[Paper Review] Non-abelian convexity by symplectic cuts

Eugene Lerman, Eckhard Meinrenken+2 more|arXiv (Cornell University)|Mar 26, 1996

Geometric and Algebraic Topology9 references4 citations

TL;DR

This paper extends non-abelian convexity theorems for moment maps to non-compact orbifolds using symplectic cuts, reducing the non-abelian case to the abelian one via geometric surgery. The key contribution is a conceptual, simplified proof of convexity and fiber connectedness results in symplectic geometry, applicable to generic symplectic quotients which are orbifolds.

ABSTRACT

In this paper we extend the results of Kirwan et alii on convexity properties of the moment map for Hamiltonian group actions, and on the connectedness of the fibers of the moment map, to the case of non-compact orbifolds. Our motivation is twofold. First, the category of orbifolds is important in symplectic geometry because, generically, the symplectic quotient of a symplectic manifold is an orbifold. Second, our proof is conceptually very simple since it reduces the non-abelian case to the abelian case.

Motivation & Objective

• To generalize Kirwan et al.'s convexity and connectedness theorems for moment maps to non-compact orbifolds.
• To address the geometric significance of orbifolds in symplectic quotients, which generically arise in Hamiltonian group actions.
• To provide a conceptually simple proof strategy by reducing the non-abelian case to the abelian case.
• To establish convexity and connectedness results for moment maps on non-compact symplectic orbifolds.
• To demonstrate the utility of symplectic cuts as a tool in non-abelian symplectic geometry.

Proposed method

• Utilizes symplectic cuts as a geometric surgery technique to decompose the symplectic manifold into simpler pieces.
• Applies the abelian convexity theorem to the resulting pieces after cutting, leveraging known results in the abelian setting.
• Constructs a global moment map on the cut manifold and analyzes its image to deduce convexity.
• Uses the structure of the moment map on orbifolds to extend results beyond compact manifolds.
• Relies on the fact that symplectic quotients are generically orbifolds, making orbifold theory essential for the generalization.
• Employs a reduction strategy: the non-abelian case is reduced to the abelian case via symplectic cutting, simplifying the topological and geometric analysis.

Experimental results

Research questions

• RQ1 Can the non-abelian convexity theorem be extended to non-compact orbifolds?
• RQ2 How does the fiber connectedness of the moment map behave in the non-compact orbifold setting?
• RQ3 Can symplectic cutting be used to reduce the non-abelian moment map problem to the abelian case?
• RQ4 What is the role of orbifolds in the structure of symplectic quotients under Hamiltonian group actions?
• RQ5 What are the implications of convexity and connectedness for moment maps on non-compact symplectic orbifolds?

Key findings

• The image of the moment map on a non-compact symplectic orbifold under a Hamiltonian group action is convex.
• The fibers of the moment map over the image are connected, extending Kirwan's connectedness theorem to the non-compact orbifold case.
• Symplectic cutting provides a method to reduce the non-abelian case to the abelian case, simplifying the proof structure.
• The results apply generically to symplectic quotients, which are typically orbifolds rather than manifolds.
• The approach establishes a conceptual framework that clarifies the geometric origin of convexity in moment map theory.
• The method preserves key topological properties such as connectedness and convexity under the cutting procedure, enabling inductive arguments.