Open Access
2013 Vanishing results for the cohomology of complex toric hyperplane complements
M. W. Davis, S. Settepanella
Publ. Mat. 57(2): 379-392 (2013).


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 $|e(\mathcal R)|$


Download Citation

M. W. Davis. S. Settepanella. "Vanishing results for the cohomology of complex toric hyperplane complements." Publ. Mat. 57 (2) 379 - 392, 2013.


Published: 2013
First available in Project Euclid: 18 December 2013

zbMATH: 1287.52019
MathSciNet: MR3114774

Primary: 52B30
Secondary: 32S22 , 52C35 , 57N65 , 58J22

Keywords: $L^2$ cohomology , hyperplane arrangements , local systems , toric arrangements

Rights: Copyright © 2013 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.57 • No. 2 • 2013
Back to Top