Vanishing results for the cohomology of complex toric hyperplane complements

TL;DR

This paper proves that the cohomology of complements of essential toric hyperplane arrangements vanishes outside the top degree under certain local coefficient systems, with explicit calculations of the top degree cohomology and Betti numbers.

Contribution

It establishes vanishing results for the cohomology of complex toric hyperplane complements with specific local coefficients, extending understanding of their topological structure.

Findings

Cohomology vanishes outside the top degree n for specified coefficients.

Top degree cohomology dimension equals the Euler characteristic.

The n-th L2 Betti number also equals the Euler characteristic.

Abstract

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 $\cn\pi$, 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 $|e(\Cal R)|$.