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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Singular semipositive metrics in non-Archimedean geometry

Authors: Sébastien Boucksom, Charles Favre and Mattias Jonsson
Journal: J. Algebraic Geom. 25 (2016), 77-139
Published electronically: August 14, 2015
MathSciNet review: 3419957
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Abstract | References | Additional Information

Abstract: Let $X$ be a smooth projective Berkovich space over a complete discrete valuation field $K$ of residue characteristic zero, endowed with an ample line bundle $L$. We introduce a general notion of (possibly singular) semipositive (or plurisubharmonic) metrics on $L$ and prove the analogue of the following two basic results in the complex case: the set of semipositive metrics is compact modulo scaling, and each semipositive metric is a decreasing limit of smooth semipositive ones. In particular, for continuous metrics, our definition agrees with the one by S.-W. Zhang. The proofs use multiplier ideals and the construction of suitable models of $X$ over the valuation ring of $K$, using toroidal techniques.

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Additional Information

Sébastien Boucksom
Affiliation: CNRS-Université Paris 6, Institut de Mathématiques, F-75251 Paris Cedex 05, France
Address at time of publication: CNRS-CMLS, École Polytechnique, 91128 Palaiseau Cedex, France
MR Author ID: 688226

Charles Favre
Affiliation: CNRS-CMLS, École Polytechnique, 91128 Palaiseau Cedex, France
MR Author ID: 641179

Mattias Jonsson
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043
MR Author ID: 631360

Received by editor(s): November 9, 2012
Received by editor(s) in revised form: September 24, 2013, October 23, 2013, and December 3, 2013
Published electronically: August 14, 2015
Additional Notes: The first author was partially supported by the ANR grants MACK and POSITIVE. The second author was partially supported by the ANR-grant BERKO and the ERC-Starting grant “Nonarcomp” No. 307856. The third author was partially supported by the CNRS and the NSF. The authors’ work was carried out at several institutions, including the IHES, IMJ, the École Polytechnique, and the University of Michigan. The authors gratefully acknowledge their support.
Article copyright: © Copyright 2015 University Press, Inc.