Sheaf of holomorphic differential forms (${\Omega }_X^p$)

On a complex manifold, ${\Omega }_X^p$ is the sheaf of holomorphic degree-$p$ forms. It is the kernel of the Dolbeault operator on ${\mathcal{A}}^{p,0}$. The Dolbeault theorem computes its sheaf cohomology using smooth forms.

A differential form of type (p, q) of type $(p,0)$ is holomorphic when its local coefficients in holomorphic coordinates are holomorphic functions, equivalently $\bar{\partial }\alpha =0$. It need not be $d$-closed on every complex manifold. Top-degree holomorphic forms are closed by bidegree. On a compact complex surface, holomorphic forms of all degrees are closed, even without a Kähler metric.

On a compact complex manifold of complex dimension $n$, a holomorphic p-form on a complex manifold of degree $n$ is $d$-closed. The form $i^{n^2}\alpha \wedge \bar{\alpha }$ is a nonnegative volume density, positive wherever $\alpha \ne 0$: in holomorphic coordinates it equals $2^n|f{|}^{2}d{x}_1\wedge d{y}_1\wedge \cdots \wedge d{x}_n\wedge d{y}_n$ for $\alpha =fd{z}_1\wedge \cdots \wedge d{z}_n$. If $\alpha =d\beta$, its integral would vanish by the Stokes theorem, contradicting this positivity. No Kähler metric is needed.