Transcendence and CM on Borcea-Voisin towers of Calabi-Yau manifolds

Abstract: This paper is a sequel to a paper by the author and Marvin D. Tretkoff (reference [49]), in which we showed the validity of a special case of a conjecture of Green, Griffths and Kerr for certain families of Calabi-Yau manifolds over Hermitian symmetric domains. These results are analogues of a celebrated theorem of Th. Schneider on the transcendence of values of the elliptic modular function, and its generalization to the context of abelian varieties by the author, Shiga and Wolfart. In the present paper, we apply related techniques to many of the examples of families of Calabi-Yau varieties with dense sets of CM fibers in the work of Rohde, and in particular to Borcea-Voisin towers. Our results fit into a broader context of transcendence theory for variations of Hodge structure of higher weight.