On descending cohomology geometrically

In this talk I will present some joint work with Jeff Achter concerning the problem of determining when the cohomology of a smooth projective variety over the rational numbers can be modeled by an abelian variety. The primary motivation is a problem posed by Barry Mazur. We provide an answer to Mazur's question in two situations. First, we show that the third cohomology group can be modeled by the cohomology of an abelian variety over the rationals provided the Chow group of points is supported on a curve. This provides an answer to Mazur's question for all rationally connected threefolds, for instance. Second, we show how a result of Beauville establishes that the middle cohomology of a fibration in quadrics over the rational numbers can be modeled by an abelian variety.