= Closed decomposable complex volume forms are integrable
If $\Omega=\theta_1\wedge\cdots\wedge\theta_n$ and $d\Omega=0$, differentiating $\theta_j\wedge\Omega=0$ gives $d\theta_j\wedge\Omega=0$. Only the $(0,2)$ component of $d\theta_j$ can survive this wedge product; multiplication by $\Omega$ is injective on that component. Thus no $d\theta_j$ has a $(0,2)$ part. The <bracket and differential-form criteria for integrability> make the induced <almost complex structure> integrable. Local decomposability is part of the hypothesis and is not implied solely by nonvanishing of $\Omega\wedge\overline\Omega$.
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