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We prove that if the Nijenhuis tensor of the almost complex structure is sufficiently small, then the components of the Hirzebruch \\(\\chi_{y}\\)-genus satisfy the inequality \\((-1)^{n-p}\\chi_{p}(X)\\geq 1\\) for all \\(p=0,1,\\cdots,n\\). In particular, this result implies the Hopf conjecture in this setting, namely that the Euler number satisfies \\((-1)^{n}\\chi(X)\\geq n+1\\). 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We prove that if the Nijenhuis tensor of the almost complex structure is sufficiently small, then the components of the Hirzebruch \\(\\chi_{y}\\)-genus satisfy the inequality \\((-1)^{n-p}\\chi_{p}(X)\\geq 1\\) for all \\(p=0,1,\\cdots,n\\). In particular, this result implies the Hopf conjecture in this setting, namely that the Euler number satisfies \\((-1)^{n}\\chi(X)\\geq n+1\\). The proof is based on new \\(L^{2}\\)-estimates for harmonic forms on the universal covering, combined with a refin"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that if the Nijenhuis tensor of the almost complex structure is sufficiently small, then the components of the Hirzebruch χ_y-genus satisfy the inequality (-1)^{n-p}χ_p(X)≥1 for all p=0,1,⋯,n. 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In particular, this result implies the Hopf conjecture in this setting, namely that the Euler number satisfies (-1)^n χ(X)≥n+1.","one_line_summary":"For compact almost Kähler manifolds with negative sectional curvature and sufficiently small Nijenhuis tensor, the Hirzebruch χ_y-genus components satisfy (-1)^{n-p} χ_p(X) ≥ 1 for all p, implying the Hopf conjecture (-1)^n χ(X) ≥ n+1.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The assumption that the Nijenhuis tensor is 'sufficiently small' (a qualitative rather than quantitative bound) so that the new L² estimates and refined vanishing theorem apply; the abstract gives no explicit threshold or verification that such smallness is compatible with negative sectional curvature on a closed manifold.","pith_extraction_headline":"If the Nijenhuis tensor is sufficiently small, then the Hirzebruch χ_y-genus of a closed almost Kähler manifold with negative sectional curvature has components satisfying (-1)^{n-p} χ_p(X) ≥ 1 for each p."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.27423/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T22:36:38.575245Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T19:15:41.039555Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"3d723c1f26b47404df51287064d6144079190e89b60f13338b0b63815065566e"},"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":"2604.27423","created_at":"2026-05-26T01:03:31.411094+00:00"},{"alias_kind":"arxiv_version","alias_value":"2604.27423v2","created_at":"2026-05-26T01:03:31.411094+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.27423","created_at":"2026-05-26T01:03:31.411094+00:00"},{"alias_kind":"pith_short_12","alias_value":"NAIHL6D6V5CL","created_at":"2026-05-26T01:03:31.411094+00:00"},{"alias_kind":"pith_short_16","alias_value":"NAIHL6D6V5CL3AWV","created_at":"2026-05-26T01:03:31.411094+00:00"},{"alias_kind":"pith_short_8","alias_value":"NAIHL6D6","created_at":"2026-05-26T01:03:31.411094+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/NAIHL6D6V5CL3AWVUN25D5OSRV","json":"https://pith.science/pith/NAIHL6D6V5CL3AWVUN25D5OSRV.json","graph_json":"https://pith.science/api/pith-number/NAIHL6D6V5CL3AWVUN25D5OSRV/graph.json","events_json":"https://pith.science/api/pith-number/NAIHL6D6V5CL3AWVUN25D5OSRV/events.json","paper":"https://pith.science/paper/NAIHL6D6"},"agent_actions":{"view_html":"https://pith.science/pith/NAIHL6D6V5CL3AWVUN25D5OSRV","download_json":"https://pith.science/pith/NAIHL6D6V5CL3AWVUN25D5OSRV.json","view_paper":"https://pith.science/paper/NAIHL6D6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2604.27423&json=true","fetch_graph":"https://pith.science/api/pith-number/NAIHL6D6V5CL3AWVUN25D5OSRV/graph.json","fetch_events":"https://pith.science/api/pith-number/NAIHL6D6V5CL3AWVUN25D5OSRV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NAIHL6D6V5CL3AWVUN25D5OSRV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NAIHL6D6V5CL3AWVUN25D5OSRV/action/storage_attestation","attest_author":"https://pith.science/pith/NAIHL6D6V5CL3AWVUN25D5OSRV/action/author_attestation","sign_citation":"https://pith.science/pith/NAIHL6D6V5CL3AWVUN25D5OSRV/action/citation_signature","submit_replication":"https://pith.science/pith/NAIHL6D6V5CL3AWVUN25D5OSRV/action/replication_record"}},"created_at":"2026-05-26T01:03:31.411094+00:00","updated_at":"2026-05-26T01:03:31.411094+00:00"}