Cohomology of configuration spaces on punctured varieties

TL;DR

This paper proves a new equivariant and mixed-Hodge-theoretic splitting theorem for the cohomology of configuration spaces on punctured varieties, and applies it to compute generating functions for elliptic curves.

Contribution

It introduces a novel equivariant and mixed-Hodge-theoretic splitting theorem for configuration spaces, extending previous cohomological relations to new mathematical structures.

Findings

Established a new splitting theorem for cohomology of configuration spaces.

Computed the generating function for mixed Hodge numbers of punctured elliptic curves.

Demonstrated the theorem's application to multi-punctured elliptic curves.

Abstract

In the theory of configuration spaces, "splitting" usually refers to the phenomenon that the configuration spaces on a manifold and those on its punctured version are closely related cohomologically. We prove a splitting theorem that is equivariant and mixed-Hodge-theoretic; both are new features in such results. As an application, we determine the generating function for the mixed Hodge numbers of the unordered configuration spaces of a multi-punctured elliptic curve.