Symplectic singularities of varieties: The method of algebraic restrictions

Abstract

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 dierential forms to singular varieties and show that this ghost is exactly the invariants of the algebraic restriction of the sym- plectic 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 geo- metric meaning. We prove 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 complete solutions of symplectic classification problem for the classical A, D, E singularities of curves, the S5 sin- gularity, and for regular union singularities.