Vanishing results for the cohomology of complex toric hyperplane complements

Abstract

Suppose $\mathcal R$ is the complement of an essential arrangement of toric hyperlanes in the complex torus $(\mathbb{C}^*)^n$ and $\pi=\pi_1(\mathcal R)$. We show that $H^*(\mathcal R;A)$ 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 $\mathcal{N}\pi$, or (c) the group ring ${\mathbb Z} \pi$. In case (a) the dimension of $H^n$ is $|e(\mathcal R)|$ where $e(\mathcal R)$ denotes the Euler characteristic, and in case (b) the $n^{\mathrm{th}}$ $\ell^2$ Betti number is also $|e(\mathcal R)|$