Almost complex manifold: Encyclopedia II - Almost complex manifold - Differential topology of almost complex manifolds

Almost complex manifold - 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 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). The almost complex structure induces the decomposition of each space of r-forms

In other words, each Ω^r(M)^C admits a decomposition into a sum of Ω^(p,q)(M), with r=p+q.

As with any direct sum, there is a canonical projection π_p,q from Ω^r(M)^C to Ω^(p,q). We also have the exterior derivative which maps Ω^r(M)^C to Ω^r+1(M)^C. Thus we may use the almost complex structure to refine the action of the exterior derivative to the forms of definite type

so that is a map which increases the holomorphic part of the type by one (takes forms of type (p,q) to forms of type (p+1,q)), and is a map which increases the antiholomorphic part of the type by one. These operators are called the Doubeault operators.

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

Adapted from the Wikipedia article "Differential topology of almost complex manifolds", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki