Integrable almost-complex structure

Definition

An almost-complex structure $J$ on a smooth $2n$ -manifold is integrable if every point has local coordinates in $\mathbb C^n$ whose ordinary multiplication by $i$ induces $J$ . Such charts have holomorphic transition maps and therefore make the manifold a complex manifold.

The Nijenhuis tensor measures the failure of the $+i$ -eigenbundle of the complexified tangent bundle to be closed under Lie brackets. The Newlander–Nirenberg theorem states that a smooth almost-complex structure is integrable exactly when this tensor vanishes.

Regularity scope

The displayed equivalence is stated here in the $C^\infty$ category. There are important finite- and low-regularity refinements, but their precise hypotheses and the regularity of the resulting coordinates require separate formulations. The bare phrase “Newlander–Nirenberg” should therefore not be used to silently promote an arbitrary continuous endomorphism $J$ to a complex structure.

In real dimension two, every smooth almost-complex structure is integrable. In higher dimensions, vanishing of the Nijenhuis tensor is a genuine differential constraint. Integrability is also stronger than the pointwise relation $J^2=-1$ : the latter alone supplies complex vector spaces on tangent spaces but not holomorphic charts.

References

August Newlander and Louis Nirenberg, “Complex Analytic Coordinates in Almost Complex Manifolds,” Annals of Mathematics 65 (1957), 391–404. DOI record.
Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005. DOI record. Relevant: Chapter 1, the Nijenhuis tensor and integrability.