K-Stability of Fano Manifolds With Not Small Alpha invariants

Abstract

Abstract. We show that any n-dimensional Fano manifold X with α(X) = n/(n + 1) and n ≥ 2 is K-stable, where α(X) is the alpha invariant of X introduced by Tian. In particular, any such X admits Kähler-Einstein metrics and the holomorphic automorphism group Aut(X) of X is finite.

Canonical and log canonical thresholds of Fano complete intersections

“…implies the existence of the Kähler-Einstein metric on F (this fact was shown for arbitrary Fano varieties, not only for complete intersections in the projective space), in fact, the inequality above implies the K -stability of F, see the most recent paper [9] on this subject. Since the property of being canonical is stronger than that of being log canonical, the claim of Conjecture 1.1 implies the existence of the Kähler-Einstein on a general Fano complete intersection of index 1.…”

Kähler-Einstein metrics: Old and New

“…[Don15]), hypothesis which holds for hypersurfaces. Fujita recently proved existence of KE metrics on all smooth hypersurfaces of degree d = n in CP n [Fuj16b]. For n = 4 the conjecture follows by (very recent!)…”

“…By definition, there exists C α such that, for any ω + √ −1∂∂ϕ > 0, Remark 4.8. If α = n n+1 , then Fujita [Fuj16b] proved K-stability. The existence of a KE metrics follows by the crucial result of [CDS15] we will describe in the last lecture.…”

Alpha-invariants and purely log terminal blow-ups

“…This was proved by Tian in [45]. In [19], this result was improved by Fujita. He proved that V admits a Kähler-Einstein metric if it is smooth and α(V )…”