Monodromy groups of regular systems on the Riemann sphere

Abstract

We consider the Riemann–Hilbert problem (or Hilbert’s 21st problem) for Fuchsian linear systems of ordinary differential equations on the Riemann sphere: Prove that for any prescribed monodromy group and poles there exists a Fuchsian system with this monodromy group and with these poles. In this setting the problem has been given a negative answer due to A. A. Bolibrukh. We give sufficient conditions for the realizability of a monodromy group by a Fuchsian system, e.g. if one of the monodromy operators in its Jordan normal form has at most one block of size 2 the rest being of size 1, or if the monodromy group is irreducible (the proof of this result has been obtained simultaneously and independently of the one of Bolibrukh). We discuss invariants of matrix groups considered as monodromy groups. We also give the codimension in the space of couples (monodromy group, poles) of the set for which the answer to the Riemann–Hilbert problem is negative, and we describe the couples on which this codimension is attained.