In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space in a coherent way. More precisely, a complex manifold has an atlas of charts to Cn, such that the change of coordinates between charts are holomorphic.
Complex manifolds can be regarded as a special case of differentiable manifolds. For example, a 1-dimensional complex manifold is geometrically a surface, known as a Riemann surface. The requirement that the transition functions be holomorphic means that unlike in the general differential case, there is no distinction between different Ck-structures for different k, since holomorphic functions are analytic, and thus any holomorphic structure is also a Ck structure, for any k ≥1.
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.
Examples of complex manifolds
- n-dimensional complex space, Cn is a complex manifold in the obvious manner.
- Complex projective space, CPn.
- Any Riemann surface including the Riemann sphere, elliptic curves, and hyperelliptic curves.
- Any complex Lie group such as GL(n,C) or SO(n,C).
- Any abelian variety has a Kähler metric.
Integrable almost-complex structures
The definition of complex manifold is by charts; one can ask whether such structures can be defined also by means of conditions on the tangent bundle, given that the tangent bundle to a complex manifold is a complex vector bundle. This is the case, but requires some result from partial differential equation theory: an integrable almost complex structure on a manifold M comes from a complex manifold structure. In other words, necessary conditions on the tangent bundle level can be given that are also sufficient for the existence of an underlying set of holomorphic charts.
Kähler and Calabi-Yau manifolds
A Kähler manifold is a manifold which simultaneously carries a Riemannian structure, a symplectic structure, and a complex structure which are all compatible in some suitable sense.