Symplectic vector space

About: Symplectic vector space is a research topic. Over the lifetime, 2048 publications have been published within this topic receiving 53456 citations.

Double points and the proper transform in symplectic geometry

Abstract: In his book, Partial Differential Relations, Gromov introduced the symplectic analogue of the complex analytic operations of blowing up and blowing down Gromov proposed, in Sect 344(D) of Partial Differential Relations, a program for resolving the singularities of symplectic immersions with symplectic crossings via blowing up, in exact analogue with the well known complex analytic technique The purpose of this note is to show that this program cannot work We show that there are symplectically immersed surfaces in symplectic 4-manifolds which do not have a symplectically embedded proper transform in any blow up of the four manifold

Moment map for the space of symplectic connections

Abstract: In this paper, we consider the space E(M,ω) of symplectic connections on a simply connected symplectic manifold (M,ω) and we build a moment map for the action of a group of symplectomorphisms of (M,ω) on E(M,ω). We also study the principal symbol of the equation defining Ricci-type symplectic connections in E(M,ω) and show that it has a kernel of dimension 1. ∗Universite Libre de Bruxelles, Campus Plaine CP 218, Bvd du Triomphe, B-1050 Brussels, Belgium ∗∗Universite de Metz, Departement de mathematiques, Ile du Saulcy, F-57045 Metz Cedex 01, France Email: mcahen@ulb.ac.be, sgutt@ulb.ac.be 1 Moment map for the space of symplectic connections A symplectic connection on a symplectic manifold (M,ω) is a torsionless linear connection ∇ on M for which the symplectic 2–form ω is parallel. To see the existence of such a connection, take ∇ to be any torsion free linear connection (for instance, the Levi Civita connection associated to a metric g on M). Consider the tensor N on M defined by ∇Xω(Y, Z) =: ω(N(X, Y ), Z) where X, Y, Z are vector fields on M (i.e. ∈ χ(M)). Since ω is closed, one has + XY Z ω(N(X, Y ), Z) = 0. Define ∇XY := ∇ 0 XY + 1 3 N(X, Y ) + 1 3 N(Y,X). Then ∇ is a symplectic connection on (M,ω). To see how (non)-unique is a symplectic connection, take ∇ symplectic; then any other linear connection reads

Symplectic subspaces of symplectic Grassmannians

Abstract: Let V be a non-degenerate symplectic space of dimension 2n over the field F and for a natural number l2 we classify subspaces S of C"l(F) where [email protected]?C"m","k(F).

Symplectic Lefschetz fibrations on S^1 x M^3

Abstract: In this paper we classify symplectic Lefschetz fibrations (with empty base locus) on a four-manifold which is the product of a three-manifold with a circle This result provides further evidence in support of the following conjecture regarding symplectic structures on such a four-manifold: if the product of a three-manifold with a circle admits a symplectic structure, then the three-manifold must fiber over a circle, and up to a self-diffeomorphism of the four-manifold, the symplectic structure is deformation equivalent to the canonical symplectic structure determined by the fibration of the three-manifold over the circle