Complex Form

The differential forms on C^n decompose into forms of type (p,q), sometimes called (p,q)-forms. For example, on C, the exterior algebra decomposes into four types:

 ^ C= ^ ^0 direct sum  ^ ^(1,0) direct sum  ^ ^(0,1) direct sum  ^ ^(1,1)
=<1> direct sum <dz> direct sum <dz^_> direct sum <dz ^ dz^_>,

where dz=dx+idy, dz^_=dx-idy, and  direct sum denotes the direct sum. In general, a (p,q)-form is the sum of terms with p dzs and q dz^_s. A k-form decomposes into a sum of (p,q)-forms, where k=p+q.

For example, the 2-forms on C^2 decompose as

 ^ ^2C^2= ^ ^(2,0) direct sum  ^ ^(1,1) direct sum  ^ ^(0,2)
=<dz_1 ^ dz_2> direct sum <dz_1 ^ dz^__1,dz_1 ^ dz^__2,dz_2 ^ dz^__1,dz_2 ^ dz^__2> direct sum <dz^__1 ^ dz^__2>.

The decomposition into forms of type (p,q) is preserved by holomorphic functions. More precisely, when f:X->Y is holomorphic and alpha is a (p,q)-form on Y, then the pullback f^*alpha is a (p,q)-form on X.

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

 Lambda^p(Lambda^(1,0)) ^ Lambda^q(Lambda^(0,1)).

The subspace Lambda^(1,0) of the complex one-forms can be identified as the +i-eigenspace of the almost complex structure J, which satisfies J^2=-I. Similarly, the -i-eigenspace is the subspace Lambda^(0,1). In fact, the decomposition of TX tensor C=TX^(1,0) direct sum TX^(0,1) determines the almost complex structure J on TX.

More abstractly, the forms into type (p,q) are a group representation of C^*, where lambda acts by multiplication by lambda^plambda^_^q.

See also

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

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Complex Form." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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