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


Thanks in advance for your help.



Relationship between $ \mathcal{D} $ - modules, Tannakian formalism and Galois theory of monodromy representations
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 $\pi\_1(X)$.


