Poincare-Cartan class and deformation quantization of Kahler manifolds

We introduce a complete invariant for Weyl manifolds, called a Poincare-Cartan class. Applying the constructions of the Weyl manifold to complex manifolds via the Poincare-Cartan class, we propose the notion of a noncommutative Kahler manifold. For a given Kahler manifold, the necessary and sufficient condition for a Weyl manifold to be a noncommutative Kahler manifold is given. In particular, there exists a noncommutative Kahler manifold for any Kahler manifold. We also construct the noncommutative version of the S-1-principal bundle over a quantizable Weyl manifold.