Semi-algebraic horizontal subvarieties of Calabi-Yau type

We study horizontal subvarieties $Z$ of a Griffiths period domain $\mathbb D$\mathrm{.}If$Z$isdefinedbyalgebraicequations,andif$Z$ 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 $\mathbb{D}$. Next we discuss the case when $Z$ is in addition of Calabi-Yau type. We classify the possible VHS of Calabi-Yau type parametrized by Hermitian symmetric domains $\mathcal{D}$ and show that they are essentially those found by Gross and Sheng-Zuo, up to taking factors of symmetric powers and certain shift operations. In the weight three case, we explicitly describe the embedding $Z\hookrightarrow \mathbb{D}$ from the perspective of Griffiths transversality and relate this description to the Harish-Chandra realization of $\mathcal{D}$ and to the Kor'anyi-Wolf tube domain description. There are further connections to homogeneous Legendrian varieties and the four Severi varieties of Zak.