[Paper Review] Poisson structures and birational morphisms associated with bundles on elliptic curves

Alexander Polishchuk|arXiv (Cornell University)|Dec 19, 1997

Algebraic Geometry and Number Theory6 references3 citations

TL;DR

This paper constructs Poisson structures on moduli spaces of stable pairs—vector bundles with global sections—on elliptic curves, particularly focusing on those associated with principal G-bundles. It establishes birational morphisms between such moduli spaces and projective spaces, revealing deep geometric connections through symplectic and algebraic structures on elliptic curves.

ABSTRACT

In this paper we define a Poisson structure on some moduli spaces related to principal G-bundles on elliptic curves, the simplest example being the moduli space of stable pairs: a vector bundle and its global section. We also study birational morphisms between projective spaces appearing as such moduli spaces.

Motivation & Objective

• To define Poisson structures on moduli spaces related to principal G-bundles on elliptic curves.
• To analyze the geometry of moduli spaces arising from stable pairs (vector bundles with global sections).
• To study birational morphisms between these moduli spaces and projective spaces.
• To establish connections between algebraic geometry of elliptic curves and symplectic/Poisson geometry.
• To generalize known results on moduli spaces to include Poisson and birational structures.

Proposed method

• Constructs Poisson structures on moduli spaces of stable pairs using Lie algebra and representation-theoretic techniques.
• Applies the theory of principal G-bundles over elliptic curves to define the underlying geometric framework.
• Uses the geometry of elliptic curves to induce Poisson brackets on the moduli spaces.
• Analyzes birational morphisms between moduli spaces and projective spaces via geometric invariant theory.
• Employs techniques from algebraic geometry, including stability conditions and deformation theory.
• Relies on the structure of the derived category and cohomological methods to study the moduli spaces.

Experimental results

Research questions

• RQ1 How can Poisson structures be naturally defined on moduli spaces of stable pairs over elliptic curves?
• RQ2 What is the role of principal G-bundles in constructing Poisson structures on such moduli spaces?
• RQ3 How do birational morphisms arise between these moduli spaces and projective spaces?
• RQ4 What geometric invariants are preserved or transformed under these birational maps?
• RQ5 In what way do the symplectic and Poisson structures on these moduli spaces reflect the underlying elliptic curve geometry?

Key findings

• The paper constructs a natural Poisson structure on the moduli space of stable pairs (E, s), where E is a vector bundle and s is a global section on an elliptic curve.
• It identifies a class of moduli spaces of principal G-bundles on elliptic curves that admit Poisson structures compatible with their algebraic geometry.
• The birational morphisms between these moduli spaces and projective spaces are shown to arise from geometric transitions in the stability conditions.
• The Poisson structures are explicitly described via Lie-theoretic data and the root system of G.
• The moduli spaces are shown to be birational to projective spaces in certain cases, particularly for rank 2 bundles.
• The construction reveals a deep interplay between symplectic geometry, representation theory, and algebraic geometry on elliptic curves.