Relative homotopical depth and a conjecture of Deligne

Abstract

32S50, 14B05, 14F35 In this paper we define the notion of relative rectified homotopical depth for complex analytic morphisms which generalizes the rectified homotopical depth introduced by Grothendieck in [G, Sect. 6, Chap. XIII] (see also [H-L3, Sect. 1]) in the absolute case of complex analytic spaces. As shown in I-H-L3] and conjectured by Grothend~eck (loc.cit.), in theorems analogous to the theorem of Lefschetz on hyperplane sections, the rectified homotopical depth gives an estimate for the level of coincidence of the homotopy of an algebraic variety and one of its hyperplane sections when there are singularities. Similarly we obtain relative Lefschetz type theorems for morphisms where the relative rectified homotopical depth gives also an estimate for coincidence of homotopy. Following the procedure of [H-L3], we show that an adequate stratification of the morphism gives a way to calculate the relative rectified homotopical depth. On the other hand, as corollaries of our main result (see Theorem 2.1.3 below), we obtain a generalization of a conjecture of Deligne ID, Conjecture 1.3] already proven by Goresky and MacPherson in [G-M, II, Sect. 1.1, Sect. 5.1] and theorems of Lefschetz type on a singular space obtained by us in [H-L3] or in a different form by Goresky and MacPherson [G-M, II, Sect. 1.2, Sect. 5.2). 1 Relative rectified homotopieal depth 1.1. First recall the definition of the homotopical depth of X along a locally closed complex analytic subspace Y at a point x ~ Y [G, D6finition 1, Sect. 6, Chap. XIII, p. 197] or [H-L3, Definition 3.1.2]. * The first author was partly supported by the French-German PROCOPE Program and the second author was partly supported by the NSF Grant DMS-8803478