Hamiltonian circle actions with almost minimal isolated fixed points

TL;DR

This paper investigates Hamiltonian circle actions on symplectic manifolds with a fixed number of isolated fixed points, revealing structural constraints and classifying the manifolds based on their fixed point data.

Contribution

It characterizes manifolds with exactly n+2 fixed points, showing n must be even, and relates fixed point weights to cohomology and Chern classes, linking to known examples.

Findings

n must be even when fixed points are exactly n+2

The particular weight determines the cohomology ring and Chern class

Manifolds are isomorphic to known examples like (7) with standard actions

Abstract

Let the circle act in a Hamiltonian fashion on a connected compact symplectic manifold $(M, \omega)$ of dimension $2n$. Then the $S^1$-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 $S^1$-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 $S^1$-action at all the fixed points. We will see that all these data are isomorphic to those of known examples, $\widetilde{G}_2(\mathbb{R}^{n+2})$ with $n\geq 2$ even, equipped with standard circle actions.