Let $X$ be a compact Kähler manifold with a Kähler form $\omega$, then from Kodaira&Spencer's paper on deformations III, p.75, the authors state that the Kähler form satisfies the Laplace equation $\Delta\omega=0$.

As we know, $\Delta\omega=0\Leftrightarrow d\omega=0,d^*\omega=0$. By definition, every Kähler form satisfies $d\omega=0$, but it is not obvious why $d^*\omega$ should also be 0. Actually, from Kodaira&Morrow's book 《complex manifolds》 p.115, they have a proof of $\Delta\omega=0$ by showing $\bar\partial\omega=0$ and $\bar\partial^*\omega=0$, then it's a result of the famous Kähler identities $\Delta=2\Delta_{\bar\partial}$, but their proof seems too complicated which involve covariant differentiation, does anybody has a simple reason why $d^*\omega$ should be 0 or $\omega\in ker\Delta$?

Let $X$ be a compact Kähler manifold with Kähler form $\omega$, then from Kodaira & Spencer's paper on deformations III, p.75, the authors state that the Kähler form satisfies the Laplace equation $\Delta\omega=0$.

As we know, $\Delta\omega=0\Leftrightarrow d\omega=0,d^*\omega=0$. By definition, every Kähler form satisfies $d\omega=0$, but it is not obvious why $d^*\omega$ should also be 0. Actually, from Kodaira & Morrow's book Complex Manifolds p.115, they have a proof of $\Delta\omega=0$ by showing $\bar\partial\omega=0$ and $\bar\partial^*\omega=0$, then it's a result of the famous Kähler identities $\Delta=2\Delta_{\bar\partial}$, but their proof seems too complicated as it involves covariant differentiation. Does anybody have a simple reason why $d^*\omega$ should be $0$, i.e. $\omega\in \ker\Delta$?