The operator is defined on a complex manifold, and is called the 'del bar operator.' The exterior derivative takes a function and yields a oneform. It decomposes as
(1)

as complex oneforms 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 welldefined on bundle sections of a holomorphic vector bundle. The reason is because a change in coordinates or trivializations is holomorphic.