Almost complex structure from a decomposable volume form ($J_{\Omega }$)

A locally decomposable complex $n$-form $\Omega$ on a real $2n$-manifold, with $\Omega \wedge \overline{\Omega }$ nowhere zero, determines an almost complex structure. The intrinsic rank-$n$ subbundle $W=\{\alpha \in T^{\ast }M\otimes \mathbb{C}:\alpha \wedge \Omega =0\}$ is locally spanned by the factors of $\Omega$. The nonvanishing condition gives $T^{\ast }M\otimes \mathbb{C}=W\oplus \overline{W}$. Set $J^{\ast }=i$ on $W$ and $J^{\ast }=-i$ on $\overline{W}$. This commutes with conjugation, so comes from a unique real bundle endomorphism with square $-I$.

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