Complex Hantzsche-Wendt manifolds

TL;DR

This paper classifies complex Hantzsche-Wendt manifolds, which are flat Kähler Calabi-Yau manifolds with specific holonomy, by describing them as quotients of elliptic curves and providing an algorithm for their classification.

Contribution

It offers a detailed description of these manifolds as quotients of elliptic curves and introduces an algorithm to classify their diffeomorphism types.

Findings

Classification of all integral holonomy representations

Algorithm for classifying diffeomorphism types

Description of manifolds as quotients of elliptic curves

Abstract

Complex Hantzsche-Wendt manifolds are flat K\"ahler manifolds with holonomy group $\mathbb{Z}_2^{n-1}\subset SU(n)$. They are important example of Calabi-Yau manifolds of abelian type. In this paper we describe them as quotients of a product of elliptic curves by a finite group $\tilde{G}$. This will allow us to classify all possible integral holonomy representations and give an algorithm classifying their diffeomorphism types.