K-Stability of Fano spherical varieties

Delcroix, Thibaut

doi:10.48550/arxiv.1608.01852

Cited by 20 publications

(50 citation statements)

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“…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.…”

Section: Aleksei Golotamentioning

confidence: 99%

“…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]. For a Fano variety X, spherical under the action of a connected reductive group G we give a formula for δ G in terms of the combinatorial data defined by X.…”

Section: Spherical Fano Varietiesmentioning

confidence: 99%

“…This formula uses the description of Val G X as the cone V in a finitedimensional vector space. The log discrepancy A X (v) identifies with a certain piecewise linear function h C on V. The function S L (v) can be expressed, following [Del16], via the moment polytope ∆ + , the Duistermaat-Heckman measure DH on ∆ + (see [Del16,Theorem 4.5]) and the vector 2ρ Q , determined by ∆ + and the root system of (G; T ). Proposition 1.4 (see Proposition 5.4 below).…”

Section: Introductionmentioning

confidence: 99%

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Delta-invariants for Fano varieties with large automorphism groups

Golota

2019

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For a polarized variety (X, L) and a closed connected subgroup G ⊂ Aut(X, L) we define a G-invariant version of the δ-threshold. We prove that for a Fano variety (X, −K X ) and a connected subgroup G ⊂ Aut(X) this invariant characterizes G-equivariant 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. ALEKSEI GOLOTAAppendix 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 smooth Fano manifold and G ⊂ 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 δ-invariant, was defined for Fano varieties by Fujita and Odaka [FO16, Definition 0.2] using log canonical thresholds of basis-type divisors. The inequality δ(X) > 1 implies uniform K-stability of X by [FO16, Theorem 0.3] and in fact turns out to be equivalent to it (see [BlJ17, Theorem B]).Since uniform K-stability forces the automorphism group of X to be finite [BBEGZ11, Theorem 5.4], we cannot use δ-invariant to study Kähler-Einstein property of Fano varieties with Aut(X) infinite. However, if we restrict to G-equivariant degenerations for a suitable subgroup G ⊂ Aut(X) (which is sufficient by Theorem 1.2), then in many examples uniform K-stability holds. Therefore it is reasonable to expect that there is a version of δ-invariant for a variety X with a reductive subgroup G ⊂ Aut(X).The main goal of the present paper is to define the δ G -invariant for a polarized variety (X, L) and a closed connected subgroup G ⊂ Aut(X, L). To do so, we follow the approach to δ-invariants and Kstability via valuations on the field of rational functions on X, developed in [Fuj16,Fuj17,BlJ17,BHJ17]. Note also that an alternative valuative criterion for K-semistability was proved in [Li15]. In Section 2 we give the nesessary definitions. The main object we consider is the space 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 ϕ : Y → X. We consider the log discrepancy A X (v) = 1 + ord E (K Y /X

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Section: Aleksei Golotamentioning

confidence: 99%

Section: Spherical Fano Varietiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Delta-invariants for Fano varieties with large automorphism groups

Golota

2019

Preprint

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“…Criteria for K-stability using combinatorial methods have been found in the toric case by Wang-Zhu [WZ04], in the case of complexity one T -varieties by Ilten-Suess [IS17], and for complexity zero varieties under general reductive groups (spherical varieties) by Delcroix [Del16].…”

Section: Introductionmentioning

confidence: 99%

$K$-polystability of smooth Fano $\text{SL}_2$-threefolds

Rogers¹

2022

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We prove the K-polystability of all smooth complex Fano threefolds admitting an effective action of SL 2 but not of a 2-torus or 3-torus. In particular, the existence of Kähler-Einstein metrics on varieties in the families (1.10), (1.15), (1.16), (1.17), (2.21), (2.27), (2.32), (3.13), (3.17), (3.25) and (4.6) of the Mori-Mukai classification of smooth Fano threefolds is proved.Theorem 2.2. [CDS15] A smooth complex Fano variety X admits a Kähler-Einstein metric if and only if (X, K −1 X ) is K-polystable. Unfortunately, since there are generally infinitely many test configurations for a given polarised variety, it is very difficult to check K-(poly/semi)stability in the general case, even accounting for the Li-Xu theorem.The α-invariant of Tian [Tia87] was for a long time one of the only practical methods to check K-(poly)stability. Recently, work of Abban-Zhuang [AZ20,AZ21] has provided other more powerful methods of verification. Otherwise, most progress on this front has come from the equivariant perspective due to the work of Datar-Székelyhidi [DS15], which we summarise now.Definition 2.5. Let X be a G-variety, and let π : L → X be a line bundle on X. We say that L is G-linearised if there is a G-action on L such that π is G-equivariant and the map π −1 (x) → π −1 (g • x) induced on the fibres is linear for all g ∈ G and all x ∈ X.Definition 2.6. Let G be a reductive algebraic group and let (X, L) be a polarised variety with a G-action on X such that L is G-linearised. A test configuration (X , L) of exponent m is G-equivariant if there is a G-action on (X , L) which commutes with the C × action and such that the isomorphisms between (X, L ⊗m ) and (X t , L t ) for t = 0 are G-equivariant. Then (X, L) is equivariantly K-(poly/semi)stable if it is K-(poly/semi)stable with respect to G-equivariant special test configurations.The main result of Datar-Székelyhidi is the following: Theorem 2.3. [DS15] Let G be a reductive algebraic group and let X be a smooth complex Fano G-variety. Then (X, K −1 X ) is equivariantly K-polystable if and only if X admits a Kähler-Einstein metric.Remark. We should mention that the result of Datar-Székelyhidi has been generalised to the singular case when G is finite by Liu-Zhu [LZ20] and for general reductive groups by Zhuang [Zhu21]. Specifically, Zhuang uses a purely algebraic argument showing (among other things) that K-polystability of a log Fano pair (X, ∆) is equivalent to G-equivariant K-polystability when G is reductive.2.2. β invariant. Here we discuss an invariant introduced by Fujita [Fuj16] and Li [Li17] which they have shown to have an intimate connection to K-stability. We must first include some preliminary definitions.

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“…Whether or not a manifold X admits such a metric is characterized by the algebreo-geometric notion of K-stability, see [1], [2]. An equivariant version of K-stability has been used in the spherical [3], and complexity-one T -variety [4] settings to obtain an effective Kähler-Einstein criterion.…”

Section: Introductionmentioning

confidence: 99%