= Dolbeault Green operator
{c}
{title2=$G_{\bar\partial}$}
On a compact <Hermitian manifold> without boundary, this operator is the inverse of the <Dolbeault Laplacian> on the orthogonal complement of its harmonic forms and is zero on the harmonic space. Elliptic theory gives $\Delta_{\bar\partial}G=G\Delta_{\bar\partial}=I-H$, with $H$ the harmonic projection, and ensures that $G$ sends smooth forms to smooth forms. It commutes with $\bar\partial$ and $\bar\partial^*$; their commutation with the Laplacian and orthogonality to its kernel give this by uniqueness of the inverse. If $\eta$ is $\bar\partial$-exact, then $\eta=\bar\partial\bar\partial^*G\eta$. On a <Kähler manifold>, the same operator serves the holomorphic Laplacian because $\Delta_\partial=\Delta_{\bar\partial}$.
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