Manifolds with holonomy U∗(2m)

We consider the geometry determined by a torsion-free affine connection whose holonomy lies in the subgroup U∗(2m) , a real form of GL(2m,C) , otherwise denoted by SL(m,H)⋅U(1) . We show in particular how examples may be generated from quaternionic Kähler or hyperkähler manifolds with a circle action.