Classical Motivation for the Riemann–hilbert Correspondence

These notes explain the equivalence between certain topological and coherent data on complex-analytic manifolds, and also discuss the phenomenon of “regular singularities” connections in the 1-dimensional case. It is assumed that the reader is familiar with the equivalence of categories between the category of locally constant sheaves of sets on a topological space X and the category of covering spaces over X, and also (for a pointed space (X,x0)) with the resulting equivalence of categories between the category of locally constant sheaves of sets on X and left π1(X,x0)-sets when X is connected and “nice” in the sense that it has a base of opens that are path-connected and simply connected. (For example, it is a hard theorem that any complex-analytic space is “nice”). These equivalences will only be used in the case of complex manifolds, but for purposes of conceptual clarity we give some initial construction without smoothness restrictions. Let us first set some conventions. As usual in mathematics, we fix an algebraic closure C of R, endowed with its unique absolute value extending that on R. We shall work with arbitrary complex-analytic spaces (the analytic counterpart to Cschemes that are locally of finite type), but the reader who is unfamiliar with this theory will not lose anything (nor run into problems in the proofs) by requiring all analytic spaces to be manifolds (in which case what we are calling a “smooth map” is a submersion by another name). For technical purposes it is important that the theory of D-modules works without smoothness restrictions. Hence, for the analytically-inclined reader who wishes to avoid smoothness restrictions we note that the global method of construction of the relative de Rham complex ΩX/S for any morphism of schemes f : X → S as in [5, IV4, §16.6] works verbatim in the complex-analytic case and is compatible with the analytification functor from locally finite type C-schemes to complex-analytic spaces. Notation and terminology. If (X,OX) is a locally ringed space and F is an OX -module then F∨ denotes H omOX (F ,OX) and F (x) denotes Fx/mxFx; we call F (x) the fiber at x (to be distinguished from the stalk Fx at x). We will freely pass between the categories of vector bundles and locally free sheaves, and so if V is a vector bundle then we may write V (x) to denote its fiber over a point x in the base space. If Σ is a set and X is a topological space then Σ denotes the associated constant sheaf of sets on X. The analytic space SpC is SpecC considered as a 0-dimensional complex manifold. Also, Z(1) denotes the kernel of exp : C→ C×; this is a free Z-module of rank 1 but it does not have a canonical basis. Finally, a local system (of sets, abelian groups, etc.) on a topological space is a locally constant sheaf (of sets, abelian groups, etc.) on the space.