ARCWISE ANALYTIC STRATIFICATION, WHITNEY FIBERING CONJECTURE AND ZARISKI EQUISINGULARITY

Adam Parusiński (1) , Laurentiu Paunescu (2)

1 LJAD - Laboratoire Jean Alexandre Dieudonné
2 School of Mathematics and statistics [Sydney]

In this paper we show Whitney's fibering conjecture in the real and complex, local analytic and global algebraic cases. For a given germ of complex or real analytic set, we show the existence of a stratifica-tion satisfying a strong (real arc-analytic with respect to all variables and analytic with respect to the parameter space) trivialization property along each stratum. We call such a trivialization arc-wise analytic and we show that it can be constructed under the classical Zariski algebro-geometric equisingularity assumptions. Using a slightly stronger version of Zariski equisingularity, we show the existence of Whitney's stratified fibration, satisfying the conditions (b) of Whitney and (w) of Verdier. Our construction is based on Puiseux with parameter theorem and a generalization of Whitney interpolation. We also give several applications of arc-wise analytic trivialization, mainly to the stratifi-cation theory and the equisingularity of analytic set and function germs. In the real algebraic case, for an algebraic family of projective varieties, we show that Zariski equisingularity implies local triviality of the weight filtration.

Mots clés: Stratifications; Whitney Fibering Conjecture.; Whitney Conditions; Zariski Equisingularity; Puiseux Theorem

Domaines: Géométrie algébrique [math.AG]

Adam Parusiński, Laurentiu Paunescu. ARCWISE ANALYTIC STRATIFICATION, WHITNEY FIBERING CONJECTURE AND ZARISKI EQUISINGULARITY. Advances in Mathematics, 2017, 309, pp.254--305. ⟨hal-01121413v2⟩