Title: Semi-algebraic horizontal subvarieties of Calabi-Yau type

Abstract: I will discuss some joint work with R. Laza. We study horizontal subvarieties of a Griffiths period domain which are defined by algebraic equations, i.e. which extend to a closed subvariety of the compact dual of the period domain (which is a projective variety). If the horizontal variety is also invariant under a large discrete subgroup in an appropriate sense, then it is a Hermitian symmetric domain equivariantly embedded in the period domain. We are especially interested in the case where the Hodge structure is of Calabi--Yau type. In this case, the relevant variations of Hodge structure are essentially those found by Gross and Sheng--Zuo, up to taking factors of symmetric powers and -- --certain shift operations. The weight three case can be quite explicitly described, and is related to the study of homogeneous Legendrian varieties.