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Hypercomplex varieties. (English) Zbl 0935.32022

A real analytic variety equipped with almost complex structures \(I,J,K\) such that \(I\circ J=-J\circ I=K\) is called an almost hypercomplex variety. It is called hypercomplex variety if there exist a pair \(I_1\), \(I_2\in\mathbb{H}\), the quaternion algebra, of inducing complex structures such that \(I_1\neq\pm I_2\) and \(I_1,I_2\) are integrable.
Equivalent definitions are given and discussed. The twistor space of a hypercomplex variety is constructed. It is proved that the normalisation of a hypercomplex variety is smooth and hypercomplex. Some applications (hypercomplex spaces, stable bundles over hyperkähler manifolds, quotients of hypercomplex varieties under finite group actions) are given.

MSC:

32Q99 Complex manifolds
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
32L25 Twistor theory, double fibrations (complex-analytic aspects)
32C05 Real-analytic manifolds, real-analytic spaces
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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