Uniform K-stability and asymptotics of energy functionals in Kähler geometry. (English) Zbl 1478.53115

Boucksom, Sébastien; Hisamoto, Tomoyuki; Jonsson, Mattias

J. Eur. Math. Soc. (JEMS) 21, No. 9, 2905-2944 (2019); erratum ibid. 24, No. 2, 735-736 (2022).

Summary: 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 [Ann. Inst. Fourier 67, No. 2, 743–841 (2017; Zbl 1391.14090)]) 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 [R. Dervan, Int. Math. Res. Not. 2016, No. 15, 4728–4783 (2016; Zbl 1405.32032); the authors, loc. cit.]. As a partial converse, we show that uniform K-stability implies coercivity of the Mabuchi functional when restricted to Bergman metrics.

MSC:

53C55	Global differential geometry of Hermitian and Kählerian manifolds
14L24	Geometric invariant theory
32P05	Non-Archimedean analysis
32Q20	Kähler-Einstein manifolds
32Q26	Notions of stability for complex manifolds

Keywords:

K-stability; Kähler geometry; canonical metrics; non-Archimedean geometry

Citations:

Zbl 1391.14090; Zbl 1405.32032

References:

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