Transcendental aspects of the Riemann-Hilbert correspondence
The purpose of the paper is to describe some properties of Riemann- Hilbert map (a map from the set of systems of differential equations to the set of representations of the fundamental group) which reflect on its essentially transcendental nature. The technique is base on Kuo-Tsai Chen expansion of the solution of a system of differential equations as a sum of iterated integrals. Let X be a compact Riemann surface of genus g. Theorem 1. Suppose X is defind over \(\bar Q.\) The standard conjecture holds for regular singular systems of equations of rank one on X. Theorem 2. The standard conjecture holds for irreducible regular singular systems of rank two on \(P^ 1-\{0,1,\infty \}.\) Theorem 3. Suppose \(s_ 1,...,s_ n\) are \(\bar Q\) rational points in \(P^ 1\). The conjectures of absoluteness I and II, and the conjecture of variation of Hodge structure, hold for unipotent regulare singular systems of equations of rank 3 on \(P^ 1-\{s_ 1,...,s_ n\}.\) Theorem 4. The Laplace transform of the integral \[ \int_{\Delta_ k\gamma}b_{i_ ki_{k-1}}(z_ k)...b_{i_ 2i_ 1}(z_ 2)b_{i_ 1i_ 0}(z_ 1)e^{t(g_{i_ k}(Q)-g_{i_ k}(z_ k)+...+g_{i_ 0}(z_ 1)-g_{i_ 0}(P))} \] has an extension with locally finite regular singularities. In theorem 5 the author states a convergence result for adding up the expansions even if the series does not terminate.
representations of the fundamental group
scientific article; zbMATH DE number 4200617