[Paper Review] CHERN CLASSES AND SYMPLECTIC CIRCLE ACTIONS.

Yunhyung Cho, Min Kyu Kim+1 more|arXiv (Cornell University)|Jan 1, 2013

Geometric and Algebraic Topology11 references3 citations

TL;DR

This paper investigates the relationship between symplectic circle actions and Chern classes on closed symplectic manifolds. It proves that if a closed symplectic manifold (M, ω) satisfies c₁(M) = λ·[ω] for some λ ∈ ℝ and admits a symplectic circle action, then λ ≥ 0; moreover, if the action is non-Hamiltonian, then λ must be zero. The results extend classical 2-dimensional results to higher dimensions.

ABSTRACT

Let (M,!) be a two dimensional closed symplectic manifold. Then it is a well-known fact that if M admits a symplectic circle action, then M is diffeomorphic to S 2 or S 1 × S 1 . For each case, any symplectic circle action on S 2 is always Hamiltonian and the first Chern class c1(S 2 ) is positively proportional to (!), and any symplectic circle action on S 1 × S 1 is non-Hamiltonian and c1(S 1 × S 1 ) = 0. For the case when (M,!) is a symplectic surface with genus greater than one, c1(M) is negatively proportional to (!). In this paper, we extended this result in higher dimensional cases. More precisely, let (M,!) be a closed symplectic manifold satisfying c1(M) = � · (!) for some real number � ∈ R. We prove that if (M,!) admits a symplectic circle action, then � ≥ 0. Also, we prove that if the action is non-Hamiltonian, thenshould be 0.

Motivation & Objective

• To generalize known results on symplectic circle actions in dimension two to higher-dimensional closed symplectic manifolds.
• To investigate the relationship between the first Chern class c₁(M) and the symplectic form ω when a symplectic circle action exists.
• To determine the sign and value of the proportionality constant λ in c₁(M) = λ·[ω] under the presence of symplectic circle actions.
• To clarify the distinction between Hamiltonian and non-Hamiltonian symplectic circle actions in terms of Chern class behavior.

Proposed method

• The analysis begins by assuming c₁(M) = λ·[ω] for some real λ, modeling the topological constraint on the manifold.
• The paper uses equivariant cohomology and properties of circle actions to analyze the fixed point set and isotropy representations.
• It applies the Atiyah-Bott-Berline-Vergne localization formula to compute integrals over the manifold in terms of fixed point data.
• The argument relies on the non-negativity of certain integrals arising from the localization formula to deduce λ ≥ 0.
• For non-Hamiltonian actions, the method shows that the absence of a moment map forces λ = 0.
• The proof leverages the fact that non-Hamiltonian actions imply trivial first Chern class in the cohomology class of ω.

Experimental results

Research questions

• RQ1 Under what conditions on the Chern class c₁(M) can a closed symplectic manifold (M, ω) admit a symplectic circle action?
• RQ2 What is the sign of the real number λ in the relation c₁(M) = λ·[ω] when a symplectic circle action exists?
• RQ3 Can a non-Hamiltonian symplectic circle action exist on a manifold with c₁(M) proportional to [ω] with λ ≠ 0?
• RQ4 How does the proportionality constant λ relate to the Hamiltonian or non-Hamiltonian nature of the action?

Key findings

• If (M, ω) is a closed symplectic manifold with c₁(M) = λ·[ω] and admits a symplectic circle action, then λ must be non-negative.
• For non-Hamiltonian symplectic circle actions, the proportionality constant λ must be exactly zero.
• The result generalizes the classical 2-dimensional case, where S² has positive λ and S¹×S¹ has λ = 0.
• In higher dimensions, the sign of λ is constrained by the existence of a symplectic circle action, with λ ≥ 0 being necessary.
• The Chern class c₁(M) cannot be negatively proportional to [ω] if a symplectic circle action exists.
• The condition λ = 0 for non-Hamiltonian actions provides a topological obstruction to such actions when c₁(M) is proportional to [ω].