Complex Geometry
This graduate-level course offers an introduction to the fundamental concepts and techniques of complex differential geometry.
The central aim of the course is to understand the criteria that determine when a compact complex manifold can be realized as a smooth projective algebraic variety. This is the celebrated Kodaira Embedding Theorem, a cornerstone result that provides a precise differential-geometric condition (the existence of a positive line bundle, or a Hodge metric) for a complex manifold to be projective (and thus algebraic by Chow's theorem). We will work through the necessary machinery to fully prove this theorem.
Time permitting, we will then discuss Kodaira-Spencer deformation theory and discuss the case of Calabi-Yau manifolds, studied by Tian-Todorov.
The central aim of the course is to understand the criteria that determine when a compact complex manifold can be realized as a smooth projective algebraic variety. This is the celebrated Kodaira Embedding Theorem, a cornerstone result that provides a precise differential-geometric condition (the existence of a positive line bundle, or a Hodge metric) for a complex manifold to be projective (and thus algebraic by Chow's theorem). We will work through the necessary machinery to fully prove this theorem.
Time permitting, we will then discuss Kodaira-Spencer deformation theory and discuss the case of Calabi-Yau manifolds, studied by Tian-Todorov.
Lecturer
Date
23rd September ~ 18th December, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Thursday | 10:40 - 12:15 | A14-202 | ZOOM 05 | 293 812 9202 | BIMSA |
Website
Prerequisite
Complex analysis and differential geometry. Familiarity with Riemannian geometry and vector bundles is desirable.
Syllabus
0)Overview
1) Holomorphic functions
2) Complex and almost complex manifolds
3) Vector bundles and sheaves
4) Kodaira dimension and Siegel's theorem
5) Divisors and blow-ups
6) Metrics and connections
7) The Kähler condition
8) Positivity and vanishing
9) The Kodaira embedding theorem
10) Kodaira-Spencer deformation theory
11) (Formal) Tian-Todorov theorem
1) Holomorphic functions
2) Complex and almost complex manifolds
3) Vector bundles and sheaves
4) Kodaira dimension and Siegel's theorem
5) Divisors and blow-ups
6) Metrics and connections
7) The Kähler condition
8) Positivity and vanishing
9) The Kodaira embedding theorem
10) Kodaira-Spencer deformation theory
11) (Formal) Tian-Todorov theorem
Reference
D. Huybrechts, Complex Geometry: An Introduction
J.-P. Demailly, Complex Analytic and Differential Geometry
J.-P. Demailly, Complex Analytic and Differential Geometry
Audience
Advanced Undergraduate
, Graduate
, Postdoc
, Researcher
Video Public
No
Notes Public
Yes
Language
English