Tropical and non-Archimedean limits of degenerating families of volume forms

Boucksom, Sébastien; Jonsson, Mattias

doi:10.5802/jep.39

Tropical and non-Archimedean limits of degenerating families of volume forms
[Limites tropicales et non archimédiennes de familles de formes volumes qui dégénèrent]

Boucksom, Sébastien ¹ ; Jonsson, Mattias ²

¹ CMLS, École polytechnique, CNRS, Université Paris-Saclay 91128 Palaiseau Cedex, France
² Department of Mathematics, University of Michigan Ann Arbor, MI 48109-1043, USA and Mathematical Sciences, Chalmers University of Technology and University of Gothenburg SE-412 96 Göteborg, Sweden

Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 87-139.

Résumé
Abstract

Nous étudions le comportement asymptotique de formes volumes dans une famille de variétés complexes compactes qui dégénèrent. Sous des conditions assez générales, nous montrons que les formes volumes convergent en un sens naturel vers une mesure du type de Lebesgue sur un certain complexe simplicial. Ceci fournit en particulier une version en théorie de la mesure d’une conjecture de Kontsevich–Soibelman et Gross–Wilson portant sur les dégénérescences maximales de variétés de Calabi-Yau.

We study the asymptotic behavior of volume forms on a degenerating family of compact complex manifolds. Under rather general conditions, we prove that the volume forms converge in a natural sense to a Lebesgue-type measure on a certain simplicial complex. In particular, this provides a measure-theoretic version of a conjecture by Kontsevich–Soibelman and Gross–Wilson, bearing on maximal degenerations of Calabi–Yau manifolds.

Reçu le : 2016-06-10
Accepté le : 2016-11-27
Publié le : 2017-01-10

MR Zbl | 7 citations dans Numdam

DOI : 10.5802/jep.39

Classification : 32Q25, 14J32, 14T05, 53C23, 32P05, 14G22
Keywords: Calabi-Yau manifolds, volume forms, degenerations, Berkovich spaces
Mot clés : Variétés de Calabi-Yau, formes volumes, dégénérescences, espaces de Berkovich

Affiliations des auteurs :

Boucksom, Sébastien ¹ ; Jonsson, Mattias ²

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     title = {Tropical and {non-Archimedean} limits of degenerating families of volume forms},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {87--139},
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Boucksom, Sébastien; Jonsson, Mattias. Tropical and non-Archimedean limits of degenerating families of volume forms. Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 87-139. doi : 10.5802/jep.39. http://www.numdam.org/articles/10.5802/jep.39/

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