Almost complex manifold
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In mathematics, an almost complex manifold is a smooth manifold equipped with smooth linear complex structure on each tangent space. The existence of this structure is a necessary, but not sufficient, condition for a manifold to be a complex manifold. That is, every complex manifold is an almost complex manifold, but not vice-versa. Almost complex structures have important applications in symplectic geometry.
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Formal definition
Let M be a smooth manifold. An almost complex structure J on M is a linear complex structure on each tangent space of the manifold, which varies smoothly on the manifold. In other words, we have a smooth tensor field J of rank (1,1) such that <math>J^2 = -1<math> when regarded as a vector bundle isomorphism J : TM → TM on the tangent bundle. A manifold equipped with an almost complex structure is called an almost complex manifold.
Since TpM must be even-dimensional for it to admit a complex structure, any manifold that admits an almost complex structure must also be even-dimensional. One can show that it must be orientable as well.
Any even dimensional vector space always admits a complex structure. Therefore an even dimensional manifold always locally admits a (1,1) rank tensor (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 almost complex structure yield a complex structure, which is then uniquely determined.
The existence of an almost complex structure on a manifold M is equivalent to a reduction of the structure group of the tangent bundle from GL(2n, R) to GL(n, C). The existence question is then a purely algebraic topological one and is fairly well understood.
A Hermitian structure on a Riemannian manifold is an almost complex structure J such that <math>g(JX,JY)=g(X,Y)<math> where g is the Riemann metric.
The Kähler form of a Hermitian structure is <math>\omega(X,Y)=g(JX,Y)<math>. Note that <math>d\omega=0<math> iff <math>\nabla J=0<math>. Note that if <math>d\omega=0<math>, then the manifold is a Kähler manifold.
Examples
The only spheres which admit almost complex structures are S2 and S6. In the case of S2, the almost complex structure comes from an honest complex structure on the Riemann sphere. The 6-sphere, S6, when considered as the set of unit norm imaginary octonions, inherits an almost complex structure from the octonion multiplication.
Differential topology of almost complex manifolds
Just as a complex structure on a vector space V allows a decomposition of VC into V+ and V-, 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-. A section of TM+ is called a vector field of type (1,0), while a section of TM- is an vector field of type (0,1). Thus J corresponds to multiplication by i on the (1,0)-vector fields of the complexified tangent bundle, and multiplication by -i on the (0,1)-vector fields.
Just as we build differential forms out of exterior powers 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), and each Ωp(M)C will decompose into a sum of Ωm,n(M), with p=m+n.
note: come back and explain this construction and subsequent decomposition better. Be careful to distinguish complex-linear duality, real-linear duality, and metric duality (no two of which are equivalent). Perhaps mention in the next section that a "holomorphic vector field" is a (1,0) vector field whose components in local holomorphic coordinates are holomorphic functions.
The exterior derivative can be extended to the complexified differential forms by linearity, and then we can decompose the exterior derivative like
- <math>d=\part+\overline{\part}<math>
where <math>\part<math> takes (m,n)-forms to (m+1,n) forms, and <math>\overline{\part}<math> takes (m,n)-forms to (m,n+1)-forms. These are called the Doubeault operators.
Note: this is not correct as it stands (the bidegree assertion implies integrability of J). needs work. see talk.
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> the almost complex structure takes the form
- <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>
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>
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 of p. Such coordinates, if they exist, are called local holomorphic coordinates for J. If M admits local holomorphic coordinates around every point then J is said to be integrable. The local holomorphic coordinates patch together to form a holomorphic atlas for M giving it the structure of a complex manifold. A complex structure can then be defined as an integrable almost complex structure.
The existence of an almost complex structure is a topological question and is relatively easy to answer. The existence of an integrable almost complex structure, on the other hand, is a much more difficult analytic question. For example, it has long been known that S6 admits an almost complex structure, but it is still an open question as to whether or not it admits an integrable complex structure.
Given an almost complex structure there are several ways for determining whether or not that structure is integrable. Let J be an almost complex structure on a manifold M. Then the Newlander-Nirenberg theorem states that the following are equivalent:
- J is integrable (i.e. M is a complex manifold)
- The Nijenhuis tensor, defined by<math>N_J(X,Y) = [X,Y] + J([J X, Y] + [X, J Y]) - [J X, J Y]\,<math>vanishes for all smooth vector fields X and Y (here [·, ·] denotes the Lie bracket of vector fields)
- The Lie bracket of two (1,0)-vector fields is again of type (1,0)
- <math>d = \partial + \bar\partial<math>
- <math>\bar\partial^2=0.<math>
Smoothness issues are important: For real-analytic J, Newlander-Nirenberg is the Frobenius theorem; for C^\infty (and less smooth) J, analysis is required (with more difficult techniques as the regularity hypothesis weakens)