The Lefschetz Type Theorem For Fundamental Group Schemes

Abstract

Let $k$ be a field, $X$ a connected scheme proper over $k$, $D⊊X$ 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 \(π(\mathcal{C}_D,x)\to π(\mathcal{C}_X,x)\) 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.

11 pages, Comments welcome!