Conjugate Dolbeault operator
= Conjugate Dolbeault operator
{title2=$\partial$}
The conjugate <Dolbeault operator> is the type-$(1,0)$ component of the <exterior derivative>: $\partial:\mathcal A^{p,q}\to\mathcal A^{p+1,q}$. <Complex conjugation> interchanges $\partial$ and $\bar\partial$. On a <complex manifold>, $d=\partial+\bar\partial$, $\partial^2=\bar\partial^2=0$, and $\partial\bar\partial+\bar\partial\partial=0$.