Algebraic Cycles

"Abstract"

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.

General information

Instructor Shane Kelly

Email shanekelly [at] math [dot] titech [dot] ac.jp

Webpage http://www.math.titech.ac.jp/~shanekelly/Cycles2018-19WS.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"

Time 月 Mon 13:20-14:50

Assessment Exercises will be given during the lectures. To get credits for Q4, it is enough to submit solutions to at least seven exercises. Exercises can be found in the notes for lecture 9, and in the textbook [MVW06]. Please send the exercise solutions for Q4 by 8th Feb. 日本語もOKです。You can also submit them in the レポートボックス in the 事務室. If you have any questions at all about anything to do with the exercises, just write me an email!

Outline

1. Introduction (10月1日)

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

体育の日 No lecture (10月8日)

2. Cohomology (10月15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]. AlgCycNotes2.pdf

3. Algebraic cycles (10月22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. Exercises: AlgCycExercises3.pdf

4. Equivalence (10月29日)

In this lecture we see other classical equivalences relations such as algebraic, numerical, and homological, and look at the codimension one case in greater detail. The references are [Mur10, Lec.2] lectures-murre.pdf, [Sch94, ¶1.3, ¶1.4, ¶1.8, ¶1.9] classical_motives.pdf, and [Rio06, Def.1.7] realizations.pdf. Exercises: AlgCycExercises4.pdf

5. Cycle maps (11月5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]. lectures-murre.pdf Lecture notes: AlgCycNotes5.pdf

6. Comparison (11月12日)

In this lecture we compare algebraic to numerical equivalence. The reference is [Mur10, Lec.4]. lectures-murre.pdf. Exercises: AlgCycExercises6.pdf.

8. Milnor Conjecture (12月13日(木)7－8時限にH335で)

In this lecture we discuss Milnor's conjecture. We begin with the question of classifying quadratic forms, motivating the study of the Witt ring, and from there move to the comparison of Milnor K-theory to Galois cohomology. This motivates Voevodsky's theory of motivic cohomology which will be developped in more detail on the second half of the course. The reference is [Dug04]. AlgCycNotes8.pdf

9. Category of finite correspondances (12月17日)

We introduce the category of finite correspondances, which is the starting point of the construction of Voevodsky's motivic cohomology. The reference is [MVW06, Lec.1]

10. Presheaves with transfers (12月24日)

We study the "colimit completion" of the category of correspondances, i.e., the category of presheaves on it. The ference is [MVW06, Lec.2]

11. Motivic cohomology (1月7日)

We introduce Voevodsky's motivic cohomology in this lecture. The reference is [MVW06, Lec.3]

--- 1月14日成人の日 No lecture 講義がありません ---

12. Weight one motivic cohomology (1月21日)

In this lecture we discuss weight one motivic cohomology. The reference is [MVW06, Lec.4]

13. Milnor K-theory (1月28日)

This lecture discusses Milnor K-theory and its relation to motivic cohomology. The reference is [MVW06, Lec.5]. See also [SV00, 3.4] which may be clearer.

14. Higher Chow groups (2月4日)15時-16時半にH340で

In this lecture we discuss the relationship between Voevodsky motivic cohomology, and Bloch's higher Chow groups. The reference is [MVW06, Lec.17, 18, 19]

15. Voevodsky's category of motives (2月4日)16時半-18時にH340で

In this lecture we construct Voevodsky's category of motives, and discuss its main properties. The reference is [Voe00]