Identification of Berezin-Toeplitz deformation quantization (Q2754268)

\textit{A. V. Karabegov} has shown in [Commun. Math. Phys. 180, 745-755 (1996; Zbl 0866.58037)] that there is a one to one correspondence between deformation quantizations with separation of variables on a Kähler manifold \((M,\omega)\) (i.e. differential star products on \(M\) such that on any chart of \(M\) left star multiplication by holomorphic functions and right star multiplication by antiholomorphic functions are point-wise multiplication of functions) and closed formal (1,1) forms \((1/\nu)\omega +\omega_0+\nu \omega_1+\). Moreover [\textit{A. V. Karabegov}, Lett. Math. Phys. 43, 347-357 (1998; Zbl 0938.53049)] these star products are classified (up to equivalence) by formal cohomology classes \((1/i\nu)\omega +H^2(M, {\mathbb C}[[\nu]])\). NEWLINENEWLINENEWLINEWhen \((M,\omega)\) is a (prequantizable) compact Kaehler manifold, it was shown by \textit{M. Schlichenmeier} [Conférence Moshé Flato (Dijon, France 1999), G. Dito and D. Sternhaimer (eds.), Kluwer Vol. 2, 289-306 (2000; Zbl 1028.53085)] that the Berezin-Toeplitz quantization map leads to a star product on \(M\) called Berezin-Toeplitz deformation quantization. In the present work, the Berezin-Toeplitz deformation quantization is proved to be a star product with separation of variables and the corresponding formal form is explicitly determined especially by use of results on Szegő kernels from \textit{L. Boutet de Monvel} and \textit{S. Sjöstrand} [Astérisque 34/35, 123-164 (1976; Zbl 0344.32010)]. The formal cohomology class of Berezin-Toeplitz deformation quantization is also obtained.