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# Complex Form

The differential forms on decompose into forms of type , sometimes called -forms. For example, on , the exterior algebra decomposes into four types:

 (1) (2)

where , , and denotes the direct sum. In general, a -form is the sum of terms with s and s. A -form decomposes into a sum of -forms, where .

For example, the 2-forms on decompose as

 (3) (4)

The decomposition into forms of type is preserved by holomorphic functions. More precisely, when is holomorphic and is a -form on , then the pullback is a -form on .

Recall that the exterior algebra is generated by the one-forms, by wedge product and addition. Then the forms of type are generated by

 (5)

The subspace of the complex one-forms can be identified as the -eigenspace of the almost complex structure , which satisfies . Similarly, the -eigenspace is the subspace . In fact, the decomposition of determines the almost complex structure on .

More abstractly, the forms into type are a group representation of , where acts by multiplication by .

Almost Complex Structure, Complex Manifold, Del Bar Operator, Dolbeault Cohomology

This entry contributed by Todd Rowland

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Rowland, Todd. "Complex Form." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ComplexForm.html