Complex space

complex-analytic space

An analytic space over the field of complex numbers $\mathbf C$. The simplest and most widely used complex space is the complex number space $\mathbf C ^ {n}$. Its points, or elements, are all possible $n$- tuples $( z _ {1} \dots z _ {n} )$ of complex numbers $z _ \nu = x _ \nu + iy _ \nu$, $\nu = 1 \dots n$. It is a vector space over $\mathbf C$ with the operations of addition

$$z + z ^ \prime = \ ( z _ {1} + z _ {1} ^ \prime \dots z _ {n} + z _ {n} ^ \prime )$$

and multiplication by a scalar $\lambda \in \mathbf C$,

$$\lambda z = \ ( \lambda z _ {1} \dots \lambda z _ {n} ),$$

as well as a metric space with the Euclidean metric

$$\rho ( z, z ^ \prime ) = \ | z - z ^ \prime | = \ \sqrt {\sum _ {\nu = 1 } ^ { n } | z _ \nu - z _ \nu ^ \prime | ^ {2} } =$$

$$= \ \sqrt {\sum _ {\nu = 1 } ^ { n } ( x _ \nu - x _ \nu ^ \prime ) ^ {2} + ( y _ \nu - y _ \nu ^ \prime ) ^ {2} } .$$

In other words, the complex number space $\mathbf C ^ {n}$ is obtained as the result of complexifying the real number space $\mathbf R ^ {2n}$. The complex number space $\mathbf C ^ {n}$ is also the topological product of $n$ complex planes $\mathbf C ^ {1} = \mathbf C$, $\mathbf C ^ {n} = \mathbf C \times \dots \times \mathbf C$.

References

[1]	B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian) Zbl 0799.32001 Zbl 0732.32001 Zbl 0732.30001 Zbl 0578.32001 Zbl 0574.30001
[2]	N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French) MR0205211 MR0205210 Zbl 0301.54002 Zbl 0301.54001 Zbl 0145.19302

Comments

A more general notion of complex space is contained in [a1]. Roughly it is as follows. Let $X$ be a Hausdorff space equipped with asheaf ${\mathcal O} _ {X}$ of local $\mathbf C$- algebras (a so-called $\mathbf C$- algebraized space). Two such spaces $( X, {\mathcal O} _ {X} )$ and $( Y, {\mathcal O} _ {Y} )$ are called isomorphic if there is a homeomorphism $f: X \rightarrow Y$ and a sheaf isomorphism $\widetilde{f} : {\mathcal O} _ {X} \rightarrow {\mathcal O} _ {Y}$( cf. [a1]). Now, a $\mathbf C$- algebraized space $( X, {\mathcal O} _ {X} )$ is called a complex manifold if it is locally isomorphic to a standard space $( D, {\mathcal O} _ {D} )$, $D \subset \mathbf C ^ {m}$ a domain, ${\mathcal O} _ {D}$ its sheaf of germs of holomorphic functions, i.e. if for every $x \in X$ there is a neighbourhood $U$ of $x$ in $X$ and a domain $D \subset \mathbf C ^ {m}$, for some $m$, so that the $\mathbf C$- algebraized spaces $( U, {\mathcal O} _ {U} )$ and $( D, {\mathcal O} _ {D} )$ are isomorphic. Let $D \subset \mathbf C ^ {m}$ be a domain and $J \subset {\mathcal O} _ {D}$ a coherent ideal. The support $A$ of the (coherent) quotient sheaf ${\mathcal O} _ {D} /J$ is a closed set in $D$, and the sheaf ${\mathcal O} _ {A} = {\mathcal O} _ {D} /J \mid _ {A}$ is a (coherent) sheaf of local $\mathbf C$- algebras. The $\mathbf C$- algebraized space $( A, {\mathcal O} _ {A} )$ is called a (closed) complex subspace of $( D, {\mathcal O} _ {D} )$( it is naturally imbedded in $( D, {\mathcal O} _ {D} )$ via the quotient sheaf mapping). A complex space $( X, {\mathcal O} _ {X} )$ is a $\mathbf C$- algebraized space that is locally isomorphic to a complex subspace, i.e. every point $x \in X$ has a neighbourhood $U$ so that $( U, {\mathcal O} _ {U} )$ is isomorphic to a complex subspace of a domain in some $\mathbf C ^ {m}$. (See also Sheaf theory; Coherent sheaf.) More on complex spaces, in particular their use in function theory of several variables and algebraic geometry, can be found in [a1]. See also Stein space; Analytic space.

References

[a1]	H. Grauert, R. Remmert, "Theory of Stein spaces" , Springer (1979) (Translated from German) MR0580152 Zbl 0433.32007