Part 2 Chow groups | Burt Totaro, UCLA

Lecture notes
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Problem set 2
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Introduction
One of the main methods of complex algebraic geometry is to think of a complex algebraic variety as a complex manifold and in particular a topological space, and look at its ordinary cohomology. Roughly speaking, that means that we think about all possible real submanifolds of the given complex variety. But an algebraic variety is more than just a topological space. In particular, it is important to ask which cohomology classes can be represented by complex algebraic subvarieties. This is the subject of the theory of algebraic cycles and in particular the Hodge conjecture, one of the famous seven “million-dollar problems” in mathematics.
The key concept in the course is the notion of the Chow groups of an algebraic variety over any field. These are groups with some of the same formal properties as homology or cohomology groups, but they are built directly from the algebraic subvarieties of a given variety. A big difference from cohomology is that Chow groups are extremely hard to compute in general. In fact, computing Chow groups for arbitrary varieties would amount to solving the Hodge conjecture and many other conjectures of algebraic geometry and number theory (such as the Birch- Swinnerton-Dyer conjecture, another million-dollar problem). Nonetheless, in this course we will see how to compute the Chow groups at least for some varieties.
Chow groups can be seen as a first step in bringing methods from homotopy theory into algebraic geometry. Some of the later steps in this direction are algebraic K-theory, motivic cohomology, and algebraic cobordism.
Problem set 1 is related to this first set of notes.
Here are some of the relevant books. For general algebraic geometry, a standard reference is R. Hartshorne, Algebraic Geometry (Springer). Many other books cover similar material, such as Ravi Vakil’s The Rising Sea, free on the web. For example, in Hartshorne, section II.6 on divisors is an excellent geometric treatment, which is the most direct background needed for Chow groups. Chapter IV on curves shows how to use divisors and line bundles to solve geometric problems. Also, Appendix A is a good short summary of the Chow ring, as covered in these notes.
W. Fulton’s Intersection Theory (Springer) is the basic reference on Chow groups. It is a densely written book, so it can be hard to read straight through. But the basic Chapter 1 on Chow groups is very readable, and you can skip around through the rest. The book has a massive number of useful examples. Finally, EIsenbud- Harris’s 3264 and All That (Cambridge), is a more elementary introduction to Chow groups and how to compute with them. It is in the PCMI library here.
Before defining Chow groups, I will discuss divisors and line bundles, which is the foundation of the theory of Chow groups. Some of this should be familiar, but it’s important to understand how these things work on singular varieties, which may be new to you.
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The 2024 Program: Motivic Homotopy Theory
Organizers: Benjamin Antieau, Northwestern University; Marc Levine, Universität Duisberg-Essen; Oliver Röndigs, Universität Osnabrück; Alexander Vishik, University of Nottingham; and Kirsten Wickelgren, Duke University
Motivic homotopy theory arose out of the work of Morel and Voevodsky in the 1990s and since then has developed into both a powerful tool for understanding many arithmetic aspects in algebra and algebraic geometry, as well as being a fascinating generalisation of classical homotopy theory with an active development in its own right.
The 2024 GSS on motivic homotopy theory will give participants an introduction to some aspects of motivic homotopy theory as well as a taste of developments in other fields that have been influenced and enabled by motivic homotopy theory. Mini-courses will include: an introduction to unstable motivic homotopy theory, a study of characteristic classes in stable motivic homotopy theory, motivic homotopy theory in enumerative geometry, and a version of Weil conjectures in motivic homotopy theory, as well as courses on recent advances in arithmetic properties of local systems, fundamental problems in Galois cohomology of fields, and aspects of G-bundles in algebraic geometry.
Prerequisites: Students should have a basic knowledge of algebraic geometry, algebraic topology, and some homotopy theory. For some of the courses, a knowledge of Galois cohomology and étale cohomology will also be helpful.
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The GSS takes place within the broader structure of PCMI, so there are many researchers at all levels in the field in attendance, as well as participants in the other PCMI programs.