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ISSN 1079-6762



On quantum de Rham cohomology theory

Authors: Huai-Dong Cao and Jian Zhou
Journal: Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 24-34
MSC (1991): Primary 53C15, 58A12, 81R05
Published electronically: April 1, 1999
MathSciNet review: 1679455
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Abstract: 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 $\wedge _h$ 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 the classical Chern-Weil theory. The quantum equivariant de Rham cohomology is defined in the similar fashion.

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Additional Information

Huai-Dong Cao
Affiliation: Department of Mathematics, Texas A&M University, College Station, TX 77843

Jian Zhou
Affiliation: Department of Mathematics, Texas A&M University, College Station, TX 77843

Received by editor(s): May 7, 1998
Published electronically: April 1, 1999
Additional Notes: Authors’ research was supported in part by NSF grants DMS-96-32028 and DMS-95-04925
Communicated by: Richard Schoen
Article copyright: © Copyright 1999 American Mathematical Society

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