Kaehler-Einstein Fano threefolds of degree 22

Cheltsov, Ivan; Shramov, Constantin

doi:10.48550/arxiv.1803.02774

Cited by 6 publications

(11 citation statements)

References 14 publications

(19 reference statements)

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“…Note that C is a G-invariant subvariety on X. Thus C is not a closed point by [CS18,Lemma 2.23]. Moreover, if F is a prime divisor on X, then A(F ) > S(F ) holds by [Fjt16,Corollary 9.3].…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

“…In [CS18], Cheltsov and Shramov considered the problem for the existence of Kähler-Einstein metrics for the above V u . If u = −1/4, then the Fano threefold is the Mukai-Umemura threefold V MU .…”

Section: Introductionmentioning

confidence: 99%

“…The main technique to prove Theorem 1.2 is the "G-valuative criterion" (see §2, [ACCFKMGSSV] and [Zhu20]), which is a G-equivariant version of [Li17, Theorem 3.7] and [Fjt19a,Theorem 1.6]. Moreover, we need deep analyses [CS18] of G-invariant curves on V u which might be possible destabilizing centers of G-invariant prime divisors over V u . Technically, the theory of quasi-log schemes [Fjn17] and the careful analysis of the volume functions (see §3) play important roles in order to show Theorem 1.2.…”

Section: Introductionmentioning

confidence: 99%

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On Fano threefolds of degree 22 after Cheltsov and Shramov

Fujita

2021

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“…Note that C is a G-invariant subvariety on X. Thus C is not a closed point by [CS18,Lemma 2.23]. Moreover, if F is a prime divisor on X, then A(F ) > S(F ) holds by [Fjt16,Corollary 9.3].…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Fano threefolds of degree 22 after Cheltsov and Shramov

Fujita

2021

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“…Appendix A] and in [CS18]. For a criterion of equivariant K-stability of spherical Fano varieties see [Del16].…”

Section: Aleksei Golotamentioning

confidence: 99%

“…This method was implemented e.g. in [Su13] for T -varieties of complexity one and in [CS18] for Fano threefolds from the V 22 family having automorphism groups G m ⋊ Z 2 (cf. [DKK17] where the additional symmetries were not used).…”

Section: 1mentioning

confidence: 99%

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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