Hypercomplex Varieties

We give a number of equivalent definitions of hypercomplex varieties and construct a twistor space for a hypercomplex variety. We prove that our definition of a hypercomplex variety (used, e. g., in alg-geom/9612013) is equivalent to a definition proposed by Deligne and Simpson, who used twistor spaces. This gives a way to define hypercomplex spaces (to allow nilpotents in the structure sheaf). We give a self-contained proof of desingularization theorem for hypercomplex varieties: a normalization of a hypercomplex variety is smooth and hypercomplex.