# Circle actions on almost complex manifolds with 4 fixed points

## TL;DR

This paper classifies the fixed point data of circle actions on compact almost complex manifolds with four fixed points, focusing on dimensions up to six, revealing specific types and extending previous results.

## Contribution

It provides a detailed classification of fixed point data for circle actions on low-dimensional almost complex manifolds with four fixed points, including new types in six dimensions.

## Key findings

- Dimension 2: manifold is a union of two 2-spheres with rotation actions.
- Dimension 4: action resembles that on a Hirzebruch surface.
- Dimension 6: six fixed point data types identified, including new potential blow-up types.

## Abstract

Let the circle act on a compact almost complex manifold $M$. In this paper, we classify the fixed point data of the action if there are 4 fixed points and the dimension of the manifold is at most 6. First, if $\dim M=2$, then $M$ is a disjoint union of rotations on two 2-spheres. Second, if $\dim M=4$, we prove that the action alikes a circle action on a Hirzebruch surface. Finally, if $\dim M=6$, we prove that six types occur for the fixed point data; $\mathbb{CP}^3$ type, complex quadric in $\mathbb{CP}^4$ type, Fano 3-fold type, $S^6 \cup S^6$ type, and two unknown types that might possibly be realized as blow ups of a manifold like $S^6$. When $\dim M=6$, we recover the result by Ahara in which the fixed point data is determined if furthermore $\mathrm{Todd}(M)=1$ and $c_1^3(M)[M] \neq 0$, and the result by Tolman in which the fixed point data is determined if furthermore the base manifold admits a symplectic structure and the action is Hamiltonian.