Distinguished models of intermediate Jacobians

Jeff Achter, Sebastian Casalaina-Martin, Charles Vial

Published 2016-11-22Version 1

We show that the image of the Abel-Jacobi map descends to an abelian variety over the field of definition, and that moreover, the Abel-Jacobi map is equivariant with respect to this model. More precisely, for a smooth projective variety defined over a base field contained in the complex numbers, we show that for any odd cohomology group, the image of the associated complex Abel-Jacobi map defined on algebraically trivial cycles, which is naturally a complex abelian variety, admits a distinguished model over the base field, such that the Abel-Jacobi map is equivariant with respect to the group of automorphisms of the complex numbers fixing the base field. 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.