Vanishing results for the cohomology of complex toric hyperplane complements

Suppose $Cal R$ is the complement of an essential arrangement of toric hyperlanes in the complex torus $(C^*)^n$ and $pi=pi_1(Cal R)$. We show that $H^*(Cal R;A)$ 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 $cnpi$, or (c) the group ring $zz pi$. In case (a) the dimension of $H^n$ is $|e(Cal R)|$ where $e(Cal R)$ denotes the Euler characteristic, and in case (b) the $n^{mathrm{th}}$ $eltwo$ Betti number is also $|e(Cal R)|$.