Quaternion Kähler flat manifolds (Q2715733)

This paper is a revised version of a talk that the author gave at the meeting ``Quaternionic structures in Mathematics and Physics'' in Rome in September 1999 and its results are contained in [\textit{I. G. Dotti} and \textit{R. J. Miatello}, Quaternion Kähler flat manifolds, Differ. Geom. Appl. 15, No. 1, 59-77 (2001; Zbl 1029.53053)]. The author gives a general method to construct quaternion Kähler compact flat manifolds which admit no Kähler structure. The construction is based on finding finite groups \(F\) which act freely on the \(4n\)-dimensional torus \(T^{4n} = \Lambda \setminus R^{4n}\) (with \(\Lambda\) a lattice in \(R^{4n}\)) endowed with the quaternion Kähler structure induced by the standard hyperkähler structure of \(R^{4n}\), in such a way that \(F \setminus T^{4n}\) becomes quaternion Kähler but its cohomology changes so that the manifold will not admit any Kähler structure. Such free actions of finite groups are constructed using Bieberbach groups (i.e crystallographic groups which are torsion-free) [\textit{L. Charlap}, Bieberbach groups and flat manifolds, Springer Verlag (1986; Zbl 0608.53001)].NEWLINENEWLINEFor the entire collection see [Zbl 0958.00032].