Generating Series of the Classes of Exotic Unordered Configuration Spaces

Abstract

The concept of exotic (ordered) configuration spaces of points in a space was proposed by Baryshnikov. He obtained formulas for the (exponential) generating series of their Euler characteristics. We explore unordered analogs of the spaces. Considering a complex quasiprojective variety, we give a formula for the generating series of the classes of these configuration spaces in the Grothendieck ring of complex quasiprojective varieties. We state the result in terms of the natural power structure over this ring. This yields formulas of generating series of additive invariants of configuration spaces like the Hodge–Deligne polynomial and the Euler characteristic.