Homework and Syllabus
Functions of a Complex Variable 2
MATH 6360 Fall 2019
Homework is due in class and must be stapled,
with your name and homework number on it, to
receive credit.
The following is a rough outline of the topics we will
cover.
| Date | Topics | Reading | Homework |
| Monday August
26 |
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 28 |
Local theory Complex and Hermitian structures. |
Section 1.2 |
HW 1 Huybrechts Section 1.2 |
| Friday August 30 |
Local
theory continued Complex and Hermitian structures continued. |
||
| Monday
September 2 |
LABOR DAY |
NO CLASS |
NO CLASS |
| Wednesday September 4 |
Local
theory continued Holomorphic functions of several variables. |
Section 1.1 |
|
| Friday September 6 |
Local
theory continued Differential forms. |
Section 1.3 |
HW 2 Huybrechts Section 1.1 |
| Monday
September 9 |
Complex manifolds Definitions and examples. |
Section 2.1 |
|
| Wednesday
September 11 |
Complex manifolds
continued Holomorphic vector bundles, line bundles, divisors. |
Sections 2.2-3 |
|
| Friday
September 13 |
Complex manifolds
continued Projective space |
Section 2.4 |
HW 3 Huybrechts Section 1.3, 2.1, 2.2 |
| Monday September 16 |
Complex
manifolds continued Blow-ups along complex submanifolds. |
Section 2.5 |
|
| Wednesday September 18 |
Complex
manifolds continued Differential calculus on complex manifolds. |
Section 2.6 |
|
| Friday September 20 |
Complex
manifolds coninued Differential calculus on complex manifolds continued. |
HW 4 Huybrechts Section 2.3, 2.4. |
|
| Monday
September 23 |
Kahler manifolds Kahler identities. |
Section 3.1 |
|
| Wednesday
September 25 |
Kahler manifolds
continued Hodge theory on Kahler manifolds. |
Section 3.2 |
|
| Friday
September 27 |
Kahler manifolds
continued Lefschetz theorems. |
Section 3.3 |
HW 5 Huybrechts Section 2.5, 2.6 |
| Monday September 30 |
Kahler
manifolds continued Formality on compact Kahler manifolds. |
Section 3.A |
|
| Wednesday October 2 |
Kahler
manifolds continued SUSY for Kahler manifolds. |
Section 3.B |
|
| Friday October 4 |
Kahler
manifolds continued Hodge structures. |
Section 3.C |
HW 6 Chapter 3 |
| Monday
October 7 |
Vector bundles Hermitian vector bundles and Serre duality. |
Section 4.1 |
|
| Wednesday
October 9 |
Vector bundles continued Connections. |
Section 4.2 |
|
| Friday
October 11 |
Vector bundles continued Curvature. |
Section 4.3 |
HW 7 Chapter 3 |
| Monday October 14 |
Vector
bundles continued Chern classes. |
Section 4.4 |
|
| Wednesday October 16 |
Vector
bundles continued The Levi-Civita connection and holonomy on complex manifolds. |
Section 4.A |
|
| Friday October 18 |
Vector
bundles continued Hermite--Einstein and Kahler--Einstein metrics. |
Section 4.B |
HW 8 Chapter 4 |
| Monday
October 21 |
Vector bundles continued Hermite--Einstein and Kahler--Einstein metrics continued. |
||
| Wednesday
October 23 |
Applications of
cohomology The Hirzebruch--Riemann--Roch theorem. |
Section 5.1 |
|
| Friday October 25 | Applications of
cohomology continued The Kodaira vanishing theorem and applications. |
Section 5.2 |
HW 9 Chapter 4 |
| Monday
October 28 |
Applications of
cohomology continued The Kodaira embedding theorem. |
Section 5.3 |
|
| Wednesday October 30 |
Applications
of cohomology continued Further topics. |
||
| Friday November 1 |
Applications
of cohomology continued Further topics. |
HW 10 Chapter 5 |
|
| Monday
November 4 |
Deformations of complex
structures The Maurer--Cartan equation. |
Section 6.1 |
|
| Wednesday
November 6 |
Deformation of complex
structures continued The Maurer--Cartan equation continued. |
||
| Friday
November 8 |
Deformation of complex
structures continued General results. |
Section 6.2 |
HW 11 Chapter 5 |
| Monday November 11 |
Deformation
of complex structures continued General results continued. |
Section 6.3 |
|
| Wednesday November 13 |
Deformation
of complex structures continued Further topics. |
||
| Friday November 15 |
Deformation
of complex structures continued Further topics. |
We will also use the
papers of M. Pflaum
and M.
Manetti. |
HW 12 Chapter 6 |
| Monday
November 18 |
Introduction to moduli
spaces Projective space, Grassmanians, moduli of smooth curves. |
||
| Wednesday
November 20 |
Topics in moduli
theory |
We will follow the appendix by O.
Garcia-Prada, in Differential
Analysis on Complex Manifolds (Third Edition),
Springer 2008. |
|
| Friday
November 22 |
Topics in moduli theory |
HW 13 Chapter 6 |
|
| November
25--29 |
THANKSGIVING BREAK |
NO CLASS |
NO CLASS |
| Monday December 2 |
Topics in moduli theory | ||
| Wednesday December 4 |
Topics in algebraic curves | ||
| Friday December 6 |
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 December 9 |
Review | ||
| Wednesday December 11 |
Review | ||
| Saturday
December 14 |
Final
Exam 4:30 PM -- 7:30 PM MATH 220
(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).