On Embedding Singular Poisson Spaces

TL;DR

This dissertation explores conditions for embedding singular Poisson spaces into smooth Poisson manifolds, revealing obstructions in certain cases and providing explicit examples of realizations.

Contribution

It introduces a new cohomological framework for understanding Poisson embeddings of singular spaces and proves nonembedding results for specific quotients.

Findings

V/ZZ_n (n odd) cannot be embedded as a Poisson subspace in certain dimensions

A refined Levi decomposition helps analyze Poisson extensions near singularities

Explicit Poisson realization of V/ZZ_3 in R^78 is constructed

Abstract

This dissertation investigates the problem of locally embedding singular Poisson spaces. Specifically, it seeks to understand when a singular symplectic quotient V/G of a symplectic vector space V by a group G \subseteq Sp_2n(R) is realizable as a Poisson subspace of some Poisson manifold (R^n,{.,.}). The local embedding problem is recast in the language of schemes and reinterpreted as a problem of extending the Poisson bracket to infinitesimal neighborhoods of an embedded singular space. Such extensions of a Poisson bracket near a singular point p of V/G are then related to the cohomology and representation theory of the cotangent Lie algebra at p. Using this framework, it is shown that the real 4-dimensional quotient V/\ZZ_n (n odd) is not realizable as a Poisson subspace of any (R^{2n+6},{.,.}), even though the underlying variety algebraically embeds into R^{2n+6}. The proof of this nonembedding result hinges on a refinement of the Levi decomposition for Poisson manifolds to partially linearize any extension with respect to the Levi decomposition of the cotangent Lie algebra of V/G at the origin. Moreover, in the case n=3, this nonembedding result is complemented by a concrete realization of V/\ZZ_3 as a Poisson subspace of R^78.