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| Embedded Surfaces for Symplectic Circle Actions |
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Citation： |
Yunhyung CHO,Min Kyu KIM,Dong Youp SUH.Embedded Surfaces for Symplectic Circle Actions[J].Chinese Annals of Mathematics B,2017,38(6):1197~1212 |
| Page view： 1269
Net amount： 1488 |
Authors： |
Yunhyung CHO; Min Kyu KIM;Dong Youp SUH |
Foundation： |
The first author was supported by the National Research
Foundation of Korea (NRF) grant funded by the Korea government
(MSIP; Ministry of Science, ICT \& Future Planning)
(No.NRF-2017R1C1B5018168), the second author was partially
supported by Gyeongin National University of Education Research Fund
and the third author was partially supported by the Basic Science
Research Program through the National Research Foundation of Korea
(NRF) Funded by the Ministry of Science, ICT & Future Planning
(No.2016R1A2B4010823). |
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| Abstract： |
The purpose of this article is to characterize symplectic and
Hamiltonian circle actions on symplectic manifolds in terms of
symplectic embeddings of Riemann surfaces. More precisely, it is
shown that (1) if $(M,\omega)$ admits a Hamiltonian $S^1$-action,
then there exists a two-sphere $S$ in $M$ with positive symplectic
area satisfying $\langle c_1(M,\omega), [S]\rangle > 0$, and (2) if
the action is non-Hamiltonian, then there exists an $S^1$-invariant
symplectic $2$-torus $T$ in $(M,\omega)$ such that $\langle
c_1(M,\omega), [T]\rangle = 0$. As applications, the authors give a
very simple proof of the following well-known theorem which was
proved by Atiyah-Bott, Lupton-Oprea, and Ono: Suppose that
$(M,\omega)$ is a smooth closed symplectic manifold satisfying
$c_1(M,\omega)=\lambda \cdot [\omega]$ for some $\lambda \in \R$ and
$G$ is a compact connected Lie group acting effectively on $M$
preserving $\omega$. Then (1) if $\lambda < 0$, then $G$ must be
trivial, (2) if $\lambda=0$, then the $G$-action is non-Hamiltonian,
and (3) if $\lambda > 0$, then the $G$-action is Hamiltonian. |
Keywords： |
Symplectic geometry, Hamiltonian action |
Classification： |
53D20, 53D05 |
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