal{V})$ is equal to $\{0\}$. By Proposition 5.9 we obtain that $\operatorname{\mathrm{Aut}}_{G}(X)$ is finite; hence, any product-type $G$-equivariant test-configuration is trivial. Therefore by Theorem 3.10 we obtain that $(X,-K_{X})$ is $G$-equivariantly $K$-stable; since the valuation cone is polyhedral, $(X,-K_{X})$ is also uniformly $G$-equivariantly $K$-stable. ∎

Example 5.11**.**

Let $G/H$ be a symmetric spherical homogeneous space with $G$ semisimple. By the result of De Concini and Procesi [dCP83, Theorem 3.1] there exists a simple compactification $X$ of $G/H$, i. e. a $G$-equivariant compactification of $G/H$ such that $X$ contains exactly one closed $G$-orbit. In fact, it follows from Proposition 5.9 that existence of a simple compactification is equivalent to finiteness of $N_{G}(H)/H$. Therefore, if $X$ is a smooth symmetric spherical Fano variety then existence of a Kähler–Einstein metric on $X$ is equivalent to the condition $\delta_{G}(X)>1$. Symmetric Fano varieties obtained as blow-ups of wonderful compactifications were described by Ruzzi [Ruz12, Theorem B], see also [Del16, Section 5.4.2] and examples therein.

6. Finite automorphism groups

In this section we show that in case of a variety $X$ with a finite group action, we can adapt Definition  2.2 to $G$-equivariant setting. Also we compare the invariant $\delta_{G}$ defined below to the $\delta$-invariant of the quotient $Y=X/G$ with the orbifold pair structure. We expect that there is an analogue of our main Theorem 3.10 for the case of a finite group $G$.

Being motivated by the results of [Su13], we would like to consider the case of a complexity-one Fano $T$-variety $X$. Let $G$ be a subgroup of the normalizer $N_{\operatorname{\mathrm{Aut}}(X)}(T)$. Then $G$ acts on the Chow quotient $Y\sim\mathbb{P}^{1}$ of $X$ by the $T$-action. Let $B=\sum_{i}(1-\frac{1}{m_{i}})[p_{i}]$ be the $G$-invariant branch $\mathbb{Q}$-divisor on $Y$ encoding the multiple fibers of the quotient fibration. Then, inspired by [Su13, Theorem 1.2], we can obtain conditions on $T\rtimes G$-equivariant $K$-stability of $X$ in terms of $\delta_{G}(Y,B)$ (defined below).

Definition 6.1**.**

Let $(X,\Delta;L)$ be a pair with a big $\mathbb{Q}$-divisor $L$. Let $G$ be a finite subgroup of the automorphism group $\mathrm{Aut}(X,\Delta;L)$. Then we define the $\alpha_{G,m}$-invariant for the pair $(X,\Delta;L)$ by

[TABLE]

Analogously, let us define

[TABLE]

Here by a $G$-orbit of an $m$-basis type divisor we mean a $\mathbb{Q}$-divisor $D\in\operatorname{\mathbb{P}}(H^{0}(X,-mL))$ which belongs to the image of the open set of $m$-basis type divisors under a surjective averaging map

[TABLE]

We define

[TABLE]

for $m\in\mathbb{N}$ sufficiently large and divisible.

Remark 6.2**.**

In the above definition we consider $G$-invariant divisors with several irreducible components (permuted by the action of $G$). Note that by lower semi-continuity of log canonical thresholds [KP17, Lemma 7.8] the infima from the definition of $\delta_{G,m}(X,\Delta,L)$ are attained on open subsets of $\operatorname{\mathbb{P}}(H^{0}(X,-mL)^{G}$. Moreover, for a Fano variety $(X,-K_{X})$ and a finite group $G\subset\operatorname{\mathrm{Aut}}(X)$ we have an isomorphism $H^{0}(X,-m|G|K_{X})\sim H^{0}(Y,-m(K_{Y}+B))$ where $Y=X/G$ is a quotient variety and $B$ is the branch divisor. Hence the above infimum can be computed using preimages of $m$-basis type divisors from $Y$. In particular, this shows the equivalence of Definition 6.1 and [LZ20, Definition 2.5].

Example 6.3**.**

Consider the case $X=\mathbb{P}^{1}$ and $G=\mathbb{Z}/m\mathbb{Z}$. Then any action of $G$ on $\mathbb{P}^{1}$ has two fixed points, so the minimal length of a $G$-orbit on $X$ is  1. Thus for any $G$-invariant divisor $D$ we have

[TABLE]

Thus we get

[TABLE]

More generally, if $G$ is a finite group acting faithfully on $\mathbb{P}^{1}$ then by the same argument we have

[TABLE]

which is equal to the minimal length of an orbit under a given $G$-action on $\mathbb{P}^{1}$ (see [CPS18, Example 2]).

We now prove the ramification formula (Proposition 1.5 from the introduction).

Proposition 6.4**.**

Let $X$ be a normal and $\mathbb{Q}$-Gorenstein variety with at most log terminal singularities and $-K_{X}$ big. Consider a finite subgroup $G\subset\mathrm{Aut}(X)$. Let us denote by $Y=X/G$ the quotient variety, by $\pi\colon X\to Y$ the quotient map and by $B=\sum(1-\frac{1}{m_{i}})B_{i}$ the branch $\mathbb{Q}$-divisor on $Y$. Then the following ramification formulas hold for $\alpha_{G}(X)$ and for $\delta_{G}(X)$:

[TABLE]

Proof.

For a $\mathbb{Q}$-divisor $D_{Y}$ on $Y$ its pullback $\pi^{*}D_{Y}$ is a $G$-invariant $\operatorname{\mathbb{Q}}$-divisor on $X$. We also define the crepant preimage $D_{X}$ of $D_{Y}$ via $\pi$ by the formula

[TABLE]

By the canonical bundle formula for a finite morphism we have $-K_{X}\operatorname{\sim_{\mathbb{Q}}}-\pi^{*}(K_{Y}+B)$, in particular, the pair $(Y,B)$ is klt. Consider a $\operatorname{\mathbb{Q}}$-divisor $D_{Y}\operatorname{\sim_{\mathbb{Q}}}-(K_{Y}+B)$. Then its pullback $\pi^{*}D_{Y}$ is a $G$-invariant $\operatorname{\mathbb{Q}}$-divisor $\operatorname{\mathbb{Q}}$-linearly equivalent to $-K_{X}$. Conversely, every $G$-invariant $\operatorname{\mathbb{Q}}$-divisor $D\sim_{\mathbb{Q}}-K_{X}$ is a pullback of a $\operatorname{\mathbb{Q}}$-divisor $D_{Y}=\pi(D)$ on $Y$. In particular, $D_{Y}\sim_{\mathbb{Q}}-(K_{Y}+B)$ is of $m$-basis type if and only if $\pi^{*}D_{Y}$ is of $m|G|$-basis type. Moreover by [Kol97, Proposition 3.16] the pair $(Y,D_{Y})$ is klt if and only if the pair $(X,D_{X})$ is klt. Therefore by Remark 6.2 we can write

[TABLE]

for all $m\in\mathbb{N}$ and thus $\delta_{G}(X)=\delta(Y,B)$. Analogously, for the $\alpha$-invariant we have

[TABLE]

as desired. ∎

Example 6.5**.**

In Example 6.3 consider the quotient map $\pi\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$ by the cyclic group $G=\mathbb{Z}/m\mathbb{Z}$. The branch divisor is equal to

[TABLE]

To compute $\alpha(\mathbb{P}^{1},B)$ we note that for any effective $D\sim_{\mathbb{Q}}-K_{\mathbb{P}^{1}}$ and for any point $p\in\mathbb{P}^{1}$ we have

[TABLE]

since $\alpha(\mathbb{P}^{1})=1/2$. Therefore for any $p$ and for any $D$ we have $\mathrm{mult}_{p}(D)\leqslant 2$. Thus for any effective $E\sim_{\mathbb{Q}}-(K_{\mathbb{P}^{1}}+B)$ we obtain that $\mathrm{mult}_{p}(E)\leqslant 2/n$. This implies that

[TABLE]

and moreover the value $1/2$ is attained; so that $\alpha(\mathbb{P}^{1},B)=1/2$. The same computation together with the fact that $\delta(\mathbb{P}^{1})=1$ shows that $\delta(\mathbb{P}^{1},B)=1$.

Remark 6.6**.**

We expect that there is a definition of $\delta_{T\rtimes G}(X)$ such that an estimate analogous to [Su13, Theorem 1.2] holds.