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On the quaternionic Heisenberg group. (English) Zbl 0613.53016
Let $$M_ Q$$ be the compact nilmanifold defined as $$M_ Q=\Gamma_ Q\setminus H_ Q$$, where $$H_ Q$$ is the quaternionic Heisenberg group and $$\Gamma_ Q$$ is its subgroup of matrices with integer coordinates. It is proved that $$M_ Q$$ does not admit either generalized Hopf or symplectic structures. However, certain semi-Kähler structures [A. Gray and L. M. Hervella, Ann. Mat. Pura Appl., IV. Ser. 123, 35-58 (1980; Zbl 0444.53032)] are constructed on $$M_ Q$$, one of them being non-Hermitian and the others Hermitian. The Frölicher spectral sequence associated to one of these structures is analyzed.
Reviewer: Z.Olszak

##### MSC:
 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 55N30 Sheaf cohomology in algebraic topology