[Paper Review] On Generalized Moment Maps for Symplectic Compact Group Actions

Pierre Sleewaegen|arXiv (Cornell University)|Apr 29, 2003

Geometric and Algebraic Topology19 references3 citations

TL;DR

This paper introduces generalized moment maps for symplectic actions of compact connected Lie groups on closed symplectic manifolds, extending McDuff's circle-valued moment map to higher-rank compact groups. It establishes a non-Hamiltonian version of the Atiyah-Guillemin-Sternberg convexity theorem for torus actions and proves that complexity-one symplectic torus actions with fixed points are Hamiltonian, using a fiber-connected moment map and reduction techniques.

ABSTRACT

A generalized moment map is proposed for arbitrary symplectic actions of compact connected Lie groups on closed symplectic manifolds, in the spirit of the circle -valued maps introduced by D. McDuff in the case of non-Hamiltonian circle actions. We study equivariance properties of generalized moments, show that they allow reduction procedures, and obtain in the torus case a version of the Atiyah-Guillemin-Sternberg convexity theorem. As illustration, we reformulate a proof of M.K. Kim that "complexity one" symplectic torus actions are Hamiltonian, and give a symplectic proof of the finiteness of certain symmetry groups of compact oriented surfaces.

Motivation & Objective

• To extend McDuff's circle-valued moment map to general compact Lie group actions on symplectic manifolds.
• To establish a non-Hamiltonian version of the Atiyah-Guillemin-Sternberg convexity theorem for torus actions.
• To prove that complexity-one symplectic torus actions with fixed points are Hamiltonian, using generalized moment maps and reduction.
• To analyze equivariance properties of generalized moment maps and their compatibility with Marsden-Weinstein reduction.
• To provide a symplectic proof of the finiteness of isometry groups on genus ≥2 surfaces using the generalized moment map framework.

Proposed method

• Define a generalized moment map μ: M → 𝒞* × (S¹)^r, where 𝒞* is a Hamiltonian moment map and (S¹)^r are McDuff moment maps for non-Hamiltonian torus actions.
• Construct an invariant integral symplectic form ω′ on M, ensuring the action is Hamiltonian relative to ω′.
• Use the decomposition G = C × T^r, where C acts Hamiltonian and T^r acts non-Hamiltonian, to define the moment map structure.
• Apply fiber-connectedness of the moment map to ensure topological control over level sets and reduced spaces.
• Utilize reduction in stages via Marsden-Weinstein reduction, adapted to non-Hamiltonian actions using affine actions on (S¹)^r.
• Leverage Sard’s theorem and local normal form near fixed points to construct regular values and identify fixed-point orbits.

Experimental results

Research questions

• RQ1 Can McDuff’s circle-valued moment map construction be generalized to higher-rank compact Lie group actions on symplectic manifolds?
• RQ2 Does the Atiyah-Guillemin-Sternberg convexity theorem hold for non-Hamiltonian torus actions via generalized moment maps?
• RQ3 Under what conditions is a symplectic torus action of complexity one necessarily Hamiltonian?
• RQ4 Can equivariance of generalized moment maps be achieved, and what are the obstructions when fixed points are absent?
• RQ5 How does the presence of fixed points influence the Hamiltonian nature of non-Hamiltonian group actions?

Key findings

• For a compact connected Lie group action on a closed symplectic manifold, a generalized moment map μ: M → 𝒞* × (S¹)^r exists, with 𝒞* a genuine Hamiltonian moment map and (S¹)^r components as McDuff moment maps.
• When G is a torus, the image of the generalized moment map satisfies μ(M) = Δ × (S¹)^r, where Δ is the convex polytope image of the Hamiltonian component, and μ is an open map.
• The first Betti number b₁(M) satisfies r ≤ b₁(M), where r is the rank of the non-Hamiltonian torus factor.
• If a fixed point exists for the T^r-action, the generalized moment map can be chosen to be equivariant under the group action.
• The non-Hamiltonian heredity property ensures that a non-Hamiltonian T^r-action remains non-Hamiltonian after T¹-reduction, preserving the structure of the moment map.
• A fiber-connected moment map μ^fc allows a reduction in stages to produce a connected 4-dimensional orbifold with a non-Hamiltonian S¹-action possessing a fixed point, contradicting McDuff’s result and proving Kim’s theorem on complexity-one actions.