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First integrals for singular holomorphic foliations with leaves of bounded volume

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Holomorphic Dynamics

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1345))

Abstract

We consider the germ of k-dimensional holomorphic foliation in ℂn with an isolated singularity at the origin. Under the assumption that the germs of the leaves have bounded k-volume, it is proved that all leaves are closed and that at least one separatrix exists. If the k-volume (or k-dimensional Hausdorff measure) of the separatrix set is also finite, the germ has a very regular structure. In particular, the leaf space is a complex analytic space. The problem is motivated by the study of singularities of complex differential equations. Illustrative examples and a partial converse are presented.

Partially supported by the National Science Foundation. Este autor agradace al Centro de Investigación del IPN y CONACYT (México) cuyo apoyo durante la visita al Centro hizo posible el presente trabajo.

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Author information

Author notes

  1. Alberto Verjovsky

    Present address: International Centre for Theoretical Physics, Strada Costiera 11, Miramare, 34100, Trieste, Italy

Authors and Affiliations

  1. Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, 20742, College Park, MD, USA

    J. C. Alexander

  2. Departamento de Matemáticas, Centro de Investigación del IPN, Apartado Postal 14-740, México 14, D. F., México

    Alberto Verjovsky

Authors
  1. J. C. Alexander
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  2. Alberto Verjovsky
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Editor information

Xavier Gomez-Mont José A. Seade Alberto Verjovski

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© 1988 Springer-Verlag

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Alexander, J.C., Verjovsky, A. (1988). First integrals for singular holomorphic foliations with leaves of bounded volume. In: Gomez-Mont, X., Seade, J.A., Verjovski, A. (eds) Holomorphic Dynamics. Lecture Notes in Mathematics, vol 1345. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0081394

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  • DOI: https://doi.org/10.1007/BFb0081394

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-50226-5

  • Online ISBN: 978-3-540-45957-6

  • eBook Packages: Springer Book Archive

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