# Foliated structure of the Kuranishi space and isomorphisms of deformation families of compact complex manifolds

Annales scientifiques de l'École Normale Supérieure (2011)

- Volume: 44, Issue: 3, page 495-525
- ISSN: 0012-9593

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topMeersseman, Laurent. "Foliated structure of the Kuranishi space and isomorphisms of deformation families of compact complex manifolds." Annales scientifiques de l'École Normale Supérieure 44.3 (2011): 495-525. <http://eudml.org/doc/272230>.

@article{Meersseman2011,

abstract = {Consider the following uniformization problem. Take two holomorphic (parametrized by some analytic set defined on a neighborhood of $0$ in $\mathbb \{C\}^p$, for some $p>0$) or differentiable (parametrized by an open neighborhood of $0$ in $\mathbb \{R\}^p$, 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.},

author = {Meersseman, Laurent},

journal = {Annales scientifiques de l'École Normale Supérieure},

keywords = {deformations of complex manifolds; foliations; uniformization},

language = {eng},

number = {3},

pages = {495-525},

publisher = {Société mathématique de France},

title = {Foliated structure of the Kuranishi space and isomorphisms of deformation families of compact complex manifolds},

url = {http://eudml.org/doc/272230},

volume = {44},

year = {2011},

}

TY - JOUR

AU - Meersseman, Laurent

TI - Foliated structure of the Kuranishi space and isomorphisms of deformation families of compact complex manifolds

JO - Annales scientifiques de l'École Normale Supérieure

PY - 2011

PB - Société mathématique de France

VL - 44

IS - 3

SP - 495

EP - 525

AB - Consider the following uniformization problem. Take two holomorphic (parametrized by some analytic set defined on a neighborhood of $0$ in $\mathbb {C}^p$, for some $p>0$) or differentiable (parametrized by an open neighborhood of $0$ in $\mathbb {R}^p$, 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.

LA - eng

KW - deformations of complex manifolds; foliations; uniformization

UR - http://eudml.org/doc/272230

ER -

## References

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