Weak Density of the Fundamental Group Scheme

TL;DR

This paper proves that for a smooth projective variety over an algebraically closed field, trivial Nori fundamental group scheme implies the triviality of the associated semistable bundle group scheme, indicating a deep connection between these invariants.

Contribution

It establishes a new criterion linking the triviality of Nori's fundamental group scheme to the triviality of the semistable bundle group scheme on smooth projective varieties.

Findings

Nori's fundamental group scheme being trivial implies no nontrivial semistable bundles of degree 0.

The triviality of the group scheme $ ext{pi}^S(X)$ is equivalent to the triviality of Nori's fundamental group scheme.

Provides a new understanding of the structure of fundamental group schemes in algebraic geometry.

Abstract

On $X$ projective smooth over an algebraically closed field, we show that if Nori's fundamental group scheme is trivial, then there are no nontrivial Nori semistable bundles of degree 0, that is the group scheme $\pi^S(X)$ studied in particular by Langer is trivial.