Kähler manifold

Remarks

In coordinates, $d\omega = 0$ holds if and only if $\dfrac{\partial h_{j\bar k}}{\partial z_l} = \dfrac{\partial h_{l\bar k}}{\partial z_j}$ for all $j, k, l$. Equivalent forms: locally $\omega = i\partial\bar\partial\varphi$ for a real smooth strictly plurisubharmonic function $\varphi$; or the complex structure is parallel for the Levi-Civita connection of the underlying Riemannian metric $\operatorname{Re} h$. A compact complex manifold is called Kähler if it admits some Kähler metric.

Example: on $\C^n$, $h_{j\bar k} = \delta_{jk}$ gives $\omega = \frac{i}{2}\sum_j dz_j \wedge d\bar z_j$, which has constant coefficients and so is closed. On a Riemann surface every Hermitian metric is Kähler, because every $3$-form on a manifold of real dimension $2$ is zero. The Fubini–Study form on $\mathbb{CP}^n$, given on $\set{z_0 \ne 0}$ by $\omega = \frac{i}{2}\partial\bar\partial \log(1 + \abs{w}^2)$ in the coordinates $w_j = z_j/z_0$, is closed because $d = \partial + \bar\partial$ and $\partial^2 = \bar\partial^2 = 0$, $\partial\bar\partial = -\bar\partial\partial$.

Non-example: on $W = \C^2 \setminus \set{0}$, the form $\omega = \frac{i}{2}\abs{z}^{-2}\sum_j dz_j \wedge d\bar z_j$ is the fundamental form of a Hermitian metric invariant under $z \mapsto \lambda z$, so it descends to a Hopf surface. It is not closed: $d\omega = d(\abs{z}^{-2}) \wedge \omega_0$ with $\omega_0 = \frac{i}{2}\sum_j dz_j \wedge d\bar z_j$, and the wedge product of a nonzero $1$-form with $\omega_0$ is nonzero in complex dimension $2$. So this metric is a Hermitian metric that is not Kähler; in fact the Hopf surface has no Kähler metric at all.