# Holomorphic Poisson structures on $C P^{n-1}$ and homogeneous Poisson structures on $C^n$

Is it correct that any holomorphic Poisson structure on $$C P^{n-1}$$ can be lifted to a homogeneous Poisson structure on $$C^n$$? By homogeneous I mean a quadratic Poisson structure of the form $$\{z_i,z_j\}=q_{ij}^{kl}z_kz_l$$ where coefficients $$q_{ij}^{kl}$$ are constants.

I suspect that this is correct but do not know any reference nor any idea of possible proof. Could you please help me with these?