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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.

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    $\begingroup$ Dear Will, Let $V$ be a PHS such that for some $(p,q)$, one has $h^{p,q} =1$ while $h^{p-1,q+1} = 0$. Then Griffiths transversality shows that $V$ cannot fit into a non-trivial variation of Hodge structures. Since there are only countably many families of algebraic varieties, we see that only countably many such $V$ are of geometric origin, so "most" such $V$ are not of 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.) $\endgroup$
    – Emerton
    Commented Nov 29, 2012 at 11:49
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    $\begingroup$ It is also why it is harder to realize the motives attached to higher weight modular forms from alternative geometric constructions (whereas we have no trouble producing elliptic curves --- which are then related to wt. 2 modular forms): the motive attached to a higher weight modular form is rigid, 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, $\endgroup$
    – Emerton
    Commented Nov 29, 2012 at 11:52
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    $\begingroup$ @Emerton: The Griffiths transversality argument certainly proves that a general Hodge structure as you specify cannot be the full weight p+q Hodge structure of a smooth proper algebraic variety. However, the OP is asking about "factors" of Hodge structures. $\endgroup$
    – Jason Starr
    Commented Nov 29, 2012 at 12:45
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    $\begingroup$ Dear Jason, Since such a factor should be motivic (by the Hodge conjecture), I think that the argument I explain extends to such factors: suppose that you have a piece of the cohomology of a variety. It is cut out by some correspondence (assuming the Hodge conjecture). There will be some locus in the moduli space of the initial variety over which this correspondence deforms, and so the motive will deform over that locus. The question then is: how many "special loci" (loci over which a particular correspondence lives) are there in a given moduli space. My sense is that there will be ... $\endgroup$
    – Emerton
    Commented Nov 29, 2012 at 19:39
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    $\begingroup$ It is probably more reasonable to ask for a Fontaine-Mazur type conjecture for Hodge structures defined over a number field. By this I mean an integral Hodge structure $H$ plus a vector space defined over a number field, say $\mathbb{Q}$, with a filtration and an isomorphism of this vector space tensored with $\mathbb{C}$ with $H \otimes \mathbb{C}$ so that the filtration goes over to the Hodge filtration. This is what one gets by combining the de Rham cohomology and the Betti cohomology of a variety over $\mathbb{Q}$. Perhaps a condition on the periods a la Grothendieck implies it's motivic. $\endgroup$
    – naf
    Commented Nov 30, 2012 at 5:39

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