The Kähler–Ricci flow and optimal degenerations

Authors
Dervan, Ruadhaí 
Székelyhidi, Gábor 

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Article
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Abstract

We prove that on Fano manifolds, the Kähler-Ricci flow produces a "most destabilising" degeneration, with respect to a new stability notion related to the H-functional. This answers questions of Chen-Sun-Wang and He.

We give two applications of this result. Firstly, we give a purely algebro-geometric formula for the supremum of Perelman's μ-functional on Fano manifolds, resolving a conjecture of Tian-Zhang-Zhang-Zhu as a special case. Secondly, we use this to prove that if a Fano manifold admits a Kähler-Ricci soliton, then the Kähler-Ricci flow converges to it modulo the action of automorphisms, with any initial metric. This extends work of Tian-Zhu and Tian-Zhang-Zhang-Zhu, where either the manifold was assumed to admit a Kähler-Einstein metric, or the initial metric of the flow was assumed to be invariant under a maximal compact group of automorphism.

Publication Date
2020-09-01
Online Publication Date
2020-09-05
Acceptance Date
2018-06-21
Keywords
4902 Mathematical Physics, 4903 Numerical and Computational Mathematics, 4904 Pure Mathematics, 49 Mathematical Sciences
Journal Title
Journal of Differential Geometry
Journal ISSN
0022-040X
Volume Title
116
Publisher
International Press of Boston