  The noncommutativity of a four-dimensional phase space is introduced from a
purely symplectic point of view. We show that there is always a coordinate map
to locally eliminate the gauge fluctuations inducing the deformation of the
symplectic structure. This uses the Moser's lemma; a refined version of the
celebrated Darboux theorem. We discuss the relation between the coordinates
change arising from Moser's lemma and the Seiberg--Witten map. As illustration,
we consider the quantum Hall systems on CP^2. We derive the action describing
the electromagnetic interaction of Hall droplets. In particular, we show that
the velocities of the edge field, along the droplet boundary, are
noncommutativity parameters-dependents.
