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The hypercomplex quotient and the quaternionic quotient. (English) Zbl 0723.53043
Quotient constructions are defined for hypercomplex and quaternionic manifolds that are analogous to those given for hyperkähler manifolds by N. J. Hitchin, A. Karlhede, U. Lindström and M. Roček [Commun. Math. Phys. 108, 535-589 (1987; Zbl 0612.53043)] and for quaternionic Kähler manifolds by K. Galicki and H. B. Lawson jun. [Math. Ann. 282, No.1, 1-21 (1988; Zbl 0628.53060)]. Poon’s metrics on \({\mathbb{C}}{\mathbb{P}}^ 2\#{\mathbb{C}}{\mathbb{P}}^ 2\) are an example of the latter. We define quaternionic complex manifolds to be manifolds with a torsion-free connection of holonomy SL(n,\({\mathbb{H}})U(1)\). In four dimensions the appropriate definition is that of a scalar-flat Kähler surface. Their basic properties are explored and a quotient construction and examples are given.
Reviewer: D.Joyce (Oxford)

53C56 Other complex differential geometry
32C99 Analytic spaces
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