The curvature of functions for a Kaehler manifold

Question

Let $(M,\omega , J)$ be a Kaehler manifold. The Poisson bracket of functions is defined as

$$\{ f,g \}= \omega (df^*, dg^*)$$

Then for $X_f =df^*$, we have the formula:

$$[ X_f, X_g ] = X_{\{ f,g \} }$$

Now, we define $\nabla_f g$:

$$ X_{\nabla_f g} = \nabla^{LC}_{X_f} X_g$$

with $\nabla^{LC}$ the Levi-Civita connection.

We have, as the torsion of $\nabla^{LC}$ is zero:

$$\nabla_f g -\nabla_g f =\{ f, g \}$$

The curvature of functions is:

$$R(f,g) h =\nabla_f \nabla_g h - \nabla_g \nabla_f h - \nabla_{\{ f,g \}}h$$

Have we good properties of this curvature of functions?