On the Yau-Tian-Donaldson conjecture for 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.