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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, "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1207.5120","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2012-07-21T10:53:10Z","cross_cats_sorted":[],"title_canon_sha256":"d20c8c5975ca76824d326e6d15d5343f27174c1fbb8d98bc638faf30fe74b001","abstract_canon_sha256":"58990f6a51040f21660d2ec58ae57db661bff4e0497775c195caac00cdc2ba08"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:34:00.706545Z","signature_b64":"Pt4Pnr+6SBVbU+ma2BwMi/B14URlXESTKimaFQUcghmmiz42cKfCClHLEn6NJ3YjI048MHu8zW+i/VpPQWsGAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2281f5ad929def8ace11f7cf267183d31cc7a5c31e7c452974df6b552dc912a8","last_reissued_at":"2026-05-18T00:34:00.705748Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:34:00.705748Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $n \\leq 4$. 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