{"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"}