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Solution to a non-Archimedean Monge-Ampère equation

Authors: Sébastien Boucksom, Charles Favre and Mattias Jonsson
Journal: J. Amer. Math. Soc. 28 (2015), 617-667
MSC (2010): Primary 32P05; Secondary 32U05
Published electronically: May 22, 2014
MathSciNet review: 3327532
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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.

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Additional Information

Sébastien Boucksom
Affiliation: CNRS–Université Pierre et Marie Curie, Institut de Mathématiques, F-75251 Paris Cedex 05, France

Charles Favre
Affiliation: CNRS–CMLS, École Polytechnique, F-91128 Palaiseau Cedex, France

Mattias Jonsson
Affiliation: Dept of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043

Received by editor(s): December 30, 2011
Received by editor(s) in revised form: March 20, 2014
Published electronically: May 22, 2014
Additional Notes: The first author was partially supported by the ANR projects MACK and POSITIVE.
The second author was supported by the ANR-grant BERKO, and by the ERC-starting grant project “Nonarcomp” no.307856.
The third author was partially supported by the CNRS and the NSF
Article copyright: © Copyright 2014 American Mathematical Society