Global pluripotential theory on hybrid spaces

Pille-Schneider, Léonard

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

Let $(X,L)$ be a polarized scheme over a Banach ring $A$. We define and study a class $\mathrm{PSH}(X,L)$ of plurisubharmonic metrics on the Berkovich analytification $X^{\mathrm{an}}$. We focus mainly on the case where $A$ is a hybrid ring of power series, so that $X^{\mathrm{an}}$ 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 $X^{\mathrm{hyb}}$. We also discuss the continuity of the family of Monge-Ampère measures associated to a continuous plurisubharmonic hybrid metric.

Soit $(X,L)$ un schéma polarisé sur un anneau de Banach $A$. Nous définissons et étudions la classe des métriques plurisousharmoniques $\mathrm{PSH}(X,L)$ sur l’analytifié de Berkovich $X^{\mathrm{an}}$. Nous nous intéressons en particulier au cas où $A$ est l’anneau hybride des séries convergentes, et $X^{\mathrm{an}}$ 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 $X^{\mathrm{hyb}}$. Nous discutons aussi de la continuité de la famille de mesures de Monge-Ampère associée à une métrique hybride plurisousharmonique continue.

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

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