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''analytic morphism''
 
''analytic morphism''
  
A morphism of analytic spaces considered as ringed spaces (cf. [[Analytic space|Analytic space]]; [[Ringed space|Ringed space]]). An analytic mapping of a space (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a0123301.png" />) into a space (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a0123302.png" />) is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a0123303.png" />, where
+
A morphism of analytic spaces considered as ringed spaces (cf. [[Analytic space|Analytic space]]; [[Ringed space|Ringed space]]). An analytic mapping of a space ( $  X , {\mathcal O} _ {X} $)  
 +
into a space ( $  X , {\mathcal O} _ {Y} $)  
 +
is a pair $  ( f _ {0} , f _ {1} ) $,  
 +
where
  
<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/a/a012/a012330/a0123304.png" /></td> </tr></table>
+
$$
 +
f _ {0} : X  \rightarrow  Y
 +
$$
  
 
is a continuous mapping, while
 
is a continuous mapping, while
  
<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/a/a012/a012330/a0123305.png" /></td> </tr></table>
+
$$
 +
f _ {1} : f _ {0}  ^ {-1} ( {\mathcal O} _ {Y} )  \rightarrow  {\mathcal O} _ {X}  $$
  
is a homomorphism of sheaves of rings on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a0123306.png" />. If the spaces are complex, an analytic mapping is also called a holomorphic mapping.
+
is a homomorphism of sheaves of rings on $  X $.  
 +
If the spaces are complex, an analytic mapping is also called a holomorphic mapping.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a0123307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a0123308.png" /> are reduced analytic spaces, the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a0123309.png" /> is completely determined by the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233010.png" /> and is the inverse mapping of the germs of functions corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233011.png" />. Thus, in this case an analytic mapping is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233012.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233013.png" /> and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233014.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233015.png" />.
+
If $  ( X, {\mathcal O} _ {X} ) $
 +
and $  ( Y, {\mathcal O} _ {Y} ) $
 +
are reduced analytic spaces, the homomorphism $  f _ {1} $
 +
is completely determined by the mapping $  f _ {0} $
 +
and is the inverse mapping of the germs of functions corresponding to $  f _ {0} $.  
 +
Thus, in this case an analytic mapping is a mapping $  f: X \rightarrow Y $
 +
such that for any $  x \in X $
 +
and for any $  \phi \in {\mathcal O} _ {f(x) }  $
 +
one has $  \phi \circ f \in {\mathcal O} _ {X} $.
  
 
A fibre of an analytic mapping
 
A fibre of an analytic mapping
  
<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/a/a012/a012330/a01233016.png" /></td> </tr></table>
+
$$
 +
= ( f _ {0} , f _ {1} ) : ( X , {\mathcal O} _ {X} )  \rightarrow \
 +
( Y , {\mathcal O} _ {Y} )
 +
$$
  
at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233017.png" /> is the analytic subspace
+
at a point $  y \in Y $
 +
is the analytic subspace
  
<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/a/a012/a012330/a01233018.png" /></td> </tr></table>
+
$$
 +
f  ^ {-1} ( y )  = ( f _ {0}  ^ {-1} ( y ) ,\
 +
{\mathcal O} _ {X} / f _ {1} ( m _ {y} ) {\mathcal O} _ {X} \mid  _ {f _ {0}  ^ {-1} ( y ) } )
 +
$$
  
of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233020.png" /> is the sheaf of germs of functions that vanish at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233021.png" />. Putting
+
of the space $  (X, {\mathcal O} _ {X} ) $,  
 +
where $  m _ {y} \in {\mathcal O} _ {y} $
 +
is the sheaf of germs of functions that vanish at the point $  y $.  
 +
Putting
  
<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/a/a012/a012330/a01233022.png" /></td> </tr></table>
+
$$
 +
d ( x )  =   \mathop{\rm dim} _ {x}  f  ^ {-1} ( f _ {0} ( x ) ) ,\ \
 +
x \in X ,
 +
$$
  
 
one obtains the inequality
 
one obtains the inequality
  
<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/a/a012/a012330/a01233023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
{ \mathop{\rm dim} } _ {x}  X  \leq    \mathop{\rm dim} _ {f _ {0}  ( x ) }  Y+d ( x ) .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233025.png" /> are reduced complex spaces, then the set
+
If $  X $
 +
and $  Y $
 +
are reduced complex spaces, then the set
  
<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/a/a012/a012330/a01233026.png" /></td> </tr></table>
+
$$
 +
X _ {l}  = \{ {x \in X } : {d ( x ) \geq  l } \}
 +
$$
  
is analytic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233027.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233028.png" />.
+
is analytic in $  X $
 +
for any $  l \geq  0 $.
  
An analytic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233029.png" /> is called flat at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233030.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233031.png" /> is a [[Flat module|flat module]] over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233032.png" />. In such a case (*) becomes an equality. An analytic mapping is called flat if it is flat at all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233033.png" />. A flat analytic mapping of complex spaces is open. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233034.png" /> is open, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233035.png" /> is smooth and all fibres are reduced, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233036.png" /> is a flat analytic mapping. The set of points of a complex or a [[Rigid analytic space|rigid analytic space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233037.png" /> at which an analytic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233038.png" /> is not flat is analytic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233039.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233041.png" /> are reduced complex spaces, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233042.png" /> has a countable base, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233043.png" /> contains a dense everywhere-open set over which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233044.png" /> is a flat analytic mapping. If an analytic mapping
+
An analytic mapping $  f = ( f _ {0} , f _ {1} ) $
 +
is called flat at a point $  x \in X $
 +
if $  {\mathcal O} _ {X,x }  $
 +
is a [[Flat module|flat module]] over the ring $  {\mathcal O} _ {Y, f _ {0}  (x) } $.  
 +
In such a case (*) becomes an equality. An analytic mapping is called flat if it is flat at all points $  x \in X $.  
 +
A flat analytic mapping of complex spaces is open. Conversely, if $  f _ {0} $
 +
is open, $  Y $
 +
is smooth and all fibres are reduced, then $  f $
 +
is a flat analytic mapping. The set of points of a complex or a [[Rigid analytic space|rigid analytic space]] $  X $
 +
at which an analytic mapping $  f $
 +
is not flat is analytic in $  X $.  
 +
If $  X $
 +
and $  Y $
 +
are reduced complex spaces, while $  X $
 +
has a countable base, then $  Y $
 +
contains a dense everywhere-open set over which $  f $
 +
is a flat analytic mapping. If an analytic mapping
  
<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/a/a012/a012330/a01233045.png" /></td> </tr></table>
+
$$
 +
f : ( X , {\mathcal O} _ {X} )  \rightarrow  ( Y , {\mathcal O} _ {Y} )
 +
$$
  
of complex spaces is flat, then the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233046.png" /> at which the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233047.png" /> is not reduced or normal is analytic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233048.png" />.
+
of complex spaces is flat, then the set of $  y \in Y $
 +
at which the fibre $  f  ^ {-1} (y) $
 +
is not reduced or normal is analytic in $  ( X, {\mathcal O} _ {X} ) $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233049.png" /> be an analytic mapping of reduced complex spaces. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233050.png" />, then there exists a stratification
+
Let $  f: X \rightarrow Y $
 +
be an analytic mapping of reduced complex spaces. If $  \mathop{\rm dim}  X < \infty $,  
 +
then there exists a stratification
  
<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/a/a012/a012330/a01233051.png" /></td> </tr></table>
+
$$
 +
\emptyset  = X ( - 1 )  \subseteq  X ( 0 )
 +
\subseteq \dots \subseteq  X ( r _ {i} )  \subseteq \dots ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233052.png" /> are analytic sets and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233053.png" /> for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233054.png" />, with the following property: Any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233055.png" /> has a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233056.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233057.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233058.png" /> is a local analytic set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233059.png" />, all irreducible components of germs of which have dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233060.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233061.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233062.png" /> is proper, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233063.png" /> is an analytic set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233064.png" />. This is a particular case of the finiteness theorem for analytic mappings.
+
where $  X (r) $
 +
are analytic sets and $  X(r) = X $
 +
for large $  r $,  
 +
with the following property: Any point $  x \in X(r) \setminus  X (r - 1) $
 +
has a neighbourhood $  U $
 +
in $  X $
 +
such that $  f ( U \cap X(r)) $
 +
is a local analytic set in $  Y $,  
 +
all irreducible components of germs of which have dimension $  r $
 +
at $  f(x) $.  
 +
If $  f $
 +
is proper, then $  f (X) $
 +
is an analytic set in $  X $.  
 +
This is a particular case of the finiteness theorem for analytic mappings.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233066.png" /> be complex spaces and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233067.png" /> be compact. Then it is possible to endow the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233068.png" /> of all analytic mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233069.png" /> with the structure of a complex space such that the mapping
+
Let $  X $,  
 +
$  Y $
 +
be complex spaces and let $  X $
 +
be compact. Then it is possible to endow the set $  { \mathop{\rm Mor} }  (X, Y) $
 +
of all analytic mappings $  f: X \rightarrow Y $
 +
with the structure of a complex space such that the mapping
  
<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/a/a012/a012330/a01233070.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Mor}  ( X , Y ) \times X  \rightarrow  Y ,
 +
$$
  
which maps the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233071.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233072.png" />, is analytic. In particular, the group of automorphisms of a compact complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233073.png" /> is a complex Lie group, acting analytically on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233074.png" />.
+
which maps the pair $  (f, x) $
 +
into $  f (x) $,  
 +
is analytic. In particular, the group of automorphisms of a compact complex space $  X $
 +
is a complex Lie group, acting analytically on $  X $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Remmert,  "Projektionen analytischer Mengen"  ''Math. Ann.'' , '''130'''  (1956)  pp. 410–441</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Remmert,  "Holomorphe und meromorphe Abbildungen komplexer Räume"  ''Math. Ann.'' , '''133'''  (1957)  pp. 328–370</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K. Stein,  , ''Colloquium for topology'' , Strasbourg  (1954)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J. Frisch,  "Points de plattitude d'une morphisme d'espaces analytiques complexes"  ''Invent. Math.'' , '''4'''  (1967)  pp. 118–138</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  G. Fisher,  "Complex analytic geometry" , Springer  (1976)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Remmert,  "Projektionen analytischer Mengen"  ''Math. Ann.'' , '''130'''  (1956)  pp. 410–441</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Remmert,  "Holomorphe und meromorphe Abbildungen komplexer Räume"  ''Math. Ann.'' , '''133'''  (1957)  pp. 328–370</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K. Stein,  , ''Colloquium for topology'' , Strasbourg  (1954)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J. Frisch,  "Points de plattitude d'une morphisme d'espaces analytiques complexes"  ''Invent. Math.'' , '''4'''  (1967)  pp. 118–138</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  G. Fisher,  "Complex analytic geometry" , Springer  (1976)</TD></TR></table>

Latest revision as of 18:47, 5 April 2020


analytic morphism

A morphism of analytic spaces considered as ringed spaces (cf. Analytic space; Ringed space). An analytic mapping of a space ( $ X , {\mathcal O} _ {X} $) into a space ( $ X , {\mathcal O} _ {Y} $) is a pair $ ( f _ {0} , f _ {1} ) $, where

$$ f _ {0} : X \rightarrow Y $$

is a continuous mapping, while

$$ f _ {1} : f _ {0} ^ {-1} ( {\mathcal O} _ {Y} ) \rightarrow {\mathcal O} _ {X} $$

is a homomorphism of sheaves of rings on $ X $. If the spaces are complex, an analytic mapping is also called a holomorphic mapping.

If $ ( X, {\mathcal O} _ {X} ) $ and $ ( Y, {\mathcal O} _ {Y} ) $ are reduced analytic spaces, the homomorphism $ f _ {1} $ is completely determined by the mapping $ f _ {0} $ and is the inverse mapping of the germs of functions corresponding to $ f _ {0} $. Thus, in this case an analytic mapping is a mapping $ f: X \rightarrow Y $ such that for any $ x \in X $ and for any $ \phi \in {\mathcal O} _ {f(x) } $ one has $ \phi \circ f \in {\mathcal O} _ {X} $.

A fibre of an analytic mapping

$$ f = ( f _ {0} , f _ {1} ) : ( X , {\mathcal O} _ {X} ) \rightarrow \ ( Y , {\mathcal O} _ {Y} ) $$

at a point $ y \in Y $ is the analytic subspace

$$ f ^ {-1} ( y ) = ( f _ {0} ^ {-1} ( y ) ,\ {\mathcal O} _ {X} / f _ {1} ( m _ {y} ) {\mathcal O} _ {X} \mid _ {f _ {0} ^ {-1} ( y ) } ) $$

of the space $ (X, {\mathcal O} _ {X} ) $, where $ m _ {y} \in {\mathcal O} _ {y} $ is the sheaf of germs of functions that vanish at the point $ y $. Putting

$$ d ( x ) = \mathop{\rm dim} _ {x} f ^ {-1} ( f _ {0} ( x ) ) ,\ \ x \in X , $$

one obtains the inequality

$$ \tag{* } { \mathop{\rm dim} } _ {x} X \leq \mathop{\rm dim} _ {f _ {0} ( x ) } Y+d ( x ) . $$

If $ X $ and $ Y $ are reduced complex spaces, then the set

$$ X _ {l} = \{ {x \in X } : {d ( x ) \geq l } \} $$

is analytic in $ X $ for any $ l \geq 0 $.

An analytic mapping $ f = ( f _ {0} , f _ {1} ) $ is called flat at a point $ x \in X $ if $ {\mathcal O} _ {X,x } $ is a flat module over the ring $ {\mathcal O} _ {Y, f _ {0} (x) } $. In such a case (*) becomes an equality. An analytic mapping is called flat if it is flat at all points $ x \in X $. A flat analytic mapping of complex spaces is open. Conversely, if $ f _ {0} $ is open, $ Y $ is smooth and all fibres are reduced, then $ f $ is a flat analytic mapping. The set of points of a complex or a rigid analytic space $ X $ at which an analytic mapping $ f $ is not flat is analytic in $ X $. If $ X $ and $ Y $ are reduced complex spaces, while $ X $ has a countable base, then $ Y $ contains a dense everywhere-open set over which $ f $ is a flat analytic mapping. If an analytic mapping

$$ f : ( X , {\mathcal O} _ {X} ) \rightarrow ( Y , {\mathcal O} _ {Y} ) $$

of complex spaces is flat, then the set of $ y \in Y $ at which the fibre $ f ^ {-1} (y) $ is not reduced or normal is analytic in $ ( X, {\mathcal O} _ {X} ) $.

Let $ f: X \rightarrow Y $ be an analytic mapping of reduced complex spaces. If $ \mathop{\rm dim} X < \infty $, then there exists a stratification

$$ \emptyset = X ( - 1 ) \subseteq X ( 0 ) \subseteq \dots \subseteq X ( r _ {i} ) \subseteq \dots , $$

where $ X (r) $ are analytic sets and $ X(r) = X $ for large $ r $, with the following property: Any point $ x \in X(r) \setminus X (r - 1) $ has a neighbourhood $ U $ in $ X $ such that $ f ( U \cap X(r)) $ is a local analytic set in $ Y $, all irreducible components of germs of which have dimension $ r $ at $ f(x) $. If $ f $ is proper, then $ f (X) $ is an analytic set in $ X $. This is a particular case of the finiteness theorem for analytic mappings.

Let $ X $, $ Y $ be complex spaces and let $ X $ be compact. Then it is possible to endow the set $ { \mathop{\rm Mor} } (X, Y) $ of all analytic mappings $ f: X \rightarrow Y $ with the structure of a complex space such that the mapping

$$ \mathop{\rm Mor} ( X , Y ) \times X \rightarrow Y , $$

which maps the pair $ (f, x) $ into $ f (x) $, is analytic. In particular, the group of automorphisms of a compact complex space $ X $ is a complex Lie group, acting analytically on $ X $.

References

[1] R. Remmert, "Projektionen analytischer Mengen" Math. Ann. , 130 (1956) pp. 410–441
[2] R. Remmert, "Holomorphe und meromorphe Abbildungen komplexer Räume" Math. Ann. , 133 (1957) pp. 328–370
[3] K. Stein, , Colloquium for topology , Strasbourg (1954)
[4] J. Frisch, "Points de plattitude d'une morphisme d'espaces analytiques complexes" Invent. Math. , 4 (1967) pp. 118–138
[5] G. Fisher, "Complex analytic geometry" , Springer (1976)
How to Cite This Entry:
Analytic mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_mapping&oldid=45176
This article was adapted from an original article by D.A. Ponomarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article