Homework and Syllabus
Functions of a Complex Variable 2
MATH 6360 Spring 2017
Homework is due in class and must be stapled, with your name and homework number on it, to receive credit.
| Date | Topics | Reading | Homework |
| Wednesday
January 18 |
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. |
|
| Friday
January 20 |
Local theory Complex and Hermitian structures. |
Section 1.2 |
HW 1 Huybrechts Section 1.2 |
| Monday January 23 |
Local
theory continued Complex and Hermitian structures continued. |
||
| Wednesday January 25 |
Local
theory continued Holomorphic functions of several variables. |
Section 1.1 |
|
| Friday January 27 |
Local
theory continued Differential forms. |
Section 1.3 |
HW 2 Huybrechts Section 1.1 |
| Monday
January 30 |
Complex manifolds Definitions and examples. |
Section 2.1 |
|
| Wednesday
February 1 |
Complex manifolds
continued Holomorphic vector bundles, line bundles, divisors. |
Sections 2.2-3 |
|
| Friday
February 3 |
Complex manifolds
continued Projective space |
Section 2.4 |
HW 3 Huybrechts Section 1.3, 2.1, 2.2 |
| Monday February 6 |
Complex
manifolds continued Blow-ups along complex submanifolds. |
Section 2.5 |
|
| Wednesday February 8 |
Complex
manifolds continued Differential calculus on complex manifolds. |
Section 2.6 |
|
| Friday February 10 |
Complex
manifolds coninued Differential calculus on complex manifolds continued. |
HW 4 Huybrechts Section 2.3, 2.4. |
|
| Monday
February 13 |
Kahler manifolds Kahler identities. |
Section 3.1 |
|
| Wednesday
February 15 |
Kahler manifolds
continued Hodge theory on Kahler manifolds. |
Section 3.2 |
|
| Friday
February 17 |
Kahler manifolds
continued Lefschetz theorems. |
Section 3.3 |
HW 5 Huybrechts Section 2.5, 2.6 |
| Monday February 20 |
Kahler
manifolds continued Formality on compact Kahler manifolds. |
Section 3.A |
|
| Wednesday February 22 |
Kahler
manifolds continued SUSY for Kahler manifolds. |
Section 3.B |
|
| Friday February 24 |
Kahler
manifolds continued Hodge structures. |
Section 3.C |
HW 6 Chapter 3 |
| Monday
February 27 |
Vector bundles Hermitian vector bundles and Serre duality. |
Section 4.1 |
|
| Wednesday
March 1 |
Vector bundles continued Connections. |
Section 4.2 |
|
| Friday
March 3 |
Vector bundles continued Curvature. |
Section 4.3 |
HW 7 Chapter 3 |
| Monday March 6 |
Vector
bundles continued Chern classes. |
Section 4.4 |
|
| Wednesday March 8 |
Vector
bundles continued The Levi-Civita connection and holonomy on complex manifolds. |
Section 4.A |
|
| Friday March 10 |
Vector
bundles continued Hermite--Einstein and Kahler--Einstein metrics. |
Section 4.B |
HW 8 Chapter 4 |
| Monday
March 13 |
Vector bundles continued Hermite--Einstein and Kahler--Einstein metrics continued. |
||
| Wednesday
March 15 |
Applications of
cohomology The Hirzebruch--Riemann--Roch theorem. |
Section 5.1 |
|
| Friday March 17 | Applications of
cohomology continued The Kodaira vanishing theorem and applications. |
Section 5.2 |
HW 9 Chapter 4 |
| Monday March 20 | Applications of
cohomology continued The Kodaira embedding theorem. |
Section 5.3 |
|
| Wednesday March 22 |
Applications
of cohomology continued Further topics. |
||
| Friday March 24 |
Applications
of cohomology continued Further topics. |
HW 10 Chapter 5 |
|
| March 27--31 | SPRING BREAK |
||
| Monday
April 3 |
Deformations of complex
structures The Maurer--Cartan equation. |
Section 6.1 |
|
| Wednesday
April 5 |
Deformation of complex
structures continued The Maurer--Cartan equation continued. |
||
| Friday
April 7 |
Deformation of complex
structures continued General results. |
Section 6.2 |
HW 11 Chapter 5 |
| Monday April 10 |
Deformation
of complex structures continued General results continued. |
Section 6.3 |
|
| Wednesday April 12 |
Deformation
of complex structures continued Further topics. |
||
| Friday April 14 | Deformation
of complex structures continued Further topics. |
We will also use the
papers of M. Pflaum
and M.
Manetti. |
HW 12 Chapter 6 |
| Monday
April 17 |
Introduction to moduli
spaces Projective space, Grassmanians, moduli of smooth curves. |
||
| Wednesday
April 19 |
Topics in moduli
theory |
We will follow the appendix by O.
Garcia-Prada, in Differential
Analysis on Complex Manifolds (Third Edition),
Springer 2008. |
|
| Friday
April 21 |
Topics in moduli theory |
HW 13 Chapter 6 |
|
| Monday April 24 |
Topics in moduli theory | ||
| Wednesday April 26 |
Topics in algebraic curves | ||
| Friday April 28 |
Classification
of algebraic surfaces Birational maps between surfaces, minimal surfaces, Kodaira dimension, and some results in the classification of surfaces. |
A. Beauville, Complex Algebraic Surfaces, Cambridge University Press, 1996. | HW 14 Chapter 6 |
| Monday May 1 |
Review | ||
| Wednesday May 3 |
Review | ||
| Friday May 5 |
Review | ||
| Monday
May 8 |
Final
Exam 1:30 PM -- 4:30 PM ECCR
116 (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).