# Hamiltonian circle actions with almost minimal isolated fixed points (2023)

## Introduction

Consider a circle action on a connected compact $2n$-dimensional symplectic manifold $\left(M,\omega \right)$ with moment map $\varphi$. By Morse theory, the fixed point set ${M}^{{S}^{1}}$ contains at least $n+1$ points. In, we study the case when ${M}^{{S}^{1}}$ consists of exactly $n+1$ isolated points. In this paper, we consider the case when ${M}^{{S}^{1}}$ consists of exactly $n+2$ isolated points, in which case, we call the action has almost minimal isolated fixed points. Such known examples are ${\stackrel{˜}{G}}_{2}\left({\mathbb{R}}^{n+2}\right)$ with $n\ge 2$ even, see Example2.1.

Let $\left(M,\omega \right)$ be a connected compact Hamiltonian ${S}^{1}$-manifold of dimension $2n$ with almost minimal isolated fixed points, and let $\varphi$ be the moment map. By Lemma2.2, the fixed points, denoted ${P}_{0},\dots ,{P}_{\frac{n}{2}-1},{P}_{\frac{n}{2}},{P}_{{\left(\frac{n}{2}\right)}^{\prime }},{P}_{\frac{n}{2}+1},\dots ,{P}_{n}$, can be labeled to satisfy $\varphi \left({P}_{0}\right)<\cdots <\varphi \left({P}_{\frac{n}{2}-1}\right)<\varphi \left({P}_{\frac{n}{2}}\right)\le \varphi \left({P}_{{\left(\frac{n}{2}\right)}^{\prime }}\right)<\varphi \left({P}_{\frac{n}{2}+1}\right)<\cdots <\varphi \left({P}_{n}\right).$In the statements of our main theorems below, we are referring to this order of moment map values of the fixed points.

For a symplectic ${S}^{1}$-manifold $M$ with isolated fixed points, in a neighborhood of each fixed point $P$, the ${S}^{1}$-action is equivalent to an ${S}^{1}$ linear action on ${T}_{P}M$, so there is a set of non-zero integers, called the weights of the ${S}^{1}$-action at the fixed point $P$.

Our main results are as follows.

Theorem 1

Let the circle act on a connected compact $2n$-dimensional symplectic manifold $\left(M,\omega \right)$ with moment map $\varphi :M\to \mathbb{R}$. Assume $\left[\omega \right]$ is a primitive integral class and the fixed point set ${M}^{{S}^{1}}$ consists of $n+2$ isolated points, denoted ${M}^{{S}^{1}}=\left\{{P}_{0},\dots ,{P}_{\frac{n}{2}-1},{P}_{\frac{n}{2}},{P}_{{\left(\frac{n}{2}\right)}^{\prime }},{P}_{\frac{n}{2}+1},\dots ,{P}_{n}\right\}$. Then for any $i,j\in \left\{0,\dots ,\frac{n}{2},{\left(\frac{n}{2}\right)}^{\prime },\dots ,n\right\}$, $\varphi \left({P}_{i}\right)-\varphi \left({P}_{j}\right)\in \mathbb{Z}$, and ${c}_{1}\left(M\right)=n\left[\omega \right]$ if and only if $\varphi \left({P}_{n-1}\right)-\varphi \left({P}_{0}\right)=\varphi \left({P}_{n}\right)-\varphi \left({P}_{1}\right)$ holds and this integer occurs as a weight of the ${S}^{1}$-action at some fixed point. In dimension $4$, for the “if” part to hold, the class $\left[\omega \right]$ needs to be chosen suitably.

Theorem 2

Let the circle act on a connected compact $2n$-dimensional symplectic manifold $\left(M,\omega \right)$ with moment map $\varphi :M\to \mathbb{R}$. Assume ${M}^{{S}^{1}}$ consists of $n+2$ isolated points, i.e.,${M}^{{S}^{1}}=\left\{{P}_{0},\dots ,{P}_{\frac{n}{2}-1},{P}_{\frac{n}{2}},{P}_{{\left(\frac{n}{2}\right)}^{\prime }},{P}_{\frac{n}{2}+1},\dots ,{P}_{n}\right\}$. Then $n\ge 2$ must be even. Assume $\left[\omega \right]$ is an integral class. If $\varphi \left({P}_{n-1}\right)-\varphi \left({P}_{0}\right)=\varphi \left({P}_{n}\right)-\varphi \left({P}_{1}\right)$ holds and this integer occurs as a weight of the ${S}^{1}$-action at some fixed point, then all the following are true.

• (1)

The integral cohomology ring of $M$ is isomorphic to that of ${\stackrel{˜}{G}}_{2}\left({\mathbb{R}}^{n+2}\right)$.

• (2)

The total Chern class of $M$ is isomorphic to that of ${\stackrel{˜}{G}}_{2}\left({\mathbb{R}}^{n+2}\right)$.

• (3)

The sets of weights of the ${S}^{1}$-action at all the fixed points on $M$ are isomorphic to those of a standard circle action on ${\stackrel{˜}{G}}_{2}\left({\mathbb{R}}^{n+2}\right)$ (as in Example2.1).

Theorem2 follows from Proposition 4.5, Proposition 5.15, Proposition 5.16.

By[8, Prop. 4.2 and Sec. 5], if the manifold $M$ in Theorem2 is Kähler, and the ${S}^{1}$-action is holomorphic, then $M$ is ${S}^{1}$-equivariantly biholomorphic and ${S}^{1}$-equivariantly symplectomorphic to ${\stackrel{˜}{G}}_{2}\left({\mathbb{R}}^{n+2}\right)$ with $n\ge 2$ even.

In[4, Theorem 6.17], Hattori studies compact almost complex ${S}^{1}$-manifold $M$ of dimension $2n$ with $n+2$ isolated fixed points. Under the assumption ${c}_{1}\left(M\right)=nx$, where $x\in {H}^{2}\left(M;\mathbb{Z}\right)$ is a generator, and under additional technical conditions, Hattori shows that the sets of weights at all the fixed points are isomorphic to those of a standard circle action on ${\stackrel{˜}{G}}_{2}\left({\mathbb{R}}^{n+2}\right)$ with $n\ge 2$ even, and he obtains that ${x}^{n}\left[M\right]=2$. In our current work, for a compact symplectic Hamiltonian ${S}^{1}$-manifold $\left(M,\omega \right)$, we prove the equivalence of the condition ${c}_{1}\left(M\right)=nx$ and the particular weight as in Theorem1, and using the particular weight as a starting point, we give methods to prove $\left(1\right)$, $\left(2\right)$ and $\left(3\right)$ in Theorem2.

In, the author studies compact Hamiltonian ${S}^{1}$-manifolds with fixed point set consisting of two connected components and almost minimal in a certain sense. Recent related works on compact Hamiltonian ${S}^{1}$-manifolds with fixed point set minimal in a certain sense include, , , , , , , .

(Video) Part 2 - Hamiltonian circle actions on compact symplectic orbifolds of dimension four

The paper is organized as follows. In Section2, we give examples of Hamiltonian ${S}^{1}$-manifolds with almost minimal isolated fixed points, prove and present some basic and key results. In Section3, we prove Theorem1. In Section4, we determine the sets of weights of the ${S}^{1}$-action at all the fixed points, proving $\left(3\right)$ of Theorem2. In Section5, we determine the integral cohomology ring and the total Chern class of the manifold, proving $\left(1\right)$ and $\left(2\right)$ of Theorem2.

## Examples and basic results

In this section, we give examples, prove and present key basic results.

Example 2.1

Let ${\stackrel{˜}{G}}_{2}\left({\mathbb{R}}^{n+2}\right)$ be the Grassmannian of oriented $2$-planes in ${\mathbb{R}}^{n+2}$, with $n\ge 2$ even. This $2n$-dimensional manifold naturally arises as a coadjoint orbit of $SO\left(n+2\right)$, hence it has a natural Kähler structure and a Hamiltonian $SO\left(n+2\right)$ action.

Consider the ${S}^{1}\subset SO\left(n+2\right)$ action on ${\stackrel{˜}{G}}_{2}\left({\mathbb{R}}^{n+2}\right)$ induced by the ${S}^{1}$ action on ${\mathbb{R}}^{n+2}={ℂ}^{\frac{n+2}{2}}$ given by $\lambda \cdot \left({z}_{0},{z}_{1},\dots ,{z}_{\frac{n}{2}}\right)=\left({\lambda }^{{b}_{0}}{z}_{0},{\lambda }^{{b}_{1}}{z}_{1},\dots ,{\lambda }^{{b}_{\frac{n}{2}}}{z}_{\frac{n}{2}}\right),$where the ${b}_{i}$’s, with $i=0,1,\dots ,\frac{n}{2}$, are mutually distinct integers.

## On ${c}_{1}\left(M\right)$ — the proof of Theorem1

In this section, we prove Theorem1.

In a symplectic ${S}^{1}$-manifold $\left(M,\omega \right)$ with isolated fixed points, if $w>0$ is a weight of the ${S}^{1}$-action at a fixed point $P$, $-w$ is a weight of the ${S}^{1}$-action at a fixed point $Q$, and $P$ and $Q$ are on the same connected component of ${M}^{{\mathbb{Z}}_{w}}$, then we say that there is a weight $w$ from $P$ to $Q$, or $w$ is a weight from $P$ to $Q$. When the signs of $w$ at $P$ and at $Q$ are clear, we also say that there is a weight $±w$ between $P$ and $Q$. It is known (see for example) that the set ${W}^{+}$ of all

## From the named weight to all the weights

In this section, using the given weight, we find the sets of weights at all the fixed points, proving $\left(3\right)$ of Theorem2.

For convenience, we state the following lemma.

Lemma 4.1

## [7, Lemma 4.1]

Let the circle act on a connected compact symplectic manifold $\left(M,\omega \right)$ with moment map $\varphi :M\to \mathbb{R}$. If ${c}_{1}\left(M\right)=k\left[\omega \right]$, then for any two fixed point set components $F$ and ${F}^{\prime }$, we have ${\Gamma }_{F}-{\Gamma }_{{F}^{\prime }}=k\left(\varphi \left({F}^{\prime }\right)-\varphi \left(F\right)\right)$, where ${\Gamma }_{F}$ and ${\Gamma }_{{F}^{\prime }}$ are respectively the sums of the weights at $F$ and ${F}^{\prime }$.

We first find the sets of weights at ${P}_{0}$, ${P}_{1}$, ${P}_{n-1}$ and ${P}_{n}$.

Lemma 4.2

Let the circle

## From the sets of weights to the integral cohomology ring and total Chern class of $M$

In this section, we use the sets of weights of the ${S}^{1}$-action to determine the integral cohomology ring and total Chern class of the manifold, proving $\left(1\right)$ and $\left(2\right)$ of Theorem2.

We will do this in a few steps.

Lemma 5.1

Let the circle act on a connected compact $2n$-dimensional symplectic manifold $\left(M,\omega \right)$ with moment map $\varphi :M\to \mathbb{R}$. Assume $\left[\omega \right]$ is a primitive integral class and ${M}^{{S}^{1}}=\left\{{P}_{0},\dots ,{P}_{\frac{n}{2}-1},{P}_{\frac{n}{2}},{P}_{{\left(\frac{n}{2}\right)}^{\prime }},{P}_{\frac{n}{2}+1}\dots ,{P}_{n}\right\}$. Let $x=\left[\omega \right]$. Then the following conditions are equivalent:

• (1)

There exist classes $y,z\in {H}^{n}\left(M;\mathbb{Z}\right)$ such that the

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