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On Verdier’s specialization formula for Chern classes. (English) Zbl 0725.57012

Let \(X_ s\) be a family of compact complex analytic varieties, where s varies over the unit disk in \({\mathbb{C}}\). J. L. Verdier [Astérisque 82-83, 149-159 (1981; Zbl 0479.14013)] has given a formula relating the Chern-MacPherson characteristic homology classes of the \(X_ s\) to those of the degenerate fiber \(X_ 0\). The present note gives an alternate proof of this formula, based on the author’s construction of the Chern classes in terms of the normal cycle. The key fact is that the normal cycle of a complex subvariety is characterized by the values it gives for the local Euler characteristic via the Gauss-Bonnet theorem.

MSC:

57R20 Characteristic classes and numbers in differential topology

Citations:

Zbl 0479.14013
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References:

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