Hodge structures not coming from geometry

Abstract

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.