Journal of Symplectic Geometry

Volume 15 (2017)

Number 4

Symplectic circle actions with isolated fixed points

Pages: 1071 – 1087



Donghoon Jang (Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Il., U.S.A.; and School of Mathematics, Korea Institute for Advanced Study, Seoul, Korea)


Consider a symplectic circle action on a closed symplectic manifold with a non-empty discrete fixed point set. 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 the sum of an even number of weights (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 $\lbrace \pm a_1 , \dotsc , \pm a_n \rbrace$ for some positive integers $a_i$, the action is Hamiltonian if the sum of three weights is never equal to zero. As applications, we recover the results for semi-free actions, and for certain circle actions on six-dimensional manifolds.

This work was partially supported by Campus Research Board Awards — University of Illinois at Urbana-Champaign. The author would like to thank the anonymous referee for helping the author to improve the exposition of this paper.

Received 14 April 2015

Accepted 10 August 2016

Published 28 November 2017