Global pluripotential theory on hybrid spaces

Léonard Pille-Schneider

doi:10.5802/jep.228

Global pluripotential theory on hybrid spaces
[Théorie du pluripotentiel global sur les espaces hybrides]

Léonard Pille-Schneider ¹

¹ Département de Mathématiques et Applications, École Normale Supérieure 45 rue d’Ulm, 75005 Paris, France

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 601-658.

Résumé
Abstract

Soit $(X, L)$ un schéma polarisé sur un anneau de Banach $A$ . Nous définissons et étudions la classe des métriques plurisousharmoniques $PSH (X, L)$ sur l’analytifié de Berkovich $X^{an}$ . Nous nous intéressons en particulier au cas où $A$ est l’anneau hybride des séries convergentes, et $X^{an}$ est l’espace hybride associé à une dégénérescence de variétés complexes $X$ . Nous démontrons alors que toute métrique plurisousharmonique sur $(X, L)$ à croissance logarithmique en zéro admet une extension plurisousharmonique canonique à l’espace hybride $X^{hyb}$ . Nous discutons aussi de la continuité de la famille de mesures de Monge-Ampère associée à une métrique hybride plurisousharmonique continue.

Let $(X, L)$ be a polarized scheme over a Banach ring $A$ . We define and study a class $PSH (X, L)$ of plurisubharmonic metrics on the Berkovich analytification $X^{an}$ . We focus mainly on the case where $A$ is a hybrid ring of power series, so that $X^{an}$ is the hybrid space associated to a degeneration of complex manifolds $X$ . We then prove that any plurisubharmonic metric on $(X, L)$ with logarithmic growth at zero admits a canonical plurisubharmonic extension to the hybrid space $X^{hyb}$ . We also discuss the continuity of the family of Monge-Ampère measures associated to a continuous plurisubharmonic hybrid metric.

Reçu le : 2022-09-12
Accepté le : 2023-03-21
Publié le : 2023-03-31

DOI : 10.5802/jep.228

Classification : 32P05, 32U05, 14D06
Keywords: Berkovich spaces, pluripotential theory, hybrid spaces
Mot clés : Espaces de Berkovich, théorie du pluripotentiel, espaces hybrides

Affiliations des auteurs :

Léonard Pille-Schneider ¹

¹ Département de Mathématiques et Applications, École Normale Supérieure 45 rue d’Ulm, 75005 Paris, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Global pluripotential theory on hybrid spaces},
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Léonard Pille-Schneider. Global pluripotential theory on hybrid spaces. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 601-658. doi : 10.5802/jep.228. https://jep.centre-mersenne.org/articles/10.5802/jep.228/

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