Complex manifold

Last updated February 08, 2024

Holomorphic Maps

In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc ^[1] in $\mathbb {C} ^{n}$ , such that the transition maps are holomorphic.

Implications of complex structure

Since holomorphic 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.

For example, the Whitney embedding theorem tells us that every smooth n-dimensional manifold can be embedded as a smooth submanifold of R²ⁿ, whereas it is "rare" for a complex manifold to have a holomorphic embedding into Cⁿ. Consider for example any compact connected complex manifold M: any holomorphic function on it is constant by the maximum modulus principle. 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.

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. 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.

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) Cⁿ gives 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 complex algebraic varieties are complex manifolds, including:

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

Simply connected

The simply connected 1-dimensional complex manifolds are isomorphic to either:

Δ, the unit disk in C
C, the complex plane
Ĉ, the Riemann 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 or open ball

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

the polydisc

\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

An almost complex structure on a real 2n-manifold is a GL(n, C)-structure (in the sense of G-structures) – that is, the tangent bundle is equipped with a linear complex structure.

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.

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 almost complex structure can be defined globally. 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 integrable complex structures the so-called Nijenhuis tensor vanishes. This tensor is defined on pairs of vector fields, X, Y by

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

For example, the 6-dimensional sphere S⁶ 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. (The question of whether it has a complex structure is known as the Hopf problem, after Heinz 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 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^0,1M and T^1,0M. 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 integrable.

Kähler and Calabi–Yau manifolds

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, 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.

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.

A Calabi–Yau manifold can be defined as a compact Ricci-flat Kähler manifold or equivalently one whose first Chern class vanishes.

Footnotes

↑ One must use the open unit disc in $\mathbb {C} ^{n}$ as the model space instead of $\mathbb {C} ^{n}$ because these are not isomorphic, unlike for real manifolds.
↑ This means that all complex projective spaces are orientable, in contrast to the real case
↑ Agricola, Ilka; Bazzoni, Giovanni; Goertsches, Oliver; Konstantis, Panagiotis; Rollenske, Sönke (2018). "On the history of the Hopf problem". Differential Geometry and Its Applications . 57: 1–9. arXiv: 1708.01068 . doi:10.1016/j.difgeo.2017.10.014. S2CID 119297359.

Related Research Articles

In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, $, equipped with a closed nondegenerate differential 2-form, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system.$

In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaves. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis.

<span class="mw-page-title-main">Calabi–Yau manifold</span> Riemannian manifold with SU(n) holonomy

In algebraic and differential geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstring theory, the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold, which led to the idea of mirror symmetry. Their name was coined by Candelas et al. (1985), after Eugenio Calabi who first conjectured that such surfaces might exist, and Shing-Tung Yau who proved the Calabi conjecture.

In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space $over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of .$

The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space $, that is, n -tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables, which the Mathematics Subject Classification has as a top-level heading.$

In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933. The terminology has been fixed by André Weil. Kähler geometry refers to the study of Kähler manifolds, their geometry and topology, as well as the study of structures and constructions that can be performed on Kähler manifolds, such as the existence of special connections like Hermitian Yang–Mills connections, or special metrics such as Kähler–Einstein metrics.

In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold M using partial differential equations. The key observation is that, given a Riemannian metric on M, every cohomology class has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric. Such forms are called harmonic.

In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not complex manifolds. Almost complex structures have important applications in symplectic geometry.

In differential geometry, a hyperkähler manifold is a Riemannian manifold $endowed with three integrable almost complex structures that are Kähler with respect to the Riemannian metric and satisfy the quaternionic relations . In particular, it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi-Yau manifolds.$

In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of n complex dimensions. They were introduced by and named after Karl Stein (1951). A Stein space is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues of affine varieties or affine schemes in algebraic geometry.

In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each (holomorphic) tangent space. One can also define a Hermitian manifold as a real manifold with a Riemannian metric that preserves a complex structure.

In differential geometry, a quaternion-Kähler manifold (or quaternionic Kähler manifold) is a Riemannian 4n-manifold whose Riemannian holonomy group is a subgroup of Sp(n)·Sp(1) for some $. Here Sp(n) is the sub-group of consisting of those orthogonal transformations that arise by left -multiplication by some quaternionic matrix, while the group of unit-length quaternions instead acts on quaternionic -space by right scalar multiplication. The Lie group generated by combining these actions is then abstractly isomorphic to .$

In mathematics, a CR manifold, or Cauchy–Riemann manifold, is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a complex vector space, or more generally modeled on an edge of a wedge.

In differential geometry, a Sasakian manifold is a contact manifold $equipped with a special kind of Riemannian metric, called a Sasakian metric.$

In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold $X$ such that the total space $E$ is a complex manifold and the projection map $π : E \to X$ is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A holomorphic line bundle is a rank one holomorphic vector bundle.

In the field of mathematics known as differential geometry, a generalized complex structure is a property of a differential manifold that includes as special cases a complex structure and a symplectic structure. Generalized complex structures were introduced by Nigel Hitchin in 2002 and further developed by his students Marco Gualtieri and Gil Cavalcanti.

In mathematics, and especially complex geometry, the holomorphic tangent bundle of a complex manifold $is the holomorphic analogue of the tangent bundle of a smooth manifold. The fibre of the holomorphic tangent bundle over a point is the holomorphic tangent space, which is the tangent space of the underlying smooth manifold, given the structure of a complex vector space via the almost complex structure of the complex manifold .$

In algebraic geometry and differential geometry, the nonabelian Hodge correspondence or Corlette–Simpson correspondence is a correspondence between Higgs bundles and representations of the fundamental group of a smooth, projective complex algebraic variety, or a compact Kähler manifold.

In differential geometry, a quaternionic manifold is a quaternionic analog of a complex manifold. The definition is more complicated and technical than the one for complex manifolds due in part to the noncommutativity of the quaternions and in part to the lack of a suitable calculus of holomorphic functions for quaternions. The most succinct definition uses the language of G-structures on a manifold. Specifically, a quaternionic n-manifold can be defined as a smooth manifold of real dimension 4n equipped with a torsion-free $-structure. More naïve, but straightforward, definitions lead to a dearth of examples, and exclude spaces like quaternionic projective space which should clearly be considered as quaternionic manifolds.$

In mathematics, hyperbolic complex space is a Hermitian manifold which is the equivalent of the real hyperbolic space in the context of complex manifolds. The complex hyperbolic space is a Kähler manifold, and it is characterised by being the only simply connected Kähler manifold whose holomorphic sectional curvature is constant equal to -1. Its underlying Riemannian manifold has non-constant negative curvature, pinched between -1 and -1/4 : in particular, it is a CAT(-1/4) space.

References

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

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[1] One must use the open unit disc in $\mathbb {C} ^{n}$ as the model space instead of $\mathbb {C} ^{n}$ because these are not isomorphic, unlike for real manifolds.

[2] This means that all complex projective spaces are orientable, in contrast to the real case

[3] Agricola, Ilka; Bazzoni, Giovanni; Goertsches, Oliver; Konstantis, Panagiotis; Rollenske, Sönke (2018). "On the history of the Hopf problem". Differential Geometry and Its Applications . 57: 1–9. arXiv: 1708.01068 . doi:10.1016/j.difgeo.2017.10.014. S2CID 119297359.

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