Abstract

Introduction

In symplectic geometry, it is often useful to consider the so-called Poisson bracket on the algebra of functions on a C ^∞ symplectic manifold M. The bracket determines, and is determined by, the symplectic form; however, many of the features of symplectic geometry are more conveniently described in terms of the Poisson bracket. When one turns to the study of symplectic manifolds in the holomorphic or algebro-geometric setting, one expects the Poisson bracket to be even more useful because of the following observation: the bracket is a purely algebraic structure, and it generalizes immediately to singular algebraic varieties and complex-analytic spaces.

The appropriate notion of singularities for symplectic algebraic varieties has been introduced recently by A. Beauville [A. Beauville, Symplectic singularities, Invent. Math. 139 (2000), 541–549.] and studied by Y. Namikawa [Y. Namikawa, Deformation theory of singular symplectic n-folds, Math. Ann. 319 (2001), 597–623.], [Y. Namikawa, Extension of 2-forms and symplectic varieties, J. reine angew. Math. 539 (2001), 123–147.]. However, the theory of singular symplectic algebraic varieties is in its starting stages; in particular, to the best of our knowledge, the Poisson methods have not been used yet. This is the goal of the present paper.