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 Kähler-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.

44 pages, accepted version

References in corpus (4)

• Riemannian geometry of Kahler-Einstein currents
• Bounding diameter of singular Kähler metric
• Openness results for uniform K-stability
• Cheeger-Colding-Tian theory for conic Kahler-Einstein metrics

Cited by in corpus (14)

• The uniform version of Yau-Tian-Donaldson conjecture for singular Fano varieties
• A quantization proof of the uniform Yau-Tian-Donaldson conjecture
• On the stability of extensions of tangent sheaves on Kähler-Einstein Fano / Calabi-Yau pairs
• On K-stability of some del Pezzo surfaces of Fano index 2
• Quasi-plurisubharmonic envelopes 1: Uniform estimates on Kähler manifolds
• Cheeger-Colding-Tian theory for conic Kahler-Einstein metrics
• ALE Calabi-Yau metrics with conical singularities along a compact divisor
• On the uniform K-stability for some asymptotically log del Pezzo surfaces
• Simply connected Sasaki-Einstein rational homology 5-spheres
• Unstable singular del Pezzo hypersurfaces with lower index
• The Partial $C^0$-estimate along a general continuity path and applications
• Existence of Kahler-Ricci solitons on smoothable Q-Fano varities
• Finiteness of $\mathbb{Q}$-Fano compactifications of semisimple group with Kähler-Einstein metrics
• Kähler-Einstein metrics with positive curvature near an isolated log terminal singularity