  The purpose of the present work is to describe a dequantization procedure for
topological modules over a deformed algebra. We define the characteristic
variety of a topological module as the common zeroes of the annihilator of the
representation obtained by setting the deformation parameter to zero. On the
other hand, the Poisson characteristic variety is defined as the common zeroes
of the ideal obtained by considering the annihilator of the deformed
representation, and only then setting the deformation parameter to zero. Using
Gabber's theorem, we show the involutivity of the characteristic variety. The
Poisson characteristic variety is indeed a Poisson subvariety of the underlying
Poisson manifold. We compute explicitly the characteristic variety in several
examples in the Poisson-linear case, including the dual of any exponential
solvable Lie algebra. In the nilpotent case, we show that any coadjoint orbit
appears as the Poisson characteristic variety of a well chosen topological
module.
