HKT manifolds with holonomy SL\((n,H)\) (Q2909671)

A hyper-Kähler with torsion (HKT) structure on an hyper-Hermitian manifold is a linear connection with totally skew-symmetric torsion preserving the hyper-Hermitian structure. In this paper, it is proved that for a HKT manifold, the holonomy of the Obata connection is contained in SL(\(n,H\)) if and only if the Lee form is an exact 1-form. Recall that the Obata connection is the unique torsion-free connection preserving the underlying hypercomplex structure. By a result of \textit{M. Verbitsky} [in: Moscow Seminar in mathematical physics, II. Providence, RI: American Mathematical Society (AMS). Translations. Series 2. American Mathematical Society 221. Advances in the Mathematical Sciences 60, 203--211 (2007; Zbl 1141.53041)], this is (for a compact HKT manifold) equivalent to the property that the HKT manifold has holomorphically trivial canonical bundle. As an application, the authors construct compact HKT manifolds with holomorphically trivial canonical bundle that are not balanced (i.e, whose Lee form does not vanish). A simple criterion for the nonexistence of HKT metrics on hypercomplex manifolds is given in terms of the Ricci-type tensors of the Obata connection.