Volume 4 (2021)
Lins-Neto, Alcides
Local transversely product singularities
Annales Henri Lebesgue, Volume 4 (2021), pp. 485-502.

Metadata

DOI10.5802/ahl.78
Keywords foliation, locally product

Abstract

In the main result of this paper we prove that a codimension one foliation of ℙ n , which is locally a product near every point of some codimension two component of the singular set, has a Kupka component. In particular, we obtain a generalization of a known result of Calvo Andrade and Brunella about foliations with a Kupka component.


References

[Bru00] Brunella, Marco Birational geometry of foliations, Monografías de Matemática, IMPA – Instituto de Matemática Pura e Aplicada, 2000 | Zbl

[Bru09] Brunella, Marco Sur les feuilletages de l’espace projectif ayant une composante de Kupka, Enseign. Math., Volume 55 (2009) no. 3-4, pp. 227-234 | DOI | MR | Zbl

[CA99] Calvo-Andrade, Omegar Foliations with a Kupka component on algebraic manifolds, Bull. Braz. Math. Soc. (N.S.), Volume 30 (1999) no. 2, pp. 183-197 | DOI | MR | Zbl

[CA09] Calvo-Andrade, Omegar Foliations of codimension greater than one with a Kupka component, Qual. Theory Dyn. Syst., Volume 8 (2009) no. 2, pp. 241-253 | DOI | MR | Zbl

[CA16] Calvo-Andrade, Omegar Foliations with a radial Kupka set on projective spaces, Bull. Braz. Math. Soc. (N.S.), Volume 47 (2016) no. 4, pp. 1071-1083 | DOI | MR | Zbl

[CJCAFP14] Corrêa Jr., Mauricio; Calvo-Andrade, Omegar; Fernández-Pérez, Arturo Highter codimension foliations and Kupka singularities (2014) (https://arxiv.org/abs/1408.7020)

[CLN94] Cerveau, Dominique; Lins Neto, Alcides Codimension-one foliations in ℂP n ,n≥3 with Kupka components, Complex analytic methods in dynamical systems. Proceedings of the congress held at Instituto de Matemática Pura e Aplicada, IMPA, Rio de Janeiro, Brazil, January 1992 (Astérisque), Volume 222, Société Mathématique de France, 1994, pp. 93-132 | Numdam | Zbl

[CLN96] Cerveau, Dominique; Lins Neto, Alcides Irreducible components of the space of holomorphic foliations of degree two in ℂℙ(n),n≥3, Ann. Math., Volume 143 (1996) no. 3, pp. 577-612 | DOI | MR | Zbl

[CLN13] Cerveau, Dominique; Lins Neto, Alcides A structural theorem for codimension-one foliations on ℙ n , n≥3, with an application to degree three foliations, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 12 (2013) no. 1, pp. 1-41 | MR | Zbl

[CLN19] Cerveau, Dominique; Lins Neto, Alcides Codimension two holomorphic foliations, J. Differ. Geom., Volume 113 (2019) no. 3, pp. 385-416 | MR | Zbl

[CS82] Camacho, Cesar; Sad, Paulo Invariant varieties through singularities of holomorphic vector fields, Ann. Math., Volume 115 (1982), pp. 579-595 | DOI | MR | Zbl

[DN13] Dan, Krishanu; Nagaraj, Donihakkalu Shankar Null correlation bundle on projective three space, J. Ramanujan Math. Soc., Volume 28A (2013), pp. 75-80 | MR | Zbl

[GML92] Gómez-Mont, Xavier; Luengo, I. Germs of holomorphic vector fields in ℂ 3 without a separatrix, Invent. Math., Volume 109 (1992) no. 2, pp. 211-219 | DOI | MR | Zbl

[LN99] Lins Neto, Alcides A note on projective Levi flats and minimal sets of algebraic foliations, Ann. Inst. Fourier, Volume 49 (1999) no. 4, pp. 1369-1385 | Numdam | MR | Zbl

[Seb97] Sebastiani, Marcos Sur l’existence de séparatrices locales des feuilletages des surfaces, Anais Acad. Brasil. Ci., Volume 69 (1997) no. 2, pp. 159-162 | Zbl

[Sei68] Seidenberg, Abraham Reduction of singularities of the differential equation Ady=Bdx, Am. J. Math., Volume 90 (1968), pp. 248-269 | DOI | Zbl