Semi-algebraic horizontal subvarieties of Calabi-Yau type

TL;DR

This paper classifies algebraic, Hermitian symmetric subvarieties of Griffiths period domains of Calabi-Yau type, linking them to known structures and describing their embeddings explicitly.

Contribution

It proves that algebraic, invariant horizontal subvarieties are Hermitian symmetric domains and classifies Calabi-Yau type VHS parametrized by these domains, extending prior classifications.

Findings

Algebraic, invariant horizontal subvarieties are Hermitian symmetric domains.

Classified VHS of Calabi-Yau type parametrized by Hermitian symmetric domains.

Explicit description of embeddings in the weight three case.

Abstract

We study horizontal subvarieties $Z$ of a Griffiths period domain $\mathbb D$. 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 $\mathbb D$. 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.