Del Bar Operator

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.