Elliptic log symplectic brackets on projective bundles

TL;DR

This paper develops a method to construct elliptic log symplectic Poisson brackets on complex projective bundles, demonstrating its effectiveness through examples including elliptic Feigin-Odesskii brackets.

Contribution

It introduces a novel construction method for elliptic log symplectic brackets using the local Torelli theorem and smoothing diagrams, expanding the understanding of such structures.

Findings

Constructed new elliptic log symplectic brackets on projective bundles.

Recovered all log symplectic cases of elliptic Feigin-Odesskii brackets.

Demonstrated the method's effectiveness through explicit examples.

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 $\mathbb{P}^{n-1}$.