On the Riemann-Hilbert problem in dimension 4
The author treats the following version of the Riemann-Hilbert problem (RHP): Which representations \(\chi :\pi _1({\mathbb CP}^1\backslash \{ a_1,\ldots ,a_n\}) \rightarrow GL(p,{\mathbb C})\) can be realized as monodromy representations of Fuchsian systems (i.e. linear systems of ordinary differential equations with logarithmic poles) on Riemann's sphere with given poles \(a_j\)? A monodromy representation is defined by \(n\) matrices \(G_j\) such that \(G_1\ldots G_n=I\); \(G_j\) defines up to conjugacy the local monodromy around the pole \(a_j\). For \(n=2\) the answer (due to Dekkers) is positive for all representations, for \(n=3\) the exhaustive answer to the RHP is given by \textit{D. V. Anosov} and \textit{A. A. Bolibruch} [The Riemann-Hilbert problem: A publication from the Steklov Institute of Mathematics. Aspects of Mathematics. 22. Braunschweig, Vieweg (1994; Zbl 0801.34002)] for both cases. The present paper gives an exhaustive answer to the RHP for \(n=4\). The 6 cases of representations for which the answer to the RHP is negative are described by means of the block upper-triangular form to which the representation can be conjugated, by the Jordan normal forms of the operators \(G_j\) and by the possible valuations of the solutions at the poles \(a_j\).
scientific article; zbMATH DE number 1496745
scientific article