{"paper":{"title":"On the K\\\"ahler structures over Quot schemes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Harish Seshadri, Indranil Biswas","submitted_at":"2014-01-29T03:46:19Z","abstract_excerpt":"Let $S^n(X)$ be the $n$-fold symmetric product of a compact connected Riemann surface $X$ of genus $g$ and gonality $d$. We prove that $S^n(X)$ admits a K\\\"ahler structure such that all the holomorphic bisectional curvatures are nonpositive if and only if $n < d$. Let ${\\mathcal Q}_X(r,n)$ be the Quot scheme parametrizing the torsion quotients of ${\\mathcal O}^{\\oplus r}_X$ of degree $n$. If $g \\geq 2$ and $n \\leq 2g-2$, we prove that ${\\mathcal Q}_X(r,n)$ does not admit a K\\\"ahler structure such that all the holomorphic bisectional curvatures are nonnegative."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.7408","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"}