Abstract
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.
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Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 39, No. 4, pp. 1–13, 2005
Original Russian Text Copyright by © A. G. Aleksandrov
The work was supported in part by the Russian Foundation for Basic Research (project 02-01-00623).
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Aleksandrov, A.G. The Index of Vector Fields and Logarithmic Differential Forms. Funct Anal Its Appl 39, 245–255 (2005). https://doi.org/10.1007/s10688-005-0046-0
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DOI: https://doi.org/10.1007/s10688-005-0046-0