Quadratic forms in Algebraic Geometry
Student No.:50
Time:Mon/Wed 10:30-12:00, 2014-02-24~2014-05-28
Instructor:Francois-Xavier Machu  [Tsinghua University]
Place:Conference Room 1, Floor 1, Jin Chun Yuan West Building
Starting Date:2014-2-24
Ending Date:2014-5-28




The theory of motives was created by Grothendieck in the mid-sixties, starting around from 1964. The purpose goal of this theory is to unify the various cohomology theories, de Rham, Betti, étale
and crystalline. A possible approach that we catch us a glimpse the study of the motives naturally is the study of the quadratic forms in Algebraic Geometry.

We start with a review on the quadratic forms, complements and quadrics at the functorial level. Then, we study the function field of a quadratic constructed from a quadric. This bring us to introduce the notion of the Chow group, its properties and computation in K-homology. Therefore, we finish with the Grothendieck's construction of the category of pure motives, and give some examples, in particular, motives of curves.

(N.B: I will remind (or introduce) the useful tools when that will be necessary.)



Basic Algebraic Geometry



Fitzgerald, function field of quadratic forms, Math Z. (178), 1981, 63-76.

Hoffmann, Isotropy of quadratic forms over the field of a quadric, Math Z. (220), 1995, 461-476.

Milnor, Algebraic Geometry and quadractic forms.