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Relative K-stability for Kähler manifolds

Published version
Peer-reviewed

Type

Article

Change log

Authors

Dervan, R 

Abstract

We study the existence of extremal Kähler metrics on Kähler manifolds. After introducing a notion of relative K-stability for Kahler manifolds, we prove that Kähler manifolds admitting extremal Kähler metrics are relatively K-stable. Along the way, we prove a general Lp lower bound on the Calabi functional involving test configurations and their associated numerical invariants, answering a question of Donaldson.

When the Kähler manifold is projective, our definition of relative K-stability is stronger than the usual definition given by Székelyhidi. In particular our result strengthens the known results in the projective case (even for constant scalar curvature Kähler metrics), and rules out a well known counterexample to the "naïve" version of the Yau-Tian-Donaldson conjecture in this setting.

Description

Keywords

4902 Mathematical Physics, 4904 Pure Mathematics, 49 Mathematical Sciences

Journal Title

Mathematische Annalen

Conference Name

Journal ISSN

0025-5831
1432-1807

Volume Title

Publisher

Springer Nature
Sponsorship
Engineering and Physical Sciences Research Council (EP/J002062/1)