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O. Calvo-Andrade
Positivity, vanishing theorems and rigidity of Codimension one Holomorphic Foliations
Annales de la faculté des sciences de Toulouse Sér. 6, 18 no. 4 (2009), p. 811-854, doi: 10.5802/afst.1225
Article PDF | Reviews MR 2590389 | Zbl 1189.32020

Résumé - Abstract

It is a known fact that the space of codimension one holomorphic foliations with singularities with a given ‘normal bundle’ has a natural structure of an algebraic variety. The aim of this paper is to consider the problem of the description of its irreducible components. To do this, we are interested in the problem of the existence of an integral factor of a twisted integrable differential 1–form defined on a projective manifold. We are going to do a geometrical analysis of the codimension one foliation associated to this form. The essential point of this paper consists in understanding the role played by a positive condition on some object associated to the foliation.

Bibliography

[1] Baum (P.), Bott (R.).— Singularities of Holomorphic Foliations, Journal on Differential Geometry 7, p. 279-342 (1972). Article |  MR 377923 |  Zbl 0268.57011
[2] Ballico (E.).— A splitting theorem for the Kupka component of a foliation of ${\mathbb{C}P}^n, n\ge 6$. Addendum to a paper by Calvo-Andrade and Soares, Ann. Inst. Fourier 45, p. 1119-1121 (1995). Cedram |  MR 1359842 |  Zbl 0831.58046
[3] Ballico (E.).— A splitting theorem for the Kupka component of a foliation of ${\mathbb{C}P}^n, n\ge 6$. Addendum to an addendum to a paper by Calvo-Andrade and Soares, Ann. Inst. Fourier 49, p. 1423-1425 (1999). Cedram |  MR 1703094 |  Zbl 0959.32037
[4] Brîzănescu (V.).— Holomorphic Vector bundles over Compact Complex Surfaces, Springer LNM 1624, (1996).  MR 1439504 |  Zbl 0848.32024
[5] Calvo (O.).— Persistência de folheações definidas por formas logaritmicas, Ph. D. Thesis, IMPA (1990).
[6] Calvo (O.).— Irreducible Components of the space of Foliations, Math. Ann. 299, p. 751-767 (1994).  MR 1286897 |  Zbl 0811.58006
[7] Calvo (O.).— Deformations of Holomorphic Foliations. Proceedings of the Workshop on Topology. Arraut, J. L., Hurder, S., Santos, N., Schweitzer, P. Eds. AMS, Contemporary Math. 161, p. 21-28 (1994).  MR 1271825
[8] Calvo (O.).— Deformations of Branched Lefschetz’s Pencils, Bull. of the Brazilian Math. Soc. 26 No. 1, p. 67-83 (1995).  MR 1339179 |  Zbl 0843.58001
[9] Calvo (O.).— Foliations with a Kupka Component on Algebraic Manifolds, Bull. of the Brazilian Math. Soc. 30 No. 2, p. 183-197 (1999).  MR 1703038 |  Zbl 1058.32023
[10] Calvo (O.), Cerveau (D.), Giraldo (L.), Lins (A.).— Irreducible components of the space of foliations associated with the affine Lie Algebra, Ergodic Theroy and Dynamical Systems 24, p. 987-1014 (2004).  MR 2085387 |  Zbl 1068.32022
[11] Calvo (O.), Giraldo (L.).— Algebraic Foliations and Actions of the Affine Group, to appear.
[12] Calvo (O.), Soares (M.).— Chern numbers of a Kupka component, Ann. Inst. Fourier 44, p. 1219-1236 (1994). Cedram |  MR 1306554 |  Zbl 0811.32024
[13] Camacho (C.), Lins (A.), Sad (P.).— Foliations with Algebraic Limit sets, Ann. of Math. 136, p. 429-446 (1992).  MR 1185124 |  Zbl 0769.57017
[14] Cerveau (D.), Déserti (J.).— Feuilletages et actions de groupes sur les espaces projectifs, Mémories de la Societé Mathématique de France No 103 (2005).  MR 2200857 |  Zbl 1107.37037
[15] Cerveau (D.), Lins (A.).— Codimension one Foliations in ${\mathbb{C}P}^n,\quad n\ge 3$, with Kupka components, in Complex Analytic Methods in Dynamical Systems. C. Camacho, A. Lins, R. Moussu, P. Sad. Eds. Astérisque 222, p. 93-133 (1994).  MR 1285387 |  Zbl 0823.32014
[16] Cerveau (D.), Lins (A.).— Irreducible components of the space of holomorphic foliations of degree two in ${\mathbb{C}P}^n,\quad n\ge 3$, Annals of Math. 143, p. 577-612 (1996).  MR 1394970 |  Zbl 0855.32015
[17] Cerveau (D.), Mattei (J. F.).— Formes intégrables holomorphes singulières, Astérisque 97 (1982).  MR 704017 |  Zbl 0545.32006
[18] Cukierman (F.), Pereira (J. V.).— Stability of foliations with split tangent sheaf, American Journal of Math. (to appear).
[19] Epstein (D.), Rosemberg (H.).— Stability of compact foliations, in Geometry and Topology, J. Palis, M. do Carmo, Springer LNM, 597, p. 151-160 (1977).  MR 501007 |  Zbl 0363.57018
[20] Gómez-Mont (X.).— Universal Families of foliations by curves, Astérisque 150-151, p. 109-129, (1987)  MR 923596 |  Zbl 0641.32014
[21] Gómez-Mont (X.), Lins (A.).— Structural Stability of foliations with a meromorphic first integral, Topology 30, p. 315-334 (1991).  MR 1113681 |  Zbl 0735.57014
[22] Gómez-Mont (X.), Muciño (J.).— Persistent cycles for foliations having a meromorphic first integral in Holomorphic Dynamics Gómez-Mont, Seade, Verjovsky Eds. Spriger LNM 1345, p. 129-162 (1987).  MR 980957 |  Zbl 0681.58032
[23] Griffiths (Ph.).— Hermitian Differential Geometry, Chern Classes and Positive Vector Bundles in Global Analysis, papers in honor of K. Kodaira. Tokyo, Princenton: University of Tokyo Press, Princenton University Press p. 185-251 (1969).  MR 258070 |  Zbl 0201.24001
[24] Griffiths (Ph.), Harris (J.).— Principles of Algebraic Geometry, Pure & Appl. Math. Wiley Interscience (1978).  MR 507725 |  Zbl 0408.14001
[25] Horrocks (G.), Mumford (D.).— A rank-2 bundle with 15000 symetries, Topology 12, p. 63-81 (1973).  MR 382279 |  Zbl 0255.14017
[26] Kobayashi (S.).— Differential Geometry of Complex Vector Bundles, Kano Memorial Lectures 5, Princeton University Press (1987).  MR 909698 |  Zbl 0708.53002
[27] Langevin (R.), Rosenberg (H.).— On stability of compact leaves and fibrations, Topology 16, p. 107-111 (1977).  MR 461523 |  Zbl 0346.57009
[28] Malgrange (B.).— Frobenius avec singularités, 1. Codimension 1, Publ. Math. IHES 46, p. 163-173 (1976). Numdam |  MR 508169 |  Zbl 0355.32013
[29] Medeiros (A.).— Structural stability of integrable differential forms. in Geormetry and Topology J. Palis, M. do Carmo Eds. Springer LNM 597, p. 395-428 (1977).  MR 451274 |  Zbl 0363.58007
[30] Muciño-Raymundo (J.).— Deformations of holomorphic foliations having a meromorphic first integral, J. für die reine und angewandte Mathematik 461, p. 189-219, 1995.  MR 1324214 |  Zbl 0816.32022
[31] Nori (M. V.).— Zariski’s conjecture and related topics, Ann. Sci. Ec. Norm. Sup. 16, p. 305-344 (1983). Numdam |  MR 732347 |  Zbl 0527.14016
[32] Okonek (Ch.), Schneider (M.), Spindler (H.).— Vector Bundles on Complex Projective spaces, Progress in Math. 3 Birkhauser (1978).  MR 561910 |  Zbl 0438.32016
[33] Paul (E.).— Etude topologique des formes logarithmiques fermées, Invent. Math. 95, p. 395-420 (1989).  MR 974909 |  Zbl 0641.57013
[34] Serre (J. P.).— Un théorème de dualité, Coment. Mat. Helv. 29, p. 9-26 (1955).  MR 67489 |  Zbl 0067.16101
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