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Let $X , Y$ be irreducible smooth complex projective varieties and $f: X \\rightarrow Y$ an algebraic morphism, such that $\\pi_1(Y)$ is virtually nilpotent and the homomorphism $f_* : \\pi_1(X) \\rightarrow\\pi_1(Y)$ is surjective. Define $$ {\\mathcal R }^f(\\pi_1(X),\\, G)\\,=\\, \\{\\rho\\, \\in\\, \\text{Hom}(\\pi_1(X),\\, G)\\, \\mid\\, A\\circ\\rho \\ \\text{ factors through }~ f_*\\}\\, , $$ $$ {\\mathcal R }^f(\\pi_1(X),\\, K)\\,=\\, \\{\\rho\\, \\in\\, \\text{Hom}(\\pi_1(X),\\, K)\\, \\mid"},"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":"1507.04568","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-07-16T13:40:29Z","cross_cats_sorted":[],"title_canon_sha256":"db0271cdb54c3d64be1f25acd62ad8a9b2cdde23939bba932223ef103404cb42","abstract_canon_sha256":"f4f9b189d0e74524a6cc041175658021260681325999a1412d9618a8747c675d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:47.002095Z","signature_b64":"EpxdznH9HaQ96yv5x8cCEJmjUpxxjz3WJq0J1I/xgcVe3tf2TJmLIwa3roJiC+s7eyZusr99rYMFG7XbMJb3Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eb93882c343cfb20a998010ad108a87ab076903c0cf663682c98dd05a3f1cc90","last_reissued_at":"2026-05-18T01:36:47.001605Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:47.001605Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Higgs bundles and representation spaces associated to morphisms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Carlos Florentino, Indranil Biswas","submitted_at":"2015-07-16T13:40:29Z","abstract_excerpt":"Let $G$ be a connected reductive affine algebraic group defined over the complex numbers, and $K\\subset G$ be a maximal compact subgroup. Let $X , Y$ be irreducible smooth complex projective varieties and $f: X \\rightarrow Y$ an algebraic morphism, such that $\\pi_1(Y)$ is virtually nilpotent and the homomorphism $f_* : \\pi_1(X) \\rightarrow\\pi_1(Y)$ is surjective. Define $$ {\\mathcal R }^f(\\pi_1(X),\\, G)\\,=\\, \\{\\rho\\, \\in\\, \\text{Hom}(\\pi_1(X),\\, G)\\, \\mid\\, A\\circ\\rho \\ \\text{ factors through }~ f_*\\}\\, , $$ $$ {\\mathcal R }^f(\\pi_1(X),\\, K)\\,=\\, \\{\\rho\\, \\in\\, \\text{Hom}(\\pi_1(X),\\, K)\\, \\mid"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.04568","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1507.04568","created_at":"2026-05-18T01:36:47.001679+00:00"},{"alias_kind":"arxiv_version","alias_value":"1507.04568v1","created_at":"2026-05-18T01:36:47.001679+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.04568","created_at":"2026-05-18T01:36:47.001679+00:00"},{"alias_kind":"pith_short_12","alias_value":"5OJYQLBUHT5S","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_16","alias_value":"5OJYQLBUHT5SBKMY","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_8","alias_value":"5OJYQLBU","created_at":"2026-05-18T12:29:05.191682+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5OJYQLBUHT5SBKMYAEFNCCFIPK","json":"https://pith.science/pith/5OJYQLBUHT5SBKMYAEFNCCFIPK.json","graph_json":"https://pith.science/api/pith-number/5OJYQLBUHT5SBKMYAEFNCCFIPK/graph.json","events_json":"https://pith.science/api/pith-number/5OJYQLBUHT5SBKMYAEFNCCFIPK/events.json","paper":"https://pith.science/paper/5OJYQLBU"},"agent_actions":{"view_html":"https://pith.science/pith/5OJYQLBUHT5SBKMYAEFNCCFIPK","download_json":"https://pith.science/pith/5OJYQLBUHT5SBKMYAEFNCCFIPK.json","view_paper":"https://pith.science/paper/5OJYQLBU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1507.04568&json=true","fetch_graph":"https://pith.science/api/pith-number/5OJYQLBUHT5SBKMYAEFNCCFIPK/graph.json","fetch_events":"https://pith.science/api/pith-number/5OJYQLBUHT5SBKMYAEFNCCFIPK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5OJYQLBUHT5SBKMYAEFNCCFIPK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5OJYQLBUHT5SBKMYAEFNCCFIPK/action/storage_attestation","attest_author":"https://pith.science/pith/5OJYQLBUHT5SBKMYAEFNCCFIPK/action/author_attestation","sign_citation":"https://pith.science/pith/5OJYQLBUHT5SBKMYAEFNCCFIPK/action/citation_signature","submit_replication":"https://pith.science/pith/5OJYQLBUHT5SBKMYAEFNCCFIPK/action/replication_record"}},"created_at":"2026-05-18T01:36:47.001679+00:00","updated_at":"2026-05-18T01:36:47.001679+00:00"}