Symplectic singularities from the Poisson point of view

TL;DR

This paper studies symplectic singularities through Poisson geometry, showing they can be decomposed into smooth symplectic pieces and analyzing their local structure and symmetries.

Contribution

It introduces a Poisson-theoretic approach to symplectic singularities, proving stratification, local product structure, and existence of dilating actions.

Findings

Symplectic singularities admit a finite stratification with smooth symplectic strata.

Locally, singularities decompose into a product of a stratum and a transversal slice.

Transversal slices are themselves symplectic singularities with a dilating $C^*$-action.

Abstract

We consider symplectic singularities in the sense of A. Beauville as examples of Poisson schemes. Using Poisson methods, we prove that a symplectic singularity admits a finite stratification with smooth symplectic strata. We also prove that in the formal neighborhood of a closed point in some stratum, the singularity is a product of the stratum and a transversal slice. The transversal slice is also a symplectic singularity, and the product decomposition is compatible with natural Poisson structures. Moreover, we prove that the transversal slice admits a $C^*$-action dilating the symplectic form.