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.
Accepted:
Published online:
DOI: 10.5802/jep.39
Keywords: Calabi-Yau manifolds, volume forms, degenerations, Berkovich spaces
Mot clés : Variétés de Calabi-Yau, formes volumes, dégénérescences, espaces de Berkovich
@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{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {87--139}, publisher = {Ecole polytechnique}, volume = {4}, year = {2017}, doi = {10.5802/jep.39}, mrnumber = {3611100}, zbl = {1401.32019}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jep.39/} }
TY - JOUR AU - Boucksom, Sébastien AU - Jonsson, Mattias TI - Tropical and non-Archimedean limits of degenerating families of volume forms JO - Journal de l’École polytechnique — Mathématiques PY - 2017 SP - 87 EP - 139 VL - 4 PB - Ecole polytechnique UR - http://www.numdam.org/articles/10.5802/jep.39/ DO - 10.5802/jep.39 LA - en ID - JEP_2017__4__87_0 ER -
%0 Journal Article %A Boucksom, Sébastien %A Jonsson, Mattias %T Tropical and non-Archimedean limits of degenerating families of volume forms %J Journal de l’École polytechnique — Mathématiques %D 2017 %P 87-139 %V 4 %I Ecole polytechnique %U http://www.numdam.org/articles/10.5802/jep.39/ %R 10.5802/jep.39 %G en %F 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/articles/10.5802/jep.39/
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