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

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• 作者：Robert J. Berman
• 刊名：Inventiones Mathematicae
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：203
• 期：3
• 页码：973-1025
• 全文大小：900 KB
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• 作者单位：Robert J. Berman (1)
  
  1. Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Göteborg, Sweden
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1297

文摘

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 \(\mathbb {Q}\)-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 \(\lambda \)-entropy functional on K-unstable Fano manifolds are also given.