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On descending cohomology geometrically

Published online by Cambridge University Press:  10 May 2017

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

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.

Type
Research Article
Copyright
© The Authors 2017 

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