  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.

Symplectic Geometry