Semialgebraic horizontal subvarieties of Calabi–Yau type

Abstract

We study horizontal subvarieties $Z$ of a Griffiths period domain $\mathbf{D}$. If $Z$ is defined by algebraic equations, and if $Z$ is also invariant under a large discrete subgroup in an appropriate sense, we prove that $Z$ is a Hermitian symmetric domain $\mathcal{D}$, embedded via a totally geodesic embedding in $\mathbf{D}$. Next we discuss the case when $Z$ is in addition of Calabi–Yau type. We classify the possible variations of Hodge structure (VHS) of Calabi–Yau type parameterized by Hermitian symmetric domains $\mathcal{D}$ and show that they are essentially those found by Gross and Sheng and Zuo, up to taking factors of symmetric powers and certain shift operations. In the weight $3$ case, we explicitly describe the embedding $Z\hookrightarrow \mathbf{D}$ from the perspective of Griffiths transversality and relate this description to the Harish-Chandra realization of $\mathcal{D}$ and to the Korányi–Wolf tube domain description. There are further connections to homogeneous Legendrian varieties and the four Severi varieties of Zak.