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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"},"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":"2604.23997","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2026-04-27T03:23:31Z","cross_cats_sorted":[],"title_canon_sha256":"8114a1bcf88ffde7f45177e9722524ca95ad20c79eeb99dd94282d36ce59060b","abstract_canon_sha256":"b0661c5fb480b061fbd38a7facd5deecd8384222c7427027ee0823520c4e9ee8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-10T01:11:00.697633Z","signature_b64":"/1svKLFv6wPhvFIx299ik67i8IAbEUFHOMs1/wAMjRBdMdRrlQpWzLy7MDl3oYoLiMV0tgUL4+OXchXolidMAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"66a4d68a421edd98de54b183c13c7c178c3246f2f49688dbc95712d34b7a0328","last_reissued_at":"2026-06-10T01:11:00.696546Z","signature_status":"signed_v1","first_computed_at":"2026-06-10T01:11:00.696546Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Birational Invariance Of Fundamental Group Schemes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Various fundamental group schemes are birationally invariant for smooth projective varieties over perfect fields.","cross_cats":[],"primary_cat":"math.AG","authors_text":"Hao Wang, Lingguang Li","submitted_at":"2026-04-27T03:23:31Z","abstract_excerpt":"Let $k$ be a field, $f \\colon X \\to Y$ a birational morphism of integral connected schemes proper over $k$ with $Y$ normal, $x \\in X(k)$ lying over $y \\in Y(k)$. 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"},"claims":{"count":3,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Various fundamental group schemes are birationally invariant for smooth projective varieties over perfect fields.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"}],"snapshot_sha256":"cc27bd6393f5d574145f0d43208523624a0dedac86c0451b328e9b5d9f7b4759"},"source":{"id":"2604.23997","kind":"arxiv","version":2},"verdict":{"id":"4c48b49a-54c8-48e3-9bb8-d3d98d17274a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T02:26:25.131862Z","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}. 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