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Let $D^b_m(X)$ (resp $D^b(X)$) denote the derived category of mixed Hodge modules (resp. constructible sheaves) on a complex variety $X$. Let

$rat\colon D^b_m(X)\to D^b(X)$

be the forgetful' functor. This is t-exact for the perverse t-structure on the right. Write $MHM(X)$ for the abelian category of mixed Hodge modules on $X$. Then $MHM(pt)$ is the category of graded polarizable mixed Hodge structures, and $rat\colon MHM(pt) \to VectorSpaces$ is the evident forgetful functor.

Now let $M,N\in D^b_m(X)$. Set

$\mathcal{H}om(M,N) = \Delta^!(\mathbb{D}M \boxtimes N)$,

where $\Delta\colon X\to X\times X$ is the diagonal map, and $\mathbb{D}$ is Verdier duality.

Let $a\colon X \to pt$ be the evident map. Then

$rat ( H^0(a_*\mathcal{H}om(M,N))) = H^0(rat(a_*\mathcal{H}om(M,N))) = Hom(rat(M), rat(N))$

and in this way we get a Hodge structure on $Hom(rat(M),rat(N))$. All functors are derived.

My question: If $M,N$ are pure pointwise pure (see Geordie Williamson's comment below), then is the induced structure on $Hom(rat(M), rat(N))$ pure?

My gut answer is no (even if $X$ is complete, the $\Delta^!$ should be messing weights up), but it would make me happier if the answer is yes!

If the answer is no, under what additional conditions (other than requiring $X$ to be smooth and complete plus $M,N$ being the constant' sheaf) can the answer be converted to yes?

I guess one could also ask the same sort of question for mixed $\ell$-adic sheaves. But I am even less familiar with that setting.

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# Mixed structures on Hom spaces induced by mixed sheaves

Let $D^b_m(X)$ (resp $D^b(X)$) denote the derived category of mixed Hodge modules (resp. constructible sheaves) on a complex variety $X$. Let

$rat\colon D^b_m(X)\to D^b(X)$

be the forgetful' functor. This is t-exact for the perverse t-structure on the right. Write $MHM(X)$ for the abelian category of mixed Hodge modules on $X$. Then $MHM(pt)$ is the category of graded polarizable mixed Hodge structures, and $rat\colon MHM(pt) \to VectorSpaces$ is the evident forgetful functor.

Now let $M,N\in D^b_m(X)$. Set

$\mathcal{H}om(M,N) = \Delta^!(\mathbb{D}M \boxtimes N)$,

where $\Delta\colon X\to X\times X$ is the diagonal map, and $\mathbb{D}$ is Verdier duality.

Let $a\colon X \to pt$ be the evident map. Then

$rat ( H^0(a_*\mathcal{H}om(M,N))) = H^0(rat(a_*\mathcal{H}om(M,N))) = Hom(rat(M), rat(N))$

and in this way we get a Hodge structure on $Hom(rat(M),rat(N))$. All functors are derived.

My question: If $M,N$ are pure, then is the induced structure on $Hom(rat(M), rat(N))$ pure?

My gut answer is no (even if $X$ is complete, the $\Delta^!$ should be messing weights up), but it would make me happier if the answer is yes!

If the answer is no, under what additional conditions (other than requiring $X$ to be smooth and complete plus $M,N$ being the constant' sheaf) can the answer be converted to yes?

I guess one could also ask the same sort of question for mixed $\ell$-adic sheaves. But I am even less familiar with that setting.