# Absolute Hodge cycles

Let $X$ is a smooth projective variety defined over a finite extension $K/\mathbf{Q}$, $\sigma : K\to\mathbf{C}$ any of the finitely many field embeddings of $K$ into the complex numbers, and call $X^{\rm an}$ the complex-analytic space associated to $$(X\times_{K,\sigma}\mathbf{C})(\mathbf{C}).$$

Is it known that all Hodge cycles on $X^{\rm an}$ are absolute Hodge cycles (in the sense of Deligne in "Absolute Hodge cycles on abelian varieties")? (beyond the case when $X$ is an abelian variety)

In what cases is this known, and in what references proved?

A couple of comments. First of all, the question of absoluteness of Hodge cycles is only interesting if there is more than one embedding if your field of definition into $\mathbb{C}$. So you really want to formulate your question for a field different from $\mathbb{Q}$, otherwise it's vacuously true. [Added: This was in response to an earlier version of the question, but it's probably still useful.]