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!

  • 1
    $\begingroup$ It is probably helpful to think about a variation of pure Hodge structure. The associated graded for the Hodge filtration not a pure variation of Hodge structure. (Think about a non-trivial variation of Hodge structure of type (1,1) given by the $H^1$ of a family elliptic curves. One wants to think about this object as being "simple", so it shouldn't break up any further.) $\endgroup$ Apr 4 '16 at 12:47
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    $\begingroup$ Maybe yes, maybe no. No using Saito's original definition, because $F$ is almost never defined over $\mathbb{Q}$. On the other hand, Sabbah and Schnell have been reworking the foundations, so that it is no longer necessary to have a $\mathbb{Q}$ "lattice" in their version. See cmls.polytechnique.fr/perso/sabbah.claude/MHMProject/mhm.html $\endgroup$ Apr 4 '16 at 14:07
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    $\begingroup$ But the associated graded for $F$ isn't a D-module, so it's obviously not a mixed Hodge module, right? Am I going crazy here? $\endgroup$
    – Ben Webster
    Apr 4 '16 at 14:29
  • $\begingroup$ Oh yea good point. So no, in either case. $\endgroup$ Apr 4 '16 at 14:38
  • $\begingroup$ I totally agree: absolutely not since the graded pieces of $F$ are no longer $D$-modules. Indeed, if my question was true, then, using that shifting we'll get that the categories of pure Hodge modules over a fixed variety are all isomorphic, no matter the weight. But this is absolutely false as can be shown, for example, using the category of pure Hodge structures of weight 0 and 1, that have a different kind of symmetry in their decompositions. However, using Tate's twisting, we get all the pure Hodge modules with even weight are isomorphic, and odd-weighted too. Is it OK? $\endgroup$
    – a_g
    Apr 18 '16 at 9:53

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