The Lefschetz Type Theorem for Fundamental Group Schemes

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 $\pi(\mathcal{C}_D,x)\longrightarrow \pi(\mathcal{C}_X,x)$ 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 $\ast\in\{S,N,EN,F, EF,Loc,ELoc,\acute{e}t,E\acute{e}t,uni\}$ over perfect fields.