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Title:  Symplectic circle actions with isolated fixed points 
Author(s):  Jang, Donghoon 
Director of Research:  Tolman, Susan 
Doctoral Committee Chair(s):  Lerman, Eugene 
Doctoral Committee Member(s):  Kerman, Ely; Leininger, Christopher J. 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  symplectic circle action
fixed points Hamiltonian circle action weights 
Abstract:  Consider a symplectic circle action on a closed symplectic manifold $M$ with nonempty isolated fixed points. Associated to each fixed point, there are welldefined nonzero 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 $\{\pm a_1, \cdots, \pm a_n\}$ 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 semifree actions, and for certain circle actions on sixdimensional manifolds. Finally, we prove that if there are exactly three fixed points, $M$ is equivariantly symplectomorphic to $\mathbb{CP}^{2}$. 
Issue Date:  20150407 
Type:  Thesis 
URI:  http://hdl.handle.net/2142/78347 
Rights Information:  Copyright 2015 Donghoon Jang 
Date Available in IDEALS:  20150722 
Date Deposited:  May 2015 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois