The uniform version of Yau-Tian-Donaldson conjecture for singular Fano varieties

TL;DR

This paper proves that uniformly K-stable singular Fano varieties admit Kähler-Einstein metrics, extending the Yau-Tian-Donaldson conjecture to singular cases using perturbative and non-Archimedean techniques.

Contribution

It establishes the uniform version of the Yau-Tian-Donaldson conjecture for singular Fano varieties by adapting existing strategies with new perturbative methods.

Findings

Uniform K-stability implies existence of Kähler-Einstein metrics for singular Fano varieties

Extension of Yau-Tian-Donaldson conjecture to singular cases

Development of perturbative and non-Archimedean techniques in the proof

Abstract

We prove the following result: if a $\mathbb{Q}$-Fano variety is uniformly K-stable, then it admits a K\"{a}hler-Einstein metric. We achieve this by modifying Berman-Boucksom-Jonsson's strategy with appropriate perturbative arguments and non-Archimedean estimates. The idea of using the perturbation is motivated by our previous paper.