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

TL;DR

This paper explores global Poisson structures on the 4-sphere, linking them to holomorphic Poisson structures on complex projective space, and extends the analysis to quaternionic projective spaces using twistor methods.

Contribution

It introduces a framework connecting Poisson structures on $S^4$ with holomorphic Poisson structures on $ ext{CP}^3$ and generalizes this to $ ext{HP}^n$ via twistor theory, providing new examples.

Findings

Poisson structures on $S^4$ form a real algebraic variety.

Established a correspondence between Poisson structures on $S^4$ and $ ext{CP}^3$.

Constructed examples related to holomorphic foliations of degree 2.

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 $\mathbb{CP}^3$.