Modification

of an analytic space

An analytic mapping $f : X \rightarrow Y$ of analytic spaces such that for certain analytic sets $S \subset X$ and $T \subset Y$ of smaller dimensions, the conditions

$$f : X \setminus S \rightarrow Y \setminus T \ \ \textrm{ is an isomorphism }$$

and

$$f ( S) = T$$

hold. A modification is also called a contraction of $S$ onto $T$. An example of a modification is a monoidal transformation.

References

 [1] H. Behnke, K. Stein, "Modifikation komplexer Mannigfaltigkeiten und Riemannschen Gebiete" Math. Ann. , 124 : 1 (1951) pp. 1–16

References

 [a1] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001
How to Cite This Entry:
Modification. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Modification&oldid=47868
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article