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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Solution to a non-Archimedean Monge-Ampère equation
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by Sébastien Boucksom, Charles Favre and Mattias Jonsson PDF
J. Amer. Math. Soc. 28 (2015), 617-667 Request permission

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
  • MR Author ID: 688226
  • Email: boucksom@math.jussieu.fr
  • Charles Favre
  • Affiliation: CNRS–CMLS, École Polytechnique, F-91128 Palaiseau Cedex, France
  • MR Author ID: 641179
  • Email: favre@math.polytechnique.fr
  • Mattias Jonsson
  • Affiliation: Dept of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043
  • MR Author ID: 631360
  • Email: mattiasj@umich.edu
  • 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
  • © Copyright 2014 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 28 (2015), 617-667
  • MSC (2010): Primary 32P05; Secondary 32U05
  • DOI: https://doi.org/10.1090/S0894-0347-2014-00806-7
  • MathSciNet review: 3327532