The differential forms on decompose into forms of type , sometimes called forms. For example, on , the exterior algebra decomposes into four types:
(1)
 
(2)

where , , and denotes the direct sum. In general, a form is the sum of terms with s and s. A form decomposes into a sum of forms, where .
For example, the 2forms on decompose as
(3)
 
(4)

The decomposition into forms of type is preserved by holomorphic functions. More precisely, when is holomorphic and is a form on , then the pullback is a form on .
Recall that the exterior algebra is generated by the oneforms, by wedge product and addition. Then the forms of type are generated by
(5)

The subspace of the complex oneforms can be identified as the eigenspace of the almost complex structure , which satisfies . Similarly, the eigenspace is the subspace . In fact, the decomposition of determines the almost complex structure on .
More abstractly, the forms into type are a group representation of , where acts by multiplication by .