On the Chern numbers for pseudo-free circle actions

Abstract

Let $(M,\psi )$ be a $(2n+1)$-dimensional oriented closed manifold equipped with a pseudo-free $S^1$-action $\psi :S^1\times M\to M$. 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 $⟨c_1(E{)}^n,[M/S^1]⟩$ 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.

24pages

References in corpus (1)

• Embedded surfaces for symplectic circle actions