Dolbeault operator ($\bar{\partial }$)

The complexified cotangent bundle decomposes into $(1,0)$ and $(0,1)$ parts. The Dolbeault operator is the type-$(0,1)$ component of the exterior derivative, $\bar{\partial }:{\Omega }^{p,q}\to {\Omega }^{p,q+1}$, and integrability gives ${\bar{\partial }}^2=0$.

Dolbeault cohomology is

The Hodge number of a compact complex manifold is $h^{p,q}={\dim }_{\mathbb{C}}{H}_{\bar{\partial }}^{p,q}(X)$.