The Walker Abel-Jacobi map descends

TL;DR

This paper introduces a new construction of the Walker Abel-Jacobi map, demonstrating its descent to any field of definition and analyzing its image via the coniveau filtration, advancing understanding of algebraic cycles.

Contribution

It provides a third, Hodge-theoretic construction of the Walker Abel-Jacobi map and proves its descent to arbitrary fields of definition.

Findings

Walker Abel-Jacobi map descends canonically to any field of definition.

The new construction offers a different perspective from previous methods.

The image of the l-adic Bloch map is characterized in terms of the coniveau filtration.

Abstract

For a complex projective manifold, Walker has defined a regular homomorphism lifting Griffiths' Abel-Jacobi map on algebraically trivial cycle classes to a complex abelian variety, which admits a finite homomorphism to the Griffiths intermediate Jacobian. Recently Suzuki gave an alternate, Hodge-theoretic, construction of this Walker Abel-Jacobi map. We provide a third construction based on a general lifting property for surjective regular homomorphisms, and prove that the Walker Abel-Jacobi map descends canonically to any field of definition of the complex projective manifold. In addition, we determine the image of the l-adic Bloch map restricted to algebraically trivial cycle classes in terms of the coniveau filtration.