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For Tannakian categories $\\mathcal{C}_X \\subset \\mathfrak{Vect}(X)$ and $\\mathcal{C}_Y \\subset \\mathfrak{Vect}(Y)$, denote by $\\pi(\\mathcal{C}_X,x)$ and $\\pi(\\mathcal{C}_Y,y)$ the corresponding Tannaka group schemes. We establish a unified Tannakian criteria for the natural homomorphism $\\pi(\\mathcal{C}_X,x)\\to \\pi(\\mathcal{C}_Y,y)$ to be an isomorphism. As applications, for a birational map $X \\dashrightarrow Y$ between smooth projec","authors_text":"Hao Wang, Lingguang Li","cross_cats":[],"headline":"Various fundamental group schemes are birationally invariant for smooth projective varieties over perfect fields.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2026-04-27T03:23:31Z","title":"The Birational Invariance Of Fundamental Group Schemes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.23997","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-08T02:26:25.131862Z","id":"4c48b49a-54c8-48e3-9bb8-d3d98d17274a","model_set":{"reader":"grok-4.3"},"one_line_summary":"Various fundamental group schemes are birationally invariant for smooth projective varieties over perfect fields.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"","strongest_claim":"For a birational map X ⇢ Y between smooth projective varieties over a perfect field k, there exists a natural isomorphism π^*(X,x) ≅ π^*(Y,y) for any * ∈ {S,N,EN,F,EF,Loc,ELoc,ét, Eét,uni}. In particular, the induced homomorphism π^str(X,x) → π^str(Y,y) is an isomorphism for any birational morphism X → Y.","weakest_assumption":"Y is normal, the schemes are integral connected and proper over k, and the Tannakian categories C_X and C_Y satisfy the general criteria making the natural homomorphism an isomorphism; for the main application the varieties must be smooth projective over a perfect field."}},"verdict_id":"4c48b49a-54c8-48e3-9bb8-d3d98d17274a"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:dece26084411d477de79c5c5b61b8a3df4060953c38f5ca799eef3b4dceb8b0e","target":"record","created_at":"2026-06-10T01:11:00Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"b0661c5fb480b061fbd38a7facd5deecd8384222c7427027ee0823520c4e9ee8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2026-04-27T03:23:31Z","title_canon_sha256":"8114a1bcf88ffde7f45177e9722524ca95ad20c79eeb99dd94282d36ce59060b"},"schema_version":"1.0","source":{"id":"2604.23997","kind":"arxiv","version":2}},"canonical_sha256":"66a4d68a421edd98de54b183c13c7c178c3246f2f49688dbc95712d34b7a0328","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"66a4d68a421edd98de54b183c13c7c178c3246f2f49688dbc95712d34b7a0328","first_computed_at":"2026-06-10T01:11:00.696546Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-10T01:11:00.696546Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/1svKLFv6wPhvFIx299ik67i8IAbEUFHOMs1/wAMjRBdMdRrlQpWzLy7MDl3oYoLiMV0tgUL4+OXchXolidMAg==","signature_status":"signed_v1","signed_at":"2026-06-10T01:11:00.697633Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.23997","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dece26084411d477de79c5c5b61b8a3df4060953c38f5ca799eef3b4dceb8b0e","sha256:875a338b3158322dfc6dd462a0a1c15acc07401c484323566bc689d9f0aaca67"],"state_sha256":"77e0304c858602e3e41dba8fee931b8ac8c34b3d98a73a2bb4b5bd3dab442764"}