Hamiltonian circle actions with almost minimal isolated fixed points

Abstract

Let the circle act in a Hamiltonian fashion on a connected compact symplectic manifold (M,ω) of dimension 2n. Then the S1-action has at least n+1 fixed points. In a previous paper, we study the case when the fixed point set consists of precisely n+1 isolated points. In this paper, we study the case when the fixed point set consists of exactly n+2 isolated points. We show that in this case n must be even. We find equivalent conditions on the first Chern class of M and a particular weight of the S1-action. We also show that the particular weight can completely determine the integral cohomology ring and the total Chern class of M, and the sets of weights of the S1-action at all the fixed points. We will see that all these data are isomorphic to those of known examples, G˜2(Rn+2) with n≥2 even, equipped with standard circle actions.