ag.algebraic geometry – Comparison of weight filtration on cohomology of complex manifold

Let $X$ be a smooth scheme of finite type over $mathbb{Z}$ (or let’s say a finitely generated $mathbb{Z}$ algebra). To each prime $p in mathbb{Z}$ we can consider the $mathbb{F}_p$ variety $$X_{mathbb{F}_p}=X times_{mathbb{Z}} mathbb{F}_p$$ and the $overline{mathbb{F}_p}$ variety $$X_{overline{mathbb{F}_p}}=X times_{mathbb{Z}} overline{mathbb{F}_p}=X_{mathbb{F}_p}times_{mathbb{F}_p} overline{mathbb{F}_p}$$ (I’m omitting spec everywhere to lighten the notations a little bit).

In this way, the etale cohomology $H^{*}(X_{overline{mathbb{F}_p}},overline{mathbb{Q}}_{ell})$ gets endowed with a Frobenius morphism and we can consider the associated increasing weight filtration $W^{i}_m$.

If one now considers $X_{mathbb{C}}=X times_{mathbb{Z}}mathbb{C}$ this is a smooth complex algebraic variety. We now from comparison theorems that $$H^{*}_{etale}(X_{mathbb{C}},overline{mathbb{Q}}_{ell})=H^{*}_{simp}(X_{mathbb{C}}(mathbb{C}),overline{mathbb{Q}}_{ell}) $$ where on the right side we have the usual simplicial cohomology.

We moreover know that for $p >>>1$ we will have $$H^{*}_{etale}(X_{mathbb{C}},overline{mathbb{Q}}_{ell}) cong H^{*}(X_{overline{mathbb{F}_p}},overline{mathbb{Q}}_{ell}) .$$ In this way, we can endow the cohomology $H^{*}_{simp}(X_{mathbb{C}}(mathbb{C}),overline{mathbb{Q}}_{ell}) $ and so the cohomology with complex coefficients $H^{*}_{simp}(X_{mathbb{C}}(mathbb{C}),mathbb{C}) $ with a weight filtration which I’m denoting $W^{*}_{m,p}$.

On the other side, Deligne’s theory of mixed Hodge structures, endowes $H^{*}_{simp}(X_{mathbb{C}}(mathbb{C}),mathbb{C})$ with another weight filtration $W^{*}_{m}$. Do these two filtrations coincide in general? Does the filtration $W_{m,p}$ depends on the prime chosen?

I know that there are some invariants which relate the two filtrations. One can define the E-polynomial for example $$E_p(X_{mathbb{C}},q)=sum_{m,k}(-1)^k dim frac{W^{k}_{m,p}}{W^{k}_{m-1,p}}q^{m} $$ and analogously $$E(X_{mathbb{C}},q)=sum_{k}(-1)^ksum_{i+j=r} h^{i,j;k}q^r $$ where $$h^{i,j;k}=dim Gr_W^{i+j} Gr_F^{i} H^{k}(X_{mathbb{C}},mathbb{C})$$ are the Hodge numbers.

One can show that $E_p(X_{mathbb{C}},q)=E(X_{mathbb{C}},q)$ using the additivity with respect to locally closed decompisition of both polynomials and the statement for projective varieties which is true as everything is pure.