I have the following (probably well-known) question: let $X$ be a regular scheme over $\mathbb Z$. Let $p$ be a prime and Let us denote the reduction of $X$ mod $p$ by $X_p$. Let also $X_{\mathbb C}$ be the corresponding complex variety. Suppose that I know everything about cohomology of $X_{\mathbb C}$ (i.e. I know the mixed Hodge structure). Is there a situation where I can use that to describe the $\ell$-adic cohomology of $X_p$ with the action of Frobenius? More precisely, are there conditions that would guarantee that all the eigen-values of Frobenius are powers of $q$ and that those powers are exactly those predicted by the Hodge structure on the cohomology of $X_{\mathbb C}$?
UPDATE: I would like to emphasize that the scheme $X$ I am talking about is not complete proper and its cohomology is not pure. However, it is not very bad either: roughly speaking nothing worse than ${\mathbb G}_m$ can appear there. In my specific situation conjecturally (I don't know how to prove this) there is a ${\mathbb G}_m$-action on $X$ which contracts it to a subscheme $Y$ (i.e. there exists a morphism ${\mathbb A}^1\times X\to X$ extending the ${\mathbb G}_m$-action such that ${ 0 } \times X$ goes to $Y$) where $Y$ is a disjoint union of locally closed subschemes of the form ${\mathbb A}^k\times {\mathbb G}_m^l$. Note that in this case the cohomology of $X$ is the same as cohomology of $Y$, so this should give you an idea how complicated the cohomology of $X$ is.
