The symplectic leaves for the elliptic Poisson bracket on projective space defined by Feigin-Odesskii and Polishchuk

TL;DR

This paper characterizes the symplectic leaves of a special elliptic Poisson structure on complex projective space, linking them to secant varieties of elliptic curves, advancing understanding of elliptic Poisson geometries.

Contribution

It explicitly describes the symplectic leaves of Feigin-Odesskii and Polishchuk's elliptic Poisson structure using secant varieties of elliptic curves.

Findings

Symplectic leaves correspond to higher secant varieties of elliptic curves.

Provides a geometric description of the Poisson structure's symplectic foliation.

Connects Poisson geometry with algebraic geometry of elliptic curves.

Abstract

This paper determines the symplectic leaves for a remarkable Poisson structure on $\mathbb{C}\mathbb{P}^{n-1}$ discovered by Feigin and Odesskii, and, independently, by Polishchuk. The Poisson bracket is determined by a holomorphic line bundle of degree $n \ge 3$ on a compact Riemann surface of genus one or, equivalently, by an elliptic normal curve $E\subseteq\mathbb{C}\mathbb{P}^{n-1}$. The symplectic leaves are described in terms of higher secant varieties to $E$.