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 
.
