The Multiplicity Polar Theorem and isolated singularities

Gaffney, Terence

doi:10.1090/S1056-3911-08-00516-X

Author: Terence Gaffney
Journal: J. Algebraic Geom. 18 (2009), 547-574
DOI: https://doi.org/10.1090/S1056-3911-08-00516-X
Published electronically: November 17, 2008
MathSciNet review: 2496457
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Abstract | References | Additional Information

Abstract: We show how the Multiplicity Polar Theorem can be used to calculate invariants which describe an “isolated singularity”. Examples include the defect of a function, which is related to the Euler obstruction, the index of a differential form, the dimension of the space of vanishing cycles of a sheaf of D-modules $M$ relative to a function $f$ at $0$, and a formula for the relative cohomology of the Milnor fiber of $f$ where $f$ has an isolated singularity on a complex analytic set with possibly non-isolated singularities. We apply the result on the defect to refine previous work on the $\mathrm {A}_f$ condition.

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Additional Information

Terence Gaffney
Affiliation: Department of Mathematics, 567 Lake Hall, Northeastern University, Boston, Massachusetts 02115
Address at time of publication: MSRI, 17 Gauss Way, Berkeley, California 94720-5070
MR Author ID: 70390
ORCID: 0000-0003-3420-0150
Email: gaff@neu.edu

Received by editor(s): April 26, 2007
Received by editor(s) in revised form: January 11, 2008
Published electronically: November 17, 2008
Article copyright: © Copyright 2008 American Mathematical Society