Let $X$ be a smooth complex algebraic variety and let $MHM(X)$ be the category of mixed Hodge modules on $X$, as defined in *(Saito, "Mixed Hodge Modules", 1990)*, *(Peters-Steenbrink, "Mixed Hodge Structures", 2008)* or *(Saito, "Mixed Hodge Modules and Applications", 1990)*, among others.

In all these references, it is said that a mixed Hodge module $M$ comes with an increasing filtration $W$, called the weight filtration, such that $Gr_W^p\,M$ is semi-simple for $p \in \mathbb{Z}$. However, no explicit mention is done to a *Hodge-type* filtration $F$ on $M$.

Nevertheless, in Saito's Mixed Hodge Modules and Applications, it is said that $MHM(X)$ is a full subcategory of the category of $(M, F, K, W)$, where $(M, F)$ is a well filtered $D_X$-module, $K$ is a rational perverse sheaf and $W$ is a pair of filtrations in $K$ and $M$ that agree under the comparison isomorphism. Therefore, it seem like $F$ could define a filtration on the mixed Hodge module $M$, maybe copied into $K \otimes \mathbb{C}$ via the comparison isomorphism. Indeed, taking $X=\left\{\star\right\}$, this $F$ should gives us the Hodge filtration on the mixed Hodge structure defined by a mixed Hodge module over $\left\{\star\right\}$. Thus, my question is:

*Is $MHM(X)$ naturally bi-filtered in the way that, for any $M \in Obj(MHM(X))$, there exists a weight filtration $W$ and a Hodge filtration $F$ of $M$ such that the "Hodge-like" pieces
$$
M_{p,q}:= Gr_F^p Gr_W^{p+q} M
$$
are (simple?) mixed Hodge modules on $X$?*

If the main question is true (or some variation), I have other questions:

- There exists a bi-shifting of $W$ and $F$? That is, given a mixed Hodge module $M$ on $X$ and $\alpha, \beta \in \mathbb{Z}$, is there another mixed Hodge module $M(\alpha, \beta)$ such that $$ (M(\alpha, \beta))_{p,q} = M_{p+\alpha, q+\beta} $$ for all $p,q \in \mathbb{Z}$?
- If that bi-shifting exists, is it natural in the sense that $$ f_*(M(p,q))=(f_*M)(p,q) $$ for any morphism of algebraic varieties $f:X \to Y$?

Thank you so much in advance!

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