Appears in collections : Geometry of singular spaces and maps / Géométrie des espaces et applications singuliers, Exposés de recherche

We study germs of singular varieties in a symplectic space. We introduce the algebraic restrictions of differential forms to singular varieties and prove the generalization of Darboux-Givental' theorem from smooth submanifolds to arbitrary quasi-homogeneous varieties in a symplectic space. Using algebraic restrictions we introduce new symplectic invariants and explain their geometric meaning. We show that a quasi-homogeneous variety $N$ is contained in a non-singular Lagrangian submanifold if and only if the algebraic restriction of the symplectic form to $N$ vanishes. The method of algebraic restriction is a powerful tool for various classification problems in a symplectic space. We illustrate this by the construction of a complete system of invariants in the problem of classifying singularities of immersed $k$-dimensional submanifolds of a symplectic 2n-manifold at a generic double point.

Keywords: symplectic manifolds - symplectic multiplicity and other invariants - Darboux-Givental's theorem - quasi-homogeneous singularities - singularities of planar curves