On descending cohomology geometrically

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.

30 pages, AMS LaTeX, shortened exposition, some results strengthened. To appear in Composito Math. Material from appendix is now in arxiv:1610.06586; results on quadrics will appear elsewhere

References in corpus (1)

• Unirational threefolds with no universal codimension 2 cycle

Cited by in corpus (10)

• Derived equivalent threefolds, algebraic representatives, and the coniveau filtration
• Normal functions for algebraically trivial cycles are algebraic for arithmetic reasons
• Motivic periods and Grothendieck arithmetic invariants
• Divisibility properties of motivic cohomology
• Parameter spaces for algebraic equivalence
• On the coniveau of rationally connected threefolds
• Albanese kernels and Griffiths groups
• On the universal regular homomorphism in codimension $2$
• Distinguished models of intermediate Jacobians
• The Walker Abel-Jacobi map descends