Symplectic Singularities from the Poisson Point of View 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 [B] and studied by Y. Namikawa [N1], [N2]. However, the theory of singular symplectic algebraic 1 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. Our results are twofold. Firstly, we prove a simple but useful structure theorem about symplectic varieties (Theorem 2.3) which says, roughly, that any symplectic variety admits a canonical stratification with a finite number of symplectic strata (in the Poisson language, a symplectic variety considered as a Poisson space has a finite number of symplectic leaves). In addition, we show that, locally near a stratum, the variety in question admits a nice decomposition into the product of the stratum itself and a transversal slice. Secondly, we study natural group actions on a symplectic variety and we prove that, again locally, a symplectic variety always admits a non-trivial action of the one-dimensional torus G m (Theorem 2.4). This is a rather strong restriction on the type of singularities a symplectic variety might have. Unfortunately, the paper is much more eclectic than we would like. Moreover , one of the two main results is seriously flawed: we were not able to show that the G m-action provided by Theorem 2.4 has positive weights. However, all the results has been known to the author for a couple of years now, and it seems that any improvement would require substantially new methods. Thus we have decided to publish the statements " as is ". Our approach, for better or for worse, is to try to use Poisson algebraic methods as much as possible, getting rid of actual geometry at an early stage. …