The operator 
is defined on a complex manifold, and is called
the 'del bar operator.' The exterior derivative 
takes a function and yields a one-form.
It decomposes as
 |
(1)
|
as complex one-forms decompose into complex form of type
 |
(2)
|
where 
denotes the direct sum. More concretely, in coordinates
,
 |
(3)
|
and
 |
(4)
|
These operators extend naturally to forms of higher degree. In general, if 
is a 
-complex form, then 
is a 
-form and 
is a 
-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. The reason is because a change in coordinates or trivializations is holomorphic.
