Chow Groups and Algebraic Cycles in Complex Algebraic Geometry - Part 4

Learn about Chow groups in algebraic geometry through this 57-minute lecture presented by Burt Totaro from UCLA at the IAS/PCMI Park City Mathematics Institute. Explore how complex algebraic varieties can be understood as complex manifolds and topological spaces, with a focus on which cohomology classes can be represented by complex algebraic subvarieties. Dive into the theory of algebraic cycles and the Hodge conjecture, one of mathematics' million-dollar problems. Master the fundamental concept of Chow groups for algebraic varieties over any field, understanding their similarities to homology/cohomology groups while grappling with their computational complexity. Connect this knowledge to broader mathematical concepts including algebraic K-theory, motivic cohomology, and algebraic cobordism. Supplementary materials include detailed lecture notes, problem sets, and recommended readings from standard texts like Hartshorne's Algebraic Geometry, Fulton's Intersection Theory, and Eisenbud-Harris's 3264 and All That.