Some examples of global Poisson structures on \(S^4\).

*(English)*Zbl 1432.53119The goal of this paper is to construct some concrete examples of global Poisson structures on the smooth manifold \(S^4\), where a global Poisson structure is a bivector \(w \in \Gamma(\wedge^2TM)\) which vanishes under the Schouten bracket \([w,w] = 0\). This paper employs the twistor method to identify a subset of real Poisson structures on \(S^4\) with the holomorphic Poisson structures on the complex manifold \(\mathbb{C}P^3\).
The holomorphic Poisson structures on \(\mathbb{C}P^3\) has been completely classified [D. Cerveau and A. Lins Neto, Ann. Math. (2) 143, No. 3, 577–612 (1996, Zbl 0855.32015); F. Loray et al., Math. Nachr. 286, No. 8–9, 921–940 (2013, Zbl 1301.37032)], thus this allows for new Poisson structures on \(S^4\) to be constructed.

The main technical result of this paper follows in two steps. First, the \(q\)-vectors on \(\mathbb{C}P^3\) are characterized as pushforwards of \(q\)-vectors on \(C^4\setminus\{0\}\) and the space of holomorphic Poisson structures are given an explicit description as a complex space with a real structure. In the second step, \(S^4\) is identified with \(\mathbb{H}P^1\) (where \(\mathbb{H}\) is Hamilton’s quaternions), and a subspace of real Poisson structure on \(\mathbb{H}P^1\) is identified with the real part of the space of holomorphic Poisson structures on \(\mathbb{C}P^3\).

In Section 5, these results are generalized to the higher-dimensional cases of \(\mathbb{C}P^n\) and \(\mathbb{H}P^m\). In Section 6, a Poisson structures on \(S^4\) is induced by a foliation of codimension-1 of degree 2 on \(\mathbb{C}P^3\) for each of the six disconnected components of the space of such folations.

The main technical result of this paper follows in two steps. First, the \(q\)-vectors on \(\mathbb{C}P^3\) are characterized as pushforwards of \(q\)-vectors on \(C^4\setminus\{0\}\) and the space of holomorphic Poisson structures are given an explicit description as a complex space with a real structure. In the second step, \(S^4\) is identified with \(\mathbb{H}P^1\) (where \(\mathbb{H}\) is Hamilton’s quaternions), and a subspace of real Poisson structure on \(\mathbb{H}P^1\) is identified with the real part of the space of holomorphic Poisson structures on \(\mathbb{C}P^3\).

In Section 5, these results are generalized to the higher-dimensional cases of \(\mathbb{C}P^n\) and \(\mathbb{H}P^m\). In Section 6, a Poisson structures on \(S^4\) is induced by a foliation of codimension-1 of degree 2 on \(\mathbb{C}P^3\) for each of the six disconnected components of the space of such folations.

Reviewer: Benjamin MacAdam (Calgary)