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Conditions de régularité et éclatements. (Regularity conditions and blowing-ups). (French) Zbl 0596.32018
One describes three types of equivalent conditions to stratify complex analytic spaces and morphisms:
1) Numerical conditions, i.e. equimultiplicity of polar varieties, either all polar varieties or only dirimants, depending on the geometrical space used in the second condition.
2) Equidimensionality of some exceptional divisors, especially divisors related to the Nash blowing-ups, and to the conormal space.
3) Differential conditions, or regularity conditions in Thom-Whitney style.
Because the Nash blowing-up contains more information than the conormal space, in the related situation the differential conditions are stronger than usual a and b conditions.
In the study of conditions related to the relative conormal space, a new condition $$b_ f$$ is defined, which, when the base space is a point, is the usual Whitney condition b; this condition is shown to be inherited by the relative dirimants for transversal projections. Equivalence between differential conditions and intersections of cycles in the grassmannian or projective cohomology are shown. These results give tools to construct canonical stratifications of a complex analytic morphism, the objects used are analytic invariants of the morphism.

##### MSC:
 32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) 32Sxx Complex singularities 32S45 Modifications; resolution of singularities (complex-analytic aspects) 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 14M15 Grassmannians, Schubert varieties, flag manifolds
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