Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities

Abstract

This is the first of a series of three papers which prove the fact that a K-stable Fano manifold admits a Kähler-Einstein metric. The main result of this paper is that a Kähler-Einstein metric with cone singularities along a divisor can be approximated by a sequence of smooth Kähler metrics with controlled geometry in the Gromov-Hausdorff sense.

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

“…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.…”

Regularity of the Kähler–Ricci flow

“…The conjecture states that if a Fano manifold M admits a Kähler-Einstein metrics if it is K-stable. Recently, solutions were provided for this conjecture in the case of Fano manifolds ( [20], also see [6,7,8]). …”