Functional Analysis and Its Applications

, Volume 39, Issue 4, pp 245–255 | Cite as

The Index of Vector Fields and Logarithmic Differential Forms

  • A. G. Aleksandrov


We introduce the notion of logarithmic index of a vector field on a hypersurface and prove that the homological index can be expressed via the logarithmic index. Then both invariants are described in terms of logarithmic differential forms for Saito free divisors, which are hypersurfaces with nonisolated singularities, and all contracting homology groups of the complex of regular holomorphic forms on such a hypersurface are computed. In conclusion, we consider the case of normal hypersurfaces, including the case of an isolated singularity, and describe the contracting homology of the complex of regular meromorphic forms with the help of the residue of logarithmic forms.

Key words

singularity vector field logarithmic differential form contracting homology logarithmic index Saito free divisor 


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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • A. G. Aleksandrov
    • 1
  1. 1.Institute of Control SciencesRASRussia

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