Homework and Syllabus
Functions of a Complex Variable 2
MATH 6360 Fall 2023
The following is a rough outline of the topics we will
cover. The dates and topics have yet to be updated, and the
topics will be updated again after the start of the course
depending on the interests of the students.
| Date | Topics | Reading | Homework |
| Monday August
28 |
Introduction to the
course, and review of complex analysis Review of results from complex analysis in one variable. |
We will be following D. Huybrechts, Complex
Geometry: an introduction, Springer 2005,
available in .pdf for free from the library. The following .pdf has a brief review of complex analysis in a single variable. |
|
| Wednesday
August 30 |
Local theory Holomorphic functions of several variables. |
Section 1.1 |
|
| Friday September 1 |
Local
theory continued Holomorphic functions of several variables (continued). |
Section 1.1 |
HW 1 Huybrechts Section 1.1 |
| Monday
September 4 |
LABOR DAY |
NO CLASS |
NO CLASS |
| Wednesday September 6 |
Local
theory continued Complex and Hermitian structures. |
Section 1.2 |
|
| Friday September 8 |
Local
theory continued Differential forms. |
Section 1.3 |
HW 2 Huybrechts Section 1.2 |
| Monday
September 11 |
Complex manifolds Definitions and examples. |
Section 2.1 |
|
| Wednesday
September 13 |
Complex manifolds
continued Holomorphic vector bundles. |
Sections 2.2 |
|
| Friday
September 15 |
Complex manifolds
continued Divisors and line bundles. |
Sections 2.3 | HW 3 Huybrechts Section 1.3, 2.1, 2.2 |
| Monday September 18 |
Complex
manifolds continued Projective space |
Section 2.4 | |
| Wednesday September 20 |
Complex
manifolds continued Blow-ups along complex submanifolds. |
Section 2.5 | |
| Friday September 22 |
Complex
manifolds coninued Differential calculus on complex manifolds. |
Section 2.6 | HW 4 Huybrechts Section 2.3, 2.4. |
| Monday
September 25 |
Kahler manifolds Kahler identities. |
Section 3.1 |
|
| Wednesday
September 27 |
Kahler manifolds
continued Hodge theory on Kahler manifolds. |
Section 3.2 |
|
| Friday
September 29 |
Kahler manifolds
continued Lefschetz theorems. |
Section 3.3 |
HW 5 Huybrechts Section 2.5, 2.6 |
| Monday October 2 |
Kahler
manifolds continued Formality on compact Kahler manifolds. |
Section 3.A |
|
| Wednesday October 4 |
Kahler
manifolds continued SUSY for Kahler manifolds. |
Section 3.B |
|
| Friday October 6 |
Kahler
manifolds continued Hodge structures. |
Section 3.C |
HW 6 Chapter 3 |
| Monday
October 9 |
Vector bundles Hermitian vector bundles and Serre duality. |
Section 4.1 |
|
| Wednesday
October 11 |
Vector bundles continued Connections. |
Section 4.2 |
|
| Friday
October 13 |
Vector bundles continued Curvature. |
Section 4.3 |
HW 7 Chapter 3 |
| Monday October 16 |
Vector
bundles continued Chern classes. |
Section 4.4 |
|
| Wednesday October 18 |
Vector
bundles continued The Levi-Civita connection and holonomy on complex manifolds. |
Section 4.A |
|
| Friday October 20 |
Vector
bundles continued Hermite--Einstein and Kahler--Einstein metrics. |
Section 4.B |
HW 8 Chapter 4 |
| Monday
October 23 |
Vector bundles continued Hermite--Einstein and Kahler--Einstein metrics continued. |
||
| Wednesday
October 25 |
Applications of
cohomology The Hirzebruch--Riemann--Roch theorem. |
Section 5.1 |
|
| Friday October 27 | Applications of
cohomology continued The Kodaira vanishing theorem and applications. |
Section 5.2 |
HW 9 Chapter 4 |
| Monday
October 30 |
Applications of
cohomology continued The Kodaira embedding theorem. |
Section 5.3 |
|
| Wednesday October 1 |
Applications
of cohomology continued Further topics. |
||
| Friday November 3 |
Applications
of cohomology continued Further topics. |
HW 10 Chapter 5 |
|
| Monday
November 6 |
Presentations |
||
| Wednesday
November 8 |
Presentations |
||
| Friday
November 10 |
Presentations |
HW 11 Chapter 5 |
|
| Monday November 13 |
Presentations |
||
| Wednesday November 15 |
Presentations |
||
| Friday November 17 |
Deformation
of complex structures continued The Maurer--Cartan equation. |
Section 6.1 | HW 12 Chapter 6 |
| November 20--24 | THANKSGIVING BREAK |
NO CLASS |
NO CLASS |
| Monday
November 27 |
Deformation of complex
structures continued General results. |
Section 6.2 | |
| Wednesday
November 29 |
Deformation of complex
structures continued General results continued. |
Section 6.2 | |
| Friday
December 1 |
Introduction to moduli
spaces Projective space, Grassmanians, moduli of smooth curves. |
HW 13 Chapter 6 |
|
| Monday December 4 |
Introduction
to moduli spaces Projective space, Grassmanians, moduli of smooth curves, continued. |
||
| Wednesday December 6 |
Introduction
to moduli spaces Moduli space of stable curves |
||
| Friday December 8 |
Introduction
to moduli spaces Moduli space of abelian varieties |
HW 14 Chapter 6 |
|
| Monday
December 11 |
Review | ||
| Wednesday
December 13 |
Review | ||
| Saturday
December 16 |
Final Exam 4:30
PM -- 7:00 PM MATH 350 (Lecture Room) |
FINAL EXAM |
I strongly encourage
everyone to use LaTeX for typing homework. If you have
a mac,
one possible easy way to get started is with texshop.
If you are using linux,
there are a number of other possible ways to go, using
emacs, ghostview, etc. If you are using windows,
you're on your own, but I'm sure there's something online.
Here is a sample homework file to use: (the .tex
file, the .bib
file, and the .pdf
file).