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Concept
Complex differential form
Résumé
In mathematics, a complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex coefficients.
Complex forms have broad applications in differential geometry. On complex manifolds, they are fundamental and serve as the basis for much of algebraic geometry, Kähler geometry, and Hodge theory. Over non-complex manifolds, they also play a role in the study of almost complex structures, the theory of spinors, and CR structures.
Typically, complex forms are considered because of some desirable decomposition that the forms admit. On a complex manifold, for instance, any complex k-form can be decomposed uniquely into a sum of so-called (p, q)-forms: roughly, wedges of p differentials of the holomorphic coordinates with q differentials of their complex conjugates. The ensemble of (p, q)-forms becomes the primitive object of study, and determines a finer geometrical structure on the manifold than the k-forms. Even finer structures exist, for example, in cases where Hodge theory applies.
Suppose that M is a complex manifold of complex dimension n. Then there is a local coordinate system consisting of n complex-valued functions z1, ..., zn such that the coordinate transitions from one patch to another are holomorphic functions of these variables. The space of complex forms carries a rich structure, depending fundamentally on the fact that these transition functions are holomorphic, rather than just smooth.
We begin with the case of one-forms. First decompose the complex coordinates into their real and imaginary parts: zj = xj + iyj for each j. Letting
one sees that any differential form with complex coefficients can be written uniquely as a sum
Let Ω1,0 be the space of complex differential forms containing only 's and Ω0,1 be the space of forms containing only 's. One can show, by the Cauchy–Riemann equations, that the spaces Ω1,0 and Ω0,1 are stable under holomorphic coordinate changes.
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