Editing Complex manifold (section)

==Almost complex structures==
{{main|Almost complex manifold}}
An [[almost complex structure]] on a real 2n-manifold is a GL(''n'', '''C''')-structure (in the sense of [[G-structure]]s) – that is, the tangent bundle is equipped with a [[linear complex structure]].

Concretely, this is an [[endomorphism]] of the [[tangent bundle]] whose square is −''I''; this endomorphism is analogous to multiplication by the imaginary number ''i'', and is denoted ''J'' (to avoid confusion with the identity matrix ''I''). An almost complex manifold is necessarily even-dimensional.

An almost complex structure is ''weaker'' than a complex structure: any complex manifold has an almost complex structure, but not every almost complex structure comes from a complex structure. Note that every even-dimensional real manifold has an almost complex structure defined locally from the local coordinate chart. The question is whether this almost complex structure can be defined globally. An almost complex structure that comes from a complex structure is called [[Frobenius_theorem_(differential_topology)|integrable]], and when one wishes to specify a complex structure as opposed to an almost complex structure, one says an ''integrable'' complex structure. For integrable complex structures the so-called [[Nijenhuis tensor]] vanishes. This tensor is defined on pairs of vector fields, ''X'', ''Y'' by

:<math>N_J(X,Y) = [X,Y] + J[JX,Y] + J[X,JY]-[JX,JY]\ .</math>

For example, the 6-dimensional [[hypersphere|sphere]] '''S'''<sup>6</sup> has a natural almost complex structure arising from the fact that it is the [[orthogonal complement]] of ''i'' in the unit sphere of the [[octonion]]s, but this is not a complex structure. (The question of whether it has a complex structure is known as the ''Hopf problem,'' after [[Heinz Hopf]].<ref>{{cite journal|last1=Agricola |first1=Ilka |author-link1=Ilka Agricola |first2=Giovanni |last2=Bazzoni |first3=Oliver |last3=Goertsches |first4=Panagiotis |last4=Konstantis |first5=Sönke |last5=Rollenske |title=On the history of the Hopf problem |arxiv=1708.01068 |journal=[[Differential Geometry and Its Applications]] |year=2018 |volume=57 |pages=1–9|doi=10.1016/j.difgeo.2017.10.014 |s2cid=119297359 }}</ref>) Using an almost complex structure we can make sense of holomorphic maps and ask about the existence of holomorphic coordinates on the manifold. The existence of holomorphic coordinates is equivalent to saying the manifold is complex (which is what the chart definition says).

Tensoring the tangent bundle with the [[complex number]]s we get the ''complexified'' tangent bundle, on which multiplication by complex numbers makes sense (even if we started with a real manifold). The eigenvalues of an almost complex structure are ±''i'' and the eigenspaces form sub-bundles denoted by ''T''<sup>0,1</sup>''M'' and ''T''<sup>1,0</sup>''M''. The [[Newlander&ndash;Nirenberg theorem]] shows that an almost complex structure is actually a complex structure precisely when these subbundles are ''involutive'', i.e., closed under the Lie bracket of vector fields, and such an almost complex structure is called [[Frobenius_theorem_(differential_topology)|integrable]].