Vanishing results for the cohomology of complex toric hyperplane complements - M. W. Davis; Simona Settepanella (Scuola Superiore Sant'Anna, Laboratory of Economics and Management (LEM), 2011)
Suppose R is the complement of an essential arrangement of toric hyperlanes in the complex torus (C*)n and pi = pi1(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 Npi, or (c) the group ring Zpi. In case (a) the dimension of Hn is the Euler characteristic, e(R), and in case (b) the nth l2 Betti number is also e(R).
Keywords: 330; hyperplane arrangements; toric arrangements; local systems; L2-cohomology