The Riemann-Hilbert Correspondence for Holonomic D-Modules on Curves

In this chapter, we define the Riemann-Hilbert functor on a Riemann surface X as a functor from the category of holonomic D-X-modules to that of Stokes-perverse sheaves. It is induced from a functor at the derived category level which is compatible with t-structures. Given a discrete set D in X, we first define the functor from the category of D-X(*D)-modules which are holonomic and have regular singularities away from D to that of Stokes-perverse sheaves on e (X) over tilde (D), and we show that it is an equivalence. We then extend the correspondence to holonomic D-X-modules with singularities on D, on the one hand, and Stokes-perverse sheaves on e (X) over tilde (D) on the other hand.