Berezin's coherent states, symbols and transform for compact Kähler manifolds.

Coherent states are quite well known, wide-spread and extremely useful tools. In this review, coherent state techniques for general quantizable compact Kähler manifolds are reviewed. Discussed are co- and contravariant Berezin symbols, Berezin-Toeplitz quantization, the Berezin transform, and related natural deformation quantizations (star products). These are the Berezin-Toeplitz, the Berezin, and the geometric quantization star product. All three star products exist in this setting and are uniquely defined. They are different but equivalent and the equivalence transformation is given. Results on the Berezin transform are used in an essential manner.