Analytic mapping
analytic morphism
A morphism of analytic spaces considered as ringed spaces (cf. Analytic space; Ringed space). An analytic mapping of a space () into a space (
) is a pair
, where
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is a continuous mapping, while
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is a homomorphism of sheaves of rings on . If the spaces are complex, an analytic mapping is also called a holomorphic mapping.
If and
are reduced analytic spaces, the homomorphism
is completely determined by the mapping
and is the inverse mapping of the germs of functions corresponding to
. Thus, in this case an analytic mapping is a mapping
such that for any
and for any
one has
.
A fibre of an analytic mapping
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at a point is the analytic subspace
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of the space , where
is the sheaf of germs of functions that vanish at the point
. Putting
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one obtains the inequality
![]() | (*) |
If and
are reduced complex spaces, then the set
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is analytic in for any
.
An analytic mapping is called flat at a point
if
is a flat module over the ring
. In such a case (*) becomes an equality. An analytic mapping is called flat if it is flat at all points
. A flat analytic mapping of complex spaces is open. Conversely, if
is open,
is smooth and all fibres are reduced, then
is a flat analytic mapping. The set of points of a complex or a rigid analytic space
at which an analytic mapping
is not flat is analytic in
. If
and
are reduced complex spaces, while
has a countable base, then
contains a dense everywhere-open set over which
is a flat analytic mapping. If an analytic mapping
![]() |
of complex spaces is flat, then the set of at which the fibre
is not reduced or normal is analytic in
.
Let be an analytic mapping of reduced complex spaces. If
, then there exists a stratification
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where are analytic sets and
for large
, with the following property: Any point
has a neighbourhood
in
such that
is a local analytic set in
, all irreducible components of germs of which have dimension
at
. If
is proper, then
is an analytic set in
. This is a particular case of the finiteness theorem for analytic mappings.
Let ,
be complex spaces and let
be compact. Then it is possible to endow the set
of all analytic mappings
with the structure of a complex space such that the mapping
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which maps the pair into
, is analytic. In particular, the group of automorphisms of a compact complex space
is a complex Lie group, acting analytically on
.
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) |
Analytic mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_mapping&oldid=12227