Logarithmic connections on principal bundles over a Riemann surface

Abstract

Let $E_G$ be a holomorphic principal $G$--bundle on a compact connected Riemann surface $X$, where $G$ is a connected reductive complex affine algebraic group. Fix a finite subset $D\, \subset\, X$, and for each $x\,\in\, D$ fix $w_x\, \in\, \text{ad}(E_G)_x$. Let $T$ be a maximal torus in the group of all holomorphic automorphisms of $E_G$. We give a necessary and sufficient condition for the existence of a $T$--invariant logarithmic connection on $E_G$ singular over $D$ such that the residue over each $x\, \in\, D$ is $w_x$. We also give a necessary and sufficient condition for the existence of a logarithmic connection on $E_G$ singular over $D$ such that the residue over each $x\, \in\, D$ is $w_x$, under the assumption that each $w_x$ is $T$--rigid.