A twist construction for manifolds with torus action is described generalizing certain T-duality examples and constructions in hypercomplex geometry. It is applied to complex, SKT, hypercomplex, and HKT manifolds to construct compact simply connected examples. In particular, we find hypercomplex manifolds that admit no compatible HKT metric, and HKT manifolds whose Obata connection has holonomy contained in .
Andrew Swann. "Twisting Hermitian and hypercomplex geometries." Duke Math. J. 155 (2) 403 - 431, 1 November 2010. https://doi.org/10.1215/00127094-2010-059