Algebraic cycles and homotopy theory. (English) Zbl 0688.14006

Lawson, H. Blaine jun.

Ann. Math. (2) 129, No. 2, 253-291 (1989).

Let \(C_{p,d}({\mathbb{P}}^ n)\) be the Chow variety of all cycles of dimension p and degree d of the complex projective space \({\mathbb{P}}^ n\). The main aim of this paper is to show that the entire homotopy structure of \(C_{p,d}({\mathbb{P}}^ n)\) stabilizes to a simple and computable one as d becomes sufficiently large. More precisely, for each \(p\geq 0\) fix a linear subspace \(L\subset {\mathbb{P}}^ n\) of dimension p, and for each \(d\geq 1\) consider the embedding \(C_{p,d}({\mathbb{P}}^ n)\subset C_{p,d+1}({\mathbb{P}}^ n)\) given by \(c\mapsto c+L\). Then it makes sense to consider \(C_ p({\mathbb{P}}^ n)=\cup_{d\geq 0}C_{p,d}({\mathbb{P}}^ n)\) endowed with the weak topology (i.e. \(F\subset C_ p({\mathbb{P}}^ n)\) is closed iff \(F\cap C_{p,d}({\mathbb{P}}^ n)\) is closed for all \(d\geq 1)\). The following result is one of the fundamental theorems proved in this paper:
Theorem. For every \(q\leq n\) one has a homotopy equivalence \(C_{n- p}({\mathbb{P}}^ n)\cong K({\mathbb{Z}},2)\times K({\mathbb{Z}},4)\times...\times K({\mathbb{Z}},2q)\), where \(K({\mathbb{Z}},2k)\) denotes the standard Eilenberg- MacLane space. - If \(q=n\) this result recovers a beautiful classical theorem of Dold-Thom concerning \(C_ 0({\mathbb{P}}^ n)\), which was in fact the starting point of the present paper.
Another result proved is the following theorem: For all n, p and d the inclusion \(C_{p,d}({\mathbb{P}}^ n)\hookrightarrow C_ p({\mathbb{P}}^ n)\) has a right homotopy inverse through dimension 2d. Several other interesting results concerning the topology of \(C_ p(X)\) (with \(X\subset {\mathbb{P}}^ n\) a closed subvariety) are also proved.

Reviewer: L.Bădescu

MSC:

14C05	Parametrization (Chow and Hilbert schemes)
55P20	Eilenberg-Mac Lane spaces
14F35	Homotopy theory and fundamental groups in algebraic geometry

Keywords:

Chow variety; homotopy equivalence; Eilenberg-MacLane space