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| In [[mathematics]], a '''complex differential form''' is a [[differential form]] on a [[manifold]] (usually a [[complex manifold]]) which is permitted to have [[complex number|complex]] coefficients.
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| 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 metric|Kähler geometry]], and [[Hodge theory]]. Over non-complex manifolds, they also play a role in the study of [[almost complex structure]]s, the theory of [[spinor]]s, and [[CR structure]]s.
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| 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'' [[exterior derivative|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.
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| ==Differential forms on a complex manifold==
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| Suppose that ''M'' is a [[complex manifold]]. Then there is a local [[coordinate system]] consisting of ''n'' complex-valued functions ''z''<sup>1</sup>,...,z<sup>''n''</sup> such that the coordinate transitions from one patch to another are [[holomorphic function]]s 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 manifold|smooth]].
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| ===One-forms===
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| We begin with the case of one-forms. First decompose the complex coordinates into their real and imaginary parts: ''z''<sup>''j''</sup>=''x''<sup>''j''</sup>+''iy''<sup>''j''</sup> for each ''j''. Letting
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| :<math>dz^j=dx^j+idy^j,\quad d\bar{z}^j=dx^j-idy^j,</math>
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| one sees that any differential form with complex coefficients can be written uniquely as a sum
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| :<math>\sum_{j=1}^n f_jdz^j+g_jd\bar{z}^j.</math>
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| Let Ω<sup>1,0</sup> be the space of complex differential forms containing only <math>dz</math>'s and Ω<sup>0,1</sup> be the space of forms containing only <math>d\bar{z}</math>'s. One can show, by the [[Cauchy-Riemann equations]], that the spaces Ω<sup>1,0</sup> and Ω<sup>0,1</sup> are stable under holomorphic coordinate changes. In other words, if one makes a different choice ''w''<sub>i</sub> of holomorphic coordinate system, then elements of Ω<sup>1,0</sup> transform [[tensor]]ially, as do elements of Ω<sup>0,1</sup>. Thus the spaces Ω<sup>0,1</sup> and Ω<sup>1,0</sup> determine complex [[vector bundle]]s on the complex manifold.
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| ===Higher degree forms===
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| 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<sup>p,q</sup> of (''p'',''q'')-forms is defined by taking linear combinations of the wedge products of ''p'' elements from Ω<sup>1,0</sup> and ''q'' elements from Ω<sup>0,1</sup>. Symbolically,
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| :<math>\Omega^{p,q}=\Omega^{1,0}\wedge\dotsb\wedge\Omega^{1,0}\wedge\Omega^{0,1}\wedge\dotsb\wedge\Omega^{0,1}</math>
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| where there are ''p'' factors of Ω<sup>1,0</sup> and ''q'' factors of Ω<sup>0,1</sup>. Just as with the two spaces of 1-forms, these are stable under holomorphic changes of coordinates, and so determine vector bundles.
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| If ''E''<sup>''k''</sup> is the space of all complex differential forms of total degree ''k'', then each element of ''E''<sup>''k''</sup> can be expressed in a unique way as a linear combination of elements from among the spaces Ω<sup>p,q</sup> with ''p''+''q''=''k''. More succinctly, there is a [[direct sum of vector bundles|direct sum]] decomposition
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| :<math>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}.</math>
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| Because this direct sum decomposition is stable under holomorphic coordinate changes, it also determines a vector bundle decomposition.
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| In particular, for each ''k'' and each ''p'' and ''q'' with ''p''+''q''=''k'', there is a canonical projection of vector bundles
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| :<math>\pi^{p,q}:E^k\rightarrow\Omega^{p,q}.</math>
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| ===The Dolbeault operators===
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| The usual exterior derivative defines a mapping of sections ''d'':''E''<sup>''k''</sup>→''E''<sup>''k+1''</sup>. Restricting this to sections of Ω<sup>''p,q''</sup>, one can show that in fact ''d'':Ω<sup>''p,q''</sup>→Ω<sup>''p''+1,''q''</sup> + Ω<sup>''p'',''q''+1</sup>.{{clarify|reason=This is an object. In fact, it exists? Equals something?|date=November 2011}} The exterior derivative does not in itself reflect the more rigid complex structure of the manifold.
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| Using ''d'' and the projections defined in the previous subsection, it is possible to define the '''Dolbeault operators''':
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| :<math>\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}</math>
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| To describe these operators in local coordinates, let
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| :<math>\alpha=\sum_{|I|=p,|J|=q}\ f_{IJ}\,dz^I\wedge d\bar{z}^J\in\Omega^{p,q}</math>
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| where ''I'' and ''J'' are [[multi-index|multi-indices]]. Then
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| :<math>\partial\alpha=\sum_{|I|,|J|}\sum_\ell \frac{\partial f_{IJ}}{\partial z^\ell}\,dz^\ell\wedge dz^I\wedge d\bar{z}^J</math>
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| :<math>\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.</math>
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| The following properties are seen to hold:
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| :<math>d=\partial+\bar{\partial}</math>
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| :<math>\partial^2=\bar{\partial}^2=\partial\bar{\partial}+\bar{\partial}\partial=0.</math>
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| These operators and their properties form the basis for [[Dolbeault cohomology]] and many aspects of [[Hodge theory]].
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| ===Holomorphic forms===
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| For each ''p'', a holomorphic ''p''-form is a holomorphic section of the bundle Ω<sup>''p,0''</sup>. In local coordinates, then, a holomorphic ''p''-form can be written in the form
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| :<math>\alpha=\sum_{|I|=p}f_I\,dz^I</math>
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| where the ''f''<sub>''I''</sub> are holomorphic functions. Equivalently, the (''p'',0)-form α is holomorphic if and only if
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| :<math>\bar{\partial}\alpha=0.</math>
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| The [[sheaf (mathematics)|sheaf]] of holomorphic ''p''-forms is often written Ω<sup>''p''</sup>, although this can sometimes lead to confusion so many authors tend to adopt an alternative notation.
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| ==See also==
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| *[[Dolbeault complex]]
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| *[[Frölicher spectral sequence]]
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| *[[Differential of the first kind]]
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| ==References==
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| * {{cite book|last=Wells|first=R.O.|authorlink=Raymond O. Wells, Jr.|title=Differential analysis on complex manifolds|year=1973|publisher=Springer-Verlag|isbn=0-387-90419-0}}
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| [[Category:Complex manifolds]]
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| [[Category:Differential forms]]
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