Cyclic coverings, Calabi-Yau manifolds and Complex multiplication

TL;DR

This paper constructs families of Calabi-Yau manifolds with dense complex multiplication fibers across various dimensions, providing explicit examples and analyzing their Hodge groups, advancing understanding of their arithmetic and geometric properties.

Contribution

It introduces a method to construct Calabi-Yau families with dense CM fibers in any dimension and provides explicit examples for genus up to 7.

Findings

Dense sets of CM fibers in constructed Calabi-Yau families

Explicit examples of CM fibers for genus ≤ 7

Analysis of generic Hodge groups of cyclic covers

Abstract

We construct families of Calabi-Yau manifolds with dense set of complex multiplication fibers in an arbitrary dimension. We will also give explicite examples of complex multiplication fibers. For this construction we use families of curves with dense set of complex multiplication fibers. In addition, we give examples of such families for each genus less or equal 7 and we study the generic Hodge groups of families of cyclic covers of the projective line.