Kähler flat manifolds

Abstract

Using a criterion of Johnson-Rees [9] we give a list of all four and six dimensional flat Kähler manifolds. We calculate their $\mathit{R}$–cohomology, including the Hodge numbers. As a corollary, we classify all flat complex manifolds of dimension 3 whose holonomy groups are subgroups of $\mathrm{SU}(3)$. Moreover, we define a family of flat Kähler manifolds which are generalizations of the oriented Hantzsche-Wendt Riemannian manifolds [14].