A flat projective variety with $D_8$-holonomy

We show explicitly that the compact flat Kähler manifold of complex dimension three with D8 holonomy studied by Dekimpe, Halenda and Szczepanski ([5] p. 367) possesses the structure of a nonsingular projective variety. This corrects a previous statement by H. Lange in [9] that the holonomy group of a hyperelliptic threefold is necessarily abelian. The study of flat Riemannian manifolds, begun by Bieberbach [2], has subsequently acquired a very extensive literature. See, for example, [3],[4],[13],[14]. In a paper published in the Tohoku Mathematical Journal [9], H. Lange investigated closed flat manifolds of real dimension six which, in addition, possess the structure of nonsingular complex projective varieties which have finite étale coverings by abelian varieties. In Lange’s terminology such varieties are called hyperelliptic three-folds. The significant claim of Lange’s paper is that the (finite) holonomy group of such a hyperelliptic three-fold is necessarily abelian. In particular, Lange claims that the dihedral group of order eight† does not occur as a holonomy group in this context. Lange’s claim is mistaken, however. In the present paper we show explicitly that the compact flat Kähler manifold of complex dimension three with D8 holonomy studied by Dekimpe, Halenda and Szczepanski ([5] p. 367) does indeed possess the structure of a nonsingular projective variety. In fact, the existence of this complex algebraic structure was previously shown, in a very general context, by the present author in the paper [7]. However, as Lange also makes a statement which explicitly claims to contradict the main result of [7] it seems appropriate, in setting the matter straight, to give a direct, and elementary, construction of the algebraic structure whose existence Lange denies. The present paper is organised as follows; in §1 we give a brief review of the theory of flat Riemannian manifolds as it pertains both to Kähler manifolds and projective varieties; in §2 we give a completely elementary criterion which guarantees that some flat Riemannian manifolds admit the structure of a nonsingular complex algebraic variety. Whilst this criterion does not immediately apply to the most general cases, it is quite sufficient to deal with all cases in which the holonomy group is D8. In §3 we construct an explicit complex algebraic structure for the Kähler manifold of Dekimpe, Halenda and Szczepanski. This can be checked by direct calculation and requires very little theory beyond an appeal to the criterion of §2. 2010 MSC Primary 53C29; Secondary 14F35, 14K02, 32J27.