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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Milnor classes of local complete intersections
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by J.-P. Brasselet, D. Lehmann, J. Seade and T. Suwa PDF
Trans. Amer. Math. Soc. 354 (2002), 1351-1371 Request permission


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:
  • D. Lehmann
  • Affiliation: Département des Sciences Mathématiques, Université de Montpellier II, 34095 Montpellier Cedex 5, France
  • Email:
  • 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:
  • T. Suwa
  • Affiliation: Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
  • Email:
  • Received by editor(s): July 15, 2000
  • Received by editor(s) in revised form: December 1, 2000
  • Published electronically: November 21, 2001
  • © Copyright 2001 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 354 (2002), 1351-1371
  • MSC (2000): Primary 57R20; Secondary 14C17, 14J17, 32S55, 58K45
  • DOI:
  • MathSciNet review: 1873009