an:04196130
Zbl 0725.14016
Hamm, Helmut A.; L?? Dung Trang
Rectified homotopical depth and Grothendieck conjectures
EN
The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. II, Prog. Math. 87, 311-351 (1990).
1990
a
14F35 14D05 32C15 14F05
Lefschetz theorems; complex analytic space; Whitney stratification; rectified homotopical depth; hyperplane section
[For the entire collection see Zbl 0717.00009.]
Let \(\emptyset \neq X\) be a reduced complex analytic space with \({\mathcal S}\) a Whitney stratification on X, where \(X_ i\) denotes the union of strata of dimension \(\leq i\). The rectified homotopical depth \(rhd_{{\mathcal S}}(X)\) of X is \(\geq n\) if, for any i and any point \(x\in X_ i\setminus X_{i-1}\), there exists a fundamental system \((U_{\alpha})\) of neighbourhoods of x in X such that, for any \(\alpha\) the pair \((U_{\alpha},U_{\alpha}\setminus X_ i)\) is (n-1-i)-connected. Then \(rhd_{{\mathcal S}}(X)\) is defined as the maximum of the integers n such that \(rhd_{{\mathcal S}}(X)\geq n\). This definition is shown to be equivalent to the ones of \textit{A. Grothendieck} [see S??minaire de g??om??trie alg??brique, SGA 2 (1962; Zbl 0159.504); Expos?? XIII, p. 27, definition 2; see also Adv. Stud. Pure Math. 2 (1968; Zbl 0197.472)]. Replacing the connectedness condition by the vanishing of \(H_ k(U_{\alpha},U_{\alpha}\setminus X_ i,{\mathbb{Z}})\) for any \(k<n-i\) the authors define the rectified homological depth of X. - A. Grothendieck conjectured that the notion of rectified homotopical (resp. homological) depth gives the right level of comparison for the homotopy (resp. homology) type between X and a hyperplane section, as stated in theorems of Lefschetz type for nonsingular varieties.
The authors give positive answers to Grothendieck's conjectures. The proofs become possible because of a handy formulation of the notion of rectified homotopical depth using Whitney stratifications shown in the paper. More general there are Lefschetz type theorems for open varieties replacing the hyperplane section by a good neighbourhood of a hyperplane section as has been done by \textit{P. Deligne} [see S??min. Bourbaki, 32e ann??e, Vol. 1979/80, Expos?? 543, Lect. Notes Math. 842, 1-10 (1981; Zbl 0478.14008)].
P.Schenzel (Halle)
Zbl 0717.00009; Zbl 0159.504; Zbl 0197.472; Zbl 0478.14008