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!
