On descending cohomology geometrically

TL;DR

This paper investigates the geometric structure of odd cohomology groups in smooth projective varieties over rationals, showing they can be modeled by abelian varieties under certain conditions, and explores rational models of the Abel--Jacobi map.

Contribution

It proves that odd cohomology groups with maximal geometric coniveau can be modeled by abelian varieties, answering Mazur's question for specific cases.

Findings

Odd cohomology groups are modeled by abelian varieties under maximal geometric coniveau.

Provides a rational model for the image of the Abel--Jacobi map in degree three.

Confirms Mazur's question for all uni-ruled threefolds.

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.