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Invariant sets near singularities of holomorphic foliations

Published online by Cambridge University Press:  21 July 2015

CÉSAR CAMACHO  and
RUDY ROSAS
CÉSAR CAMACHO
Affiliation:
IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brazil email [email protected]
RUDY ROSAS
Affiliation:
Pontificia Universidad Católica del Perú, Av Universitaria 1801, Lima, Peru Instituto de Matemática y Ciencias Afines, Jr. los biólogos 245, Lima, Peru email [email protected]
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Abstract

Consider a complex one-dimensional foliation on a complex surface near a singularity $p$. If ${\mathcal{I}}$ is a closed invariant set containing the singularity $p$, then ${\mathcal{I}}$ contains either a separatrix at $p$ or an invariant real three-dimensional manifold singular at $p$.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 8 , December 2016 , pp. 2408 - 2418
Copyright
© Cambridge University Press, 2015 

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References

Camacho, C.. Quadratic forms and holomorphic foliations on singular surfaces. Math. Ann. 282 (1988), 177–184.Google Scholar
Camacho, C., Lins, A. and Sad, P.. Topological invariants and equidesingularization for holomorphic vector fields. J. Differential Geom. 20 (1984), 143–174.Google Scholar
Camacho, C. and Sad, P.. Invariant varieties through singularities of holomorphic vector fields. Ann. of Math.  (2) 115(3) (1982), 579–595.Google Scholar
Du Val, P.. On alsolute and non-absolute singularities of algebraic surfaces. Rev. Fac. Sci. Univ. Istanbul. Ser. A 91 (1944), 159–215.Google Scholar
Hukuhara, M., Kimura, T. and Matuda, T.. Equations Différentielles Ordinaires du Premier Ordre dans le Champ Complexe (Publications of the Mathematical Society of Japan, 7) . Mathematical Society of Japan, Tokyo, 1961.Google Scholar
Laufer, H.. Normal Two-Dimensional Singularities (Annals of Mathematics Studies, 71) . Princeton Uiversity Press, Princeton, NJ, 1971.Google Scholar
Loray, F.. Pseudo-groupe d’une singularité de feuilletage holomorphe en dimension deux. Prépublication IRMAR, ccsd-00016434, 2005. Available at https://hal.archives-ouvertes.fr/hal-00016434.Google Scholar
Marín, D. and Mattei, J.-F.. Incompressibilité des feuilles des germes de feuilletages holomorphes singuliers. Ann. Sci. Éc. Norm. Supér (4) 41 (2008), 855–903.Google Scholar
Marín, D. and Mattei, J.-F.. Monodromy and topological classification of germs of holomorphic foliations. Ann. Sci. Éc. Norm. Supér. (4) 3 (2012), 405–445.Google Scholar
Marín, D. and Mattei, J.-F.. Topology of singular holomorphic foliations along a compact divisor. J. Singul. 9 (2014), 122–150. Proceedings of Algebraic Methods in Geometry (Guanajuato, 2011).Google Scholar
Mumford, D.. The topology of normal singularities of an algebraic surface and a criterion for simplicity. Publ. Math. Inst. Hautes Études Sci. 9 (1961), 5–22.Google Scholar
Ortiz-Bobadilla, L., Rosales-Gonzáles, E. and Voronin, S. M.. On Camacho–Sad’s Theorem about the existence of a separatrix. Internat. J. Math. 21(11) (2010), 1413–1420.Google Scholar