= Green operator potential for ddbar exactness
{c}
{title2=$\beta=\bar\partial^*\partial^*G_{\bar\partial}^2\alpha$}
For a $d$-closed pure-type form $\alpha$ orthogonal to <harmonic forms> on a <compact> <Kähler manifold>, let $G_{\bar\partial}$ be the inverse of the <Dolbeault Laplacian> on the harmonic complement, zero on the harmonic space. The <Kähler Laplacian identity> makes it commute with $\partial$ and $\bar\partial$. Thus $\alpha=\bar\partial\bar\partial^*G_{\bar\partial}\alpha$ and $G_{\bar\partial}\alpha=\partial\partial^*G_{\bar\partial}^2\alpha$. The mixed anticommutator $\bar\partial^*\partial+\partial\bar\partial^*=0$ gives $\alpha=\partial\bar\partial\beta$. This is an explicit potential for the <ddbar lemma>, with one lower degree of each type; using the inverse of the <Hodge Laplacian> instead multiplies the displayed expression by four.
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