On a compact manifold $\Delta_{\bar\partial}(f)=0$ implies $f$ constant

Question

Let $X$ be a complex hermitian manifold, moreover assume that $X$ is compact. I don't understand why the following proposition is true:

If $f$ is a global $\bar\partial$-harmonic function on $X$ (that is: $f\in C^\infty (X)$ and $\Delta_{\bar\partial}(f)=0$), then $f$ is constant.

Can you give a proof of this? I'd like to understand the point where we use the compactness of $X$.

Answer 1

Hint: Show that this implies that $f$ is holomorphic. What do you know about holomorphic functions on compact manifolds?