Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches \boldmath2𝜋 and completion of the main proof

Abstract

This is the third and final article in a series which prove the fact that a K-stable Fano manifold admits a Kähler-Einstein metric. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle approaches 2 π \pi . We also put all our technical results together to complete the proof of the main theorem.

K-Stability of Fano Manifolds With Not Small Alpha invariants

“…Thus X is K-stable. The remaining assertions are proved from [CDS15a,CDS15b,CDS15c], [Tia15] and [Mat57] (see also [OS12,§1]). …”

“…On the other hand, it has been known that a Fano manifold X admits Kähler-Einstein metrics if and only if X is K-polystable by the works [DT92, Tia97, Don02, Don05, CT08, Sto09, Mab08, Mab09,Ber16] and [CDS15a,CDS15b,CDS15c,Tia15]. In this article, we will focus on the conditions K-stability and K-semistability; K-stability is stronger than K-polystability and K-polystability is stronger than K-semistability.…”

Kähler–Einstein metrics along the smooth continuity method

“…In particular we obtain a new proof of the result of Chen-Donaldson-Sun [20], without using metrics with conical singularities. At the same time our arguments are analogous to those in [20], using also the adaptation of some of those ideas to the smooth continuity method in [42]. A key advantage of the smooth continuity path is that it allows one to work in a G-equivariant setting.…”

“…The Yau-Tian-Donaldson conjecture [53,46,26], confirmed recently by Chen-Donaldson-Sun [17,18,19,20], says that M admits a Kähler-Einstein metric if and only if it is K-stable. In general it seems to be intractable at present to check K-stability since in principle one must study an infinite number of possible degenerations of M to Q-Fano varieites.…”

K-semistable Fano manifolds with the smallest alpha invariant

“…It has been known that a Fano manifold X (i.e., a smooth Q-Fano variety) admits Kähler-Einstein metrics if and only if X is K-polystable by the works [DT92, Tia97, Don02, Don05, CT08, Sto09, Mab08, Mab09,Ber16] and [CDS15a,CDS15b,CDS15c,Tia15].…”