On K-polystability of cscK manifolds with transcendental cohomology class

TL;DR

This paper establishes that the existence of a cscK metric on a Kähler manifold implies geodesic K-polystability, linking geometric stability notions and extending results to non-projective cases with transcendental classes.

Contribution

It proves that cscK metrics imply geodesic K-polystability for Kähler manifolds, including non-projective cases, and connects this to algebraic K-polystability and equivariant stability.

Findings

Existence of cscK metric implies geodesic K-polystability.

Geodesic K-polystability implies algebraic K-polystability.

Geodesic K-polystability implies equivariant K-polystability.

Abstract

In this paper we study K-polystability of arbitrary (possibly non-projective) compact K\"ahler manifolds admitting holomorphic vector fields. As a main result, we show that existence of a constant scalar curvature K\"ahler (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. Hence 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 K-energy 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 previous work and proves that existence of a cscK (or extremal) K\"ahler metric implies equivariant K-polystability (resp. relative K-stability).