Almost-complex structure

Definition

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

J:TM\to TM

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. This is pointwise linear-algebraic data varying smoothly; no coordinate integrability is included.

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.

Maps preserving this structure are the almost-complex, $J$ -holomorphic, or pseudoholomorphic maps. Their tangent maps intertwine the two almost-complex structures.

Examples and constraints

The standard multiplication-by- $i$ map on $\mathbb C^n\cong\mathbb R^{2n}$ is an almost-complex structure. A symplectic manifold admits compatible almost-complex structures, although none is selected by the symplectic form alone. By contrast, an odd-dimensional manifold cannot carry an almost-complex structure because a real vector space with an endomorphism squaring to $-1$ has even dimension.

References

Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005. DOI record. Relevant: Chapter 1, complex and almost-complex structures.
Dusa McDuff and Dietmar Salamon, J-Holomorphic Curves and Symplectic Topology, 2nd ed., AMS, 2012. DOI record. Relevant: Chapter 2, almost-complex structures and $J$ -holomorphic maps.