On the Cohomology Ring of Flat Manifolds with a Special Structure

A Riemannian manifold is said to be Kähler if the holonomy group is contained in U(n). It is quaternion Kähler if the holonomy group is contained in Sp(n)Sp(1). It is known that quaternion Kähler manifolds of dimension ≥ 8 are Einstein, so the scalar curvature s splits these manifolds according to whether s > 0, s = 0 or s < 0. Ricci flat quaternion Kähler manifolds include hyperkähler manifolds, that is, those with holonomy group contained in Sp(n). Such a manifold can be characterized by the existence of a pair of integrable, anticommuting complex structures, compatible with the Riemannian metric, and parallel with respect to the Levi-Civita connection (see [Be], for instance). The simplest model of hyperkähler manifolds is provided by R with the standard flat metric and a pair J,K of orthogonal anticommuting complex structures. This hyperkähler structure descends to the 4n-torus TΛ := Λ\R, for any lattice Λ in R. IfMΓ = Γ\R is a compact flat manifold such that the holonomy action of F = Λ\Γ centralizes (resp. normalizes) the algebra generated by J,K, then MΓ inherits a hyperkähler (resp. quaternion Kähler) structure. In [DM] (see also [JR] and [BDM]) we described a doubling construction for Bieberbach groups which allows to give rather simple examples of quaternion Kähler flat manifolds which admit no Kähler structure. The purpose of the present paper is to study the real cohomology ring of low dimensional compact flat manifolds endowed with one of these special structures. In particular, we will determine the structure of this ring in the case of all 4dimensional Kähler flat manifolds and all 8-dimensional compact flat kyperkähler manifolds. We shall make use of the known classification of space groups in dimension 4, given in [BBNWZ], and of the classification of flat hyperkähler 8-manifolds due to L. Whitt ([Wh]). It turns out that the integral holonomy groups of hyperkähler 8-manifolds are obtained by doubling the holonomy groups of the Kähler flat 4-manifolds and as a consequence we will show that the cohomology ring is an exterior algebra in generators of degree one and two. In [Sa], [Sa2] and [Sa3], Salamon obtains a family of linear relations among the Betti numbers of general hyperkähler manifolds (see Remark 4.2). In Section 5 we give several examples (5.1 5.3) showing that these relations may not hold in the quaternion Kähler case.