# Foliated Structure of The Kuranishi Space and Isomorphisms of Deformation Families of Compact Complex Manifolds

Abstract : Consider the following uniformization problem. Take two holomorphic (parametrized by some analytic set defined on a neighborhood of $0$ in $\Bbb C^p$, for some $p>0$) or differentiable (parametrized by an open neighborhood of $0$ in $\Bbb R^p$, 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.
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Journal articles

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https://hal.archives-ouvertes.fr/hal-00403954
Submitted on : Friday, December 29, 2017 - 11:47:02 AM
Last modification on : Thursday, October 11, 2018 - 1:46:04 PM

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### Identifiers

• HAL Id : hal-00403954, version 2
• ARXIV : 0907.2511

### Citation

Laurent Meersseman. Foliated Structure of The Kuranishi Space and Isomorphisms of Deformation Families of Compact Complex Manifolds. Annales Scientifiques de l'École Normale Supérieure, Elsevier Masson, 2011, 44 (3), pp.495-525. ⟨hal-00403954v2⟩

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