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{{Short description|Property of a differential manifold that includes complex structures}}
{{No footnotes|date=June 2020}}
In the field of [[mathematics]] known as [[differential geometry]], a '''generalized complex structure''' is a property of a [[differential manifold]] that includes as special cases a [[linear complex structure|complex structure]] and a [[symplectic structure]]. Generalized complex structures were introduced by [[Nigel Hitchin]] in 2002 and further developed by his students [[Marco Gualtieri]] and [[Gil Cavalcanti]].
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===The generalized tangent bundle===
Consider an [[Manifold|''N''-manifold]] ''M''. The [[tangent bundle]] of ''M'', which will be denoted '''T''', is the [[vector bundle]] over ''M'' whose fibers consist of all [[tangent vector]]s to ''M''. A [[fiber bundle#Sections|section]] of '''T''' is a [[vector field]] on ''M''. The [[cotangent bundle]] of ''M'', denoted '''T'''<sup>*</sup>, is the vector bundle over ''M'' whose sections are [[differential form|one-forms]] on ''M''.
In [[complex geometry]] one considers structures on the tangent bundles of manifolds. In [[symplectic geometry]] one is instead interested in [[Exterior algebra#
The fibers are endowed with a natural [[inner product]] with [[Metric signature
:<math>\langle X+\xi,Y+\eta\rangle=\frac{1}{2}(\xi(Y)+\eta(X)).</math>
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:<math>\langle {\mathcal J}(X+\xi),{\mathcal J}(Y+\eta)\rangle=\langle X+\xi, Y+\eta \rangle.</math>
Like in the case of an ordinary [[almost complex structure]], a generalized almost complex structure is uniquely determined by its <math>\sqrt{-1}</math>-[[Vector bundle#Operations on vector bundles|eigenbundle]], i.e. a subbundle <math>L</math> of the complexified generalized tangent bundle <math>(\mathbf{T}\oplus\mathbf{T}^*)\otimes\
given by
:<math>L=\{X+\xi\in (\mathbf{T}\oplus\mathbf{T}^*)\otimes\
Such subbundle ''L'' satisfies the following properties:
{{ordered list | list-style-type=lower-roman
▲(ii) ''L'' is '''maximal isotropic''', i.e. its complex [[rank (linear algebra)|rank]] equals ''N'' and <math>\langle\ell, \ell' \rangle =0</math> for all <math>\ell,\ell'\in L.</math>
Vice versa, any subbundle ''L'' satisfying (i), (ii) is the <math>\sqrt{-1}</math>-eigenbundle of a unique generalized almost complex structure, so that the properties (i), (ii) can be considered as an alternative definition of generalized almost complex structure.
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where <math>\mathcal{L}_X</math> is the [[Lie derivative]] along the vector field ''X'', ''d'' is the [[exterior derivative]] and ''i'' is the [[Exterior algebra#The interior product or insertion operator|interior product]].
===
A '''generalized complex structure''' is a generalized almost complex structure such that the space of smooth sections of ''L'' is closed under the Courant bracket.
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Given a pair <math>(\mathbf{E}, \varepsilon)</math> one can construct a maximally isotropic subbundle <math>L(\mathbf{E}, \varepsilon)</math> of <math>\mathbf{T} \oplus \mathbf{T}^*</math> as follows. The elements of the subbundle are the [[formal sum]]s <math>X+\xi</math> where the [[vector field]] ''X'' is a section of '''E''' and the one-form ''ξ'' restricted to the [[dual space]] <math>\mathbf{E}^*</math> is equal to the one-form <math>\varepsilon(X).</math>
To see that <math>L(\mathbf{E}, \varepsilon)</math> is isotropic, notice that if ''Y'' is a section of '''E''' and <math>\xi</math> restricted to <math>\mathbf{E}^*</math> is <math>\varepsilon(X)</math> then <math>\xi(Y) =\varepsilon(X,Y),</math> as the part of <math>\xi</math> orthogonal to <math>\mathbf{E}^*</math> annihilates ''Y''.
:<math>\langle X+\xi,Y+\eta\rangle=\frac{1}{2}(\xi(Y)+\eta(X))=\frac{1}{2}(\varepsilon(Y,X)+\varepsilon(X,Y))=0</math>
and so <math>L(\mathbf{E}, \varepsilon)</math> is isotropic. Furthermore, <math>L(\mathbf{E}, \varepsilon)</math> is maximal because there are <math>\dim(\mathbf{E})</math> (complex) dimensions of choices for <math>\mathbf{E},</math> and <math>\varepsilon</math> is unrestricted on the [[complement (complexity)|complement]] of <math>\mathbf{E}^*,</math> which is of (complex) dimension <math>n-\dim(\mathbf{E}).</math> Thus the total (complex) dimension
===Type===
The '''type''' of a maximal isotropic subbundle <math>L(\mathbf{E}, \varepsilon)</math> is the real dimension of the subbundle that annihilates '''E'''. Equivalently it is 2''N'' minus the real dimension of the [[projection (mathematics)|projection]] of <math>L(\mathbf{E}, \varepsilon)</math> onto the tangent bundle '''T'''. In other words, the type of a maximal isotropic subbundle is the codimension of its projection onto the tangent bundle. In the complex case one uses the complex dimension and the type is sometimes referred to as the '''complex type'''. While the type of a subbundle can in principle be any integer between 0 and 2''N'', generalized almost complex structures cannot have a type greater than ''N'' because the sum of the subbundle and its complex conjugate must be all of <math>(\mathbf{T} \oplus \mathbf{T}^*) \otimes \
The type of a maximal isotropic subbundle is [[Invariant (mathematics)|invariant]] under [[diffeomorphisms]] and also under shifts of the [[
:<math>X+\xi\longrightarrow X+\xi+i_XB</math>
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===Real index===
The real index ''r'' of a maximal isotropic subspace ''L'' is the complex dimension of the [[intersection (set theory)|intersection]] of ''L'' with its complex conjugate. A maximal isotropic subspace of <math>(\mathbf{T} \oplus \mathbf{T}^*) \otimes \
==Canonical bundle==
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===Generalized almost complex structures===
The canonical bundle is a one complex dimensional subbundle of the bundle <math>\mathbf{\Lambda}^* \mathbf{T} \otimes \
A spinor is said to be a '''pure spinor''' if it is annihilated by half
Given a generalized almost complex structure, one can also determine a pure spinor up to multiplication by an arbitrary [[complex function]]. These choices of pure spinors are defined to be the sections of the canonical bundle.
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===Regular point===
Define the subbundle '''E''' of the complexified tangent bundle <math>\mathbf{T} \otimes \
:<math>E\cap\overline{E}=\Delta\otimes\
for some subbundle Δ. A point which has an [[open set|open]] [[neighborhood (mathematics)|neighborhood]] in which the dimension of the fibers of Δ is constant is said to be a '''regular point'''.
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===Darboux's theorem===
{{main|Darboux's theorem}}
Every regular point in a generalized complex manifold has an open neighborhood which, after a diffeomorphism and shift of the B-field, has the same generalized complex structure as the [[Cartesian product]] of the [[Linear complex structure|complex vector space]] <math>\
===Local holomorphicity===
Near non-regular points, the above [[classification theorem]] does not apply. However, about any point, a generalized complex manifold is, up to diffeomorphism and B-field, a product of a symplectic manifold with a generalized complex manifold which is of complex type at the point, much like Weinstein's theorem for the local structure of [[Poisson manifold]]s. The remaining question of the local structure is: what does a generalized complex structure look like near a point of complex type? In fact, it will be induced by a holomorphic [[Poisson manifold|Poisson structure]].
==Examples==
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===Complex manifolds===
The space of complex differential forms <math>\mathbf{\Lambda}^* \mathbf{T} \otimes \
(''n'', 0)-forms are pure spinors, as they are annihilated by antiholomorphic tangent vectors and by holomorphic one-forms. Thus this line bundle can be used as a canonical bundle to define a generalized complex structure. Restricting the annihilator from <math>(\mathbf{T} \oplus \mathbf{T}^*) \otimes \
As only half of a basis of vector fields are holomorphic, these complex structures are of type ''N''. In fact complex manifolds, and the manifolds obtained by multiplying the pure spinor bundle defining a complex manifold by a complex, <math>\partial</math>-closed (2,0)-form, are the only type ''N'' generalized complex manifolds.
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Some of the almost structures in generalized complex geometry may be rephrased in the language of [[G-structure]]s. The word "almost" is removed if the structure is integrable.
The bundle <math>(\mathbf{T} \oplus \mathbf{T}^*) \otimes \
:<math>\frac{O(2n,2n)}{U(n,n)}.</math>
A [[generalized Kähler structure|generalized almost Kähler structure]] is a pair of [[commutative operation|commuting]] generalized complex structures such that minus the product of the corresponding tensors is a positive definite metric on <math>(\mathbf{T} \oplus \mathbf{T}^*) \otimes \
Finally, a generalized almost Calabi-Yau metric structure is a further reduction of the structure group to <math>SU(n) \times SU(n).</math>
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==References==
*
*{{cite thesis |last=Gualtieri
*{{cite journal |last=Gualtieri
*{{cite journal |last=Graña
*
{{String theory topics |state=collapsed}}
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