Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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 $S$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 57R20, 14C17, 14J17, 32S55, 58K45

Retrieve articles in all journals with MSC (2000): 57R20, 14C17, 14J17, 32S55, 58K45


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
Email: jseade@matem.unam.mx

T. Suwa
Affiliation: Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
Email: suwa@math.sci.hokudai.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02846-X
PII: S 0002-9947(01)02846-X
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