We present a collection of examples borrowed from celestial mechanics and projective dynamics. In these examples symplectic structures with singularities arise naturally from regularization transformations, Appell’s transformation or classical changes like McGehee coordinates, which end up blowing up the symplectic structure or lowering its rank at certain points. The resulting geometrical stru...

A symplectic semitoric manifold is a symplectic 4-manifold endowed with a Hamiltonian (S1×R)-action satisfying certain conditions. The goal of this paper is to construct a new symplectic invariant of symplectic semitoric manifolds, the helix, and give applications. The helix is a symplectic analogue of the fan of a nonsingular complete toric variety in algebraic geometry, that takes into accoun...

The time t maps of Hamiltonian flows are symplectic. The order n Taylor series approximation with respect to initial conditions of such a map is symplectic through terms of order n. Given an order n Hamiltonian symplectic map, there are a variety of procedures, called symplectification methods, which produce exactly symplectic maps with Taylor series that agree with the initial Taylor map throu...

Generalized complex structures were introduced as a common format for discussing both symplectic and complex manifolds, but the most interesting examples are hybrid objects – part symplectic and part complex. One such class of examples consists of holomorphic Poisson surfaces, but in [5],[6] Cavalcanti and Gualtieri also construct generalized complex 4-manifolds with similar features which are ...

We develop an approach to affine symplectic invariant geometry of Lagrangian surfaces by the method of moving frames. The fundamental invariants of elliptic Lagrangian immersions in affine symplectic four-space are derived together with their integrability equations. The invariant setup is applied to discuss the question of symplectic applicability for elliptic Lagrangian immersions. Explicit e...

An invariant for symplectic involutions on central simple algebras of degree divisible by 4 over fields of characteristic different from 2 is defined on the basis of Rost’s cohomological invariant of degree 3 for torsors under symplectic groups. We relate this invariant to trace forms and show how its triviality yields a decomposability criterion for algebras with symplectic involution.

We establish the uniqueness of the symplectic 4-manifolds which admit low degree symplectic embeddings into CP 3. We also discuss the uniqueness of the fundamental group of the complement of such embeddings into arbitrary symplectic 6-manifolds. 2000 Mathematics Subject Classification: Primary 57R17; Secondary 57R95.

The paradigmatic result in symplectic toric geometry is the paper of Delzant that classifies compact connected symplectic manifolds with effective completely integrable torus actions, the so called (compact) symplectic toric manifolds. The moment map induces an embedding of the quotient of the manifold by the torus action into the dual of the Lie algebra of the torus; its image is a simple unim...

The natural sum operation for symplectic manifolds is defined by gluing along codimension two submanifolds. Specifically, let X be a symplectic 2n-manifold with a symplectic (2n − 2)submanifold V . Given a similar pair (Y, V ) with a symplectic identification V = V and a complex anti-linear isomorphism between the normal bundles of V and V , we can form the symplectic sum Z = X# V=V Y . This no...

The space of symplectic connections on a symplectic manifold is a symplectic affine space. M. Cahen and S. Gutt showed that the action of the group of Hamiltonian diffeomorphisms on this space is Hamiltonian and calculated the moment map. This is analogous to, but distinct from, the action of Hamiltonian diffeomorphisms on the space of compatible almost complex structures that motivates study o...