Complex manifold

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Complex manifolds are topological manifolds with model space whose map change homeomorphisms are even biholomorphic . These objects are studied in differential geometry and function theory. Its definition is analogous to the definition of the differentiable manifold , but in contrast to the differentiable manifolds, not every complex manifold can be embedded in the.


Let be a topological Hausdorff space which satisfies the second axiom of countability . Furthermore, let it be a natural number.

Complex atlas

A map of complex dimension n is an open subset together with a homeomorphism


So a card is a 2-tuple .

A complex atlas (dimension ) is a set of such maps such that

applies, with the property that for every two cards , the card change images

are biholomorphic.

Complex structure

A complex structure is a maximally complex atlas with regard to inclusion. Every complex atlas is contained in exactly one complex structure, namely in the union of all atlases that are equivalent to it . Two complex atlases are equivalent if their union set is also a complex atlas (i.e. if all map change maps between the two atlases are biholomorphic).

Note: Alternatively, a complex structure can also be defined as an equivalence class with regard to this concept of equivalence.

Complex manifold

If one now provides such a complex structure, one speaks of a complex manifold . More precisely, a 2-tuple is a complex manifold of dimension if a complex structure of dimension is on . The cards from are then also referred to as cards of the complex manifold.

Holomorphic functions, structural grain

A function is holomorphic in if for a card with the function of an in holomorphic function is. Because of the above compatibility condition, this condition is independent of the selected card. A function is called holomorphic on an open subset if it is holomorphic at every point .

As a structure sheaf of complex manifold is sheaf of holomorphic functions referred to. is a small space .


  • Every complex manifold of the dimension can also be understood as a smooth manifold of the dimension .
  • The space of the holomorphic function from M to contains only the constant function if M is compact . Therefore one is interested in whether a complex manifold is holomorphically separable .
  • Compact, complex manifolds cannot be embedded in the.


Almost complex manifolds

A weakening of the notion of complex manifold is the notion of almost complex manifold . While complex manifolds look locally like complex space, almost complex manifolds only do this "infinitesimally", that is, the tangent spaces are (in a mutually compatible way) complex vector spaces. In order to make a real vector space complex, one has to determine what the product of a vector with the imaginary unit should be. In the case of the tangent space, this is the task of the mapping .

Almost complex structure

An almost complex structure on a smooth manifold is a smooth map with the property that the restriction to the tangent space to each point is a bijective linear map, which

Fulfills. (This corresponds to equality .)

Almost complex manifold

An almost complex manifold is a smooth manifold together with an almost complex structure .


  • Let and be two almost complex manifolds with the respective almost complex structures and . A continuously differentiable mapping is called holomorphic (or pseudo-holomorphic) if the push forward of is compatible with the almost complex structures of and , that is, it must apply.

  • A complex manifold is automatically an almost complex one. Due to the complex structure, the tangential spaces become complex vector spaces and an almost complex structure is defined by for . Conversely, an almost complex manifold does not in general need to have a complex structure. If, however, there is an atlas with maps whose target area is a complex vector space and which are holomorphic in the sense of the almost complex structure, then this atlas is a complex atlas that induces the almost complex structure. Complex manifolds can therefore also be defined as almost complex manifolds that have a holomorphic atlas.
  • In the real two-dimensional (i.e. in the complex one-dimensional) every almost complex manifold is a complex manifold, i.e. a Riemann surface . This can be shown by solving the Beltrami equation .