The Lefschetz Type Theorem for Fundamental Group Schemes

TL;DR

This paper establishes Tannakian criteria for fundamental group schemes of schemes over fields and proves isomorphism results under positivity assumptions, extending Lefschetz type theorems.

Contribution

It provides new Tannakian criteria for fundamental group schemes and demonstrates isomorphisms under positivity conditions, generalizing Lefschetz theorems.

Findings

Criteria for faithful flatness, closed immersion, or isomorphism of fundamental group homomorphisms.

Proves isomorphism of fundamental group schemes under positivity assumptions.

Extends Lefschetz type theorems to fundamental group schemes over perfect fields.

Abstract

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.