{"paper":{"title":"Scalar curvature and uniruledness on projective manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV","math.DG"],"primary_cat":"math.AG","authors_text":"Bun Wong, Gordon Heier","submitted_at":"2012-06-12T16:03:27Z","abstract_excerpt":"It is a basic tenet in complex geometry that {\\it negative} curvature corresponds, in a suitable sense, to the absence of rational curves on, say, a complex projective manifold, while {\\it positive} curvature corresponds to the abundance of rational curves. In this spirit, we prove in this note that a projective manifold $M$ with a K\\\"ahler metric with positive total scalar curvature is uniruled, which is equivalent to every point of $M$ being contained in a rational curve. We also prove that if $M$ possesses a K\\\"ahler metric of total scalar curvature equal to zero, then either $M$ is unirule"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.2576","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":""},"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"}