Line arrangements and direct products of free groups

Abstract

We show that if the fundamental groups of the complements of two line arrangements in the complex projective plane are isomorphic to the same direct product of free groups, then the complements of the arrangements are homotopy equivalent. For any such arrangement $\mathcal{A}$, we also construct an arrangement ${\mathcal{A}}^{\prime }$ such that ${\mathcal{A}}^{\prime }$ is a complexified-real arrangement, the intersection lattices of the arrangements are isomorphic, and the complements of the arrangements are diffeomorphic.