Let $M$ be a complex manifold with the complex structure $J$ and let $h$ be an Hermitian metric, associated with the $(1, 1)$-form $\Omega$. (which means that $\Omega(X, Y) = h(JX, Y))$
It's known that $\omega$ is Kahler if and only if $\nabla J$ = 0, i.e. $J$ is parallel with respect to the Levi-Civita connection on $M$.
How one can prove the equivalent proposition, namely, $\Omega$ is Kahler if and only if $\nabla \Omega = 0$, where $\nabla$ is the Levi-Civita connection which is compatible with $\operatorname{Re}(h)$ (the real part of $h$). Equivalently, that means that the parallel transport is unitary, i.e. preserves the Hermitian structure.
It seems that one should differentiate the definition $\Omega(X, Y) = h(JX, Y)$ and apply some kind of a known identity (to use the symmetries/e.t.c.) but i cannot see a way to proceed that. Are there any ways to prove the proposed statement: (1) independently (2) having the fact that $\nabla J = 0$?
In any case, does the question above somehow relate to the fact that the Chern connection and Levi-Civita connection coincide?