Transcendental aspects of the Riemann-Hilbert correspondence
type: Item
instance of: scholarly article
title: Transcendental aspects of the Riemann-Hilbert correspondence
zbMATH Open document ID: 0727.34007
author: Carlos Simpson
published in: Illinois Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 34M99
Mathematics Subject Classification ID: 34A25
Mathematics Subject Classification ID: 14H55
zbMATH DE Number: 4200617
zbMATH Keywords: Riemann-Hilbert correspondences
zbMATH Keywords: representations of the fundamental group
zbMATH Keywords: Kuo-Tsai Chen expansion
zbMATH Keywords: iterated integrals
reviewed by: Vasil G. Angelov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4200617