On Quantum de Rham Cohomology Theory

We define quantum exterior product wedge_h and quantum exterior differential d_h on Poisson manifolds (of which symplectic manifolds are an important class of examples). Quantum de Rham cohomology, which is a deformation quantization of de Rham cohomology, is defined as the cohomology of d_h. We also define quantum Dolbeault cohomology. A version of quantum integral on symplectic manifolds is considered and the correspoding quantum Stokes theorem is proved. We also derive quantum hard Lefschetz theorem. By replacing d by d_h and wedge by wedge_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 classical Chern-Weil theory. Quantum equivariant de Rham cohomology is defined in the similar fashion.