= Almost complex structure from a decomposable volume form
{title2=$J_\Omega$}
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 $W=\{\alpha\in T^*M\otimes\mathbb C:\alpha\wedge\Omega=0\}$ is locally spanned by the factors of $\Omega$. The nonvanishing condition gives $T^*M\otimes\mathbb C=W\oplus\overline W$. Set $J^*=i$ on $W$ and $J^*=-i$ on $\overline W$. This commutes with conjugation, so comes from a unique real bundle endomorphism with square $-I$.
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