# Analytic mapping

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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 is a continuous mapping, while 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 at a point is the analytic subspace of the space , where is the sheaf of germs of functions that vanish at the point . Putting one obtains the inequality (*)

If and are reduced complex spaces, then the set 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 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 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 .

How to Cite This Entry:
Analytic mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_mapping&oldid=12227
This article was adapted from an original article by D.A. Ponomarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article