Absolute Hodge cycles

Question

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?

Answer 1

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.]

I think there are not that many examples beyond abelian varieties where it is known. It is true for things like products of curves, cubic hypersurfaces of dimension at most 6, moduli of vector bundles over a curve etc. The key point is all of the examples are close to abelian varieties in some motivic sense. For further details, see Deligne-Milne, Tannakian categories; André, Pour une theorie inconditionelle des motifs; my paper, Motivation for Hodge cycles.