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.
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
One can show, by the Cauchy–Riemann equations, that the spaces Ω1,0 and Ω0,1 are stable under holomorphic coordinate changes.
Just as with the two spaces of 1-forms, these are stable under holomorphic changes of coordinates, and so determine vector bundles.
If Ek is the space of all complex differential forms of total degree k, then each element of Ek can be expressed in a unique way as a linear combination of elements from among the spaces Ωp,q with p + q = k. More succinctly, there is a direct sum decomposition Because this direct sum decomposition is stable under holomorphic coordinate changes, it also determines a vector bundle decomposition.
In particular, for each k and each p and q with p + q = k, there is a canonical projection of vector bundles The usual exterior derivative defines a mapping of sections
via The exterior derivative does not in itself reflect the more rigid complex structure of the manifold.
On compact Kähler manifolds a global form of the local
It is a consequence of Hodge theory, and states that a complex differential form which is globally
-exact (in other words, whose class in de Rham cohomology is zero) is globally
In local coordinates, then, a holomorphic p-form can be written in the form where the
Equivalently, and due to the independence of the complex conjugate, the (p, 0)-form α is holomorphic if and only if The sheaf of holomorphic p-forms is often written Ωp, although this can sometimes lead to confusion so many authors tend to adopt an alternative notation.