On the cohomology of the Milnor fibre of a hyperplane arrangement

TL;DR

This paper explores the cohomology of Milnor fibres of hyperplane arrangements, linking monodromy, Hodge structures, and group actions, with complete results for rank two unitary reflection groups and partial insights for others.

Contribution

It provides new sum formulas and methods to analyze the monodromy and Hodge structures of Milnor fibre cohomology, especially for low-rank reflection groups.

Findings

Complete cohomology descriptions for rank two unitary reflection groups.

Sum formulas for equivariant weight polynomials and related invariants.

Insights into the spectrum and Hodge structure via monodromy eigenspaces.

Abstract

We investigate the cohomology of the Milnor fibre of a reflection arrangement as a module for the group $\Gamma$ generated by the reflections, together with the cyclic monodromy. Although we succeed completely only for unitary reflection groups of rank two, we establish some general results which relate the isotypic componenents of the monodromy on the cohomology, to the Hodge structure and to the cohomology degree. Using eigenspace theory for reflection groups, we prove some sum formulae for additive functions such as the equivariant weight polynomial and certain polynomials related to the Euler characteristic, such as the Hodge-Deligne polynomials. We also use monodromy eigenspaces to determine the spectrum in some cases, which in turn throws light on the Hodge structure of the cohomology. These methods enable us to compute the complete story, including the representation of $\Gamma$ on the Hodge components in each cohomology degree, for some groups of low rank.