Berezin-type quantization on even-dimensional compact manifolds

In this article we show that a Berezin-type quantization can be achieved on a compact even dimensional manifold M^2d by removing a skeleton M₀ of lower dimension such that what remains is diffeomorphic to R^2d (cell decomposition) which we identify with C^d and embed in CP^d. A local Poisson structure and Berezin-type quantization are induced from CP^d. Thus we have a Hilbert space with a reproducing kernel. The symbols of bounded linear operators on the Hilbert space have a star product which satisfies the correspondence principle outside a set of measure zero. This construction depends on the diffeomorphism. One needs to keep track of the global holonomy and hence the cell decomposition of the manifold. As an example, we illustrate this type of quanitzation of the torus. We exhibit Berezin-Toeplitz quantization of a complex manifold in the same spirit as above.