On Fano threefolds of degree 22 after Cheltsov and Shramov

Abstract

It has been known that nonsingular Fano threefolds of Picard rank one with the anti-canonical degree 22 admitting faithful actions of the multiplicative group form a one-dimensional family. Cheltsov and Shramov showed that all but two of them admit Kähler-Einstein metrics. In this paper, we show that the remaining Fano threefolds also admit Kähler-Einstein metrics.

Extremal metrics on the total space of destabilising test configurations

“…In the family 1.10 of Fano threefolds, there exists a one-parameter family of distinct Fano manifolds with Aut 0 (X 0 ) ∼ = C * acting non trivially on their deformation space (see for example [22] for the explicit description of their deformation space near the Mukai-Umemura threefold). All of these admit Kähler-Einstein metrics, by works of Cheltsov and Shramov [10, Corollary 1.7] and Fujita [24,Theorem 1.2]. Similarly, we have a completely analogous picture in the family 2.21, where there also is a oneparameter family of distinct Fano manifolds with Aut 0 (X 0 ) ∼ = C * .…”