Semifree Hamiltonian circle actions on 6-dimensional symplectic manifolds with non-isolated fixed point set

Abstract

Let $(M,\omega )$ be a 6-dimensional closed symplectic manifold with a symplectic $S^1$-action with $M^{S^1}\ne \varnothing$ and $\dim M^{S^1}\le 2$. 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 any reduced space with respect to $\mu$. It means that if the action is non-Hamiltonian, then $b_2^+(M_{\red}) \geq 2$. 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 $k\ne 2$ and we construct some examples of 6-dimensional semifree Hamiltonian $S^1$-manifolds such that $M^{S^1}$ contains $k$ surfaces of positive genera for $k=0$ and 4. Examples with $k=1$ and 3 were given in \cite{L2}.

32 pages, no figures, Proof of Theorem 1.2 revised, incorrect Example 7.12 removed

References in corpus (1)

• Classifying semi-free Hamiltonian $S^1$-manifolds

Cited by in corpus (4)

• The log-concavity conjecture on semifree symplectic S^1-manifolds with isolated fixed points
• Log-concavity and symplectic flows
• Hamiltonian $S^1$ actions with Isolated Fixed Points on 6-Dimensional Symplectic Manifolds
• An integer optimization problem for non-Hamiltonian periodic flows