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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.

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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).

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It should be added that at the moment a reasonable theory of Stokes structures exists only in dimension 1 (Sabbah mentions some ideas in higher dimensions but there is no theory yet). –  Alexander Braverman Dec 18 '10 at 19:10

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