The Kähler–Ricci flow and optimal degenerations
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Authors
Dervan, Ruadhaí
Székelyhidi, Gábor
Publication Date
2020-09-01Journal Title
Journal of Differential Geometry
ISSN
0022-040X
Publisher
International Press of Boston
Volume
116
Issue
1
Type
Article
Metadata
Show full item recordCitation
Dervan, R., & Székelyhidi, G. (2020). The Kähler–Ricci flow and optimal degenerations. Journal of Differential Geometry, 116 (1) https://doi.org/10.4310/jdg/1599271255
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.
Identifiers
External DOI: https://doi.org/10.4310/jdg/1599271255
This record's URL: https://www.repository.cam.ac.uk/handle/1810/280503
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