Cycles evanescents, sections planes et conditions de Whitney. II.

*(French)*Zbl 0532.32003
Singularities, Summer Inst., Arcata/Calif. 1981, Proc. Symp. Pure Math. 40, Part 2, 65-103 (1983).

[For the entire collection see Zbl 0509.00008. For part I see B. Teissier, Astérisque 7-8 (1973), 285-392 (1974; Zbl 0295.14003).]

Let \(X\) be a complex-analytic space and \(X=\cup_{a\in A}X_ a\) a partition of \(X\) into nonsingular constructible subspaces. The general problem considered inthe paper under review is to state several conditions of algebraic, combinatorial, homotopical and topological nature, which are equivalent to the fact that this partition is a Whitney stratification of \(X\). A result of Thom and Mather asserts that in a Whitney stratification of \(X\) one has topological triviality along strata. Zariski studied from the algebraic point of view the particular case where \(X\) is a hypersurface such that its singular locus is smooth and of codimension one. More precisely, he found algebraic conditions ensuring that the smooth locus and the singular locus of \(X\) be a Whitney stratification of \(X\). Later Teissier gave a numerical criterion for the regularity of the same partition of a hypersurface whose singular locus is nonsingular. If the singular locus of the hypersurface is of codimension one, Zariski and Lê D. T. showed that the topological triviality of \(X\) along the singular locus implies the Whitney regularity condition. If the singular locus of the hypersurface \(X\) is nonsingular but of arbitrary codimension such a result is no longer true in general, and Teissier found a necessary and sufficient condition such that the smooth and the singular locus of \(X\) be a Whitney stratification of \(X\). Namely, this condition regards the local topological triviality when one cuts out \(X\) with a general flag of nonsingular subspaces containing the singular locus of \(X\). The main result of this paper extends the latter result for hypersurfaces and gives in particular a ”correct” converse of a suitable improvement of the Thom-Mather topological triviality theorem.

Let \(X\) be a complex-analytic space and \(X=\cup_{a\in A}X_ a\) a partition of \(X\) into nonsingular constructible subspaces. The general problem considered inthe paper under review is to state several conditions of algebraic, combinatorial, homotopical and topological nature, which are equivalent to the fact that this partition is a Whitney stratification of \(X\). A result of Thom and Mather asserts that in a Whitney stratification of \(X\) one has topological triviality along strata. Zariski studied from the algebraic point of view the particular case where \(X\) is a hypersurface such that its singular locus is smooth and of codimension one. More precisely, he found algebraic conditions ensuring that the smooth locus and the singular locus of \(X\) be a Whitney stratification of \(X\). Later Teissier gave a numerical criterion for the regularity of the same partition of a hypersurface whose singular locus is nonsingular. If the singular locus of the hypersurface is of codimension one, Zariski and Lê D. T. showed that the topological triviality of \(X\) along the singular locus implies the Whitney regularity condition. If the singular locus of the hypersurface \(X\) is nonsingular but of arbitrary codimension such a result is no longer true in general, and Teissier found a necessary and sufficient condition such that the smooth and the singular locus of \(X\) be a Whitney stratification of \(X\). Namely, this condition regards the local topological triviality when one cuts out \(X\) with a general flag of nonsingular subspaces containing the singular locus of \(X\). The main result of this paper extends the latter result for hypersurfaces and gives in particular a ”correct” converse of a suitable improvement of the Thom-Mather topological triviality theorem.

Reviewer: L.Bădescu

##### MSC:

32S60 | Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) |

32Sxx | Complex singularities |

32S05 | Local complex singularities |