Semialgebraic horizontal subvarieties of calabi-yau type

We study horizontal subvarieties Z of a Griffiths period domain 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 D, embedded via a totally geodesic embedding in 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 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{heavy black curved downwards and rightwards arrow}D from the perspective of Griffiths transversality and relate this description to the Harish-Chandra realization of D and to the Korányi-Wolf tube domain description. There are further connections to homogeneous Legendrian varieties and the four Severi varieties of Zak.