{"paper":{"title":"The Lefschetz Type Theorem For Fundamental Group Schemes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Langer positivity assumptions ensure that fundamental group schemes of ample divisors are isomorphic to those of the ambient schemes.","cross_cats":[],"primary_cat":"math.AG","authors_text":"Lingguang Li, Niantao Tian","submitted_at":"2026-04-21T15:04:53Z","abstract_excerpt":"Let $k$ be a field, $X$ a connected scheme proper over $k$, $D\\subsetneq X$ an ample effective connected divisor, $x\\in D(k)$. For Tannakian categories $\\mathcal{C}_X$ and $\\mathcal{C}_D$ whose objects consist of vector bundles on $X$ and $D$ respectively, we establish general Tannakian criteria for the natural homomorphism \\(\\pi(\\mathcal{C}_D,x)\\to \\pi(\\mathcal{C}_X,x)\\) to be faithfully flat, a closed immersion, or an isomorphism. As applications, under Langer type positivity assumptions, we prove that \\(\\pi^{\\ast}(D,x)\\longrightarrow \\pi^{\\ast}(X,x)\\) is an isomorphism for $\\ast\\in\\{S,N,EN,"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"under Langer type positivity assumptions, we prove that π∗(D,x)⟶π∗(X,x) is an isomorphism for ∗∈{S,N,EN,F, EF,Loc,ELoc,ét,Eét,uni} over perfect fields.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The Langer-type positivity assumptions on the divisor D that are invoked for the isomorphism statements.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Under Langer-type positivity assumptions the fundamental group scheme of an ample divisor D is isomorphic to that of X for many variants including etale, unipotent, and local versions over perfect fields.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Langer positivity assumptions ensure that fundamental group schemes of ample divisors are isomorphic to those of the ambient schemes.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"bca3f5ac8666faa80b973527b23e823be3da0f01966bffb5dfcc7d1d1dc06a4c"},"source":{"id":"2604.19546","kind":"arxiv","version":3},"verdict":{"id":"907e3d4b-e47d-42a6-b73a-bd477eeddb0a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T01:54:02.650228Z","strongest_claim":"under Langer type positivity assumptions, we prove that π∗(D,x)⟶π∗(X,x) is an isomorphism for ∗∈{S,N,EN,F, EF,Loc,ELoc,ét,Eét,uni} over perfect fields.","one_line_summary":"Under Langer-type positivity assumptions the fundamental group scheme of an ample divisor D is isomorphic to that of X for many variants including etale, unipotent, and local versions over perfect fields.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The Langer-type positivity assumptions on the divisor D that are invoked for the isomorphism statements.","pith_extraction_headline":"Langer positivity assumptions ensure that fundamental group schemes of ample divisors are isomorphic to those of the ambient schemes."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.19546/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-20T02:49:02.471029Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"0f6ea496a2ceb61d506fa4be1d0dcc341263f4e2344b4d1d0eaf844aa121c149"},"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"}