Skip to main content
SpringerLink
Log in

On the index of a holomorphic vector field tangent to a singular variety

  • Published:
Boletim da Sociedade Brasileira de Matemática - Bulletin/Brazilian Mathematical Society Aims and scope Submit manuscript

Abstract

In this article we define and compute an index for a holomorphic vector field on a (possibly singular) subvariety of a complex manifold, provided the subvariety is a local complete intersection. This index reduces to the usual Poincaré-Hopf index in case the subvariety is smooth, and is equal more generally to the index defined in [GSV] and [SS].

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

  • [A] V. I. Arnold:Normal forms of functions in the neighbourhood of degenerate critical points, Russian Mathematical Surveys,29: 2 (1974), 10–50.

    Google Scholar 

  • [B 1] R. Bott:Vector fields and characteristic numbers, Michigan Math. J.14 (1967), 231–244.

    Google Scholar 

  • [B 2] R. Bott:A residue formula for holomorphic vector fields, J. of Differential Geometry,1 (1967), 311–330.

    Google Scholar 

  • [B 3] R. Bott:Lectures on characteristic classes and foliations, Springer Lecture Notes,279, (1972).

  • [BB 1] P. Baum and R. Bott:On the zeroes of holomorphic vector fields, Essays on Topology and related topics (Mémoires dédiés à Georges de Rham), Springer (1970), 29–47.

  • [BB 2] P. Baum and R. Bott:Singularities of holomorphic foliations, J. of Differential Geometry,7 (1982), 279–342.

    Google Scholar 

  • [BG] C. Bonatti and X. Gomez-Mont:The Index of holomorphic vector fields on singular varieties I, Astérisque,222, (1994).

  • [G] X. Gomez-Mont:An algebraic formula for the index of a vector field on a variety with an isolated singularity, preprint.

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

    Google Scholar 

  • [KT] F. Kamber and P. Tondeur:Foliated Bundles and Characteristic Classes, Lecture Notes in Mathematics,493, Springer-Verlag, New York, Heidelberg, Berlin, 1975.

    Google Scholar 

  • [L] D. Lehmann:Résidus des sous variétés invariantes d'un feuilletage singulier, Ann. Inst. Fourier,41 (1991), 211–258.

    Google Scholar 

  • [LS] D. Lehmann and T. Suwa:Residues of holomorphic vector fields relative to singular invariant subvarieties, to appear in J. of Differential Geometry.

  • [SS] J. Seade and T. Suwa:A residue formula for the index of a holomorphic flow, to appear in Math. Ann.

Download references

Author information

Authors and Affiliations

  1. GETODIM, CNRS, UA 1407, Université de Montpellier II, 34095, Montpellier cedex, France

    Daniel Lehmann

  2. Departmento de Matemática, UFMG, 31270-90, Belo Horizonte, Brasil

    Marcio Soares

  3. Department of Mathematics, Hokkaido University, 060, Sapporo, Japan

    Tatsuo Suwa

Authors
  1. Daniel Lehmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marcio Soares
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tatsuo Suwa
    View author publications

    You can also search for this author in PubMed Google Scholar

About this article

Cite this article

Lehmann, D., Soares, M. & Suwa, T. On the index of a holomorphic vector field tangent to a singular variety. Bol. Soc. Bras. Mat 26, 183–199 (1995). https://doi.org/10.1007/BF01236993

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01236993

Keywords

Search

Navigation