 Solution

For the Lefschetz operator of a Kähler manifold and its adjoint $\Lambda$, the Kähler identities includeThey make the mixed anticommutators in the expansion of ${\Delta }_d$ vanish and imply ${\Delta }_{\partial }={\Delta }_{\bar{\partial }}$. Expanding $d=\partial +\bar{\partial }$ therefore proves the Kähler Laplacian identityThus ${\Delta }_d\alpha =0$ exactly when ${\Delta }_{\bar{\partial }}\alpha =0$.