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 version of quantum integral on symplectic manifolds is considered and the corresponding 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 objects in differential geometry, e.g. quantum curvature. The quantum characteristic classes are then studied along the lines of the classical Chern-Weil theory. The quantum equivariant de Rham cohomology is defined in the similar fashion.