Math 894: Abelian Varieties Reading Course, Fall 2023

Instructor

Dr. Katrina Honigs, khonigs@sfu.ca, Office SC K 10506

Meetings

2:30-4:20 pm Fridays, AQ5015

Grading

  • Lectures: 80%
  • Participation: 20%

    • Prerequisites

    I recommend Math 818 or equivalent.

    Goal and format

  • Each meeting, one of the participants will lecture on a part of a text. Lecturers should plan for a lecture of 50 minutes. Our time slot will allow for overrun and discussion.
  • We will fix the assignment of topics and the lecture schedule on the first meeting. I handed out a list of tentative lecture topics at the first NTAG meeting (available on request).
  • Abelian varieties are very commonly encountered in the study of algebraic geometry. The goal of the course is to give an introduction to their structure. The start of the course will primarily follow the outline in sections 1 and 3 of Debarre's notes, as well as (parts of) Chapter 1 of Milne's notes, both linked below. Additional topics may be added, I welcome suggestions.

    • Lecture schedule:

    Date Lecturer Lecture Topic Text references
    Sept 15 Katrina Introduction to Abelian Varieties Milne: Start of Ch. 1, §1 on group varieties
    Debarre 1.1 on Chevalley's structure theorem
    Sept 22 Pijush Isogenies Lombardo: Ch. 1, §5,7
    Sept 29 Carl The rigidity lemma
    abelian varieties are abelian    		 Debarre: Lemma 3 and Prop. 4
    Milne Ch. 1, §1 (the rest of it)
    Oct 6 Sharon Rational maps into abelian varieties Milne Ch. 1, §3
    Oct 13 Ahmad Abelian varieties are projective Debarre p. 3, Milne Ch. 1, §6
    Oct 20 Pijush Theorem of square, cube
    Definition of φ_L, dual abelian variety as its image     		Debarre, §3.1
    Milne Ch. 1 §5, §8 bottom of p. 34 only
    Oct 27 Carl Dual abelian variety as moduli space,
    Poincaré bundle    		 Milne Ch. 1 §8
    Nov 3 Ahmad Complex tori Lombardo: Ch. 1, §4
    Nov 10 Damara Tate modules Milne: middle of p.44 - 45 (part of Ch. 1 §10)
    Not in notes: kernel of isogeny vs cokernel of map on Tate modules
    Nov 17 Sharon Endomorphism rings are division algebras
    Finite rank of Neron-Severi group    		 Milne: middle of p.42-43, 46-50 (part of Ch. 1 §10)
    Nov 24 Katrina Last session: TBD
    Dec 1 No Meeting

    Reading Sources

  • Milne "Abelian Varieties" course notes
  • Debarre "Two or three things I know about abelian varieties" lecture notes
  • Lombardo "Abelian varieties" lecture notes