On the Yau-Tian-Donaldson conjecture for singular Fano varieties

TL;DR

This paper proves the Yau-Tian-Donaldson conjecture for a class of singular Fano varieties with specific resolution properties, establishing a link between K-polystability and the existence of Kähler-Einstein metrics.

Contribution

It extends the Yau-Tian-Donaldson conjecture to certain singular Fano varieties, broadening the scope from smooth cases to those with controlled singularities.

Findings

K-polystable singular Fano varieties admit Kähler-Einstein metrics.

The result applies to varieties with log smooth resolutions and non-positive discrepancies.

Extends known results from smooth to certain singular Fano varieties.

Abstract

We prove the Yau-Tian-Donaldson's conjecture for any $\mathbb{Q}$-Fano variety that has a log smooth resolution of singularities such that the discrepancies of all exceptional divisors are non-positive. In other words, if such a Fano variety is K-polystable, then it admits a K\"{a}hler-Einstein metric. This extends the previous result for smooth Fano varieties to this class of singular $\mathbb{Q}$-Fano varieties, which include those admitting crepant log resolutions.