K-polystability of Q-Fano varieties admitting Kahler-Einstein metrics
Artikel i vetenskaplig tidskrift, 2016

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.

scalar curvature

stability

geometry

polytopes

Mathematics

geodesic

rays

monge-ampere equations

continuity

bundles

stable varieties

manifolds

Författare

Robert Berman

Chalmers, Matematiska vetenskaper, Matematik

Göteborgs universitet

Inventiones Mathematicae

0020-9910 (ISSN) 1432-1297 (eISSN)

Vol. 203 3 973-1025

Ämneskategorier

Geometri

DOI

10.1007/s00222-015-0607-7