Complex manifold

Indifferential geometryandcomplex geometry,acomplex manifoldis amanifoldwith acomplex structure,that is anatlasofchartsto theopen unit discin thecomplex coordinate space$\mathbb {C} ^{n}$,such that thetransition mapsareholomorphic.

Holomorphic Maps

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

Implications of complex structure

Sinceholomorphic functionsare much more rigid thansmooth functions,the theories ofsmoothand complex manifolds have very different flavors:compactcomplex manifolds are much closer toalgebraic varietiesthan to differentiable manifolds.

For example, theWhitney embedding theoremtells us that every smoothn-dimensional manifold can beembeddedas a smooth submanifold ofR²ⁿ,whereas it is "rare" for a complex manifold to have a holomorphic embedding intoCⁿ.Consider for example anycompactconnected complex manifoldM:any holomorphic function on it is constant bythe maximum modulus principle.Now if we had a holomorphic embedding ofMintoCⁿ,then the coordinate functions ofCⁿwould restrict to nonconstant holomorphic functions onM,contradicting compactness, except in the case thatMis just a point. Complex manifolds that can be embedded inCⁿare calledStein manifoldsand 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)Cⁿgives an orientation, as biholomorphic maps are orientation-preserving).

Examples of complex manifolds

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

Smooth complex algebraic varieties

Smooth complexalgebraic varietiesare complex manifolds, including:

• Complex vector spaces.
• Complex projective spaces,Pⁿ(C).
• ComplexGrassmannians.
• ComplexLie groupssuch as GL(n,C) or Sp(n,C).

Simply connected

Thesimply connected1-dimensional complex manifolds are isomorphic to either:

• Δ, the unitdiskinC
• C,the complex plane
• Ĉ,theRiemann sphere

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

Disc vs. space vs. polydisc

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

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

$\left\{z\in \mathbb {C} ^{n}:\|z\|<1\right\}.$

• thepolydisc

$\left\{z=(z_{1},\dots,z_{n})\in \mathbb {C} ^{n}:\vert z_{j}\vert <1\ \forall j=1,\dots,n\right\}.$

Almost complex structures

Main article:Almost complex manifold

Analmost complex structureon 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 anendomorphismof thetangent bundlewhose 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 isweakerthan 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 anintegrablecomplex structure. For integrable complex structures the so-calledNijenhuis tensorvanishes. This tensor is defined on pairs of vector fields,X,Yby

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

For example, the 6-dimensionalsphereS⁶has a natural almost complex structure arising from the fact that it is theorthogonal complementofiin 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.) 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 numberswe get thecomplexifiedtangent 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 ±iand the eigenspaces form sub-bundles denoted byT^0,1MandT^1,0M.TheNewlander–Nirenberg theoremshows 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.

Kähler and Calabi–Yau manifolds

One can define an analogue of aRiemannian metricfor 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. If the skew symmetric part of such a metric issymplectic,i.e. closed and nondegenerate, then the metric is calledKähler.Kähler structures are much more difficult to come by and are much more rigid.

Examples ofKähler manifoldsinclude smoothprojective varietiesand more generally any complex submanifold of a Kähler manifold. TheHopf manifoldsare 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 numberis one, so by theHodge theory,it cannot be Kähler.

ACalabi–Yau manifoldcan be defined as a compactRicci-flatKähler manifold or equivalently one whose firstChern classvanishes.

See also

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

References

• Kodaira, Kunihiko(17 November 2004).Complex Manifolds and Deformation of Complex Structures.Classics in Mathematics. Springer.ISBN3-540-22614-1.