Elliptic log symplectic brackets on projective bundles

Abstract: Let $\mathsf{X}$ be the product of a complex projective space and a polydisc. We study Poisson brackets on $\mathsf{X}$ that are log symplectic, that is, generically symplectic and such that the inverse two-form has only first order poles. We propose a method of constructing such Poisson brackets that additionally are elliptic, in a precise sense. Our method relies on the local Torelli theorem for log symplectic manifolds of Pym, Schedler and the author, and uses combinatorics of smoothing diagrams. We demonstrate effectiveness of the method on a series of examples, recovering, in particular, all log symplectic cases of elliptic Feigin-Odesskii Poisson brackets $q_{n,k}$ on $\mathbb{P}^{n-1}$.