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Let $X$ and $Y$ be two algebraic varieties over $\mathbb{C}$. I am interested in the cohomology of $\operatorname{Hom}(X_{top},Y_{top}),$ where I am taking homomorphisms between the underlying topological spaces rather than morphisms of algebraic varieties. Can I naturally attach a mixed Hodge structure to this cohomology?

I don't have a precise formulation of what I want from this mixed Hodge structure; I'm hoping for something like a spectral sequence converging to the cohomology whose morphisms are morphisms of mixed Hodge structures coming from $X$ and $Y$. I am really interested in analogous situations where I am considering some other Weil cohomology theory, but there my question is even less precise.

EDIT: I'm really interested in the case where $X$ and $Y$ are over some finite field, and I want a Galois rep which could be called the cohomology of $\operatorname{Hom}(X_{top},Y_{top})$ (as well as a computational procedure for computing it.) Maybe Jason's suggestion of etale homotopy theory in the comments is the way to go?

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  • $\begingroup$ If $X$ is $\mathbb{P}^1$, there is a natural mixed Hodge structure first constructed by John Morgan, with further work by Hain, Deligne, and Navarro Aznar. $\endgroup$
    – Jason Starr
    Commented Dec 31, 2016 at 13:47
  • $\begingroup$ @JasonStarr: Thanks for the references! I vaguely remember you mentioning this to me at some point but couldn't recall what precisely you said. Do you mind me sending you an email with some further details? $\endgroup$
    – dhy
    Commented Dec 31, 2016 at 13:57
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    $\begingroup$ (or why not posting details as an answer, so we can all learn something?) $\endgroup$
    – Yosemite Sam
    Commented Dec 31, 2016 at 18:08
  • $\begingroup$ @YosemiteSam: I could elaborate, but it concerns a much more specialized question than the one asked here... $\endgroup$
    – dhy
    Commented Dec 31, 2016 at 18:30
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    $\begingroup$ @JasonStarr Makes sense. I was just thinking more about whether the construction you mentioned generalizes to arbitrary Hom-spaces. I'm saying it can't unless their rational cohomology can be computed from the rational homotopy types of $X$ and $Y$. $\endgroup$
    – Will Sawin
    Commented Dec 31, 2016 at 21:02

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