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Is there a relationship between $ \mathcal{D} $ - modules, Tannakian formalism and Galois theory of monodromy representations ?

Thanks in advance for your help.

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    $\begingroup$ Welcome to Mathoverflow. You may want to add to your question to get a good answer. $\endgroup$
    – Chris Ramsey
    Commented Apr 4, 2016 at 17:15
  • $\begingroup$ One thing you can say is that D-modules finitely generated over the structure sheaf are the same as flat vector bundles which are the same as representations of the fundamental group. $\endgroup$
    – Daniel Barter
    Commented Apr 4, 2016 at 18:28

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I don't know what level of depth you want, so here is the short version. Given a smooth complex variety $X$, as Daniel Barter points out, the categories of $\mathcal{O}_X$-coherent $D$-modules, vector bundles with integrable connections, and complex representations of $\pi_1(X)$ are equivalent via monodromy. Moreover, with their natural tensor structures, they are neutral Tannakian. It follows that the Tannakian group for the first category is isomorphic to the pro algebraic hull of $\pi_1(X)$.

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  • $\begingroup$ Can a similar statement be said in the slightly more general context of regular singular $D$-modules? $\endgroup$
    – Saal Hardali
    Commented Mar 12, 2017 at 10:58
  • $\begingroup$ Yes, I expect so. $\endgroup$
    – Donu Arapura
    Commented Mar 12, 2017 at 16:27
  • $\begingroup$ What would this statement look like? $\endgroup$
    – Saal Hardali
    Commented Mar 12, 2017 at 17:20

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