{"paper":{"title":"Sharp Weitzenb\\\"{o}ck and PIC2 Estimates from Sectional-Scalar Curvature Pinching","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.DG","authors_text":"Jian Ge","submitted_at":"2026-07-20T17:51:14Z","abstract_excerpt":"Let $V$ be an $n$-dimensional Euclidean vector space, ,where $n\\ge 4$, and $\\ell = \\lfloor\\frac{n}{2}\\rfloor$. We prove the sharp pointwise estimate\n  \\[\n  q_2(E) \\ge -\\frac{2(\\ell -1)}{3\\ell} \\mathrm{Scal}(E) \\mathrm{Id}_{\\Lambda^2V^*}\n  \\]\n  for every algebraic curvature tensor $E$ on $V$ with nonnegative sectional curvature. Applying this estimate to the decomposition $\\operatorname{Rm}_{g}=K_{\\min}I+E$, we obtain the vanishing of $H^2(M; \\mathbb{R})$ under a dimension-dependent strict sectional-scalar curvature pinching condition. At the weak endpoint, all harmonic two-forms are parallel. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.18216","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.18216/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}