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

• 成果类型: Article
• 署名作者: Berman, Robert J.
• 署名单位: Chalmers University of Technology; University of Gothenburg
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-015-0607-7
• 发表日期: 2016
• 页码: 973-1025
• 关键词: stability; continuity; CURVATURE; polytopes; LIMITS

摘要

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.