{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:NPQ27CWJYOOKMHFVH627RRCJIJ","short_pith_number":"pith:NPQ27CWJ","schema_version":"1.0","canonical_sha256":"6be1af8ac9c39ca61cb53fb5f8c449425cadc506eb72b2e5938257233ff54c2a","source":{"kind":"arxiv","id":"2608.08193","version":1},"attestation_state":"computed","paper":{"title":"The sharp volume gap for K\\\"ahler manifolds with positive Ricci curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.MG"],"primary_cat":"math.DG","authors_text":"Chi Li, Kewei Zhang, Minghao Miao","submitted_at":"2026-08-08T15:37:47Z","abstract_excerpt":"We prove a sharp volume gap estimate: if an $n$-dimensional compact K\\\"ahler manifold $(X, \\omega)$ satisfies $\\mathrm{Ric}(\\omega)\\ge (n+1)\\omega$ and $X\\not\\cong \\mathbb{P}^n$, then $\\mathrm{vol}(X, \\omega)\\le \\frac{2n^n}{(n+1)^n}\\mathrm{vol}(\\mathbb{P}^n,\\omega_{\\mathrm{FS}})=\\frac{2^{n+1} \\, \\pi^n \\, n^n}{(n+1)^n}$. Moreover $\\mathrm{vol}(X, \\omega)= \\frac{2^{n+1} \\, \\pi^n \\, n^n}{(n+1)^n}$ occurs if and only if $(X, \\omega)$ is biholomorphically isometric to the K\\\"ahler-Einstein metric on the quadric hypersurface $Q^n$ or on the product $\\mathbb{P}^1\\times \\mathbb{P}^{n-1}$. We also obta"},"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":"2608.08193","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-08-08T15:37:47Z","cross_cats_sorted":["math.AG","math.MG"],"title_canon_sha256":"f30544ef5e724a84a0cbf4258af9a1649a4eec595c5f15bbec9b535b9f1486c5","abstract_canon_sha256":"246cb1b77eb356aeb0ec06f48e2d314e1381d1a4db203b7b7509b09f07329c5b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-11T01:21:56.231648Z","signature_b64":"98+KRdzR4+kHPbOABPUuCwEiu0sPhfFm9yiO1EoBo6xCWrCd1/hmW2pY9bjA9gjzPs04J7FwrAeI5OH3YpE5Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6be1af8ac9c39ca61cb53fb5f8c449425cadc506eb72b2e5938257233ff54c2a","last_reissued_at":"2026-08-11T01:21:56.229045Z","signature_status":"signed_v1","first_computed_at":"2026-08-11T01:21:56.229045Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The sharp volume gap for K\\\"ahler manifolds with positive Ricci curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.MG"],"primary_cat":"math.DG","authors_text":"Chi Li, Kewei Zhang, Minghao Miao","submitted_at":"2026-08-08T15:37:47Z","abstract_excerpt":"We prove a sharp volume gap estimate: if an $n$-dimensional compact K\\\"ahler manifold $(X, \\omega)$ satisfies $\\mathrm{Ric}(\\omega)\\ge (n+1)\\omega$ and $X\\not\\cong \\mathbb{P}^n$, then $\\mathrm{vol}(X, \\omega)\\le \\frac{2n^n}{(n+1)^n}\\mathrm{vol}(\\mathbb{P}^n,\\omega_{\\mathrm{FS}})=\\frac{2^{n+1} \\, \\pi^n \\, n^n}{(n+1)^n}$. Moreover $\\mathrm{vol}(X, \\omega)= \\frac{2^{n+1} \\, \\pi^n \\, n^n}{(n+1)^n}$ occurs if and only if $(X, \\omega)$ is biholomorphically isometric to the K\\\"ahler-Einstein metric on the quadric hypersurface $Q^n$ or on the product $\\mathbb{P}^1\\times \\mathbb{P}^{n-1}$. We also obta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.08193","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/2608.08193/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2608.08193","created_at":"2026-08-11T01:21:56.229934+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.08193v1","created_at":"2026-08-11T01:21:56.229934+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.08193","created_at":"2026-08-11T01:21:56.229934+00:00"},{"alias_kind":"pith_short_12","alias_value":"NPQ27CWJYOOK","created_at":"2026-08-11T01:21:56.229934+00:00"},{"alias_kind":"pith_short_16","alias_value":"NPQ27CWJYOOKMHFV","created_at":"2026-08-11T01:21:56.229934+00:00"},{"alias_kind":"pith_short_8","alias_value":"NPQ27CWJ","created_at":"2026-08-11T01:21:56.229934+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/NPQ27CWJYOOKMHFVH627RRCJIJ","json":"https://pith.science/pith/NPQ27CWJYOOKMHFVH627RRCJIJ.json","graph_json":"https://pith.science/api/pith-number/NPQ27CWJYOOKMHFVH627RRCJIJ/graph.json","events_json":"https://pith.science/api/pith-number/NPQ27CWJYOOKMHFVH627RRCJIJ/events.json","paper":"https://pith.science/paper/NPQ27CWJ"},"agent_actions":{"view_html":"https://pith.science/pith/NPQ27CWJYOOKMHFVH627RRCJIJ","download_json":"https://pith.science/pith/NPQ27CWJYOOKMHFVH627RRCJIJ.json","view_paper":"https://pith.science/paper/NPQ27CWJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.08193&json=true","fetch_graph":"https://pith.science/api/pith-number/NPQ27CWJYOOKMHFVH627RRCJIJ/graph.json","fetch_events":"https://pith.science/api/pith-number/NPQ27CWJYOOKMHFVH627RRCJIJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NPQ27CWJYOOKMHFVH627RRCJIJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NPQ27CWJYOOKMHFVH627RRCJIJ/action/storage_attestation","attest_author":"https://pith.science/pith/NPQ27CWJYOOKMHFVH627RRCJIJ/action/author_attestation","sign_citation":"https://pith.science/pith/NPQ27CWJYOOKMHFVH627RRCJIJ/action/citation_signature","submit_replication":"https://pith.science/pith/NPQ27CWJYOOKMHFVH627RRCJIJ/action/replication_record"}},"created_at":"2026-08-11T01:21:56.229934+00:00","updated_at":"2026-08-11T01:21:56.229934+00:00"}