Bordifications of hyperplane arrangements and their curve complexes

The complement of an arrangement of hyperplanes in Cn has a natural bordification to a manifold with corners formed by removing (or “blowing up”) tubular neighborhoods of the hyperplanes and certain of their intersections. When the arrangement is the complexification of a real simplicial arrangement, the bordification closely resembles Harvey's bordification of moduli space. We prove that the faces of the universal cover of the bordification are parameterized by the simplices of a simplicial complex C, the vertices of which are the irreducible “parabolic subgroups” of the fundamental group of the arrangement complement. So, the complex C plays a similar role for an arrangement complement as the curve complex does for moduli space. Also, in analogy with curve complexes and with spherical buildings, we prove that C has the homotopy type of a wedge of spheres. Our results apply in particular to spherical Artin groups, where the associated arrangement is a reflection arrangement of a finite Coxeter group.