In [GM], a family of parabolic Higgs bundles on $CP^1$ has been constructed and identified with a moduli space of hyperpolygons. Our aim here is to give a canonical alternative construction of this family. This enables us to compute the Higgs symplectic form for this family and show that the isomorphism of [GM] is a symplectomorphism.
The aim of this paper is to describe how to obtain residue-type formulas for push-forwards in equivariant cohomology, using the Jeffrey-Kirwan nonabelian localization theorem and the related result of Guillemin and Kalkman. This paper relates the description of equivariant push-forwards for homogeneous spaces of classical Lie groups with the action of the maximal torus with symplectic geometry, using the example of projective spaces and classical Grassmannians.
In this paper, we study the symplectic cohomologies and symplectic harmonic forms which introduced by Tseng and Yau. Based on this, we get if $(M^{2n},ω)$ is a compact symplectic parabolic manifold which satisfies the hard Lefschetz property, then its Euler number satisfies the inequality $(-1)^nχ(M)\geq 0$.
Fix a compact 4-dimensional manifold with self-dual 2nd Betti number one and with a given symplectic form. This article proves the following: The Frechet space of tamed almost complex structures as defined by the given symplectic form has an open and dense subset whose complex structures are compatible with respect to a symplectic form that is cohomologous to the given one. The theorem is proved by constructing the new symplectic form by integrating over a space of currents that are defined by pseudo-holomorphic curves.
Given an ascending sequence of weak symplectic Banach manifolds on which the Darboux theorem is true, we can ask about conditions under which the Darboux Theorem is also true on the direct limit. We will show in general, without very strong conditions, the answer is negative. In particular we give an example of an ascending weak symplectic Banach manifolds on which the Darboux Theorem is true but not on the direct limit. In a second part, we illustrate this discussion in the context of an ascending sequences of Sobolev manifolds of loops in symplectic finite dimensional manifolds. This context…
In this paper we study whether symplectic toric manifolds are symplectically cohomologically rigid. Here we say that symplectic cohomological rigidity holds for some family of symplectic manifolds if the members of that family can be distinguished by their integral cohomology rings and the cohomology classes of their symplectic forms. We show how toric degenerations can be used to produce the symplectomorphisms necessary to answer this question. As a consequence we prove that symplectic cohomological rigidity holds for the family of symplectic Bott manifolds with rational symplectic form whose…
In this note we study the behavior of symplectic cohomology groups under symplectic deformations. Moreover, we show that for compact almost-Kähler manifolds $(X,J,g,ω)$ with $J$ $\mathcal{C}^\infty$-pure and full the space of de Rham harmonic forms is contained in the space of symplectic-Bott-Chern harmonic forms. Furthermore, we prove that the second non-HLC degree measures the gap between the de Rham and the symplectic-Bott-Chern harmonic forms.
Let $(S^2,ω)$ be a symplectic sphere, and let $τ\colon S^2 \to S^2$ be an anti-symplectic involution of $(S^2,ω)$. We consider the product $(S^2,ω) \times (S^2,ω)$ endowed with the anti-symplectic involution $τ\times τ$, and study the space of monotone anti-invariant symplectic forms on this four-manifold. We show that this space is disconnected. In addition, during the course of the proof, we produce a diffeomorphism of the grassmannian (2,4) which induces the identity map on all homology and homotopy groups, but which is not homotopic to the identity.
In this paper, we give a new method to construct a compact symplectic manifold which does not satisfy the hard Lefschetz property. Using our method, we construct a simply connected compact Kähler manifold $(M,J,ω)$ and a symplectic form $σ$ on $M$ which does not satisfy the hard Lefschetz property, but is symplectically deformation equivalent to the Kähler form $ω$. As a consequence, we can give an answer to the question posed by Khesin and McDuff as follows. According to symplectic Hodge theory, any symplectic form $ω$ on a smooth manifold $M$ defines \textit{symplectic harmonic forms} on $M$…
We construct an infinite family of odd-symplectic forms (also known as Hamiltonian structures) on the 3-sphere that do not admit a symplectic cobordism to the standard contact structure on the 3-sphere. This answers in the negative a question raised by Joel Fish motivated by the search for minimal characteristic flows.
Symplectic forms on the space of embedded symplectic surfaces and their reductions
Let (M,ω) be a symplectic manifold , and (Σ,σ) a closed connected symplectic 2-manifold. We construct a weakly symplectic form {ω^{D}}_{(Σ, σ)} on the space of immersions Σ \to M that is a special case of Donaldson's form. We show that the restriction of {ω^{D}}_{(Σ,σ)} to any orbit of the group of Hamiltonian symplectomorphisms through a symplectic embedding (Σ,σ) \to (M,ω) descends to a weakly symplectic form ω^ D_{\red} on the quotient by Sympl(Σ,σ), and that the obtained symplectic space is a symplectic quotient of the subspace of symplectic embeddings S_{e} (Σ,σ) with respect to the Sympl…
We prove that the generalized symplectic capacities recognize objects in symplectic categories whose objects are of the form $(M, ω)$, such that $M$ is a compact and 1-connected manifold, $ω$ is an exact symplectic form on $M$, and there exists a boundary component of $M$ with negative helicity. The set of generalized symplectic capacities is thus a complete invariant for such categories. This answers a question by Cieliebak, Hofer, Latschev, and Schlenk. It appears to be the first result concerning this question, except for recognition results for manifolds of dimension 2, ellipsoids, and pol…
Let $M$ be a symplectic manifold and $G$ a connected, compact Lie group acting on $M$ in a Hamiltonian way. In this paper, we study the equivariant cohomology of $M$ represented by basic differential forms, and relate it to the cohomology of the Marsden-Weinstein reduced space via certain residue formulae using resolution of singularities and the stationary phase principle. In case that $ M $ is a compact, symplectic manifold or the co-tangent bundle of a $G$-manifold, similar residue formulae were derived by Jeffrey, Kirwan et al. for general equivariantly closed forms and by Ramacher for bas…
In this paper, we prove a splitting formula for the Maslov-type indices of symplectic paths induced by the splitting of the nullity in weak symplectic Hilbert space. Then we give a direct proof of iteration formulae for the Maslov-type indices of symplectic paths.
Using dynamical stability of symplectic curvature flow, we show that on a compact Calabi-Yau manifold, any small symplectic deformation of a Kähler form remains Kähler.
We deform the moment map picture on the space of symplectic connections on a symplectic manifold. To do that, we study a vector bundle of Fedosov star product algebras on the space of symplectic connections. We describe a natural formal connection on this bundle adapted to the star product algebras on the fibers. We study its curvature and show the star product trace of the curvature is a formal symplectic form on the space of symplectic connections. The action of Hamiltonian diffeomorphisms on symplectic connections preserves the formal symplectic structure and we show the star product trace …
This paper concerns the action of linear symplectomorphisms on linear symplectic forms by conjugation in even dimensions. We prove that pfaffian and $-\frac{1}{2}\operatorname{tr}(JA)$ (sum function) of $A$ are invariants on the action. We use these invariants to provide a complete description of the orbit space in dimension four. In addition, we investigate the geometric shapes of the individual orbits in dimension four. In symplectic geometry, our classification result in dimension four provides a necessary condition for two symplectic forms on $\mathbb{R}^{4}$ to be intertwined by symplecto…
A famous result of Jurgen Moser states that a symplectic form on a compact manifold cannot be deformed within its cohomology class to an inequivalent symplectic form. It is well known that this does not hold in general for noncompact symplectic manifolds. The notion of Eliashberg-Gromov convex ends provides a natural restricted setting for the study of analogs of Moser's symplectic stability result in the noncompact case, and this has been significantly developed in work of Cieliebak-Eliashberg. Retaining the end structure on the underlying smooth manifold, but dropping the convexity and compl…
This article details a construction of symplectic foliations on 3-dimensional orientable riemannian manifolds from harmonic forms; and how it suggests a topological approach to Poisson's equation and newtonian gravity.
Given a projective sequence of Banach bundles, each one provided with a of weak symplectic form, we look for conditions under which, the corresponding sequence of weak symplectic forms gives rise to weak symplectic form on the projective limit bundle. Then we apply this results to the tangent bundle of a projective limit of Banach manifolds. This naturally leads to ask about conditions under which the Darboux Theorem is also true on the projective limit of Banach manifolds. We will give some necessary and some sufficient conditions so that such a result is true. Then we discuss why, in general…
We launch the study of the tropicalization of the symplectic Grassmannian, that is, the space of all linear subspaces that are isotropic with respect to a fixed symplectic form. We formulate tropical analogues of several equivalent characterizations of the symplectic Grassmannian and determine all implications between them. In the process, we show that the Plücker and symplectic relations form a tropical basis if and only if the rank is at most 2. We provide plenty of examples that show that several features of the symplectic Grassmannian do not hold after tropicalizing. We show exactly when d…
In this expository note, we explain the so-called Van den Bergh functor, which enables the formalization of the Kontsevich-Rosenberg principle, whereby a structure on an associative algebra has geometric meaning if it induces standard geometric structures on its representation spaces. Crawley-Boevey, Etingof and Ginzburg proved that bi-symplectic forms satisfy this principle; this implies that bi-symplectic algebras can be regarded as noncommutative symplectic manifolds. In this note, we use the Van den Bergh functor to give an alternative proof.
