Algebraic Cycles and Representation Theory

Fall Semester 2015

Instructor: Matt Kerr
Office: Cupples I, Room 114
e-mail: matkerr [at]
Office Hours: Friday 4-5, and by appointment

Course Outline:

Part I. Algebraic cycles and their Hodge-theoretic invariants
A. Cycle groups and equivalence relations
B. Cycle-class maps and the Hodge Conjecture
C. The Abel-Jacobi map and normal functions
D. Higher Abel-Jacobi maps and Bloch-Beilinson filtrations

Part II. Higher Chow cycles and motivic cohomology
A. Bloch's construction and generalized Abel-Jacobi maps
B. Borel's theorem and the algebraic K-theory of fields
C. Beilinson's conjectures on special values of L-functions
D. The Milnor regulator and applications

We will study major results and conjectures in the theory of algebraic cycles and higher Chow cycles. Part I will focus on usual cycles, concentrating on normal functions and approaches to the Hodge Conjecture. For instance, we expect to include results of Hazama and Schoen on abelian varieties, the Griffiths-Green theorem, Bloch-Beilinson filtrations, and recent work of Kerr's FRG group. Part II makes a natural shift toward higher Chow cycles, regulator and Abel-Jacobi maps, Borel's theorem and construction of these cycles. If time allows, we will mention some connections to mirror symmetry and Feynman integrals. Prerequisites include Math 5031-5032 and some acquaintance with representation theory (as for example Mumford-Tate groups will be introduced and used from the beginning). The course also will use elements of my previous course on Hodge Theory.

Prerequisites: Math 5022, 5032, 5042, or permission of the instructor.

Class Schedule:

Lectures are on Monday, Wednesday and Friday from 1-2 in Cupples I Rm. 113. I will occasionally reschedule classes due to travel, usually to a Thursday (or Tuesday) afternoon.

First class is Monday Aug. 24 and last class is Friday Dec. 4 (with holidays Sept. 7, Oct. 16, Nov. 25, and Nov. 27).

Final Exam: no exam; a project/presentation will substitute for final (see below)


I will collect some homework based on exercises in the notes a couple of times per month.

Problem Set #1 due Monday Sept. 14
Problem Set #2 due Wednesday Sept. 30
Problem Set #3 due Monday Oct. 19

More important will be your mini-project. The options will be unveiled in October and we'll discuss these one-on-one. The idea will be to explore topics and/or do computations not covered (or maybe only mentioned) in the lectures. You'll prepare a short writeup and give a 10-15 minute talk at the end.

Lecture Notes:

Will be scanned and posted here as I write them. The hope is that this makes taking notes optional.

I. Algebraic cycles and their Hodge-theoretic invariants
A. Cycle groups
1. Operations on cycles
2. Equivalence relations
3. The Chow group
B. Cycle-class maps and the Hodge Conjecture
1. Cycle classes and Lefschetz (1,1)
2. Mumford-Tate groups of Hodge structures
3. The theorem of Hazama and Murty
4. Mumford-Tate domains and period mappings
5. Weil-Hodge classes and Schoen's construction
6. Algebraicity of Hodge loci
7. Absolute Hodge classes and Voisin's strategy
C. The Abel-Jacobi map and normal functions
1. Extensions of MHS and the Deligne cycle class
2. Cycles modulo algebraic equivalence
3. Infinite generation of the Griffiths group
4. Admissible normal functions and their invariants
5. Cycles on projective hypersurfaces
6. Homogeneous VHS and cycles on abelian varieties
7. Normal functions and the Hodge Conjecture
D. Higher Abel-Jacobi maps and filtrations on Chow groups
1. Mumford's theorem on zero-cycles
2. Spreads and Bloch-Beilinson filtrations
3. Higher cycle- and AJ-classes: examples
II. Higher Chow groups and motivic cohomology
A. Bloch's construction
1. K0 and algebraic cycles
2. Milnor K-theory and relative 0-cycles
3. Higher Chow groups and their properties
4. Moving lemmas for higher cycles
5. Abel-Jacobi maps for higher Chow cycles
B. The Borel regulator
1. K-theory of number fields
2. Sketch of Borel's theorem
3. Applications of Borel's theorem
C. The Beilinson conjectures
1. Statements and motivation
2. CM elliptic curves
3. The Eisenstein symbol

Reference articles:

Links to articles related to each section that are freely available online, and which you are free to ignore.

(I.A) (Murre) (Samuel) (Lewis)
(I.B) (Kerr/Pearlstein) (Moonen, Wallach) (Murty, Ribet) (Schoen) (CDK) (Voisin, Deligne)
(I.C) (Harris, Bloch) (Ceresa, Nori, Totaro) (Griffiths, Green, Clemens) (Zucker, GG, BFNP)

Reference Books:

These should now be on reserve at the library.

[Vo] Claire Voisin, Hodge theory and complex algebraic geometry (tr. Schneps), v. 1 & 2; Cambridge
Online access(!): (volume I) (volume II)

[Le] James D. Lewis, A survey of the Hodge conjecture; CRM

[PS] Chris Peters and Joseph Steenbrink, Mixed Hodge structures; Springer

[GGK] M. Green, P. Griffiths and M. Kerr, Mumford-Tate groups and domains: their geometry and arithmetic; Princeton
Online access

[Fu] William Fulton, Intersection theory (2nd Ed.); Springer

[FH] William Fulton and Joseph Harris, Representation theory: a first course; Springer

Grading Policy:

The project, at 60%, is the main component of your grade; take-home midterm and HW both count 20%.