$p$-adic étale cohomology groups are not $\mathbb{C}_p$-admissible

Question

It is stated in Caruso - An introduction to $p$-adic period rings (the remarks following equation (2)) that the $p$-adic étale cohomology groups of an algebraic variety $X$ over a finite extension $K$ of $\mathbb{Q}_p$ are not $\mathbb{C}_p$-admissible. In fact, $\mathbb{C}_p\otimes_{\mathbb{Q}_p}H_\text{ét}^r(X_{\overline{K}},\mathbb{Q}_p)$ is isomorphic to the graded module of $\mathbb{C}_p\otimes_K H_{\text{dR}}^r(X)$ for the de Rham filtration, but not to $\mathbb{C}_p\otimes_K H_{\text{dR}}^r(X)$ itself. Is there a reference for this result or a quick proof using a characterization of $\mathbb{C}_p$-admissibility (at least to prove that they are not $\mathbb{C}_p$-admissible) ?

Answer 1

The characterization of $\mathbb C_p$-admissibility in the notes is that the action of the inertia group factors through a finite quotient.

A standard calculation in étale cohomology shows that for $\mathbb P^1$, the action of the Galois group on $H^2( \mathbb P^1_{\overline{K}}, \mathbb Q_p)$ is by the inverse of the $p$-adic cyclotomic character.

Since the $p$-adic cyclotomic character does not have the finite inertia property, this shows that it fails for at least one variety.