Criterion for logarithmic connections with prescribed residues

TL;DR

This paper establishes a criterion for the existence of logarithmic connections with prescribed residues on holomorphic vector bundles over Riemann surfaces, extending classical results by considering rigid residues.

Contribution

It provides a necessary and sufficient condition for the existence of such connections when residues are rigid, generalizing the Weil-Atiyah theorem.

Findings

Criteria for logarithmic connections with prescribed residues

Extension of classical theorems to rigid residues

Conditions under which such connections exist

Abstract

A theorem of Weil and Atiyah says that a holomorphic vector bundle $E$ on a compact Riemann surface $X$ admits a holomorphic connection if and only if the degree of every direct summand of $E$ is zero. Fix a finite subset $S$ of $X$, and fix an endomorphism $A(x) \in \text{End}(E_x)$ for every $x \in S$. It is natural to ask when there is a logarithmic connection on $E$ singular over $S$ with residue $A(x)$ at every $x \in S$. We give a necessary and sufficient condition for it under the assumption that the residues $A(x)$ are rigid.