# Difference between revisions of "Normal analytic space"

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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 $ 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) | + | 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: | ||

− | + | 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 | + | 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 | + | 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 | + | 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 | + | 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 | + | 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 | + | 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 | + | 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 | + | 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=23913