AI 中文总结
研究闭连通2n维辛流形上辛\(T^{n - 1}\)作用，证明其为哈密顿作用的充要条件是轨道可缩，推广相关结果。当轨道迷向时得到更强结论，还证明了此类作用可分解，最后提出关于其拓扑的若干开放性问题。
AI 中文摘要
我们证明，在一个闭的连通2n维辛流形上的辛\(T^{n - 1}\)作用是哈密顿的，当且仅当它的轨道是可缩的。这推广了Lalonde - McDuff - Polterovich在四维流形上的结果以及McDuff和Kim关于不动点存在性的定理。当轨道是迷向时，我们证明了该结果的一个更强变体，这意味着某些六维流形上的辛圆作用不能扩展为辛\(T^2\)作用。此外，我们证明具有迷向轨道的辛\(T^{n - 1}\)作用总是可分解为一个极大哈密顿作用和一个局部自由作用。最后我们提出了几个关于辛环面作用拓扑的开放性问题。
英文摘要
We prove that a symplectic $T^{n-1}$ action on a closed connected $2n$-dimensional symplectic manifold is Hamiltonian if and only if its orbits are contractible. This generalizes a result of Lalonde--McDuff--Polterovich on four-manifolds and theorems of McDuff and Kim on existence of fixed points. When the orbits are isotropic, we prove a stronger variant of this result, which implies non-extendability of certain symplectic circle actions on 6-manifolds to symplectic $T^2$ actions. Moreover, we prove that a symplectic $T^{n-1}$ action with isotropic orbits always splits into a maximal Hamiltonian action and a locally-free action. We end by posing several open questions on the topology of symplectic torus actions.
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