Almost Complex Manifold - Differential Topology of Almost Complex Manifolds

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− (the eigenspaces of J corresponding to +i and −i, respectively), 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 a 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 Ωr(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 πp,q from Ωr(M)C to Ω(p,q). We also have the exterior derivative d which maps Ωr(M)C to Ωr+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 Dolbeault operators.

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

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