In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in , such that the transition maps are holomorphic.
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.
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, Pn(C). Complex Grassmannians. Complex Lie groups such as GL(n, C) or Sp(n, C). Similarly, the quaternionic analogs of these are also complex manifolds.
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 . the unit disc or open ball
the polydisc
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.