On Fano threefolds of degree 22 after Cheltsov and Shramov

TL;DR

This paper proves that all nonsingular Fano threefolds of degree 22 with Picard rank one admit Kähler-Einstein metrics, completing the classification of their metric properties.

Contribution

It demonstrates that the remaining two Fano threefolds of degree 22 also admit Kähler-Einstein metrics, extending previous results.

Findings

All degree 22 Fano threefolds with Picard rank one admit Kähler-Einstein metrics.

Completes the classification of such Fano threefolds regarding their metric properties.

Supports the broader conjecture on the existence of Kähler-Einstein metrics for Fano varieties.

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 K\"ahler-Einstein metrics. In this paper, we show that the remaining Fano threefolds also admit K\"ahler-Einstein metrics.