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Uniform K-stability and asymptotics of energy functionals in Kahler geometry

Journal article
Authors S. Boucksom
T. Hisamoto
Mattias Jonsson
Published in Journal of the European Mathematical Society
Volume 21
Issue 9
Pages 2905-2944
ISSN 1435-9855
Publication year 2019
Published at Department of Mathematical Sciences
Pages 2905-2944
Language en
Links dx.doi.org/10.4171/jems/894
Keywords K-stability, Kahler geometry, canonical metrics, non-Archimedean geometry, complex monge-ampere, scalar curvature, einstein metrics, continuity, bounds, Mathematics, oll w, 1967, mathematische zeitschrift, v95, p87
Subject categories Mathematics

Abstract

Consider a polarized complex manifold (X, L) and a ray of positive metrics on L defined by a positive metric on a test configuration for (X, L). For many common functionals in Kahler geometry, we prove that the slope at infinity along the ray is given by evaluating the non-Archimedean version of the functional (as defined in our earlier paper [BHJ17]) at the non-Archimedean metric on L defined by the test configuration. Using this asymptotic result, we show that coercivity of the Mabuchi functional implies uniform K-stability, as defined in [Der 15, BHJ17]. As a partial converse, we show that uniform K-stability implies coercivity of the Mabuchi functional when restricted to Bergman metrics.

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