Symplectic circle actions with isolated fixed points

Jang, Donghoon

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https://hdl.handle.net/2142/78347 Copy

Description

Title

Symplectic circle actions with isolated fixed points

Author(s)

Jang, Donghoon

Issue Date

2015-04-07

Director of Research (if dissertation) or Advisor (if thesis)

Tolman, Susan

Doctoral Committee Chair(s)

Lerman, Eugene

Committee Member(s)

• Kerman, Ely
• Leininger, Christopher J.

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

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

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 $\mathbb{CP}^{2}$.

Graduation Semester

2015-5

Type of Resource

text

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

Copyright and License Information

Copyright 2015 Donghoon Jang

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