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EULER OBSTRUCTION AND DEFECTS OF FUNCTIONS ON SINGULAR VARIETIES

Published online by Cambridge University Press:  23 July 2004

J.-P. BRASSELET
Affiliation:
Institut de Mathématiques de Luminy, UMR 6206 CNRS, Campus de Luminy – Case 907, 13288 Marseille Cedex 9, Francejpb@iml.univ-mrs.fr
D. MASSEY
Affiliation:
Department of Mathematics, Northeastern University, 567 Lake Hall, Boston, MA 02115, USAdmassey@neu.edu
A. J. PARAMESWARAN
Affiliation:
Tata Institute of Fundamental Research, Homi Bhaba Road, Colaba, Mumbai, Indiaparam@math.tifr.res.in
J. SEADE
Affiliation:
Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México, Apartado Postal 273-3, CP 62210, Cuernavaca, Morelos, Mexicojseade@matem.unam.mx
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Abstract

Several authors have proved Lefschetz type formulas for the local Euler obstruction. In particular, a result of this type has been proved that turns out to be equivalent to saying that the local Euler obstruction, as a constructible function, satisfies the local Euler condition (in bivariant theory) with respect to general linear forms. The purpose of the paper is to determine what prevents the local Euler obstruction from satisfying the local Euler condition with respect to functions which are singular at the considered point. This is measured by an invariant (or ‘defect’) of such functions. An interpretation of this defect is given in terms of vanishing cycles, which allows it to be calculated algebraically. When the function has an isolated singularity, the invariant can be defined geometrically, via obstruction theory. This invariant unifies the usual concepts of the Milnor number of a function and the local Euler obstruction of an analytic set.

Type
Notes and Papers
Copyright
The London Mathematical Society 2004

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Footnotes

This research was partially supported by the Cooperation Programs France–México CNRS/CONACYT and France–India CNRS/NBHM, and by CONACYT grant G36357-E.