Solution to a non-Archimedean Monge-Amp\`ere equation

Abstract

Let X be a smooth projective Berkovich space over a complete discrete valuation field K of residue characteristic zero, and assume that X is defined over a function field admitting K as a completion. Let further m be a positive measure on X and L be an ample line bundle such that the mass of m is equal to the degree of L. Then we show the existence a continuous semipositive metric whose associated measure is equal to m in the sense of Zhang and Chambert-Loir. This we do under a technical assumption on the support of m, which is, for instance, fulfilled if the support is a finite set of divisorial points. Our method draws on analogues of the variational approach developed to solve complex Monge-Amp\`ere equations on compact K\"ahler manifolds by Berman, Guedj, Zeriahi and the first named author, and of Ko{\l}odziej's continuity estimates. It relies in a crucial way on the compactness properties of singular semipositive metrics, as defined and studied in a companion article.