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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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##### References:
 [1] Alekseevsky, D.V.; Marchiafava, S., Quaternionic structures on a manifold and subordinated structures, Annali di matematica pura ed applicata, 171, 205-273, (1996) · Zbl 0968.53033 [2] Bergery, L.Bérard; Ochiai, T., On some generalizations of the construction of twistor spaces, (), 52-58 [3] Besse, A.L., Einstein manifolds, (), 3. Folge · Zbl 1147.53001 [4] Glazebrook, J.F.; Kamber, F.W.; Pedersen, H.; Swann, A., Foliation reduction and self-duality, (), 219-249 [5] Gray, A.; Gray, A., A note on manifolds whose holonomy group is a subgroup of sp(n) · sp(1), Mich. math. J., Mich. math. J., 17, 409-128, (1970) · Zbl 0177.50001 [6] Joyce, D., Compact quaternionic and hypercomplex manifolds, J. differential geom., 35, 743-761, (1992) · Zbl 0735.53050 [7] Marchiafava, S.; Romani, G., Sui fibrati con struttura quaternionale generalizzata, Annali di matematica pura ed applicata, 107, 131-157, (1976) · Zbl 0356.55003 [8] Obata, M., Affine connections on manifolds with almost complex, quaternion or Hermitian structure, Jap. J. math., 26, 43-79, (1956) [9] Pedersen, H.; Swann, A.F., Riemannian submersions, four-manifolds and Einstein-Weyl geometry, (), 381-399 · Zbl 0742.53014 [10] Poon, Y.S., Compact self-dual manifolds with positive scalar curvature, J. differential geom., 24, 97-132, (1986) · Zbl 0583.53054 [11] Salamon, S.M., Differential geometry of quaternionic manifolds, Ann. scient. éc. norm. sup., 19, 31-55, (1986) · Zbl 0616.53023 [12] Steenrod, N., The topology of fibre bundles, () · Zbl 0054.07103 [13] Swann, A.F., Hyperkähler and quaternionic Kähler geometry, Math. ann., 289, 421-450, (1991) · Zbl 0711.53051 [14] Wolf, J.A., Complex homogeneous contact manifolds and quaternionic symmetric spaces, J. math. mech., 14, 1033-1047, (1965) · Zbl 0141.38202
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