The KSBA compactification for the moduli space of degree two K3 pairs

Inspired by the ideas of the minimal model program, Shepherd-Barron, Koll\'ar, and Alexeev have constructed a geometric compactification for the moduli space of surfaces of log general type. In this paper, we discuss one of the simplest examples that fits into this framework: the case of pairs (X,H) consisting of a degree two K3 surface X and an ample divisor H. Specifically, we construct and describe explicitly a geometric compactification $\bar{P_2}$ for the moduli of degree two K3 pairs. This compactification has a natural forgetful map to the Baily-Borel compactification of the moduli space $F_2$ of degree two K3 surfaces. Using this map and the modular meaning of $\bar{P_2}$, we obtain a better understanding of the geometry of the standard compactifications of $F_2$.Comment: 45 pages, 4 figures, 2 table

Semi-algebraic horizontal subvarieties of Calabi-Yau type

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.Comment: 53 pages, final version, to appear in Duke Math. J.; changes from v3: new references added; changes from v2: for Hermitian VHS of CY 3-fold type with real multiplication, we discuss the case SU(3,3) for arbitrary totally real number fields; the case SO^*(12) is discussed in arXiv:1301.2582; changes from v1: some inaccuracies corrected, Section 3 substantially expande