An Example of Rank 2 Poisson Structure which is Stable Under Deformations

TL;DR

This paper investigates the stability of a specific rank 2 Poisson structure under deformations, showing that for generic foliations, the associated components of Poisson structures and foliations coincide.

Contribution

It demonstrates the stability of a particular Poisson structure of rank 2 in projective space under deformations for generic foliations.

Findings

The irreducible component of Poisson structures containing the given structure coincides with that of the foliations.

Stability of the Poisson structure under deformations is established for generic cases.

The result links the geometry of Poisson structures with foliation theory in projective space.

Abstract

Let $\Pi$ be a rank $2$ Poisson Structure in the Projective Space defined by the dimension $2$ foliation $\mathcal{F}$ in the pull-back component. We prove that for a generic choice of $\mathcal{F}$, the irreducible component of the Poisson structures containing $\Pi$ and the irreducible component of Foliations containing $\mathcal{F}$ coincides.