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Analytic Varieties Invariant by Holomorphic Foliations and Pfaff Systems

In: Handbook of Geometry and Topology of Singularities VI: Foliations

Author

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  • Maurício Corrêa

    (Dipartimento di Matematica, Università degli Studi di Bari
    Universidade Federal de Minas Gerais-UFMG)

Abstract

In this work we shall present a survey on problems and results on singular holomorphic foliations and Pfaff systems on complex manifolds assuming that these objects possess invariant analytic varieties. We will focus on recent results which have been motivated by classical works of Darboux, Poincaré and Painlevé on the problem of algebraic integration of singular polynomial differential equations. We present results on Poincaré and Painlevé problem of bounding the degree and the genus of analytic varieties invariant by holomorphic foliations and Pfaff systems. We shall discuss the general ideas of the theory of integrability characterizing the existence of meromorphic first integrals for complex analytic Pfaff equations.

Suggested Citation

  • Maurício Corrêa, 2024. "Analytic Varieties Invariant by Holomorphic Foliations and Pfaff Systems," Springer Books, in: Felipe Cano & José Luis Cisneros-Molina & Lê Dũng Tráng & José Seade (ed.), Handbook of Geometry and Topology of Singularities VI: Foliations, chapter 0, pages 123-150, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-54172-8_4
    DOI: 10.1007/978-3-031-54172-8_4
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