What is an explicit example a pure polarizable finite-dimensional $\mathbb{Q}$-Hodge structure that is not a subquotient of the cohomology of a scheme smooth proper over $\mathbb{C}$? What if replace "a scheme smooth proper over $\mathbb{C}$" with "closed complex manifold bimeromorphic to a closed Kaehler manifold"?

Note that we are asking for an explicit example, not "Hilbert scheme blah blah countable blah blah".

If it helps, assume any famous conjecture you want to assume.


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The existence of non-motivic (not coming from algebraic geometry) polarized Hodge structures is currently non-constructive. For period domains which are not Hermitian symmetric, motivic Hodge structures are contained in countably infinite proper submanifolds, so in a sense most Hodge structures in the domain are not motivic. However the explicit construction of a single example is still an open problem.


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