Title: A Maximal Family of Borcea-Voisin Calabi-Yau 3-folds.
Abstract: We will discuss Rohde's construction of a maximal family of Calabi-Yau 3-folds with a maximal automorphism, arising from the restricted family of marked K3 surfaces with a non-symplectic Z3-action. We will then indicate how the existence of this maximal automorphism gives some constancy results in the variations of Hodge structure, and how this implies that members of this family cannot satisfy Morrison's formulation of the mirror conjecture.
Finally, we will investigate how the methods described here can extend to constructing Calabi-Yau 3-folds over a proper base with desirable properties. In particular, we will discuss some possible solutions to this problem by considering certain compact Shimura varieties as moduli of abelian varieties with quaternionic multiplication.