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 2.23 

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 Blowups 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 LeviCivita connection and holonomy on complex manifolds. 
Section 4.A 

Friday March 10 
Vector
bundles continued HermiteEinstein and KahlerEinstein metrics. 
Section 4.B 
HW 8 Chapter 4 
Monday
March 13 
Vector bundles continued HermiteEinstein and KahlerEinstein metrics continued. 

Wednesday
March 15 
Applications of
cohomology The HirzebruchRiemannRoch 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 2731  SPRING BREAK 

Monday
April 3 
Deformations of complex
structures The MaurerCartan equation. 
Section 6.1 

Wednesday
April 5 
Deformation of complex
structures continued The MaurerCartan 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.
GarciaPrada, 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).