TITLE: Proof of: "holomorphic forms are closed on compact Kähler manifolds"
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: On any Kahler manifold, you have that holomorphic forms are harmonic because of the identity $\Delta = 2\Delta_{\overline{\partial}}$.
Moreover in the compact case an harmonic form $\omega $ it is also closed because 
$$ 0 = \langle\Delta\omega,\omega\rangle = || d \omega||^2 + ||d^*\omega||^2 $$
(this is not true in the non-compact case as $d^*$ is just the formal adjoint)
it follows that holomorphic forms are closed.