MODULAR CLASS OF EVEN SYMPLECTIC MANIFOLDS (original) (raw)

The modular class of a regular Poisson manifold and the Reeb class of its symplectic foliation

Comptes Rendus Mathematique, 2003

We show that, for any regular Poisson manifold, there is an injective natural linear map from the first leafwise cohomology space into the first Poisson cohomology space which maps the Reeb class of the symplectic foliation to the modular class of the Poisson manifold. The Riemannian interpretation of those classes will permit us to show that a regular Poisson manifold whose symplectic foliation is of codimension one is unimodular if and only if its symplectic foliation is Riemannian foliation. It permit us also to construct examples of unimodular Poisson manifolds and other which are not unimodular. Finally, we prove that the first leafwise cohomology space is an invariant of Morita equivalence.

The modular class of a regular Poisson manifold and the Reeb invariant of its symplectic foliation

arXiv (Cornell University), 2002

We show that, for any regular Poisson manifold, there is an injective natural linear map from the first leafwise cohomology space into the first Poisson cohomology space which maps the Reeb class of the symplectic foliation to the modular class of the Poisson manifold. The Riemannian interpretation of those classes will permit us to show that a regular Poisson manifold whose symplectic foliation is of codimension one is unimodular if and only if its symplectic foliation is Riemannian foliation. It permit us also to construct examples of unimodular Poisson manifolds and other which are not unimodular. Finally, we prove that the first leafwise cohomology space is an invariant of Morita equivalence.

SG ] 1 0 M ay 2 01 7 An Invitation to Singular Symplectic Geometry

2017

In this paper we analyze in detail a collection of motivating examples to consider bsymplectic forms and folded-type symplectic structures. In particular, we provide models in Celestial Mechanics for every b-symplectic structure. At the end of the paper, we introduce the odd-dimensional analogue to b-symplectic manifolds: b-contact manifolds.

An invitation to singular symplectic geometry

International Journal of Geometric Methods in Modern Physics, 2018

In this paper, we analyze in detail a collection of motivating examples to consider [Formula: see text]-symplectic forms and folded-type symplectic structures. In particular, we provide models in Celestial Mechanics for every [Formula: see text]-symplectic structure. At the end of the paper, we introduce the odd-dimensional analogue to [Formula: see text]-symplectic manifolds: [Formula: see text]-contact manifolds.

A characterization of graded symplectic structures

Differential Geometry and its Applications, 1992

We give a characterization of graded symplectic forms by studying the module of derivations of a graded sheaf. When the graded sheaf is the sheaf of differentiable forms on the underlying manifold M, we find canonical liftings from metrics on TM to odd symplectic forms, and from symplectic forms on M and metrics on TM to even symplectic forms. These graded symplectic forms give rise to canonical Poisson brackets on the graded manifold.

Symplectic forms on moduli spaces of flat connectionso n 2-manifolds

AMS/IP Studies in Advanced Mathematics

Let G be a compact connected semisimple Lie group. We extend the techniques of Weinstein [W] to give a construction in group cohomology of symplectic forms ω on 'twisted' moduli spaces of representations of the fundamental group π of a 2-manifold Σ (the smooth analogues of Hom(π 1 (Σ), G)/G) and on relative character varieties of fundamental groups of 2-manifolds. We extend this construction to exhibit a symplectic form on the extended moduli space [J1] (a Hamiltonian G-space from which these moduli spaces may be obtained by symplectic reduction), and compute the moment map for the action of G on the extended moduli space.