# Topics on Abelian Varieties

An episode of GAGLeS organized by Raymond Cheng during the spring of 2018.

The first half of this seminar aims to develop the basic theory of abelian varieties, mainly following Mumford's beautiful book mentioned below. The second half of this seminar will be devoted to topics chosen depending on the tastes of participants. Some suggestions include:

• properties of the derived categories of abelian varieties;
• moduli spaces of abelian varieties and their compactifications;
• theta functions and their relation to moduli;
• Serre–Tate deformation theory for abelian varieties in positive characteristic;
• Honda–Tate theory for abelian varieties over finite fields;
• $p$-adic uniformizatin of abelian varieties over non-archimedean fields;
• etc.

## References

Abelian varieties are heavily studied objects and there are many great references for the subject. Some notable general references on abelian varieties include:
1. David Mumford, Abelian Varieties.

This book is a beautiful classic. Mumford treats abelian varieties first from a complex analytic point of view, before moving onto an old-style variety-theoretic manner, before finally dealing with the modern scheme-theoretic language.

2. Christina Birkenhake and Herbert Lange, Complex abelian varieties.

If there is something about complex analytic abelian varieties you would like to know, this book probably contains it. In particular, this book is a rich source on theta functions.

3. Bas Edixhoven, Gerard van der Geer, and Ben Moonen, Abelian Varieties.

This is book-in-progress which, along with the basic theory of abelian varieties, touches on many interesting arithmetic topics. See also Ben Moonen's research page for (possibly) updated veresions of chapters of this book.

4. James S. Milne, Abelian Varieties.

Notes on abelian varieties by Milne. I don't think any more has to be said.

5. Brian Conrad with notes by Tony Feng, Math 249C: Abelian Varieties.

These are live-$\TeX$ed notes for a topics course on abelian varieties taught by Brian Conrad at Stanford in 2015.

6. Bhargav Bhatt with notes by Matt Stevenson, Math 731: Topics in Algebraic Geometry I – Abelian Varieties.

These are notes from a topics course on abelian varieties taught by Bhargav Bhatt at the University of Michigan during the Fall of 2017. Beyond the basic theory of abelian varieties, these notes discuss derived properties of abelian varieties, such as the Fourier–Mukai equivalence between the derived category of an abelain variety and its dual.

References for topics will be added as they are decided upon.

## Schedule

We meet Wednesdays in Mathematics Room 507 between 2:30PM and 4:00PM.
01/17
Raymond Cheng,
Introduction and Organization.
[§4, Mumford], [§§1.2–2.1, Bhatt], and [§§1.1, 1.5–1.7, Conrad].
01/24
Semon Rezchnikov,
Line bundles on abelian varieties: Theorem of the Cube and applications.
[§§6 and 10, Mumford], [§§6 and 9, Bhatt], and [§2.1, EvdGM].
01/31
Song Yu,
More line bundles and projectivity of abelian varieties.
[§§6 and 10, Mumford], [§§10 and 11.1, Bhatt], and [§§2.2–2.3, EvdGM].
02/07
Raymond Cheng,
Duality of abelian varieties.
[§I.8, Milne], [§§7 and 13, Mumford], [§§12.2, 13, 14, and 15.1, Bhatt], and [§6.3, EvdGM].
02/14
Dmitrii Pirozhkov,
Derived categories of abelian varieties.
[Lecture 24, Schnell], [Mukai], and [§§16.2, 17, and 18.1, Bhatt].
02/21
Carl Lian,
Cohomology of line bundles on abelian varieties.
[§16–17, Mumford], [§§19.1–20.1 Bhatt], and [§§7.4 and 9, EvdGM].
02/28
Monica Marinescu,
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03/07
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03/14
No Seminar.
Spring Recess.
03/21
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03/28
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04/04
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04/11
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04/18
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04/25
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05/02
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