Let the circle act in a Hamiltonian fashion on a connected compact symplectic manifold $(M, ω)$ of dimension $2n$. Then the $S^1$-action has at least $n+1$ fixed points. In a previous paper, we study the case when the fixed point set consists of precisely $n+1$ isolated points. In this paper, we study the case when the fixed point set consists of exactly $n+2$ isolated points. We show that in this case $n$ must be even. We find equivalent conditions on the first Chern class of $M$ and a particular weight of the $S^1$-action. We also show that the particular weight can completely determine the integral cohomology ring and the total Chern class of $M$, and the sets of weights of the $S^1$-action at all the fixed points. We will see that all these data are isomorphic to those of known examples, $\widetilde{G}_2(\mathbb{R}^{n+2})$ with $n\geq 2$ even, equipped with standard circle actions.
Let $(M,ω)$ be an eight-dimensional closed symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points. In this article, we will show that the Betti numbers of $M$ are unimodal, i.e. $b_0(M) \leq b_2(M) \leq b_4(M)$.
Let $M$ be a symplectic manifold, equipped with a semifree symplectic circle action with a finite, nonempty fixed point set. We show that the circle action must be Hamiltonian, and $M$ must have the equivariant cohomology and Chern classes of $(P^1)^n$.
We show that almost complex circle actions with exactly three fixed points do not exist in dimension 8 and present an infinite series of 6-dimensional manifolds possessing an almost complex circle action with exactly two fixed points.
Consider a symplectic circle action on a closed symplectic manifold with non-empty isolated fixed points. Associated to each fixed point, there are well-defined non-zero integers, called weights. We prove that the action is Hamiltonian if the sum of an odd number of weights is never equal to zero (the weights may be taken at different fixed points). Moreover, we show that if $\dim M=6$, or if $\dim M=2n \leq 10$ and each fixed point has weights $\{\pm a_1, \cdots, \pm a_n\}$ for some positive integers $a_i$, it is enough to consider the sum of three weights. As applications, we recover the res…
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.
Following the idea of Lusztig, Atiyah-Hirzebruch and Kosniowski, we note that the Dolbeault-type operators on compact, almost-complex manifolds are rigid. When the circle action has isolated fixed points, this rigidity result will produce many identities concerning the weights on the fixed points. In particular, it gives a criterion to detemine whether or not a symplectic circle action with isolated fixed points is Hamiltonian. As applications, we simplify the proofs of some known results related to symplectic circle actions, due to Godinho, Tolman-Weitsman and Pelayo-Tolman, and generalize so…
Assume $(M, ω)$ is a connected, compact 6 dimensional symplectic manifold equipped with a semi-free Hamiltonian circle action, such that the fixed point set consists of isolated points or compact orientable surfaces. We restrict attention to the case $\dim$ $H^2(M)<3$. We give a complete list of the possible manifolds, determine their equivariant cohomology ring and equivariant Chern classes. Some of these manifolds are classified up to diffeomorphism. We also show the existence for a few cases.
Motivated by recent works on Hamiltonian circle actions satisfying certain minimal conditions, in this paper, we consider Hamiltonian circle actions satisfying an almost minimal condition. More precisely, we consider a compact symplectic manifold $(M, ω)$ admitting a Hamiltonian circle action with fixed point set consisting of two connected components $X$ and $Y$ satisfying $\dim(X)+\dim(Y)=\dim(M)$. Under certain cohomology conditions, we determine the circle action, the integral cohomology rings of $M$, $X$ and $Y$, and the total Chern classes of $M$, $X$, $Y$, and of the normal bundles of $…
Let $M$ be a closed Fano symplectic manifold with a semifree Hamiltonian circle action with isolated maximum. We compute the Gromov width and the Hofer-Zehnder capacity of $M$ using a moment map.
For every compact almost complex manifold (M,J) equipped with a J-preserving circle action with isolated fixed points, a simple algebraic identity involving the first Chern class is derived. This enables us to construct an algorithm to obtain linear relations among the isotropy weights at the fixed points. Suppose that M is symplectic and the action is Hamiltonian. If the manifold satisfies an extra "positivity condition" this algorithm determines a family of vector spaces which contain the admissible lattices of weights. When the number of fixed points is minimal, this positivity condition is…
Let $(M,J)$ be a compact, connected, almost complex manifold of dimension $2n$ endowed with a $J$-preserving circle action with isolated fixed points. In this note we analyse the `geography problem' for such manifolds, deriving equations relating the Chern numbers to the index $k_0$ of $(M,J)$. We study the symmetries and zeros of the Hilbert polynomial associated to $(M,J)$, which imply many rigidity results for the Chern numbers when $k_0\neq 1$. We apply these results to the category of compact, connected symplectic manifolds. A long-standing question posed by McDuff and Salamon, also known…
Let $(M, ω)$ be a 6-dimensional closed symplectic manifold with a symplectic $S^1$-action with $M^{S^1} \neq \emptyset$ and $\dim M^{S^1} \leq 2$. Assume that $ω$ is integral with a generalized moment map $μ$. We first prove that the action is Hamiltonian if and only if $b_2^+(M_{\red})=1$, where $M_{\red}$ is any reduced space with respect to $μ$. It means that if the action is non-Hamiltonian, then $b_2^+(M_{\red}) \geq 2$. Secondly, we focus on the case when the action is semifree and Hamiltonian. We prove that if $M^{S^1}$ consists of surfaces, then the number $k$ of fixed surfaces with po…
Let $M$ be a compact oriented simply-connected manifold of dimension at least 8. Assume $M$ is equipped with a torsion-free semi-free circle action with isolated fixed points. We prove $M$ has a perfect invariant Morse-Smale function. The major ingredient in the proof is a new cancellation theorem for the invariant Morse theory.
Let the circle act holomorphically on a compact Kähler manifold $M$ of complex dimension $n$ with moment map $ϕ\colon M\to\R$. Assume the critical set of $ϕ$ consists of 3 connected components, the extrema being isolated points. We show that $M$ is equivariantly biholomorphic to $\CP^n$, where $n\geq 2$, or to $\Tilde G_2(\R^{n+2})$, the Grassmannian of oriented 2-planes in $\R^{n+2}$, where $n\geq 3$, with a standard circle action; we also show that $M$ is equivariantly symplectomorphic to $\CP^n$, where $n\geq 2$, or to $\Tilde G_2(\R^{n+2})$, where $n\geq 3$, with a standard circle action.
Let $(M,ω_M)$ be a six dimensional closed monotone symplectic manifold admitting an effective semifree Hamiltonian $S^1$-action. We show that if the minimal (or maximal) fixed component of the action is an isolated point, then $(M,ω_M)$ is $S^1$-equivariant symplectomorphic to some Kähler Fano manifold $(X,ω_X, J)$ with a certain holomorphic $\mathbb{C}^*$-action. We also give a complete list of all such Fano manifolds and describe all semifree $\mathbb{C}^*$-actions on them specifically.
This paper studies symplectic manifolds that admit semi-free circle actions with isolated fixed points. We prove, using results on the Seidel element due to McDuff and Tolman, that the (small) quantum cohomology of a $2n$ dimensional manifold of this type is isomorphic to the (small) quantum cohomology of a product of $n$ copies of $\bb{P}^1$. This generalizes a result due to Tolman and Witsman.
The unimodality conjecture posed by Tolman in the conference `Moment maps in Various Geometry" in 2005 states that if (M,w) is a 2n-dimensional smooth compact symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points, then the sequence of Betti numbers is unimodal. Recently, the author and M. Kim proved that the unimodality holds in eight-dimensional cases by using equivariant cohomology theory. In this paper, we generalize the idea in \cite{CK} to an arbitrary dimensional case. Also, we prove the conjecture in arbitrary dimension with an assumption that a m…
Let $(M, ω)$ be a connected, compact 6-dimensional symplectic manifold equipped with a semi-free Hamiltonian $S^1$ action such that the fixed point set consists of isolated points or surfaces. Assume dim $H^2(M)<3$, in \cite{L}, we defined a certain invariant of such spaces which consists of fixed point data and twist type, and we divided the possible values of these invariants into six ``types''. In this paper, we construct such manifolds with these ``types''. As a consequence, we have a precise list of the values of these invariants.
A Hamiltonian circle action on a compact symplectic manifold is known to be a closed geodesic with respect to the Hofer metric on the group of Hamiltonian diffeomorphisms. If the momentum map attains its minimum or maximum at an isolated fixed point with isotropy weights not all equal to plus or minus one, then this closed geodesic can be deformed into a loop of shorter Hofer length. In this paper we give a lower bound for the possible amount of shortening, and we give a lower bound for the index ("number of independent shortening directions"). If the minimum or maximum is attained along a sub…
Let $(M,ω)$ be a $2n$-dimensional smooth compact symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points and let $μ: M \rightarrow \R$ be a corresponding moment map. Let $Λ_{2k}$ be the set of all fixed points of index $2k$. In this paper, we will show that if $μ$ is constant on $Λ_{2k}$ for each $k$, then $(M,ω)$ satisfies the hard Lefschetz property. In particular, if $(M,ω)$ admits a self-indexing moment map, i.e. $μ(p) = 2k$ for every $p \in Λ_{2k}$ and $k=0,1,\cdots,n,$ then $(M,ω)$ satisfies the hard Lefschetz property.
In this paper, we consider Sjamaar's holomorphic slice theorem, the birational equivalence theorem of Guillemin and Sternberg, and a number of important standard constructions that work for Hamiltonian circle actions in both the symplectic category and the Kähler category: reduction, cutting, and blow-up. In each case, we show that the theory extends to Hamiltonian circle actions on complex manifolds with tamed symplectic forms. (At least, the theory extends if the fixed points are isolated.) Our main motivation for this paper is that the first author needs the machinery that we develop here t…
It is shown that the signature of a manifold with a symplectic circle action having only isolated fixed points, equals the alternating sum of the Novikov numbers corresponding to the cohomology class of the generalized moment map. The same is true for more general fixed point sets.
The question of what conditions guarantee that a symplectic $S^1$ action is Hamiltonian has been studied for many years. In a 1998 paper, Sue Tolman and Jonathon Weitsman proved that if the action is semifree and has a non-empty set of isolated fixed points then the action is Hamiltonian. Furthermore, in a 2010 paper Cho, Hwang, and Suh proved in the 6-dimensional case that if we have $b_2^+=1$ at a reduced space at a regular level $λ$ of the circle valued moment map, then the action is Hamiltonian. In this paper, we will use this to prove that certain 6-dimensional symplectic actions which ar…
In this article we are concerned with how to compute the cohomology ring of a symplectic quotient by a circle action using the information we have about the cohomology of the original manifold and some data at the fixed point set of the action. Our method is based on the Tolman-Weitsman theorem which gives a characterization of the kernel of the Kirwan map. First we compute a generating set for the kernel of the Kirwan map for the case of product of compact connected manifolds such that the cohomology ring of each of them is generated by a degree two class. We assume the fixed point set is iso…
We study the problem of determining which diffeomorphism classes of Kähler manifolds admit a Hamiltonian circle action. Our main result is the following: Let $M$ be a closed symplectic manifold, diffeomorphic to a complete intersection with complex dimension $4k$, having a Hamiltonian circle action such that each component of the fixed point set is an isolated fixed point or has dimension $2 \mod 4$. Then $M$ is diffeomorphic to $\mathbb{CP}^{4k}$, a quadric $Q \subset \mathbb{CP}^{4k+1}$ or an intersection of two quadrics $Q_1 \cap Q_2 \subset \mathbb{CP}^{4k+2}$.
