Differential form of type (p, q) (${\Omega }^{p,q}(X)$)

The complexified cotangent bundle of a complex manifold splits asA differential form has type $(p,q)$ when it is locally a sum of terms containing $p$ factors $d{z}^j$ and $q$ factors $d{\bar{z}}^k$.

A differential form of type (p, q) with equal holomorphic and antiholomorphic degrees is real when it is fixed by complex conjugation. For such a form, $\overline{\bar{\partial }\eta }=\partial \eta$, so $\bar{\partial }\eta =0$ implies $d\eta =0$. The converse follows from the distinct types of $\partial \eta$ and $\bar{\partial }\eta$.

Complex conjugation interchanges the bidegrees of complex differential forms:It also interchanges the two type components of the exterior derivative, so $\bar{\partial }\bar{\alpha }=\overline{\partial \alpha }$ and $\partial \bar{\alpha }=\overline{\bar{\partial }\alpha }$.