MaMath-138 - Complex geometry
Course Info
The lectures will be every Wednesday from 11:30-13:00 in Maison du Nombre 1.020. You can email me at nathaniel(dot)sagman(at)uni(dot)lu or nathaniels1729(at)gmail(dot)com.
Formally, we'll have office hours on Monday's from 3-4 PM in my office, MNO E06 0615-170. You're also welcome to just come to my office at any time and if I'm available we can chat.
IMPORTANT: The lecture on May 24 will be done via zoom. The lecture on May 31 is being moved to June 2, 11:30-12:00 (room TBA).
Resources on mental health at the university: see here and here.
Course catalogue description
The aim of the lecture course is to give an introduction to complex geometry. In particular the course will treat many examples. After the course the students should understand Hermitian vector bundles and the basics of Kahler manifolds and the Hodge decomposition. The students should be acquainted with the basic theorems and examples.
What the course is shaping up to be about
Review/necessary background on real manifolds.
Basics on complex manifolds.
Kähler manifolds (assuming the Hodge theorem without proof).
For assessments, there will be a) 2 homework sets, and b) final presentations or written reports, depending on students' preferences. Homeworks will be worth 30% of the final grade, and the final presentation/report worth 70%.
Course Materials
Homeworks
Homework 1 . Due April 26.
Homework 2. Due May 31.
Final project
Lecture notes
Notes should be posted a day or two after each lecture. The notes may contain more details than given in the lecture. I provide some additional references, although there are usually things in the notes that don't come from the references.
Lecture 1. Helpful references: Chapter 0 in Krantz, Chapters 1.1 and 2.1 in Huybretchs, Chapter 1.1 in Wells.
Lecture 2. Helpful references: Chapter 2.1 in Huybrechts, Chapter 1.1 in Wells.
Lecture 3. Helpful references: Chapter 2.1 in Huybrechts, Chapter 1.1 in Wells, Chapters 4-5 in Forster.
Lecture 4. Helpful references: Chapters 1.1, 2.1, and 2.2 in Huybrechts, Chapters 1.1 and 1.2 in Wells, Chapters 3 and 10 in Lee (for content on the tangent bundle for real manifolds).
Lecture 5. Helpful references: Chapters 2.2, 2.3, and 2.4 in Huybrechts, Chapter 1.2 in Wells, Chapters 3 and 10 in Lee.
Lecture 6. Helpful references: Chapters 3, 11, 12, and 14, and 17 in Lee. This wikipedia article on wedge products. This math overflow post on exterior derivatives. The formula for the wedge product of matrices is missing a factor of 1/2. I wrote up a small note saying more about wedge products and explaining how to get the right formula.
Lecture 7. Helpful references: Chapters 15 and 16 in Lee.
Lecture 8. Helpful references: Chapters 1.2 and 2.6 in Huybrechts, Chapters 1 and 2 in Moroianu (main reference), Chapter 1.3 in Wells. See example 2.7 in Ballman for more about the almost complex structure on S^6. Math overflow post about eigenbundles (i.e., our T^{1,0}M).
Lecture 9. Helpful references: Chapters 1.2 and 2.6 (main reference) in Huybrechts, Chapters 1 and 2 in Moroianu, Chapter 1.3 in Wells, Chapter 13 in Lee. Mathoverflow post about diffeomorphic complex manifolds with different Dolbeaut cohomology.
Lecture 10 (content of last page will be next class). Helpful references: Chapter 3.1 in Huybrechts, Chapters 4-7 in Moroianu. Mathoverflow post about compact Kähler manifolds that are not projective--the example uses that the dimension of an algebraic variety is the transcendence degree of its function field.
Lecture 11. Helpful references: Chapter 3.1 in Huybrechts, Chapter 5 in Moroianu. Wikipedia page for quaternionic manifold.
Lecture 12. Helpful references: Chapter 1.2 and Appendix A in Huybrechts, Chapter 8 in Moroianu.
Lecture 13. Helpful references: Chapters 1.2, 3.1, and 3.2 in Huybrechts, Chapter 5 in Wells.
Lecture 14. Helpful references: Chapter 2.3 in Huybrechts and Chapter 29 in Forster. For more on the perspective shared on Chern classes, one can see Milnor and Stasheff's book "Characteristic classes."
Literature
I do not plan to follow one text, and will mainly draw from the sources below.
Huybrechts: Complex geometry.
Wells: Differential Analysis on Complex Manifolds.
Moroianu: Lectures on Kähler geometry.
Some other texts that may be helpful:
Lee: Introduction to smooth manifolds.
Griffiths and Harris: Principles of Algebraic Geometry.
Demailly: Complex Analytic and Differential Geometry.
Ballman: Lectures on Kähler manifolds.
Krantz: Function Theory of Several Complex Variables.
Forster: Lectures on Riemann Surfaces.
Chern: Complex Manifolds without Potential Theory.
Voisin: Hodge Theory and Complex Algebraic Geometry I.
Szekelyhidi: An Introduction to Extremal Kähler metrics.