Journal of Symplectic Geometry

Volume 17 (2019)

Number 1

On the Chern numbers for pseudo-free circle actions

Pages: 1 – 40



Byung Hee An (Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang, South Korea)

Yunhyung Cho (Department of Mathematics Education, Sungkyunkwan University, Seoul, South Korea)


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.

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