Some New Examples of Compact Inhomogeneous Hypercomplex Manifolds

We announce the construction of new families of compact, irreducible, inhomogeneous, hypercomplex manifolds which are not locally conformally hyperkähler. We obtain, for all $n >2$ and all $n$-tuples of non-zero real numbers ${\bold p}=(p_1,\dots,p_n),$ a hypercomplex structure $\{{\cal I}^a(\bold p)\}_{a=1,2,3}$ on the Stiefel manifold of $2$-planes in $\bbc^n.$ We 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. Furthermore, these hypercomplex structures are inhomogeneous with the exception of the classical homogeneous spaces obtained when all the $p_i$’s are equal. Finally, countably many of our examples admit discrete hypercomplex quotients by an action of the cyclic group of order $k$ and we analyze the topology of these non-simply connected examples. Full details and proofs appear in our paper [BGM3].

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Published 1 January 1994