 Real (p, p)-form

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 $\bar{\partial }\eta$.