Harmonic orthogonality criterion for ddbar exactness ($\alpha \perp {\mathcal{H}}^{p,q}⟺\alpha \in \mathrm{im}(\partial \bar{\partial })$)

If $\alpha$ is a $d$-closed $(p,q)$-form on a compact Kähler manifold, then $\alpha$ is ddbar lemma exact if and only if it is orthogonal to every harmonic differential form of type $(p,q)$. The forward implication follows by integration by parts. For the reverse implication, Dolbeault Hodge decomposition first makes $\alpha$ $\bar{\partial }$-exact, and the ddbar lemma then makes it $\partial \bar{\partial }$-exact.