Deformations of transversely holomorphic flows on spheres and deformations of Hopf manifolds. (English) Zbl 0582.32026

Let \(a=(a_ 1,...,a_ n)\) be a sequence of complex numbers such that Re \(a_ i<0\) for all i. The set \({\mathfrak g}_ a\) of all holomorphic vector fields X on \({\mathbb{C}}^ n\) commuting with \(\sum a_ i\partial /\partial z_ i\) is a Lie subalgebra of the Lie algebra of holomorphic vector fields on \({\mathbb{C}}^ n\); \(X\in {\mathfrak g}_ a\) are called a-resonant. A vector field \(X\in {\mathfrak g}_ a\) generates a global holomorphic flow on \({\mathbb{C}}^ n\) and induces a holomorphic foliation \(F_ X\) on \({\mathbb{C}}^ n\setminus \{0\}\). If X is sufficiently near its linear part, the leaves of \(F_ X\) are transversal to the unit sphere \(S^{2n-1}\) in \({\mathbb{C}}^ n\) and hence their intersections with \(S^{2n-1}\) define a transversely holomorphic foliation \(F^ 0_ X\) on \(S^{2n-1}.\)
The author proves the following theorem: Let U be a small neighborhood of 0 in a vector subspace of \({\mathfrak g}_ a\) complementary to the vector subspace spanned by [X,\({\mathfrak g}_ a]\) and X. Then, the family \(\{F^ 0_{X+Y}| Y\in U\}\) is a versal deformation of \(F^ 0_ X.\)
The author also obtains a versal deformation of the Hopf manifold in the appendix.
Reviewer: A.Morimoto


32G07 Deformations of special (e.g., CR) structures
57R30 Foliations in differential topology; geometric theory
32G05 Deformations of complex structures
Full Text: Numdam EuDML


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