On K-stability of finite covers

Abstract

We show that certain Galois covers of K-semistable Fano varieties are K-stable. We use this to give some new examples of Fano manifolds admitting Kähler-Einstein metrics, including hypersurfaces, double solids and threefolds.

Positivity of the CM line bundle for families of K-stable klt Fano varieties

“…Then, equidimensionality of f pre follows from [36,Prop 34], as X t = d i=1 H i is a union of lc-centers of codimension d, and [36,Prop 34] states that an lccenter is contained in the intersection of d Q-Cartier divisors of coefficient 1 from the boundary (here the H i ), then the codimension of the lc-center has to be at least d. As X pre , pre is klt, X pre is Cohen-Macayulay, an hence by the above shown equidimensionality, f pre is indeed flat. Additionally, [36,Prop 34] tells us that locally there is a finite cover where the pullbacks of the H i become simple normal crossing. This shows that d i=1 H i is reduced, otherwise the intersection of the above pullbacks would be non-reduced.…”

On toric geometry and K-stability of Fano varieties

“…Since X is K-polystable, Theorem 2.1 implies that the general members of the 3 deformation families of smooth Fano 3-folds V 12 , MM 2−6 and MM 3−1 are K-semistable. This was already known: the general member of the deformation family MM 2−6 is K-semistable by Theorem 2.1 and [24], and every member of the family MM 3−1 is K-semistable by [24]. A general member of the deformation family of V 12 is Ksemistable [9].…”

K-stability of Fano varieties via admissible flags

“…(3) [Der16] double covers of P n branched over some smooth divisor D of degree d ≥ n + 1. (4) cyclic covers π : X → Y of degree s (where Y ⊆ P n+1 is a smooth hypersurface of degree m) branched along some smooth divisor D ∈ |dH| (where H is the hyperplane class) with…”

“…More recently, [SZ19] discovered another K-stability criterion in the particular case of birationally superrigid Fano varieties; and, as an application [Zhu18] proved that Fano complete intersections of index one and large dimension are K-stable. However, both criteria apply exclusively to certain Fano varieties of index one and except in a few sporadic cases it is unclear how to attack the problem when the required conditions in neither criterion are satisfied; see for example [AGP06,Der16,SS17,LX19].…”