Differential Equations on the Complex Sphere and the Riemann-Hilbert Problem

Abstract

Let a differential field K with a derivation f↦f′ be given. A differential module over K has been defined as a K-vector space M of finite dimension together with a map ∂: M→M satisfying the rules: ∂(m₁+m₂)=∂(m₁)+∂(m₂), and ∂(fm)=f′m+f∂(m). In this definition, one refers to the chosen derivation of K. We want to introduce the more general concept of connection, which avoids this choice. The advantage is that one can perform constructions, especially for the Riemann-Hilbert problem, without reference to local parameters. To be more explicit, consider the field K=C(z) of the rational functions on the complex sphere P=C∪{∞}. The derivations that we have used are $\frac{d}{{dt}}$ and t^N ${t^N}\frac{d}{{dt}}$ where t is a local parameter on the complex sphere (say t is z−a or 1/z or an even more complicated expression). The definition of connection (in its various forms) requires other concepts such as (universal) differentials, analytic and algebraic vector bundles, and local systems. We will introduce those concepts and discuss the properties that interest us here.