Dolbeault operator
= Dolbeault operator
{title2=$\bar\partial$}
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 $\bar\partial^2=0$.
Together with its $(1,0)$ counterpart, the <Dolbeault operator> decomposes the <exterior derivative> on complex differential forms.