Hodge theory and varieties defined over subfields of the complex numbers

This question is related to the question: Is there a $k$-structure for Hodge modules over a $k$-variety?.

Suppose $K$ is a subfield of $\mathbb{C}$ and $M$ is a holonomic $D$-module "of geometric origin" on a smooth scheme $X/K$. By Saito's theory, the base-changed $D$-module $M_{\mathbb{C}}$ on $X\underset{\operatorname{Spec}(K)}\times \operatorname{Spec}(\mathbb{C})$ carries a natural mixed Hodge module structure.

An ill-formed question: how does the $K$-rational structure on $M_{\mathbb{C}}$ interact with the Hodge structures?

Here are some precise incarnations of that question. Saito's theory endows $M_{\mathbb{C}}$ with the weight filtration $W$ (by $D$-submodules) and a Hodge filtration (compatible with the filtration on differential operators). Are $W$ and $F$ defined over $K$? Or, a seemingly weaker question: does the action of $\operatorname{Gal}(\mathbb{C}/F)$ on (the mixed Hodge module of) de Rham cohomology of $M_{\mathbb{C}}$ preserve the induced filtrations on de Rham cohomology? (This is readily verified when $M=\mathcal{O}_X$).

• Dear Moosbrugger, I wouldn't like to try and prove it, but if $M$ is of geometric origin over $K$, then I think that both the Hodge and the weight filtrations should be defined over $K$. Regards, Matthew – Emerton Sep 28 '11 at 6:46
• I think at least the Hodge filtration should be. The weight filtration is of a somewhat more Betti-origin, so I wouldn't be so surprised to find an example where it's not geometric in this way. But it's quite non-trivial to extract anything like this from Saito -- the set-up uses polarizations (and therefore the real numbers) in a quite non-trivial way. – Moosbrugger Sep 28 '11 at 12:19
• I've retagged it. I hope this will give more visibility to the question. – Leo Alonso Oct 7 '11 at 8:36

Take a look at section 1 of M. Saito, Arithmetic mixed sheaves, Inventiones 2001. Part of the data for an object in his category $MHM(X/K)$ is a bifiltered $D$-module defined over $K$. So it seems that your question has a positive answer. But as always with this stuff, it is easy to overlook something. If it's really important to you, you should ask Saito directly (he's not on Mathoverflow as far as I know). Alternatively, as Matthew Emerton has suggested, it may be simpler to work out the cases you need by hand.
If you think about where the weight filtration on cohomology comes from in the case of constant coefficients, it comes from compactifying the variety via a normal crossings divisor at infinity. If the variety is defined over $K$, so should this compacitifcation be (at least, we should be able to take one defined over $K$), and hence (computing cohomology de Rham-wise) we should get that, on de Rham cohomology, the weight filtration is defined over $K$. Extending the intuition gained from this special case to the general context of mixed Hodge modules, I would guess that for objects of geometric origin, the weight filtration should be defined over $K$.