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Some examples of global Poisson structures on $$S^4$$. (English) Zbl 1432.53119
The 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.
##### MSC:
 53D17 Poisson manifolds; Poisson groupoids and algebroids 37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
##### Keywords:
Poisson structure; twistor method; holomorphic foliation
##### Citations:
Zbl 0855.32015; Zbl 1301.37032
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