Advances in Mathematical Physics
Volume 2010 (2010), Article ID 927280, 38 pages
Review Article

Berezin-Toeplitz Quantization for Compact Kähler Manifolds. A Review of Results

Mathematics Research Unit, University of Luxembourg, FSTC, 6, rue Coudenhove-Kalergi 1359 Luxembourg-Kirchberg, Luxembourg

Received 30 December 2009; Accepted 5 March 2010

Academic Editor: S. T. Ali

Copyright © 2010 Martin Schlichenmaier. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This article is a review on Berezin-Toeplitz operator and Berezin-Toeplitz deformation quantization for compact quantizable Kähler manifolds. The basic objects, concepts, and results are given. This concerns the correct semiclassical limit behaviour of the operator quantization, the unique Berezin-Toeplitz deformation quantization (star product), covariant and contravariant Berezin symbols, and Berezin transform. Other related objects and constructions are also discussed.