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 10 lectures (so at least 10 exercises, but you are welcome to submit as many solutions as you want).

Please submit the exercise solutions via email by 23rd July.

日本語でもOKです。

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, 5月21日)

In this lecture we discuss pullback, pushforward, and intersection of cycles, adequate equivalence relations, and introduce the Chow groups. The reference for cycles is [Mur10, Lec.1].
2026AlgCycNotes3.pdf

--- 5月7日 振替授業日（月曜授業実施） No lecture ---

--- 5月14日 授業延期 — 山本修司先生の多重ゼータ値集中講義（午後3時〜5時、123号室）のため ---

4. Cycle maps (5月28日)

In this lecture we construct the cycle class map towards de Rham cohomology, allowing us to state the Hodge conjecture.
AlgCycNotes4.pdf

5. Classical motives (6月4日)

In this lecture we introduce the classical category of motives, Weil cohomology theories, homological, and numerical equivalence. study numerical equivalence. We state the Standard Conjecture D. The reference is [Scholl].

6. Numerical Equivalence (6月11日)

In this lecture we discuss Jannsen's proof that the category of classical motives is semi-simple if and only if the equivalence relation is numerical equivalence. The reference is Jannsen's article.

7. Standard Conjectures (6月18日)

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


The above lectures will surely run long. There are slots scheduled for

(6月25日), (7月2日),(7月9日)


for this.