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

Abstract

Let \(X\) be a compact connected Riemann surface and \(G\) a connected reductive complex affine algebraic group. Given a holomorphic principal \(G\)-bundle \(E_G\) over \(X\), we construct a \(C^\infty \) Hermitian structure on \(E_G\) together with a \(1\)-parameter family of \(C^\infty \) automorphisms \(\{F_t\}_{t\in \mathbb R }\) of the principal \(G\)-bundle \(E_G\) with the following property: Let \(\nabla ^t\) be the connection on \(E_G\) corresponding to the Hermitian structure and the new holomorphic structure on \(E_G\) constructed using \(F_t\) from the original holomorphic structure. As \(t\rightarrow -\infty \), the connection \(\nabla ^t\) converges in \(C^\infty \) Frechet topology to the connection on \(E_G\) given by the Hermitian–Einstein connection on the polystable principal bundle associated to \(E_G\). In particular, as \(t\rightarrow -\infty \), the curvature of \(\nabla ^t\) converges in \(C^\infty \) Frechet topology to the curvature of the connection on \(E_G\) given by the Hermitian–Einstein connection on the polystable principal bundle associated to \(E_G\). The family \(\{F_t\}_{t\in \mathbb R }\) is constructed by generalizing the method of [6]. Given a holomorphic vector bundle \(E\) on \(X\), in [6] a \(1\)-parameter family of \(C^\infty \) automorphisms of \(E\) is constructed such that as \(t\rightarrow -\infty \), the curvature converges, in \(C^0\) topology, to the curvature of the Hermitian–Einstein connection of the associated graded bundle.