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Fix a stable degree-$n$ rank-$k$ bundle $\mathcal{F}$ on a complex ellipticcurve for (coprime) $1\le k<n\ge 3$. We identify the symplectic leaves of thePoisson structure introduced independently by Polishchuk and Feigin-Odesskii on$\mathbb{P}^{n-1}\cong \mathbb{P}\mathrm{Ext}^1(\mathcal{F},\mathcal{O})$ asprecisely 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 / combinatorialterms, involving their Harder-Narasimhan polygons introduced by Shatz.

Symplectic leaves in projective spaces of bundle extensions