Vanishing results for the cohomology of complex toric hyperplane complements

Suppose R is the complement of an essential arrangement of toric hyperlanes in the complex torus and ? = ?1(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?, or (c) the group ring Z?. 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)|.