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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 $|e(\\Cal R)|$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1111.2866","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2011-11-11T21:12:27Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"d234a5e2be86bf5236eed0e0e84deca7156f91d26eaa0f20ac541fbba116a1ae","abstract_canon_sha256":"312d111932931965edafbe3fc45ea771b9b82c427d81756a619499de60335134"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:47:05.972575Z","signature_b64":"7WW366aKPyOPL4qThgDmf+BSdCwg9DpnovqeMQ+tOSI+Eh3aNLawx3PdJGa15d8+pftlh0zM06coF3vepMtgAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2f218b766fa447d562decb230a36aa02e724b697a8746499df475473721687e3","last_reissued_at":"2026-05-18T02:47:05.972143Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:47:05.972143Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1111.2866","created_at":"2026-05-18T02:47:05.972203+00:00"},{"alias_kind":"arxiv_version","alias_value":"1111.2866v2","created_at":"2026-05-18T02:47:05.972203+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.2866","created_at":"2026-05-18T02:47:05.972203+00:00"},{"alias_kind":"pith_short_12","alias_value":"F4QYW5TPURD5","created_at":"2026-05-18T12:26:28.662955+00:00"},{"alias_kind":"pith_short_16","alias_value":"F4QYW5TPURD5KYW6","created_at":"2026-05-18T12:26:28.662955+00:00"},{"alias_kind":"pith_short_8","alias_value":"F4QYW5TP","created_at":"2026-05-18T12:26:28.662955+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/F4QYW5TPURD5KYW6ZMRQUNVKAL","json":"https://pith.science/pith/F4QYW5TPURD5KYW6ZMRQUNVKAL.json","graph_json":"https://pith.science/api/pith-number/F4QYW5TPURD5KYW6ZMRQUNVKAL/graph.json","events_json":"https://pith.science/api/pith-number/F4QYW5TPURD5KYW6ZMRQUNVKAL/events.json","paper":"https://pith.science/paper/F4QYW5TP"},"agent_actions":{"view_html":"https://pith.science/pith/F4QYW5TPURD5KYW6ZMRQUNVKAL","download_json":"https://pith.science/pith/F4QYW5TPURD5KYW6ZMRQUNVKAL.json","view_paper":"https://pith.science/paper/F4QYW5TP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1111.2866&json=true","fetch_graph":"https://pith.science/api/pith-number/F4QYW5TPURD5KYW6ZMRQUNVKAL/graph.json","fetch_events":"https://pith.science/api/pith-number/F4QYW5TPURD5KYW6ZMRQUNVKAL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/F4QYW5TPURD5KYW6ZMRQUNVKAL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/F4QYW5TPURD5KYW6ZMRQUNVKAL/action/storage_attestation","attest_author":"https://pith.science/pith/F4QYW5TPURD5KYW6ZMRQUNVKAL/action/author_attestation","sign_citation":"https://pith.science/pith/F4QYW5TPURD5KYW6ZMRQUNVKAL/action/citation_signature","submit_replication":"https://pith.science/pith/F4QYW5TPURD5KYW6ZMRQUNVKAL/action/replication_record"}},"created_at":"2026-05-18T02:47:05.972203+00:00","updated_at":"2026-05-18T02:47:05.972203+00:00"}