The theory of algebraic cycles aims to understand all subvarieties of a given algebraic variety. (A cycle means a finite linear combination of subvarieties with integer coefficients.) The subject includes several of the hardest problems in algebraic geometry, such as the Hodge conjecture. An important aspect of the subject is the use of ideas and methods from topology.
In particular, we will define the Chow groups of algebraic cycles on a given algebraic variety over a field. (The divisor class group is the special case of codimension-1 cycles.) Chow groups have the formal properties of a homology theory, but (unlike the ordinary homology of a complex variety) they are generated by algebraic subvarieties. One main result is the construction of the intersection product on the Chow groups of a smooth variety.
The second part of the course considers some of the main conjectures about the behavior of Chow groups, such as the Bloch-Beilinson conjectures. Along the way, we will discuss some of the unexpected developments of the subject, including Mumford's theorem that Chow groups can be "infinite-dimensional", and Griffiths's example showing that algebraic and homological equivalence of cycles can be different.
Instructor: Burt Totaro.