# Complex manifold

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. ${\ displaystyle \ mathbb {C} ^ {n}}$${\ displaystyle \ mathbb {C} ^ {n}}$

## Definitions

Let be a topological Hausdorff space which satisfies the second axiom of countability . Furthermore, let it be a natural number. ${\ displaystyle M}$${\ displaystyle n}$

### Complex atlas

A map of complex dimension n is an open subset together with a homeomorphism${\ displaystyle U \ subset M}$

${\ displaystyle \ phi \ colon U \ to \ phi (U) \ subset \ mathbb {C} ^ {n}}$.

So a card is a 2-tuple . ${\ displaystyle (U, \ phi)}$

A complex atlas (dimension ) is a set of such maps such that ${\ displaystyle A}$${\ displaystyle n}$

${\ displaystyle M = \ bigcup _ {(U, \ phi) \ in A} U}$

applies, with the property that for every two cards , the card change images ${\ displaystyle (U, \ phi)}$${\ displaystyle (V, \ psi) \ in A}$

${\ displaystyle \ phi \ circ \ psi ^ {- 1} \ colon \ \ psi (U \ cap V) \ to \ phi (U \ cap V)}$

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. ${\ displaystyle M}$${\ displaystyle (M, S)}$${\ displaystyle n}$${\ displaystyle S}$${\ displaystyle n}$${\ displaystyle M}$${\ displaystyle S}$

### 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 . ${\ displaystyle f \ colon M \ to \ mathbb {C}}$${\ displaystyle x \ in M}$${\ displaystyle (U, \ phi)}$${\ displaystyle x \ in U}$${\ displaystyle f \ circ \ phi ^ {- 1} \ colon \ phi (U) \ to \ mathbb {C}}$${\ displaystyle \ phi (x)}$ ${\ displaystyle U \ subset M}$${\ displaystyle x \ in U}$

As a structure sheaf of complex manifold is sheaf of holomorphic functions referred to. is a small space . ${\ displaystyle {\ mathcal {O}} _ {M}}$${\ displaystyle M}$${\ displaystyle (M, {\ mathcal {O}} _ {M})}$

## properties

• Every complex manifold of the dimension can also be understood as a smooth manifold of the dimension .${\ displaystyle n}$${\ displaystyle 2n}$
• The space of the holomorphic function from M to contains only the constant function if M is compact . Therefore one is interested in whether a complex manifold is holomorphically separable .${\ displaystyle {\ mathcal {O}} (M)}$${\ displaystyle \ mathbb {C}}$
• Compact, complex manifolds cannot be embedded in the.${\ displaystyle \ mathbb {C} ^ {n}}$

## 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 . ${\ displaystyle \ mathrm {i}}$${\ displaystyle T_ {p} M}$${\ displaystyle J_ {p}}$

### 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 ${\ displaystyle M}$ ${\ displaystyle J \ colon TM \ to TM}$${\ displaystyle J_ {p}: = J | _ {T_ {p} M}}$${\ displaystyle p \ in M}$

${\ displaystyle J_ {p} \ circ J_ {p} = - \ mathrm {id}.}$

Fulfills. (This corresponds to equality .) ${\ displaystyle \ mathrm {i} ^ {2} = - 1}$

### Almost complex manifold

An almost complex manifold is a smooth manifold together with an almost complex structure . ${\ displaystyle M}$${\ displaystyle M}$

### properties

• 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.${\ displaystyle M}$${\ displaystyle N}$${\ displaystyle J_ {M}}$${\ displaystyle J_ {N}}$${\ displaystyle f \ colon M \ to N}$ ${\ displaystyle df \ colon TM \ to TN}$${\ displaystyle f}$${\ displaystyle M}$${\ displaystyle N}$
${\ displaystyle df \ circ J_ {M} = J_ {N} \ circ df}$
• A complex manifold is automatically an almost complex one. Due to the complex structure, the tangential spaces become complex vector spaces and an almost complex structure is defined by for . Conversely, an almost complex manifold does not in general need to have a complex structure. If, however, there is an atlas with maps whose target area is a complex vector space and which are holomorphic in the sense of the almost complex structure, then this atlas is a complex atlas that induces the almost complex structure. Complex manifolds can therefore also be defined as almost complex manifolds that have a holomorphic atlas.${\ displaystyle Jv: = \ mathrm {i} v}$${\ displaystyle v \ in TM}$
• 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 .