Namespaces
Variants
Actions

Difference between revisions of "Real-analytic space"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (tex encoded by computer)
m (fixing spaces)
 
Line 16: Line 16:
 
Let  $  V _ {a} $
 
Let  $  V _ {a} $
 
be the germ at a point  $  a $
 
be the germ at a point  $  a $
of a real-analytic subset of  $  \mathbf R  ^ {n} $(
+
of a real-analytic subset of  $  \mathbf R  ^ {n} $ (cf. [[Analytic set|Analytic set]]). This defines the germ at  $  a $
cf. [[Analytic set|Analytic set]]). This defines the germ at  $  a $
+
of a complex-analytic subset  $  \widetilde{V} _ {a} $
of a complex-analytic subset  $  {\widetilde{V} _ {a} $
 
 
of the space  $  \mathbf C  ^ {n} $
 
of the space  $  \mathbf C  ^ {n} $
with the following equivalent properties: 1)  $  \widetilde{V} _ {a} $
+
with the following equivalent properties: 1)  $  \widetilde{V}_ {a} $
 
is the intersection of all germs of complex-analytic sets containing  $  V _ {a} $;  
 
is the intersection of all germs of complex-analytic sets containing  $  V _ {a} $;  
 
2) if  $  {\mathcal O} _ {V _ {a}  } $
 
2) if  $  {\mathcal O} _ {V _ {a}  } $
 
is the analytic algebra of the germ  $  V _ {a} $,  
 
is the analytic algebra of the germ  $  V _ {a} $,  
 
then  $  {\mathcal O} _ {V _ {a}  } \otimes \mathbf C $
 
then  $  {\mathcal O} _ {V _ {a}  } \otimes \mathbf C $
is the analytic algebra of the germ  $  \widetilde{V} _ {a} $.  
+
is the analytic algebra of the germ  $  \widetilde{V}_ {a} $.  
The germ  $  \widetilde{V} _ {a} $
+
The germ  $  \widetilde{V}_ {a} $
 
is said to be the complexification of the germ  $  V _ {a} $,  
 
is said to be the complexification of the germ  $  V _ {a} $,  
 
and  $  V _ {a} $
 
and  $  V _ {a} $
is said to be the real part of the germ  $  \widetilde{V} _ {a} $.  
+
is said to be the real part of the germ  $  \widetilde{V}_ {a} $.  
 
Analogously, for any coherent real-analytic countably-infinite space  $  X $
 
Analogously, for any coherent real-analytic countably-infinite space  $  X $
 
it is possible to construct the complexification  $  \widetilde{X}  $,  
 
it is possible to construct the complexification  $  \widetilde{X}  $,  

Latest revision as of 16:15, 11 February 2022


An analytic space over the field $ \mathbf R $ of real numbers. Unlike in the case of complex-analytic spaces, the structure sheaves of real-analytic spaces need not be coherent (cf. Coherent sheaf). Real-analytic spaces are said to be coherent if their structure sheaves are coherent. All real-analytic manifolds (i.e. smooth real-analytic spaces) are coherent real-analytic spaces.

Let $ V _ {a} $ be the germ at a point $ a $ of a real-analytic subset of $ \mathbf R ^ {n} $ (cf. Analytic set). This defines the germ at $ a $ of a complex-analytic subset $ \widetilde{V} _ {a} $ of the space $ \mathbf C ^ {n} $ with the following equivalent properties: 1) $ \widetilde{V}_ {a} $ is the intersection of all germs of complex-analytic sets containing $ V _ {a} $; 2) if $ {\mathcal O} _ {V _ {a} } $ is the analytic algebra of the germ $ V _ {a} $, then $ {\mathcal O} _ {V _ {a} } \otimes \mathbf C $ is the analytic algebra of the germ $ \widetilde{V}_ {a} $. The germ $ \widetilde{V}_ {a} $ is said to be the complexification of the germ $ V _ {a} $, and $ V _ {a} $ is said to be the real part of the germ $ \widetilde{V}_ {a} $. Analogously, for any coherent real-analytic countably-infinite space $ X $ it is possible to construct the complexification $ \widetilde{X} $, which is a complex-analytic space. $ X $ will then have a fundamental system of neighbourhoods in $ \widetilde{X} $ which are Stein spaces (cf. Stein space).

The theory of coherent real-analytic spaces is similar to the theory of complex Stein spaces. Global sections of any coherent analytic sheaf of modules $ F $ on a coherent real-analytic countably-infinite space $ X $ generate modules of germs of its sections at any point of $ X $, and all groups $ H ^ {q} ( X, F ) $ vanish if $ q \geq 1 $.

For any finite-dimensional coherent real-analytic countably-infinite space $ ( X, {\mathcal O} _ {X} ) $ there exists a morphism

$$ f = ( f _ {0} , f _ {1} ): ( X, {\mathcal O} _ {X} ) \rightarrow \ ( \mathbf R ^ {n} , {\mathcal O} _ {\mathbf R ^ {n} } ) $$

such that $ f _ {0} $ is a proper one-to-one mapping of $ X $ into a coherent subspace in $ \mathbf R ^ {n} $, while $ f $ is an imbedding at the smooth points of $ X $. In particular, any (Hausdorff and countably-infinite) real-analytic manifold is isomorphic to a real-analytic submanifold in $ \mathbf R ^ {n} $. For a reduced coherent real-analytic space $ X $ the set of classes of isomorphic real-analytic principal fibre bundles with a real structure Lie group $ G $, admitting complexification, and base $ X $ is in one-to-one correspondence with the set of classes of isomorphic topological principal fibre bundles with the same structure group $ G $.

References

[1] A. Tognoli, "Some results in the theory of real analytic spaces" M. Jurchesan (ed.) , Espaces Analytiques (Bucharest 1969) , Acad. Roumanie (1971) pp. 149–157

Comments

References

[a1] H. Cartan, "Variétés analytiques réelles et variétés analytiques complexes" Bull. Soc. Math. France , 85 (1957) pp. 77–99
[a2] F. Bruhat, H. Cartan, "Sur la structure des sous-ensembles analytiques réels" C.R. Acad. Sci. Paris , 244 (1957) pp. 988–900
[a3] F. Bruhat, H. Cartan, "Sur les composantes irréductibles d'un sous-ensemble" C.R. Acad. Sci. Paris , 244 (1957) pp. 1123–1126
[a4] F. Bruhat, H. Whitney, "Quelques propriétés fondamentales des ensembles analytiques-réels" Comm. Math. Helv. , 33 (1959) pp. 132–160
[a5] R. Narasimhan, "Introduction to the theory of analytic spaces" , Lect. notes in math. , 25 , Springer (1966)
[a6] H. Grauert, R. Remmert, "Theory of Stein spaces" , Springer (1979) (Translated from German)
How to Cite This Entry:
Real-analytic space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Real-analytic_space&oldid=52048
This article was adapted from an original article by D.A. Ponomarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article