Fuchsian systems with completely reducible monodromy
The author proves that any monodromy representation \(\chi\) of the fundamental group of the punctured Riemann sphere can be realized as a direct summand of the monodromy representation \(\chi_f=\chi\oplus\tilde\chi\) of a Fuchsian system. In particular, this means that the decomposability (complete reducibility) of the monodromy representation of a Fuchsian system does not imply the corresponding decomposability of the system itself into a direct sum of Fuchsian systems. To prove this assertion the author explicitly constructs a monodromy representation \(\tilde\chi\) and the extended representation \(\chi_f=\chi\oplus\tilde\chi\) in such a way that a pair \((F^{\Lambda,S},\nabla^{\Lambda,S})\) of a vector bundle \(F\) with the logarithmic connection \(\nabla\) corresponding to \(\chi_f\) is stable. Now it is enough to observe that the Riemann-Hilbert (RH) problem for the extended representation is solvable due to a theorem by Bolibrukh. Additionally, the author proves that if \(\chi=\chi_1\oplus\chi_2\) is the monodromy representation of a Fuchsian system where the spectra of the generators of the representations \(\chi_1\) and \(\chi_2\) do not intersect, then the latter representations can be realized as the monodromies of Fuchsian systems. This theorem together with the known fact that any B-representation (i.e., a reducible representation whose generators can be reduced to a Jordan block such that the product of its eigenvalues over all points is not \(1\)) is a counterexample to the RH problem allow the author to construct counterexamples to the RH problem in dimensions \(p\geq4\) with \(n\geq3\) singular points.
scientific article; zbMATH DE number 5628211