  This paper is devoted, first of all, to give a complete unified proof of the
Characterization Theorem for compact generalized $p-$K\"ahler manifolds
(Theorem 3.2). The proof is based on the classical duality between "closed"
positive forms and "exact" positive currents. In the last part of the paper we
approach the general case of non compact complex manifolds, where "exact"
positive forms seem to play a more significant role than "closed" forms. In
this setting, we state the appropriate characterization theorems and give some
interesting applications.
