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 dim(C) X=1.