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: for each j. Letting
dzj=dxj+idyj, d\bar{z}j=dxj-idyj,
j\right). | |
\sum | |
jd\bar{z} |
Let Ω1,0 be the space of complex differential forms containing only
dz
d\bar{z}
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,
\Omegap,q=\underbrace{\Omega1,0\wedge...b\wedge\Omega1,0
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
Ek=\Omegak,0 ⊕ \Omegak-1,1 ⊕ ...b ⊕ \Omega1,k-1 ⊕ \Omega0,k=oplusp+q=k\Omegap,q.
In particular, for each k and each p and q with, there is a canonical projection of vector bundles
\pip,q:Ek → \Omegap,q.
The usual exterior derivative defines a mapping of sections
d:\Omegar\to\Omegar+1
d(\Omegap,q)\subseteqoplusr\Omegar,s
Using d and the projections defined in the previous subsection, it is possible to define the Dolbeault operators:
\partial=\pip+1,q\circd:\Omegap,q → \Omegap+1,q, \bar{\partial}=\pip,q+1\circd:\Omegap,q → \Omegap,q+1
\alpha=\sum|I|=p,|J|=q fIJdzI\wedged\bar{z}J\in\Omegap,q
\partial\alpha=\sum|I|,|J|\sum\ell
\partialfIJ | |
\partialz\ell |
dz\ell\wedgedzI\wedged\bar{z}J
\bar{\partial}\alpha=\sum|I|,|J|\sum\ell
\partialfIJ | |
\partial\bar{z |
\ell}d\bar{z}\ell\wedgedzI\wedged\bar{z}J.
The following properties are seen to hold:
d=\partial+\bar{\partial}
\partial2=\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
d
The Poincaré lemma for
\bar\partial
\partial
\partial\bar\partial
d
\partial\bar\partial
\partial\bar\partial
\partial\bar\partial
d
\partial\bar\partial
For each p, a holomorphic p-form is a holomorphic section of the bundle Ωp,0. In local coordinates, then, a holomorphic p-form can be written in the form
\alpha=\sum|I|=p
I | |
f | |
Idz |
where the
fI
\bar{\partial}\alpha=0.
. P. Griffiths . Phillip Griffiths . J. Harris . Joe Harris (mathematician) . Principles of Algebraic Geometry . Wiley Classics Library . Wiley Interscience . 1994 . 0-471-05059-8 . 23–25 .
. Raymond O. Wells Jr.. Differential analysis on complex manifolds. 1973. Springer-Verlag. 0-387-90419-0.
. Claire Voisin. Hodge Theory and Complex Algebraic Geometry I. 2008. Cambridge University Press. 978-0521718011.