2026 (Current Year) Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Geometry G
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Nobuhiro Honda
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Class
- -
- Course Code
- MTH.B503
- Number of credits
- 100
- Course offered
- 2026
- Offered quarter
- 3Q
- Syllabus updated
- Mar 5, 2026
- Language
- English
Syllabus
Course overview and goals
This course provides an overview of the fundamental concepts in differential geometry and complex geometry, and discusses more advanced topics such as the relationship between Ricci curvature and topological structure. Particular emphasis will be placed on Ricci-flat compact Kähler manifolds. Depending on the progress of the course, basic properties of K3 surfaces may also be covered.
Course description and aims
Upon completion of the course, students will be able to:
・explain the distinctive properties of compact Kähler manifolds;
・describe the fundamental properties of line bundles with positive curvature;
・analyze the basic properties of compact Kähler surfaces;
・situate special Riemannian manifolds—such as Kähler and Ricci-flat Kähler manifolds—within the framework of holonomy theory;
・explain the Calabi conjecture and its geometric consequences.
Keywords
Chern class, Curvature, Kahler 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
Standard lecture course
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | harmonic theory on compact Riemannian manifold |
Details will be provided in class. |
| Class 2 | connection on complex vector bundle, curvature, Chern class |
Details will be provided in class. |
| Class 3 | Kahler manifold and Hodge decomposition 1 |
Details will be provided in class. |
| Class 4 | Kahler manifold and Hodge decomposition 2 |
Details will be provided in class. |
| Class 5 | compact Kahler surface |
Details will be provided in class. |
| Class 6 | holonomy group |
Details will be provided in class. |
| Class 7 | Ricci curvature and holonomy representation, Calabi conjecture |
Details will be provided in class. |
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 contents afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
None
Reference books, course materials, etc.
P. Griffiths, J. Harris, "Principles of Algebraic Geometry"（Wiley-Interscience）
Evaluation methods and criteria
Homework assignments (100％)
Related courses
- MTH.B301 ： Geometry I
- MTH.B202 ： Introduction to Topology II
- MTH.B302 ： Geometry II
- MTH.B341 ： Topology
- MTH.B504 ： Advanced topics in Geometry H
Prerequisites
Basic theory on smooth manifolds and Riemannian geometry is assumed.
Other
To be announced.