Xiaolong Liu
Xiaolong Liu's Blog
I'm interested in higher dimensional algebraic geometry, birational geometry, moduli space of varieties and bundles and derived categories of sheaves.

Reading Program on Intersection Theory and Motives

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This is my plan of the reading program of the intersection theory and Chow motives.

-1. The timeline

Maybe 12/09/2022 – ???.

0. Preliminaries we need

Including the basic theory of schemes and cohomology of coherent sheaves (such as R. Hartshorne’s AG chapter 2,3), the basic theory of curves and a little bit surface theory.

1. The basic theory of algebraic cycles, Chow groups and intersection theory

Here are my notes about examples and gaps in book:

1.1. Some Gaps and Examples in Intersection Theory by Fulton chapter 1-6;

1.2. Some Gaps and Examples in Intersection Theory by Fulton chapter 7-9;

1.3. Some Gaps and Examples in Intersection Theory by Fulton chapter 10-13;

1.4. Some Gaps and Examples in Intersection Theory by Fulton chapter 14-16;

1.5. Need to add.

  • More References. 3264 and All That, A Second Course in Algebraic Geometry by David Eisenbud and Joe Harris. (see 3264 and All That.

2. The basic theory of étale cohomology

Here are my notes (in Chinese): EtaleCoh.

  • Lectures on Etale Cohomology - J.S. Milne. I will read the first part of this notes (that is, before the proof of Weil’s conjecture).
  • More References.

Etale Cohomology Theory by Lei Fu;

Etale Cohomology by Milne;

An Introduction to Etale Cohomology by Donu Arapura;

Lecture Notes for Étale Cohomology I by Jens Franke;

Étale Cohomology by B. Conrad;

LECTURE NOTES ON ÉTALE COHOMOLOGY by JOHANNES ANSCHÜTZ;

3. The basic theory of pure motives and Motivic Cohomology

3.1. Pure motives

  • More References. Mixed Motives by Marc Levine. (see Mixed Motives).

3.2. Motivic Cohomology

  • I will follow the course in BIMSA: Motivic Cohomology. See HomePage (for more courses, see 2023BIMSA).

Some basic information of the course:

Lecturer Date Weekday Time Zoom ID Password
Nanjun Yang 2023-03-15 ~ 2023-06-28 Wed 09:50 - 12:15 537 192 5549 BIMSA

Introduction:

Motivic cohomology, originated from Deligne, Beilinson and Lichtenbaum and developed by Voevodsky, is a kind of cohomology theory on schemes. It admits comparison with étale cohomology of powers of roots of unity (Beilinson-Lichtenbaum), together with higher Chow groups, and relates to K-theory by Atiyah-Hirzebruch spectral sequence. In this lecture, we establish the category of motives in which the motivic cohomologies are realized. We explain its relationship with Milnor K-theory and Chow group, as well as the theory of cycle modules. Furthermore, we introduce cancellation theorem, Gysin triangle, projective bundle formula, BB-decomposition and duality.

References:

C. Mazza, V. Voevodsky, C. Weibel, Lecture Notes on Motivic Cohomology, American Mathematical Society, Providence, RI, for the Clay Mathematics Institute, Cambridge, MA (2006).

M. Rost, Chow Groups with Coefficients, Documenta Mathematica (1996), Volume: 1, page 319-393

$\infty$. More things need to add

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