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.
日期
2025年09月23日 至 12月18日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周二,周四 | 10:40 - 12:15 | A14-202 | ZOOM 05 | 293 812 9202 | BIMSA |
网站
修课要求
Complex analysis and differential geometry. Familiarity with Riemannian geometry and vector bundles is desirable.
课程大纲
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
参考资料
D. Huybrechts, Complex Geometry: An Introduction
J.-P. Demailly, Complex Analytic and Differential Geometry
J.-P. Demailly, Complex Analytic and Differential Geometry
听众
Advanced Undergraduate
, Graduate
, 博士后
, Researcher
视频公开
不公开
笔记公开
公开
语言
英文