Sebastian Casalaina

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
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
Section 4.2

Friday October 11
Vector bundles continued
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
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

Wednesday December 11

Saturday December 14
Final Exam 4:30 PM -- 7:30 PM MATH 220 (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).