Sebastian Casalaina

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
Section 4.2

Friday March 3
Vector bundles continued
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

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

Wednesday May 3

Friday May 5

Monday May 8
Final Exam 1:30 PM -- 4:30 PM ECCR 116 (Lecture Room)


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).