## Introduction

Consider a circle action on a connected compact $2n$-dimensional symplectic manifold $(M,\omega )$ with moment map $\varphi $. By Morse theory, the fixed point set ${M}^{{S}^{1}}$ contains at least $n+1$ points. In[7], 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 ${\tilde{G}}_{2}\left({\mathbb{R}}^{n+2}\right)$ with $n\ge 2$ even, see Example2.1.

Let $(M,\omega )$ 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* $(M,\omega )$ *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* $(M,\omega )$ *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*${\tilde{G}}_{2}\left({\mathbb{R}}^{n+2}\right)$*.**(2)**The total Chern class of*$M$*is isomorphic to that of*${\tilde{G}}_{2}\left({\mathbb{R}}^{n+2}\right)$*.**(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*${\tilde{G}}_{2}\left({\mathbb{R}}^{n+2}\right)$*(as in**Example*2.1*).*

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 ${\tilde{G}}_{2}\left({\mathbb{R}}^{n+2}\right)$ 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 $x\in {H}^{2}(M;\mathbb{Z})$ 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 ${\tilde{G}}_{2}\left({\mathbb{R}}^{n+2}\right)$ 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 $(M,\omega )$, 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[9], 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[13], [12], [11], [8], [7], [10], [3], [5].

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.

## Section snippets

## Examples and basic results

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

**Example 2.1**

Let ${\tilde{G}}_{2}\left({\mathbb{R}}^{n+2}\right)$ 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 $SO(n+2)$, hence it has a natural Kähler structure and a Hamiltonian $SO(n+2)$ action.

Consider the ${S}^{1}\subset SO(n+2)$ action on ${\tilde{G}}_{2}\left({\mathbb{R}}^{n+2}\right)$ induced by the ${S}^{1}$ action on ${\mathbb{R}}^{n+2}={\u2102}^{\frac{n+2}{2}}$ 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 $(M,\omega )$ 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** $\pm w$ **between** $P$ **and** $Q$. It is known (see[4] 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* $(M,\omega )$ *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* $(M,\omega )$ *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*$y,z\in {H}^{n}(M;\mathbb{Z})$*such that the*

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