Homology Using Chow Varieties

We introduce "Lawson homology groups" LrH2r+i(Xf2i) associated to an arbitrary projective algebraic variety X over an algebraically closed field k of characteristic p > 0 and a prime I ^ p. Our work is directly inspired by recent work of Blaine Lawson (cf. [5, 6]), consisting in part of an algebraization of Lawson's geometric ideas and analytic arguments. The Lawson homology group LQH^X, Z/) is the /th etale /-adic homology group of X, whereas Lr//2r(XZ/) is the group of algebraic r-cycles on X modulo algebraic equivalence (see Theorem 5 below). More generally, LrH2r+i(X,2i) should be viewed as an /-adic homology group of X involving r algebraic dimensions and i topological dimensions." As we describe below, these groups are interesting algebraic invariants with good properties which should prove useful in the study of algebraic cycles. Moreover, the author and Barry Mazur construct maps LrH2r+i(X, Z/) -> Lr-iH2r+i(X, Z/) whose iterates determine the cycle map relating algebraic cycles to etale homology. We gratefully thank Blaine Lawson for sharing his recent results with us while still in their formative stages. We also acknowledge our great debt to Ofer Gabber whose insights were essential to our early understanding of Lawson's work. Proofs of results announced below, as well as statements and proofs of further results being developed in collaboration with Blaine Lawson and Barry Mazur, will appear elsewhere.