Almost Complex Manifold Complex Structure On A Real Vector Space

For any real vector space V, a complex structure on V is an R-linear map J such that J²=-1. Then we may extend J by linearity to the complexification of V,

because C is algebraically closed, J is guaranteed to have eigenvalues which satisfy λ², namely +/-i. Thus we may write V^C=V⁺+V^-, where V⁺ and V^- are the eigenspaces of +i and -i, respectively. Complex conjugation provides a congugate-linear isomorphism over C between V⁺ and V^-, and thus they have the same complex dimension. Thus if n is the complex dimension of V⁺, then 2n is the complex dimension of V^C, and so 2n is also the real dimension of V. Thus if a vector space admits a complex structure, it must be even dimensional.

2 Formal definition

Let M be a smooth manifold. An almost complex structure on M is a 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 when regarded as a vector bundle isomorphism J : TM→TM on the tangent bundle. Since T_pM must be even dimensional, one concludes that M is even-dimensional if it admits an almost complex structure. A manifold equipped with an almost complex structure is called an almost complex manifold.

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 J_p²=-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.

3 Differential topology of almost complex manifolds

Just as a complex structure on a vector space V allows a decomposition of V^C into V⁺ and V^-, so an almost complex structure on M allows a decomposition of the complexified tangent bundle TM^C (which is the vector bundle of complexified tangent spaces at each point) into TM⁺ and TM^-. A section of TM⁺ is called a holomorphic vector field while a section of TM^- is an antiholomorphic vector field. Thus J corresponds to multiplication by i on the holomorphic vector fields of the complexified tangent bundle, and multiplication by -i on the antiholomorphic 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 Ω^p(M)^C will decompose into a sum of Ω^m,n(M), with p=m+n.

note: come back and explain this construction and subsequent decomposition better

The exterior derivative can be extended to the complexified differential forms by linearity, and then we can decompose the exterior derivative like

where takes (m,n)-forms to (m+1,n) forms, and takes (m,n)-forms to (m,n+1)-forms. These are called the Doubeault operators.

4 Relation to complex manifolds

A generic almost complex structure does not necessarily define a complex structure on a manifold. Moreover, there are examples of almost complex manifolds which do not admit complex structures. In particular, there are almost complex structures which cannot be obtained as a deformations of a complex structure.

For example, the 6-sphere, S⁶, when considered as the set of unit norm imaginary octonions, inherits an almost complex structure from the octonion multiplication. S⁶, however, does not admit a complex structure.

There are several equivalent conditions for determining whether an almost complex structure is one induced by a full complex structure.

(A) The manifold admits an atlas of charts to Cⁿ. Such an atlas is a complex structure. A manifold with a complex structure necessarily admits an almost complex structure.

(B) The almost complex structure is integrable if and only if the Lie bracketA lie bracket can refer to: Lie algebra Lie derivative. of two holomorphic vector fields is again holomorphic. An integral almost complex structure is induced by a unique complex structure.

(C) The obstruction to the existence of a complex structure can be codified in a rank (1,2) tensor, called the Nijenhuis tensor, defined by

If this tensor vanishes for all smooth vector fieldIn mathematics a vector field is a construction in vector calculus which associates a vector to every point in Euclidean space. Vector fields are often used in physics to model for example the speed and direction of a moving fluid throughout space, or thes X and Y (and here [·, ·] denotes the Lie bracketIn mathematics, a Lie derivative is a derivation on the algebra of smooth functions over a manifold M''. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by : A B ≡ B A . of vector fields) then the almost complex structure is globally defined and so determines a complex structure.

(D) If we have , then the almost complex structure is integrable.

(E) On a Riemannian manifoldIn Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. This allows one to define of various notions as the length, an almost complex structure lets one choose a Hermitian metricA hermitian metric on a complex vector bundle E over a smooth manifold M is a positive-definite, hermitian inner product on each fiber E that varies smoothly with the point p in M''. An important special case is that of a hermitian metric on the complexif. If J is preserved by parallel transportIn mathematics, a parallel transport on a manifold M with specified connection is a way to transport vectors along smooth curves, in such a way that they stay "parallel" with respect to the given connection. A field on a smooth curve is called parallel if for the Levi-Civita connectionIn Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). The Fundamental theorem of Riemannian geometry stat for this Hermitian metric is covariantly constant, then it is also integrable, and so we have a complex manifold with Riemannian metric. Moreover, this defines a Kähler structureIn mathematics, a Kahler manifold is a complex manifold which also carries a Riemannian metric and a symplectic form on the underlying real manifold in such a way that the three structures (complex, Riemannian, and symplectic) are all mutually compatible. on the manifold.

5 Related article

Chern class

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