Vector bundles trivialized by proper morphisms and the fundamental group scheme

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.

Local Systems on Proper Algebraic V -manifolds

“…When X is smooth, the extra information given by the surjectivewhere-etale property allows us to transfer analytic constructions from Z • back to X, to get things like the definition and existence of harmonic metrics. It might be possible to descend these things along proper surjective hypercoverings too, and in that way get around Theorem 5.4 entirely, but that would require a much more detailed study of descent for bundles along proper surjective maps, a subject discussed in [13] [14].…”

Vector Bundles Trivialized by Proper Morphisms and the Fundamental Group Scheme, II

“…A The main result of [BdS10] relates property (T) to the more sophisticated notion of essential finiteness.…”

“…The present work is a continuation of [BdS10], giving some applications of the main result in [BdS10] which throw light on the nature of the fundamental group scheme of Nori [No76] for a smooth projective variety.…”

“…Let X be a smooth projective variety over an algebraically closed field k. The fundamental group scheme of X is the affine group scheme obtained from the (Tannakian) category of essentially finite vector bundles on X (see Definition 3). The main theorem of [BdS10] says that a vector bundle E over X is essentially finite if and only if there is a proper k-scheme Y and a surjective morphism f : Y −→ X such that f * E is trivial. As an application of this theorem, we prove the following : Theorem 1.…”

Vector bundles over normal varieties trivialized by finite morphisms

“…Then it has been subsequently proved by Balaji and Parameswaran [1, Section 6] for Y smooth and projective of any dimension provided g is separable. Then finally Biswas and Dos Santos [2] have given a different proof: for any finite and surjective g : X → Y, with Y smooth and projective over k, they first explain how to reduce to the case of curves ([2, Section 3]) by means of the Grothendieck-Lefschetz theorem for the S-fundamental group scheme, then in loc. cit.…”