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

Abstract

A Poisson structure is a bivector whose Schouten bracket vanishes. We study a global Poisson structure on $S^4$ associated with a holomorphic Poisson structure on $\mathbf{CP}^3$. The space of such Poisson structures on $S^4$ is realised as a real algebraic variety in the space of holomorphic Poisson structures on $\mathbf{CP}^3$. We generalize the result to the higher dimensional case $\mathbf{HP}^n$ by the twistor method. It is known that a holomorphic Poisson structure on $\mathbf{CP}^3$ 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 $S^4$ associated with these components.