Monodromy group for a strongly semistable principal bundle over a curve

TLDR: In this paper, an affine group scheme X defined over the field k as well as a principal X-bundle S1755069607000151inline1 over the curve X is given by a xs0211A-graded neutral Tannakian category built out of all strongly semistable vector bundles over X.

Abstract: Let X be a geometrically irreducible smooth projective curve defined over a field k. Assume that X has a k-rational point; fix a k-rational point x e X. From these data we construct an affine group scheme X defined over the field k as well as a principal X-bundle S1755069607000151inline1 over the curve X. The group scheme X is given by a xs0211A-graded neutral Tannakian category built out of all strongly semistable vector bundles over X. The principal bundle S1755069607000151inline1 is tautological. Let G be a linear algebraic group, defined over k, that does not admit any nontrivial character which is trivial on the connected component, containing the identity element, of the reduced center of G. Let E G be a strongly semistable principal G-bundle over X. We associate to E G a group scheme M defined over k, which we call the monodromy group scheme of E G , and a principal M-bundle E M over X, which we call the monodromy bundle of E G . The group scheme M is canonically a quotient of X, and E M is the extension of structure group of S1755069607000151inline1. The group scheme M is also canonically embedded in the fiber Ad(E G ) x over x of the adjoint bundle.