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

  • Sébastien Boucksom

    École Polytechnique, Palaiseau, France
  • Tomoyuki Hisamoto

    Nagoya University, Japan
  • Mattias Jonsson

    University of Michigan, Ann Arbor, USA, Chalmers University of Technology and University of Gothenburg, Sweden
Uniform K-stability and asymptotics of energy functionals in Kähler geometry cover
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Abstract

Consider a polarized complex manifold and a ray of positive metrics on defined by a positive metric on a test configuration for . 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 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.

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Cite this article

Sébastien Boucksom, Tomoyuki Hisamoto, Mattias Jonsson, Uniform K-stability and asymptotics of energy functionals in Kähler geometry. J. Eur. Math. Soc. 21 (2019), no. 9, pp. 2905–2944

DOI 10.4171/JEMS/894