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
DOI:
https://doi.org/10.1090/S0894-0347-2014-00806-7
Published electronically:
May 22, 2014
MathSciNet review:
3327532
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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
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American Mathematical Society