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)
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as complex one-forms decompose into complex form of type
(2)
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where denotes the direct sum. More concretely, in coordinates ,
(3)
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and
(4)
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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.