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We define the quantum exterior product $\wedge_h$ and quantum exterior differential $d_h$ 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 $\wedge$ by $\wh$ 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.

Copyright 1999 American Mathematical Society

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Article Info

• ERA Amer. Math. Soc. 05 (1999), pp. 24-34
• Publisher Identifier: S 1079-6762(99)00056-6
• 1991 Mathematics Subject Classification. Primary 53C15, 58A12, 81R05
• Key words and phrases.
• Received by the editors May 07, 1998
• Posted on April 1, 1999
• Communicated by Richard Schoen
• Comments (When Available)

Huai-Dong Cao

Department of Mathematics, Texas A\&M University, College Station, TX 77843

E-mail address: cao@math.tamu.edu

Jian Zhou

Department of Mathematics, Texas A\&M University, College Station, TX 77843

E-mail address: zhou@math.tamu.edu