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| In [[differential geometry]], a '''complex manifold''' is a [[manifold]] with an [[atlas (topology)|atlas]] of [[chart (topology)|charts]] to the [[open unit disk]]<ref>One must use the open unit disk in '''C'''<sup>''n''</sup> as the model space instead of '''C'''<sup>''n''</sup> because these are not isomorphic, unlike for real manifolds.</ref> in '''C'''<sup>''n''</sup>, such that the [[transition map]]s are [[holomorphic]].
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| The term '''complex manifold''' is variously used to mean a complex manifold in the sense above (which can be specified as an '''integrable''' complex manifold), and an [[almost complex manifold]].
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| ==Implications of complex structure==
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| Since [[holomorphic function]]s are much more rigid than [[smooth function]]s, the theories of smooth and complex manifolds have very different flavors: compact complex manifolds are much closer to [[algebraic variety|algebraic varieties]] than to differentiable manifolds.
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| For example, the [[Whitney embedding theorem]] tells us that every smooth manifold can be embedded as a smooth submanifold of '''R'''<sup>''n''</sup>, whereas it is "rare" for a complex manifold to have a holomorphic embedding into '''C'''<sup>''n''</sup>. Consider for example any [[compact space|compact]] connected complex manifold ''M'': any holomorphic function on it is [[locally constant]] by [[Liouville's theorem (complex analysis)|Liouville's theorem]]. Now if we had a holomorphic embedding of ''M'' into '''C'''<sup>''n''</sup>, then the coordinate functions of '''C'''<sup>''n''</sup> would restrict to nonconstant holomorphic functions on ''M'', contradicting compactness, except in the case that ''M'' is just a point. Complex manifolds that can be embedded in '''C'''<sup>''n''</sup> are called [[Stein manifold]]s and form a very special class of manifolds including, for example, smooth complex affine algebraic varieties.
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| The classification of complex manifolds is much more subtle than that of differentiable manifolds. For example, while in dimensions other than four, a given topological manifold has at most finitely many [[smooth structure]]s, a topological manifold supporting a complex structure can and often does support uncountably many complex structures. [[Riemann surface]]s, two dimensional manifolds equipped with a complex structure, which are topologically classified by the [[genus (mathematics)|genus]], are an important example of this phenomenon. The set of complex structures on a given orientable surface, modulo biholomorphic equivalence, itself forms a complex algebraic variety called a [[moduli space]], the structure of which remains an area of active research.
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| Since the transition maps between charts are biholomorphic, complex manifolds are, in particular, smooth and canonically oriented (not just [[orientable]]: a biholomorphic map to (a subset of) '''C'''<sup>''n''</sup> gives an orientation, as biholomorphic maps are orientation-preserving).
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| ==Examples of complex manifolds==
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| * [[Riemann surface]]s.
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| * The Cartesian product of two complex manifolds.
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| * The inverse image of any noncritical value of a holomorphic map.
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| ===Smooth complex algebraic varieties===
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| Smooth complex [[algebraic varieties]] are complex manifolds, including:
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| * Complex vector spaces.
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| * [[Complex projective space]]s,<ref>This means that all complex projective spaces are ''orientable'', in contrast to the real case</ref> '''P'''<sup>''n''</sup>('''C''').
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| * Complex [[Grassmannian]]s.
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| * Complex [[Lie groups]] such as GL(''n'', '''C''') or Sp(''n'', '''C''').
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| Similarly, the [[quaternions|quaternionic]] analogs of these are also complex manifolds.
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| ===Simply connected===
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| The [[simply connected]] 1-dimensional complex manifolds are isomorphic to either:
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| * Δ, the unit disk in '''C'''
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| * '''C''', the complex plane
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| * '''Ĉ''', the [[Riemann sphere]]
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| Note that there are inclusions between these as
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| Δ ⊆ '''C''' ⊆ '''Ĉ''', but that there are no non-constant maps in the other direction, by
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| [[Liouville's theorem (complex analysis)|Liouville's theorem]].
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| ==Disk vs. space vs. polydisk==
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| The following spaces are different as complex manifolds, demonstrating the more rigid geometric character of complex manifolds (compared to smooth manifolds):
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| * complex space '''C'''<sup>''n''</sup>.
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| * the unit disk or [[open ball]]
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| ::<math>\left \{ z \in \mathbf{C}^n \ : \ \|z\| < 1 \right \}.</math>
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| * the [[polydisk]]
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| ::<math>\left \{ z=(z_1, z_2, \dots, z_n) \in \mathbf{C}^n \ : \ \vert z_i \vert < 1, \mbox{ for all } i = 1,\dots,n \right \}.</math>
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| ==Almost complex structures==
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| {{main|Almost complex manifold}}
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| An [[almost complex structure]] on a real manifold is a GL(''n'', '''C''')-structure (in the sense of [[G-structure]]s) – that is, the tangent bundle is equipped with a [[linear complex structure]].
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| Concretely, this is an [[endomorphism]] of the [[tangent bundle]] whose square is −''I''; this endomorphism is analogous to multiplication by the imaginary number ''i'', and is denoted ''J'' (to avoid confusion with the identity matrix ''I''). An almost complex manifold is necessarily even dimensional.
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| An almost complex structure is ''weaker'' than a complex structure: any complex manifold has an almost complex structure, but not every almost complex structure comes from a complex structure. Note that every even dimensional real manifold has an almost complex structure defined locally from the local coordinate chart. The question is whether this complex structure can be defined globally. An almost complex structure that comes from a complex structure is called [[Frobenius_theorem_(differential_topology)|integrable]], and when one wishes to specify a complex structure as opposed to an almost complex structure, one says an ''integrable'' complex structure. For integrable complex structures the so-called Nijenhuis tensor vanishes. This tensor is defined on pairs of vector fields, ''X'', ''Y'' by
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| :''N''<sub>''J''</sub>(''X'', ''Y'') = [''X'', ''Y''] + ''J''[''JX'', ''Y''] + ''J''[''X'', ''JY''] − [''JX'', ''JY''].
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| For example, the 6-dimensional [[hypersphere|sphere]] '''S'''<sup>6</sup> has a natural almost complex structure arising from the fact that it is the [[orthogonal complement]] of ''i'' in the unit sphere of the [[octonion]]s, but this is not a complex structure. (It is not currently known whether or not the 6-sphere has a complex structure.) Using an almost complex structure we can make sense of holomorphic maps and ask about the existence of holomorphic coordinates on the manifold. The existence of holomorphic coordinates is equivalent to saying the manifold is complex (which is what the chart definition says).
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| Tensoring the tangent bundle with the complex numbers we get the ''complexified'' tangent bundle, on which multiplication by complex numbers makes sense (even if we started with a real manifold). The eigenvalues of an almost complex structure are ±''i'' and the eigenspaces form sub-bundles denoted by ''T''<sup>0,1</sup>''M'' and ''T''<sup>1,0</sup>''M''. The [[Newlander–Nirenberg theorem]] shows that an almost complex structure is actually a complex structure precisely when these subbundles are ''involutive'', i.e., closed under the Lie bracket of vector fields, and such an almost complex structure is called [[Frobenius_theorem_(differential_topology)|integrable]].
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| == Kähler and Calabi–Yau manifolds ==
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| One can define an analogue of a [[Riemannian metric]] for complex manifolds, called a [[Hermitian metric]]. Like a Riemannian metric, a Hermitian metric consists of a smoothly varying, positive definite inner product on the tangent bundle, which is Hermitian with respect to the complex structure on the tangent space at each point. As in the Riemannian case, such metrics always exist in abundance on any complex manifold. If the skew symmetric part of such a metric is [[Symplectic geometry|symplectic]], i.e. closed and nondegenerate, then the metric is called Kähler. Kähler structures are much more difficult to come by and are much more rigid.
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| Examples of Kähler manifolds include smooth [[projective varieties]] and more generally any complex submanifold of a Kähler manifold. The Hopf manifolds are examples of complex manifolds that are not Kähler. To construct one, take a complex vector space minus the origin and consider the action of the group of integers on this space by multiplication by exp(''n''). The quotient is a complex manifold whose first [[Betti number]] is one, so by the [[Hodge theory]], it cannot be Kähler.
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| A [[Calabi–Yau manifold]] can be defined as a compact Ricci-flat Kähler manifold or equivalently one whose first [[Chern class]] vanishes.
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| == See also ==
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| * [[Quaternionic manifold]]
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| * [[Real-complex manifold]]
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| ==Footnotes==
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| <references/>
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| ==References==
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| * {{Cite book|last=Kodaira
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| |first=Kunihiko
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| |authorlink=Kunihiko Kodaira
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| |title=Complex Manifolds and Deformation of Complex Structures
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| |series=Classics in Mathematics
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| |publisher=Springer
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| |isbn=3-540-22614-1
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| }}
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| {{DEFAULTSORT:Complex Manifold}}
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| [[Category:Complex manifolds| ]]
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Not much to say about me at all.
Enjoying to be a member of wmflabs.org.
I just hope Im useful in one way .
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