Normal analytic space

An analytic space the local rings of all points of which are normal, that is, are integrally-closed integral domains. A point $ x $ of an analytic space $ X $ is said to be normal (one also says that $ X $ is normal at $ x $) if the local ring $ {\mathcal O} _ {X,x} $ is normal. In a neighbourhood of such a point the space has a reduced and irreducible model. Every simple (non-singular) point is normal. The simplest example of a normal analytic space is an analytic manifold.

In what follows the (complete non-discretely normed) ground field $ k $ is assumed to be algebraically closed. In this case the most complete results on normal analytic spaces have been obtained (see [1]) and a normalization theory has been constructed [2] that gives a natural link between arbitrary reduced analytic spaces and normal analytic spaces. Let $ N ( X) $ be the set of points of an analytic space $ X $ that are not normal and let $ S ( X) $ be the set of singular points of $ X $ (cf. Singular point). Then:

1) $ N ( X) $ and $ S ( X) $ are closed analytic subspaces of $ X $, and $ N ( X) \subset S ( X) $;

2) for $ x \in X \setminus N ( X) $,

$$ \mathop{\rm dim} _ {x} S ( X) \leq \mathop{\rm dim} _ {x} X - 2 $$

(that is, a normal analytic space is smooth in codimension 1);

3) if $ X $ is a complete intersection at $ x $ and if the above inequality holds, then $ X $ is normal at that point.

A normalization of a reduced analytic space $ X $ is a pair $ ( \widetilde{X} , v) $, where $ \widetilde{X} $ is a normal analytic space and $ v: \widetilde{X} \rightarrow X $ is a finite surjective analytic mapping inducing an isomorphism of the open sets

$$ \widetilde{X} \setminus v ^ {-1} ( N ( X)) \rightarrow X \setminus N ( X). $$

The normalization is uniquely determined up to an isomorphism, that is, if $ ( \widetilde{X} _ {1} , v _ {1} ) $ and $ ( \widetilde{X} _ {2} , v _ {2} ) $ are two normalizations,

$$ \begin{array}{rcr} \widetilde{X} _ {1} & \mathop \rightarrow \limits ^ \phi &\widetilde{X} _ {2} \\ {} _ {v _ {1} } \searrow &{} &\swarrow _ {v _ {2} } \\ {} & X &{} \\ \end{array} $$

then there exists a unique analytic isomorphism $ \phi : \widetilde{X} _ {1} \rightarrow \widetilde{X} _ {2} $ such that the diagram commutes. The normalization exists and has the following properties. For every point $ x \in X $ the set of irreducible components of $ X $ at $ x $ is in one-to-one correspondence with $ v ^ {-1} ( x) $. The fibre at $ x \in X $ of the direct image $ v _ {*} ( {\mathcal O} _ {\widetilde{X} } ) $ of the structure sheaf $ {\mathcal O} _ {\widetilde{X} } $ is naturally isomorphic to the integral closure of the ring $ {\mathcal O} _ {X,x} $ in its complete ring of fractions.

The concept of a normal analytic space over $ \mathbf C $ can be introduced in terms of analytic continuation of holomorphic functions [3]. Namely, a reduced complex space is normal if and only if Riemann's first theorem on the removal of singularities holds for it: If $ U \subset X $ is an open subset and $ A \subset U $ is a closed analytic subset not containing irreducible components of $ U $, then any function that is holomorphic on $ U \setminus A $ and locally bounded on $ U $ has a unique analytic continuation to a holomorphic function on $ U $. For normal complex spaces Riemann's second theorem on the removal of singularities also holds: If $ \mathop{\rm codim} _ {x} A \geq 2 $ at every point $ x \in A $, then the analytic continuation in question is possible without the requirement that the function is bounded. A reduced complex space $ X $ is normal if and only if for every open set $ U \subset X $ the restriction mapping of holomorphic functions

$$ \Gamma ( U, {\mathcal O} _ {X} ) \rightarrow \ \Gamma ( U \setminus S ( X), {\mathcal O} _ {X} ) $$

is bijective. The property of being normal can also be phrased in the language of local cohomology — it is equivalent to $ H _ {S ( X) } ^ {1} {\mathcal O} _ {X} = 0 $ (see [5]). For any reduced complex space $ X $ one can define the sheaf $ \widetilde{\mathcal O} _ {X} $ 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 $ \widetilde{\mathcal O} _ {X,x} $ is finite as an $ {\mathcal O} _ {X,x} $-module and equal to the integral closure of $ {\mathcal O} _ {X,x} $ in its complete ring of fractions. In other words, $ {\mathcal O} tilde _ {X} = v _ {*} ( {\mathcal O} _ {\widetilde{X} } ) $, where $ v: \widetilde{X} \rightarrow X $ 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 $ \mathbf C ^ {n} $ (see [3], [8]).

A reduced complex space $ X $ is a Stein space if and only if its normalization $ \widetilde{X} $ has this property (see [4]). To normal complex spaces one can extend the concept of a Hodge metric (see Kähler metric). Kodaira's projective imbedding theorem [6] 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 [7], [1]).

References

[1]	S.S. Abhyankar, "Local analytic geometry" , Acad. Press (1964) MR0175897 Zbl 0205.50401
[2]	C. Houzel, "Géométrie analytique locale I" , Sem. H. Cartan Ann. 13 1960/61 , 2 (1963) pp. Exp. 18–21 Zbl 0121.15906
[3]	H. Grauert, R. Remmert, "Komplexe Räume" Math. Ann. , 136 (1958) pp. 245–318 MR0103285 Zbl 0087.29003
[4]	R. Narasimhan, "A note on Stein spaces and their normalisations" Ann. Scuola Norm. Sup. Pisa , 16 (1962) pp. 327–333 MR0153870
[5]	Y.T. Siu, G. Trautmann, "Gap sheaves and extensions of coherent analytic subsheaves" , Springer (1971) MR0287033
[6]	H. Grauert, "Ueber Modifikationen und exzeptionelle analytische Mengen" Math. Ann. , 146 (1962) pp. 331–368 Zbl 0178.42702 Zbl 0173.33004
[7]	O. Zariski, P. Samuel, "Commutative algebra" , 2 , Springer (1960) MR0120249 Zbl 0121.27801
[8]	B.A. Fuks, "Theory of analytic functions of several complex variables" , 1 , Amer. Math. Soc. (1963) (Translated from Russian) MR0174786 MR0168793 Zbl 0138.30902
[a1]	H. Whitney, "Complex analytic varieties" , Addison-Wesley (1972) pp. Chapt. 8 MR0387634 Zbl 0265.32008