# Math 632: Algebraic Geometry II

### Winter 2011

Course meets: Tuesdays and Thursdays 10:00-11:30, 4096 East Hall

Text: Hodge Theory and Complex Algebraic Geometry I, Claire Voisin, ISBN 978-0-521-80260-1

I will be following the text fairly loosely. In particular, I want to make sure that you come out of this course with a fair amount of experience working with sheaves, and with some basic examples of complex algebraic varieties, which means I will spend more time on these topics than Voisin does, and get to them earlier. I also expect to follow the book more closely as the term goes on, since I have fewer strong opinions about how the later material should be taught.

Office Hours: 2844 East Hall, Monday 10-12 and Thursday 2:30-4:00. Also, please feel free to knock on my door or e-mail me at any time.

Professor: David E Speyer, 2844 East Hall, speyer@umich.edu

Course homepage: http://www.math.lsa.umich.edu/~speyer/632.html

Level: Graduate students with some familiarity with manifolds, complex analysis and algebraic topology.

Student work expected: I will give problem sets every week, due Tuesdays in class. I will also require students to take turns serving as scribe for the course, meaning taking TeXed notes on what I have said that day. Homework Policy: You are welcome to consult each other provided (1) you list all people and sources who aided you, or whom you aided and (2) you write-up the solutions independently, in your own language.

Scribing: I will require students to take turns signing up to record, in TeXed format, what I said in lecture. The due date for turning notes is one week after the lecture, sent to me by e-mail. I highly encourage you to be faster, as this will both be easier for you and more helpful for you fellow students. You may want to bring a laptop to class and try live TeXing (some useful tips here). I will edit these notes before posting them.

If you are scribing and are confused about some point from the lecture, please let me know so that I can work with you. Of course, this also applies if you are confused and are not scribing!

There are currently 14 students registered for a 28 lecture class. A few of these will be review classes which will not need scribes, and I will scribe the first lecture myself. If all of you remain for the whole term, then most of you will scribe twice. I expect a more likely estimate is that you will wind up scribing 3–4 times.

For your convenience, this LaTeX template provides some convenient formatting and macros. I will add useful macros throughout the term as I think of them. If you are a TeX wizard who would rather use some other template, this is fine. If you have never learned how to use LaTeX, then you really need to, as it is the standard tool of all mathematical publishing and will be crucial to anyone pursuing a career as a mathematician. Please alert me to your situation, and I will help you. A useful website for getting help with LaTeX difficulties is tex.stackexchange.com.

Note that these notes will be posted publicly online, in both TeX and PDF form, with your name attached to them, and that I may edit them to remove errors in my lectures or your understanding of them. You give me permission to do this, but otherwise retain copyright to your notes.

## Course Notes and Rough Syllabus

Jan 6 Introduction
Notes TeX PDF. Scribe: David Speyer
Problem Set 1 distributed

### Sheaves and DeRham cohomology

Jan 11 Review of Differential Forms
Notes TeX PDF. Scribe: Giwan Kim

Jan 13 Sheaves
Notes TeX PDF. Scribe: Tengren Zhang

Jan 18 Sheaf Cohomology I
Notes TeX PDF Scribe:Pedro Acosta
Problem Set 2 distributed

Jan 20 Sheaf Cohomology II and de Rham Cohomology
Notes TeX PDF Scribe:Hunter Brooks

### Dolbeault cohomology

Jan 25 More Sheaf Cohomology, Complex differential forms, (p,q) forms, the del-bar operator
Notes: TeX, PDF Scribe: Kevin Carde
Problem Set 3 distributed

Jan 27 Dolbeault's Lemma
Notes: TeX, PDF Scribe: Emily Clader

Feb 1 Cohomology vanishes on polydiscs
Notes: TeX, PDF Scribe: Yi Su
Problem Set 4 distributed

Feb 3 How close are we to computing topological cohomology?
Notes: TeX, PDF Scribe: Chris Fraser

Feb 8 Cartan's theorems: The analytic part
Notes: TeX, PDF Scribe: Adam Kaye
Problem Set 5 distributed

Feb 10 Cartan's theorems: The formal part
Notes: TeX, PDF Scribe: Geoffery Scott

Feb 15 Catch up and review
Notes: TeX, PDF Scribe: Justin Campbell
Problem Set 6 distributed — Note unusual due date

### Vector bundles and associated tools

Feb 17 Vector bundles
Notes: TeX, PDF Scribe: Brooke Ullery

Feb 22 Connections
Notes: TeX, PDF Scribe: Xin Zhou

Feb 24 Bilinear forms on vector bundles
Notes: TeX, PDF Scribe: Kevin Carde

SPRING BREAK

### Kahler manifolds and the Hodge theorems

March 8 The Hodge star; the Hodge theorem for real manifolds
Notes: TeX, PDF Scribe: Pedro Acosta
Problem Set 7 distributed

March 10 The Hodge theorem on complex manifolds, Poincare and Serre duality
Notes: TeX, PDF Scribe: Adam Kaye

March 15 Kahler manifolds
Notes: TeX, PDF Scribe: Justin Campbell
Problem Set 8 distributed

March 17 The Kahler identities
Notes: TeX, PDF Scribe: Chris Fraser

March 22 The Hodge decomposition
Notes: TeX, PDF Scribe: Hunter Brooks
Problem Set 9 distributed

March 24 More Hodge decomposition, the dee-deebar lemma
Notes: TeX, PDF Scribe: Giwan Kim

### Applications of the Kahler condition

March 29 Lefschetz decomposition, starting Hodge index
Notes: TeX, PDF Scribe: Brooke Ullery
Problem Set 10 distributed

March 31 Hodge Index: Reformulations and Corollaries
Notes: TeX, PDF Scribe: Tengren Zhang

April 5 Line bundles, Divisors
Notes: TeX, PDF Scribe: Xin Zhou
Problem Set 11 distributed

April 7 More on line bundles and their curvature
Notes: TeX, PDF Scribe: Justin Campbell

April 12 Serre and Kodaira vanishing
Problem Set 12 distributed
Notes: TeX, PDF Scribe: Hunter Brooks

### Algebraic deRham Theory

April 14 Hypercohomology
Notes: TeX, PDF Scribe: Kevin Carde

April 19 The Zariski topology and GAGA
Notes: TeX, PDF Scribe: Giwan Kim