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For the nonnegative Hodge Laplacian ${\Delta }_d^{\mathrm{H}}=d{d}^{\ast }+d^{\ast }d$, the Kähler Laplacian identity gives ${\Delta }_d^{\mathrm{H}}=2{\Delta }_{\bar{\partial }}$. If $u$ is a holomorphic function, then $\bar{\partial }u=0$, while ${\bar{\partial }}^{\ast }u=0$ by degree. ThereforeThe divergence-of-gradient convention printed in part (b) is ${\Delta }_d=-{\Delta }_d^{\mathrm{H}}$ on functions. The sign changes no kernel, so the requested harmonicity of holomorphic functions on a Kähler manifold isHolomorphic functions are harmonic without any compactness assumption. This is a local identity from the Kähler Laplacian identity, not an integration argument requiring a compact manifold.