Symplectic singularities of varieties: The method of algebraic restrictions

We study germs of singular varieties in a symplectic space. In [A1], V. Arnol’d discovered so called ‘‘ghost’’ symplectic invariants which are induced purely by singularity. We introduce algebraic restrictions of di¤erential forms to singular varieties and show that this ghost is exactly the invariants of the algebraic restriction of the symplectic form. This follows from our generalization of Darboux-Givental’ theorem from non-singular submanifolds to arbitrary quasi-homogeneous varieties in a symplectic space. Using algebraic restrictions we introduce new symplectic invariants and explain their geometric meaning. We prove that a quasi-homogeneous variety N is contained in a nonsingular 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 complete solutions of symplectic classification problem for the classical A, D, E singularities of curves, the S5 singularity, and for regular union singularities.