Height Pairing on Graded Pieces of a Bloch-Beilinson Filtration

Abstract

For a smooth projective variety defined over a number field, Beilinson (and independently Bloch) constructed a `height' pairing under very reasonable assumptions and with a number of conjectural properties. A folklore conjecture related to this pairing states that the rational Griffiths Abel-Jacobi map is injective. But for a smooth projective variety defined over a field of finite transcendence degree over a number field, the folklore conjecture is no longer valid. Instead we have the concept of a conjectural Bloch-Beilinson filtration, a candidate for which was given by James Lewis. Under some standard assumptions, the main point of this thesis is to generalize the height pairing to the graded pieces of this candidate Bloch-Beilinson filtration using cohomological machinery.