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 $|e(Cal R)|$.

Vanishing results for the cohomology of complex toric hyperplane complements

Settepanella S
2013-01-01

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 $cnpi$, 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 $|e(Cal R)|$.
57
2
379
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hyperplane arrangements, toric arrangements, local sys- tems, L2-cohomology.
Davis M W; Settepanella S
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/1843496
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