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 2.23 

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

Friday October 18 
Vector
bundles continued HermiteEinstein and KahlerEinstein metrics. 
Section 4.B 
HW 8 Chapter 4 
Monday
October 21 
Vector bundles continued HermiteEinstein and KahlerEinstein metrics continued. 

Wednesday
October 23 
Applications of
cohomology The HirzebruchRiemannRoch 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 MaurerCartan equation. 
Section 6.1 

Wednesday
November 6 
Deformation of complex
structures continued The MaurerCartan 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.
GarciaPrada, in Differential
Analysis on Complex Manifolds (Third Edition),
Springer 2008. 

Friday
November 22 
Topics in moduli theory 
HW 13 Chapter 6 

November
2529 
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).