Generating series of classes of exotic un-ordered configuration spaces

TL;DR

This paper develops formulas for the generating series of classes of exotic un-ordered configuration spaces on complex varieties, extending previous work on ordered spaces and providing tools for computing various invariants.

Contribution

It introduces equations for the generating series of un-ordered exotic configuration spaces in the Grothendieck ring, generalizing Baryshnikov's ordered case to un-ordered varieties.

Findings

Derived equations for the classes in the Grothendieck ring

Provided formulas for additive invariants like Hodge–Deligne polynomial

Extended the theory to un-ordered configuration spaces on complex varieties

Abstract

A notion of exotic (ordered) configuration spaces of points on a space $X$ was suggested by Yu.~Baryshnikov. He gave equations for the (exponential) generating series of the Euler characteristics of these spaces. Here we consider un-ordered analogues of these spaces. For $X$ being a complex quasiprojective variety, we give equations for the generating series of classes of these configuration spaces in the Grothendieck ring $K_0({\rm{Var}_{\mathbb{C}}})$ of complex quasiprojective varieties. The answer is formulated in terms of the (natural) power structure over the ring $K_0({\rm{Var}_{\mathbb{C}}})$. This gives equations for the generating series of additive invariants of the configuration spaces such as the Hodge--Deligne polynomial and the Euler characteristic.