{"paper":{"title":"Hirzebruch $\\chi_{y}$-genus of compact almost K\\\"{a}hler manifold with negative sectional curvature","license":"http://creativecommons.org/licenses/by/4.0/","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.","cross_cats":[],"primary_cat":"math.DG","authors_text":"Pan Zhang, Teng Huang","submitted_at":"2026-04-30T04:53:47Z","abstract_excerpt":"Let \\((X,J,\\omega)\\) be a closed \\(2n\\)-dimensional almost K\\\"{a}hler manifold with negative sectional curvature. 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. In particular, this result implies the Hopf conjecture in this setting, namely that the Euler number satisfies (-1)^n χ(X)≥n+1.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"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.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"faacddd89bc52face9adc221e8f72d06af5cf22b5013244f36703845f92bb7e0"},"source":{"id":"2604.27423","kind":"arxiv","version":2},"verdict":{"id":"ff20ba1b-7b08-4ee9-81e5-0dea686a8c79","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T08:49:12.474834Z","strongest_claim":"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. 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"}