Poisson structures on twistor spaces of hyperkähler and HKT manifolds

We characterize HKT structures in terms of a nondegenerate complex Poisson bivector on a hypercomplex manifold. We extend the characterization to the twistor space. After considering the flat case in detail, we show that the twistor space of a hyperkähler manifold admits a holomorphic Poisson structure. We briefly mention the relation to quaternionic and hypercomplex deformations on tori and K3 surfaces.