On the Chern numbers for pseudo-free circle actions

Let (M, ψ) be a (2n + 1)-dimensional oriented closed manifold with a pseudo-free S¹-action ψ: S¹ × M → M. We first define a local data L(M, ψ) of the action ψ which consists of pairs (C, (p(C); −→q (C))) where C is an exceptional orbit, p(C) is the order of isotropy subgroup of C, and^−→q (C) ∈ (Z^×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 〈c₁ (E)ⁿ, [M/S¹ ]〉 modulo Z in terms of the local data, where E = M ×_S 1 C is the associated complex line orbibundle over M/S¹ . Also, we illustrate several applications to various problems arising in equivariant symplectic topology.