{"paper":{"title":"Families of smooth Fano fourfolds of Picard rank 1 without Bott vanishing","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Jiahe Wang","submitted_at":"2026-06-11T15:19:16Z","abstract_excerpt":"We show that $\\chi(X,T_X)<0$ for the currently known families of smooth Fano fourfolds of Picard rank $1$ and index $1$. Combining this with the known Picard rank $1$ index $> 1$ cases, we show that among all currently known smooth Fano fourfolds of Picard rank $1$, the only variety satisfying Bott vanishing is the projective space. By a result of Kawakami--Totaro, the existence of an endomorphism of degree greater than 1 implies Bott vanishing. Therefore, among the currently known smooth Fano fourfolds of Picard rank $1$, any variety admitting an endomorphism of degree greater than 1 must be "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.13466","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/2606.13466/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"}