# Comparison for cycle class maps for de Rham and etale cohomology via p-adic Hodge theory

Let $$K$$ be a p-adic local field, $$X$$ a smooth projective variety over $$Spec(K)$$, $$CH^k(X)$$ the Chow group of pure codimension $$k$$. Then there are cycle maps

$$cl^X_{DR}:CH^k(X)\to H^{2k}_{DR}(X/K)$$ and $$cl^X_{et}:CH^k(X)\to H^{2k}_{et}(X,\mathbb{Q}_p)$$.

There is also a comparison morphism $$\alpha$$ which comes from p-adic Hodge theory:

$$\alpha\colon B_{dR} \otimes_K H^*_{dR}(X/K) \cong B_{dR}\otimes_{\mathbb{Q}_p} H^*_{et}(X\times_K \overline{K},\mathbb{Q}_p).$$

Question:

(1) Are these two cycle class maps related via the comparison morphism $$\alpha$$? This would be analogous to what happens in the complex case: (Perhaps Fontaine's $$t\in B_{dR}$$ appears?)

(2) If (1) is too difficult, I would be interested in just the case where $$X=Y\times Spec(K)$$ where $$Y$$ is defined over a number field $$E$$ and $$K$$ is a $$p$$-adic completion of $$E$$.

• Is there a reason you wrote the first cycle class map having $Z^k(X)$ instead of $\operatorname{CH}^k(X)$ as its domain? – R. van Dobben de Bruyn May 7 '18 at 19:58
• @JesseSilliman: that paper looks a bit fancy. I feel that this shouldn't involve studying Chern classes on higher $K$-theory. My strategy would be similar, though: replace $\operatorname{CH}^*(X)$ by $K_0(X)$ (they agree after tensoring with $\mathbb Q$), and use the splitting principle to reduce to the case of $c_1(\mathscr L)$ for $\mathscr L$ a line bundle. For this case an actual argument is needed, though. – R. van Dobben de Bruyn May 7 '18 at 21:30
• I think that one can deduce that $\alpha$ respects the cycle class maps up to a scalar purely from the functorial properties of the comparison isomorphism. For $X=\mathbb{P}^1$ the element $\alpha(1\otimes c_1^{dR}(O(1)))$ is stable under the Galois action. Hence, $\alpha(1\otimes c_1^{dR}(O(1)))=t\otimes a\cdot c_1^{\'et}(O(1))$ for some $a\in\mathbb{Q}_p$. For any curve $C$ considering a finite cover $C\to\mathbb{P}^1$ we get that $\alpha$ is the multiplication by $at$ on the second cohomology of $C$. – SashaP May 9 '18 at 16:33
• Finally, for any line bundle $\mathcal{L}$ on a smooth proper $X$ we can conclude that $\alpha(1\otimes c_1^{dR})(\mathcal{L})=t^{-1}\otimes a\cdot c_1^{\'et}(\mathcal{L})$ since the class of $\mathcal{L}$ in the Neron-Severi group modulo torsion is completely determined by the restriction of $\mathcal{L}$ to all curves. However, to compute $a$ it is necessary to look at the actual construction of the comparison isomorphism which you have in mind. – SashaP May 9 '18 at 16:33