{"paper":{"title":"Motives and the Hodge Conjecture for moduli spaces of pairs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Andr\\'e Oliveira, Jonathan S\\'anchez, Vicente Mu\\~noz","submitted_at":"2012-07-21T10:53:10Z","abstract_excerpt":"Let $C$ be a smooth projective curve of genus $g\\geq 2$ over $\\mathbb C$. Fix $n\\geq 1$, $d\\in {\\mathbb Z}$. A pair $(E,\\phi)$ over $C$ consists of an algebraic vector bundle $E$ of rank $n$ and degree $d$ over $C$ and a section $\\phi \\in H^0(E)$. There is a concept of stability for pairs which depends on a real parameter $\\tau$. Let ${\\mathfrak M}_\\tau(n,d)$ be the moduli space of $\\tau$-polystable pairs of rank $n$ and degree $d$ over $C$. Here we prove that for a generic curve $C$, the moduli space ${\\mathfrak M}_\\tau(n,d)$ satisfies the Hodge Conjecture for $n \\leq 4$. For obtaining this, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.5120","kind":"arxiv","version":3},"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"}