Del Bar Operator

The operator partial^_ is defined on a complex manifold, and is called the 'del bar operator.' The exterior derivative d takes a function and yields a one-form. It decomposes as


as complex one-forms decompose into complex form of type

 Lambda^1=Lambda^(1,0) direct sum Lambda^(0,1),

where  direct sum denotes the direct sum. More concretely, in coordinates z_k=x_k+iy_k,




These operators extend naturally to forms of higher degree. In general, if alpha is a (p,q)-complex form, then partialalpha is a (p+1,q)-form and partial^_alpha is a (p,q+1)-form. The equation partial^_f=0 expresses the condition of f being a holomorphic function. More generally, a (p,0)-complex form alpha is called holomorphic if partial^_alpha=0, 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.

See also

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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Cite this as:

Rowland, Todd. "Del Bar Operator." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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