Dolbeault operator

Remarks

In a chart, with the Wirtinger derivatives $\partial / \partial \bar z_k = \frac{1}{2}(\partial / \partial x_k + i \, \partial / \partial y_k)$ for $z_k = x_k + i y_k$, $$\bar\partial \Bigl(\sum_{I, J} f_{IJ} \, dz_I \wedge d\bar z_J\Bigr) = \sum_{I, J} \sum_{k=1}^n \frac{\partial f_{IJ}}{\partial \bar z_k} \, d\bar z_k \wedge dz_I \wedge d\bar z_J.$$ The other component of $d$ is written $\partial$, so $d = \partial + \bar\partial$, and $d^2 = 0$ gives $\bar\partial^2 = 0$, $\partial^2 = 0$ and $\partial \bar\partial + \bar\partial \partial = 0$. The splitting of $d$ into two components uses that the complex structure comes from holomorphic charts. The cohomology of $\bar\partial$ on $\Omega^{p, \bullet}(X)$ is the Dolbeault cohomology $H^{p,q}_{\bar\partial}(X)$, and the same construction works for forms with values in a holomorphic vector bundle.

Example: on $X = \C$ with coordinate $z$, let $f(z) = z^2$. Then $\partial f / \partial \bar z = 0$, so $\bar\partial f = 0$, as for every holomorphic function.

Non-example: the function $g(z) = \bar z$ on $\C$ is not holomorphic, and indeed $\bar\partial g = \frac{\partial \bar z}{\partial \bar z} \, d\bar z = d\bar z \ne 0$.