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


32G05 Deformations of complex structures
57R30 Foliations in differential topology; geometric theory
32J99 Compact analytic spaces
Full Text: DOI Numdam EuDML


[1] M. ARTIN, On the solutions of analytic equations, Inventiones Math., 5 (1968), 277-291. · Zbl 0172.05301
[2] J.-L. CATHELINEAU, Déformations équivariantes d’espaces analytiques complexes compacts, Ann. Sc. Ec. Norm. Sup., 11 (1978), 391-406. · Zbl 0401.32010
[3] A. DOUADY, Déformations régulières, Sém. M. Cartan, 13e année, (1960-1961), n° 3. · Zbl 0156.42802
[4] A. DOUADY, Le problème des modules pour LES variétés analytiques complexes, Sém. Bourbaki, 17e année (1964-1965), n° 277. · Zbl 0191.38002
[5] A. DOUADY, Le problème des modules pour LES sous-espaces analytiques compactes d’un espace analytique donné, Ann. Inst. Fourier, Grenoble, 16-1 (1966), 1-95. · Zbl 0146.31103
[6] A. DOUADY, Le problème des modules locaux pour LES espaces ℂ-analytiques compacts, Ann. Sc. Ec. Norm. Sup., 7 (1974), 569-602. · Zbl 0313.32036
[7] T. DUCHAMP, M. KALKA, Deformation theory for holomorphic foliations, J. Diff. Geom., 14 (1979), 317-337. · Zbl 0451.57015
[8] T. DUCHAMP, M. KALKA, Holomorphic foliations and deformations of the Hopf foliation, Pacific J. of Math., 112 (1984), 69-81. · Zbl 0501.57010
[9] J. FRENKEL, Cohomologie non abélienne et espaces fibrés, Bull. Soc. Math. de France, 85 (1957), 135-220. · Zbl 0082.37702
[10] J. GIRBAU, A. HAEFLIGER, D. SUNDARARAMAN, On deformations of transversely holomorphic foliations, Journal für die reine and angew. Math., 345 (1983), 122-147. · Zbl 0538.32015
[11] J. GIRBAU, M. NICOLAU, Deformations of holomorphic foliations and transversely holomorphic foliations, Research Notes in Math., 131 (1985), 162-173. · Zbl 0648.57013
[12] R. GODEMENT, Topologie algébrique et théorie des faisceaux, Hermann, Paris, 1964.
[13] X. GOMEZ-MONT, Transverse deformations of holomorphic foliations, Contemporary Math., 58, part I, 127-139. · Zbl 0607.32014
[14] A. HAEFLIGER, Deformations of transversely holomorphic flows on spheres and deformations of Hopf manifolds, Compositio Math., 55 (1985), 241-251. · Zbl 0582.32026
[15] G. R. KEMPF, Deformations of symmetric products. Proceedings of the 1978 Stony Brook Conference, Ann. of Math. Studies, 97 (1981), 319-342. · Zbl 0465.14013
[16] K. KODAIRA, D. C. SPENCER, On deformation of complex analytic structures. I and II, Ann. of Math., 67 (1958), 328-466. · Zbl 0128.16901
[17] K. KODAIRA, D. C. SPENCER, Multifoliate structures, Ann. of Math., 74 (1961), 52-100. · Zbl 0123.16401
[18] M. KURANISHI, Deformations of compact complex manifolds, Montreal, 1971. · Zbl 0382.32014
[19] B. MALGRANGE, Analytic spaces. L’enseignement Math., t. XIV (1968), 1-28. · Zbl 0165.40501
[20] J. MORROW, K. KODAIRA, Complex manifolds, New York, 1971. · Zbl 0325.32001
[21] L. NIRENBERG, A complex Frobenius theorem, Seminar of Analytic Functions, Inst. for Advanced Study, Princeton, (1957), 172-189. · Zbl 0099.37502
[22] B. SHIFFMAN, A. J. SOMMESSE, Vanishing theorems on complex manifolds, Progress in Math., Birkhäuser, Vol. 56 (1985). · Zbl 0578.32055
[23] J. J. WAVRIK, Obstructions to the existence of a space of moduli. Papers in Honour of K. Kodaira, Princeton Univ. Press, (1969), 403-414. · Zbl 0191.38003
[24] J. J. WAVRIK, A theorem of completenes for families of compact analytic spaces, Trans. of the A.M.S., 163 (1972), 147-155. · Zbl 0205.38803
[25] J. WEHLER, Versal deformations of Hopf surfaces, Journal für die reine and angew. Math., 345 (1987), 122-147.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.