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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
Degree Granting Institution:University of Illinois at Urbana-Champaign
Subject(s):symplectic circle action
fixed points
Hamiltonian circle action
Abstract: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 $\{\pm a_1, \cdots, \pm a_n\}$ 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 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 $\mathbb{CP}^{2}$.
Issue Date:2015-04-07
Rights Information:Copyright 2015 Donghoon Jang
Date Available in IDEALS:2015-07-22
Date Deposited:May 2015

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