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Let $X_0$ be a smooth projective variety defined over a number field $k$.

Let $\sigma : k\to\mathbf{C}$ be one of the finitely many field embeddings of $k$ into the complex numbers, and call $X := (X_0)\otimes_{k,\sigma}\mathbf{C}$. Also call $\overline{X} := X_{\overline{k}}$.

The Artin comparison in $\ell$-adic cohomology gives a canonical isomorphism:

$$a^j: H^j(X,\mathbf{Q}(p))\otimes\mathbf{Q}_{\ell} \simeq H^j(\overline{X},\mathbf{Q}_{\ell}(p)).$$

Question 1. Is $a^{2p}$ compatible with cycle class maps? (a reference?)

The right side carries an action of $\text{Gal}(\overline{k}/k)$.

Let $H^j_{\rm dR}(X_0/k)$ be the algebraic de Rham cohomology of $X_0$. We have: $$H^j_{\rm dR}(X/\mathbf{C}) = H^j_{\rm dR}(X_0/k)\otimes_k\mathbf{C}\supset H^j_{\rm dR}(X_0/k).$$

On the other hand, we have the Grothendieck comparison:

$$g^j : H^j_{\rm dR}(X/\mathbf{C})(p)\simeq H^j(X,\mathbf{C}(p))$$ (where Tate twists on de Rham cohomology are trivial).

Now call $V_0 := g^j(H^j_{\rm dR}(X_0/k))\cap H^j(X,\mathbf{Q}(p))$.

Question 2. Is $a^j(V_0)$ Galois invariant?

Question 3. More generally, if $V$ is the intersection of the image of $H^j_{\rm dR}(X_0/k)\otimes \overline{k}$ under $g^j$, with $H^j(X,\mathbf{Q}(p))$, what is the interaction between the Galois action on $V$ through $H^j_{\rm dR}(X_0/k)\otimes \overline{k}$ and the Galois action on $H^j(\overline{X},\mathbf{Q}_{\ell})$? Do they agree? If so, the answer to Question 2 is yes.

Improve this question
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    $\begingroup$ For Q1, since you don't say how you're defining cycle class maps on the analytic side but the comparison isomorphism is compatible with excision sequences and cup products, is the main content of that question the compatibility with top-degree trace maps on ${\rm{H}}^{2d}_c(Y, \mathbf{Q}_{\ell}(d))$ for smooth separated $Y$ of finite type with pure dimension $d$ in the absence of properness? For Q3, typically ${\rm{H}}^j(\overline{X},\mathbf{Q}_{\ell})$ has no nonzero vectors with open Galois stabilizer (e.g., already for elliptic curves!), so the answer is negative. Similar issue impacts Q2. $\endgroup$
    – nfdc23
    Feb 28, 2018 at 17:57
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    $\begingroup$ For the compatibility of topological and algebro-geometric trace maps through the comparison isomorphism in the smooth (not necessarily proper) case, I'm not aware of any reference but I worked it out for myself when I was teaching myself etale cohomology many years ago and it didn't require any rocket science: just careful consideration of definitions and a fair amount of time and energy to relate the topological side to the fibration method that is a definition on the algebro-geometric side (it reduces to smooth proper curves, which is a fun calculation requiring some thought). $\endgroup$
    – nfdc23
    Feb 28, 2018 at 18:03
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    $\begingroup$ There's an explanation of Q1 in Section 1 of Deligne, Hodge cycles on abelian varieties, LNM 900 (which can be found on the web). $\endgroup$
    – anon
    Feb 28, 2018 at 19:13

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