A finiteness result on representations of the profinite étale Nori fundamental group

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 ${\pi }_1^N(X,x)\to {\mathrm{GL}}_n$, where ${\pi }_1^N(X,x)$ is the Nori 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 Gasbarri. We also discuss analogues of this finiteness result in characteristic $p>0$.

arXiv comment11 pages. Accepted version. Comments welcome