 Dolbeault operator

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 ${\bar{\partial }}^2=0$.

Together with its $(1,0)$ counterpart, the Dolbeault operator decomposes the exterior derivative on complex differential forms.