Distinguished models of intermediate Jacobians

TL;DR

This paper constructs a functorial model of the intermediate Jacobian over the base field, linking Abel-Jacobi maps, Galois representations, and Hilbert schemes, and addresses a question of Mazur.

Contribution

It introduces a new functorial model of the intermediate Jacobian over the base field, connecting it with the Albanese variety and Hilbert schemes, and answers Mazur's question.

Findings

The Abel-Jacobi map admits a functorial model over the base field.

The Galois representation of the cohomology matches the coniveau filtration.

The model is dominated by the Albanese variety of a Hilbert scheme component.

Abstract

We show that the image of the Abel-Jacobi map admits functorially a model over the field of definition, with the property that the Abel-Jacobi map is equivariant with respect to this model. The cohomology of this abelian variety over the base field is isomorphic as a Galois representation to the deepest part of the coniveau filtration of the cohomology of the projective variety. Moreover, we show that this model over the base field is dominated by the Albanese variety of a product of components of the Hilbert scheme of the projective variety, and thus we answer a question of Mazur. We also recover a result of Deligne on complete intersections of Hodge level one.