Mathematics – Symplectic Geometry
Scientific paper
2010-05-03
Mathematics
Symplectic Geometry
32 pages, no figures, Theorem 1.3 revised, incorrect Example 7.12 removed
Scientific paper
Let $(M,\omega)$ be a 6-dimensional closed symplectic manifold with a symplectic $S^1$-action with $M^{S^1} \neq \emptyset$ and $\textrm{dim} M^{S^1} <= 2$. Assume that $\omega$ is integral with a generalized moment map $\mu$. We first prove that the action is Hamiltonian if and only if $b_2^+(M_{red}) = 1$, where $M_{red}$ is the reduced space with respect to $\mu$. It means that if the action is non-Hamiltonian, then $b_2^+(M_{red}) >= 2$. Secondly, we focus on the case when the action is semifree and Hamiltonian. We prove that if $M^{S^1}$ consists of surfaces, then the number $k$ of fixed surfaces with positive genera is at most four. In particular, if the extremal fixed surfaces are spheres, then $k$ is at most one. Finally, we prove that $k \neq 2$ and we construct some examples of 6-dimensional semifree Hamiltonian $S^1$-manifold such that $M^{S^1}$ contains $k$ surfaces of positive gerera for $k = 0, 1, 3$ and 4.
Cho Yunhyung
Hwang Taekgyu
Suh Dong Youp
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