# K-polystability of Q-Fano varieties admitting Kahler-Einstein metrics

**Inventiones Mathematicae**(0020-9910). Vol. 203 (2016), 3, p. 973-1025.

[Artikel, refereegranskad vetenskaplig]

It is shown that any, possibly singular, Fano variety X admitting a Kahler-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 Kahler-Einsteinmetrics with edge-cone singularities. Applications to geodesic stability, bounds on the Ricci potential and Perelman's lambda-entropy functional on K-unstable Fano manifolds are also given.

**Nyckelord: **monge-ampere equations, scalar curvature, stable varieties, geodesic, rays, stability, bundles, manifolds, continuity, polytopes, geometry, Mathematics

Denna post skapades 2016-05-03. Senast ändrad 2016-07-01.

CPL Pubid: 235848