The Kähler–Ricci flow and optimal degenerations

Dervan, R. and Székelyhidi, G. (2020) The Kähler–Ricci flow and optimal degenerations. Journal of Differential Geometry, 116(1), pp. 187-203. (doi: 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.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Dervan, Dr Ruadhaí
• Authors: Dervan, R., and Székelyhidi, G.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Journal of Differential Geometry
• Publisher: International Press
• ISSN: 0022-040X
• ISSN (Online): 1945-743X
• Published Online: 05 September 2020
• Copyright Holders: Copyright © 2020 Lehigh University
• First Published: First published in Journal of Differential Geometry 116(1):187-203
• Publisher Policy: Reproduced in accordance with the publisher copyright policy