Hyperplane arrangements and tensor product invariants

TL;DR

This paper develops explicit formulas for cohomology maps in hyperplane arrangements and explores their applications in invariant theory, revealing Hodge structures and conformal blocks in the context of local system cohomology.

Contribution

It provides an explicit algebraic de Rham representative for cohomology maps and applies these results to determine the Hodge structure of invariants in hyperplane arrangement complements.

Findings

Explicit algebraic de Rham representatives for cohomology maps.

Invariants acquire mixed Hodge structures over cyclotomic fields.

Hodge filtration characterizes conformal blocks in cohomology.

Abstract

In the first part of this paper, we consider, in the context of an arbitrary hyperplane arrangement, the map between compactly supported cohomology to the usual cohomology of a local system. A formula (i.e., an explicit algebraic de Rham representative) for a generalized version of this map is obtained. These results are applied in the second part to invariant theory: Schechtman and Varchenko connect invariant theoretic objects to the cohomology of local systems on complements of hyperplane arrangements. The first part of this paper is then used, following and completing arguments of Looijenga, to determine the image of invariants in cohomology. In suitable cases (e.g., corresponding to positive integral levels), the space of invariants is shown to acquire a mixed Hodge structure over a cyclotomic field. We investigate the Hodge filtration on the space of invariants, and characterize the subspace of conformal blocks in Hodge theoretic terms.