{"paper":{"title":"Siu's curvature positivity and $L^2$ extension theorems for $(p,q)$-forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.DG"],"primary_cat":"math.CV","authors_text":"Gang Huang","submitted_at":"2026-07-25T08:11:36Z","abstract_excerpt":"In this paper, we introduce Siu's curvature operator \\(A^E_{p,q}\\) for vector-bundle-valued differential forms on K\\\"ahler manifolds. When $p=n$, this operator reduces to the classical Akizuki--Nakano curvature operator. We first characterize the semipositivity of \\(A^E_{p,q}\\) in terms of an optimal \\(L^2\\)-estimate condition for the \\(\\bar\\partial\\)-operator, and then prove an Ohsawa--Takegoshi-type extension theorem for \\(E\\)-valued \\((p,q)\\)-forms under the curvature condition \\(A^E_{p,q+1}\\geq0\\), using a new twisted basic estimate adapted to this setting. As an application, we prove the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.23094","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/2607.23094/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"}