Rectified Homotopical Depth and Grothendieck Conjectures

Abstract

In SGA 2 ([G]), A. Grothendieck introduced the notion of rectified homotopical (resp. homological) depth. He conjectured that it gives the level of comparison for the homotopy type (resp. the homology) between a complex algebraic variety and a hyperplane section, as stated in theorems of Lefschetz type for singular algebraic varieties. In the case of non-singular varieties, the rectified homotopical (resp. homological) depth equals the complex dimension of the variety. But in the case of local complete inter-sections, one can show that this is still true. In fact, using the comparison theorem of Grothendieck as formulated by Mebkhout for \( \mathcal{D} \)-modules in [Me], the constant sheaf \( \underline C \) of complex numbers on a variety which is locally a complete intersection is perverse and one can prove that the constant sheaf \( \underline C \) of complex numbers on the variety is perverse if and only if the rectified homological depth for the rational homology equals the complex dimension of the variety. So the rectified homological depth for the rational homology measures how far the constant sheaf \( \underline C \) of complex numbers on the variety is from being perverse. In this paper we give a positive answer to the conjecture of Grothendieck. Actually, we prove all the conjectures given by Grothendieck on this theme in SGA 2, except Conjecture A, which is obviously incorrect as stated, but can be easily corrected.