## Tohoku Mathematical Journal 2010 March SECOND SERIES VOL. 62, NO. 1

 Tohoku Math. J. 62 (2010), 29-44

Title INDEX FORMULA FOR MACPHERSON CYCLES OF AFFINE ALGEBRAIC VARIETIES

Author Jörg Schürmann and Mihai Tibăr

(Received October 17, 2008, revised July 28, 2009)
Abstract. 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.

2000 Mathematics Subject Classification. Primary 14C25; Secondary 14C17, 14R25, 32S60, 14D06, 32S20.

Key words and phrases. Characteristic classes, constructible function, affine polar varieties, Euler obstruction, index theorem, characteristic cycles, stratified Morse theory.