On the Chern numbers for pseudo-free circle actions

Let $(M,\psi )$ be a $(2n + 1)$-dimensional oriented closed manifold with a pseudo-free $S^1$-action $\psi : S^1 \times M \to M$. We first define a 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 $\overrightarrow{q} (C) \in {(\mathbb{Z}^{\times}_{p(C)})}^n$ 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.

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The authors thank anonymous referee for their endurance and kindness to improve the paper. The first author was supported by IBS-R003-D1. The second author is supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP; Ministry of Science, ICT & Future Planning) (NRF-2017R1C1B5018168). This project was initiated when two authors were affiliated to Institute for Basic Science Center for Geometry and Physics (IBS-CGP).

Received 5 February 2016

Accepted 14 June 2018

Published 23 May 2019