Wick quantisation of a symplectic manifold

The notion of the Wick star-product is covariantly introduced for a general symplectic manifold equipped with two transverse polarisations. Along the lines of Fedosov method, the explicit procedure is given to construct the Wick symbols on the manifold. The cohomological obstruction is identified to the equivalence between the Wick star-product and the Fedosov one. In particular in the K"ahler case, the Wick star-product is shown to be equivalent the Weyl one, iff the manifold is a Calabi-Yau one.