Embedded Surfaces for Symplectic Circle Actions

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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 < c(1)(M, omega.)), [S])> > 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 < c(1)(M, omega), [T]> = 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, = lambda center dot[omega] for some A E R and G is a compact connected Lie group acting effectively on M preserving w. 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.
Publisher
SHANGHAI SCIENTIFIC TECHNOLOGY LITERATURE PUBLISHING HOUSE
Issue Date
2017-11
Language
English
Article Type
Article
Citation

CHINESE ANNALS OF MATHEMATICS SERIES B, v.38, no.6, pp.1197 - 1212

ISSN
0252-9599
DOI
10.1007/s11401-017-1031-7
URI
http://hdl.handle.net/10203/240251
Appears in Collection
MA-Journal Papers(저널논문)
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