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


[1] V. Arnold : Chapitres supplémentaires de la théorie des équations différentielles ordinaires . Moscou: Editions Mir (1980) traduction française. · Zbl 0455.34001
[2] C. Borcea : Some remarks on deformations of Hopf manifolds . Rev. Roum. Math. pures et appl. 26 (1981) 1287-1294. · Zbl 0543.32010
[3] N. Brouchlinskaia : Finiteness theorem for familities of vector fields in the neighbourhood of a Poincaré-type singularity . Funct. Anal. 5 (1971), english translation 177-181. · Zbl 0243.34008 · doi:10.1007/BF01078121
[4] A. Douady : Séminaire H. Cartan , exp. 3 (1960 -1961). |
[5] T. Duchamp and M. Kalka , Deformation theory for holomorphic foliations . J. Diff. Geom. 14 (1979) 317-337. · Zbl 0451.57015 · doi:10.4310/jdg/1214435099
[6] T. Duchamp and M. Kalka : Holomorphic foliations and deformations of the Hopf foliation , preprint. · Zbl 0501.57010 · doi:10.2140/pjm.1984.112.69
[7] J. Girbau , A. Haefliger and D. Sundararaman : On deformations of transversely holomorphic foliations , J. für die reine und ang. Math. 345 (1983) 122-147. · Zbl 0538.32015 · doi:10.1515/crll.1983.345.122
[8] K. Kodaira and D.C. Spencer : On deformation of complex analytic structures, I, II, III . Ann. of Math. 67 (1958) 328-466 and 71 (1960) 43-76. · Zbl 0128.16902 · doi:10.2307/1969879
[9] J. Wehler : Versal deformation of Hopf surfaces. J. für die reine und ang. Math. 328 (1981) 22-32. · Zbl 0459.32009 · doi:10.1515/crll.1981.328.22
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