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
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 |
Normal analytic space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_analytic_space&oldid=55909