{"paper":{"title":"On the volume of K-semistable Fano manifolds","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AG","authors_text":"Chi Li, Minghao Miao","submitted_at":"2025-06-20T18:35:27Z","abstract_excerpt":"We prove that the anti-canonical volume of an $n$-dimensional K-semistable Fano manifold that is not $\\mathbb{P}^n$ is at most $2n^n$. Moreover, the volume is equal to $2n^n$ if and only if $X\\cong \\mathbb{P}^1\\times \\mathbb{P}^{n-1}$ or $X$ is a smooth quadric hypersurface $Q\\subset \\mathbb{P}^{n+1}$. Our proof is based on a new connection between K-semistability and minimal rational curves."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.17420","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.17420/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"}