Abstract. Let E be a principal G-bundle over a smooth projective curve over an algebraically closed field k, where G is a reductive linear algebraic group over k. We construct a canonical reduction of E. The uniqueness of canonical reduction is proved under the assumption that the characteristic of k is zero. Under a mild assumption on the characteristic, the uniqueness is also proved when the characteristic of k is positive.

Abstract. Let X be a smooth projective variety defined over an algebraically closed field, and let L be an ample line bundle over X. We prove that for any smooth hypersurface D on X in the complete linear system |L ⊗d |, the inclusion map D ֒→ X induces an isomorphism of fundamental group schemes, provided d is sufficiently large and dim X ≥ 3. If dim X = 2, and d is sufficiently large, then the induced homomorphism of fundamental group schemes remains surjective. We give an example to show that the homomorphism of fundamental group schemes induced by the inclusion map of a reduced ample curve in a smooth projective surface is not surjective in general.

In this article we explicitly compute the number of maximal subbundles of rank k of a generically stable bundle of rank r and degree d over a smooth projective curve C of genus g ≥ 2 over C, when the dimension of the quot scheme of maximal subbundles is zero. Our method is to describe the this number purely in terms of the Gromov invariants of the Grassmannian and then use the formula of Vafa and Intriligator to compute them.

In this paper we use Weil conjectures (Deligne's theorem) to calculate the Betti numbers of the moduli spaces of semi-stable parabolic bundles on a curve. The quasi parabolic analogue of the Siegel formula, together with the method of Harder-Narasimhan filtration gives us a recursive formula for the Poincaré polynomials of the moduli. We solve the recursive formula by the method of Zagier, to give the Poincaré polynomial in a closed form. We also give explicit tables of Betti numbers in small rank, and genera.

No abstract