Relation between holonomic D-modules and perverse sheaves

Question

Given a smooth complex algebraic variety, the Riemann-Hilbert-correspondence tells us, that the category of perverse sheaves is equivalent to the category of regular, holonomic D-modules.

However not every interesting holonomic D-module is regular. For example the solution sheaves of all the $D_{\mathbb A^1}$-modules $\mathbb C[x]e^{\chi x}$ are isomorphic to the constant sheaf and only for $\chi=0$ our module $\mathbb C[x]e^{\chi x}$ is regular.

So my question is, is there an analogue of the Riemann-Hilbert correspondence if we replace regular by something else (and perhaps also perverse sheaves by something else)?

For example in the above example one could do the following: One could fix a $\chi$ and tensor first with $\mathbb C[x] e^{-\chi x} $, before applying the deRham functor. This gives an equivalence between perverse sheaves and holonomic modules with "$e^{-\chi x}$-like " singularities.

Answer 1

The answer is yes but it's not easy. You need additional data to describe the irregular part of your connexion. These are known as Stokes structures. Very loosely it's a filtration of your sheaf of solutions according to their growth in a given sector. Very recently, Claude Sabbah has written lecture notes on the subject (arXiv:0912.2762).

Answer 2

I wish to add a bit more reference on this question.

1. For dimension 1, it can be found in Malgrange's book "equations differentielles a coefficients polynomiaux" P60 Theorem (3.1), that (honomic D module on a curve) is equivalent to the category of (local system outside of singularities, stokes structure on the singularities, some compactibility condition desribed by vanishing cycles). Part I of Sabbah's book "an introduction to stokes structure" also gives a nice introduction.

2. For higher dimension, Part II of Sabbah's book discusses the case of "good" meromorphic connections.