Volume 22, issue 6 (2018)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Kähler–Ricci flow, Kähler–Einstein metric, and K–stability

Xiuxiong Chen, Song Sun and Bing Wang

Geometry & Topology 22 (2018) 3145–3173
Abstract

We prove the existence of a Kähler–Einstein metric on a K–stable Fano manifold using the recent compactness result on Kähler–Ricci flows. The key ingredient is an algebrogeometric description of the asymptotic behavior of Kähler–Ricci flow on Fano manifolds. This is in turn based on a general finite-dimensional discussion, which is interesting on its own and could potentially apply to other problems. As one application, we relate the asymptotics of the Calabi flow on a polarized Kähler manifold to K–stability, assuming bounds on geometry.

Keywords
Kähler Ricci flow, convergence, uniqueness, Fano manifold, K–stability
Mathematical Subject Classification 2010
Primary: 53C25, 53C44
Secondary: 14J45
References
Publication
Received: 19 October 2015
Revised: 27 February 2018
Accepted: 27 March 2018
Published: 23 September 2018
Proposed: Simon Donaldson
Seconded: Frances Kirwan, Bruce Kleiner
Authors
Xiuxiong Chen
School of Mathematics
University of Science and Technology of China
Hefei, Anhui
China
Department of Mathematics
Stony Brook University
Stony Brook, NY
United States
http://www.math.stonybrook.edu/~xiu/
Song Sun
Department of Mathematics
University of California, Berkeley
Berkeley, CA
United States
Department of Mathematics
Stony Brook University
Stony Brook, NY
United States
https://math.berkeley.edu/people/faculty/song-sun
Bing Wang
School of Mathematics
University of Science and Technology of China
Hefei, Anhui
China
Department of Mathematics
University of Wisconsin-Madison
Madison, WI
United States
http://www.math.wisc.edu/~bwang/