Openness of uniform K-stability in the Kähler cone

Abstract

We prove continuity results for new stability thresholds related to uniform K-stability and deduce that uniform K-stability is an open condition in the Kähler cone of any compact Kähler manifold, thus establishing an algebro-geometric counterpart to a classical result of LeBrun-Simanca for constant scalar curvature (cscK) metrics. This settles a folklore conjecture in the field, and in particular implies openness of uniform K-stability for smooth polarized varieties. Moreover, it strengthens evidence supporting the uniform version of the Yau-Tian-Donaldson conjecture for arbitrary polarizations, including the case of irrational polarizations and non-projective Kähler manifolds. As a key tool we introduce a new norm on test configurations and establish estimates for non-archimedean energy functionals in terms of this norm. This leads to new characterizations of uniform K-stability by restricting to test configurations that satisfy certain uniform bounds. As a byproduct we obtain continuity results for a stability threshold related to non-archimedean entropy and deduce openness of uniform J-stability, as well as openness of J-stability in the projective case.