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Symplectic leaves in projective spaces of bundle extensions

Alexandru Chirvasitu

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.

Symplectic leaves in projective spaces of bundle extensions

Abstract

Fix a stable degree- rank- bundle on a complex elliptic curve for (coprime) . We identify the symplectic leaves of the Poisson structure introduced independently by Polishchuk and Feigin-Odesskii on as precisely the loci classifying extensions with fitting into a fixed isomorphism class, verifying a claim of Feigin-Odesskii. We also classify the bundles which do fit into such extensions in geometric / combinatorial terms, involving their Harder-Narasimhan polygons introduced by Shatz.
Paper Structure (2 sections, 18 theorems, 38 equations)

This paper contains 2 sections, 18 theorems, 38 equations.

Table of Contents

  1. Homological and symplectic leaves
  2. Relevant middle terms

Key Result

Theorem 1

Fix coprime $1\le k<n\ge 3$ and a stable degree-$n$ rank-$k$ bundle on the complex elliptic curve $E$. The symplectic leaves of the bundle Poisson structure of pl98 on eq:p0p1 are the non-empty spaces via the usual gm_halg_2e_2003 correspondence between $\operatorname {Ext}^1$ and extensions. $\blacksquare$

Theorems & Definitions (38)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 1.1
  • Lemma 1.4
  • Proof 1
  • Lemma 1.5
  • Proof 2
  • Lemma 1.6
  • Proof 3
  • ...and 28 more