Openness of uniform K-stability in the K\"ahler 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 K\"ahler cone of any compact K\"ahler 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 K\"ahler 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.
Keywords
Cite
@article{arxiv.2011.14806,
title = {Openness of uniform K-stability in the K\"ahler cone},
author = {Zakarias Sjöström Dyrefelt},
journal= {arXiv preprint arXiv:2011.14806},
year = {2022}
}
Comments
Withdrawn for now due to a gap in the proof of Corollary 3.22, needed for the main result. We still hope to fill the gap or get a replacement for this statement in the future. Until then, parts of the paper still hold without changes, e.g. openness of uniform J-stability and all estimates regarding energy. This may appear in a reworked future version, or elsewhere. Comments are much appreciated