Cohomology of the Milnor fibre of a hyperplane arrangement with symmetry

Abstract

We prove some general results concerning the cohomology of theMilnor fibre of a hyperplane arrangement, and apply them to the case when the arrangement has some symmetry properties, particularly the case of the set of reflecting hyperplanes of a unitary reflection group. We relate the isotypic components of the monodromy action on the cohomology to the cohomology degree and to the mixed Hodge structure of the cohomology. We also use monodromy eigenspaces to determine the spectrum in some cases, which in turn throws further light on the equivariant Hodge structure of the cohomology and on the determination of the equivariant Hodge-Deligne polynomials. When the arrangement is the set of reflecting hyperplanes of a unitary reflection group, then using eigenspace theory for reflection groups, we prove 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. This leads to a case-free determination of the Euler characteristic in this case, answering a question of Denham-Lemire. We also give an alternative formula for the spectrum of an arrangement which permits its computation in low dimensions, and we provide several examples of such computations.