{"paper":{"title":"Vanishing results for the cohomology of complex toric hyperplane complements","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.AT","authors_text":"M. W. Davis, S. Settepanella","submitted_at":"2011-11-11T21:12:27Z","abstract_excerpt":"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 $|e(\\Cal R)|$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.2866","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}