  We generalize various symplectic reduction techniques to the context of the
optimal momentum map. Our approach allows the construction of symplectic point
and orbit reduced spaces purely within the Poisson category under hypotheses
that do not necessarily imply the existence of a momentum map. We construct an
orbit reduction procedure for canonical actions on a Poisson manifold that
exhibits an interesting interplay with the von Neumann condition previously
introduced by the author in his study of singular dual pairs. This condition
ensures that the orbits in the momentum space of the optimal momentum map (we
call them polar reduced spaces) admit a presymplectic structure that
generalizes the Kostant--Kirillov--Souriau symplectic structure of the
coadjoint orbits in the dual of a Lie algebra. Using this presymplectic
structure, the optimal orbit reduced spaces are symplectic with a form that
satisfies a relation identical to the classical one obtained by Marle, Kazhdan,
Kostant, and Sternberg for free Hamiltonian actions on a symplectic manifold.
In the symplectic case we provide a necessary and sufficient condition for the
polar reduced spaces to be symplectic. In general, the presymplectic polar
reduced spaces are foliated by symplectic submanifolds that are obtained
through a generalization to the optimal context of the so called Sjamaar
Principle, already existing in the theory of Hamiltonian singular reduction. We
use these ideas in the construction of a family of presymplectic homogeneous
manifolds and of its symplectic foliation and we show that these reduction
techniques can be implemented in stages in total analogy with the case of free
globally Hamiltonian proper actions.
