MAT 545: Complex Geometry (Fall 2024)

About the course

“Complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaves. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis.” (Wikipedia)

Please see the syllabus for additional information about the course, including university-wide policies. Your grade will be determined based on homework and class participation.

Time and location

We meet on Tuesday and Thursday, 12:30–1:50 pm, in Physics P–129.

My office hours are Friday, 9:30am–12:30pm, in Mathematics 4–110.

Lecture notes

We are mostly going to follow my lecture notes from several years ago; however, I plan to spend a bit less time on Kähler manifolds and a bit more time on Stein manifolds and coherent sheaves.

Schedule

The topics covered in each class will appear below. Click on a link to download the notes for that particular class.

Week	Dates	Topic
1	Aug 27	Overview, holomorphic functions
2	Sep 3	Germs, Weierstrass theorems
	Sep 5	Analytic sets, implicit mapping theorem, geometric spaces
3	Sep 10	Complex manifolds, examples
	Sep 12	Blowing up a point, vector bundles
4	Sep 17	Tangent bundle, complex submanifolds
	Sep 19	Submanifold theorem, differential forms, type
5	Sep 24	Poincaré lemma, integration
	Sep 26	Riemannian and hermitian manifolds
6	Oct 1	Sheaves and cohomology
	Oct 3	Cech cohomology and Dolbeault cohomology
7	Oct 8	Linear differential operators and the fundamental theorem
	Oct 10	Harmonic theory
8	Oct 17	Complex harmonic theory, Kähler manifolds
9	Oct 22	Kähler identities
	Oct 24	Representation theory, Lefschetz decomposition
10	Oct 29	Weil's identity, Proof of Kähler identities
	Oct 31	Hodge decomposition, Hard Lefschetz theorem, Bilinear relations
11	Nov 5	Hodge index theorem, Examples, Hypersurfaces in projective space
	Nov 7	Holomorphic vector bundles, Chern connection
12	Nov 12	Holomorphic line bundles
	Nov 14	Harmonic theory for line bundles, Kodaira's vanishing theorem
13	Nov 19	Maps to projective space, Kodaira's embedding theorem
	Nov 21	Proof of Kodaira's embedding theorem, Riemann's criterion
14	Nov 26	Kähler versus projective manifolds, Levi extension theorem
15	Dec 3	Chow's theorem, coherent analytic sheaves
	Dec 5	Grauert's theorem, Stein manifolds
16	Dec 10	Oka principle, embedding theorem for Stein manifols

Homework assignments

During most weeks, I will be collecting written homework; we may also talk about some problems in class. For each assignment, please write up your solutions nicely and hand them in by the due date. Please staple your papers together and put your name on the first page.

Number	Due Date	Assignment
1	Sep 10	PDF file
2	Sep 17	PDF file
3	Sep 26	PDF file
4	Oct 10	PDF file
5	Oct 22	PDF file
6	Oct 31	PDF file
7	Nov 11	PDF file
8	Nov 21	PDF file
9	Dec 5	PDF file