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On deformation of holomorphic foliations. (English) Zbl 0659.32019

Given a non-singular holomorphic foliation \({\mathcal F}\) on a compact manifold M we analyze the relationship between the versal spaces K and \(K^{tr}\) of deformations of \({\mathcal F}\) as a holomorphic foliation and as a transversely holomorphic foliation respectively. With this purpose, we prove the existence of a versal unfolding of \({\mathcal F}\) parametrized by an analytic space \(K^ f\) isomorphic to \(\pi^{-1}(0)\times \Sigma\) where \(\Sigma\) is smooth and \(\pi\) : \(K\to K^{tr}\) is the forgetful map. The map \(\pi\) is shown to be an epimorphism in two situations: (i) if \(H^ 2(M,\Theta^ f_{{\mathcal F}})=0\), where \(\Theta^ f_{{\mathcal F}}\) is the sheaf of germs of holomorphic vector fields tangent to \({\mathcal F}\), and (ii) if there exists a holomorphic foliation \({\mathcal F}^ t\) transverse and supplementary to \({\mathcal F}\). When the conditions (i) and (ii) are both fulfilled then \(K\cong K^ f\times K^{tr}\).
Reviewer: J.Girbau

MSC:

32G05 Deformations of complex structures
57R30 Foliations in differential topology; geometric theory
32J99 Compact analytic spaces
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