I am trying to understand (Saito's?) category of mixed Hodge modules as a category (i.e. I am not interested in its construction, just in properties of objects and morphisms). I would be grateful for any nice references for this (though I have some texts already); yet currently I am interested in the following question.


For algebraic varieties over a subfield $k$ of the field of complex numbers, can one define certain mixed Hodge modules with some $k$-structure that would be related with the $k$-structure on the De Rham cohomology of $k$-varieties? Please tell me also if such a structure could be obtained from the usual 'complex' Hodge modules, or if the presence of a $k$-structure 'does not affect morphisms significantly'.



Is there a $k$-structure for  Hodge modules over a $k$-variety?
I think that the answer is "yes". If you denote by $MFW(X)$ (resp. $MFW(X\_\mathbb{C})$) the category of regular holonomic $D$-modules on $X$ (resp. $X\_\mathbb{C}$) with a good filtration $F$ and a finite filtration $W$, then there are obvious functors $MFW(X)\rightarrow MFW(X\_\mathbb{C})$ and $MHM(X\_\mathbb{C})\rightarrow MFW(X\_\mathbb{C})$, where $MHM(X\_\mathbb{C})$ is the category of mixed Hodge modules on $X\_\mathbb{C}$. So you can consider the fiber product of $MFW(X)$ and $MHM(X\_\mathbb{C})$ over $MFW(X\_\mathbb{C})$, and indeed things should work nicely for this category, at least if you believe Saito (the construction is taken from his paper "On the formalism of mixed sheaves", [here](http://arxiv.org/abs/math/0611597); it's example 1.8(ii)).


(I don't think that this paper of Saito has appeared in any peer-reviewed journal, so you should check whatever you want to take from it. It's a pity because it's a nice reference.)


