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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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up vote 5 down vote accepted

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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