Properties of manifolds with skew-symmetric torsion and special holonomy

In this paper we provide examples of hypercomplex manifolds which do not carry HKT structures, thus answering a question in Grantcharov and Poon (Comm. Math. Phys. 213 (2000) 19). We also prove that the existence of an HKT structure is not stable under small deformations. Similarly we provide examples of compact complex manifolds with vanishing first Chern class which do not admit a Hermitian structure whose Bismut connection has restricted holonomy in SU(n), thus providing a counter-example to the conjecture in Gutowski et al. (Deformations of generalized calibrations and compact non-Kähler manifolds with vanishing first Chern class, math.DG/0205012, Asian J. Math., to appear). Again we prove that such a property is not stable under small deformations.