## 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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### References:

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