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

Complex manifolds of dimension 1 are called Riemann surfaces . This should not be confused with Riemannian manifolds .

Every complex manifold of the dimension can also be understood as a smooth manifold of the dimension . ${\ displaystyle n}$ ${\ displaystyle 2n}$

Every complex manifold is orientable .

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

Examples

The vector space and open subset of it. ${\ displaystyle \ mathbb {C} ^ {n}}$
General Stein's manifolds
Complex projective spaces ${\ displaystyle \ mathbb {C} P ^ {n}}$
Riemann surfaces such as the Riemann number sphere , the Jacobi variety and the dotted complex plane.
Kahler manifolds

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 .

literature

Klaus Fritzsche , Hans Grauert : From Holomorphic Functions to Complex Manifolds (= Graduate Texts in Mathematics . 213). Springer, New York NY et al. 2002, ISBN 0-387-95395-7 .