Symplectic Circle Actions Research Articles

Remarks on symplectic circle actions, torsion and loops

Remarks on symplectic circle actions, torsion and loops

Non-Hamiltonian Actions With Fewer Isolated Fixed Points

Abstract In an earlier paper, the second author resolved a question of McDuff by constructing a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly 32 fixed points. In this paper, we improve on this example by reducing the number of fixed points. More concretely, we construct a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly $2k$ fixed points for any $k \geq 5$.

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, it is shown that (1) if (M, ω) admits a Hamiltonian S1-action, then there exists a two-sphere S in M with positive symplectic area satisfying ‹c1(M, ω), [S]› > 0, and (2) if the action is non-Hamiltonian, then there exists an S1-invariant symplectic 2-torus T in (M, ω) such that ‹c1(M, ω), [T]› = 0. As applications, the authors give a very simple proof of the following well-known theorem which was proved by Atiyah-Bott, Lupton-Oprea, and Ono: Suppose that (M, ω) is a smooth closed symplectic manifold satisfying c1(M, ω) = λ·[ω] for some λ ∈ R and G is a compact connected Lie group acting effectively on M preserving ω. Then (1) if λ 0, then the G-action is Hamiltonian.

Non-Hamiltonian actions with isolated fixed points

We construct a non-Hamiltonian symplectic circle action on a closed, connected, six-dimensional symplectic manifold with exactly 32 fixed points.

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 to construct a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly 32 fixed points; this answers an open question in symplectic geometry. However, we also believe that the setting we work in is intrinsically interesting and elucidates the key role played by the following fact: the moment image of e t ⋅ x e^t \cdot x increases as t ∈ R t \in \mathbb {R} increases.

Poisson manifolds of compact types (PMCT 1)

Abstract This is the first in a series of papers dedicated to the study of Poisson manifolds of compact types (PMCTs). This notion encompasses several classes of Poisson manifolds defined via properties of their symplectic integrations. In this first paper we establish some fundamental properties and constructions of PMCTs. For instance, we show that their Poisson cohomology behaves very much like the de Rham cohomology of a compact manifold (Hodge decomposition, non-degenerate Poincaré duality pairing, etc.) and that the Moser trick can be adapted to PMCTs. More important, we find unexpected connections between PMCTs and symplectic topology: PMCTs are related with the theory of Lagrangian fibrations and we exhibit a construction of a non-trivial PMCT related to a classical question on the topology of the orbits of a free symplectic circle action. In subsequent papers, we will establish deep connections between PMCTs and integral affine geometry, Hamiltonian G-spaces, foliation theory, orbifolds, Lie theory and symplectic gerbes.

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 results for semi-free actions, and for certain circle actions on six-dimensional manifolds.

Circle actions in 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.

Circle-valued momentum maps for symplectic periodic flows

We give a detailed proof of the well-known classical fact that every symplectic circle action on a compact manifold admits a circle-valued momentum map relative to some symplectic form. This momentum map is Morse-Bott-Novikov and each connected component of the fixed point set has even index. These proofs do not seem to have appeared elsewhere.

The rigidity of Dolbeault-type operators and symplectic circle actions

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 determine 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 some of them to more general cases.

Fixed points of symplectic periodic flows

AbstractThe 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 $\frac {1}{2}\,{\dim M}+1$ 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 sum of the action weights at the point. We complement this result with less sharp lower bounds on the number of fixed points, under no assumptions; from a dynamical systems viewpoint, our results imply that there is no symplectic periodic flow with exactly one or two equilibrium points on a compact manifold of dimension at least eight.

Semifree symplectic circle actions on 4-orbifolds

A theorem of Tolman and Weitsman states that all symplectic semifree circle actions with isolated fixed points on compact symplectic manifolds must be Hamiltonian and have the same equivariant cohomology and Chern classes of ( C P 1 ) n (\mathbb {C}P^1)^n equipped with the standard diagonal circle action. In this paper, we show that the situation is much different when we consider compact symplectic orbifolds. Focusing on 4 4 -orbifolds with isolated cone singularities, we show that such actions, besides being Hamiltonian, can now be obtained from either S 2 × S 2 S^2\times S^2 or a weighted projective space, or a quotient of one of these spaces by a finite cyclic group, by a sequence of special weighted blow-ups at fixed points. In particular, they can have any number of fixed points.

A C-symplectic free 𝑆¹-manifold with contractible orbits and 𝐂𝐀𝐓=\frac12𝐃𝐈𝐌

An interesting question in symplectic geometry concerns whether or not a closed symplectic manifold can have a free symplectic circle action with orbits contractible in the manifold. Here we present a c-symplectic example, thus showing that the problem is truly geometric as opposed to topological. Furthermore, we see that our example is the only known example of a c-symplectic manifold having non-trivial fundamental group and Lusternik-Schnirelmann category precisely half its dimension.

Seiberg–Witten vanishing theorem for S1-manifolds with fixed points

We show that the Seiberg-Witten invariant is zero for all smooth 4-manifolds with b + >1 that admit circle actions having at least one fixed point. We also show that all symplectic 4-manifolds that admit (possibly nonsymplectic) circle actions with fixed points are rational or ruled, and thus admit a symplectic circle action.

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 [Kl], [KM], [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.

On semifree symplectic circle actions with isolated fixed points

Let M be a symplectic manifold, equipped with a semi-free symplectic circle action with a finite, non-empty 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 .

Homotopy theory and circle actions on symplectic manifolds

Introduction. Traditionally, soft techniques of algebraic topology have found much use in the world of hard geometry. In recent years, in particular, the subject of symplectic topology (see [McS]) has arisen from important and interesting connections between symplectic geometry and algebraic topology. In this paper, we will consider one application of homotopical methods to symplectic geometry. Namely, we shall discuss some aspects of the homotopy theory of circle actions on symplectic manifolds. Because this paper is meant to be accessible to both geometers and topologists, we shall try to review relevant ideas in homotopy theory and symplectic geometry as we go along. We also present some new results (e.g. see Theorem 2.12 and §5) which extend the methods reviewed earlier. This paper then serves two roles: as an exposition and survey of the homotopical approach to symplectic circle actions and as a first step to extending the approach beyond the symplectic world.

Birational equivalences of vortex moduli

We construct a finite-dimensional Kähler manifold with a holomorphic, symplectic circle action whose symplectic reduced spaces may be identified with the τ-vortex moduli spaces (or τ-stable pairs). The Morse theory of the circle action induces natural birational maps between the reduced spaces for different values of τ which in the case of rank two bundles can be canonically resolved in a sequence of blow-ups and blow-downs.

A Duistermaat-Heckman formula for symplectic circle actions

A Duistermaat-Heckman formula for symplectic circle actions Get access Jonathan Weitsman Jonathan Weitsman Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 1993, Issue 12, 1993, Pages 309–312, https://doi.org/10.1155/S1073792893000352 Published: 01 June 1993 Article history Published: 01 June 1993 Received: 28 September 1993