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)