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Transactions of the American Mathematical Society

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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
Published electronically: November 21, 2001
MathSciNet review: 1873009
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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

D. Lehmann
Affiliation: Département des Sciences Mathématiques, Université de Montpellier II, 34095 Montpellier Cedex 5, France

J. Seade
Affiliation: Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, Circuito Exterior, México 04510 D.F., México

T. Suwa
Affiliation: Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan

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