I'm looking for an elementary proof (if any) of the following
Statement: holomorphic forms are closed on compact Kähler manifolds.
Any classical reference(s) would be welcome as well as the names of the mathematicians to whom this result should be attributed.
Thanks.
Let $\alpha$ be a holomorphic $p$-form, i.e. $\alpha$ is a $(p, 0)$-form which satisfies $\bar{\partial}\alpha = 0$. Note that $\bar{\partial}^*\alpha = 0$, so $\alpha$ is $\bar{\partial}$-harmonic, i.e. $\Delta_{\bar{\partial}}\alpha = 0$. If the manifold is compact and Kähler, then $\Delta_{\bar{\partial}} = \Delta_{\partial}$ so $\Delta_{\partial}\alpha = 0$ and hence $\partial\alpha = 0$. Therefore $d\alpha = \partial\alpha + \bar{\partial}\alpha = 0$, so $\alpha$ is closed.