For any compact connected Lie group $G$, we study the Hamiltonian sum of two compact Hamiltonian group $G$-manifolds $(X^+,ω^+,μ^+)$ and $(X^-,ω^-,μ^-)$ with a common codimension 2 Hamiltonian submanifold $Z$ of the opposite equivariant Euler classes of the normal bundles. We establish that the symplectic reduction of the Hamiltonian sum agrees with the symplectic sum of the reduced symplectic manifolds. We also compare the equivariant first Chern class of the Hamiltonian sum with the equivariant first Chern classes of $X^\pm$.
We show that a generic Hamiltonian diffeomorphism on a closed symplectic manifold which is symplectically aspherical has at least the stable Morse number of fixed points - this is in line with a conjecture by Arnold.
Let the circle act effectively in a Hamiltonian fashion on a compact symplectic manifold $(M, ω)$. Assume that the fixed point set $M^{S^1}$ has exactly two components, $X$ and $Y$, and that $\dim(X) + \dim(Y) +2 = \dim(M)$. We first show that $X$, $Y$ and $M$ are simply connected. Then we show that, up to $S^1$-equivariant diffeomorphism, there are finitely many such manifolds in each dimension. Moreover, we show that in low dimensions, the manifold is unique in a certain category. We use techniques from both areas of symplectic geometry and geometric topology.
A simple Hamiltonian manifold is a closed connected symplectic manifold equipped with a Hamiltonian action of a torus T with moment map Phi: M-->t^*, such that the fixed set M^T has exactly two connected components, denoted M_0 and M_1. We study the differential and symplectic geometry of simple Hamiltonian manifolds, including a large number of examples.
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 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.
This paper is devoted to the study of symplectic manifolds and their connection with Hamiltonian dynamical systems. We review some properties and operations on these manifolds and see how they intervene when studying the complete integrability of these systems, with detailed proofs. Several explicit calculations for which references are not immediately available are given. These results are exemplified by applications to some Hamiltonian dynamical systems.
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.
We show that the quotient associated to a quasi-Hamiltonian space has a symplectic structure even when 1 is not a regular value of the momentum map: it is a disjoint union of symplectic manifolds of possibly different dimensions, which generalizes a result of Alekseev, Malkin and Meinrenken. We illustrate this theorem with the example of representation spaces of surface groups. As an intermediary step, we show that the isotropy submanifolds of a quasi-Hamiltonian space are quasi-Hamiltonian spaces themselves.
This paper defines symplectic scale manifolds based on Hofer-Wysocki-Zehnder's scale calculus. We introduce Hamiltonian vector fields and flows on these by narrowing down sc-smoothness to what we denote by strong sc-smoothness, a concept which effectively formalizes the desired smoothness properties for Hamiltonian functions. We show the concept to be invariant under sc-smooth symplectomorphisms, whence it is compatible with Hofer's scale manifolds. We develop and verify the theory at the hand of the free Schrödinger equation.
In this work, we prove a compactness theorem on the space of all Hamiltonian stationay Lagrangian submanifolds in a compact symplectic manifold with uniform bounds on area and total extrinsic curvature.
In this work, we show that an autonomous dynamical system defined by a nonvanishing vector field on an orientable three-dimensional manifold is globally bi-Hamiltonian if and only if the first Chern class of the normal bundle of the given vector field vanishes. Furthermore, the bi-Hamiltonian structure is globally compatible if and only if the Bott class of the complex codimension one foliation defined by the given vector field vanishes.
Let K be a connected Lie group and M a Hamiltonian K-manifold. In this paper, we introduce the notion of convexity of M. It implies that the momentum image is convex, the moment map has connected fibers, and the total moment map is open onto its image. Conversely, the three properties above imply convexity. We show that most Hamiltonian manifolds occuring "in nature" are convex (e.g., if M is compact, complex algebraic, or a cotangent bundle). Moreover, every Hamiltonian manifold is locally convex. This is an expanded version of section 2 of my paper dg-ga/9712010 on Weyl groups of Hamiltonian…
A (quasi-)Hamiltonian manifold is called multiplicity free if all of its symplectic reductions are 0-dimensional. In this paper, we classify multiplicity free Hamiltonian actions for (twisted) loop groups or, equivalently, multiplicity free (twisted) quasi-Hamiltonian manifolds for simply connected compact Lie groups. As a result we recover old and find new examples of these structures.
We lift a Hamiltonian loop on a symplectic manifold to a Hamiltonian loop on the symplectic one-point blow up of a symplectic manifold. Then we use Weinstein's morphism to show that the lifted Hamiltonian loop has infinite order on the fundamental group of the group of Hamiltonian diffeomorphisms of the blown up manifold.
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.
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 …
We prove that $π_1(\text{Ham}(M))$ contains an infinite cyclic subgroup, where $\text{Ham}(M)$ is the Hamiltonian group of the one point blow up of ${\Bbb C}P^3$. We give a sufficient condition for the group $π_1(\text{Ham}(M))$ to contain an infinite cyclic subgroup, when $M$ is a general toric manifold.
Let $(M, ω)$ be a connected, compact symplectic manifold equipped with a Hamiltonian $S^1$ action. We prove that, as fundamental groups of topological spaces, $π_1(M)=π_1(\hbox{minimum})=π_1(\hbox{maximum})=π_1(M_{red})$, where $M_{red}$ is the symplectic quotient at any value in the image of the moment map $ϕ$.
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 study analysis over infinite dimensional manifolds consisted by sequences of almost Kähler manifolds. In particular we develop moduli theory of pseudo holomorphic curves into the spaces with high symmetry. As applications, we study Hamiltonian dynamics over the infinite dimensional manifolds, and induce some dynamical properties of Hamiltonian diffeomorphisms on such spaces.
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 $ϕ$.
In this paper we introduce the concept of characteristic number that are proven to be useful in the study of the combinatorics of graph cohomology. We claim that it is a good combinatorial counterpart for geometric Betti numbers. We then use this concept and tools built along the way to study Hamiltonian GKM manifolds whose moment maps are in general position. We prove some connectivity properties of the their GKM graphs and show an upper bound of their second Betti numbers, which allows us to conclude that these manifolds, in the case of dimension 8 and 10, have non-decreasing even Betti numb…
In this paper the Gromov-Witten invariants on a class of noncompact symplectic manifolds are defined by combining Ruan-Tian's method with that of McDuff-Salamon. The main point of the arguments is to introduce a method dealing with the transversality problems in the case of noncompact manifolds. Moreover, the techniques are also used to study the topological rigidity of Hamiltonian loops with compact support on a class of noncompact symplectic manifolds.
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 classify all momentum polytopes of rank one for multiplicity free quasi-Hamiltonian $K$-manifolds for simple and simply connected Lie groups $K$ by using the methods developed in a recent paper by F. Knop. This leads to lots of new concrete examples of multiplicity free quasi-Hamiltonian manifolds or equivalently, Hamiltonian loop group actions.
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.
We compute the sheaf of automorphisms of a multiplicity free Hamiltonian manifold over its momentum polytope and show that its higher cohomology groups vanish. Together with a theorem of Losev, arXiv:math/0612561, this implies a conjecture of Delzant: a compact multiplicity free Hamiltonian manifold is uniquely determined by its momentum polytope and its principal isotropy group.
In this paper we describe a method to establish when a symplectic manifold $M$ with semi-free Hamiltonian $S^{1}$-action is unique up to isomorphism (equivariant symplectomorphism). This will rely on a study of the symplectic topology of the reduced spaces. We prove that if the reduced spaces satisfy a rigidity condition, then the manifold $M$ is uniquely determined by fixed point data. In particular we can prove that there is a unique family up to isomorphism of 6-dimensional symplectic manifolds with semi-free $S^1$-action and isolated fixed points.
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.
We study the dynamics of Hamiltonian diffeomorphisms on convex symplectic manifolds. To this end we first establish the Piunikhin-Salamon-Schwarz isomorphism between the Floer homology and the Morse homology of such a manifold, and then use this isomorphism to construct a biinvariant metric on the group of compactly supported Hamiltonian diffeomorphisms analogous to the metrics constructed by Viterbo and Schwarz. These tools are then applied to prove and reprove results in Hamiltonian dynamics. Our applications comprise a uniform lower estimate for the slow entropy of a compactly supported Ham…
The main result of this note is that every closed Hamiltonian S^1 manifold is uniruled, i.e. it has a nonzero Gromov--Witten invariant one of whose constraints is a point. The proof uses the Seidel representation of π_1 of the Hamiltonian group in the small quantum homology of M as well as the blow up technique recently introduced by Hu, Li and Ruan. It applies more generally to manifolds that have a loop of Hamiltonian symplectomorphisms with a nondegenerate fixed maximum. Some consequences for Hofer geometry are explored. An appendix discusses the structure of the quantum homology ring of un…
We construct absolute and relative versions of Hamiltonian Floer homology algebras for strongly semi-positive compact symplectic manifolds with convex boundary, where the ring structures are given by the appropriate versions of the pair-of-pants products. We establish the absolute and relative Piunikhin-Salamon-Schwarz isomorphisms between these Floer homology algebras and the corresponding absolute and relative quantum homology algebras. As a result, the absolute and relative analogues of the spectral invariants on the group of compactly supported Hamiltonian diffeomorphisms are defined.
In this paper we construct six-dimensional compact non-Kähler Hamiltonian circle manifolds which satisfy the strong Lefschetz property themselves but nevertheless have a non-Lefschetz symplectic quotient. This provides the first known counter examples to the question whether the strong Lefschetz property descends to the symplectic quotient. We also give examples of Hamiltonian strong Lefschetz circle manifolds which have a non-Lefschetz fixed point submanifold. In addition, we establish a sufficient and necessary condition for a finitely presentable group to be the fundamental group of a stron…
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.
A classical theorem of Frankel for compact Kähler manifolds states that a Kähler S^1-action is Hamiltonian if and only if it has fixed points. We prove a metatheorem which says that when Hodge theory holds on non-compact manifolds, then Frankel's theorem still holds. Finally, we present several concrete situations in which the assumptions of the metatheorem hold.
We study Hamiltonian diffeomorphisms of closed symplectic manifolds with non-contractible periodic orbits. In a variety of settings, we show that the presence of one non-contractible periodic orbit of a Hamiltonian diffeomorphism of a closed toroidally monotone or toroidally negative monotone symplectic manifold implies the existence of infinitely many non-contractible periodic orbits in a specific collection of free homotopy classes. The main new ingredient in the proofs of these results is a filtration of Floer homology by the so-called augmented action. This action is independent of capping…
We address the problem of computing the fundamental group of a symplectic $S^1$-manifold for non-Hamiltonian actions on compact manifolds, and for Hamiltonian actions on non-compact manifolds with a proper moment map. We generalize known results for compact manifolds equipped with a Hamiltonian $S^1$-action. Several examples are presented to illustrate our main results.
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 prove that the group of Hamiltonian automorphisms of a symplectic 4-manifold contains only finitely many conjugacy classes of maximal compact tori with respect to the action of the full symplectomorphism group. We also extend to rational and ruled manifolds a result of Kedra which asserts that, if $M$ is a simply connected symplectic 4-manifold with $b_{2}\geq 3$, and if $\widetilde{M}_δ$ denotes a blow-up of $M$ of small enough capacity $δ$, then the rational cohomology algebra of the Hamiltonian group of $\widetilde{M}_δ)$ is not finitely generated. Both results are based on the fact that…
Let $(M,ω)$ be an almost symplectic manifold ($ω$ is a non degenerate, not closed, 2-form). We say that a vector field $X$ of $M$ is locally Hamiltonian if $L_Xω=0,d(i(X)ω)=0$, and it is Hamiltonian if, furthermore, the 1-form $i(X)ω$ is exact. Such vector fields were considered in a 2007 paper by F. Fasso and N. Sansonetto, under the name of strongly Hamiltonian, and a corresponding action-angle theorem was proven. Almost symplectic manifolds may have few, non-zero, Hamiltonian vector fields or even none. Therefore, it is important to have examples and it is our aim to provide such examples h…
A covariant Hamiltonian description of Palatini's gravity on manifolds with boundary is presented. Palatini's gravity appears as a gauge theory satisfying a constraint in a certain topological limit. This approach allows the consideration of non-trivial topological situations. The multisymplectic framework for first-order covariant Hamiltonian field theories on manifolds with boundary, developed in [Ib15], enables analysis of the system at the boundary. The reduced phase space of the system is determined to be a symplectic manifold with a distinguished isotropic submanifold corresponding to th…
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 study the topology of Hamiltonian-minimal Lagrangian submanifolds N in C^m constructed from intersections of real quadrics in a work of the first author. This construction is linked via an embedding criterion to the well-known Delzant construction of Hamiltonian toric manifolds. We establish the following topological properties of N: every N embeds as a submanifold in the corresponding moment-angle manifold Z, and every N is the total space of two different fibrations, one over the torus T^{m-n} with fibre a real moment-angle manifold R, and another over a quotient of R by a finite group wi…
We prove that when Hodge theory survives on non-compact symplectic manifolds, a compact symplectic Lie group action having fixed points is necessarily Hamiltonian, provided the associated almost complex structure preserves the space of harmonic one-forms. For example, this is the case for complete Kähler manifolds for which the symplectic form has an appropriate decay at infinity. This extends a classical theorem of Frankel for compact Kähler manifolds to complete non-compact Kähler manifolds.
