The Lefschetz Type Theorem for Fundamental Group Schemes
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910168301699072 |
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| author | Li, Lingguang Tian, Niantao |
| author_facet | Li, Lingguang Tian, Niantao |
| contents | 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 \(π(\mathcal{C}_D,x)\longrightarrow π(\mathcal{C}_X,x)\) to be faithfully flat, a closed immersion, or an isomorphism. As applications, under Langer type positivity assumptions, we prove that \(π^{\ast}(D,x)\longrightarrow π^{\ast}(X,x)\) is an isomorphism for $\ast\in\{S,N,EN,F, EF,Loc,ELoc,\acute{e}t,E\acute{e}t,uni\}$ over perfect fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_19546 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Lefschetz Type Theorem for Fundamental Group Schemes Li, Lingguang Tian, Niantao Algebraic Geometry 14F35, 18M25, 14J60, 14L15 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 \(π(\mathcal{C}_D,x)\longrightarrow π(\mathcal{C}_X,x)\) to be faithfully flat, a closed immersion, or an isomorphism. As applications, under Langer type positivity assumptions, we prove that \(π^{\ast}(D,x)\longrightarrow π^{\ast}(X,x)\) is an isomorphism for $\ast\in\{S,N,EN,F, EF,Loc,ELoc,\acute{e}t,E\acute{e}t,uni\}$ over perfect fields. |
| title | The Lefschetz Type Theorem for Fundamental Group Schemes |
| topic | Algebraic Geometry 14F35, 18M25, 14J60, 14L15 |
| url | https://arxiv.org/abs/2604.19546 |