K-semistability of cscK manifolds with transcendental cohomology class

TL;DR

This paper proves that cscK manifolds with transcendental cohomology are K-semistable and, under certain conditions, are uniformly K-stable, extending stability results beyond polarised manifolds using recent advances in K-energy properness.

Contribution

It generalizes K-stability results to transcendental cohomology classes and establishes a formula linking Donaldson-Futaki invariant to K-energy slopes in the Kähler setting.

Findings

cscK manifolds with transcendental classes are K-semistable

finite automorphism group implies uniform K-stability

relation between Donaldson-Futaki invariant and K-energy slope

Abstract

We prove that constant scalar curvature K\"ahler (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 K\"ahler setting, a formula relating the (generalised) Donaldson-Futaki invariant to the asymptotic slope of the K-energy along weak geodesic rays.