Using a criterion of Johnson-Rees [9] we give a list of all four and six dimensional flat Ka ¨hler manifolds. We calculate their R-cohomology, including the Hodge numbers. As a corollary, we classify all flat complex manifolds of dimension 3 whose holonomy groups are subgroups of SUð3Þ. Moreover, we define a family of flat Ka ¨hler manifolds which are generalizations of the oriented Hantzsche-Wendt Riemannian manifolds [14].

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.Comment: 19 pages, 1 tabl

In this article we show how to calculate the group of automorphisms of the flat Kähler manifolds. Moreover we are interested in the problem of classification such manifolds up to biholomorphism. We consider these problems from two points of view. The first one treats the automorphism group as a subgroup of the group of affine transformations, while in the second one we analyze it using automorphisms of complex tori. This leads us to the analogues of the Bieberbach theorems in the complex case. We end with some examples, which in particular show that in general the finiteness of the automorphism group depends not only on the fundamental group of a flat manifold.