{"paper":{"title":"Bott vanishing for Fano 3-folds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Burt Totaro","submitted_at":"2023-02-16T08:28:13Z","abstract_excerpt":"Bott proved a strong vanishing theorem for sheaf cohomology on projective space, namely that $H^j(X,\\Omega^i_X\\otimes L)=0$ for every $j>0$, $i\\geq 0$, and $L$ ample. This holds for toric varieties, but not for most other varieties.\n  We classify the smooth Fano 3-folds that satisfy Bott vanishing. There are many more than expected.\n  Along the way, we conjecture that for every projective birational morphism $\\pi\\colon X\\to Y$ of smooth varieties, and every line bundle $A$ on $X$ that is ample over $Y$, the higher direct image sheaf $R^j\\pi_*(\\Omega^i_X\\otimes A)$ is zero for every $j>0$ and $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2302.08142","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/2302.08142/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"}