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

TL;DR

This paper proves a finiteness property for the set of isomorphism classes of certain representations of Nori's fundamental group scheme over a specific class of algebraic varieties, answering a previously open question.

Contribution

It establishes that for a smooth projective variety over a sub-$p$-adic field, only finitely many essentially finite bundles of a fixed rank exist, confirming a conjecture by C. Gasbarri.

Findings

Finiteness of isomorphism classes of representations of Nori's fundamental group scheme

Finiteness of essentially finite bundles of fixed rank

Answers a question posed by C. Gasbarri

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 $\pi_{1}^{EF}(X,x)\rightarrow \mathrm{GL}_{n}$, 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.