Complex differential form

We begin with the case of one-forms. First decompose the complex coordinates into their real and imaginary parts: z^j = x^j + iy^j for each j. Letting

${\displaystyle dz^{j}=dx^{j}+idy^{j},\quad d{\bar {z}}^{j}=dx^{j}-idy^{j},}$

one sees that any differential form with complex coefficients can be written uniquely as a sum

${\displaystyle \sum _{j=1}^{n}\left(f_{j}dz^{j}+g_{j}d{\bar {z}}^{j}\right).}$

Let Ω^1,0 be the space of complex differential forms containing only ${\displaystyle dz}$'s and Ω^0,1 be the space of forms containing only ${\displaystyle d{\bar {z}}}$'s. One can show, by the Cauchy–Riemann equations, that the spaces Ω^1,0 and Ω^0,1 are stable under holomorphic coordinate changes. In other words, if one makes a different choice w_i of holomorphic coordinate system, then elements of Ω^1,0 transform tensorially, as do elements of Ω^0,1. Thus the spaces Ω^0,1 and Ω^1,0 determine complex vector bundles on the complex manifold.

The wedge product of complex differential forms is defined in the same way as with real forms. Let p and q be a pair of non-negative integers ≤ n. The space Ω^p,q of (p, q)-forms is defined by taking linear combinations of the wedge products of p elements from Ω^1,0 and q elements from Ω^0,1. Symbolically,

${\displaystyle \Omega ^{p,q}=\underbrace {\Omega ^{1,0}\wedge \dotsb \wedge \Omega ^{1,0}} _{p{\text{ times}}}\wedge \underbrace {\Omega ^{0,1}\wedge \dotsb \wedge \Omega ^{0,1}} _{q{\text{ times}}}}$

where there are p factors of Ω^1,0 and q factors of Ω^0,1. Just as with the two spaces of 1-forms, these are stable under holomorphic changes of coordinates, and so determine vector bundles.

If E^k is the space of all complex differential forms of total degree k, then each element of E^k can be expressed in a unique way as a linear combination of elements from among the spaces Ω^p,q with p + q = k. More succinctly, there is a direct sum decomposition

${\displaystyle E^{k}=\Omega ^{k,0}\oplus \Omega ^{k-1,1}\oplus \dotsb \oplus \Omega ^{1,k-1}\oplus \Omega ^{0,k}=\bigoplus _{p+q=k}\Omega ^{p,q}.}$

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

${\displaystyle \pi ^{p,q}:E^{k}\rightarrow \Omega ^{p,q}.}$

The usual exterior derivative defines a mapping of sections ${\displaystyle d:\Omega ^{r}\to \Omega ^{r+1}}$ via

${\displaystyle d(\Omega ^{p,q})\subseteq \bigoplus _{r+s=p+q+1}\Omega ^{r,s}}$

The exterior derivative does not in itself reflect the more rigid complex structure of the manifold.

Using d and the projections defined in the previous subsection, it is possible to define the Dolbeault operators:

${\displaystyle \partial =\pi ^{p+1,q}\circ d:\Omega ^{p,q}\rightarrow \Omega ^{p+1,q},\quad {\bar {\partial }}=\pi ^{p,q+1}\circ d:\Omega ^{p,q}\rightarrow \Omega ^{p,q+1}}$

To describe these operators in local coordinates, let

${\displaystyle \alpha =\sum _{|I|=p,|J|=q}\ f_{IJ}\,dz^{I}\wedge d{\bar {z}}^{J}\in \Omega ^{p,q}}$

where I and J are multi-indices. Then

${\displaystyle \partial \alpha =\sum _{|I|,|J|}\sum _{\ell }{\frac {\partial f_{IJ}}{\partial z^{\ell }}}\,dz^{\ell }\wedge dz^{I}\wedge d{\bar {z}}^{J}}$

${\displaystyle {\bar {\partial }}\alpha =\sum _{|I|,|J|}\sum _{\ell }{\frac {\partial f_{IJ}}{\partial {\bar {z}}^{\ell }}}d{\bar {z}}^{\ell }\wedge dz^{I}\wedge d{\bar {z}}^{J}.}$

The following properties are seen to hold:

${\displaystyle d=\partial +{\bar {\partial }}}$

${\displaystyle \partial ^{2}={\bar {\partial }}^{2}=\partial {\bar {\partial }}+{\bar {\partial }}\partial =0.}$

These operators and their properties form the basis for Dolbeault cohomology and many aspects of Hodge theory.

On a star-shaped domain of a complex manifold the Dolbeault operators have dual homotopy operators [1] that result from splitting of the homotopy operator for ${\displaystyle d}$.[1] This is a content of the Poincare lemma on a complex manifold.

The Poincaré lemma for ${\displaystyle {\bar {\partial }}}$ and ${\displaystyle \partial }$ can be improved further to the local ${\displaystyle \partial {\bar {\partial }}}$-lemma, which shows that every ${\displaystyle d}$-exact complex differential form is actually ${\displaystyle \partial {\bar {\partial }}}$-exact. On compact Kähler manifolds a global form of the local ${\displaystyle \partial {\bar {\partial }}}$-lemma holds, known as the ${\displaystyle \partial {\bar {\partial }}}$-lemma. It is a consequence of Hodge theory, and states that a complex differential form which is globally ${\displaystyle d}$-exact (in other words, whose class in de Rham cohomology is zero) is globally ${\displaystyle \partial {\bar {\partial }}}$-exact.

For each p, a holomorphic p-form is a holomorphic section of the bundle Ω^p,0. In local coordinates, then, a holomorphic p-form can be written in the form

${\displaystyle \alpha =\sum _{|I|=p}f_{I}\,dz^{I}}$

where the ${\displaystyle f_{I}}$ are holomorphic functions. Equivalently, and due to the independence of the complex conjugate, the (p, 0)-form α is holomorphic if and only if

${\displaystyle {\bar {\partial }}\alpha =0.}$

The sheaf of holomorphic p-forms is often written Ω^p, although this can sometimes lead to confusion so many authors tend to adopt an alternative notation.