 Integral pairing of complex top forms

For a smooth form $\theta =gd{z}_1\wedge \cdots \wedge d{z}_n$ of type $(n,0)$, the complex orientation gives $i^{n^2}\theta \wedge \overline{\theta }=2^n|g{|}^{2}d{x}_1\wedge d{y}_1\wedge \cdots \wedge d{x}_n\wedge d{y}_n$. Consequently its integral on a compact complex manifold is positive unless $\theta$ vanishes identically. Applying Stokes theorem to $d(\psi \wedge \overline{d\psi })$ proves that a holomorphic differential form of degree $n-1$ on a compact complex $n$-manifold is closed, without requiring a Kähler metric.