Foliated structure of the Kuranishi space and isomorphisms of deformation families of compact complex manifolds
[Structure feuilletée de l'espace de Kuranishi et isomorphismes de familles de déformations de variétés compactes complexes]
Meersseman, Laurent
Annales scientifiques de l'École Normale Supérieure, Tome 44 (2011), p. 495-525 / Harvested from Numdam
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Considérons le problème d’uniformisation suivant. Prenons deux familles de déformation holomorphes (paramétrées par un ensemble analytique défini dans un voisinage de 0 dans ℂ p pour p>0) ou différentiables (paramétrées par un voisinage de 0 dans ℝ p pour p>0) de variétés compactes complexes. Supposons-les ponctuellement isomorphes, c’est-à-dire que, pour tout point t de l’espace des paramètres, la fibre en t de la première famille est biholomorphe à la fibre en t de la deuxième famille. Sous quelle(s) condition(s) les deux familles sont-elles localement isomorphes en 0 ? Dans cet article, nous donnons une condition suffisante dans le cas de familles holomorphes. Nous montrons ensuite que, de façon surprenante, la condition n’est pas suffisante dans le cas des familles différentiables. Nous décrivons également plusieurs types de contre-exemples et donnons quelques éléments de classifications de ces contre-exemples. Ces résultats reposent sur une étude géométrique de l’espace de Kuranishi d’une variété compacte complexe.

Consider the following uniformization problem. Take two holomorphic (parametrized by some analytic set defined on a neighborhood of 0 in ℂ p , for some p>0) or differentiable (parametrized by an open neighborhood of 0 in ℝ p , for some p>0) deformation families of compact complex manifolds. Assume they are pointwise isomorphic, that is for each point t of the parameter space, the fiber over t of the first family is biholomorphic to the fiber over t of the second family. Then, under which conditions are the two families locally isomorphic at 0? In this article, we give a sufficient condition in the case of holomorphic families. We show then that, surprisingly, this condition is not sufficient in the case of differentiable families. We also describe different types of counterexamples and give some elements of classification of the counterexamples. These results rely on a geometric study of the Kuranishi space of a compact complex manifold.

Publié le : 2011-01-01
DOI : https://doi.org/10.24033/asens.2148
Classification:  32G07,  57R30
Mots clés: déformations de variétés complexes, feuilletages, uniformisation 
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     author = {Meersseman, Laurent},
     title = {Foliated structure of the Kuranishi space and isomorphisms of deformation families of compact complex manifolds},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     volume = {44},
     year = {2011},
     pages = {495-525},
     doi = {10.24033/asens.2148},
     mrnumber = {2839457},
     zbl = {1239.32012},
     language = {en},
     url = {http://dml.mathdoc.fr/item/ASENS_2011_4_44_3_495_0}
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Meersseman, Laurent. Foliated structure of the Kuranishi space and isomorphisms of deformation families of compact complex manifolds. Annales scientifiques de l'École Normale Supérieure, Tome 44 (2011) pp. 495-525. doi : 10.24033/asens.2148. http://gdmltest.u-ga.fr/item/ASENS_2011_4_44_3_495_0/

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