Delta-invariants for Fano varieties with large automorphism groups

Abstract

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

Basis divisors and balanced metrics

“…Let T C ∼ = (C * ) r be a complex torus acting effectively and holomorphically on X. Assume that the action of T C lifts to L. By Blum-Jonsson and Golota [10,28], to compute δ(L) it is enough to only consider T C -invariant divisorial valuations. However the same fact is not known at level m: Problem 2.10.…”

The Calabi Problem for Fano Threefolds

“…Note that the number δ G (X, ∆; L) differs from δ G (X, ∆; L), and δ G (X, ∆; L) also differs from the counter-part of the number δ G (X, ∆; L) defined in [84].…”

K-stability of Fano varieties via admissible flags

“…One can also define local stability thresholds δ x (X) at some x ∈ X by taking log canonical thresholds around the point x so that the global invariant δ(X) is the minimum of the local ones δ x (X); see Subsection 2.2. It is again a challenging problem to find the precise value of these invariants, unless the variety has a large group of automorphisms [BJ20, Gol19,ZZ20]; see [PW18,CZ18] for some estimates on del Pezzo surfaces.…”