The Lefschetz Type Theorem For Fundamental Group Schemes

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 $\pi ({\mathcal{C}}_D,x)\to \pi ({\mathcal{C}}_X,x)$ to be faithfully flat, a closed immersion, or an isomorphism. As applications, under Langer type positivity assumptions, we prove that ${\pi }^{\ast }(D,x)⟶{\pi }^{\ast }(X,x)$ is an isomorphism for $\ast \in \{S,N,EN,F,EF,Loc,ELoc,\acute{e}t,E\acute{e}t,uni\}$ over perfect fields.