Almost complex structures have important applications in symplectic geometry.
An almost complex structure J on M is a linear complex structure (that is, a linear map which squares to −1) on each tangent space of the manifold, which varies smoothly on the manifold.
Therefore, an even dimensional manifold always admits a (1, 1)-rank tensor pointwise (which is just a linear transformation on each tangent space) such that Jp2 = −1 at each point p. Only when this local tensor can be patched together to be defined globally does the pointwise linear complex structure yield an almost complex structure, which is then uniquely determined.
The existence question is then a purely algebraic topological one and is fairly well understood.
The 6-sphere, S6, when considered as the set of unit norm imaginary octonions, inherits an almost complex structure from the octonion multiplication; the question of whether it has a complex structure is known as the Hopf problem, after Heinz Hopf.
[2] Just as a complex structure on a vector space V allows a decomposition of VC into V+ and V− (the eigenspaces of J corresponding to +i and −i, respectively), so an almost complex structure on M allows a decomposition of the complexified tangent bundle TMC (which is the vector bundle of complexified tangent spaces at each point) into TM+ and TM−.
As with any direct sum, there is a canonical projection πp,q from Ωr(M)C to Ω(p,q).
There are several other criteria which are equivalent to the vanishing of the Nijenhuis tensor, and which therefore furnish methods for checking the integrability of an almost complex structure (and in fact each of these can be found in the literature): Any of these conditions implies the existence of a unique compatible complex structure.
The existence of an almost complex structure is a topological question and is relatively easy to answer, as discussed above.
The existence of an integrable almost complex structure, on the other hand, is a much more difficult analytic question.
For example, it is still not known whether S6 admits an integrable almost complex structure, despite a long history of ultimately unverified claims.
Suppose M is equipped with a symplectic form ω, a Riemannian metric g, and an almost complex structure J.
Since ω and g are nondegenerate, each induces a bundle isomorphism TM → T*M, where the first map, denoted φω, is given by the interior product φω(u) = iuω = ω(u, •) and the other, denoted φg, is given by the analogous operation for g. With this understood, the three structures (g, ω, J) form a compatible triple when each structure can be specified by the two others as follows: In each of these equations, the two structures on the right hand side are called compatible when the corresponding construction yields a structure of the type specified.
The bundle on M whose sections are the almost complex structures compatible to ω has contractible fibres: the complex structures on the tangent fibres compatible with the restriction to the symplectic forms.
Using elementary properties of the symplectic form ω, one can show that a compatible almost complex structure J is an almost Kähler structure for the Riemannian metric ω(u, Jv).
Nigel Hitchin introduced the notion of a generalized almost complex structure on the manifold M, which was elaborated in the doctoral dissertations of his students Marco Gualtieri and Gil Cavalcanti.
An ordinary almost complex structure is a choice of a half-dimensional subspace of each fiber of the complexified tangent bundle TM.
A generalized almost complex structure is a choice of a half-dimensional isotropic subspace of each fiber of the direct sum of the complexified tangent and cotangent bundles.
In both cases one demands that the direct sum of the subbundle and its complex conjugate yield the original bundle.
If furthermore this half-dimensional space is the annihilator of a nowhere vanishing pure spinor then M is a generalized Calabi–Yau manifold.