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Hypercomplex structures associated to quaternionic manifolds. (English) Zbl 0920.53018
If \(M\) is a quaternionic manifold and \(P\) is an \(S^1\)-instanton over \(M\), then D. D. Joyce [J. Differ. Geom. 35, 743-761 (1992; Zbl 0735.53050)] constructed a hypercomplex manifold we call \({\mathcal P}(M)\) over \(M\). These hypercomplex manifolds admit a \(U(2)\)-action of a special type permuting the complex structures. We show that up to double covers, all such hypercomplex manifolds arise in this way. Examples, including that of a hypercomplex structure on \(SU(3)\), show the necessity of including double covers of \({\mathcal P}(M)\).

MSC:
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
57S15 Compact Lie groups of differentiable transformations
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
53C56 Other complex differential geometry
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