Comparison of weight filtration on cohomology of complex manifold

Question

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 $$E(X_{\mathbb{C}},q)=\sum_{k}(-1)^k\sum_{i+j=r} h^{i,j;k}q^r $$ 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.

Answer 1

Yes, the $\ell$-adic weight filtration is compatible with the weight filtration in mixed Hodge theory under the comparison isomorphism. These facts go back to Deligne, and are described in his announcement Poids dans la cohomologie des variétiés algébriques ICM 1974. Finding a detailed proof is bit harder though... Added remarks You can take a look at Huber's Mixed motives and their realizations in derived categories. Although she proves something more general, which involves quite a bit of overhead. Unfortunately, I don't know of a simple complete account in the literature for what you are asking about. So why don't I simple do it here. Since you ask in the comments about the case when $X$ is projective, let me just focus on that.

Supppose that $X$ is a complex projective variety. Choose finitely generated field of definition $K$. By resolution of singularities, one can construct a smooth projective simplicial scheme $\pi_\bullet:X_\bullet\to X$ which satisfies cohomological descent. See Brian Conrad's notes on cohomological descent for a detailed construction. Then one has a spectral sequence $$E_1^{pq} = H^q(X_{p})\Rightarrow H^{p+q}(X)$$ for either "Betti" or $\ell$-adic cohomology. In the first case, the filtration on the abuttment is the weight filtration for the MHS, essentially by construction (cf Deligne Hodge III). In the second case, this is a spectral sequence of $Gal(\bar K/K)$-modules. The term $E_1^{pq}$ is pure of weight $q$ by the Weil conjectures, so the same holds for $E_\infty^{pq} = Gr_W^pH^{p+q}(X,\mathbb{Q}_\ell)$. That's it.