Is there a $k$-structure for Hodge modules over a $k$-variety?

Question

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'.

Answer 1

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; 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.)

Answer 2

When $X$ is a $k$-variety the category $MHM(X_{\mathbb{C}})$ of mixed Hodge modules on $X\otimes_k \mathbb{C}$ doesn't remember the $k$-structure. For example you have $Ext^1_{MHM(X_{\mathbb{C}})}(Q_X, Q_X(1)) = \mathbb{C}(X)^\times \otimes_{\mathbb{Z}} \mathbb{Q}$. In a category of mixed Hodge modules with de Rham $k$-structure this group would be $k(X)^\times \otimes_{\mathbb{Z}} \mathbb{Q}$ instead.