Editing Almost complex manifold

{{Short description|Smooth manifold}}
In [[mathematics]], an '''almost complex manifold''' is a [[smooth manifold]] equipped with a smooth [[linear complex structure]] on each [[tangent space]]. Every [[complex manifold]] is an almost complex manifold, but there are almost complex manifolds that are not complex manifolds. Almost complex structures have important applications in [[symplectic geometry]].

The concept is due to [[Charles Ehresmann]] and [[Heinz Hopf]] in the 1940s.<ref>{{cite journal|author1-last=Van de Ven|author1-first=A.|title=On the Chern numbers of certain complex and almost complex manifolds|journal=[[Proceedings of the National Academy of Sciences]]|volume=55|issue=6|pages=1624–1627|date=June 1966|doi=10.1073/pnas.55.6.1624|pmid=16578639|pmc=224368|bibcode=1966PNAS...55.1624V|doi-access=free}}</ref>

== Formal definition ==
Let ''M'' be a smooth manifold. 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. In other words, we have a [[smooth function|smooth]] [[tensor field]] ''J'' of [[Tensor#Tensor degree|degree]] {{nowrap|(1, 1)}} such that <math>J^2=-1</math> when regarded as a [[vector bundle]] [[isomorphism]] <math>J\colon TM\to TM</math> on the [[tangent bundle]]. A manifold equipped with an almost complex structure is called an '''almost complex manifold'''.

If ''M'' admits an almost complex structure, it must be even-dimensional. This can be seen as follows. Suppose ''M'' is ''n''-dimensional, and let {{nowrap|''J'' : ''TM'' → ''TM''}} be an almost complex structure. If {{nowrap|1=''J''{{i sup|2}} = −1}} then {{nowrap|1=(det ''J'')<sup>2</sup> = (−1){{sup|''n''}}}}. But if ''M'' is a real manifold, then {{nowrap|det ''J''}} is a real number – thus ''n'' must be even if ''M'' has an almost complex structure. One can show that it must be [[orientable manifold|orientable]] as well.

An easy exercise in [[linear algebra]] shows that any even dimensional vector space admits a linear complex structure. Therefore, an even dimensional manifold always admits a {{nowrap|(1, 1)}}-rank tensor ''pointwise'' (which is just a linear transformation on each tangent space) such that {{nowrap|1=''J''{{sub|''p''}}{{sup|2}} = −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 possibility of this patching, and therefore existence of an almost complex structure on a manifold ''M'' is equivalent to a [[reduction of the structure group]] of the tangent bundle from {{nowrap|GL(2''n'', '''R''')}} to {{nowrap|GL(''n'', '''C''')}}. The existence question is then a purely [[algebraic topology|algebraic topological]] one and is fairly well understood.

==Examples==
For every integer n, the flat space '''R'''<sup>2''n''</sup> admits an almost complex structure. An example for such an almost complex structure is (1 ≤ ''j'', ''k'' ≤ 2''n''): <math>J_{jk} = -i\delta_{j,k-1} </math> for odd ''j'', <math>J_{jk} = i\delta_{j,k+1} </math> for even ''j''.

The only [[sphere]]s which admit almost complex structures are '''S'''<sup>2</sup> and '''S'''<sup>6</sup> ({{harvtxt|Borel|Serre|1953}}). In particular, '''S'''<sup>4</sup> cannot be given an almost complex 
structure (Ehresmann and Hopf). In the case of '''S'''<sup>2</sup>, the almost complex structure comes from an honest complex structure on the [[Riemann sphere]]. The 6-sphere, '''S'''<sup>6</sup>, when considered as the set of unit norm imaginary [[octonion]]s, inherits an almost complex structure from the octonion multiplication; the question of whether it has a [[#Integrable almost complex structures|complex structure]] is known as the ''Hopf problem,'' after [[Heinz Hopf]].<ref>{{cite journal|last1=Agricola |first1=Ilka |authorlink1=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>

== Differential topology of almost complex manifolds ==
Just as a complex structure on a vector space ''V'' allows a decomposition of ''V''<sup>'''C'''</sup> into ''V''<sup>+</sup> and ''V''<sup>−</sup> (the [[eigenspace]]s of ''J'' corresponding to +''i'' and −''i'', respectively), so an almost complex structure on ''M'' allows a decomposition of the complexified tangent bundle ''TM''<sup>'''C'''</sup> (which is the vector bundle of complexified tangent spaces at each point) into ''TM''<sup>+</sup> and ''TM''<sup>−</sup>. A section of ''TM''<sup>+</sup> is called a [[vector field]] of type (1, 0), while a section of ''TM''<sup>−</sup> is a vector field of type (0, 1). Thus ''J'' corresponds to multiplication by [[Imaginary unit|''i'']] on the (1,&nbsp;0)-vector fields of the complexified tangent bundle, and multiplication by −''i'' on the (0,&nbsp;1)-vector fields.

Just as we build [[differential form]]s out of [[exterior power]]s of the [[cotangent bundle]], we can build exterior powers of the complexified cotangent bundle (which is canonically isomorphic to the bundle of dual spaces of the complexified tangent bundle). The almost complex structure induces the decomposition of each space of ''r''-forms

:<math>\Omega^r(M)^\mathbf{C}=\bigoplus_{p+q=r} \Omega^{(p,q)}(M). \, </math>

In other words, each Ω<sup>''r''</sup>(''M'')<sup>'''C'''</sup> admits a decomposition into a sum of Ω<sup>(''p'',&nbsp;''q'')</sup>(''M''), with ''r''&nbsp;=&nbsp;''p''&nbsp;+&nbsp;''q''.

As with any [[direct sum of vector bundles|direct sum]], there is a canonical projection π<sub>''p'',''q''</sub> from Ω<sup>''r''</sup>(''M'')<sup>'''C'''</sup> to Ω<sup>(''p'',''q'')</sup>. We also have the [[exterior derivative]] ''d'' which maps Ω<sup>''r''</sup>(''M'')<sup>'''C'''</sup> to Ω<sup>''r''+1</sup>(''M'')<sup>'''C'''</sup>. Thus we may use the almost complex structure to refine the action of the exterior derivative to the forms of definite type

:<math>\partial=\pi_{p+1,q}\circ d</math>
:<math>\overline{\partial}=\pi_{p,q+1}\circ d</math>

so that <math>\partial</math> is a map which increases the holomorphic part of the type by one (takes forms of type (''p'',&nbsp;''q'') to forms of type (''p''+1, ''q'')), and <math>\overline{\partial}</math> is a map which increases the antiholomorphic part of the type by one. These operators are called the [[Dolbeault operator]]s.

Since the sum of all the projections must be the [[identity function|identity map]], we note that the exterior derivative can be written

:<math>d=\sum_{r+s=p+q+1} \pi_{r,s}\circ d=\partial + \overline{\partial} + \cdots .</math>

== Integrable almost complex structures ==
Every [[complex manifold]] is itself an almost complex manifold. In local holomorphic coordinates <math>z^\mu = x^\mu + i y^\mu</math> one can define the maps

:<math>J\frac{\partial}{\partial x^\mu} = \frac{\partial}{\partial y^\mu} \qquad J\frac{\partial}{\partial y^\mu} = -\frac{\partial}{\partial x^\mu}</math>

(just like a counterclockwise rotation of π/2) or

:<math>J\frac{\partial}{\partial z^\mu} = i\frac{\partial}{\partial z^\mu} \qquad J\frac{\partial}{\partial \bar{z}^\mu} = -i\frac{\partial}{\partial \bar{z}^\mu}.</math>

One easily checks that this map defines an almost complex structure. Thus any complex structure on a manifold yields an almost complex structure, which is said to be 'induced' by the complex structure, and the complex structure is said to be 'compatible with' the almost complex structure.

The converse question, whether the almost complex structure implies the existence of a complex structure is much less trivial, and not true in general. On an arbitrary almost complex manifold one can always find coordinates for which the almost complex structure takes the above canonical form at any given point ''p''. In general, however, it is not possible to find coordinates so that ''J'' takes the canonical form on an entire [[neighborhood (topology)|neighborhood]] of ''p''. Such coordinates, if they exist, are called 'local holomorphic coordinates for J'. If ''M'' admits local holomorphic coordinates for ''J'' around every point then these patch together to form a [[Holomorphic function|holomorphic]] [[atlas (topology)|atlas]] for ''M'' giving it a complex structure, which moreover induces ''J''. ''J'' is then said to be '[[Frobenius theorem (differential topology)|integrable]]'. If ''J'' is induced by a complex structure, then it is induced by a unique complex structure.

Given any linear map ''A'' on each tangent space of ''M''; i.e., ''A'' is a tensor field of rank (1,&nbsp;1), then the '''Nijenhuis tensor''' is a tensor field of rank (1,2) given by

:<math> N_A(X,Y) = -A^2[X,Y]+A([AX,Y]+[X,AY]) -[AX,AY]. \, </math>

or, for the usual case of an almost complex structure ''A=J'' such that <math> J^2=-Id </math>,

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

The individual expressions on the right depend on the choice of the smooth vector fields ''X'' and ''Y'', but the left side actually depends only on the pointwise values of ''X'' and ''Y'', which is why ''N''<sub>''A''</sub> is a tensor. This is also clear from the component formula

:<math> -(N_A)_{ij}^k=A_i^m\partial_m A^k_j -A_j^m\partial_mA^k_i-A^k_m(\partial_iA^m_j-\partial_jA^m_i).</math>

In terms of the [[Frölicher–Nijenhuis bracket]], which generalizes the Lie bracket of vector fields, the Nijenhuis tensor ''N<sub>A</sub>'' is just one-half of [''A'',&nbsp;''A''].

The '''Newlander–Nirenberg theorem''' states that an almost complex structure ''J'' is integrable if and only if ''N<sub>J</sub>''&nbsp;=&nbsp;0. The compatible complex structure is unique, as discussed above. Since the existence of an integrable almost complex structure is equivalent to the existence of a complex structure, this is sometimes taken as the definition of a complex structure.

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):

*The Lie bracket of any two (1,&nbsp;0)-vector fields is again of type (1,&nbsp;0)
*<math>d = \partial + \bar\partial</math>
*<math>\bar\partial^2=0.</math>

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 '''S'''<sup>6</sup> admits an integrable almost complex structure, despite a long history of ultimately unverified claims. Smoothness issues are important. For [[real-analytic]] ''J'', the Newlander–Nirenberg theorem follows from the [[Frobenius theorem (differential topology)|Frobenius theorem]]; for ''C''<sup>∞</sup> (and less smooth) ''J'', analysis is required (with more difficult techniques as the regularity hypothesis weakens).

== Compatible triples ==
Suppose ''M'' is equipped with a [[symplectic form]] ''ω'', a [[Riemannian metric]] ''g'', and an almost complex structure ''J''. Since ''ω'' and ''g'' are [[Degenerate form|nondegenerate]], each induces a bundle isomorphism ''TM → T*M'', where the first map, denoted ''φ''<sub>''ω''</sub>, is given by the [[interior product]] ''φ''<sub>''ω''</sub>(''u'')&nbsp;=&nbsp;''i''<sub>''u''</sub>''ω''&nbsp;=&nbsp;''ω''(''u'',&nbsp;•) and the other, denoted ''φ''<sub>''g''</sub>, 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:
*''g''(''u'', ''v'') = ''ω''(''u'', ''Jv'')
*ω(''u'', ''v'') = ''g''(''Ju'', ''v'')
*''J''(''u'') = (''φ''<sub>''g''</sub>)<sup>−1</sup>(''φ''<sub>''ω''</sub>(''u'')).
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. For example, ''ω'' and ''J'' are compatible if and only if ''ω''(•, ''J''•) is a Riemannian metric. 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 manifold|almost Kähler structure]] for the Riemannian metric ''ω''(''u'', ''Jv''). Also, if ''J'' is integrable, then (''M'', ''ω'', ''J'') is a [[Kähler manifold]].

These triples are related to the [[Unitary group#2-out-of-3 property|2 out of 3 property of the unitary group]].

== Generalized almost complex structure ==
[[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 [[Linear subspace|subspace]] of each fiber of the complexified [[tangent bundle]] ''TM''. A generalized almost complex structure is a choice of a half-dimensional [[Isotropic manifold|isotropic]] subspace of each fiber of the [[direct sum of vector bundles|direct sum]] of the complexified tangent and [[cotangent bundle]]s. In both cases one demands that the direct sum of the [[subbundle]] and its [[complex conjugate]] yield the original bundle.

An almost complex structure integrates to a complex structure if the half-dimensional subspace is closed under the [[Lie derivative|Lie bracket]]. A generalized almost complex structure integrates to a [[generalized complex structure]] if the subspace is closed under the [[Courant bracket]]. If furthermore this half-dimensional space is the annihilator of a nowhere vanishing [[pure spinor]] then ''M'' is a [[generalized Calabi–Yau manifold]].

== See also ==
* {{annotated link|Almost quaternionic manifold}}
* {{annotated link|Chern class}}
* {{annotated link|Frölicher–Nijenhuis bracket}}
* {{annotated link|Kähler manifold}}
* {{annotated link|Poisson manifold}}
* {{annotated link|Rizza manifold}}
* {{annotated link|Symplectic manifold}}

== References ==
{{reflist}}

* {{Cite journal | doi=10.2307/1970051 | last1=Newlander | first1=August | last2=Nirenberg | first2=Louis | authorlink2=Louis Nirenberg | title=Complex analytic coordinates in almost complex manifolds | mr=0088770 | year=1957 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=65 | issue=3 | pages=391–404 | jstor=1970051}}
* {{cite book |last=Cannas da Silva |first=Ana | authorlink=Ana Cannas da Silva |title=Lectures on Symplectic Geometry |publisher=Springer |year=2001 |isbn=3-540-42195-5 }} Information on compatible triples, Kähler and Hermitian manifolds, etc.
* {{cite book |authorlink=Raymond O. Wells Jr. |last=Wells |first=Raymond O. |title=Differential Analysis on Complex Manifolds |publisher=Springer-Verlag |location=New York |year=1980 |isbn=0-387-90419-0 }} Short section which introduces standard basic material.
* {{Cite book | last1=Rubei | first1=Elena |  title=Algebraic Geometry, a concise dictionary | publisher=Walter De Gruyter | location=Berlin/Boston | isbn=978-3-11-031622-3 | year=2014}}
* {{cite journal |last1=Borel |first1=Armand | authorlink1=Armand Borel | last2=Serre |first2=Jean-Pierre| authorlink2 = Jean-Pierre Serre |title=Groupes de Lie et puissances réduites de Steenrod |journal=[[American Journal of Mathematics]] |volume=75 |issue=3 |year=1953 |pages=409–448 |jstor=2372495|mr=0058213|doi=10.2307/2372495 }}

{{Manifolds}}

[[Category:Smooth manifolds]]