Non-Archimedean balanced metrics and their application to totally degenerate abelian varieties

Abstract

For a polarized complex manifold with discrete automorphism group, it is known that if the first Chern class admits a cscK metric, then the balanced metrics, which are characterized in terms of the algebro-geometric notion of Chow stability, approximate this cscK metric. In this paper, we study a non-Archimedean analogue of this phenomenon. In particular, we prove that such an analogue holds for polarized totally degenerate abelian varieties. As an application, we also show that, for a totally degenerating family of polarized abelian varieties, the validity of this non-Archimedean analogue yields a uniform estimate for the Calabi--Yau metrics on fibers sufficiently close to the degenerate fiber.