Factorization of the Abel-Jacobi maps

TL;DR

This paper provides a direct proof that the Abel--Jacobi map $\\psi^{p}$ for codimension $p$ cycles factors through a regular homomorphism, as originally shown by Walker using Lawson homology. The construction builds the Walker map $\\tilde{\\psi}^{p}$ by lifting through the coniveau filtration $N^{p-1}H^{2p-1}(X, \mathbb{Z}(p))$ and its intermediate Jacobian, employing boundary maps in the Ext groups of mixed Hodge structures and taking direct limits. It is established that $\\tilde{\\psi}^{p}$ is regular, surjective onto $J\bigl(N^{p-1}H^{2p-1}(X, \mathbb{Z}(p))\bigr)$, and factors the Abel--Jacobi map through $A^{p}(X)$, thereby yielding a universal regular morphism in this setting. By avoiding the full Lawson homology framework, the argument clarifies the role of coniveau and mixed Hodge structures and connects to Murre’s universality questions, with consequences for the coniveau filtration and torsion phenomena in intermediate Jacobians.

Abstract

As an application of the theory of Lawson homology and morphic cohomology, Walker proved that the Abel-Jacobi map factors through another regular homomorphism. In this note, we give a direct proof of the theorem.

Theorems & Definitions (7)

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• proof : Proof of Theorem \ref{['UAJw']}