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An [[Analytic space|analytic space]] the local rings of all points of which are normal, that is, are integrally-closed integral domains. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n0674001.png" /> of an analytic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n0674002.png" /> is said to be normal (one also says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n0674003.png" /> is normal at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n0674004.png" />) if the local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n0674005.png" /> 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|analytic manifold]].
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In what follows the (complete non-discretely normed) ground field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n0674006.png" /> is assumed to be algebraically closed. In this case the most complete results on normal analytic spaces have been obtained (see [[#References|[1]]]) and a normalization theory has been constructed [[#References|[2]]] that gives a natural link between arbitrary reduced analytic spaces and normal analytic spaces. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n0674007.png" /> be the set of points of an analytic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n0674008.png" /> that are not normal and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n0674009.png" /> be the set of singular points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740010.png" /> (cf. [[Singular point|Singular point]]). Then:
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1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740012.png" /> are closed analytic subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740013.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740014.png" />;
+
An [[Analytic space|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|analytic manifold]].
  
2) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740015.png" />,
+
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 [[#References|[1]]]) and a normalization theory has been constructed [[#References|[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|Singular point]]). Then:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740016.png" /></td> </tr></table>
+
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);
 
(that is, a normal analytic space is smooth in codimension 1);
  
3) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740017.png" /> is a complete intersection at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740018.png" /> and if the above inequality holds, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740019.png" /> is normal at that point.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740020.png" /> is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740022.png" /> is a normal analytic space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740023.png" /> is a finite surjective analytic mapping inducing an isomorphism of the open sets
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740024.png" /></td> </tr></table>
+
$$
 +
\widetilde{X}  \setminus  v  ^ {-} 1 ( N ( X))  \rightarrow  X \setminus  N ( X).
 +
$$
  
The normalization is uniquely determined up to an isomorphism, that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740026.png" /> are two normalizations,
+
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,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740027.png" /></td> </tr></table>
+
$$
  
then there exists a unique analytic isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740028.png" /> such that the diagram commutes. The normalization exists and has the following properties. For every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740029.png" /> the set of irreducible components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740030.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740031.png" /> is in one-to-one correspondence with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740032.png" />. The fibre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740033.png" /> of the direct image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740034.png" /> of the structure sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740035.png" /> is naturally isomorphic to the integral closure of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740036.png" /> in its complete ring of fractions.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740037.png" /> can be introduced in terms of analytic continuation of holomorphic functions [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740038.png" /> is an open subset and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740039.png" /> is a closed analytic subset not containing irreducible components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740040.png" />, then any function that is holomorphic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740041.png" /> and locally bounded on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740042.png" /> has a unique analytic continuation to a holomorphic function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740043.png" />. For normal complex spaces Riemann's second theorem on the removal of singularities also holds: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740044.png" /> at every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740045.png" />, then the analytic continuation in question is possible without the requirement that the function is bounded. A reduced complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740046.png" /> is normal if and only if for every open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740047.png" /> the restriction mapping of holomorphic functions
+
The concept of a normal analytic space over $  \mathbf C $
 +
can be introduced in terms of analytic continuation of holomorphic functions [[#References|[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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740048.png" /></td> </tr></table>
+
$$
 +
\Gamma ( U, {\mathcal O} _ {X} )  \rightarrow \
 +
\Gamma ( U \setminus  S ( X), {\mathcal O} _ {X} )
 +
$$
  
is bijective. The property of being normal can also be phrased in the language of local cohomology — it is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740049.png" /> (see [[#References|[5]]]). For any reduced complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740050.png" /> one can define the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740051.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740052.png" /> is finite as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740053.png" />-module and equal to the integral closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740054.png" /> in its complete ring of fractions. In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740055.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740056.png" /> is the normalization mapping.
+
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 [[#References|[5]]]). For any reduced complex space $  X $
 +
one can define the sheaf $  {\mathcal O}  tilde _ {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 $  {\mathcal O}  tilde _ {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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740057.png" /> (see [[#References|[3]]], [[#References|[8]]]).
+
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 [[#References|[3]]], [[#References|[8]]]).
  
A reduced complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740058.png" /> is a [[Stein space|Stein space]] if and only if its normalization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067400/n06740059.png" /> has this property (see [[#References|[4]]]). To normal complex spaces one can extend the concept of a Hodge metric (see [[Kähler metric|Kähler metric]]). Kodaira's projective imbedding theorem [[#References|[6]]] carries over to compact normal spaces with such a metric.
+
A reduced complex space $  X $
 +
is a [[Stein space|Stein space]] if and only if its normalization $  \widetilde{X}  $
 +
has this property (see [[#References|[4]]]). To normal complex spaces one can extend the concept of a Hodge metric (see [[Kähler metric|Kähler metric]]). Kodaira's projective imbedding theorem [[#References|[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|Normal scheme]]). For algebraic varieties over a complete non-discretely normed field the two concepts are the same (see [[#References|[7]]], [[#References|[1]]]).
 
In algebraic geometry one examines analogues of normal analytic spaces: normal algebraic varieties (see [[Normal scheme|Normal scheme]]). For algebraic varieties over a complete non-discretely normed field the two concepts are the same (see [[#References|[7]]], [[#References|[1]]]).
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S.S. Abhyankar, "Local analytic geometry" , Acad. Press (1964) {{MR|0175897}} {{ZBL|0205.50401}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> C. Houzel, "Géometrie analytique locale I" , ''Sem. H. Cartan Ann. 13 1960/61'' , '''2''' (1963) pp. Exp. 18–21 {{MR|}} {{ZBL|0121.15906}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> H. Grauert, R. Remmert, "Komplexe Räume" ''Math. Ann.'' , '''136''' (1958) pp. 245–318 {{MR|0103285}} {{ZBL|0087.29003}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> R. Narasimhan, "A note on Stein spaces and their normalisations" ''Ann. Scuola Norm. Sup. Pisa'' , '''16''' (1962) pp. 327–333 {{MR|0153870}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> Y.T. Siu, G. Trautmann, "Gap sheaves and extensions of coherent analytic subsheaves" , Springer (1971) {{MR|0287033}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> H. Grauert, "Ueber Modifikationen und exzeptionelle analytische Mengen" ''Math. Ann.'' , '''146''' (1962) pp. 331–368 {{MR|}} {{ZBL|0178.42702}} {{ZBL|0173.33004}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''2''' , Springer (1960) {{MR|0120249}} {{ZBL|0121.27801}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> B.A. Fuks, "Theory of analytic functions of several complex variables" , '''1''' , Amer. Math. Soc. (1963) (Translated from Russian) {{MR|0174786}} {{MR|0168793}} {{ZBL|0138.30902}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S.S. Abhyankar, "Local analytic geometry" , Acad. Press (1964) {{MR|0175897}} {{ZBL|0205.50401}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> C. Houzel, "Géometrie analytique locale I" , ''Sem. H. Cartan Ann. 13 1960/61'' , '''2''' (1963) pp. Exp. 18–21 {{MR|}} {{ZBL|0121.15906}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> H. Grauert, R. Remmert, "Komplexe Räume" ''Math. Ann.'' , '''136''' (1958) pp. 245–318 {{MR|0103285}} {{ZBL|0087.29003}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> R. Narasimhan, "A note on Stein spaces and their normalisations" ''Ann. Scuola Norm. Sup. Pisa'' , '''16''' (1962) pp. 327–333 {{MR|0153870}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> Y.T. Siu, G. Trautmann, "Gap sheaves and extensions of coherent analytic subsheaves" , Springer (1971) {{MR|0287033}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> H. Grauert, "Ueber Modifikationen und exzeptionelle analytische Mengen" ''Math. Ann.'' , '''146''' (1962) pp. 331–368 {{MR|}} {{ZBL|0178.42702}} {{ZBL|0173.33004}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''2''' , Springer (1960) {{MR|0120249}} {{ZBL|0121.27801}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> B.A. Fuks, "Theory of analytic functions of several complex variables" , '''1''' , Amer. Math. Soc. (1963) (Translated from Russian) {{MR|0174786}} {{MR|0168793}} {{ZBL|0138.30902}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Whitney, "Complex analytic varieties" , Addison-Wesley (1972) pp. Chapt. 8 {{MR|0387634}} {{ZBL|0265.32008}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Whitney, "Complex analytic varieties" , Addison-Wesley (1972) pp. Chapt. 8 {{MR|0387634}} {{ZBL|0265.32008}} </TD></TR></table>

Revision as of 08:03, 6 June 2020


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,

$$ then there exists a unique analytic isomorphism $ \phi : \widetilde{X} _ {1} \rightarrow \widetilde{X} _ {2} $ such that the diagram commutes. The normalization exists and has the following properties. For every point $ x \in X $ the set of irreducible components of $ X $ at $ x $ is in one-to-one correspondence with $ v ^ {-} 1 ( x) $. The fibre at $ x \in X $ of the direct image $ v _ {*} ( {\mathcal O} _ {\widetilde{X} } ) $ of the structure sheaf $ {\mathcal O} _ {\widetilde{X} } $ is naturally isomorphic to the integral closure of the ring $ {\mathcal O} _ {X,x} $ in its complete ring of fractions. The concept of a normal analytic space over $ \mathbf C $ can be introduced in terms of analytic continuation of holomorphic functions [[#References|[3]]]. Namely, a reduced complex space is normal if and only if Riemann's first theorem on the removal of singularities holds for it: If $ U \subset X $ is an open subset and $ A \subset U $ is a closed analytic subset not containing irreducible components of $ U $, then any function that is holomorphic on $ U \setminus A $ and locally bounded on $ U $ has a unique analytic continuation to a holomorphic function on $ U $. For normal complex spaces Riemann's second theorem on the removal of singularities also holds: If $ \mathop{\rm codim} _ {x} A \geq 2 $ at every point $ x \in A $, then the analytic continuation in question is possible without the requirement that the function is bounded. A reduced complex space $ X $ is normal if and only if for every open set $ U \subset X $ the restriction mapping of holomorphic functions $$ \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 $ {\mathcal O} tilde _ {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 $ {\mathcal O} tilde _ {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

Comments

References

[a1] H. Whitney, "Complex analytic varieties" , Addison-Wesley (1972) pp. Chapt. 8 MR0387634 Zbl 0265.32008
How to Cite This Entry:
Normal analytic space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_analytic_space&oldid=48007
This article was adapted from an original article by D.N. Akhiezer (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article