  We introduce new finite-dimensional cohomologies on symplectic manifolds.
Each exhibits Lefschetz decomposition and contains a unique harmonic
representative within each class. Associated with each cohomology is a
primitive cohomology defined purely on the space of primitive forms. We
identify the dual currents of lagrangians and more generally coisotropic
submanifolds with elements of a primitive cohomology, which dualizes to a
homology on coisotropic chains.
