Chow motives and applications to quadratic forms

Wintersemester 2024/25

Time and place: Fridays, 12-14 in Room B252.
In this seminar, we will first give an overview of the algebraic theory of quadratic forms. Then we will introduce Chow groups and motives of algebraic varieties and complute them for smooth projective quadrics in several cases. Finally, we will see applications of algebraic cycles and motives to algebraic and geometric problems of quadratic forms.

Information: The seminar will be conducted in English. The first meeting will be on Friday, 18th of October. The seminar is intended for Master students (prerequisites: algebra, commutative algebra and algebraic geometry).
Please write me an email if you want to attend the seminar and give a talk.

Preliminary programm:

1. Introduction. Distribution of the talks. 18th of October.

2. Overview of Witt theory and Pfister froms. 25th of October. References: [1], Chapter 2, 10.1; [3], Chapter 1-2.

3. Function field of quadrics, higher Witt indices of quadratic forms . 8th of November. References: [3], Chapter 3-4.

4. Hoffmann's Separation Theorem . 15th of November. References: [3], Chapter 5.2. Notes

5. Chow groups, properties, A^1-invariance.. 22nd of November. References: overview from [2], [4].

6. Chow group of cellular varieties. Examples: projective space, slit quadric. 29th of November. References: [2], Section 66 and 68.

7. Category of Chow motives. Properties, examples. . 6th of December. References: [2], Chapter 12.

8. Chow motives of quadrics (split, isotropic cases). 13th of December. References: [2], Chapter 12, 17.

9. Motivic decomposition of Pfister forms . 20th of December. Notes

10. Motivic decomposition of excellent forms . 10th of January. Notes

11. Hasse principle for Rost motives . 17th of January.

12. Quadratic forms of dimension 10 in I^3 . 24th of January.

13. TBA . 31st of January.

14. TBA . 7th of February.

References for talks 1-8:
1. T. Y. Lam, Introduction to Quadratic forms over fields.
2. R. Elman, N. Karpenko, A. Merkurjev, The algebraic theory of quadratic forms.
3. B. Kahn, Formes quadratiques sur un corps (available online, in french).
4. W. Fulton, Intersection theory.