2021 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Geometry G1
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Nobuhiro Honda
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Fri
- Class
- -
- Course Code
- MTH.B507
- Number of credits
- 100
- Course offered
- 2021
- Offered quarter
- 3Q
- Syllabus updated
- Jul 10, 2025
- Language
- English
Syllabus
Course overview and goals
Overviewing basic notions in differential geometry and complex geometry, some advanced topics such as relevance of Ricci curvature with the topology of manifolds will be treated. Basic properties of K3 surfaces might be also included.
Course description and aims
・To understand special properties of compact Kaehler manifolds.
・To understand basic properties of "positive" line bundle over complex manifolds
・To understand basic properties of compact Kaehler surfaces
・To understand "special" Riemannian manifolds such as Kaehler manifolds and Ricci flat Kaehler manifolds, in terms of holonomy group
・To understand Calabi conjecture and its consequences
Keywords
Chern class, curvature, kaehler manifold, positive line bundle, harmonic form, Ricci form, holonomy group, Hodge decomposition, Calabi conjecture, Bochner principle
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
regular lecture course
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | harmonic theory on compact Riemannian manifolds | Definitions and properties |
| Class 2 | connections, curvature and Chern class of complex vector bundle | Definitions and properties |
| Class 3 | Kaehler manifolds and Hodge decomposition 1 | Definitions and properties |
| Class 4 | Kaehler manifolds and Hodge decomposition 2 | Definitions and properties |
| Class 5 | compact Kaehler surfaces | Definitions and properties |
| Class 6 | holonomy group | Definitions and properties |
| Class 7 | Ricci curvature and holonomy group, Calabi conjecture | Definitions and properties |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
No textbook
Reference books, course materials, etc.
D. Joyce, "Compact manifolds with special holonomy", Oxford University Press
A. Besse, "Einstein manifolds" Springer
P. Griffiths, J. Harris, "Principles of Algebraic Geometry" Wiley-Interscience
R. O. Wells, "Differential analysis on complex manifolds" Springer Graduate Texts in Mathematics
Evaluation methods and criteria
Homework assignments (100％)
Related courses
- MTH.B202 ： Introduction to Topology II
- MTH.B301 ： Geometry I
- MTH.B302 ： Geometry II
Prerequisites
This course is not entirely introductory. We expect audience to be familiar with basic notions in differential geometry and complex geometry to some extent.