A. G. Aleksandrov, “The Index of Vector Fields and Logarithmic Differential Forms”, Funktsional. Anal. i Prilozhen., 39:4 (2005), 1–13; Funct. Anal. Appl., 39:4 (2005), 245

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Funktsional'nyi Analiz i ego Prilozheniya, 2005, Volume 39, Issue 4, Pages 1–13
DOI: https://doi.org/10.4213/faa81 (Mi faa81)

This article is cited in 13 scientific papers (total in 13 papers)

The Index of Vector Fields and Logarithmic Differential Forms

A. G. Aleksandrov

Institute of Control Sciences, Russian Academy of Sciences

Full-text PDF (235 kB) Citations (13)

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DOI: https://doi.org/10.4213/faa81

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.

Keywords: singularity, vector field, logarithmic differential form, contracting homology, logarithmic index, Saito free divisor.

Received: 03.03.2004

English version:
Functional Analysis and Its Applications, 2005, Volume 39, Issue 4, Pages 245–255
DOI: https://doi.org/10.1007/s10688-005-0046-0

Bibliographic databases:

Document Type: Article

UDC: 515.16+512.7

Language: Russian

Citation: A. G. Aleksandrov, “The Index of Vector Fields and Logarithmic Differential Forms”, Funktsional. Anal. i Prilozhen., 39:4 (2005), 1–13; Funct. Anal. Appl., 39:4 (2005), 245–255

Citation in format AMSBIB

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https://doi.org/10.4213/faa81

https://www.mathnet.ru/eng/faa/v39/i4/p1

This publication is cited in the following 13 articles:

Diogo da Silva Machado, “Residue formula for logarithmic foliations along a divisor with isolated singularities and applications”, Trans. Amer. Math. Soc., 2025
Maurício Corrêa, Handbook of Geometry and Topology of Singularities VI: Foliations, 2024, 123
José Luis Cisneros-Molina, Agustín Romano-Velázquez, “Indices of vector fields for mixed functions”, Bol. Soc. Mat. Mex., 30:3 (2024)
Alexander G. Aleksandrov, “The Poincaré Index on Singular Varieties”, J, 5:3 (2022), 380
Alexander G. Aleksandrov, “The Poincaré Index and Its Applications”, Universe, 8:4 (2022), 223
Correa M., Machado Diogo da Silva, “Residue Formulas For Logarithmic Foliations and Applications”, Trans. Am. Math. Soc., 371:9 (2019), 6403–6420
Nabeshima K., Tajima Sh., “Computation Methods of Logarithmic Vector Fields Associated to Semi-Weighted Homogeneous Isolated Hypersurface Singularities”, Tsukuba J. Math., 42:2 (2018), 191–231
Aleksandrov A.G., “the Poincaré Index and the Chi(Y)-Characteristic of Hirzebruch”, Complex Var. Elliptic Equ., 61:2 (2016), 166–212
A. G. Aleksandrov, “The Index of Differential Forms on Complete Intersections”, Funct. Anal. Appl., 49:1 (2015), 1–14
Aleksandrov A.G., “The Topological Index of Vector Fields at Quasihomogeneous Complete Intersections”, C. R. Math., 350:19-20 (2012), 911–916
Brasselet J.-P., Seade J., Suwa T., Vector fields on singular varieties, Lecture Notes in Math., 1987, Springer-Verlag, Berlin, 2009, xx+225 pp.
A.G. Aleksandrov, A.A. Castro, V.A. Gruzman, “Control of evolutionary processes, topological index and deformation theory”, IFAC Proceedings Volumes, 41:2 (2008), 13293
Aleksandr G. Aleksandrov *, “Logarithmic differential forms, torsion differentials and residue”, Complex Variables, Theory and Application: An International Journal, 50:7-11 (2005), 777

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Functional Analysis and Its Applications

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References:	96

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