Regularity of Weak Minimizers of the K-Energy and Applications to Properness and K-Stability
Journal article, 2020

Let (X,ω)(X,ω) be a compact Kähler manifold and HH the space of Kähler metrics cohomologous to ωω. If a csck metric exists in HH, we show that all finite energy minimizers of the extended K-energy are smooth csck metrics, partially confirming a conjecture of Y.A. Rubinstein and the second author. As an immediate application, we obtain that the existence of a csck metric in HH implies J-properness of the K-energy, thus confirming one direction of a conjecture of Tian. Exploiting this properness result we prove that an ample line bundle (X,L)(X,L) admitting a csck metric in c1(L)c1(L) is KK-polystable. When the automorphism group is finite, the properness result, combined with a result of Boucksom-Hisamoto-Jonsson, also implies that  (X,L)(X,L) is uniformly K-stable.

Author

Robert Berman

Chalmers, Mathematical Sciences, Algebra and geometry

T. Darvas

University of Maryland

Hoang Chinh Lu

Scuola Normale Superiore di Pisa

Published in

Annales Scientifiques de lEcole Normale Superieure

0012-9593 (ISSN)

Vol. 53 Issue 2 p. 267-289

Categorizing

Subject Categories (SSIF 2011)

Computational Mathematics

Geometry

Mathematical Analysis

Identifiers

DOI

10.24033/asens.2422

More information

Latest update

11/30/2020