Rigidity of Poisson Structures

We study germs of analytic Poisson structures which are suitable perturbations of a quasihomogeneous Poisson structure in a neighborhood of the origin of $\mathbb R^n$ or $\mathbb C^n$, a fixed point of the Poisson structures. We define a “diophantine condition” relative to the quasihomogeneous initial part $\mathcal L$ which ensures that such a good perturbation of $\mathcal L$ which is formally conjugate to $\mathcal L$ is also analytically conjugate to it.