Symplectic leaves in projective spaces of bundle extensions

Abstract

Fix a stable degree-$n$ rank-$k$ bundle $\mathcal{F}$ on a complex elliptic curve for (coprime) $1\le k<n\ge 3$. We identify the symplectic leaves of the Poisson structure introduced independently by Polishchuk and Feigin-Odesskii on $\mathbb{P}^{n-1}\cong \mathbb{P}\mathrm{Ext}^1(\mathcal{F},\mathcal{O})$ as precisely the loci classifying extensions $0\to \mathcal{O}\to \mathcal{E}\to \mathcal{F}\to 0$ with $\mathcal{E}$ fitting into a fixed isomorphism class, verifying a claim of Feigin-Odesskii. We also classify the bundles $\mathcal{E}$ which do fit into such extensions in geometric / combinatorial terms, involving their Harder-Narasimhan polygons introduced by Shatz.