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 $C^\infty$ Fréchet 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 $C^\infty$ Fréchet 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.