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On descending cohomology geometrically

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

Published online by Cambridge University Press:  10 May 2017

Jeffrey D. Achter ,
Sebastian Casalaina-Martin  and
Charles Vial
Jeffrey D. Achter
Affiliation:
Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA email [email protected]
Sebastian Casalaina-Martin
Affiliation:
Department of Mathematics, University of Colorado, Campus Box 395, Boulder, CO 80309, USA email [email protected]
Charles Vial
Affiliation:
DPMMS, University of Cambridge, Cambridge CB3 0WB, UK email [email protected] Current address:Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany
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Abstract

In this paper, motivated by a problem posed by Barry Mazur, we show that for smooth projective varieties over the rationals, the odd cohomology groups of degree less than or equal to the dimension can be modeled by the cohomology of an abelian variety, provided the geometric coniveau is maximal. This provides an affirmative answer to Mazur’s question for all uni-ruled threefolds, for instance. Concerning cohomology in degree three, we show that the image of the Abel–Jacobi map admits a distinguished model over the rationals.

Keywords

abelian varietiesGalois representationsalgebraic cyclesHodge theoryAbel–Jacobi map

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
Compositio Mathematica , Volume 153 , Issue 7 , July 2017 , pp. 1446 - 1478
Copyright
© The Authors 2017 

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