Almost-complex structure

An almost-complex structure on a smooth manifold $M$ is a smooth vector-bundle endomorphism

such that $J^2=-\operatorname{id}_{TM}$. Thus each real tangent space becomes a complex vector space by declaring multiplication by $i$ to be $J$. In particular, $M$ must have even real dimension.

Every complex manifold has a canonical almost-complex structure. An arbitrary almost-complex structure need not come from complex coordinate charts; when it does, it is integrable.