Stable vector bundles and the Riemann–Hilbert problem on a Riemann surface

The paper is devoted to holomorphic vector bundles with logarithmic connections on a compact Riemann surface and the applications of the results obtained to the question of solvability of the Riemann–Hilbert problem on a Riemann surface. We give an example of a representation of the fundamental group of a Riemann surface with four punctured points which cannot be realized as the monodromy representation of a logarithmic connection with four singular points on a semistable bundle. For an arbitrary pair of a bundle and a logarithmic connection on it we prove an estimate for the slopes of the associated Harder–Narasimhan filtration quotients. In addition, we present results on the realizability of a representation as a direct summand in the monodromy representation of a logarithmic connection on a semistable bundle of degree zero.