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On the Chern numbers for pseudo-free circle actions
Journal of Symplectic Geometry ( IF 0.6 ) Pub Date : 2019-01-01 , DOI: 10.4310/jsg.2019.v17.n1.a1
Byung Hee An 1 , Yunhyung Cho 2
Affiliation  

Let $(M,\psi)$ be a $(2n+1)$-dimensional oriented closed manifold equipped with a pseudo-free $S^1$-action $\psi : S^1 \times M \rightarrow M$. We first define a \textit{local data} $\mathcal{L}(M,\psi)$ of the action $\psi$ which consists of pairs $(C, (p(C) ; \overrightarrow{q}(C)))$ where $C$ is an exceptional orbit, $p(C)$ is the order of isotropy subgroup of $C$, and $\overrightarrow{q}(C) \in (\mathbb{Z}_{p(C)}^{\times})^n$ is a vector whose entries are the weights of the slice representation of $C$. In this paper, we give an explicit formula of the Chern number $\langle c_1(E)^n, [M/S^1] \rangle$ modulo $\mathbb{Z}$ in terms of the local data, where $E = M \times_{S^1} \mathbb{C}$ is the associated complex line orbibundle over $M/S^1$. Also, we illustrate several applications to various problems arising in equivariant symplectic topology.

中文翻译:

关于伪自由圆动作的陈数

令 $(M,\psi)$ 是一个 $(2n+1)$ 维定向闭流形,配备一个伪自由 $S^1$-action $\psi : S^1 \times M \rightarrow M$ . 我们首先定义动作 $\psi$ 的 \textit{local data} $\mathcal{L}(M,\psi)$,它由对 $(C, (p(C) ; \overrightarrow{q}( C)))$ 其中 $C$ 是异常轨道,$p(C)$ 是 $C$ 的各向同性子群的阶数,$\overrightarrow{q}(C) \in (\mathbb{Z}_ {p(C)}^{\times})^n$ 是一个向量,其条目是 $C$ 的切片表示的权重。在本文中,我们根据本地数据给出了陈数的显式公式 $\langle c_1(E)^n, [M/S^1] \rangle$ modulo $\mathbb{Z}$,其中 $ E = M \times_{S^1} \mathbb{C}$ 是在 $M/S^1$ 上的相关复线轨道。此外,我们还说明了对等变辛拓扑中出现的各种问题的几种应用。
更新日期:2019-01-01
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