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?


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"

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    $\begingroup$ Welcome to mathoverflow! $\endgroup$ – Alexander Chervov Jan 20 at 17:49

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