Localisation de syst�mes diff�rentiels, stratifications de Whitney et condition de Thom

Refinements of Milnor's fibration theorem for complex singularities

“…It would be interesting to study geometric and topological properties of the 4-manifolds one gets in this way, by considering the link of the hypersurface in C 3 defined by the real part of a holomorphic function with an isolated critical point. For example, for the map (z 1 , z 2 , z 3 ) f → z 2 1 + z 3 2 + z 5 3 , the corresponding 4-manifold is the double of the famous E 8 manifold with boundary Poincaré's homology 3-sphere.…”

Variation of the Milnor Fibration in Pencils of Hypersurface Singularities

“…Thus, the union of the ®bres of the relative conormal space at the points of VW, that is, the ®bre at 0 of the composition W ± t W , is contained in the ®nite union of the conormal spaces of the strata of S, which are known to be n-dimensional. Since any ®bre of W ± t W at a regular value of W is n-dimensional (as the conormal space to the ®bre of W at this point), and since the dimension of the ®bres of W ± t W is a lower semi-continuous function, this composition is equidimensional, and thus open, over any suf®ciently small neighbourhood of 0 in C (a) p 0 Any germ f : C n 1 ; 0 3 C; 0 of a function has no blowing up (any Whitney strati®cation of the zero locus V f is in fact a good strati®cation; see [2]). …”

Multiplier ideals, V-filtrations and transversal sections

“…Note that the µ-constantness is equivalent to the Thom a f -condition (see [25] and 4.8 below) and the latter is weaker than the Whitney (b) condition, see [4]. It is not clear whether Corollary 1.4 holds assuming only the a f -condition without the Whitney (b) condition.…”