K-semistability of cscK manifolds with transcendental cohomology class

Abstract

We prove that constant scalar curvature Kähler (cscK) manifolds with transcendental cohomology class are K-semistable, naturally generalising the situation for polarised manifolds. Relying on a very recent result by R. Berman, T. Darvas and C. Lu regarding properness of the K-energy, it moreover follows that cscK manifolds with finite automorphism group are uniformly K-stable. As a main step of the proof we establish, in the general Kähler setting, a formula relating the (generalised) Donaldson-Futaki invariant to the asymptotic slope of the K-energy along weak geodesic rays.

Added remark 1.2 and Theorem A b) on uniform K-stability, and updated the abstract accordingly. Added proof of Proposition 2.3. To appear in J. Geom. Anal

References in corpus (1)

• Tian's properness conjectures and Finsler geometry of the space of Kahler metrics

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