Monodromy group for a strongly semistable principal bundle over a curve, II

2006/01/31 by Indranil Biswas, A. J. Parameswaran, Biswas, Indranil +1
Mathematics · #14H60 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14H60

paper · pdf · doi:10.48550/arxiv.math/0601768

This final version includes strengthening of the result by referee's comments. K-Theory (to appear)

arxiv created 2007/02/07 · arxiv updated 2009/12/01

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∈ X. From these data we construct an affine group scheme \mathcal G_X defined over the field k as well as a principal \mathcal G_X-bundle E_\mathcal G_X over the curve X. The group scheme \mathcal G_X is given by a \mathbb Q--graded neutral Tannakian category built out of all strongly semistable vector bundles over X. The principal bundle E_\mathcal G_X 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 \mathcal G_X, and E_M is the extension of structure group of E_\mathcal G_X. The group scheme M is also canonically embedded in the fiber \rm Ad(E_G)_x over x of the adjoint bundle.

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