Verbitsky, Misha Hypercomplex varieties. (English) Zbl 0935.32022 Commun. Anal. Geom. 7, No. 2, 355-396 (1999). 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. Reviewer: G.Pfister (Kaiserslautern) Cited in 13 Documents 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.) Keywords:hyperkähler manifolds; hypercomplex variety; twistor space PDFBibTeX XMLCite \textit{M. Verbitsky}, Commun. Anal. Geom. 7, No. 2, 355--396 (1999; Zbl 0935.32022) Full Text: DOI arXiv