Symplectic circle actions with isolated fixed points

TL;DR

This paper investigates conditions under which symplectic circle actions with isolated fixed points are Hamiltonian, providing new criteria involving sums of weights and applying these to specific dimensions and cases.

Contribution

It establishes new criteria for Hamiltonian actions based on weight sums and extends previous results to certain low-dimensional cases and specific weight configurations.

Findings

Action is Hamiltonian if sum of an odd number of weights is never zero.

In 6-dimensional case, or with specific weight structures, sum of three weights suffices.

Results recover known cases for semi-free actions and certain 6-manifold actions.

Abstract

Consider a symplectic circle action on a closed symplectic manifold with non-empty isolated fixed points. Associated to each fixed point, there are well-defined non-zero integers, called weights. We prove that the action is Hamiltonian if the sum of an odd number of weights is never equal to zero (the weights may be taken at different fixed points). Moreover, we show that if $\dim M=6$, or if $\dim M=2n \leq 10$ and each fixed point has weights $\{\pm a_1, \cdots, \pm a_n\}$ for some positive integers $a_i$, it is enough to consider the sum of three weights. As applications, we recover the results for semi-free actions, and for certain circle actions on six-dimensional manifolds.