7 - Moduli Space of Semistable G-Bundles Over a Smooth Curve

Summary

Using some general result of Grothendieck on the existence of quot schemes, we construct the coarse moduli space M(r, d) for rank-r and degree-d vector bundles on a smooth projective curve ?, which consists of S-equivalence classes of semistable vector bundles of rank r and degree d. The construction proceeds via the Geometric Invariant Theory. The moduli space M(r, d) is an irreducible, normal projective variety with rational singularities. Moreover, the subset consisting of stable vector bundles is an open subset, which bijectively parameterizes the isomorphism classes of stable vector bundles. This subset provides the coarse moduli space of stable vector bundles. We extend the above results for vector bundles to G-bundles over? and even more generally to equivariant G-bundles. It is achieved by taking an embedding of G into the general linear group GL(r) and realizing a G-bundle as a rank-r vector bundle together with a reduction of the structure group to G.