Solution to a non-Archimedean Monge-Ampère 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 $\mu$ be a positive measure on $X$ and $L$ be an ample line bundle such that the mass of $\mu$ is equal to the degree of $L$. We prove the existence of a continuous semipositive metric whose associated measure is equal to $\mu$ in the sense of Zhang and Chambert-Loir. We do this under a technical assumption on the support of $\mu$, 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 $C^0$-estimates. It relies in a crucial way on the compactness properties of singular semipositive metrics, as defined and studied in a companion article.