Milnor classes of local complete intersections

Brasselet, J.-P.; Lehmann, D.; Seade, J.; Suwa, T.

doi:10.1090/S0002-9947-01-02846-X

Milnor classes of local complete intersections
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by J.-P. Brasselet, D. Lehmann, J. Seade and T. Suwa

Trans. Amer. Math. Soc. 354 (2002), 1351-1371

DOI: https://doi.org/10.1090/S0002-9947-01-02846-X

Published electronically: November 21, 2001

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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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