Complex Form

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 2-forms 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 one-forms, by wedge product and addition. Then the forms of type  are generated by

(5)

The subspace  of the complex one-forms 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 .