On K-Polystability of cscK Manifolds with Transcendental Cohomology Class
Artikel i vetenskaplig tidskrift, 2020
In this paper we study K-polystability of arbitrary (possibly non-projective) compact Miller manifolds admitting holomorphic vector fields. As a main result we show that existence of a constant scalar curvature Kahler (cscK) metric implies geodesic K-polystability, in a sense that is expected to be equivalent to K-polystability in general. In particular, in the spirit of an expectation of Chen-Tang  we show that geodesic K-polystability implies algebraic K-polystability for polarized manifolds, so our main result recovers a possibly stronger version of results of Berman-Darvas-Lu  in this case. As a key part of the proof we also study subgeodesic rays with singularity type prescribed by singular test configurations and prove a result on asymptotics of the Kenergy functional along such rays. In an appendix by R. Dervan it is moreover deduced that geodesic K-polystability implies equivariant K-polystability. This improves upon the results of  and proves that existence of a cscK (or extremal) Miller metric implies equivariant K-polystability (resp. relative K-stability).