On the quaternionic Heisenberg group. (English) Zbl 0613.53016

Cordero, Luis A.; Fernández, Marisa; de León, Manuel

Boll. Unione Mat. Ital., VII. Ser., A 1, 31-37 (1987).

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

Keywords:

almost Hermitian manifold; nilmanifold; quaternionic Heisenberg group; symplectic structures; Frölicher spectral sequence