The Poincaré polynomial of an mp arrangement

Abstract

Let A = {A i } i ∈ I be an mp arrangement in a complex algebraic variety X with corresponding complement Q(A) = X\U i ∈ I A i and intersection poset L(A). Examples of such arrangements are hyperplane arrangements and toral arrangements, i.e., collections of codimension 1 subtori, in an algebraic torus. Suppose a finite group Γ acts on X as a group of automorphisms and stabilizes the arrangement {A i } i ∈ I setwise. We give a formula for the graded character of Γ on the cohomology of Q(A) in terms of the graded character of r on the cohomology of certain subvarieties in L(A).