A structural theorem for codimension-one foliations on n , n3, with an application to degree-three foliations
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 1, pp. 1-41.

Let be a codimension-one foliation on n : for each point p n we define 𝒥(,p) as the order of the first non-zero jet j p k (ω) of a holomorphic 1-form ω defining at p. The singular set of is sing()={p n |𝒥(,p)1}. We prove (main Theorem 1.2) that a foliation satisfying 𝒥(,p)1 for all p n has a non-constant rational first integral. Using this fact we are able to prove that any foliation of degree-three on n , with n3, is either the pull-back of a foliation on 2 , or has a transverse affine structure with poles. This extends previous results for foliations of degree at most two.

Publié le :
Classification : 37F75, 34M45
Cerveau, Dominique 1 ; Lins Neto, Alcides 2

1 Institut Universitaire de France & IRMAR Campus de Beaulieu 35042, Rennes Cedex, France
2 Instituto de Matemática Pura e Aplicada Estrada Dona Castorina 110 Rio de Janeiro, Brasil
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Cerveau, Dominique; Lins Neto, Alcides. A structural theorem for codimension-one foliations on $\protect \mathbb{P}^n$, $n\ge 3$, with an application to degree-three foliations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 1, pp. 1-41. http://www.numdam.org/item/ASNSP_2013_5_12_1_1_0/

[1] W. Barth, Fortsetzung, meromorpher Funktionen in Tori und komplex-projektiven Räumen, Invent. Math. 5 (1968), 42–62. | EuDML | MR | Zbl

[2] M. Brunella, “Birational Geometry of Foliations”, Publicações Matemáticas do IMPA. Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2004. | MR | Zbl

[3] M. Brunella, Sur les feuilletages de l’espace projectif ayant une composante de Kupka, Enseign. Math. 55 (2009), 1–8. | MR | Zbl

[4] G. Casale, Suites de Godbillon-Vey et intégrales premières, C. R. Math. Acad. Sci. Paris 335 (2002), 1003–1006. | MR | Zbl

[5] O. Calvo-Andrade, Foliations with a Kupka component on algebraic manifolds, Bol. Soc. Brasil. Mat. (N.S.) 30 (1999), 183–197. | MR | Zbl

[6] O. Calvo-Andrade, Foliations with a radial Kupka set on projective spaces, (2008), preprint. | MR

[7] C. Camacho, A. Lins Neto and P. Sad, Topological invariants and equidesingularization for holomorphic vector fields, J. Differential Geom. 20 (1984), 143–174. | MR | Zbl

[8] C. Camacho, A. Lins Neto and P. Sad, Foliations with algebraic limit sets, Ann. of Math. 136 (1992), 429–446. | MR | Zbl

[9] C. Camacho and P. Sad, Invariant varieties through singularities of holomorphic vector fields, Ann. of Math. 115 (1982), 579–595. | MR | Zbl

[10] D. Cerveau and A. Lins Neto, Irreducible components of the space of holomorphic foliations of degree two in P(n), n3, Ann. of Math. (2) 143 (1996), 577–612. | MR | Zbl

[11] D. Cerveau and A. Lins Neto, Codimension-one foliations in P(n), n3, with Kupka components, Astérisque 222 (1994), 93–132. | MR | Zbl

[12] D. Cerveau, A. Lins-Neto, F. Loray, J. V. Pereira and F. Touzet, Complex codimension-one singular foliations and Godbillon-Vey sequences, Mosc. Math. J. 7 (2007), 21–54. | MR | Zbl

[13] D. Cerveau and J.-F. Mattei, “Formes intégrables holomorphes singulières”, Astérisque, Vol. 97, Société Mathématique de France, Paris, 1982. | Numdam | MR | Zbl

[14] C. Godbillon, “Feuilletages. Études géométriques. With a preface by G. Reeb”, Progress in Mathematics, Vol. 98, Birkhäuser Verlag, Basel, 1991. | MR | Zbl

[15] I. Kupka, The singularities of integrable structurally stable Pfaffian forms, Proc. Nat. Acad. Sci. USA 52 (1964), 1431–1432. | MR | Zbl

[16] A. Lins Neto, A note on projective Levi flats and minimal sets of algebraic foliations, Ann. Inst. Fourier (Grenoble) 49 (1999), 1369–1385. | EuDML | Numdam | MR | Zbl

[17] F. Loray, A preparation theorem for codimension-one foliations, Ann. of Math. (2) 163 (2006), 709–722. | MR | Zbl

[18] J. Martinet and J.-P. Ramis, Problème de modules pour des équations différentielles non lineaires du premier ordre, Publ. Math. Inst. Hautes Étud. Sci. 55 (1982), 63–124. | EuDML | Numdam | MR | Zbl

[19] J.-F. Mattei, Modules de feuilletages holomorphes singuliers. I. Équisingularité, Invent. Math. 103 (1991), 297–325. | EuDML | MR | Zbl

[20] J.-F. Mattei and R. Moussu, Holonomie et intégrales premires, Ann. Sci. École Norm. Sup. 13 (1980), 469–523. | EuDML | Numdam | MR | Zbl

[21] R. Meziani, Classification analytique d’équations différentielles ydy+=0 et espace de modules, Bol. Soc. Brasil. Mat. (N.S.) 27 (1996), 23–53. | MR | Zbl

[22] H. Rossi, Continuation of subvarieties of projective varieties, Amer. J. Math. 91 (1969), 565–575. | MR | Zbl

[23] B. A. Scárdua, Transversely affine and transversely projective holomorphic foliations, Ann. Sci. École Norm. Sup. 30 (1997), 169–204. | EuDML | Numdam | MR | Zbl