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Algebraic uniqueness of Kähler–Ricci flow limits and optimal degenerations of Fano varieties

Jiyuan Han and Chi Li

Geometry & Topology 28 (2024) 539–592
Abstract

We prove that for any Fano manifold X, the special –test configuration that minimizes the HNA–functional is unique and has a K–semistable –Fano central fiber (W,ξ). Moreover there is a unique K–polystable degeneration of (W,ξ). As an application, we confirm the conjecture of Chen, Sun and Wang about the algebraic uniqueness for Kähler–Ricci flow limits on Fano manifolds, which implies that the Gromov–Hausdorff limit of the flow does not depend on the choice of initial Kähler metrics. The results are achieved by studying algebraic optimal degeneration problems via new functionals for real valuations over –Fano varieties, which are analogous to the minimization problem for normalized volumes.

Keywords
Fano varieties, optimal degeneration, Kähler–Ricci flow, Hamilton–Tian conjecture, K–stability, test configuration
Mathematical Subject Classification
Primary: 14J45, 32Q26, 53E30
References
Publication
Received: 14 April 2021
Revised: 1 June 2022
Accepted: 2 July 2022
Published: 13 March 2024
Proposed: Gang Tian
Seconded: Simon Donaldson, John Lott
Authors
Jiyuan Han
Westlake University
Hangzhou
China
Chi Li
Department of Mathematics
Rutgers University
Piscataway, NJ
United States

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