Skip to main content
SpringerLink
Log in

Differentiable Invariants of Holomorphic Foliations

  • Published:
Bulletin of the Brazilian Mathematical Society, New Series Aims and scope Submit manuscript

Abstract

In this paper we study differentiable equivalences of germs of singular holomorphic foliations in dimension two. We prove that the Camacho–Sad indices are invariant by such equivalences. We also prove that the Baum–Bott index is a differentiable invariant for some classes of foliations. As a corollary we show that generic degree two holomorphic foliations of \({\mathbb {P}}^2\) are differentiably rigid.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Brunella, M.: Some remarks on indices of holomorphic vector fields. Publ. Mat. 41(2), 527–544 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • Brunella, M.: Birational Geometry of Foliations. IMPA Monographs Series (2015)

  • Camacho, C., Sad, P.: Invariant varieties through singularities of holomorphic vector fields. Ann. Math. (2) 115(3), 579–595 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  • Camacho, C., Lins Neto, A., Sad, P.: Topological invariants and equidesingularization for holomorphic vector fields. J. Differ. Geom. 20, 143–174 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  • Fernández-Pérez, A., Mol, R.: Residue-type indices and holomorphic foliations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19(3), 1111–1134 (2019)

  • Genzmer, Y., Mol, R.: Local polar invariants and the Poincaré problem in the dicritical case. J. Math. Soc. Jpn. 70(4), 1419–1451 (2018)

    Article  MATH  Google Scholar 

  • Gómez-Mont, X., Seade, J., Verjovsky, A.: The index of a holomorphic flow with an isolated singularity. Math. Ann. 291, 737–751 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  • Ilyashenko, Y., Moldavskis, V.: Total rigidity of generic quadratic vector fields. Mosc. Math. J. 11(3), 521–530, 630 (2011)

  • Lins Neto, A.: Fibers of the Baum–Bott map for foliations of degree two on P2. Bull. Braz. Math. Soc. (N.S.) 43(1), 129–169 (2012)

  • Lins Neto, A., Azevedo Scárdua, B.: Folheaçoes algébricas complexas. Projeto Euclides, Instituto de Matemática Pura e Aplicada (1997)

  • Lins Neto, A., Pereira, J.V.: The generic rank of the Baum–Bott map for foliations of the projective plane. Compos. Math. 142(6), 1549–1586 (2006)

  • Marín, D., Mattei, J.-F.: Monodromy and topological classification of germs of holomorphic foliations. Ann. Sci. Éc. Norm. Supér. 45, 405–445 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Marín, D., Mattei, J.-F., Salem, E.: Topological moduli space for germs of holomorphic foliations. Int. Res. Math. Not. 23, 9228–9292 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Mol, R., Rosas, R.: Differentiable equisingularity of holomorphic foliations. J. Singul. 19, 76–96 (2019)

    MathSciNet  MATH  Google Scholar 

  • Rosas, R.: The differentiable invariance of the algebraic multiplicity of a holomorphic vector field. J. Differ. Geom. 83(2), 337–376 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Rosas, R.: The C1 invariance of the algebraic multiplicity of a holomorphic vector field. Ann. Inst. Fourier (Grenoble) 60(6), 2115–2135 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Rosas, R.: Bilipschitz invariants for germs of holomorphic foliations. Int. Math. Res. Not. IMRN 11, 3425–3472 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Seidenberg, A.: Reduction of singularities of the differential equation $Ady=B dx$. Am. J. Math. 90, 248–269 (1968)

    Article  MATH  Google Scholar 

  • Suwa, T.: Indices of Vector Fields and Residues of Holomorphic Singular Foliations. Hermann (1998)

  • Teyssier, L.: Germs de feuilletages présentables du plan complexe. Bull. Braz. Math. Soc. 46, 275–329 (2015)

  • Zariski, O.: On the topology of algebroid singularities. Am. J. Math. 54, 453–465 (1932)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Pontificia Universidad Católica del Perú, Av Universitaria, 1801, Lima, Peru

    Rudy Rosas

Authors
  1. Rudy Rosas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rudy Rosas.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The author was supported by the Vicerrectorado the Investigación de la Pontificia Universidad Católica del Perú

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rosas, R. Differentiable Invariants of Holomorphic Foliations. Bull Braz Math Soc, New Series 53, 1107–1130 (2022). https://doi.org/10.1007/s00574-022-00297-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00574-022-00297-6

Keywords

Search

Navigation