Non-Archimedean Green’s functions and Zariski decompositions

Sébastien Boucksom; Mattias Jonsson

doi:10.5802/crmath.579

Article de recherche - Géométrie algébrique

Non-Archimedean Green’s functions and Zariski decompositions
[Fonctions de Green non-archimédiennes et décompositions de Zariski]

Sébastien Boucksom ¹ ; Mattias Jonsson ²

¹ Sorbonne Université and Université Paris Cité, CNRS, IMJ-PRG, F-75005 Paris, France
² Dept of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA

Comptes Rendus. Mathématique, Volume 362 (2024) no. S1, pp. 5-42.

Résumé
Abstract

Nous étudions l’équation de Monge–Ampère non-archimédienne sur une variété projective lisse sur un corps de valuation discrète ou triviale. Tout d’abord, nous donnons un exemple de fonction de Green, associée à une valuation divisorielle, qui n’est pas $ℚ$ -PL (i.e. pas une fonction modèle, dans le cas de valuation discrète). Ensuite, nous produisons un exemple de fonction dont la mesure de Monge–Ampère est à support dans un complexe dual, mais qui n’est invariante par la rétraction associée à aucun modele snc. Ceci répond négativement à une question de Burgos Gil et al. Nos exemples sont basés sur des constructions géométriques de Cutkosky et Lesieutre, et sont produits par changement de base à partir de fonctions de Green sur un corps trivialement valué ; cette théorie nous permet d’encoder de façon efficace la décomposition de Zariski de toute classe pseudo-effective.

We study the non-Archimedean Monge–Ampère equation on a smooth projective variety over a discretely or trivially valued field. First, we give an example of a Green’s function, associated to a divisorial valuation, which is not $ℚ$ -PL (i.e. not a model function in the discretely valued case). Second, we produce an example of a function whose Monge–Ampère measure is a finite atomic measure supported in a dual complex, but which is not invariant under the retraction associated to any snc model. This answers a question by Burgos Gil et al. in the negative. Our examples are based on geometric constructions by Cutkosky and Lesieutre, and arise via base change from Green’s functions over a trivially valued field; this theory allows us to efficiently encode the Zariski decomposition of a pseudoeffective numerical class.

Reçu le : 2023-03-31
Accepté le : 2023-09-28
Publié le : 2024-06-06

DOI : 10.5802/crmath.579

Affiliations des auteurs :

Sébastien Boucksom ¹ ; Mattias Jonsson ²

¹ Sorbonne Université and Université Paris Cité, CNRS, IMJ-PRG, F-75005 Paris, France
² Dept of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {S\'ebastien Boucksom and Mattias Jonsson},
     title = {Non-Archimedean {Green{\textquoteright}s} functions and {Zariski} decompositions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {5--42},
     publisher = {Acad\'emie des sciences, Paris},
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     number = {S1},
     year = {2024},
     doi = {10.5802/crmath.579},
     language = {en},
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Sébastien Boucksom; Mattias Jonsson. Non-Archimedean Green’s functions and Zariski decompositions. Comptes Rendus. Mathématique, Volume 362 (2024) no. S1, pp. 5-42. doi : 10.5802/crmath.579. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.579/

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