The Uniform Version of Yau–Tian–Donaldson Conjecture for Singular Fano Varieties

Abstract

Let X be a Q-Fano variety and Aut(X) 0 be the identity component of the automorphism group of X. Let G denote a connected reductive subgroup of Aut(X) 0 . We prove that if X is G-uniformly K-stable, then it admits a Kähler-Einstein metric. The converse of this result holds true if G is a maximal torus of Aut(X) 0 , or is equal to Aut(X) 0 itself. These results give (equivariantly uniform) versions of Yau-Tian-Donaldson conjecture for arbitrary singular Fano varieties. A key new ingredient is a valuative criterion for the G-uniform K-stability.

Canonical Kähler metrics and stability of algebraic varieties

“…The proof of Theorem 3.8 is much more technical than [7] because we need to overcome the difficulties caused by singularities. The first key idea is to use an approximation approach initiated in [49]. Consider the log resolution µ : X ′ → X as in section 1.3 and re-organize (1.4) as:…”

On K-stability of Fano weighted hypersurfaces

“…Since X is K-stable, we see that X admits a Kähler-Einstein metric (cf. [DK01, Section 6], [LTW19], [Li19]).…”

Global pluripotential theory over a trivially valued field

“…For an excellent introduction, see the book by V. Guedj and A. Zeriahi [77]. This theory lies at the heart of the variational approach to complex Monge-Ampère equations [11], which proved particularly fruitful in relation to Kähler-Einstein metrics, and ultimately led to a new proof of the Yau-Tian-Donaldson (YTD) conjecture for Fano manifolds [12], later generalized to (log terminal) Fano varieties [89,91].…”