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

Question

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?

Answer 1

It was proven by Alesha Bondal in his MPIM preprint MPI/93 (Thm. on p.11) and on the paper of Sasha Polishchuk ("Algebraic Geometry of Poisson brackets", J. Math. Sci.,vol.94 no.5, 1997, thm. 12.1 Both proofs are very simple and use only "Poisson differential calculus"