47 research outputs found
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
for the moduli of degree two K3 pairs. This compactification has a natural
forgetful map to the Baily-Borel compactification of the moduli space of
degree two K3 surfaces. Using this map and the modular meaning of ,
we obtain a better understanding of the geometry of the standard
compactifications of .Comment: 45 pages, 4 figures, 2 table
Semi-algebraic horizontal subvarieties of Calabi-Yau type
We study horizontal subvarieties of a Griffiths period domain . If is defined by algebraic equations, and if is also invariant
under a large discrete subgroup in an appropriate sense, we prove that is a
Hermitian symmetric domain , embedded via a totally geodesic
embedding in . Next we discuss the case when is in addition of
Calabi-Yau type. We classify the possible VHS of Calabi-Yau type parametrized
by Hermitian symmetric domains 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 from the perspective of Griffiths
transversality and relate this description to the Harish-Chandra realization of
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