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Morphismes analytiques stratifiés sans éclatement et cycles évanescents. (French) Zbl 0542.32005
Astérisque 101-102, 286-319 (1983).
Let $$f:X\to Y$$ be a morphism between complex reduced analytic spaces, and let S be a complex analytic stratification for f. The author gives the following definition: (f,S) is without blowing-up if the stratification of Y satisfies Whitney’s conditions, the stratification of X satisfies $$A_ f$$-Thom’s condition, and if, moreover, over each stratum in Y, the restriction of the upstairs stratification satisfies Whitney’s conditions. - This notion gives a class of morphisms for which Thom’s second isotopy lemma allows to build a vanishing cycles’ theory. Moreover this class is stable by base change.
The main theorem of this article is: Let $$f:X\to Y$$ be a proper map between complex reduced spaces, and let S be a complex analytic stratification of f. Then there exist a proper sequence of blowing-up’s $$\pi:\tilde Y\to Y$$, and a complex analytic stratification $$\tilde S$$ of $$\tilde f=\pi^*(f),$$ compatible with S, such that (\~f,\~S) is without blowing-up. The proof of this theorem, which is a finiteness result, uses Hironaka’s flattening theorem and Hironaka’s criterium for Thom’s condition [H. Hironaka, Am. J. Math. 97, 503-547 (1975; Zbl 0307.32011) and Real and complex Singul., Proc. Nordic Summer Sch., Symp. Math., Oslo 1976, 199-265 (1977; Zbl 0424.32004)]. - The author gives also a result of the same type without properness assumption. For the proof, the local flattening from H. Hironaka, M. Lejeune-Jalabert and B. Teissier is then used [Astérisque 7/8 (1973), 441-463 (1974; Zbl 0287.14007)].
For the entire collection see [Zbl 0515.00021].
Reviewer: D.Barlet

##### MSC:
 32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) 32C15 Complex spaces