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.

notof geometric origin. (This is why in modularity theorems it is easier to treat the case when $V$ does not have such gaps, and why e.g. Sato--Tate was proved for elliptic cruves before it was proved for higher weight forms.) – Emerton Nov 29 '12 at 11:49rigid, by the preceding Griffiths transversality argument, and so there is no family of motives of which is a member (whereas elliptic curves are easy to write down, since there is a family of them depending on parameters, and we can just choose rational values of the parameters). Regards, – Emerton Nov 29 '12 at 11:52