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

Abstract

Abstract To any projective pair ( X , B ) {(X,B)} equipped with an ample ℚ {\mathbb{Q}} -line bundle L (or even any ample numerical class), we attach a new invariant β ⁢ ( μ ) ∈ ℝ {\beta(\mu)\in\mathbb{R}} , defined on convex combinations μ 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 β-invariant, we define divisorial (semi)stability, and show that divisorial semistability implies ( X , B ) {(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 ) {(X,L)} , as considered by Chi Li.