A note on certain Tannakian group schemes

Abstract

In this note, we prove that theF-fundamental group schemeis a birational invariant for smooth projective varieties. We prove that theF-fundamental group scheme is naturally a quotient of the Nori fundamentalgroup scheme. For elliptic curves, it turns out that theF-fundamental groupscheme and the Nori fundamental group scheme coincide. We also consideran extension of the Nori fundamental group scheme in positive characteristicusing semi-essentially finite vector bundles, and prove that in this way, wedo not get a non-trivial extension of the Nori fundamental group scheme forelliptic curves, unlike in characteristic zero.