S. Subramanian

Monodromy group for a strongly semistable principal bundle over a curve

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.

Flat holomorphic connections on principal bundles over a projective manifold

Let G be a connected complex linear algebraic group and R u (G) its unipotent radical. A principal G-bundle E G over a projective manifold M will be called polystable if the associated principal G/R u (G)-bundle is so. A G-bundle E G over M is polystable with vanishing characteristic classes of degrees one and two if and only if E G admits a at holomorphic connection with the property that the image in G/R u (G) of the monodromy of the connection is contained in a maximal compact subgroup of G/R u (G).

Essentially finite vector bundles on varieties with trivial tangent bundle

Let X be a smooth projective variety, defined over an algebraically closed field of positive characteristic, such that the tangent bundle TX is trivial. Let F X : X → X be the absolute Frobenius morphism of X. We prove that for any n ≥ 1, the n―fold composition F n X is a torsor over X for a finite group—scheme that depends on n. For any vector bundle E → X, we show that the direct image (F n X ) * E is essentially finite (respectively, F—trivial) if and only if E is essentially finite (respectively, F―trivial).

Principal Bundles Over Finite Fields

Let M be an irreducible smooth projective variety defined over . Let ϖ(M, x 0) be the fundamental group scheme of M with respect to a base point x 0. Let G be a connected semisimple linear algebraic group over . Fix a parabolic subgroup P ⊊ G, and also fix a strictly antidominant character χ of P. Let E G → M be a principal G-bundle such that the associated line bundle E G (χ) → E G /P is numerically effective. We prove that E G is given by a homomorphism ϖ(M, x 0) → G. As a consequence, there is no principal G-bundle E G → M such that degree(ϕ*E G (χ)) > 0 for every pair (Y, ϕ), where Y is an irreducible smooth projective curve, and ϕ: Y → E G /P is a nonconstant morphism.

Parabolic ample bundles III: Numerically effective vector bundles

In this continuation of [Bi2] and [BN], we define numerically effective vector bundles in the parabolic category. Some properties of the usual numerically effective vector bundles are shown to be valid in the more general context of numerically effective parabolic vector bundles.