# 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.