hyperplane arrangements, toric arrangements, local systems, L2-cohomology.

Suppose R is the complement of an essential arrangement of toric hyperlanes in the complex torus and &pi = π1(R). We show that H*(R;A) vanishes except in the top degree n when A is one of the following systems of local coefficients: (a) a system of nonresonant coefficients in a complex line bundle, (b) the von Neumann algebra N&pi, or (c) the group ring Zπ. In case (a) the dimension of Hn is the Euler characteristic, e(R), and in case (b) the nth l2 Betti number is also |e(R)|.