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.

References in corpus (1)

• Criterion for logarithmic connections with prescribed residues

Cited by in corpus (1)

• Criterion for existence of a logarithmic connection on a principal bundle over a smooth complex projective variety