Tropical and non-Archimedean limits of degenerating families of volume forms
Journal de l’École polytechnique - Mathématiques, Volume 4  (2017), p. 87-139

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.

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.

Received : 2016-06-10
Accepted : 2016-11-27
Published online : 2017-01-10
DOI : https://doi.org/10.5802/jep.39
Classification:  32Q25,  14J32,  14T05,  53C23,  32P05,  14G22
Keywords: Calabi-Yau manifolds, volume forms, degenerations, Berkovich spaces
@article{JEP_2017__4__87_0,
     author = {Boucksom, S\'ebastien and Jonsson, Mattias},
     title = {Tropical and non-Archimedean limits of degenerating families of volume forms},
     journal = {Journal de l'\'Ecole polytechnique - Math\'ematiques},
     publisher = {Ecole polytechnique},
     volume = {4},
     year = {2017},
     pages = {87-139},
     doi = {10.5802/jep.39},
     mrnumber = {3611100},
     zbl = {1401.32019},
     language = {en},
     url = {http://www.numdam.org/item/JEP_2017__4__87_0}
}
Boucksom, Sébastien; Jonsson, Mattias. Tropical and non-Archimedean limits of degenerating families of volume forms. Journal de l’École polytechnique - Mathématiques, Volume 4 (2017) , pp. 87-139. doi : 10.5802/jep.39. http://www.numdam.org/item/JEP_2017__4__87_0/

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