View source for Analytic mapping

''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

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is a continuous mapping, while

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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.

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" />.

A fibre of an analytic mapping

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at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012330/a01233017.png" /> is the analytic subspace

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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

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one obtains the inequality

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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

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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" />.

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

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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" />.

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

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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.

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

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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" />.

====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>