Approximate Hermitian-Einstein connections on principal bundles over a compact Riemann surface

Let be a compact connected Riemann surface and a connected reductive complex affine algebraic group. Given a holomorphic principal -bundle over , we construct a Hermitian structure on together with a -parameter family of automorphisms of the principal -bundle with the following property: Let be the connection on corresponding to the Hermitian structure and the new holomorphic structure on constructed using from the original holomorphic structure. As , the connection converges in Fr,chet topology to the connection on given by the Hermitian-Einstein connection on the polystable principal bundle associated to . In particular, as , the curvature of converges in Fr,chet topology to the curvature of the connection on given by the Hermitian-Einstein connection on the polystable principal bundle associated to . The family is constructed by generalizing the method of [6]. Given a holomorphic vector bundle on , in [6] a -parameter family of automorphisms of is constructed such that as , the curvature converges, in topology, to the curvature of the Hermitian-Einstein connection of the associated graded bundle.