Quantization of a symplectic manifold associated to a manifold with projective structure

Let X be a complex manifold equipped with a projective structure P. There is a holomorphic principal C*-bundle LP' over X associated with P. We show that the holomorphic cotangent bundle of the total space of LP' equipped with the Liouville symplectic form has a canonical deformation quantization. This generalizes the construction in the work of and Ben-Zvi and Biswas [''A quantization on Riemann surfaces with projective structure,'' Lett. Math. Phys. 54, 73 (2000)] done under the assumption that dimC X=1.