Tohoku Mathematical Journal

Index formula for MacPherson cycles of affine algebraic varieties

Jörg Schürmann and Mihai Tibăr

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We give explicit MacPherson cycles for the Chern-MacPherson class of a closed affine algebraic variety $X$ and for any constructible function $\alpha$ with respect to a complex algebraic Whitney stratification of $X$.

We define generalized degrees of the global polar varieties and of the MacPherson cycles and we prove a global index formula for the Euler characteristic of $\alpha$. Whenever $\alpha$ is the Euler obstruction of $X$, this index formula specializes to the Seade-Tibăr-Verjovsky global counterpart of the Lê-Teissier formula for the local Euler obstruction.

Article information

Tohoku Math. J. (2), Volume 62, Number 1 (2010), 29-44.

First available in Project Euclid: 31 March 2010

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Zentralblatt MATH identifier

Primary: 14C25: Algebraic cycles
Secondary: 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15] 14R25: Affine fibrations [See also 14D06] 32S60: Stratifications; constructible sheaves; intersection cohomology [See also 58Kxx] 14D06: Fibrations, degenerations 32S20: Global theory of singularities; cohomological properties [See also 14E15]

Characteristic classes constructible function affine polar varieties Euler obstruction index theorem characteristic cycles stratified Morse theory


Schürmann, Jörg; Tibăr, Mihai. Index formula for MacPherson cycles of affine algebraic varieties. Tohoku Math. J. (2) 62 (2010), no. 1, 29--44. doi:10.2748/tmj/1270041025.

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