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Assume that the GIT quotient $M/\\!\\!/G$ is a nonempty set. We prove that the homomorphism of algebraic fundamental groups $\\pi_1(M)\\, \\longrightarrow\\, \\pi_1(M/\\!\\!/G)$, induced by the rational map $M\\, \\longrightarrow\\, M/\\!\\!/G$, is an isomorphism.\n  If $k\\,=\\, \\mathbb C$, then we show that the above rational map $M\\, \\longrightarrow \\, M/\\!\\!/G$"},"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":"1410.5156","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2014-10-20T05:08:15Z","cross_cats_sorted":[],"title_canon_sha256":"5e7c9b3b380064a34103817fdbe48f430a64c550b6e63bd46c9b2cf1723134b2","abstract_canon_sha256":"2f8af16dcaaf6456bdb8f992277569300f22824f3cd39f1feaea0c46a511ff8a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:39:46.726588Z","signature_b64":"811E+UAlZWN7MdTnd8e2NSw6sikYv6dvYDOWhFfcLXYmEMIVqGBqDTFOy8DNc+zK4qWDFB0j3sVunxMD+O1VDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"956b072d76e21fe675df2e533976f2c8a67182c26e22687ab3c46be245228cd7","last_reissued_at":"2026-05-18T02:39:46.726061Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:39:46.726061Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fundamental group of a geometric invariant theoretic quotient","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"A. J. Parameswaran, Amit Hogadi, Indranil Biswas","submitted_at":"2014-10-20T05:08:15Z","abstract_excerpt":"Let $M$ be an irreducible smooth projective variety, defined over an algebraically closed field, equipped with an action of a connected reductive affine algebraic group $G$, and let ${\\mathcal L}$ be a $G$--equivariant very ample line bundle on $M$. Assume that the GIT quotient $M/\\!\\!/G$ is a nonempty set. We prove that the homomorphism of algebraic fundamental groups $\\pi_1(M)\\, \\longrightarrow\\, \\pi_1(M/\\!\\!/G)$, induced by the rational map $M\\, \\longrightarrow\\, M/\\!\\!/G$, is an isomorphism.\n  If $k\\,=\\, \\mathbb C$, then we show that the above rational map $M\\, \\longrightarrow \\, M/\\!\\!/G$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.5156","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":"1410.5156","created_at":"2026-05-18T02:39:46.726163+00:00"},{"alias_kind":"arxiv_version","alias_value":"1410.5156v1","created_at":"2026-05-18T02:39:46.726163+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.5156","created_at":"2026-05-18T02:39:46.726163+00:00"},{"alias_kind":"pith_short_12","alias_value":"SVVQOLLW4IP6","created_at":"2026-05-18T12:28:49.207871+00:00"},{"alias_kind":"pith_short_16","alias_value":"SVVQOLLW4IP6M5O7","created_at":"2026-05-18T12:28:49.207871+00:00"},{"alias_kind":"pith_short_8","alias_value":"SVVQOLLW","created_at":"2026-05-18T12:28:49.207871+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/SVVQOLLW4IP6M5O7FZJTS5XSZC","json":"https://pith.science/pith/SVVQOLLW4IP6M5O7FZJTS5XSZC.json","graph_json":"https://pith.science/api/pith-number/SVVQOLLW4IP6M5O7FZJTS5XSZC/graph.json","events_json":"https://pith.science/api/pith-number/SVVQOLLW4IP6M5O7FZJTS5XSZC/events.json","paper":"https://pith.science/paper/SVVQOLLW"},"agent_actions":{"view_html":"https://pith.science/pith/SVVQOLLW4IP6M5O7FZJTS5XSZC","download_json":"https://pith.science/pith/SVVQOLLW4IP6M5O7FZJTS5XSZC.json","view_paper":"https://pith.science/paper/SVVQOLLW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1410.5156&json=true","fetch_graph":"https://pith.science/api/pith-number/SVVQOLLW4IP6M5O7FZJTS5XSZC/graph.json","fetch_events":"https://pith.science/api/pith-number/SVVQOLLW4IP6M5O7FZJTS5XSZC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SVVQOLLW4IP6M5O7FZJTS5XSZC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SVVQOLLW4IP6M5O7FZJTS5XSZC/action/storage_attestation","attest_author":"https://pith.science/pith/SVVQOLLW4IP6M5O7FZJTS5XSZC/action/author_attestation","sign_citation":"https://pith.science/pith/SVVQOLLW4IP6M5O7FZJTS5XSZC/action/citation_signature","submit_replication":"https://pith.science/pith/SVVQOLLW4IP6M5O7FZJTS5XSZC/action/replication_record"}},"created_at":"2026-05-18T02:39:46.726163+00:00","updated_at":"2026-05-18T02:39:46.726163+00:00"}