Let's compute the Levi_civita connection in a Kähler manifold. To simplify notation, we'll denote $\partial_{j}=\dfrac{\partial}{\partial z^j}$, $\partial_{\overline{j}}=\dfrac{\partial}{\partial \bar{z}^j}$.

The connection is described by the Christoffel symbols:

Since $\nabla J=0$, $\nabla_v(Jw)=(\nabla_v J)(w)+J(\nabla_v w)=J(\nabla_v w)$. Then,

and therefore $\Gamma_{jk}^{\bar{l}}=\Gamma_{j\bar{k}}^l=\Gamma_{\bar{j}k}^{\bar{l}}=\Gamma_{\bar{j}\bar{k}}^l=0$, and due to the symmetry, also $\Gamma_{j\bar{k}}^{\bar{l}}=\Gamma_{\bar{j}k}^{l}=0$.

On the other hand,

and in the end