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Milnor numbers and classes of local complete intersections. (English) Zbl 1021.32012

Summary: Let \(V\) be the set of zeros for a section of a holomorphic vector bundle on a complex manifold. When \(S\) is a connected component of the singular locus of \(V\), the authors define a new set of invariants, the Milnor class \(\mu_*(V,S)\) of \(V\) at \(S\). Each \(\mu_i(V,S)\) belongs to \(H_{2i}(S,\mathbb{Z})\) for \(0\leq i\leq \dim S\). \(\mu_0\) coincides with the Milnor number when the latter is defined and the authors call it the generalized Milnor number. When \(S\) is smooth and \(V\) satisfies a condition slightly weaker than the Whitney condition along \(S\), they prove a Lefschetz type theorem for Milnor classes (Corollary 5.13 and Remark 5.14). The main result is that the difference between the Schwartz-MacPherson class and the Fulton-Johnson class of \(V\) is the sum of the Milnor classes of \(V\) along the connected components of the singular locus of \(V\) (Theorem 5.2). The paper ends with two worked out examples.

MSC:

32S50 Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14J17 Singularities of surfaces or higher-dimensional varieties
57R20 Characteristic classes and numbers in differential topology
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References:

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