Solution to a non-Archimedean Monge-Ampère equation

Boucksom, Sébastien¹ ; Favre, Charles² ; Jonsson, Mattias³

¹ CNRS–Université Pierre et Marie Curie, Institut de Mathématiques, F-75251 Paris Cedex 05, France
² CNRS–CMLS, École Polytechnique, F-91128 Palaiseau Cedex, France
³ Dept of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043

Journal of the American Mathematical Society, Tome 28 (2015) no. 3, pp. 617-667

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-2014-00806-7

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