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

Reboulet, Rémi

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

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 ${\widehat{ℰ}}^1\left(L\right)$ 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 $X_{\mathrm{K}}$, $L_{\mathrm{K}}$ over the discretely-valued field $\mathrm{K}=ℂ\left(\right(t\left)\right)$ of complex Laurent series, we show that the space ${ℰ}^1\left(L_{\mathrm{K}}^{\mathrm{an}}\right)$ of non-Archimedean finite-energy metrics over $L_{\mathrm{K}}^{\mathrm{an}}$ embeds isometrically and geodesically into ${\widehat{ℰ}}^1\left(L\right)$, and characterize its image. This generalizes previous work of Berman-Boucksom-Jonsson, treating the trivially-valued case.

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 ${\widehat{ℰ}}^1\left(L\right)$ 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 $X_{\mathrm{K}}$, $L_{\mathrm{K}}$ sur le champ discrètement valué $\mathrm{K}=ℂ\left(\right(t\left)\right)$ des séries de Laurent complexes, nous montrons que l’espace ${ℰ}^1\left(L_{\mathrm{K}}^{\mathrm{an}}\right)$ des métriques non archimédiennes d’énergie finie sur $L_{\mathrm{K}}^{\mathrm{an}}$ s’immerge isométriquement et géodésiquement dans ${\widehat{ℰ}}^1\left(L\right)$, et nous caractérisons son image. Ceci généralise un travail précédent de Berman-Boucksom-Jonsson, traitant le cas trivialement valué.

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

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