Embedded surfaces for symplectic circle actions

Abstract

The purpose of this article is to characterize symplectic and Hamiltonian circle actions on symplectic manifolds in terms of symplectic embeddings of Riemann surfaces. More precisely, it is shown that (1) if (M, ω) admits a Hamiltonian S¹-action, then there exists a two-sphere S in M with positive symplectic area satisfying ‹c₁(M, ω), [S]› > 0, and (2) if the action is non-Hamiltonian, then there exists an S¹-invariant symplectic 2-torus T in (M, ω) such that ‹c₁(M, ω), [T]› = 0. As applications, the authors give a very simple proof of the following well-known theorem which was proved by Atiyah-Bott, Lupton-Oprea, and Ono: Suppose that (M, ω) is a smooth closed symplectic manifold satisfying c₁(M, ω) = λ·[ω] for some λ ∈ R and G is a compact connected Lie group acting effectively on M preserving ω. Then (1) if λ < 0, then G must be trivial, (2) if λ = 0, then the G-action is non-Hamiltonian, and (3) if λ > 0, then the G-action is Hamiltonian.