Mixed Hodge structures of configuration spaces

TL;DR

This paper investigates the symmetric group actions on the cohomology of configuration spaces using mixed Hodge theory, providing explicit calculations of equivariant Hodge polynomials for compactified configuration spaces.

Contribution

It introduces a novel approach combining mixed Hodge theory and symmetric functions to analyze group actions on configuration space cohomology, including a motivic version.

Findings

Computed the S_n-equivariant Hodge polynomial of X[n].

Established a motivic version of the symmetric group action analysis.

Applied mixed Hodge theory to configuration spaces with symmetric group actions.

Abstract

The symmetric group S_n acts freely on the configuration space of n distinct points in a quasi-projective variety. In this paper, we study the induced action of the symmetric group S_n on the de Rham cohomology of this space, using mixed Hodge theory, combined with methods from the theory of symmetric functions. (We prove a motivic version of this as well.) As an application of our results, we calculate the S_n-equivariant Hodge polynomial of the Fulton-MacPherson compactification X[n] of the configuration space.