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complex differential form
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.
==Differential forms on a complex manifold==
Suppose that ''M'' is a complex manifold. Then there is a local coordinate system consisting of ''n'' complex-valued functions ''z''1,...,z''n'' 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.
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