The Riemann–Hilbert Correspondence for Holonomic $$\mathcal{D}$$ -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 (Forumala Presented). -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 (Forumala Presented). in X, we first define the functor from the category of (Forumala Presented). -modules which are holonomic and have regular singularities away from D to that of Stokes-perverse sheaves on (Forumala Presented)., and we show that it is an equivalence. We then extend the correspondence to holonomic (Forumala Presented). -modules with singularities on D, on the one hand, and Stokes-perverse sheaves on (Forumala Presented). on the other hand.