River Chiang
We consider compact connected six dimensional symplectic manifolds with Hamiltonian SU(2) or SO(3) actions with cyclic principal stabilizers. We classify such manifolds up to equivariant symplectomorphisms.
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We determine conditions under which two Hamiltonian torus actions on a symplectic manifold $M$ are homotopic by a family of Hamiltonian torus actions, when $M$ is a toric manifold and when $M$ is a coadjoint orbit.
We give a complete characterization of Hamiltonian actions of compact Lie groups on exact symplectic manifolds with proper momentum maps. We deduce that every Hamiltonian action of a compact Lie group on a contractible symplectic manifold with a proper momentum map is globally linearizable.
In this paper we generalize constructions of non-commutative integrable systems to the context of weakly Hamiltonian actions on Poisson manifolds. In particular we prove that abelian weakly Hamiltonian actions on symplectic manifolds split into Hamiltonian and non-Hamiltonian factors, and explore generalizations in the Poisson setting.
In this paper, we establish a certain inequality in terms of Betti numbers of a closed Hamiltonian $S^1$-manifold with isolated fixed points.
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.
In this paper we generalize to coisotropic actions of compact Lie groups a theorem of Guillemin on deformations of Hamiltonian structures on compact symplectic manifolds. We show how one can reconstruct from the moment polytope the symplectic form on the manifold.
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. $b_0(M) \leq b_2(M) \leq b_4(M)$.
In this paper, we study the hard Lefschetz property of a symplectic manifold which admits a Hamiltonian torus action. More precisely, let $(M,ω)$ be a 6-dimensional compact symplectic manifold with a Hamiltonian $T^2$-action. We will show that if the moment map image of $M$ is a GKM-graph and if the graph is index-increasing, then $(M,ω)$ satisfies the hard Lefschetz property.
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…
Finite Group Actions on Quasi-Hamiltonian Spaces
本文研究有限群作用下的准哈密顿空间性质，并应用于平坦连接模空间理论。
In this paper we develop an abstract setup for hamiltonian group actions as follows: Starting with a continuous 2-cochain $ω$ on a Lie algebra $h$ with values in an $h$-module $V$, we associate subalgebras $sp(h,ω) \supeq ham(h,ω)$ of symplectic, resp., hamiltonian elements. Then $ham(h,ω)$ has a natural central extension which in turn is contained in a larger abelian extension of $sp(h,ω)$. In this setting, we study linear actions of a Lie group $G$ on $V$ which are compatible with a homomorphism $g \to ham(h,ω)$, i.e. abstract hamiltonian actions, corresponding central and abelian extensions…
Special Hamiltonian $S^1$-actions on symplectic 4-manifolds
本文研究了满足特定条件的辛四流形上的哈密顿 $S^1$-作用，证明不存在紧致空间，并展示所有精确空间均为斯蒂恩表面。
Classification of momentum proper exact Hamiltonian group actions and the equivariant Eliashberg cotangent bundle conjecture
本文通过哈密顿G-模型函子在动量正合子范畴上的限制，分类了任意复杂度的动量正合精确哈密顿群作用，并作为极端情况证明了等变Eliashberg余切丛猜想及可缩流形上的相关结论。
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.
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.
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 $Q_1 \cap Q_2 \subset \mathbb{CP}^{4k+2}$.
We show that for a Hamiltonian action of a compact torus $G$ on a compact, connected symplectic manifold $M$, the $G$-equivariant cohomology is determined by the residual $S^1$ action on the submanifolds of $M$ fixed by codimension-1 tori. This theorem allows us to compute the equivariant cohomology of certain manifolds, which have pieces that are four-dimensional or smaller. We give several examples of the computations that this allows.
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 $…
This paper determines a condition that is necessary and sufficient for a metaplectic-c prequantizable symplectic manifold with an effective Hamiltonian torus action to admit an equivariant metaplectic-c prequantization. The condition is evaluated at a fixed point of the momentum map, and is shifted from the one that is known for equivariant prequantization line bundles. Given a metaplectic-c prequantized symplectic manifold with a Hamiltonian energy function, the author previously proposed a condition under which a regular value of the function should be considered a quantized energy level of …
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。
Given a Hamiltonian action of a proper symplectic groupoid (for instance, a Hamiltonian action of a compact Lie group), we show that the transverse momentum map admits a natural constant rank stratification. To this end, we construct a refinement of the canonical stratification associated to the Lie groupoid action (the orbit type stratification, in the case of a Hamiltonian Lie group action) that seems not to have appeared before, even in the literature on Hamiltonian Lie group actions. This refinement turns out to be compatible with the Poisson geometry of the Hamiltonian action: it is a Poi…
In this paper we define invariants of Hamiltonian group actions for central regular values of the moment map. The key hypotheses are that the moment map is proper and that the ambient manifold is symplectically aspherical. The invariants are based on the symplectic vortex equations. Applications include an existence theorem for relative periodic orbits, a computation for circle actions on a complex vector space, and a theorem about the relation between the invariants introduced here and the Seiberg--Witten invariants of a product of a Riemann surface with a two-sphere.
In this note we prove the following theorem: Let $G$ be a compact Lie group acting on a compact symplectic manifold $M$ in a Hamiltonian fashion. If $L$ is an $l$-dimensional closed invariant submanifold of $M$, on which the $G$-action is locally free then the fundamental class $[L]$ is trivial in $H_l(M,{\mathbb Q})$. We also prove similar results for lower homology groups of $L$, in case the group $G$ is a finite product of copies of $S^1$ and SU(2). The key ingredients of the proofs are Kirwan's theorem that Hamiltonian spaces are equivariantly formal and symplectic reduction.
In this paper we first describe the geometry of the Newton polyhedra of polynomials invariant under certain linear Hamiltonian circle actions. From the geometry of the polyhedra, various Poisson structures on the orbit spaces of the actions are derived and Poisson embeddings into model spaces, for the orbit spaces, are constructed. The Poisson structures, on respective source and model space, are compatible even for the minimum possible (embedding) dimension of the model spaces. This is, in particular, important since it is still an open question if, in general, there exist finite dimensional …
The main contribution of this manuscript is a local normal form for Hamiltonian actions of Poisson-Lie groups $K$ on a symplectic manifold equipped with an $AN$-valued moment map, where $AN$ is the dual Poisson-Lie group of $K$. Our proof uses the delinearization theorem of Alekseev which relates a classical Hamiltonian action of $K$ with $\mathfrak{k}^*$-valued moment map to a Hamiltonian action with an $AN$-valued moment map, via a deformation of symplectic structures. We obtain our main result by proving a ``delinearization commutes with symplectic quotients'' theorem which is also of indep…
Hamiltonian torus actions and the unimodality of odd Betti numbers
研究具有非平凡哈密顿环面作用的闭辛流形奇贝蒂数序列是否单峰，通过逐渐加强对称假设，在维度6、8和10下为该问题提供肯定证据。
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.
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 …
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.
Hamiltonian group actions in cosymplectic geometry
研究余辛流形上哈密顿群作用，通过辛加厚过程，将其提升为辛流形上哈密顿辛作用，建立相关理论桥梁，得出凸性定理等余辛类似结果，还应用于b-辛流形临界超曲面的哈密顿几何。
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…
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…
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。
Suppose given an holomorphic and Hamiltonian action of a compact torus $T$ on a polarized Hodge manifold $M$. Assume that the action lifts to the quantizing line bundle, so that there is an induced unitary representation of $T$ on the associated Hardy space. If in addition the moment map is nowhere zero, for each weight $\boldsymbolν$ the $\boldsymbolν$-th isotypical component in the Hardy space of the polarization is finite-dimensional. Assuming that the moment map is transverse to the ray through $\boldsymbolν$, we give a gometric interpretation of the isotypical components associated to the…
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…
In this paper we study the size of the fixed point set of a Hamiltonian diffeomorphism on a closed symplectic manifold which is both rational and weakly monotone. We show that there exists a non-trivial cycle of fixed points whenever the action spectrum is smaller, in a certain sense, than required by the Ljusternik-Schirelman theory. For instance, in the aspherical case, we prove that when the number of points in the action spectrum is less than or equal to the cup length of the manifold, then the cohomology of the fixed point set must be non-trivial. This is a consequence of a more general r…
In this paper we detect new restrictions on the Maslov class of displaceable Lagrangian submanifolds of symplectic manifolds which are symplectically aspherical. These restrictions are established using action selectors for Hamiltonian flows. In particular, we construct and utilize a new action selector for the flows of a special class of Hamiltonian functions which arises naturally in the study of Hamiltonian paths which minimize the Hofer length functional.
Let G be an algebraic group over a field F of characteristic zero, with Lie algebra g=Lie(G). The dual space g^* equipped with the Kirillov bracket is a Poisson variety and each irreducible G-invariant subvariety X\subset g^* carries the induced Poisson structure. We prove that there is a family of algebraically independent polynomial functions {f_1,...f_l} on X, which pairwise commute with respect to the Poisson bracket and such that l=(dim X+tr.deg F(X)^G)/2. We also discuss several applications of this result to complete integrability of Hamiltonian systems on symplectic Hamiltonian G-varie…
Flexibility of the adjoint action of the group of Hamiltonian diffeomorphisms
本文证明闭连通辛流形上哈密顿微分同胚群伴随作用的灵活性：任意零均值光滑函数可表示为非零函数轨道中元素的有限和，且项数受一致范数控制，为Banyaga定理提供有界无穷小版本。
We discuss BFV deformation quantization of singular symplectic quotient spaces in the special case of linear Hamiltonian torus actions. In particular, we show that the Koszul complex on the moment map of an effective linear Hamiltonian torus action is acyclic. We rephrase the nonpositivity condition of Arms, Gotay and Jennings for linear Hamiltonian torus actions. It follows that reduced spaces of such actions admit continuous star products.
In this paper, we prove that the two well-known natural normalizations of Hamiltonian functions on the symplectic manifold $(M,ω)$ canonically relates the action spectra of different normalized Hamiltonians on {\it arbitrary} symplectic manifolds $(M,ω)$. The natural class of normalized Hamiltonians consists of those whose mean value is zero for the closed manifold, and those which are compactly supported in $\text{Int} M$ for the open manifold. We also study the effect of the action spectrum under the $π_1$ of Hamiltonian diffeomorphism group. This forms a foundational basis for our study of …
We outline the construction of invariants of Hamiltonian group actions on symplectic manifolds. These invariants can be viewed as an equivariant version of Gromov-Witten invariants. They are derived from solutions of a PDE involving the Cauchy-Riemann operator, the curvature of a connection, and the moment map.
Kähler complexity one Hamiltonian $T$-manifolds have trivial paintings
本文研究Kähler复杂度为一的辛$T$-流形，证明其具有平凡绘画，并展示此类流形在辛同胚下的分类条件。
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 $ϕ$.
Let G be a connected reductive group over an algebraically closed field K of characteristic 0, X an affine symplectic variety equipped with a Hamiltonian action of G. Further, let * be a G-invariant Fedosov star-product on X such that the Hamiltonian action is quantized. We establish an isomorphism between the center of the associative algebra K[X][[h]]^G and the algebra of formal power series with coefficients in the Poisson center of K[X]^G.
We develop differential and symplectic geometry of differentiable Deligne-Mumford stacks (orbifolds) including Hamiltonian group actions and symplectic reduction. As an application we construct new examples of symplectic toric DM stacks as symplectic quotients of C^NxBG, where G is a finite non-abelian group.
The convexity package for Hamiltonian actions on conformal symplectic manifolds
本文证明了共形辛流形上Lee型哈密顿作用的动量映射凸性定理，建立了共形辛几何中Kirwan凸性定理的类比。
Hamiltonian actions on 0-shifted cosymplectic groupoids
本文研究了微分堆栈上的0-移位余辛结构，发展了余辛作用的动量映射理论，并给出了Morse-Bott李群状映射的例子。
We construct all possible Hamiltonian torus actions for which all the non-empty reduced spaces are two dimensional (and not single points) and the manifold is connected and compact, or, more generally, the moment map is proper as a map to a convex set. This construction completes the classification of tall complexity one spaces.
Stacky Hamiltonian actions and symplectic reduction
本文引入了étale李群叠在étale辛叠上的哈密顿作用概念，并在该背景下证明了Kirwan凸性定理、Meyer-Marsden-Weinstein辛约化定理及Duistermaat-Heckman定理的版本。
Suppose that a compact and connected Lie group $G$ acts on a complex Hodge manifold $M$ in a holomorphic and Hamiltonian manner, and that the action linearizes to a positive holomorphic line bundle $A$ on $M$. Then there is an induced unitary representation on the associated Hardy space and, if the moment map of the action is nowhere vanishing, the corresponding isotypical components are all finite dimensional. We study the asymptotic concentration behavior of the corresponding equivariant Szegö kernels near certain loci defined by the moment map.
A trivial bundle of regular connected invariant manifolds of a completely integrable Hamiltonian system can be provided with action-angle coordinates.
We show that the actions and indexes of fixed points of a Hamiltonian diffeomorphism with finitely many periodic points must satisfy certain relations, provided that the quantum cohomology of the ambient manifold meets an algebraic requirement satisfied for projective spaces, Grassmannians and many other manifolds. We also refine a previous result on the Conley conjecture for negative monotone symplectic manifolds, due to the second and third authors, and show that a Hamiltonian diffeomorphism of such a manifold must have simple periodic orbits of arbitrarily large period whenever its fixed po…
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.
We construct Hermitian representations of Lie algebroids and associated unitary representations of Lie groupoids by a geometric quantization procedure. For this purpose we introduce a new notion of Hamiltonian Lie algebroid actions. The first step of our procedure consists of the construction of a prequantization line bundle. Next, we discuss a version of Kähler quantization suitable for this setting. We proceed by defining a Marsden-Weinstein quotient for our setting and prove a ``quantization commutes with reduction'' theorem. We explain how our geometric quantization procedure relates to a …
For a Hamiltonian, proper and free action of a Lie group $G$ on a Dirac manifold $(M,L)$, with a regular moment map $μ:M\to \mathfrak{g}^*$, the manifolds $M/G$, $μ^{-1}(0)$ and $μ^{-1}(0)/G$ all have natural induced Dirac structures. If $(M,L)$ is an integrable Dirac structure, we show that $M/G$ is always integrable, but $μ^{-1}(0)$ and $μ^{-1}(0)/G$ may fail to be integrable, and we describe the obstructions to their integrability.
A Hamiltonian action of a complex torus on a symplectic complex manifold is said to be {\it multiplicity free} if a general orbit is a lagrangian submanifold. To any multiplicity free Hamiltonian action of a complex torus $T\cong (\C^\times)^n$ on a Stein manifold $X$ we assign a certain 5-tuple consisting of a Stein manifold $Y$, an étale map $Y\to \t^*$, a set of divisors on $Y$ and elements of $H^2(Y,\Z)^{\oplus n}, H^2(Y,\C)$. We show that $X$ is uniquely determined by this invariants. Furthermore, we describe all 5-tuples arising in this way.
The main theme of this paper is the connection between the existence of infinitely many periodic orbits for a Hamiltonian system and the behavior of its action or index spectrum under iterations. We use the action and index spectra to show that any Hamiltonian diffeomorphism of a closed, rational manifold with zero first Chern class has infinitely many periodic orbits and that, for a general rational manifold, the number of geometrically distinct periodic orbits is bounded from below by the ratio of the minimal Chern number and half of the dimension. These generalizations of the Conley conject…
Hamiltonian symplectic actions of tori on compact symplectic manifolds have been extensively studied in the past thirty years, and a number of classifications have been achieved, for instance in the case that the acting torus is $n$-dimensional and the symplectic manifold is $2n$-dimensional. In this case the $n$-dimensional orbits are Lagrangian, so it is natural to wonder whether there are interesting classes of symplectic actions with Lagrangian orbits, and that are not Hamiltonian. It turns out that there are many such classes which contain for example the Kodaira variety, and which can be…
For symplectic group actions which are not Hamiltonian there are two ways to define reduction. Firstly using the cylinder-valued momentum map and secondly lifting the action to any Hamiltonian cover (such as the universal cover), and then performing symplectic reduction in the usual way. We show that provided the action is free and proper, and the Hamiltonian holonomy associated to the action is closed, the natural projection from the latter to the former is a symplectic cover. At the same time we give a classification of all Hamiltonian covers of a given symplectic group action. The main prop…
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