Poincaré–Cartan Class and Deformation Quantization of Kähler Manifolds

Abstract:

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