Mathematical Research Letters

Volume 24 (2017)

Number 3

K-stability for Kähler manifolds

Pages: 689 – 739

DOI: http://dx.doi.org/10.4310/MRL.2017.v24.n3.a5

Authors

Ruadhaí Dervan (DPMMS, Centre for Mathematical Sciences, University of Cambridge, United Kingdom; and and Université Libre de Bruxelles, Belgium)

Julius Ross (DPMMS, Centre for Mathematical Sciences, University of Cambridge, United Kingdom)

Abstract

We formulate a notion of K-stability for Kähler manifolds, and prove one direction of the Yau–Tian–Donaldson conjecture in this setting. More precisely, we prove that the Mabuchi functional being bounded below (resp. coercive) implies K-semistability (resp. uniformly K-stable). In particular this shows that the existence of a constant scalar curvature Kähler metric implies K-semistability, and K-stability if one assumes the automorphism group is discrete. We also show how Stoppa’s argument holds in the Kähler case, giving a simpler proof of this K-stability statement.

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Paper received on 10 March 2016.