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