Some examples of global Poisson structures on S⁴

Abstract

A Poisson structure is a bivector whose Schouten bracket vanishes. We study a global Poisson structure on S⁴ associated with a holomorphic Poisson structure on CP³. The space of such Poisson structures on S⁴ is realised as a real algebraic variety in the space of holomorphic Poisson structures on CP³. We generalize the result to the higher dimensional case HPⁿ by the twistor method. It is known that a holomorphic Poisson structure on CP³ corresponds to a codimension one holomorphic foliation and the space of these foliations of degree 2 has six components. In this paper we provide examples of Poisson structures on S⁴ associated with these components.