The morphic Abel–Jacobi map

Abstract

The morphic Abel–Jacobi map is the analogue of the classical Abel–Jacobi map one obtains by using Lawson and morphic (co)homology in place of the usual singular (co)homology. It thus gives a map from the group of $r$-cycles on a complex variety that are algebraically equivalent to zero to a certain ‘Jacobian’ built from the Lawson homology groups viewed as inductive limits of mixed Hodge structures. In this paper, we define the morphic Abel–Jacobi map, establish its foundational properties, and then apply these results to the study of algebraic cycles. In particular, we show the classical Abel–Jacobi map (when restricted to cycles algebraically equivalent to zero) factors through the morphic version, and show that the morphic version detects cycles that cannot be detected by its classical counterpart; that is, we give examples of cycles in the kernel of the classical Abel–Jacobi map that are not in the kernel of the morphic version. We also investigate the behavior of the morphic Abel–Jacobi map on the torsion subgroup of the Chow group of cycles algebraically equivalent to zero modulo rational equivalence.