Compact hypercomplex and quaternionic manifolds. (English) Zbl 0735.53050

Joyce, Dominic David

J. Differ. Geom. 35, No. 3, 743-761 (1992).

Hypercomplex manifolds and quaternionic manifolds are manifolds with a \(GL(n,{\mathbb{H}})\)- and a \(GL(n,{\mathbb{H}}){\mathbb{H}}^*\)-structure respectively, preserved by a torsion-free connection. A hypercomplex manifold thus has three anticommuting integrable complex structures \(I_ 2\), \(I_ 2\), \(I_ 3\). Hypercomplex and quaternionic manifolds are generalizations of hyperkähler and quaternionic Kähler manifolds, which occur in the theory of special holonomy groups of Riemannian manifolds. The author presents two constructions for these two types of manifolds. The first construction involves modifying a given hypercomplex or quaternionic manifold acted on by a group \(G\) by ’twisting’ by a \(G\)-invariant \(G\)- connection satisfying a curvature condition, to get another hypercomplex or quaternionic manifold. The second construction uses the structure theory of Lie groups to construct homogeneous hypercomplex and quaternionic manifolds. Both constructions yield new compact examples of these manifolds.

Reviewer: D.D.Joyce (Oxford)

MSC:

53C55	Global differential geometry of Hermitian and Kählerian manifolds
53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)

Keywords:

hypercomplex manifolds; quaternionic manifolds; integrable complex structures; holonomy groups; G-connection; Lie groups