# Normal analytic space

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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, "Géometrie 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