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===Courant bracket===
In ordinary complex geometry, an [[almost complex structure]] is [[Foliation#Foliations and integrability|integrable]] to a [[linear complex structure|complex structure]] if and only if the [[Lie derivative|Lie bracket]] of two sections of the [[Holomorphic function|holomorphic]] subbundle is another section of the holomorphic subbundle.
In generalized complex geometry one is not interested in vector fields, but rather in the formal sums of vector fields and one-forms. A kind of Lie bracket for such formal sums was introduced in 1990 and is called the [[Courant bracket]] which is defined by
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===Complex manifolds===
The space of complex differential forms '''Λ<sup>*</sup>T'''<math>\otimes</math>'''C''' has a complex conjugation operation given by complex conjugation in '''C'''. This allows one to define [[Holomorphic function|holomorphic]] and [[antiholomorphic]] one-forms and (''m, n'')-forms, which are homogeneous polynomials in these one-forms with ''m'' holomorphic factors and ''n'' antiholomorphic factors. In particular, all (''n,0'')-forms are related locally by multiplication by a complex function and so they form a complex line bundle.
(''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 ('''T'''<math>\oplus</math>
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