A non-Archimedean approach to K-stability, II: divisorial stability and openness

TL;DR

This paper introduces a new non-Archimedean invariant for projective pairs, establishing divisorial stability as an open condition and linking it to uniform K-stability through valuations and filtrations.

Contribution

It extends the Dervan-Legendre invariant to a broader setting, defines divisorial stability, and proves its relation to K-stability and openness in polarization.

Findings

Divisorial semistability implies sublc singularities.

Divisorial stability is an open condition in polarization.

Divisorial stability is equivalent to uniform K-stability via valuations and filtrations.

Abstract

To any projective pair $(X,B)$ equipped with an ample $\mathbb{Q}$-line bundle $L$ (or even any ample numerical class), we attach a new invariant $\beta(\mu)\in\mathbb{R}$, defined on convex combinations $\mu$ of divisorial valuations on $X$, viewed as point masses on the Berkovich analytification of $X$. The construction is based on non-Archimedean pluripotential theory, and extends the Dervan-Legendre invariant for a single valuation--itself specializing to Li and Fujita's valuative invariant in the Fano case, which detects K-stability. Using our $\beta$-invariant, we define divisorial (semi)stability, and show that divisorial semistability implies $(X,B)$ is sublc (i.e. its log discrepancy function is non-negative), and that divisorial stability is an open condition with respect to the polarization $L$. We also show that divisorial stability implies uniform K-stability in the usual sense of (ample) test configurations, and that it is equivalent to uniform K-stability with respect to all norms/filtrations on the section ring of $(X,L)$, as considered by Chi Li.