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Complex analytic varieities
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For every point the set of irreducible components of at is in one-to-one correspondence with. The fibre at of the direct image of the structure sheaf is naturally isomorphic to the integral closure of the ring in its complete ring of fractions. The concept of a normal analytic space over can be introduced in terms of analytic continuation of holomorphic functions . Namely, a reduced complex space is normal if and only if Riemann's first theorem on the removal of singularities holds for it: If is an open subset and is a closed analytic subset not containing irreducible components of , then any function that is holomorphic on and locally bounded on has a unique analytic continuation to a holomorphic function on.
For normal complex spaces Riemann's second theorem on the removal of singularities also holds: If at every point , then the analytic continuation in question is possible without the requirement that the function is bounded. A reduced complex space is normal if and only if for every open set the restriction mapping of holomorphic functions.
The property of being normal can also be phrased in the language of local cohomology — it is equivalent to see . For any reduced complex space one can define the sheaf of rings of germs of weakly holomorphic functions, that is, functions satisfying the conditions of Riemann's first theorem.
It turns out that the ring is finite as an -module and equal to the integral closure of in its complete ring of fractions. In other words, , where is the normalization mapping. A normal complex space can also be characterized in the following manner: A complex space is normal if and only if every point of it has a neighbourhood that admits an analytic covering onto a domain of see  , .
A reduced complex space is a Stein space if and only if its normalization has this property see . Kodaira's projective imbedding theorem  carries over to compact normal spaces with such a metric. In algebraic geometry one examines analogues of normal analytic spaces: normal algebraic varieties see Normal scheme. For algebraic varieties over a complete non-discretely normed field the two concepts are the same see  , .
Mongodi, Some applications of metric currents to complex analysis, Man. Ruget, A propos des cycles analytiques de dimension infinie, Inv. Stolzenberg, Uniform approximation on smooth curves, Acta Math. Wermer, The hull of a curve in Cn, Ann.
Extendable cohomologies for complex analytic varieties
Williamson, L. Janos, Constructing metrics with the Heine-Borel property, Proc. Export Citation. User Account Log in Register Help. Search Close Advanced Search Help. My Content 1 Recently viewed 1 Non compact boundaries Show Summary Details. More options ….
Complex Manifolds. Open Access.
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