### The morphic Abel-Jacobi map, by Mark E. Walker

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 one. 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.

Mark E. Walker <mwalker@math.unl.edu>