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Girbau, Joan; Nicolau, Marcel
On deformations of holomorphic foliations. Annales de l'institut Fourier, 39 no. 2 (1989), p. 417-449
Full text djvu | pdf | Reviews MR 91b:32021 | Zbl 0659.32019

stable URL: http://www.numdam.org/item?id=AIF_1989__39_2_417_0

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Abstract

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

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