For an algebraic variety, algebraic cycles are used to define an intrinsic cohomology theory. The idea is to use the subvarieties of the algebraic variety to define a free abelian group, and then to refine this construction suitably to obtain a well-behaved cup product, imposing various linear-algebraic conditions in the process. This perspective forms part of Grothendieck’s vision of a motivic cohomology theory.

Focusing on smooth and projective varieties over the complex numbers, the aim of this talk will be twofold. First, I will introduce algebraic cycles, examples, and various maps connecting them to singular and de Rham cohomology groups, as well as a particular bilinear pairing called the height pairing. This part will be a survey. Next, I will introduce my current topic of research (joint with Irene Spelta): the study of the limit of height pairings for a family of smooth projective varieties. I will explore this topic through a particularly interesting algebraic cycle called the Ceresa cycle. (Joint work with Irene Spelta.)