Vector bundles trivialized by proper morphisms and the fundamental group scheme

TL;DR

This paper characterizes essentially finite vector bundles on a smooth projective variety over an algebraically closed field as those becoming trivial after a proper surjective pullback, linking geometric trivializations to fundamental group schemes.

Contribution

It proves that essentially finite vector bundles are exactly those trivialized by some proper surjective morphism, clarifying their geometric and group-theoretic nature.

Findings

Essentially finite vector bundles coincide with those trivialized by proper surjective morphisms.

The category of these bundles is equivalent to representations of a pro-finite group scheme.

Provides a geometric criterion for the trivialization of vector bundles via proper morphisms.

Abstract

Let X be a smooth projective variety defined over an algebraically closed field k. Nori constructed a category of vector bundles on X, called essentially finite vector bundles, which is reminiscent of the category of representations of the fundamental group (in characteristic zero). In fact, this category is equivalent to the category of representations of a pro--finite group scheme which controls all finite torsors. We show that essentially finite vector bundles coincide with those which become trivial after being pulled back by some proper and surjective morphism to X.