Kahler-Einstein metrics on symmetric Fano T-varieties

TL;DR

This paper establishes a connection between the global log canonical thresholds of symmetric Fano T-varieties and their quotients, applying Tian's criterion to prove the existence of Kahler-Einstein metrics on certain Fano varieties.

Contribution

It introduces a method to relate thresholds of T-varieties to their quotients and demonstrates the existence of Kahler-Einstein metrics on specific Fano threefolds.

Findings

Proves existence of Kahler-Einstein metrics on certain Fano varieties.

Shows some Fano threefolds are Kahler-Einstein but can deform without such metrics.

Provides a new approach to studying Kahler-Einstein metrics via quotient thresholds.

Abstract

We relate the global log canonical threshold of a variety with torus action to the global log canonical threshold of its quotient. We apply this to certain Fano varieties and use Tian's criterion to prove the existence of Kahler-Einstein metrics on them. In particular, we obtain simple examples of Fano threefolds being Kahler-Einstein but admitting deformations without Kahler-Einstein metric.