A NON-ARCHIMEDEAN APPROACH TO K-STABILITY, II: DIVISORIAL STABILITY AND OPENNESS

Overview & Synopsis

To any projective pair (X, B) equipped with an ample Q-line bundle L (or even any ample numerical class), we attach a new invariant β(µ) ∈ 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 βinvariant, 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).