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DISTINGUISHED MODELS OF INTERMEDIATE JACOBIANS

Part of: Abelian varieties and schemes Cycles and subschemes Arithmetic algebraic geometry

Published online by Cambridge University Press:  08 June 2018

Jeffrey D. Achter ,
Sebastian Casalaina-Martin  and
Charles Vial
Jeffrey D. Achter
Affiliation:
Colorado State University, Department of Mathematics, Fort Collins, CO 80523, USA (j.achter@colostate.edu)
Sebastian Casalaina-Martin
Affiliation:
University of Colorado, Department of Mathematics, Boulder, CO 80309, USA (casa@math.colorado.edu)
Charles Vial
Affiliation:
Universität Bielefeld, Fakultät für Mathematik, Postfach 100131, D-33501, Germany (vial@math.uni-bielefeld.de)
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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 1.

Keywords

14-XX Algebraic Geometry

MSC classification

Primary: 14K30: Picard schemes, higher Jacobians 14C25: Algebraic cycles 14C30: Transcendental methods, Hodge theory , Hodge conjecture 11G10: Abelian varieties of dimension $> 1$ 11G35: Varieties over global fields
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 3 , May 2020 , pp. 891 - 918
Copyright
© Cambridge University Press 2018

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Footnotes

The first author was partially supported by grants from the NSA (H98230-14-1-0161, H98230-15-1-0247 and H98230-16-1-0046). The second author was partially supported by a Simons Foundation Collaboration Grant for Mathematicians (317572) and NSA grant H98230-16-1-0053. The third author was supported by EPSRC Early Career Fellowship EP/K005545/1.

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