# 206A, Fall 2022

### Professor

Elham Izadi ; AP&M 6240 ; 858-534-2638 ;
eizadi@math.ucsd.edu ;

Office hours: 11:00-12:00 Tuesdays and Thursdays in AP&M 5218, I
would very much appreciate if you could let me know in advance if
you are planning to come to office hours;

Lectures: Tuesday, Thursday 9:30-10:50 AP&M 5402
### Course description

We will learn about some applications of Hodge theory.
### Prerequisites

203C
### References

Hodge Theory and Complex Algebraic Geometry, Volumes I and II, by Claire Voisin

Complex Geometry An Introduction, by Daniel Huybrechts

Principles of Algebraic Geometry, by Phillip Griffiths and Joseph Harris

A Survey of the Hodge Conjecture, by James Lewis

Period Mappings and Period Domains, by James Carlson, Stefan Müller-Stach and Christiaan Peters

Mixed Hodge Structures, by Christiaan Peters and Joseph Steenbrink (beware mistakes)

Mumford-Tate Groups and Domains: Their Geometry and Arithmetic, by Phillip Griffiths, Mark Green and Matthew Kerr

Hodge Theory, by Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths and Lê Dũng Tráng

A course in Hodge Theory, by Hossein Movasati

Abelian varieties, by David Mumford

Complex tori and abelian varieties, by Olivier Debarre

Complex abelian varieties, by Christina Birkenhake and Herbert Lange

More references can be found in the notes below

### Presentations

This course will feature student
presentations. It is well known that the best way to learn
something is to explain it to others. When we present something to
others, we have to think about it in many different ways and look
at it from different angles. Giving presentations is very
difficult, which is why, when you give presentations, I will not
judge your performance, but will help you learn
the material in the best way possible and also help you learn
how to give presentations. This is a skill that will be very useful in
many different contexts, regardless of what your career goals
are. Any comments will not affect your grade, but will help you learn.

Please prepare your presentations carefully. In particular, please
write detailed typed notes for your presentations. It is usually a good
idea to have a rehearsal with some of your friends/classmates
before your presentation.
### Some class notes:

Preliminary notes
### Topics for presentations:

(1) Period domains

(2) Mumford-Tate groups and domains

(3) Challenge: What is an admissible normal function?

(4) Deligne cohomology and the normal function of a primitive Hodge class

(5) Some deformation theory

(6) The proof of Voisin's formula
### Note that some adjustment to the above might be necessary during the quarter.

Elham Izadi
Last modified: Thu Oct 20 13:44:08 PDT 2022