On the Yau‐Tian‐Donaldson Conjecture for Singular Fano Varieties

Abstract

We prove the Yau‐Tian‐Donaldson conjecture for any ℚ‐Fano variety that has a log smooth resolution of singularities such that a negative linear combination of exceptional divisors is relatively ample and the discrepancies of all exceptional divisors are nonpositive. 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 ℚ‐Fano varieties, which includes all ℚ‐factorial ℚ‐Fano varieties that admit crepant log resolutions. © 2020 Wiley Periodicals LLC

Some observations on the dimension of Fano K-moduli

“…The general small deformation X t of X l is then a K-polystable variety which is also Kähler-Einstein by [9]. Moreover the second Betti number gets bigger and bigger as l goes to infinity: indeed, smoothing out an A l−1 -singularity introduces a chain of S 2 of length l − 1, giving distinct homological classes.…”

Higher regularity for singular Kähler-Einstein metrics

“…An early result in this direction is Kobayashi [25] on orbifold Kähler-Einstein metrics, while a definitive existence result for a large class of singularities was obtained by Eyssidieux-Guedj-Zeriahi [19]. These works focus on the case of non-positive Ricci curvature, however recently Li-Tian-Wang [26] extended Chen-Donaldson-Sun's solution [5,6,7,8] of the Yau-Tian-Donaldson conjecture to general Q-Fano varieties. As a result we now have several sources of singular Kähler-Einstein manifolds on normal varieties.…”

Quasi-plurisubharmonic envelopes 1: Uniform estimates on Kähler manifolds

“…An alternative pluripotential variational approach has been developed by Berman-Boucksom-Jonsson in [BBJ21], based on finite energy classes studied in [GZ07] and variational tools obtained in [BBGZ13]. This approach has been pushed one step further by Li-Tian-Wang who have settled the case of singular Fano varieties [LTW20].…”