 ddc lemma

On a compact Kähler manifold, if a differential form $\alpha$ is $d$-closed and $d^c$-exact, then it is $d{d}^c$-exact: there is a form $\beta$ two degrees lower such that $\alpha =d{d}^c\beta$. This real-form statement is equivalent, after decomposing by type, to the ddbar lemma.