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For example, we prove that if $E$ is an ample vector bundle over a compact K\\\"ahler manifold $X$, $S^kE\\ts \\det E$ is both Nakano-positive and dual-Nakano-positive for any $k\\geq 0$. Moreover, $H^{n,q}(X,S^kE\\ts \\det E)=H^{q,n}(X,S^kE\\ts \\det E)=0$ for any $q\\geq 1$. In particular, if $(E,h)$ is a Griffiths-positive vector bundle, the naturally induced Hermitian vector bundle $(S"},"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":"1006.1465","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-06-08T06:41:26Z","cross_cats_sorted":["math.AG","math.CV"],"title_canon_sha256":"a28344902024e75d30722587eb9287a05087e5fe945a11b7dee5a486f7439cb2","abstract_canon_sha256":"9566f3124edc84992cb280ce96c1c5e18510c8313d9ab1eb753d13143e6a3f0c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:25:28.329058Z","signature_b64":"bjznXqVL/yf5Cw0CiwMrFbLEcUyEgrfmcJwD9s++yvhQckVWwxwMqbgm3kDrifwMaXm97KlkBlHMzj9KM4eJCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9c84c99210e6f6bb0ffbe9f0102d2ca08941571628938a11765e6c1f7679150e","last_reissued_at":"2026-05-18T04:25:28.328338Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:25:28.328338Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Positivity and vanishing theorems for ample vector bundles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.CV"],"primary_cat":"math.DG","authors_text":"Kefeng Liu, Xiaofeng Sun, Xiaokui Yang","submitted_at":"2010-06-08T06:41:26Z","abstract_excerpt":"In this paper, we study the Nakano-positivity and dual-Nakano-positivity of certain adjoint vector bundles associated to ample vector bundles. As applications, we get new vanishing theorems about ample vector bundles. For example, we prove that if $E$ is an ample vector bundle over a compact K\\\"ahler manifold $X$, $S^kE\\ts \\det E$ is both Nakano-positive and dual-Nakano-positive for any $k\\geq 0$. Moreover, $H^{n,q}(X,S^kE\\ts \\det E)=H^{q,n}(X,S^kE\\ts \\det E)=0$ for any $q\\geq 1$. In particular, if $(E,h)$ is a Griffiths-positive vector bundle, the naturally induced Hermitian vector bundle $(S"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.1465","kind":"arxiv","version":3},"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":"1006.1465","created_at":"2026-05-18T04:25:28.328457+00:00"},{"alias_kind":"arxiv_version","alias_value":"1006.1465v3","created_at":"2026-05-18T04:25:28.328457+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1006.1465","created_at":"2026-05-18T04:25:28.328457+00:00"},{"alias_kind":"pith_short_12","alias_value":"TSCMTEQQ433L","created_at":"2026-05-18T12:26:15.391820+00:00"},{"alias_kind":"pith_short_16","alias_value":"TSCMTEQQ433LWD73","created_at":"2026-05-18T12:26:15.391820+00:00"},{"alias_kind":"pith_short_8","alias_value":"TSCMTEQQ","created_at":"2026-05-18T12:26:15.391820+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/TSCMTEQQ433LWD735HYBALJMUC","json":"https://pith.science/pith/TSCMTEQQ433LWD735HYBALJMUC.json","graph_json":"https://pith.science/api/pith-number/TSCMTEQQ433LWD735HYBALJMUC/graph.json","events_json":"https://pith.science/api/pith-number/TSCMTEQQ433LWD735HYBALJMUC/events.json","paper":"https://pith.science/paper/TSCMTEQQ"},"agent_actions":{"view_html":"https://pith.science/pith/TSCMTEQQ433LWD735HYBALJMUC","download_json":"https://pith.science/pith/TSCMTEQQ433LWD735HYBALJMUC.json","view_paper":"https://pith.science/paper/TSCMTEQQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1006.1465&json=true","fetch_graph":"https://pith.science/api/pith-number/TSCMTEQQ433LWD735HYBALJMUC/graph.json","fetch_events":"https://pith.science/api/pith-number/TSCMTEQQ433LWD735HYBALJMUC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TSCMTEQQ433LWD735HYBALJMUC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TSCMTEQQ433LWD735HYBALJMUC/action/storage_attestation","attest_author":"https://pith.science/pith/TSCMTEQQ433LWD735HYBALJMUC/action/author_attestation","sign_citation":"https://pith.science/pith/TSCMTEQQ433LWD735HYBALJMUC/action/citation_signature","submit_replication":"https://pith.science/pith/TSCMTEQQ433LWD735HYBALJMUC/action/replication_record"}},"created_at":"2026-05-18T04:25:28.328457+00:00","updated_at":"2026-05-18T04:25:28.328457+00:00"}