Some new examples of compact inhomogeneous hypercomplex manifolds (Q1902186)

A smooth manifold \(M\) is said to be a hypercomplex manifold if there exist three complex structures \(I\), \(J\), and \(K\) on \(M\), satisfying the algebra of imaginary quaternions. Hyperkähler manifolds are examples of hypercomplex manifolds. The Hopf manifolds \(S^{4n + 3} \times S^1\) are examples of hypercomplex manifolds that are not Kähler, but they are locally conformally hyperkähler. In [J. Reine Angew. Math. 455, 183-220 (1994)] the authors gave a new class of compact, locally conformally hyperkähler manifolds by replacing \(S^{4n+3}\) by any 3-Sasakian manifold. However, there were no examples of inhomogeneous hypercomplex manifolds. In this paper, the authors present the explicit construction of new classes of compact, irreducible, hypercomplex manifolds which are not locally conformally hyperkähler and almost all of which are inhomogeneous. For all \(n\)-tuples of non-zero real numbers \(p = (p_1,p_2,\dots,p_n)\) they obtain a hypercomplex structure \(\{I^\alpha(p)\}_{\alpha = 1,2,3}\) on the Stiefel manifold of 2-planes in \(\mathbb{C}^n\). The authors determine the Lie algebra of infinitesimal hypercomplex automorphisms for each structure and show that among these structures there are uncountable families of pairwise inequivalent ones. Most of the presented examples admit discrete hypercomplex quotients by an action of the cyclic group of order \(k\). The authors analyze the topology of these non-simply connected examples.