Is there a conjecture, or known result, describing which integral Hodge structures are composition factors in the Hodge structure on the cohomology groups of smooth proper algebraic varieties over $\mathbb C$?
Are there pure Hodge structures which fail to have geometric origin for surprising reasons? For instance, such a Hodge structure should certainly be polarizable.
Motivation: Fontaine-Mazur gives the conjectural conditions for a Galois representation to come from an algebraic variety. This, together with the Tate conjecture, would, if proven, tell us a lot about the relationship between motives and their Galois representations. An analogue of Fontaine-Mazur, together with the Hodge conjecture, would tell us a similarly large amount about the relationship between motives and their Hodge structures.