K-polystability of Q-Fano varieties admitting Kähler-Einstein metrics

Abstract

It is shown that any, possibly singular, Fano variety X admitting a Kähler-Einstein metric is K-polystable, thus confirming one direction of the Yau-Tian-Donaldson conjecture in the setting of Q-Fano varieties equipped with their anti-canonical polarization. The proof is based on a new formula expressing the Donaldson-Futaki invariants in terms of the slope of the Ding functional along a geodesic ray in the space of all bounded positively curved metrics on the anti-canonical line bundle of X . One consequence is that a toric Fano variety X is K-polystable iff it is K-polystable along toric degenerations iff 0 is the barycenter of the canonical weight polytope P associated to X . The results also extend to the logarithmic setting and in particular to the setting of Kähler-Einstein metrics with edge-cone singularities. Applications to geodesic stability, bounds on the Ricci potential and Perelman's λ-entropy functional on K -unstable Fano manifolds are also given.

On K-stability of finite covers

“…(2) It is known that the existence of a constant scalar curvature Kähler metric implies K-stability provided the automorphism group of the polarised variety is finite [15,33], and that the two are equivalent for Fano varieties X with L = −K X (in this case, the metric is Kähler-Einstein) [5,8,36]. The Yau-Tian-Donaldson conjecture states that the existence of a constant scalar curvature Kähler metric is equivalent to K-polystability.…”

“…In most examples, we take the base X to either be projective space or a product of projective spaces. Since these admit Kähler-Einstein metrics, they are K-polystable [5]. It is an open problem to give an algebro-geometric proof of K-polystability of projective space.…”

“…One of the foremost results in complex geometry in recent years is the proof of the YauTian-Donaldson conjecture on Fano manifolds, which states that a Fano manifold admits a Kähler-Einstein metric if and only if it is K-polystable, which is a completely algebro-geometric notion [5,8,36]. While this result is of great theoretical interest, there is currently only one known way of directly proving a Fano variety is K-stable [28].…”

K-Stability of Fano Manifolds With Not Small Alpha invariants

“…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.…”

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].…”