Given a degeneration of projective complex manifolds with meromorphic singularities, and a relatively ample line bundle on , we study spaces of plurisubharmonic metrics on , with particular focus on (relative) finite-energy conditions. We endow the space of relatively maximal, relative finite-energy metrics with a -type distance given by the Lelong number at zero of the collection of fiberwise Darvas -distances. We show that this metric structure is complete and geodesic. Seeing and as schemes , over the discretely-valued field of complex Laurent series, we show that the space of non-Archimedean finite-energy metrics over embeds isometrically and geodesically into , 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 avec des singularités méromorphes, et un fibré en droites relativement ample sur , nous étudions des espaces de métriques plurisousharmoniques sur , avec une attention particulière aux conditions (relatives) d’énergie finie. Nous munissons l’espace des métriques relativement maximales d’énergie finie d’une distance de type donnée par le nombre de Lelong en de la famille des distances de Darvas fibre à fibre. Nous montrons que cette structure métrique est complète et géodésique. En considérant et comme des schémas , sur le champ discrètement valué des séries de Laurent complexes, nous montrons que l’espace des métriques non archimédiennes d’énergie finie sur s’immerge isométriquement et géodésiquement dans , et nous caractérisons son image. Ceci généralise un travail précédent de Berman-Boucksom-Jonsson, traitant le cas trivialement valué.
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Keywords: Berkovich spaces, complex manifolds, pluripotential theory, degenerations
Mot clés : Espaces de Berkovich, variétés complexes, théorie du pluripotentiel, dégénérescences
Rémi Reboulet 1
@article{JEP_2023__10__659_0, author = {R\'emi Reboulet}, title = {The space of finite-energy metrics over a degeneration of complex manifolds}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {659--701}, publisher = {\'Ecole polytechnique}, volume = {10}, year = {2023}, doi = {10.5802/jep.229}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.229/} }
TY - JOUR AU - Rémi Reboulet TI - The space of finite-energy metrics over a degeneration of complex manifolds JO - Journal de l’École polytechnique — Mathématiques PY - 2023 SP - 659 EP - 701 VL - 10 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.229/ DO - 10.5802/jep.229 LA - en ID - JEP_2023__10__659_0 ER -
%0 Journal Article %A Rémi Reboulet %T The space of finite-energy metrics over a degeneration of complex manifolds %J Journal de l’École polytechnique — Mathématiques %D 2023 %P 659-701 %V 10 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.229/ %R 10.5802/jep.229 %G en %F JEP_2023__10__659_0
Rémi Reboulet. The space of finite-energy metrics over a degeneration of complex manifolds. Journal de l’École polytechnique — Mathématiques, Volume 10 (2023), pp. 659-701. doi : 10.5802/jep.229. https://jep.centre-mersenne.org/articles/10.5802/jep.229/
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