Editing Almost complex manifold (section)

== 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>