Kähler Laplacian identity (${\Delta }_d=2{\Delta }_{\bar{\partial }}$)

On a Kähler manifold, expanding $d=\partial +\bar{\partial }$ and using the Kähler identities makes the mixed anticommutators vanish and gives ${\Delta }_{\partial }={\Delta }_{\bar{\partial }}$. Therefore

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 ${\bar{\partial }}^{\ast }u=0$ 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.