Symplectic circle actions with isolated fixed points

Consider a symplectic circle action on a closed symplectic manifold $M$ with non-empty isolated fixed points. Associated to each fixed point, there are well-defined non-zero integers, called \emph{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 \{\pm a₁, \cdots, \pm a_n\} 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. Finally, we prove that if there are exactly three fixed points, $M$ is equivariantly symplectomorphic to \mathbb{CP}².

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2015

Author dc:creator

• Jang, Donghoon

Contributors dc:contributor

• Tolman, Susan
• Lerman, Eugene
• Kerman, Ely
• Leininger, Christopher J.

• symplectic circle action
• fixed points
• Hamiltonian circle action
• weights

Statement dc:rights

• Copyright 2015 Donghoon Jang

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/78347

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/78347