The del bar operator 
on a complex manifold is the 
component of the exterior
derivative 
.
On functions, the exterior
derivative yields a one-form and decomposes as
 |
(1)
|
as complex one-forms decompose into complex forms of type
 |
(2)
|
where 
denotes the direct sum. More concretely, in coordinates 
,
 |
(3)
|
and
 |
(4)
|
These operators extend naturally to differential forms of higher degree. In general, if 
is a 
-complex form, then 
is a 
-complex form and 
is a 
-complex form. The equation 
expresses the condition of 
being a holomorphic function.
More generally, a 
-complex form 
is called holomorphic if 
, in which case its coefficients,
as written in a coordinate chart, are holomorphic
functions.
The del bar operator is also well-defined on bundle sections of a holomorphic vector bundle. This is because a change in coordinates or trivializations is holomorphic.
