I'm new here. I hope to do it right!
I am interested in studying mixed Hodge structures over complex algebraic surfaces and their generalizations.
Let us take a smooth complex variety $X$ and a variation of Hodge structures $\mathbb{V}$ over $X$ of weight $k$. That is, $\mathbb{V}$ is a local system of complex vector spaces with a finite decreasing filtration $\left\{\mathcal{F}^p\right\}$ of the holomorphic vector bundle $\mathbb{V} \otimes_\mathbb{C} \mathcal{O}_X$ defining a pure Hodge structure of weight $k$ on each stalk and such that, for the associated flat connection $\nabla$ over $\mathbb{V} \otimes \mathcal{O}_X$ whose horizontal sections are $\mathbb{V}$ we have $\nabla(\mathcal{F}^p) \subseteq \mathcal{F}^{p-1} \otimes \Omega_X^1$.
My question is: Is there a natural (i.e. functorial) way of obtaining a (mixed?) Hodge structure in the sheaf cohomology $H^p(X, \mathbb{V})$?
Related to this problem, Griffiths proved that, given a proper morphism $f: X \to S$ of maximal rank between complex varieties, with $X$ birrationally equivalent to a compact Kähler manifold, then, the relative de Rham cohomology sheaf
$$
H^p(X/S) :=R^pf_*\underline{\mathbb{C}} \otimes \mathcal{O}_S
$$
underlies a variation of Hodge structures compatible with the pure Hodge structure in the stalk $\left(H^p(X/S)\right)_s \cong H^p(f^{-1}(s), \mathbb{C})$ obtained as subvariety of a variety birrationally equivalent to a compact Kähler manifold.
Another try was to mimic Deligne's proof of existence of mixed Hodge structures on the cohomology of a smooth complex variety via the complex of logarithmic forms. However, it is not not clear what auxiliar resultion take when substituying the de Rham resolution $\underline{\mathbb{C}} \to \Omega^*_X$ by a resolution of $\mathcal{F}$.
Thank you so much in advance!
Best Answer
The answer to your question is yes (provided the VHS is polarized). This is due to Morihiko Saito. It is implicitly contained his two long papers on (mixed) Hodge modules, and there is an explicit statement in his note "Mixed Hodge modules and admissible variations" Compte Rendus 1989. When the base is a curve, it goes back to Zucker, "Hodge theory with degenerating coefficients" Annals 1979; and even earlier due to Deligne when the base is projective (unpublished, but see Zucker). For a VHS of geometric origin there is a different construction due to me, "The Leray spectral sequence is motivic" Invent. 2005, but the MHS is the same as Saito's.