General quantization scheme for symplectic manifolds which admit Kähler structures

We will consider arbitrary holomorphic families of Kähler structures on a fixed prequantizable symplectic manifold and provided the corresponding family of Kähler quantisations for level k form a vector bundle over the chosen family of Kähler structures, we will for large enough k construct a large family of Hitchin connections in this bundle. We will also consider a natural family of Hermitian structures on the bundle of quantum spaces and understand which (unique) Hitchin connection is compatible with a given one of these Hermitian structures. Once we have selected either a Hermitian structure or a Hitchin connection, for the given family, we will provide a quantisation construction for an certain algebra of quantizable observables (sub-algebra of the real analytic functions), which are parallel w.r.t. to the chosen Hitchin connection. If we have a subgroup of the symplectomorphisms, which preserves the family of Kähler structures and whose action has been lifted to the given prequantum line bundle and the family Kähler structures is equivariant under this subgroup, then both the Hermitian structures and the Hitchin connections can be chosen such that they are invariant under this subgroup and the quantisation of the are equivariant under the action of this subgroup.