On deformations of holomorphic foliations

Abstract

Given a non-singular holomorphic foliation ℱ on a compact manifold M we analyze the relationship between the versal spaces K and K tr of deformations of ℱ as a holomorphic foliation and as a transversely holomorphic foliation respectively. With this purpose, we prove the existence of a versal unfolding of ℱ parametrized by an analytic space K f isomorphic to π -1 (0)×Σ where Σ is smooth and π : K→K tr is the forgetful map. The map π is shown to be an epimorphism in two situations: (i) if H 2 (M,Θ ℱ f )=0, where Θ ℱ f is the sheaf of germs of holomorphic vector fields tangent to ℱ, and (ii) if there exists a holomorphic foliation ℱ ⋔ transverse and supplementary to ℱ. When the conditions (i) and (ii) are both fulfilled then K≅K f ×K tr .