On quantum de Rham cohomology theory

We define the quantum exterior product boolean AND(h) and quantum exterior differential dh on Poisson manifolds. The quantum de Rham cohomology, which is a deformation quantization of the de Rham cohomology, is defined as the cohomology of d(h). We also define the quantum Dolbeault cohomology. A ver sion of quantum integral on symplectic manifolds is considered and the corr esponding quantum Stokes theorem is stated. We also derive the quantum hard Lefschetz theorem. By replacing d by d(h) and boolean AND by boolean AND(h ) in the usual definitions, we define many quantum analogues of important o bjects in differential geometry, e.g. quantum curvature. The quantum charac teristic classes are then studied along the lines of the classical Chern-We il theory. The quantum equivariant de Rham cohomology is defined in the sim ilar fashion.