The Riemann–Hilbert Correspondence for Holonomic \(\mathcal{D}\)-Modules on Curves

  • Claude Sabbah
Chapter
First Online:
Part of the Lecture Notes in Mathematics book series (LNM, volume 2060)

Abstract

In this chapter, we define the Riemann–Hilbert functor on a Riemann surface X as a functor from the category of holonomic \({\mathcal{D}}_{X}\)-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 \({\mathcal{D}}_{X}({_\ast}D)\)-modules which are holonomic and have regular singularities away from D to that of Stokes-perverse sheaves on \(\widetilde{X}(D)\), and we show that it is an equivalence. We then extend the correspondence to holonomic \({\mathcal{D}}_{X}\)-modules with singularities on D, on the one hand, and Stokes-perverse sheaves on \(\underline{\widetilde{X}}(D)\) on the other hand.

Keywords

Riemann Surface Local System Holomorphic Vector Bundle Perverse Sheave Regular Singularity 
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Authors and Affiliations

  • Claude Sabbah
    • 1
  1. 1.CNRS École polytechniqueCentre de mathématiques Laurent SchwartzPalaiseauFrance

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