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I do not know much about mixed Hodge modules. I would like to ask: Let $X$ be a smooth connected algebraic complex variety, with a chosen point. Could one describe smooth mixed Hodge modules on $X$ as some representations of the fundamental group? Of course, I ask because I have in mind other cases - for example, smooth perverse sheaves will be the same as representations of the fundamental group in finite-dimensional vector spaces.

Thank you, Sasha

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    $\begingroup$ My expectation is that there will be no positive answer to your question. The data of a smooth (pure) Hodge module (aka variation of Hodge structure) involves a filtration on the holomorphic sections of the underlying vector bundle. This filtration is not flat, but rather satisfies the Griffiths transversality condition $\nabla \mathcal F^n \subseteq \mathcal F^{n-1}\otimes \Omega^1$. It doesn't look like it is possible to express these data and conditions in terms of the monodromy. $\endgroup$
    – Sam Gunningham
    Commented Apr 8, 2014 at 16:24
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    $\begingroup$ The answer is pretty much what Sam & Dan are saying. While I don't want to self-advertise, you can take a look at front.math.ucdavis.edu/0902.4252 for a bit more info. $\endgroup$
    – Donu Arapura
    Commented Apr 8, 2014 at 17:21
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    $\begingroup$ It is important to specify whether one considers an integral, real or complex mixed Hodge structure. For example, the integral lattice in a variation of pure Hodge structures of weight 1 is the same as a family of smooth curves which can be interesting (even over a disc). $\endgroup$
    – Geordie Williamson
    Commented Apr 8, 2014 at 20:25

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The short answer is no. The best you can say is the following: there is a functor rat from mixed Hodge modules to perverse sheaves, and for smooth $X$, $\mathrm{rat}(\mathscr M)$ is a local system iff $\mathscr M$ is smooth. But a smooth mixed Hodge module contains more information than its underlying local system (think about the case when $X$ is a point!). The precise extra data you need on a local system to define a mixed Hodge module is that of an admissible variation of mixed Hodge structure, see Steenbrink and Zucker: Variation of mixed Hodge structure I.

Edit: On second thought, maybe I want to change my answer in a different direction. The category of smooth mixed Hodge modules on $X$ is neutral tannakian, with the basepoint you wisely fixed providing the fiber functor. Then such mixed Hodge modules are in fact the same as representations of some horribly complicated pro-algebraic group. Maybe this is not the kind of answer you were after, but there it is.

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