Wenhan Dai

Abelian Varieties (I)

Student seminar for the 3+X program.

The goal of this learning seminar is to study the basics of abelian varieties. The pre-requisite is Chapters 2–3 of Hartshorne’s algebraic geometry, but this seminar can also be taken while reading Hartshorne’s book. Some familiarity with elliptic curves will be helpful to understand some of the materials. Each lecture will be two hours long.

The textbook we are going to use is Mumford’s book. Another complementary reference is Milne’s online note. Mumford’s book is mostly about the geometry (i.e. abelian varieties over algebraically closed fields), while Milne’s note has more about the arithmetic (e.g. abelian varieties over finite fields or number fields). Moreover, Milne’s note has a chapter about Jacobian varieties.

• Time: Every Friday, 7:30-9:30pm.
• Location: School of Math 1418 / Jia-Yi-Bing Building.
• Instructor: Liang Xiao and Xinyi Yuan.

Announcements

About logistics -

1. (10/28) Starting from the first meeting in November, free dinners will be provided; if needed, please collect it directly outside the door of Room 77201 (reminder: the classes will still be held on the 2nd floor of the Jia-Yi-Bing building, not in 77201).
2. (10/7) WATCH: As of October 14, our meeting place was changed to the second floor of the Jia-Yi-Bing Building instead of the SMS 1418.

My collated notes are available -

(Thanks to Bin Zhao’s previous lecture series at YMSC in 2020.)

• Part I: analytic theory, Lecture 1-3 (Mum, §1-3); September 9-30.
• Part II: algebraic theory via varieties, Lecture 4-8 (Mum, §4-8); October 7 - November 4.
• Part III: algebraic theory via schemes, Lecture 10-12 (Mum, §11-13); November 18 - December 2.

Syllabus (modified)

• (9/9) Liang Xiao: Introduction and organization. Outline the materials and talk about distribution of the lectures to students. (notes)
• (9/16) Wenhan Dai: [Mum, §1] This starts the complex theory of abelian varieties. (notes)
• (9/23) Xiaolong Hu: [Mum §2]. (Wenhan absent; no notes for this week.)
• (9/30) Suichu Zhang: [Mum §3]. A nice exercise is to work out the analytic construction of abelian varieties with complex multiplication (cf. [Mil2, Prop. 10.2]). (notes)
• (10/7) Di Wu: [Mum §4]. This section takes less time than the usual. (notes)
• (10/14) Jiaming Zhou: [Mum §5]. (live-TeXed notes)
• (10/21) Dingchang Zhou: [Mum §6]. (Wenhan took a trip; no notes for this week. See his collated notes instead.)
• (10/28) Kehao Cheng: [Mum §7]. (ibid.)
• (11/4) Yiyang Chang: [Mum §8]. (collated notes, speaker’s handout)
• (11/11) Shun Yin: [Mum §9-10]. (notes; have not been finished; to be continued next time)
• (11/18) Tianwei Gao: [Mum §11]. (Resumed online. collated notes)
• (11/25) Wenxuan Qi: [Mum §12]. (Online. collated notes, speaker’s handout)
• (12/2) Haoda Li: [Mum §13]. (collated notes)
• (12/9) Xingzhu Fang: [Mum §14]. (notes)
• (12/16) Huishi Yu: [Mum §15]. (speaker’s handout)

References

1. [Mum] D. Mumford: Abelian varieties. Corrected reprint of the second (1974) edition.
2. [Mil] J. Milne: Abelian varieties. Course note, available here.
3. [Mil2] J. Milne: Introduction to Shimura varieties. Course note, available here.