On the generalized Riemann–Hilbert problem with irregular singularities

We study the generalized Riemann–Hilbert problem, which extends the classical Riemann–Hilbert problem to the case of irregular singularities. The problem is stated in terms of generalized monodromy data which include the monodromy representation, the Stokes matrices and the true Poincaré rank at each singular point. We give sufficient conditions for the existence of a linear differential system with such data. These conditions are in particular fulfilled when the monodromy representation is irreducible, as in the classical case. We solve the problem almost completely in dimension two and three. Our results have applications in differential Galois theory. We give sufficient conditions for a given linear algebraic group G to be the differential Galois group over of a differential system with the smallest possible number of singularities, and with singularities all Fuchsian but one, at which the Poincaré rank is minimal.