On the Chern numbers for pseudo-free circle actions

TL;DR

This paper derives an explicit formula for Chern numbers of manifolds with pseudo-free circle actions using local data, with applications to equivariant symplectic topology.

Contribution

It introduces a method to compute Chern numbers modulo integers based on local data of pseudo-free circle actions, providing new tools for symplectic topology.

Findings

Explicit formula for Chern number modulo integers in terms of local data.

Application of the formula to problems in equivariant symplectic topology.

Framework for analyzing pseudo-free circle actions on manifolds.

Abstract

Let $(M,\psi)$ be a $(2n+1)$-dimensional oriented closed manifold equipped with a pseudo-free $S^1$-action $\psi : S^1 \times M \rightarrow M$. 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.