  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 $M_0$ 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.
