Analytic mapping

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)