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
References [BT82] Bott, Tu, "Differential forms in algebraic topology"
[Gro69] Grothendieck, "Standard conjectures on algebraic cycles"
[Jan92] Jannsen, "Motives, numerical equivalence, and semi-simplicity"
[Mur10] Murre, "Lectures on algebraic cycles and Chow groups" pdf
Bibliography
Full bibliography [ARS95] Auslander, Reiten, Smalø, "Representation theory of Artin algebras"
[And17] André, "Groupes de Galois motiviques et périodes"
[Ayo14] Ayoub, "A guide to (étale) motivic sheaves"
[Ayo14b] Ayoub, "Periods and the conjectures of Grothendieck and Kontsevich-Zagier"
[Ayo17] Ayoub, "Motives and algebraic cycles: a selection of conjectures and open questions" pdf
[BT82] Bott, Tu, "Differential forms in algebraic topology"
[Bei10] Beilinson, "Remarks on Grothendieck's standard conjectures" pdf
[Blo80] Bloch, "Lectures on algebraic cycles"
[CD19] Cisinski, Déglise, "Triangulated categories of mixed motives"
[Con03] Conrad, "Cohomological descent"
[Dem71] Demazure, "Motifs des variétés algébriques"
[FV00] Friedlander, Voevodsky, "Bivariant cycle cohomology"
[Ful97] Fulton, "Young tableaux: with applications to representation theory and geometry"
[Ful98] Fulton, "Intersection theory"
[GH78] Griffiths, Harris, "Principles of algebraic geometry"
[Gro69] Grothendieck, "Standard conjectures on algebraic cycles"
[Har92] Harris, "Algebraic geometry: a first course"
[Har77] Hartshorne, "Algebraic geometry"
[Hat02] Hatcher, "Algebraic topology" pdf
[Hir64] Hironaka, "Resolution of singularities of an algebraic variety over a field of characteristic zero"
[Jan87] Jantzen, "Representations of algebraic groups"
[Jan92] Jannsen, "Motives, numerical equivalence, and semi-simplicity"
[Jan94] Jannsen, "Motivic sheaves and filtrations on Chow groups"
[Kim05] Kimura, "Chow groups are finite dimensional, in some sense"
[Kle68] Kleiman, "Algebraic cycles and the Weil conjectures"
[Lam91] Lam, "A first course in noncommutative rings"
[Mac98] Mac Lane, "Categories for the working mathematician"
[Mat86] Matsumura, "Commutative ring theory"
[Mil06] Milne, "Elliptic curves" pdf
[Mil12] Milne, "Motives: Grothendieck's dream" pdf
[Mil80] Milne, "Etale cohomology"
[Mor06] Morel, "Rational stable splitting of Grassmannians and the rational motivic sphere spectrum"
[Mum88] Mumford, "The red book of varieties and schemes"
[Mur10] Murre, "Lectures on algebraic cycles and Chow groups" pdf
[SV00] Suslin, Voevodsky, "Relative cycles"
[Sam58] Samuel, "Relations d'équivalence en géométrie algébrique"
[Sha13] Shafarevich, "Basic algebraic geometry 1: varieties in projective space"
[Sta] The Stacks Project website
[Str95] Street, "Ideals, radicals, and structure of additive categories"
[Voe00] Voevodsky, "Triangulated categories of motives over a field" pdf
[Voe10] Voevodsky, "Unstable motivic homotopy categories in Nisnevich and cdh-topologies"
[Voi02] Voisin, "Hodge theory and complex algebraic geometry I"
[Voi13] Voisin, "Symplectic involutions of K3 surfaces act trivially on CH_0"
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.
2026AlgCycNotes4.pdf

5. Classical motives (6月4日, 6月11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.
2026AlgCycNotes5.pdf

6. Numerical Equivalence (6月18日, 6月25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.
2026AlgCycNotes6.pdf

7. Standard Conjectures (7月2日, 7月9日)

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