Moving Algebraic Cycles of Bounded Degree, by Eric M. Friedlander and H. Blaine Lawson

The authors prove the following parameterized version of the classical Chow Moving Lemma. Let $X$ be a smooth projective variety of dimension $n$ over an infinite field $k$ and let $r,s$ be non-negative integers with $r+s \geq n$. Let ${\cal C}_s(X)$ denote the Chow monoid of effective algebraic $s$-cycles on $X$ and let $Z_s(X)$ denote the group of all algebraic $s$-cycles on $X$. For any $d > 0$, there exists a Zariski neighborhood $U$ of $\{ 0, 1 \} \subset {\bf A}^1$ and an algebraic homotopy $$\Psi: {\cal C}_s(X) \times U \to {\cal C}_s(X)^2$$ with the property that this homotopy induces the identity on $Z_s(X)$ at $t=0$ and at $t=1$ sends any effective $s$-cycle of degree $\leq d$ to a pair of effective $s$-cycles each of which intersect properly every effective $r$-cycle on $X$ of degree $\leq d$.

In a forthcoming paper, this result is used by the authors to prove a duality theorem relating ``morphic cohomology'' (defined in terms of algebraic cocycles) and Lawson homology. In this present paper, this result is shown to imply that intersection is a well defined pairing on Chow groups and to realize ``concretely'' the intersection pairing on cycle spaces constructed in a recent paper by the first author and Ofer Gabber.

Eric M. Friedlander <>
H. Blaine Lawson <>