A covariant Poisson deformation quantization with separation of variables up to the third order

We give a simple formula for the operator C-3 of the standard deformation quantization with separation of variables on a Kahler manifold M. Unlike C-1 and C-2, this operator cannot be expressed in terms of the Kahler-Poisson tensor on M. We modify C-3 to obtain a covariant deformation quantization with separation of variables up to the third order which is expressed in terms of the Poisson tensor on M and can thus be defined on an arbitrary complex manifold endowed with a Poisson bivector field of type (1,1).