Complex manifold - Citizendia

In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space in a coherent way. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. More precisely, a complex manifold has an atlas of charts to the open unit disk^[1] in $\mathbb{C}^n$ , such that the change of coordinates between charts are holomorphic. For other uses of "atlas" see Atlas (disambiguation. In Mathematics, particularly topology an atlas describes how For other uses of "atlas" see Atlas (disambiguation. In Mathematics, particularly topology an atlas describes how Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane

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, as discussed below.

1 Implications of complex structure
2 Examples of complex manifolds
- 2.1 Smooth complex algebraic varieties
- 2.2 Simply connected
3 Disk vs. Space vs. Polydisk
4 Almost complex structures
5 Kähler and Calabi-Yau manifolds
6 See also
7 Footnotes
8 References

Implications of complex structure

Since complex analytic functions are much more rigid than smooth functions, the theories of smooth and complex manifolds have very different flavors: compact complex manifolds are much closer to algebraic varieties than to differentiable manifolds. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety

For example, the Whitney embedding theorem tells us that every smooth manifold can be embedded as a smooth submanifold of Rⁿ, whereas it is "rare" for a complex manifold to have a holomorphic embedding into Cⁿ. In Mathematics, particularly in Differential topology,there are two Whitney embedding theorems The strong Whitney embedding theorem states that any Consider for example any compact, connected complex manifold M: any holomorphic function on it is locally constant by Liouville's theorem. In Mathematics, a function f from a Topological space A to a set B is called locally constant, Iff In Complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded Entire function must be constant Now if we had a holomorphic embedding of M into Cⁿ, then the coordinate functions of Cⁿ 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 Cⁿ are called Stein manifolds and form a very special class of manifolds including, for example, smooth complex affine algebraic varieties. In Mathematics, a Stein manifold in the theory of Several complex variables and Complex manifolds is a complex Submanifold of the Vector

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 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 the genus, are an important example of this phenomenon. In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional 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. In Algebraic geometry, a moduli space is a geometric space (usually a scheme or an Algebraic stack) whose points represent algebro-geometric objects of

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) $\mathbb{C}^n$ gives an orientation, as biholomorphic maps are orientation-preserving). A surface S in the Euclidean space R 3 is orientable if a two-dimensional figure (for example) cannot be moved around the surface and back

Examples of complex manifolds

Riemann surfaces. In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional
The Cartesian product of two complex manifolds.
The inverse image of any noncritical value of a holomorphic map.

Smooth complex algebraic varieties

Smooth complex algebraic varieties are complex manifolds, including:

Complex vector spaces. This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety
Complex projective spaces^[2], $\mathbb{CP}^n$ . In Mathematics, complex projective space, P ( C n +1 P n ( C) or CP n
Complex Grassmannians. In Mathematics, a Grassmannian is a space which parameterizes all Linear subspaces of a Vector space V of a given Dimension.
Complex Lie groups such as GL(n,C) or Sp(n,C). In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group

Similarly, the quaternionic analogs of these are also complex manifolds. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician

Simply connected

The simply connected 1-dimensional complex manifolds are:

$Δ$ , the unit disk in $\mathbb{C}$
$\mathbb{C}$ , the complex plane
$\hat{\mathbb{C}}$ , the Riemann sphere

Note that there are inclusions between these as $\Delta \subset \mathbb{C} \subset \hat{\mathbb{C}}$ , but that there are no non-constant maps in the other direction, by Liouville's theorem. In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be In Mathematics, the Riemann sphere is a way of extending the plane of Complex numbers with one additional Point at infinity, in a way that In Complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded Entire function must be constant

Disk vs. Space vs. Polydisk

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

the unit disk or open ball, $\{ z \in \mathbb{C}^n \mid \lVert z \rVert < 1\}$
complex space $\mathbb{C}^n$
the polydisk $\{ z=(z_1, z_2, \dots, z_n) \in {\mathbb{C}}^n \mid \vert z_i \vert < 1, \mbox{ for all } i = 1,\dots,n \}$

Almost complex structures

For more details on this topic, see Almost complex manifold. In Mathematics, a ball is the inside of a Sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions and for metric In the theory of functions of Several complex variables, a branch of Mathematics, a polydisc is a Cartesian product of discs More specifically In Mathematics, an almost complex manifold is a Smooth manifold equipped with smooth Linear complex structure on each Tangent space.

An almost complex structure on a real manifold is a $\mathrm{GL}_n(\mathbb{C})$ -structure (in the sense of G-structures). In Mathematics, an almost complex manifold is a Smooth manifold equipped with smooth Linear complex structure on each Tangent space. In Differential geometry, a G -structure on an n - Manifold M, for a given Structure group G, is a G

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). In Mathematics, an endomorphism is a Morphism (or Homomorphism) from a mathematical object to itself In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the An almost complex manifold is necessarily even dimensional.

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. An almost complex structure that comes from a complex structure is called integrable, and when one wishes to specify a complex structure as opposed to an almost complex structure, one says an integrable complex structure.

For example, the 6 dimensional sphere has a natural almost complex structure arising from the fact that it is the orthogonal complement of i in the unit sphere of the octonions, but this is not a complex structure. In Mathematics, an n -sphere is a generalization of an ordinary Sphere to arbitrary Dimension. In the mathematical fields of Linear algebra and Functional analysis, the orthogonal complement W^\bot of a subspace W In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real (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).

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 $\pm i$ and the eigenspaces form sub-bundles denoted by $T 0,1 M$ and $T 1,0 M$ . The Newlander-Niremberg theorem shows that an almost complex structure is actually a complex structure precisely when these subbundles are involutive, i. In Mathematics, an almost complex manifold is a Smooth manifold equipped with smooth Linear complex structure on each Tangent space. e. , closed under the Lie bracket of vector fields, and such an almost complex structure is called integrable.

The Nijenhuis tensor is defined on pairs of vector fields,

X, Y

N J (X, Y) = [X, Y] + J [J X, Y] + J [X, J Y] - [J X, J Y]

Kähler and Calabi-Yau manifolds

One can define an analogue of a Riemannian metric for complex manifolds, called a Hermitian metric. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Mathematics, a Hermitian manifold is the complex analog of a Riemannian manifold. 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, i. Symplectic geometry is a branch of differential topology/geometry which studies Symplectic manifolds that is Differentiable manifolds equipped with a e. closed and nondegenerate, then the metric is called Kähler. Kähler structures are much more difficult to come by and are much more rigid.

Examples of Kähler manifolds include smooth projective varieties and more generally any complex submanifold of a Kähler manifold. This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety 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 theorem, it cannot be Kähler. In Algebraic topology, the Betti number of a Topological space is in intuitive terms a way of counting the maximum number of cuts that can be made without dividing Hodge theorem may refer to Hodge theory Hodge index theorem

A Calabi-Yau manifold is a compact Ricci-flat Kähler manifold or equivalently one whose first Chern class vanishes. In mathematics Calabi&ndashYau manifolds are compact Kähler manifolds whose Canonical bundle is trivial In Mathematics, in particular in Algebraic topology and differential geometry, the Chern classes are a particular type of Characteristic class

Footnotes

^ One must use the open unit disk in $\mathbb{C}^n$ as the model space instead of $\mathbb{C}^n$ because these are not isomorphic, unlike for real manifolds. In Mathematics, a CR manifold is a Differentiable manifold together with a geometric structure modeled on that of a real Hypersurface in a Complex
^ This means that all complex projective spaces are orientable, in contrast to the real case

References

Kodaira, Kunihiko. Complex Manifolds and Deformation of Complex Structures, Classics in Mathematics. Springer. ISBN 3540226141.

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