Chern classes of proalgebraic varieties and motivic measures

Abstract

Michael Gromov has recently initiated what he calls ``symbolic algebraic geometry", in which objects are proalgebraic varieties: a proalgebraic variety is by definition the projective limit of a projective system of algebraic varieties. In this paper we construct Chern--Schwartz--MacPherson classes of proalgebraic varieties, by introducing the notion of ``proconstructible functions " and "$\chi$-stable proconstructible functions" and using the Fulton-MacPherson's Bivariant Theory. As a "motivic" version of a $\chi$-stable proconstructible function, $\Ga$-stable constructible functions are introduced. This construction naturally generalizes the so-called motivic measure and motivic integration. For the Nash arc space $\Cal L(X)$ of an algebraic variety $X$, the proconstructible set is equivalent to the so-called cylinder set or constructible set in the arc space.