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Indifferential geometry andcomplex geometry, acomplex manifold or acomplex analytic manifold is amanifold with acomplex structure, that is anatlas ofcharts to theopen unit disc^[1] in thecomplex coordinate space${\displaystyle {\mathbb{C}}^n}$, such that thetransition maps areholomorphic.

The term "complex manifold" is variously used to mean a complex manifold in the sense above (which can be specified as anintegrable complex manifold) or analmost complex manifold.

Sinceholomorphic functions are much more rigid thansmooth functions, the theories ofsmooth and complex manifolds have very different flavors:compact complex manifolds are much closer toalgebraic varieties than to differentiable manifolds. In particular, while complex manifold and complex-analytic manifold are the same,smooth manifold andreal-analytic manifold are not the same.

For example, theWhitney embedding theorem tells us that every smoothn-dimensional manifold can beembedded as a smooth submanifold ofR²ⁿ, whereas it is "rare" for a complex manifold to have a holomorphic embedding intoCⁿ. Consider for example anycompact connected complex manifoldM: any holomorphic function on it is constant bythe maximum modulus principle. Now if we had a holomorphic embedding ofM intoCⁿ, then the coordinate functions ofCⁿ would restrict to nonconstant holomorphic functions onM, contradicting compactness, except in the case thatM is just a point. Complex manifolds that can be embedded inCⁿ are calledStein manifolds and form a very special class of manifolds including, for example, smooth complex affine algebraic varieties.

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 manysmooth structures, a topological manifold supporting a complex structure can and often does support uncountably many complex structures.Riemann surfaces, two dimensional manifolds equipped with a complex structure, which are topologically classified by thegenus, 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 amoduli space, the structure of which remains an area of active research.

Since the transition maps between charts are biholomorphic, complex manifolds are, in particular, smooth and canonically oriented (not justorientable: a biholomorphic map to (a subset of)Cⁿ gives an orientation, as biholomorphic maps are orientation-preserving).

• Riemann surfaces.
• Calabi–Yau manifolds.
• The Cartesian product of two complex manifolds.
• The inverse image of any noncritical value of a holomorphic map.

Smooth complexalgebraic varieties are complex manifolds, including:

• Complex vector spaces.
• Complex projective spaces,^[2]Pⁿ(C).
• ComplexGrassmannians.
• ComplexLie groups such as GL(n,C) or Sp(n,C).

Thesimply connected 1-dimensional complex manifolds are isomorphic to either:

• Δ, the unitdisk inC
• C, the complex plane
• Ĉ, theRiemann sphere

Note that there are inclusions between these asΔ ⊆C ⊆Ĉ, but that there are no non-constant holomorphic maps in the other direction, byLiouville's theorem.

The following spaces are different as complex manifolds, demonstrating the more rigid geometric character of complex manifolds (compared to smooth manifolds):

• complex space${\displaystyle {\mathbb{C}}^n}$.
• the unit disc oropen ball

• thepolydisc

Analmost complex structure on a real 2n-manifold is a GL(n,C)-structure (in the sense ofG-structures) – that is, the tangent bundle is equipped with alinear complex structure.

Concretely, this is anendomorphism of thetangent bundle whose square is −I; this endomorphism is analogous to multiplication by the imaginary numberi, and is denotedJ (to avoid confusion with the identity matrixI). An almost complex manifold is necessarily even-dimensional.

An almost complex structure isweaker 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 almost complex structure can be defined globally. An almost complex structure that comes from a complex structure is calledintegrable, and when one wishes to specify a complex structure as opposed to an almost complex structure, one says anintegrable complex structure. For integrable complex structures the so-calledNijenhuis tensor vanishes. This tensor is defined on pairs of vector fields,X,Y by

${\displaystyle N_J(X,Y)=[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY].}$

For example, the 6-dimensionalsphereS⁶ has a natural almost complex structure arising from the fact that it is theorthogonal complement ofi in the unit sphere of theoctonions, but this is not a complex structure. (The question of whether it has a complex structure is known as theHopf problem, afterHeinz Hopf.^[3]) 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).

Tensoring the tangent bundle with thecomplex numbers we get thecomplexified 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 byT^0,1M andT^1,0M. TheNewlander–Nirenberg theorem shows that an almost complex structure is actually a complex structure precisely when these subbundles areinvolutive, i.e., closed under the Lie bracket of vector fields, and such an almost complex structure is calledintegrable.

One can define an analogue of aRiemannian metric for complex manifolds, called aHermitian 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, i.e. a Hermitian structure is not rigid.

A Hermitian form can be decomposed into a real part and a complex part as${\displaystyle h(X,Y)=g(X,Y)+i\omega (X,Y)}$. The real part is a Riemannian metric, and the complex part isskew-symmetric. If the skew-symmetric part issymplectic, i.e.closed andnondegenerate, then the metric is calledKähler. Kähler structures are rare and are much more rigid.

Examples ofKähler manifolds include smoothprojective varieties and more generally any complex submanifold of a Kähler manifold. TheHopf 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 firstBetti number is one, so by theHodge theory, it cannot be Kähler.

ACalabi–Yau manifold can be defined as a compactRicci-flat Kähler manifold or equivalently one whose firstChern class vanishes.

• Complex dimension
• Complex analytic variety
• Quaternionic manifold
• Real-complex manifold

• Kodaira, Kunihiko (17 November 2004).Complex Manifolds and Deformation of Complex Structures. Classics in Mathematics. Springer.ISBN 3-540-22614-1.

• Complex manifolds
• Differential geometry

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