Circle actions on almost complex manifolds with isolated fixed points

TL;DR

This paper extends results on circle actions from symplectic to almost complex manifolds, showing that with three fixed points the manifold must be four-dimensional, and also explores cases with fewer fixed points.

Contribution

It generalizes previous symplectic results to almost complex manifolds and classifies fixed point scenarios with one, two, or three fixed points.

Findings

Manifolds with three fixed points are four-dimensional.

Classifies fixed point cases with one and two fixed points.

Extends symplectic circle action results to almost complex manifolds.

Abstract

The author proved that if the circle acts symplectically on a compact, connected symplectic manifold $M$ with three fixed points, then $M$ is equivariantly symplectomorphic to some standard action on $\mathbb{CP}^2$. In this paper, we extend the result to a circle action on an almost complex manifold; if the circle acts on a compact, connected almost complex manifold $M$ with exactly three fixed points, then $\dim M=4$. Moreover, we deal with the cases of one fixed point and two fixed points.