Solution to a non-Archimedean Monge-Ampère equation

Abstract

Let X X be a smooth projective Berkovich space over a complete discrete valuation field K K of residue characteristic zero, and assume that X X is defined over a function field admitting K K as a completion. Let further μ \mu be a positive measure on X X and L L be an ample line bundle such that the mass of μ \mu is equal to the degree of L L . We prove the existence of a continuous semipositive metric whose associated measure is equal to μ \mu in the sense of Zhang and Chambert-Loir. We do this under a technical assumption on the support of μ \mu , which is, for instance, fulfilled if the support is a finite set of divisorial points. Our method draws on analogs of the variational approach developed to solve complex Monge-Ampère equations on compact Kähler manifolds by Berman, Guedj, Zeriahi, and the first named author, and of Kołodziej’s C 0 C^0 -estimates. It relies in a crucial way on the compactness properties of singular semipositive metrics, as defined and studied in a companion article.