Quantization in polarization via transverse Poincaré-Birkhoff-Witt theorem

We construct, for every prequantizable symplectic manifold $(M,ω,L)$ equipped with a nonsingular polarization, a deformation quantization $(C_M^\infty[[\hbar]],\star)$ together with a filtered sheaf of subalgebras $C_{M,\hbar}^{(<\infty)}$ that admits an exact action on polarized sections of $L^{\ot