= Harmonic orthogonality criterion for ddbar exactness
{title2=$\alpha\perp\mathcal H^{p,q}\Longleftrightarrow\alpha\in\operatorname{im}(\partial\bar\partial)$}
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.
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