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Semialgebraic horizontal subvarieties of Calabi-Yau type. (English) Zbl 1435.14011

Summary: 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.

MSC:

14D07 Variation of Hodge structures (algebro-geometric aspects)
32G20 Period matrices, variation of Hodge structure; degenerations
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
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