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# 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.

Almost Complex Structure, Analytic Function, Cauchy-Riemann Equations, Complex Manifold, Complex Form, Differential k-Form, Dolbeault Cohomology, Dolbeault Operators, Holomorphic Function, Holomorphic Vector Bundle

This entry contributed by Todd Rowland

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Rowland, Todd. "Del Bar Operator." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/DelBarOperator.html