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.