Moduli space of complex structures (Q844401)

This paper is motivated by the understanding of configurations of points \(s_1,\dots, s_m\in \mathbb{P}^1(\mathbb{C})\), for which the Riemann-Hilbert monodromy problem admits solutions. For this, the author investigates connections between the following three spaces: the space of complex structures on the Riemann surface \(\mathbb{P}^1(\mathbb{C})\setminus\{s_1, \dots, s_m\}\), the space of Fuchsian systems of differential equations with singularities at the points \(\{s_1,\dots, s_m\}\) and the configuration space of \(n\)-gons in the three-dimensional space. He computes the Euler characteristics of the corresponding compactified manifolds with a method based on the signature formulas of the topological degree of a map. The author establishes a connection between the configuration of singular points for which the corresponding Riemann-Hilbert monodromly problem admits a solution and the configuration of singular points with no solution for Riemann-Hilbert problem.