{"paper":{"title":"K\\\"ahler-Ricci Flow on Projective Bundles over K\\\"ahler-Einstein Manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.CV"],"primary_cat":"math.DG","authors_text":"Frederick Tsz-Ho Fong","submitted_at":"2011-04-20T02:21:21Z","abstract_excerpt":"We study the K\\\"ahler-Ricci flow on a class of projective bundles $\\mathbb{P}(\\mathcal{O}_\\Sigma \\oplus L)$ over compact K\\\"ahler-Einstein manifold $\\Sigma^n$. Assuming the initial K\\\"ahler metric $\\omega_0$ admits a U(1)-invariant momentum profile, we give a criterion, characterized by the triple $(\\Sigma, L, [\\omega_0])$, under which the $\\mathbb{P}^1$-fiber collapses along the K\\\"ahler-Ricci flow and the projective bundle converges to $\\Sigma$ in Gromov-Hausdorff sense. Furthermore, the K\\\"ahler-Ricci flow must have Type I singularity and is of $(\\C^n \\times \\mathbb{P}^1)$-type. This genera"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.3924","kind":"arxiv","version":2},"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"}