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

Introduction

Consider a circle action on a connected compact $2 n$ -dimensional symplectic manifold $(M, ω)$ with moment map $ϕ$ . 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 ${\tilde{G}}_{2} (R^{n + 2})$ with $n \geq 2$ even, see Example2.1.

Let $(M, ω)$ be a connected compact Hamiltonian $S^{1}$ -manifold of dimension $2 n$ with almost minimal isolated fixed points, and let $ϕ$ be the moment map. By Lemma2.2, the fixed points, denoted $P_{0}, \dots, P_{\frac{n}{2} - 1}, P_{\frac{n}{2}}, P_{{(\frac{n}{2})}^{'}}, P_{\frac{n}{2} + 1}, \dots, P_{n}$ , can be labeled to satisfy $ϕ (P_{0}) < \dots < ϕ (P_{\frac{n}{2} - 1}) < ϕ (P_{\frac{n}{2}}) \leq ϕ (P_{{(\frac{n}{2})}^{'}}) < ϕ (P_{\frac{n}{2} + 1}) < \dots < ϕ (P_{n}) .$ 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 $2 n$ -dimensional symplectic manifold $(M, ω)$ with moment map $ϕ : M \to R$ . Assume $[ω]$ is a primitive integral class and the fixed point set $M^{S^{1}}$ consists of $n + 2$ isolated points, denoted $M^{S^{1}} = {P_{0}, \dots, P_{\frac{n}{2} - 1}, P_{\frac{n}{2}}, P_{{(\frac{n}{2})}^{'}}, P_{\frac{n}{2} + 1}, \dots, P_{n}}$ . Then for any $i, j \in {0, \dots, \frac{n}{2}, {(\frac{n}{2})}^{'}, \dots, n}$ , $ϕ (P_{i}) - ϕ (P_{j}) \in Z$ , and $c_{1} (M) = n [ω]$ if and only if $ϕ (P_{n - 1}) - ϕ (P_{0}) = ϕ (P_{n}) - ϕ (P_{1})$ 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 $[ω]$ needs to be chosen suitably.

Theorem 2

Let the circle act on a connected compact $2 n$ -dimensional symplectic manifold $(M, ω)$ with moment map $ϕ : M \to R$ . Assume $M^{S^{1}}$ consists of $n + 2$ isolated points, i.e., $M^{S^{1}} = {P_{0}, \dots, P_{\frac{n}{2} - 1}, P_{\frac{n}{2}}, P_{{(\frac{n}{2})}^{'}}, P_{\frac{n}{2} + 1}, \dots, P_{n}}$ . Then $n \geq 2$ must be even. Assume $[ω]$ is an integral class. If $ϕ (P_{n - 1}) - ϕ (P_{0}) = ϕ (P_{n}) - ϕ (P_{1})$ 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 ${\tilde{G}}_{2} (R^{n + 2})$ .
(2)
The total Chern class of $M$ is isomorphic to that of ${\tilde{G}}_{2} (R^{n + 2})$ .
(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 ${\tilde{G}}_{2} (R^{n + 2})$ (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 ${\tilde{G}}_{2} (R^{n + 2})$ with $n \geq 2$ even.

In[4, Theorem 6.17], Hattori studies compact almost complex $S^{1}$ -manifold $M$ of dimension $2 n$ with $n + 2$ isolated fixed points. Under the assumption $c_{1} (M) = n x$ , where $x \in H^{2} (M; Z)$ 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 ${\tilde{G}}_{2} (R^{n + 2})$ with $n \geq 2$ even, and he obtains that $x^{n} [M] = 2$ . In our current work, for a compact symplectic Hamiltonian $S^{1}$ -manifold $(M, ω)$ , we prove the equivalence of the condition $c_{1} (M) = n x$ and the particular weight as in Theorem1, and using the particular weight as a starting point, we give methods to prove $(1)$ , $(2)$ and $(3)$ 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].

(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 $(3)$ of Theorem2. In Section5, we determine the integral cohomology ring and the total Chern class of the manifold, proving $(1)$ and $(2)$ of Theorem2.

Section snippets

Examples and basic results

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

Example 2.1

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

Consider the $S^{1} \subset S O (n + 2)$ action on ${\tilde{G}}_{2} (R^{n + 2})$ induced by the $S^{1}$ action on $R^{n + 2} = ℂ^{\frac{n + 2}{2}}$ given by $λ \cdot (z_{0}, z_{1}, \dots, z_{\frac{n}{2}}) = (λ^{b_{0}} z_{0}, λ^{b_{1}} z_{1}, \dots, λ^{b_{\frac{n}{2}}} z_{\frac{n}{2}}),$ where the $b_{i}$ ’s, with $i = 0, 1, \dots, \frac{n}{2}$ , are mutually distinct integers.

On $c_{1} (M)$ — the proof of Theorem1

In this section, we prove Theorem1.

In a symplectic $S^{1}$ -manifold $(M, ω)$ 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 $M^{Z_{w}}$ , 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 $(3)$ 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 $(M, ω)$ with moment map $ϕ : M \to R$ . If $c_{1} (M) = k [ω]$ , then for any two fixed point set components $F$ and $F^{'}$ , we have $Γ_{F} - Γ_{F^{'}} = k (ϕ (F^{'}) - ϕ (F))$ , where $Γ_{F}$ and $Γ_{F^{'}}$ are respectively the sums of the weights at $F$ and $F^{'}$ .

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

Lemma 4.2

(Video) Unlinked fixed points of Hamiltonian...spectral invariants - Sobhan Seyfaddini

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 $(1)$ and $(2)$ of Theorem2.

We will do this in a few steps.

Lemma 5.1

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

(1)
There exist classes $y, z \in H^{n} (M; Z)$ such that the

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

Introduction

Section snippets

Examples and basic results

On $c_{1} (M)$ — the proof of Theorem1

From the named weight to all the weights

[7, Lemma 4.1]

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

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