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 $π=π_1(\Cal R)$. 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 $\cnπ$, or (c) the group ring $\zz π$. 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 $|e(\Cal R)|$.

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