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Vanishing results for the cohomology of complex toric hyperplane complements

Publicacions Matematiques, ISSN: 0214-1493, Vol: 57, Issue: 2, Page: 379-392
2013
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Suppose R is the complement of an essential arrangement of toric hyperlanes in the complex torus (ℂ) and π = π(R). We show that H(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 Nπ, or (c) the group ring ℤπ. In case (a) the dimension of H is |e(R)| where e(R) denotes the Euler characteristic, and in case (b) the n l Betti number is also |e(R)|.

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