Let $X$ be a complex compact Kähler manifold, $\Omega_X$ its cotangent bundle and $D$ the Chern connection.

It is, I believe, a standard fact that parallel $k$-forms $\alpha \in \mathcal{A}^k(X)$, i.e. $D(\alpha)=0$, are closed, see for example

https://math.stackexchange.com/questions/1672761/riemannian-manifold-alpha-in-omegapm-parallel-implies-alpha-is-close?rq=1

It is even true that parallel forms are harmonic, see for example

https://math.stackexchange.com/questions/479991/geometric-interpretation-parallel-forms-are-harmonic

I have found in a book the statement that "any form $\alpha \in H^0(X, \Omega_X^k)$ is parallel". Could someone explain why this is the case? I know that these forms are harmonic with respect to any Kähler metric. How do I show that they are also parallel?

Let $X$ be a complex compact Kähler manifold, $\Omega_X$ its cotangent bundle and $D$ the Chern connection.

It is, I believe, a standard fact that parallel $k$-forms $\alpha \in \mathcal{A}^k(X)$, i.e. $D(\alpha)=0$, are closed, see for example

https://math.stackexchange.com/questions/1672761/riemannian-manifold-alpha-in-omegapm-parallel-implies-alpha-is-close?rq=1

It is even true that parallel forms are harmonic, see for example

https://math.stackexchange.com/questions/479991/geometric-interpretation-parallel-forms-are-harmonic

I have found in a book the statement that "any form $\alpha \in H^0(X, \Omega_X^k)$ is parallel". Could someone explain why this is the case? I know that these forms are harmonic with respect to any Kähler metric. How do I show that they are also parallel?

If needed, we can assume that $X$ is simply-connected, but I would not see how that helps apart from the fact that one only has to consider $k>1$.