 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 $\Omega \wedge \overline{\Omega }$.