Criterion for logarithmic connections with prescribed residues

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.