Symplectic circle actions with isolated fixed points

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 $\dim M=2n\le 10$ 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.

Cited by in corpus (5)

• Circle actions on almost complex manifolds with isolated fixed points
• Circle actions on almost complex manifolds with 4 fixed points
• Circle actions on 8-dimensional almost complex manifolds with 4 fixed points
• Non-Hamiltonian actions with fewer isolated fixed points
• Circle actions on four dimensional almost complex manifolds with discrete fixed point sets