  Hodge theory associates to a smooth projective variety over $\mathbb{C}$ a
piece of linear algebra information, called a $\mathbb{Q}$-Hodge structure.
Conversely, it is a natural question which abstract $\mathbb{Q}$-Hodge
structures arise from the cohomology of a smooth projective complex variety, or
more generally, from a pure motive over $\mathbb{C}$. By a classical argument
involving Griffiths transversality and a Baire category argument, it is well
known that there are many Hodge structures which do not come from geometry in
this sense. However, the argument is not constructive, and does not seem to
give a criterion to decide whether a given Hodge structure comes from geometry.
We formulate an intrinsic condition on a $\mathbb{Q}$-Hodge structure that we
expect to be satisfied for all Hodge structures coming from geometry. We prove
that this expectation follows from the conjunction of two fundamental
conjectures in Hodge theory and transcendence theory: the conjecture that Hodge
cycles are motivated and Andr\'e's generalized Grothendieck period conjecture.
By doing so, we exhibit explicit examples of $\mathbb{Q}$-Hodge structures
which should not come from geometry.
