Calabi-Yau Varieties via Cyclic Covers, and Complex Hyperbolic Structures for their Moduli Spaces

In this paper we mainly study Calabi--Yau varieties that arise as triple covers of products of projective lines branched along simple normal crossing divisors.

For some of those families of Calabi--Yau varieties, the period maps factor through arithmetic quotients of complex hyperbolic balls.

We give a classification of such examples.

One of the families was previously studied by Voisin, Borcea and Rohde.

For these ball-type cases, we will show arithmeticity of the monodromy groups.

These ball quotients are all commensurable to ball quotients in Deligne--Mostow theory.

As a byproduct, we prove some commensurability relations among arithmetic groups in Deligne--Mostow theory.