Embedded surfaces for symplectic circle actions

TL;DR

This paper characterizes symplectic and Hamiltonian circle actions on symplectic manifolds via symplectic embeddings of Riemann surfaces, providing new insights and simplified proofs of classical theorems relating to symplectic geometry and group actions.

Contribution

It introduces a novel characterization of circle actions using symplectic surfaces and offers a simplified proof of a classical classification theorem for symplectic manifolds with group actions.

Findings

Existence of invariant symplectic 2-spheres or tori characterizes Hamiltonian and non-Hamiltonian actions.

Provides a simple proof of a classification theorem for symplectic manifolds with Lie group actions based on the first Chern class.

Clarifies the relationship between the first Chern class and the nature of 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, we will show that (1) if $(M,\omega)$ admits a Hamiltonian $S^1$-action, then there exists an $S^1$-invariant symplectic $2$-sphere $S$ in $(M,\omega)$ such that $\langle c_1(M), [S] \rangle > 0$, and (2) if the action is non-Hamiltonian, then there exists an $S^1$-invariant symplectic $2$-torus $T$ in $(M,\omega)$ such that $\langle c_1(M), [T] \rangle = 0$. As applications, we will give a very simple proof of the following well-known theorem which was proved by Atiyah-Bott \cite{AB}, Lupton-Oprea \cite{LO}, and Ono \cite{O2} : suppose that $(M,\omega)$ is a smooth closed symplectic manifold satisfying $c_1(TM)=\lambda \cdot [\omega]$ for some $\lambda \in \R$ and let $G$ be a compact connected Lie group acting effectively on $M$ preserving $\omega$. Then (1) if $\lambda < 0$, then $G$ must be trivial, (2) if $\lambda=0$, then the $G$-action is non-Hamiltonian, and (3) if $\lambda > 0$, then the $G$-action is Hamiltonian.