= Harmonicity of holomorphic functions on a Kähler manifold
{title2=$\bar\partial u=0\Longrightarrow\Delta_du=0$}
The <Kähler Laplacian identity> $\Delta_d^{\mathrm H}=2\Delta_{\bar\partial}$ implies that every <holomorphic function> on a <Kähler manifold> is a <harmonic function>. On functions, $\bar\partial u=0$ and $\bar\partial^*u=0$ by degree, so both terms of the <Dolbeault Laplacian> vanish. Compactness is unnecessary. Changing the sign convention for the scalar Laplacian does not change this conclusion.
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