Hodge Theory of Calabi-Yau Varieties

Abstract

We study Calabi-Yau varieties from the perspective of Hodge theory. First, we review constructions in Hodge theory, toric geometry, and $\cD$-modules.Next, we complete the period map associated with a variation of polarized Hodge structure arising from a two-dimensional geometric family of Hodge type $(1,2,2,1)$. This is the second known example of a completion of a period map in a dimension higher than one, with a non-Hermitian symmetric Mumford-Tate domain. The completion technique we use is the theory of Kato-Usui spaces, while the calculation of monodromy matrices is aided by mirror symmetry.We then present a method for computing the generic degree of a period map defined on a quasi-projective surface. As an application, we explicitly compute the generic degree of three period maps underlying families of Calabi-Yau threefolds arising from toric hypersurfaces. As a consequence, we show that the generic Torelli theorem holds in these cases.