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Complex manifolds can be regarded as a special case of real manifolds of twice the dimension. For example, a 1-dimensional complex manifold is geometrically a surface, known as a Riemann surface.

The theory of complex manifolds has a much different flavor than that of real manifolds, since complex analytic functions are much more rigid than smooth functions. For example, by the Whitney embedding theorem, every real manifold can be embedded as a submanifold of Rⁿ, while it is rare for a complex manifold to be a (complex) submanifold of Cⁿ. Consider for example any compact complex manifold M: any entire function on it must be locally constant, by the extension to several complex variables of Liouville's theorem. This means that M cannot be embedded in Cⁿ unless it has dimension 0. Complex manifolds which can be embedded in Cⁿ (which are necessarily noncompact) are known as Stein manifolds.

One can define an analogue of a Riemannian metric for complex manifolds, called a Kähler metric. Again, unlike the case of real manifolds (which always have Riemannian metrics), it is unusual for a complex manifold to have a Kähler metric.

1 Examples of complex manifolds

• n-dimensional complex space, Cⁿ is a complex manifold in the obvious manner.
• Complex projective spaceIn mathematics, complex projective space or CP n is the projective space of (complex) lines in C n 1. The case n 1 gives the Riemann sphere (also called the complex projective line), and the case n 2 the complex projective plane. CP n is a complex manifol, CPⁿ.
• Any Riemann surface including the Riemann sphereIn mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. It consists of the complex plane plus the point at infinity : This is just the one-point compactification of the complex plane, also known as the extended complex, elliptic curveIn mathematics, elliptic curves are defined by certain cubic (the superscript exponent is three, a. third degree) equations. They have been used in the proof of Fermat's last theorem and they also find applications in cryptography (for details, see the ars, and hyperelliptic curveIn algebraic geometry, a hyperelliptic curve (over the complex numbers) is an algebraic curve given by an equation of the form : where f(x is a polynomial of degree n > 4 with n distinct roots. A hyperelliptic function is a function from the function fiels.
• Any complex Lie groupIn mathematics, a Lie group (pronounced "lee", named after Sophus Lie) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. Lie groups are important in mathematical ana such as GL(n,C) or SO(n,C).
• Any abelian variety has a Kähler metric.