# Logarithmic connections on principal bundles over a Riemann surface

## TL;DR

This paper establishes criteria for the existence of specific logarithmic connections on principal bundles over Riemann surfaces, considering invariance and residue conditions, advancing the understanding of connections with prescribed singularities.

## Contribution

It provides necessary and sufficient conditions for the existence of $T$-invariant and general logarithmic connections with prescribed residues on principal bundles over Riemann surfaces.

## Key findings

- Criteria for $T$-invariant logarithmic connections established.
- Conditions for general logarithmic connections with residues given.
- Residue conditions depend on $T$-rigidity of elements.

## 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.