Hypercomplex varieties

We give a number of equivalent definitions of hypercomplex varieties and co nstruct a twister space for a hypercomplex variety. We prove that our defin ition of a hypercomplex variety (used, e. g., in alg-geom 9612013) is equiv alent to a definition proposed by Deligne and Simpson, who used twister spa ces. 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 varie ty is smooth and hypercomplex.