Topological equivalence of holomorphic foliation germs of rank 1 with isolated singularity in the Poincaré domain
Annales de l'Institut Fourier, Volume 69 (2019) no. 2, pp. 561-590.

We show that the topological equivalence class of holomorphic foliation germs of rank 1 with an isolated singularity of Poincaré type is determined by the topological equivalence class of the real intersection foliation of the (suitably normalized) foliation germ with a sphere centered in the singularity. We use this Reconstruction Theorem to completely classify topological equivalence classes of plane holomorphic foliation germs of Poincaré type and discuss a conjecture on the classification in dimension 3.

Nous démontrons que la classe d’équivalence topologique des germes de feuilletages holomorphiques de rang 1 avec une singularité isolée de type Poincaré est déterminée par la classe d’équivalence topologique du feuilletage réel d’intersection du germe du feuilletage (normalisé) avec une sphère centrée dans la singularité. Nous utilisons ce Theorème de Reconstruction afin de classifier complètement les classes d’équivalence topologique des germes de feuilletages holomorphiques planes de type Poincaré et nous discutons une conjecture sur la classification en dimension 3.

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DOI: 10.5802/aif.3251
Classification: 32S65, 58K45
Keywords: holomorphic foliation germs, isolated singularity, topological equivalence, Poincaré domain
Mot clés : germes des feuilletages holomorphes, singularité isolée, equivalence topologique, domaine de Poincaré

Eckl, Thomas 1; Lönne, Michael 2

1 University of Liverpool Dept. of Mathematical Sciences Liverpool, L69 7ZL (UK)
2 Universität Bayreuth Mathematisches Institut Universitätsstr. 30 95447 Bayreuth (Germany)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Eckl, Thomas; Lönne, Michael. Topological equivalence of holomorphic foliation germs of rank $1$ with isolated singularity in the Poincaré domain. Annales de l'Institut Fourier, Volume 69 (2019) no. 2, pp. 561-590. doi : 10.5802/aif.3251. https://aif.centre-mersenne.org/articles/10.5802/aif.3251/

[1] Arnolʼd, Vladimir I. Remarks on singularities of finite codimension in complex dynamical systems, Funkts. Anal. Prilozh., Volume 3 (1969) no. 1, pp. 1-6 | MR | Zbl

[2] Arnolʼd, Vladimir I. Geometrical methods in the theory of ordinary differential equations, Grundlehren der Mathematischen Wissenschaften, 250, Springer, 1983 | MR | Zbl

[3] Brieskorn, Egbert; Knörrer, Horst Plane algebraic curves, Modern Birkhäuser Classics, Birkhäuser, 1986 | DOI | Zbl

[4] Brunella, Marco; Sad, Paulo Holomorphic foliations in certain holomorphically convex domains of 2 , Bull. Soc. Math. Fr., Volume 123 (1995) no. 4, pp. 535-546 | DOI | MR | Zbl

[5] Camacho, Cesar; Kuiper, Nicolaas H.; Palis, Jacob The topology of holomorphic flows with singularity, Publ. Math., Inst. Hautes Étud. Sci. (1978) no. 48, pp. 5-38 | DOI | MR | Zbl

[6] Camacho, Cesar; Sad, Paulo Topological classification and bifurcations of holomorphic flows with resonances in C 2 , Invent. Math., Volume 67 (1982) no. 3, pp. 447-472 | MR | Zbl

[7] Farb, Benson; Margalit, Dan A primer on mapping class groups, Princeton Mathematical Series, 49, Princeton University Press, 2012 | MR | Zbl

[8] Guckenheimer, John Hartman’s theorem for complex flows in the Poincaré domain, Compos. Math., Volume 24 (1972), pp. 75-82 | Zbl

[9] Hirsch, Morris W. Differential topology, Graduate Texts in Mathematics, 33, Springer, 1994 (Corrected reprint of the 1976 original) | MR | Zbl

[10] Ilyashenko, Yulij; Yakovenko, Sergei Lectures on analytic differential equations, Graduate Studies in Mathematics, 86, American Mathematical Society, 2008 | MR | Zbl

[11] Ito, Toshikazu A Poincaré-Bendixson type theorem for holomorphic vector fields, Sūrikaisekikenkyūsho Kōkyūroku (1994) no. 878, pp. 1-9 Singularities of holomorphic vector fields and related topics (Japanese) (Kyoto, 1993) | MR | Zbl

[12] Ito, Toshikazu; Scárdua, Bruno On holomorphic foliations transverse to spheres, Mosc. Math. J., Volume 5 (2005) no. 2, pp. 379-397 | MR | Zbl

[13] Limón, Beatriz; Seade, José Morse theory and the topology of holomorphic foliations near an isolated singularity, J. Topol., Volume 4 (2011) no. 3, pp. 667-686 | DOI | MR | Zbl

[14] Marín, David; Mattei, Jean-François Monodromy and topological classification of germs of holomorphic foliations, Ann. Sci. Éc. Norm. Supér., Volume 45 (2012) no. 3, pp. 405-445 | DOI | MR | Zbl

[15] Stöcker, Ralph; Zieschang, Heiner Algebraische Topologie, Mathematische Leitfäden, B. G. Teubner, 1994 | Zbl

[16] Warner, Frank W. Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, 94, Springer, 1983 | MR | Zbl

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