Some examples of global Poisson structures on $S^4$

Abstract

A Poisson structure is represented by a bivector whose Schouten bracket vanishes. We study a global Poisson structure on $S^4$ associated with a holomorphic Poisson structure on ${\mathbb{CP}}^3$. The space of the Poisson structures on $S^4$ is a real algebraic variety in the space of holomorphic Poisson structures on ${\mathbb{CP}}^3$. We generalize it to ${\mathbb{HP}}^n$ by using the twistor method. Furthermore, we provide examples of Poisson structures on $S^4$ associated with codimension one holomorphic foliations of degree 2 on ${\mathbb{CP}}^3$.