Fundamental group of Symp(M,w) with no circle action

On semifree symplectic circle actions with isolated fixed points

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 $(P^1)^n$.

An example of circle actions on symplectic Calabi-Yau manifolds with non-empty fixed points

Let $(X,σ,J)$ be a compact Kähler Calabi-Yau manifold equipped with a symplectic circle action. By Frankel's theorem \cite{F}, the action on $X$ is non-Hamiltonian and $X$ does not have any fixed point. In this paper, we will show that a symplectic circle action on a compact non-Kähler symplectic Calabi-Yau manifold may have a fixed point. More precisely, we will show that the symplectic $S^1$-manifold constructed by D. McDuff \cite{McD} has the vanishing first Chern class. This manifold has the Betti numbers $b_1 = 3$, $b_2 = 8$, and $b_3 = 12$. In particular, it does not admit any Kähler str…

奇异辛同胚与切触圆周作用

Exotic symplectomorphisms and contact circle actions

该研究利用Floer同调方法，建立了标准辛球上奇异辛同胚的不存在性与标准切触球面上自由切触圆周作用的拓扑约束的关联，还推广至Liouville域情形，应用于辛映射类群与切触同胚群基本群的研究。

Classification of six dimensional monotone symplectic manifolds admitting semifree circle actions III

In this paper, we complete the classification of six-dimensional closed monotone symplectic manifolds admitting semifree Hamiltonian $S^1$-actions. We also show that every such manifold is $S^1$-equivariantly symplectomorphic to some Käahler Fano manifold with a certain holomorphic Hamiltonian circle action.

Homologically trivial symplectic cyclic actions need not extend to Hamiltonian circle actions

We give examples of symplectic actions of a cyclic group, inducing a trivial action on homology, on four-manifolds that admit Hamiltonian circle actions, and show that they do not extend to Hamiltonian circle actions. Our work applies holomorphic methods to extend combinatorial tools developed for circle actions to study cyclic actions.

Circle and torus actions on equal symplectic blow-ups of CP^2

A manifold obtained by k simultaneous symplectic blow-ups of CP^2 of equal sizes epsilon (where the size of CP^1 \subset CP^2 is one) admits an effective two dimensional torus action if k <= 3 and admits an effective circle action if (k-1)epsilon < 1. We show that these bounds are sharp if 1/epsilon is an integer.

Symplectic forms invariant under free circle actions on 4--manifolds

Let $M^4$ be a smooth compact manifold with a free circle action generated by a vector field $X.$ Then for any invariant symplectic form $ω$ on $M$ the contracted form $ι_X ω$ is nondegenerate. Using the map $ω--> ι_X ω$ and the related map to $H^1(M / S^1, R)$ we study the topology of the space $S_{inv}$ of invariant symplectic forms. In particular we give a description of $π_0 S_{inv}(M^4)$ in terms of the unit Thurston ball.

Free circle actions with contractible orbits on symplectic manifolds

We prove that closed symplectic four-manifolds do not admit any smooth free circle actions with contractible orbits, without assuming that the actions preserve the symplectic forms. In higher dimensions such actions by symplectomorphisms do exist, and we give explicit examples based on a construction of Fernandez, Gray and Morgan. We also prove that the only compact complex surfaces admitting free circle actions with contractible orbits are primary Hopf surfaces.

The topology of symplectic circle bundles

We consider circle bundles over compact three-manifolds with symplectic total spaces. We show that the base of such a space must be irreducible or the product of the two-sphere with the circle. We then deduce that such a bundle admits a symplectic form if and only if it admits one that is invariant under the circle action in three special cases: namely if the base is Seifert fibered, has vanishing Thurston norm, or if the total space admits a Lefschetz fibration.

Loops in the fundamental group of $\mathrm{Symp} (\mathbb C\mathbb P^2\#\,5\overline{ \mathbb C\mathbb P}\,\!^2)$ which are not represented by circle actions

We study generators of the fundamental group of the group of symplectomorphisms $\mathrm{Symp}({\mathbb C\mathbb P}^2\#\,5\overline{\mathbb C\mathbb P}\,\!^2, ω)$ for some particular symplectic forms. It was observed by J. Kȩdra that there are many symplectic 4-manifolds $(M, ω)$, where $M$ is neither rational nor ruled, that admit no circle action and $π_1 (\mathrm{Ham} (M,ω))$ is nontrivial. On the other hand, it follows from previous results that the fundamental group of the group $\mathrm{Symp}_h({\mathbb C\mathbb P}^2\#\,k\,\overline{\mathbb C\mathbb P}\,\!^2, ω)$, of symplectomorphisms t…

4-流形上辛非哈密顿圆作用的分类

Classification of symplectic non-Hamiltonian circle actions on 4-manifolds

通过定义一组不变量并证明其完备性，对紧连通辛4-流形上的辛非哈密顿圆作用进行等变辛同胚分类，并构造空间实现所有有效值。

Symplectic 4-manifolds with a free circle action

The content of this unpublished paper is subsumed in successive work of the authors, in particular arXiv:1102.0820, arXiv:1102.0821 and arXiv:1205.2434.

Symplectic circle actions with isolated 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…

Classification of six dimensional monotone symplectic manifolds admitting semifree circle actions I

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.

Circle actions on symplectic four-manifolds

We complete the classification of Hamiltonian torus and circle actions on symplectic four-dimensional manifolds. Following work of Delzant and Karshon, Hamiltonian circle and 2-torus actions on any fixed simply connected symplectic four-manifold were characterized by Karshon, Kessler and Pinsonnault. What remains is to study the case of Hamiltonian actions on blowups of S^2-bundles over a Riemann surface of positive genus. These do not admit 2-torus actions. In this paper, we characterize Hamiltonian circle actions on them. We then derive combinatorial results on the existence and counting of …

Construction of symplectic structures on 4-manifolds with a free circle action

Let $M$ be a closed 4-manifold with a free circle action. If the orbit manifold $N^3$ satisfies an appropriate fibering condition, then we show how to represent a cone in $H^2(M;\R)$ by symplectic forms. This generalizes earlier constructions by Thurston, Bouyakoub and Fernández-Gray-Morgan. In the case that $M$ is the product 4-manifold $S^1\times N$ our construction complements the results of \cite{FV08} (arXiv:0805:1234 [math.GT]) and allows us to completely determine the symplectic cone of such 4-manifolds. The content of this paper partly overlaps with the content of the unpublished prepr…

On the construction of certain 6-dimensional symplectic manifolds with Hamiltonian circle actions

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.

Embedded surfaces for symplectic circle actions

The purpose of this article is to characterize symplectic and Hamiltonian circle actions on symplectic manifolds in terms of symplectic embeddings of Riemann surfaces. More precisely, we will show that (1) if $(M,ω)$ admits a Hamiltonian $S^1$-action, then there exists an $S^1$-invariant symplectic $2$-sphere $S$ in $(M,ω)$ such that $\langle c_1(M), [S] \rangle > 0$, and (2) if the action is non-Hamiltonian, then there exists an $S^1$-invariant symplectic $2$-torus $T$ in $(M,ω)$ such that $\langle c_1(M), [T] \rangle = 0$. As applications, we will give a very simple proof of the following we…

Tame Circle Actions

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…

关于辛圆作用、挠与环的注记

Remarks on symplectic circle actions, torsion and loops

本文研究闭辛流形上辛环的通量，证明轨道可缩时通量消失，由此得到通量群消失结果、辛圆作用哈密顿性的充分条件及辛挠应用。

Hirzebruch genera of manifolds equipped with a Hamiltonian circle action

In the paper we describe obstructions for the existence of symplectic and Hamiltonian symplectic circle actions on closed compact manifolds in terms of Hirzebruch genera and relations between differential and homotopic invariants of such manifolds.

Classification of six dimensional monotone symplectic manifolds admitting semifree circle actions

Let $(M,ω_M)$ be a six dimensional closed monotone symplectic manifold admitting an effective semifree Hamiltonian $S^1$-action. We show that $(M,ω_M)$ 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.

Classification of six dimensional monotone symplectic manifolds admitting semifree circle actions II

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 maximal and the minimal fixed component are both two dimensional, then $(M,ω_M)$ is $S^1$-equivariantly symplectomorphic to some Kähler Fano manifold $(X, ω_X, J)$ equipped with a certain holomorphic Hamiltonian $S^1$-action. We also give a complete list of all such Fano manifolds together with an explicit description of the corresponding $S^1$-actions.

Symplectic quotients and representability: the circle action case

Let $S^1$ act on a symplectic manifold in a Hamiltonian fashion with momentum map $Ψ$. Fix a value $a$ of $Ψ$. There is a question of whether the symplectic quotient at $a$ is diffeomorphic to the orbit space of some proper Lie group action. We prove under mild assumptions that this only occurs if the symplectic quotient is diffeomorphic to an effective orbifold. This, in turn, only occurs if $a$ is a regular value, or there is at most one positive weight or at most one negative weight.

Circle actions on almost complex manifolds with isolated fixed points

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.

Cohomology rings of symplectic quotients by circle actions

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…

A limit of toric symplectic forms that has no periodic Hamiltonians

We calculate the Riemann-Roch number of some of the pentagon spaces defined in [Klyachko,Kapovich-Millson,HK1]. Using this, we show that while the regular pentagon space is diffeomorphic to a toric variety, even symplectomorphic to one under arbitrarily small perturbations of its symplectic structure, it does not admit a symplectic circle action. In particular, within the cohomology classes of symplectic structures, the subset admitting a circle action is not closed.

辛圈作用在具有接触类型的边界的流形上

Symplectic circle actions on manifolds with contact type boundary

本文研究具有接触类型边界的辛流形上的辛圈作用，证明了Morse-Bott理论的核心思想仍适用，从而推广了闭合情形下的多项结果，包括辛群作用的哈密顿性、流形拓扑结构及边界连通性等。

Asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions

We derive a complete asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions on arbitrary symplectic manifolds, characterizing the coefficients in the expansion as integrals over the symplectic strata of the corresponding Marsden-Weinstein reduced space and distributions on the Lie algebra. The obtained coefficients involve singular contributions of the lower-dimensional strata related to numerical invariants of the fixed-point set.

On Generalized Moment Maps for Symplectic Compact Group Actions

A generalized moment map is proposed for arbitrary symplectic actions of compact connected Lie groups on closed symplectic manifolds, in the spirit of the circle -valued maps introduced by D. McDuff in the case of non-Hamiltonian circle actions. We study equivariance properties of generalized moments, show that they allow reduction procedures, and obtain in the torus case a version of the Atiyah-Guillemin-Sternberg convexity theorem. As illustration, we reformulate a proof of M.K. Kim that "complexity one" symplectic torus actions are Hamiltonian, and give a symplectic proof of the finiteness …

Hamiltonian circle actions on eight dimensional manifolds with minimal fixed sets

Consider a Hamiltonian circle action on a closed $8$-dimensional symplectic manifold $M$ with exactly five fixed points, which is the smallest possible fixed set. In their paper, L. Godinho and S. Sabatini show that if $M$ satisfies an extra "positivity condition" then the isotropy weights at the fixed points of $M$ agree with those of some linear action on $\mathbb{CP}^4$. Therefore, the (equivariant) cohomology rings and the (equivariant) Chern classes of $M$ and $\mathbb{CP}^4$ agree; in particular, $H^*(M;\mathbb{Z}) \simeq \mathbb{Z}[y]/y^5$ and $c(TM) = (1+y)^5$. In this paper, we prove …

Lagrangian Circle Actions

We consider paths of Hamiltonian diffeomorphism preserving a given compact monotone Lagrangian in a symplectic manifold that extend to an $S^1$--Hamiltonian action. We compute the leading term of the associated Lagrangian Seidel element. We show that such paths minimize the Lagrangian Hofer length. Finally we apply these computations to Lagrangian uniruledness and to give a nice presentation of the Quantum cohomology of real lagrangians in Fano symplectic toric manifolds.

Symplectic 4--manifolds with K = 0 and the Lubotzky alternative

In this paper we use the Lubotzky alternative for finitely generated linear groups to determine which 4-manifolds admitting a free circle action can be endowed with a symplectic structure with trivial canonical class. The content of this paper partly overlaps with the content of the unpublished preprint "Symplectic 4-manifolds with a free circle action" (arXiv:0801.1313 [math.GT]).

On fundamental groups of symplectically aspherical manifolds II: abelian groups

We describe all abelian groups which can appear as the fundamental groups of closed symplectically aspherical manifolds. The proofs use the theory of symplectic Lefschetz fibrations.

Semifree Hamiltonian circle actions on 6-dimensional symplectic manifolds with non-isolated fixed point set

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…

Hamiltonian circle actions with minimal fixed sets

Consider an effective Hamiltonian circle action on a compact symplectic $2n$-dimensional manifold $(M, ω)$. Assume that the fixed set $M^{S^1}$ is {\em minimal}, in two senses: it has exactly two components, $X$ and $Y$, and $\dim(X) + \dim(Y) = \dim(M) - 2$. We prove that the integral cohomology ring and Chern classes of $M$ are isomorphic to either those of $\CP^n$ or (if $n \neq 1$ is odd) to those of $\Gt_2(\R^{n+2})$, the Grassmannian of oriented two-planes in $\R^{n+2}$. In particular, $H^i(M;\Z) = H^i(\CP^n;\Z)$ for all $i$, and the Chern classes of $M$ are determined by the integral co…

The fundamental group of symplectic manifolds with Hamiltonian SU(2) or SO(3) actions

Let $(M, ω)$ be a connected compact symplectic manifold equipped with a Hamiltonian SU(2) or SO(3) action. We prove that, as fundamental group of topological spaces, $π_1(M)=π_1(M_{red})$, where $M_{red}$ is the symplectic quotient at any value of the moment map $ϕ$.

Topological properties of Hamiltonian circle actions

This paper studies Hamiltonian circle actions, i.e. circle subgroups of the group Ham(M,\om) of Hamiltonian symplectomorphisms of a closed symplectic manifold (M,\om). Our main tool is the Seidel representation of π_1(\Ham(M,\om)) in the units of the quantum homology ring. We show that if the weights of the action at the points at which the moment map is a maximum are sufficiently small then the circle represents a nonzero element of π_1(\Ham(M,\om)). Further, if the isotropy has order at most two and the circle contracts in \Ham(M,\om) then the homology of M is invariant under an involution. …

Maximal tori in the contactomorphism groups of circle bundles over Hirzebruch surfaces

In a recent preprint Yael Karshon showed that there exist non-conjugate tori in a group of symplectomorphisms of a Hirzebruch surface. She counted them in terms of the cohomology class of the symplectic structure. We show that a similar phenomenon exists in the contactomorphism groups of pre-quantum circle bundles over Hirzebruch surfaces. Note that the contact structures in question are fillable. This may be contrasted with an earlier paper where we showed that there are infinitely many non-conjugate tori in the contactomorphism groups of certain overtwisted lens spaces (Contact cuts, Israel …

Triviality of symplectic SU(2)-actions on homology

Lalonde and McDuff showed that the natural action of the rational homology of the group of Hamiltonian diffeomorphisms of a closed symplectic manifold $(M, ω)$ on the rational homology groups $H_*(M,{\mathbb Q})$ is trivial. In this note, given a symplectic action of SU(2), $ϕ:SU(2)\times M \to M$, we will construct a symplectic fiber bundle $P_ϕ\to {\mathbb CP}^2$ with fiber $(M,ω)$ and use it to construct the chains, which bound the images of the homology cycles under the trace map given by the SU(2)-action. It turns out that the natural chains bounded by the SU(2)-orbits in $M$ are puncture…

Tight, not semi--fillable contact circle bundles

Extending our earlier results, we prove that certain tight contact structures on circle bundles over surfaces are not symplectically semi--fillable, thus confirming a conjecture of Ko Honda.

Unimodality of the Betti numbers for Hamiltonian circle action with isolated fixed points

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. $b_0(M) \leq b_2(M) \leq b_4(M)$.

Hamiltonian circle actions with fixed point set almost minimal

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 $…

具有更少孤立不动点的非 Hamilton 作用

Non-Hamiltonian actions with fewer isolated fixed points

本文改进此前含32个不动点的非Hamilton辛圆周作用构造，证明对任意k≥5，都可在闭连通六维辛流形上构造恰有2k个不动点的此类作用。

Seiberg-Witten vanishing theorem for $S^1$-manifolds with fixed points

In this paper we show that the Seiberg--Witten invariant is zero for all smooth 4--manifolds with $b_+{>}1$ which admit circle actions that have at least one fixed point. Furthermore, we show that all symplectic 4--manifolds which admit circle actions with fixed points are rational or ruled, and thus admit a symplectic circle action.

由圆作用的奇异约化得到的黎曼-罗赫公式

A Riemann-Roch formula for singular reductions by circle actions

该研究针对允许矩映射取奇异值的可量子化哈密顿S¹-流形，推导了Guillemin-Sternberg原理下的不变黎曼-罗赫数公式，得到了仅依赖辛数据的奇异辛不变量相关结果。

The fundamental group of symplectic manifolds with Hamiltonian Lie group actions

Let $(M, ω)$ be a connected, compact symplectic manifold equipped with a Hamiltonian $G$ action, where $G$ is a connected compact Lie group. Let $ϕ$ be the moment map. In \cite{L}, we proved the following result for $G=S^1$ action: as fundamental groups of topological spaces, $π_1(M)=π_1(M_{red})$, where $M_{red}$ is the symplectic quotient at any value of the moment map $ϕ$. In this paper, we generalize this result to other connected compact Lie group $G$ actions. We also prove that the above fundamental group is isomorphic to that of $M/G$. We briefly discuss the generalization of the first …

哈密顿圆作用、不变超曲面与复射影空间

Hamiltonian circle action, invariant hypersurface and the complex projective space

本文研究带孤立不动点哈密顿圆作用的2n维闭辛流形，证明当存在S¹-不变辛超曲面使流形去除该超曲面为同调胞腔且n不模4余3时，流形与超曲面为具标准陈类的同伦复射影空间，其不动点集的S¹-表示与标准线性作用一致，是相关猜想结果的变换群类比。

Hofer-Zehnder capacity and a Hamiltonian circle action with noncontractible orbits

Let $(M,ω)$ be an aspherical symplectic manifold, which is closed or convex. Let $U$ be an open set in $M$, which admits a circle action generated by an autonomous Hamiltonian $H \in C^\infty(U)$, such that each orbit of the circle action is not contractible in $M$. Under these assumptions, we prove that the Hofer-Zehnder capacity of $U$ is bounded by the Hofer norm of $H$. The proof uses a variant of the energy-capacity inequality, which is proved by the theory of action selectors.

有理直纹面上 Hamilton 圆周作用的中心化子

Centralizers of Hamiltonian circle actions on rational ruled surfaces

本文研究两类有理直纹面上 Hamilton $S^1$ 作用的等变辛微分同胚群，通过近复结构分层及 Delzant、Karshon 分类确定其同伦型为环面或两环面的同伦推出。

在有效圆作用不变下的超辛结构

Hypersymplectic Structures Invariant Under an Effective Circle Action

本文证明了在有效 $S^1$-作用不变假设下，紧致4-流形上的超辛结构可形变至超凯勒三元组，且证明了该流形微分同胚于 $\mathbb{T}^4$。

Circle actions and geometric quantisation

The aim of this article is to present unifying proofs for results in geometric quantisation with real polarisations by exploring the existence of symplectic circle actions. It provides an extension of Rawnsley's results on the Kostant complex, and gives an alternative proof for Sniatycki's and Hamilton's theorems; as well as, a partial result for the focus-focus contribution to geometric quantisation.

Fixed points of symplectic periodic flows

The study of fixed points is a classical subject in geometry and dynamics. If the circle acts in a Hamiltonian fashion on a compact symplectic manifold M, then it is classically known that there are at least 1 + dim(M)/2 fixed points; this follows from Morse theory for the momentum map of the action. In this paper we use Atiyah-Bott-Berline-Vergne (ABBV) localization in equivariant cohomology to prove that this conclusion also holds for symplectic circle actions with non-empty fixed sets, as long as the Chern class map is somewhere injective -- the Chern class map assigns to a fixed point the …

Certain circle actions on Kaehler manifolds

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.