Manifolds with holonomy $$U^*(2m)$$ U ∗ ( 2 m )

Abstract

We consider the geometry determined by a torsion-free affine connection whose holonomy lies in the subgroup \(U^*(2m)\), a real form of \(GL(2m,\mathbf {C})\), otherwise denoted by \(SL(m,\mathbf {H}) \cdot U(1)\). We show in particular how examples may be generated from quaternionic Kahler or hyperkahler manifolds with a circle action.