Constructive solvability conditions for the Riemann-Hilbert problem (Q2388173)

The Riemann-Hilbert problem is to construct a Fuchsian system on the Riemann sphere with a given monodromy \(\chi:\pi_1(\overline{\mathbb C} \setminus\{a_1, \dots , a_n\}, z_0) \to \text{GL}(p,\mathbb C)\) and given singular points \(a_1, \dots ,a_n\). In 1989, A.~A.~Bolibrukh gave an example of a monodromy and singular points for which the Riemann-Hilbert problem cannot be solved. However, the following question arises: For what monodromy representations and sets of singular points can a Fuchsian system be constructed? Various sufficient conditions for the solvability of the Riemann-Hilbert problem have been obtained. The general approach to studying this question consists in constructing all possible holomorphic bundles with logarithmic connections over the Riemann sphere that have a given monodromy and set of singular points. The objective of this paper is to construct a finite algorithm for determining whether there exists a stable pair of bundles and connections with given monodromy representation. The class of representations for which this algorithm makes it possible to determine whether there exists a Fuchsian system with given monodromy representation (i.e., a solution of the Riemann-Hilbert problem) is larger than that to which the existing sufficient conditions apply. The author also studies the existence of a similar algorithm for semistable pairs. This question is interesting too, because such an algorithm makes it possible to verify the necessary solvability condition for the Riemann-Hilbert problem and to construct counterexamples.