A finiteness result on representations of Nori's fundamental group scheme

Abstract

Let $(X,x)$ be a pointed geometrically connected smooth projective variety over a sub-$p$-adic field $K$. For any given rank $n$, we prove that there are only finitely many isomorphism classes of representations ${\pi }_1^{EF}(X,x)\to {\mathrm{GL}}_n$, where ${\pi }_1^{EF}(X,x)$ is Nori's fundamental group of essentially finite bundles. Equivalently, there are only finitely many isomorphism classes of essentially finite bundles of rank $n$. This answers a question from C.Gasbarri.

The proof for Lemma 2.3 in the previous version is incorrect. In the new version the same assertion appears as Proposition 2.7 with corrected proof. The article has been reorganized. 9 pages. Comments welcome!