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

TL;DR

This paper extends transcendence results to families of Calabi-Yau manifolds, especially Borcea-Voisin towers, linking complex multiplication and transcendence in higher-dimensional Hodge structures.

Contribution

It applies techniques from previous transcendence results to Borcea-Voisin towers, advancing understanding of CM fibers and their transcendence properties in Calabi-Yau families.

Findings

Proved special cases of Green-Griffiths-Kerr conjecture for Calabi-Yau families.

Established dense sets of CM fibers in Borcea-Voisin towers.

Connected transcendence theory with higher-weight Hodge structures.

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.