Algebraic cycles

General information

Algebraic cycles are a central theme in algebraic geometry, appearing in places such as Abel’s Theorem, The Riemann-Roch Theorem, enumerative geometry, higher K-theory, motivic cohomology, and the Hodge conjecture. In this course we develop some basic ideas, and review some of these applications.

Instructor Shane Kelly
Email shanekelly [at] g.ecc [dot] u-tokyo [dot] ac.jp
Webpage https://www.ms.u-tokyo.ac.jp/~kelly/Course2026AlgCyc/Cycles2026.html
Main References [MVW06] Mazza, Voevodsky, Weibel, "Lecture notes on motivic cohomology" pdf
[Mur10] Murre, "Lectures on algebraic cycles and Chow groups" pdf
Other References [Ayo17] Ayoub, "Motives and algebraic cycles: a selection of conjectures and open questions" pdf
[Blo80] Bloch, "Lectures on algebraic cycles"
[Dug04] Dugger, "Notes on the Milnor conjectures" pdf
[Ful84] Fulton, "Intersection theory"
[Har77] Hartshorne, "Algebraic geometry"
[Man68] Manin, "Correspondences, motifs and monoidal transformations"
[Rio06] Riou, "Realization functors" pdf
[Sch94] Scholl, "Classical motives" pdf
[SV00] Suslin, Voevodsky, "Bloch-Kato conjecture and motivic cohomology with finite coefficients," pdf
[Voe00] Voevodsky, "Triangulated categories of motives over a field" pdf
[Voi06a] Voisin, "Hodge Theory and Complex Algebraic Geometry, Vol. I"
[Voi06b] Voisin, "Hodge Theory and Complex Algebraic Geometry, Vol. II"
Room Graduate School of Mathmatical Science Bldg. , Room 118.
Time Thurs(木) 14:55–16:40
Assessment Exercises will be given during the lectures. To pass the course, it is enough to submit solutions to at least one exercise from each of the first 10 lectures (so at least 10 exercises, but you are welcome to submit as many solutions as you want).

If you have any questions at all about anything to do with the exercises, please write me an email!

Outline

1. Introduction (4月9日)

In this lecture we give an outline of the course.
2026AlgCycNotes0.pdf
2026AlgCycNotes1.pdf

2. Cohomology (4月16日, 4月23日)

In this lecture we develop some basics about classical cohomology theories, specifically singular and de Rham, which will be used over the next few weeks. The reference is [Voi06].
2026AlgCycNotes2.pdf

3. Algebraic cycles (4月30日)

In this lecture we discuss pullback, pushforward, and intersection of cycles, adequate equivalence relations, and introduce the Chow groups. The reference is [Mur10, Lec.1]
lectures-murre.pdf.

--- 5月7日 No lecture 講義がありません ---

4. Cycle maps (5月14日)

In this lecture we construct the cycle class map towards de Rham cohomology, allowing us to state the Hodge conjecture. We further discuss intermediate Jacobians and the Abel-Jacobi map, and possibly Deligne cohomology if there is time. The reference is [Mur10, Lec.3].

5. Numerical Equivalence (5月21日)

In this lecture we study numerical equivalence. We state the Standard Conjecture D. We discuss Jannsen's proof that the category of classical motives is semi-simple if and only if the equivalence relation is numerical equivalence.

6. Standard Conjectures (5月28日)

In this lecture we state the Standard Conjectures A, B, and C, and various implications among them.