Suppose that $(M,g,J)$ is a complex manifold such that $g$ is a Hermitian metric. Then the Kahler form can be written $$\omega = i \sum_{j,k} g_{j \bar{k}}dz^j \wedge d \bar{z}^k$$
However, $\omega$ is also a symplectic form. Then shouldn't Darboux's theorem imply that there is a coordinate chart $U$ so that $$\omega|_U= i \sum_{j =1}^m dz^j \wedge d \bar{z}^j$$
However, this would imply that every Kahler manifold is locally flat which is certainly false. I think I am misunderstanding what vector space $\omega$ is supposed to be a symplectic form for. Can somebody explain why Darboux's theorem doesn't seem to apply even though $\omega$ is a symplectic form for $M$?