Journal of the European Mathematical Society

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Volume 21, Issue 9, 2019, pp. 2905–2944
DOI: 10.4171/JEMS/894

Published online: 2019-05-24

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

Sébastien Boucksom[1], Tomoyuki Hisamoto[2] and Mattias Jonsson[3]

(1) École Polytechnique, Palaiseau, France
(2) Nagoya University, Japan
(3) University of Michigan, Ann Arbor, USA, Chalmers University of Technology and University of Gothenburg, Sweden

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 Kähler 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 [Der15, BHJ17]. As a partial converse, we show that uniform K-stability implies coercivity of the Mabuchi functional when restricted to Bergman metrics.

Keywords: K-stability, Kähler geometry, canonical metrics, non-Archimedean geometry

Boucksom Sébastien, Hisamoto Tomoyuki, Jonsson Mattias: Uniform K-stability and asymptotics of energy functionals in Kähler geometry. J. Eur. Math. Soc. 21 (2019), 2905-2944. doi: 10.4171/JEMS/894