Homological index for 1-forms and a Milnor number for isolated singularities

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

Autorschaft

  • Wolfgang Ebeling
  • Sabir M. Gusein-Zade
  • José Seade

Organisationseinheiten

Externe Organisationen

  • Lomonosov Moscow State University
  • Universidad Nacional Autónoma de México (UNAM)
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Details

OriginalspracheEnglisch
Seiten (von - bis)895-905
Seitenumfang11
FachzeitschriftINT J MATH
Jahrgang15
Ausgabenummer9
PublikationsstatusVeröffentlicht - Nov. 2004

Abstract

We introduce a notion of a homological index of a holomorphic 1-form on a germ of a complex analytic variety with an isolated singularity, inspired by Gómez-Mont and Greuel. For isolated complete intersection singularities it coincides with the index defined earlier by two of the authors. Subtracting from this index another one, called radial, we get an invariant of the singularity which does not depend on the 1-form. For isolated complete intersection singularities this invariant coincides with the Milnor number. We compute this invariant for arbitrary curve singularities and compare it with the Milnor number introduced by Buchweitz and Greuel for such singularities.

ASJC Scopus Sachgebiete

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Homological index for 1-forms and a Milnor number for isolated singularities. / Ebeling, Wolfgang; Gusein-Zade, Sabir M.; Seade, José.
in: INT J MATH, Jahrgang 15, Nr. 9, 11.2004, S. 895-905.

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

Ebeling W, Gusein-Zade SM, Seade J. Homological index for 1-forms and a Milnor number for isolated singularities. INT J MATH. 2004 Nov;15(9):895-905. doi: 10.1142/S0129167X04002624
Ebeling, Wolfgang ; Gusein-Zade, Sabir M. ; Seade, José. / Homological index for 1-forms and a Milnor number for isolated singularities. in: INT J MATH. 2004 ; Jahrgang 15, Nr. 9. S. 895-905.
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abstract = "We introduce a notion of a homological index of a holomorphic 1-form on a germ of a complex analytic variety with an isolated singularity, inspired by G{\'o}mez-Mont and Greuel. For isolated complete intersection singularities it coincides with the index defined earlier by two of the authors. Subtracting from this index another one, called radial, we get an invariant of the singularity which does not depend on the 1-form. For isolated complete intersection singularities this invariant coincides with the Milnor number. We compute this invariant for arbitrary curve singularities and compare it with the Milnor number introduced by Buchweitz and Greuel for such singularities.",
keywords = "1-forms, De Rham complex, Homological index, Index, Isolated singularity, Milnor number, Radial index",
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note = "Funding information: The research was partially supported by the DFG programme “Global methods in complex geometry” (Eb 102/4–2). The second author also was partially supported by the grants RFBR–04–01–00762, INTAS–00–259; the third author had partial support by CONACYT grant G36357/E.",
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AU - Ebeling, Wolfgang

AU - Gusein-Zade, Sabir M.

AU - Seade, José

N1 - Funding information: The research was partially supported by the DFG programme “Global methods in complex geometry” (Eb 102/4–2). The second author also was partially supported by the grants RFBR–04–01–00762, INTAS–00–259; the third author had partial support by CONACYT grant G36357/E.

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N2 - We introduce a notion of a homological index of a holomorphic 1-form on a germ of a complex analytic variety with an isolated singularity, inspired by Gómez-Mont and Greuel. For isolated complete intersection singularities it coincides with the index defined earlier by two of the authors. Subtracting from this index another one, called radial, we get an invariant of the singularity which does not depend on the 1-form. For isolated complete intersection singularities this invariant coincides with the Milnor number. We compute this invariant for arbitrary curve singularities and compare it with the Milnor number introduced by Buchweitz and Greuel for such singularities.

AB - We introduce a notion of a homological index of a holomorphic 1-form on a germ of a complex analytic variety with an isolated singularity, inspired by Gómez-Mont and Greuel. For isolated complete intersection singularities it coincides with the index defined earlier by two of the authors. Subtracting from this index another one, called radial, we get an invariant of the singularity which does not depend on the 1-form. For isolated complete intersection singularities this invariant coincides with the Milnor number. We compute this invariant for arbitrary curve singularities and compare it with the Milnor number introduced by Buchweitz and Greuel for such singularities.

KW - 1-forms

KW - De Rham complex

KW - Homological index

KW - Index

KW - Isolated singularity

KW - Milnor number

KW - Radial index

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UR - https://arxiv.org/abs/math/0307239

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