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Addendum to the article “Global pluripotential theory over a trivially valued field”
Sébastien Boucksom1; Mattias Jonsson2
1 CNRS–CMLS École Polytechnique F-91128 Palaiseau Cedex France
2 Dept of Mathematics University of Michigan Ann Arbor, MI 48109-1043 USA
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 33 (2024) no. 2, pp. 405-417.
complements Global pluripotential theory over a trivially valued field

This note is an addendum to the paper “Global pluripotential theory over a trivially valued field” by the present authors, in which we prove two results. Let X be an irreducible projective variety over an algebraically closed field field k, and assume that k has characteristic zero, or that X has dimension at most two. We first prove that when X is smooth, the envelope property holds for any numerical class on X. Then we prove that for X possibly singular and for an ample numerical class, the Monge–Ampère energy of a bounded function is equal to the energy of its usc regularized plurisubharmonic envelope.

Cette note est un appendice au papier « Global pluripotential theory over a trivially valued field » par les présents auteurs, dans lequel nous prouvons deux résultats. Soit X une variété projective irréductible sur un corps algébriquement clos k, et supposons que k est de caractéristique nulle, ou que X est de dimension au plus deux. Nous prouvons d’abord que, lorsque X est lisse, la propriété d’enveloppe est valable pour toute classe numérique sur X. Ensuite, nous prouvons que, pour X possiblement singulier et pour toute classe numérique ample, l’énergie de Monge–Ampère de toute fonction bornée est égale à celle de son enveloppe plurisousharmonique régularisée.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1775

Sébastien Boucksom 1; Mattias Jonsson 2

1 CNRS–CMLS École Polytechnique F-91128 Palaiseau Cedex France
2 Dept of Mathematics University of Michigan Ann Arbor, MI 48109-1043 USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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Sébastien Boucksom; Mattias Jonsson. Addendum to the article “Global pluripotential theory over a trivially valued field”. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 33 (2024) no. 2, pp. 405-417. doi : 10.5802/afst.1775. https://afst.centre-mersenne.org/articles/10.5802/afst.1775/

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[3] Sébastien Boucksom; Mattias Jonsson Global pluripotential theory over a trivially valued field, Ann. Fac. Sci. Toulouse, Math., Volume 31 (2022) no. 3, pp. 647-836 | DOI | Numdam | Zbl

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[5] Sébastien Boucksom; Mattias Jonsson A non-Archimedean approach to K-stability, I: Metric geometry of spaces of test configurations and valuations, Ann. Inst. Fourier (2024) (online first, 99 pages, https://doi.org/10.5802/aif.3668)

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