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, ..., 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.
One-forms We begin with the case of one-forms. First decompose the complex coordinates into their real and imaginary parts: for each
j. Letting :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 :\sum_{j=1}^n\left(f_jdz^j+g_jd\bar{z}^j\right). Let Ω1,0 be the space of complex differential forms containing only dz's and Ω0,1 be the space of forms containing only d\bar{z}'s. One can show, by the
Cauchy–Riemann equations, that the spaces Ω1,0 and Ω0,1 are stable under holomorphic coordinate changes. In other words, if one makes a different choice
wi of holomorphic coordinate system, then elements of Ω1,0 transform
tensorially, as do elements of Ω0,1. Thus the spaces Ω0,1 and Ω1,0 determine complex
vector bundles on the complex manifold.
Higher-degree forms 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 Ωp,q of (
p,
q)-forms is defined by taking linear combinations of the wedge products of
p elements from Ω1,0 and
q elements from Ω0,1. Symbolically, :\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 Ω1,0 and
q factors of Ω0,1. 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 . More succinctly, there is a
direct sum decomposition :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 , there is a canonical projection of vector bundles :\pi^{p,q}:E^k\rightarrow\Omega^{p,q}.
The Dolbeault operators The usual
exterior derivative defines a mapping of sections d: \Omega^{r} \to \Omega^{r+1} via : 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: :\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 :\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 :\partial\alpha=\sum_{I,J}\sum_\ell \frac{\partial f_{IJ}}{\partial z^\ell}\,dz^\ell\wedge dz^I\wedge d\bar{z}^J :\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: :d=\partial+\bar{\partial} :\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 that result from splitting of the
homotopy operator for d. This is a content of the
Poincaré lemma on a complex manifold. The Poincaré lemma for \bar \partial and \partial can be improved further to the
local \partial \bar \partial-lemma, which shows that every d-exact complex differential form is actually \partial \bar \partial-exact. On compact
Kähler manifolds a global form of the local \partial \bar \partial-lemma holds, known as the
\partial \bar \partial-lemma. It is a consequence of
Hodge theory, and states that a complex differential form which is globally d-exact (in other words, whose class in
de Rham cohomology is zero) is globally \partial \bar \partial-exact.
Holomorphic forms 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 :\alpha=\sum_{|I|=p}f_I\,dz^I where the 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 :\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. == See also ==