The space of finite-energy metrics over a degeneration of complex manifolds

Rémi Reboulet

doi:10.5802/jep.229

The space of finite-energy metrics over a degeneration of complex manifolds
[L’espace des métriques d’énergie finie sur une dégénérescence de variétés complexes]

Rémi Reboulet ¹

¹ Department of mathematical sciences, Chalmers University of Technology 412 96 Gothenburg, Sweden

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 659-701.

Résumé
Abstract

Étant donné une dégénérescence de variétés projectives complexes $X \to 𝔻^{*}$ avec des singularités méromorphes, et un fibré en droites relativement ample $L$ sur $X$ , nous étudions des espaces de métriques plurisousharmoniques sur $L$ , avec une attention particulière aux conditions (relatives) d’énergie finie. Nous munissons l’espace ${\hat{ℰ}}^{1} (L)$ des métriques relativement maximales d’énergie finie d’une distance de type $d_{1}$ donnée par le nombre de Lelong en $0$ de la famille des distances de Darvas $d_{1}$ fibre à fibre. Nous montrons que cette structure métrique est complète et géodésique. En considérant $X$ et $L$ comme des schémas $X_{K}$ , $L_{K}$ sur le champ discrètement valué $K = ℂ ((t))$ des séries de Laurent complexes, nous montrons que l’espace $ℰ^{1} (L_{K}^{an})$ des métriques non archimédiennes d’énergie finie sur $L_{K}^{an}$ s’immerge isométriquement et géodésiquement dans ${\hat{ℰ}}^{1} (L)$ , et nous caractérisons son image. Ceci généralise un travail précédent de Berman-Boucksom-Jonsson, traitant le cas trivialement valué.

Given a degeneration of projective complex manifolds $X \to 𝔻^{*}$ with meromorphic singularities, and a relatively ample line bundle $L$ on $X$ , we study spaces of plurisubharmonic metrics on $L$ , with particular focus on (relative) finite-energy conditions. We endow the space ${\hat{ℰ}}^{1} (L)$ of relatively maximal, relative finite-energy metrics with a $d_{1}$ -type distance given by the Lelong number at zero of the collection of fiberwise Darvas $d_{1}$ -distances. We show that this metric structure is complete and geodesic. Seeing $X$ and $L$ as schemes $X_{K}$ , $L_{K}$ over the discretely-valued field $K = ℂ ((t))$ of complex Laurent series, we show that the space $ℰ^{1} (L_{K}^{an})$ of non-Archimedean finite-energy metrics over $L_{K}^{an}$ embeds isometrically and geodesically into ${\hat{ℰ}}^{1} (L)$ , and characterize its image. This generalizes previous work of Berman-Boucksom-Jonsson, treating the trivially-valued case.

Reçu le : 2023-03-13
Accepté le : 2023-03-22
Publié le : 2023-03-31

DOI : 10.5802/jep.229

Classification : 32U05, 32Q15, 14E99
Keywords: Berkovich spaces, complex manifolds, pluripotential theory, degenerations
Mot clés : Espaces de Berkovich, variétés complexes, théorie du pluripotentiel, dégénérescences

Affiliations des auteurs :

Rémi Reboulet ¹

¹ Department of mathematical sciences, Chalmers University of Technology 412 96 Gothenburg, Sweden

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Rémi Reboulet. The space of finite-energy metrics over a degeneration of complex manifolds. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 659-701. doi : 10.5802/jep.229. https://jep.centre-mersenne.org/articles/10.5802/jep.229/

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