Milnor classes of local complete intersections
Authors:
J.-P. Brasselet, D. Lehmann, J. Seade and T. Suwa
Journal:
Trans. Amer. Math. Soc. 354 (2002), 1351-1371
MSC (2000):
Primary 57R20; Secondary 14C17, 14J17, 32S55, 58K45
DOI:
https://doi.org/10.1090/S0002-9947-01-02846-X
Published electronically:
November 21, 2001
MathSciNet review:
1873009
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Let $V$ be a compact local complete intersection defined as the zero set of a section of a holomorphic vector bundle over the ambient space. For each connected component $S$ of the singular set $\operatorname {Sing}(V)$ of $V$, we define the Milnor class $\mu _{*}(V,S)$ in the homology of $S$. The difference between the Schwartz-MacPherson class and the Fulton-Johnson class of $V$ is shown to be equal to the sum of $\mu _{*}(V,S)$ over the connected components $S$ of $\operatorname {Sing}(V)$. This is done by proving Poincaré-Hopf type theorems for these classes with respect to suitable tangent frames. The $0$-degree component $\mu _{0}(V,S)$ coincides with the Milnor numbers already defined by various authors in particular situations. We also give an explicit formula for $\mu _{*}(V,S)$ when $S$ is a non-singular component and $V$ satisfies the Whitney condition along $S$.
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Additional Information
J.-P. Brasselet
Affiliation:
Institut de Mathématiques de Luminy, UPR 9016 CNRS, Campus de Luminy - Case 907, 13288 Marseille Cedex 9, France
Email:
jpb@iml.univ-mrs.fr
D. Lehmann
Affiliation:
Département des Sciences Mathématiques, Université de Montpellier II, 34095 Montpellier Cedex 5, France
Email:
lehmann@darboux.math.univ-montp2.fr
J. Seade
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, Circuito Exterior, México 04510 D.F., México
MR Author ID:
157790
Email:
jseade@matem.unam.mx
T. Suwa
Affiliation:
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
Email:
suwa@math.sci.hokudai.ac.jp
Received by editor(s):
July 15, 2000
Received by editor(s) in revised form:
December 1, 2000
Published electronically:
November 21, 2001
Article copyright:
© Copyright 2001
American Mathematical Society